Dantzig’s Number

Tobias Dantzig Number: The Language of Science Pi Press  2005.

Edited by Joseph Mazur. Foreword by Barry Mazur. Foreword, Notes, Afterword and Further Readings © 2005 by Pearson Education, Inc.© 1930, 1933, 1939, and 1954 by the Macmillan Company.

[The emphasis is in the original, apart from the colouring, which encodes my own views: you might like to form your own.]


“Algebra is the offer made by the devil to the mathematician. The devil says: I will give you this powerful machine, it will answer any question you like. All you need to do is give me your soul: give up geometry and you will have this marvelous machine.”
(Atiyah, Sir Michael. Special Article: Mathematics in the 20th Century. Page 7. Bulletin of the London Mathematical Society, 34 (2002) 1–15.)

Barry Mazur

Part One: Evolution of the Number Concept

CHAPTER 3: Number-lore

No two branches of mathematics present a greater contrast than arithmetic and the Theory of Numbers.
The great generality and simplicity of its rules makes arithmetic accessible to the dullest mind. …
On the other hand, the theory of numbers is by far the most difficult of all mathematical disciplines. It is true that the statement of its problems is so simple that even a child can understand what is at issue. But, the methods used are so individual that uncanny ingenuity and the greatest skill are required to find a proper avenue of approach. Here intuition is given free play. Most of the properties known have been discovered by a sort of induction. Statements held true for centuries have been later proved false, and to this day there are problems which have challenged the power of the greatest mathematicians and still remain unsolved.
Arithmetic is the foundation of all mathematics, pure or applied. It is the most useful of all sciences, and there is, probably, no other branch of human knowledge which is more widely spread among the masses. On the other hand, the theory of numbers is the branch of mathematics which has found the least number of applications. Not only has it so far [1930] remained without influence on technical progress, but even in the domain of pure mathematics it has always occupied an isolated position, only loosely connected with the general body of the science.

CHAPTER 4: The Last Number

What is there in mathematics that makes it the acknowledged model of the sciences called exact, and the ideal of the newer sciences which have not yet achieved this distinction? It is, indeed, the avowed ambition of the younger investigators at least, in such fields as biology or the social sciences, to develop standards and methods which will permit these to join the ever-growing ranks of sciences which have already accepted the domination of mathematics.
Mathematics is not only the model along the lines of which the exact sciences are striving to design their structure; mathematics is the cement which holds this structure together. A problem, in fact, is not considered solved until the studied phenomenon has been formulated as a mathematical law. Why is it believed that only mathematical processes can lend to observation, experiment, and speculation that precision, that conciseness, that solid certainty which the exact sciences demand?
When we analyze these mathematical processes we find that they rest on the two concepts: Number and Function; that Function itself can in the ultimate be reduced to Number; that the general concept of Number rests in turn on the properties we ascribe to the natural sequence: one, two, three ….
It is then in the properties of the whole numbers that we may hope to find the clue to this implicit faith in the infallibility of mathematical reasoning!

Returning to our problem, suppose that we have examined our premises and have found them free from contradictions. Then we say that our conclusion is logically flawless. If, however, this conclusion does not agree with the observed facts, we know that the assumptions we have made do not fit the concrete problem to which they were applied. There is nothing wrong with the tailoring of the suit. If it bulges in some spots and cracks in others, it is the fault of the fitter.

CHAPTER 6: The Unutterable

God created the integers, the rest is the work of man.
—Leopold Kronecker

[There] can be little doubt that [Pythagoras] and his disciples attached the greatest importance to [his theorem]; for therein they saw the inherent union between geometry and arithmetic, a new confirmation of their dictum: “Number rules the universe.” But the triumph was short-lived. …

Says Proclos:

“It is told that those who first brought out the irrationals from concealment into the open perished in shipwreck, to a man. For the unutterable and the formless must needs be concealed. And those who uncovered and touched this image of life were instantly destroyed and shall remain forever exposed to the play of the eternal waves.”

CHAPTER 7: This Flowing World

Our first naïve impression of Nature and matter is that of continuity. Be it a piece of metal or a volume of liquid, we invariably conceive it as divisible into infinity, and ever so small a part of it appears to us to possess the same properties as the whole.
—David Hilbert

CHAPTER 8: The Art of Becoming

No more fiction for us: we calculate; but that we may calculate, we had to make fiction first.”

CHAPTER 12: The Two Realities

We have found a strange footprint on the shores of the unknown. We have devised profound theories, one after another, to account for its origin. At last, we have succeeded in reconstructing the creature that made the footprint. And lo! it is our own.”
—A. S. Eddington [Space Time and Gravitation. 1920]

I have come to the end of my narrative. It was my object to survey the present status of the science of number in the light of its past; so it would be proper in the concluding chapter of such a survey to take a glimpse into the future. … There remains the ever-present: the issue of reality. This issue … is the philosopher’s chief preoccupation today. And so I realize fully that by selecting reality as the theme of this concluding chapter, I am encroaching on a field foreign to my training, foreign to my outlook. …

… Mathematical achievement shall be measured by standards which are peculiar to mathematics. These standards are independent of the crude reality of our senses. They are:

  • freedom from logical contradictions,
  • the generality of the laws governing the created form,
  • the kinship which exists between this new form and those that have preceded it.

The mathematician may be compared to a designer of garments, who is utterly oblivious of the creatures whom his garments may fit. …

To this day the central problems of philosophy smack of theology. It seems to me that what philosophy lacks most is a principle of relativity.
A principle of relativity is just a code of limitations: it defines the boundaries wherein a discipline shall move and frankly admits that there is no way of ascertaining whether a certain body of facts is the manifestation of the observata, or the hallucination of the observer.

The man of science will act as if this world were an absolute whole controlled by laws independent of his own thoughts or acts; but whenever he discovers a law of striking simplicity or one of sweeping universality or one which points to a perfect harmony in the cosmos, he will be wise to wonder what rôle his mind has played in the discovery, and whether the beautiful image he sees in the pool of eternity reveals the nature of this eternity, or is but a reflection of his own mind.

The terms used by the mathematician are, after all, words and belong to the limited vocabulary by means of which man from the earliest days had endeavored to express his thoughts, both mathematical and non-mathematical. Some of these terms, such as geometry and calculus, have lost their original double meaning and are understood by everybody in the specific sense that they have acquired in mathematical practice. Others, however, such as logical and illogical, rational and irrational, finite and infinite, real and imaginary, have to this day retained their multiple meaning. To the mathematician, who rarely ventures into the realm of metaphysics, these words have a very specific and quite unambiguous meaning; to the philosopher who uses these terms as his stock in trade they have also a very specific but an entirely different meaning; to the man who is neither philosopher nor mathematician these words have a general and rather vague significance.

[The] further we progress in our knowledge of the physical world, or in other words the further we extend our perceptual world by means of scientific instruments, the more we find our concept of infinity incompatible with this physical world in deed as well as in principle.
Since then the conception of infinity is not a logical necessity and since, far from being sanctified by experience, all experience protests its falsity, it would seem that the application of the infinite to mathematics must be condemned in the name of reality.

…. .But after this revision has been effected, what little remained of mathematics after this purging process has been consummated would be in perfect consonance with reality.
    Would it? That is the question, and this question is tantamount to another: “What is reality?” …

…. Stripped of all its metaphysical irrelevancies and free of philosophical jargon is this description by Poincaré: “What we call objective reality is, in the last analysis, what is common to many thinking beings and could be common to all.” In spite of its vagueness, in spite of the obvious weakness of the phrase “what could be common to all,” this is the nearest we can get to this intuitive idea of reality which we all seem to possess.

[Dantzig had been a student of Poincaré before feeling to America.]

…. Counting presupposes the human ability to classify various perceptions under the same head and to endow the class with a name; it presupposes the ability to match two collections, element for element, and to associate these collections with a number-word, which is but the model for a given plurality; it presupposes the ability to order these models into a sequence and to evolve a syntax which will permit an indefinite extension of these number-words. In short, the counting process postulates the existence of a language, an institution which transcends the subjective reality or the immediate perceptions of any individual.
If then this subjective reality be taken as criterion of what is valid in mathematics, we should be compelled not only to condemn the infinite process and all it implies, but to scrap the counting procedure as well.

Of the absolute and immutable world which exists outside our consciousness we know only through theological speculations: accepting it or rejecting it are alike futile to a natural philosophy. …  Such speculations are tremendously fascinating in that they allow free rein to our power of resolving our sensations into their constituents, and then regarding the concept as a synthesis of these arch-sensations. But to accept such a synthesis as reality, as the reality, has, to my way of thinking, one fatal defect: it postulates the existence of an individual intellect; whereas the very process of coordinating these sensations involves thought, which is impossible without the vehicle language, which in turn implies an organized exchange of impressions, which in turn presupposes a collective existence for human beings, some form of social organization.
The only reality that can be taken as a criterion of validity is not that absolute, immutable reality which exists outside of our consciousness and is therefore pure metaphysics, nor that arch-reality which the physiologist and the psychologist manage to isolate by means of painstaking experiments; it is rather that objective reality which is common to many and could be common to all. And that reality is not a collection of frozen images, but a living, growing organism.

[The] question: what reality shall we ascribe to number? is meaningless, because there is no reality without number, as there is no reality without space or without time.
And so neither in the subjective nor yet in the objective world can we find a criterion for the reality of the number concept, because the first contains no such concept, and the second contains nothing that is free of the concept.
How then can we arrive at a criterion? Not by evidence, for the dice of evidence are loaded. Not by logic, for logic has no existence independent of mathematics: it is only one phase of this multiphased necessity that we call mathematics. How then shall mathematical concepts be judged? They shall not be judged! Mathematics is the supreme judge; from its decisions there is no appeal. We cannot change the rules of the game, we cannot ascertain whether the game is fair. We can only study the player at his game; not, however, with the detached attitude of a bystander, for we are watching our own minds at play.

We have attached a phantom to a fiction, which had this advantage over the phantom that it was a familiar fiction. But it had not always been familiar; there was a time when this too caused bewilderment and restlessness, until we attached it to a still more primeval illusion, which, in turn, had been rendered concrete through centuries of habit.

The reality of today was but an illusion yesterday. The illusion survived because it helped to organize and systematize and guide our experience and therefore was useful to the life of the race. Such is my interpretation of the words of Nietzsche:

“We hold mere falsity no ground for rejecting a judgment. The issue is: to what extent has the conception preserved and furthered the life of the race? The falsest conceptions,—and to these belong our synthetic judgments a priori,—are also those which are the most indispensable. Without his logical fictions, without measuring reality in a fictitious absolute and immutable world, without the perpetual counterfeiting of the universe by number, man could not continue to live. The renunciation of all false judgment would mean a renunciation, a negation of life.”

Experimental evidence and logical necessity do not exhaust the objective world which we call reality. There is a mathematical necessity which guides observation and experiment, and of which logic is only one phase. The other phase is that intangible, vague thing which escapes all definition, and is called intuition. And so, to return to the fundamental issue of the science of number: the infinite. The concept of infinity is not an experiential nor a logical necessity; it is a mathematical necessity.

… There are many [by-paths through which the Quest of the Absolute has taken man]: simplicity, uniformity, homogeneity, regularity, causality are … manifestations of this mathematical intuition. For it is mathematical intuition that urges the mind on to follow the mirage of the absolute and so enriches the intellectual heritage of the race; but when further pursuit of the mirage would endanger this heritage, it is mathematical intuition that halts the mind in its flight, while it whispers slyly: “How strangely the pursued resembles the pursuer!”

Part Two: Problems, Old and New

[1953. “Part Two should not be construed as a commentary on the original text, but as an integrated story of the development of method and argument in the field of number.” – Dantzig – Preface to the Fourth Edition.]

APPENDIX D On Principles and Arguments

“Mathematicians do not deal in objects, but in relations between objects; thus, they are free to replace some objects by others so long as the relations remain unchanged. Content to them is irrelevant: they are interested in form only.”

The Terms “Possible” and “Impossible” in Geometry

The terms “possible” or “impossible,” as applied to geometrical construction, have no absolute significance: we must stipulate in each case the equipment by means of which the construction is to be executed.

The Measurable and the Commensurate

… To the early Pythagoreans every triangle was a rational triangle, because they held that all things measurable were commensurate. This last dictum seems to them as incontrovertible as any axiom; and when they proclaimed that number ruled the universe, they meant by number integer, for the very conception that magnitudes might exist which were not directly amenable to integers was alien to their outlook as well as to their experience.

Some modern interpreters of mathematical thought have been inclined to dismiss the ideas of the early Pythagoreans as naïve notions of a bygone age. And yet in the eyes of the individual who uses mathematical tools in his daily work—and his name today is legion—but to whom mathematics is but a means to an end, and never an end in itself, these notions are neither obsolete nor naïve. For such numbers as are of practical significance to him result either from counting or from measuring, and are, therefore, either integers or rational fractions. To be sure, he may have learned to use with comparative facility symbols and terms which allude to the existence of non-rational entities, but this phraseology is to him but a useful turn of speech. In the end, the rational number emerges as the only magnitude that can be put to practical use.

This individual would feel far more at home among the early Pythagoreans than among their more rigorous successors. He would willingly embrace their credo that all things measurable are commensurate. Indeed, he would be at a loss to understand why a principle so beautiful in its simplicity was so wantonly dismissed. And, in the end, the mathematician would be forced to concede that the principle was abandoned not because it contradicted experience, but because it was found to be incompatible with the axioms of geometry.

Time and the Continuum

… To reduce a physical phenomenon to number without destroying its stream-like character— such is the Herculean task of the mathematical physicist; and, in a broad sense, geometry too should be viewed as but a branch of physics.

Mathematics and Reality

Classical science assigned to man an exceptional position in the scheme of things: he was capable of detaching himself from the ties which chained him to the universal mechanism, and of appraising this latter in true perspective. … The book of nature lay open before his eyes; he had but to decipher the code in which it was written, and his faculties were equal to the task.
This code was rational: the immutable order that was man’s to contemplate was governed by rational laws; the universe had been designed on patterns which human reason would have devised, had it been entrusted with the task; the structure of the universe was reducible to a rational discipline; its code of laws could be deduced from a finite body of premises by means of the syllogisms of formal logic. These premises derived their validity not from speculation but from experience, which alone could decide the merit of a theory. [Speculation constantly gained] by contact with the firm reality of experience.
The mathematical method reflected the universe. It had the power to produce an inexhaustible variety of rational forms. Among these was that cosmic form which some day may embrace the universe in a single sweep. By successive approximations science would eventually attain this cosmic form, for with each successive step it was getting nearer and nearer to it. The very structure of mathematics guaranteed this asymptotic approach, since every successive generalization embraced a larger portion of the universe, without ever surrendering any of the previously acquired territory.

Mathematics and experiment reign more firmly than ever over the new physics ,but an all-pervading skepticism has affected their validity. Man’s confident belief in the absolute validity of the two methods has been found to be of an anthropomorphic origin; both have been found to rest on articles of faith.
Mathematics would collapse like a house of cards were it deprived of the certainties that man may safely proceed as though he possessed an unlimited memory, and an inexhaustible life lay ahead of him. It is on this assumption that the validity of infinite processes is based, and these processes dominate mathematical analysis. But this is not all: arithmetic itself would lose its generality were this hypothesis refuted, for our concept of whole number is inseparable from it; and so would geometry and mechanics. This catastrophe would in turn uproot the whole edifice of the physical sciences.
The validity of experience rests on our faith that the future will resemble the past. We believe that because in a series of events which appear to us similar in character a certain tendency has manifested itself, this tendency reveals permanence, and that this permanence will be the more assured for the future, the more uniformly and regularly it has been witnessed in the past. And yet this validity of inference, on which all empirical knowledge is based, may rest on no firmer foundation than the human longing for certainty and permanence.
And this unbridgeable chasm between our unorganized experience and systematic experiment! Our instruments of detection and measurement, which we have been trained to regard as refined extensions of our senses, are they not like loaded dice, charged as they are with preconceived notions concerning the very things which we are seeking to determine? Is not our scientific knowledge a colossal, even though unconscious, attempt to counterfeit by number the vague and elusive world disclosed to our senses? Color, sound, and warmth reduced to frequencies of vibrations, taste and odor to numerical subscripts in chemical formulae, are these the reality that pervades our consciousness?
In this, then, modern science differs from its classical predecessor: it has recognized the anthropomorphic origin and nature of human knowledge. Be it determinism or rationality, empiricism or the mathematical method, it has recognized that man is the measure of all things, and that there is no other measure.

The End.



[Readers] of Number should be aware that although few of the prize problems mentioned in Number have been solved, the past 50 years of attempts at solving problems like them have given us a higher—much higher—comprehension of the things we do when we do mathematics. We now see it all coming from that one great and stable unifying source—the thing that is mathematics. This viewpoint was unavailable to Dantzig and other mathematicians working in the first half of the twentieth century.
We know also—just as Dantzig did back in 1954—that great theorems of mathematics tidily unveil themselves in one branch to cast teasing silhouettes on delicate curtains separating others. Perhaps some curtains will gently separate in the breeze of the next 50 years.

Joseph Mazur

Further Readings

Lest the conclusion of the afterword be too enigmatic, Mazur recommends:

  • Whitehead, Alfred North. An Introduction to Mathematics. New York: Henry Holt, 1939.
  • Russell, Bertrand.
    • Introduction to Mathematical Philosophy. London: George Allen and Unwin, 1919.
    • Our Knowledge of the External World. Chicago: Open Court, 1914. If you can get your hands on this book, it is well worth going through Chapters 3, 4, and 5.
    • The Principles of Mathematics. London: George Allen & Unwin, 1956. This book was written at the turn of the twentieth century, but it is still one of the best, most clear accounts of the philosophy of mathematics that can be found.
  • Stewart, Ian. Concepts of Modern Mathematics. New York: Dover, 1995. This is precisely about what the title says it is about. If you have ever read other books by this author, you will know that the reading will be clear, concise, accurate, current, and lucid.

My Comments

With only a very few quibbles, I find this a remarkably insightful work.

As a mathematician working with individuals and institutions who see mathematics as a tool and perhaps as a branch of physics or psychology, much of my professional and social life has consisted as a kind of armed truce in which we agree to disagree on the matters that Dantzig discusses, particularly those I have coloured.

Dantzig largely reflects my own views, which have been informed by later work (such as Turing’s).

  • I don’t agree that functions can be reduced to numbers (ch. 4).
  • I recognize that model theory – as developed since Dantzig’s time – now provides a logically sound alternative basis for science and more broadly for ‘principled activity’ in the face of those challenges and uncertainties that Dantzig outlines.
  • I recognize some of the practical implications of these. (Which motivates my blog.)

I had not appreciated the extent to which Poincaré  had anticipated much of the findings that I associate with Whitehead et al (as in Mazur’s recommended reading). In my experience, where mathematical issues seem to me to have been implicated in various crises and disasters, a reading of this book might provide some useful insights to policy and decision-makers whose habits of thought resemble those of Dantzig’s stereotypical ‘man of science’, and that this might be more effective than trying to get them to appreciate the somewhat more challenging modern stuff (e.g., post Turing).

The conclusion to part one seems particularly important. I paraphrase it as:

Simplicity, uniformity, homogeneity, regularity, causality are not to be found ‘out there’ in some supposed ‘objective’ reality, but are human constructs. Their particular formulations,  relevance and reliability in particular cases ought to be considered mathematically (as well as experimentally and psychologically).

For good or ill,  intuition – sound or otherwise – urges us on to ‘progress’. To reduce the dangers, one should ensure that intuition that is at least as sound and active is on the look-out and urging caution, and that mathematical intuition (besides physical and psychological) is given due weight.

It seems to me that most individuals in most fields would go along with that, leading us to question how to judge soundness and what weight is ‘due’, and how we might collaborate in this.

More may follow. Meanwhile, see my blog:

Dave Marsay

Who thinks probability is just a number? A plea.

Many people think – perhaps they were taught it – that it is meaningful to talk about the unconditional probability of ‘Heads’ (I.e. P(Heads)) for a real coin, and even that there are logical or mathematical arguments to this effect. I have been collecting and commenting on works which have been – too widely – interpreted in this way, and quoting their authors in contradiction. De Finetti seemed to be the only example of a respected person who seemed to think that he had provided such an argument. But a friendly economist has just forwarded a link to a recent work that debunks this notion, based on wider  reading of his work.

So, am I done? Does anyone have any seeming mathematical sources for the view that ‘probability is just a number’ for me to consider?

I have already covered:

There are some more modern authors who make strong claims about probability, but – unless you know different – they rely on the above, and hence do not need to be addressed separately. I do also opine on a few less well known sources: you can search my blog to check.

Dave Marsay

Law of Great Numbers: Keynes’ Treatise

Keynes’ Treatise on Probability discusses ‘the law of great numbers’, now more familiar as ‘the law of large numbers’, at some length. Roughly speaking, this is that in the long-run, sample frequencies tendreasonably fast (depending on your assumptions) to probabilities. 

Ch. XXVIII The Law of Great Numbers

Within the part dealing with statistical inference, Keynes says of Poisson’s introduction of the ‘law’:

This is the language of exaggeration; it is also extremely vague. But it is exciting; it seems to open up a whole new field to scientific investigation; and it has had a great influence on subsequent thought. Poisson seems to claim that, in the whole field of chance and variable occurrence, there really exists, amidst the apparent disorder, a discoverable system. Constant causes are always at work and assert themselves in the long run, so that each class of event does eventually occur in a definite proportion of cases. It is not clear how far Poisson’s result is due to à priori reasoning, and how far it is a natural law based on experience; but it is represented as displaying a certain harmony between natural law and the à priori reasoning of probabilities.”

On applications of the supposed law, Keynes notes:

The existence of numerous instances of the Law of Great Numbers, or of something of the kind, is absolutely essential for the importance of Statistical Induction. Apart from this the more precise parts of statistics, the collection of facts for the prediction of future frequencies and associations, would be nearly useless. But the ‘Law of Great Numbers’ is not at all a good name for the principle which underlies Statistical Induction. The ‘Stability of Statistical Frequencies’ would be a much better name for it. The former suggests, as perhaps Poisson intended to suggest, but what is certainly false, that every class of event shows statistical regularity of occurrence if only one takes a sufficient number of instances of it. It also encourages the method of procedure, by which it is thought legitimate to take any observed degree of frequency or association, which is shown in a fairly numerous set of statistics, and to assume with insufficient investigation that, because the statistics are numerous, the observed degree of frequency is therefore stable. Observation shows that some statistical frequencies are, within narrower or wider limits, stable. But stable frequencies are not very common, and cannot be assumed lightly.

Ch. XXIX The Use of A Priori Probabilities for the Prediction of Statistical frequency … 

Bernoulli’s Theorem [concerning the variability of sample proportions] is generally regarded as the central theorem of statistical probability. It embodies the first attempt to deduce the measures of statistical frequencies from the measures of individual probabilities, and …out of it the conception first arose of general laws amongst masses of phenomena, in spite of the uncertainty of each particular case. But, as we shall see, the theorem is only valid subject to stricter qualifications, than have always been remembered, and in conditions which are the exception, not the rule.

… Thus Bernoulli’s Theorem is only valid if our initial data are of such a character that additional knowledge, as to the proportion of failures and successes in one part of a series of cases is altogether irrelevant to our expectation as to the proportion in another part.

Such a condition is very seldom fulfilled. If our initial probability is partly founded upon experience, it is clear that it is liable to modification in the light of further experience. It is, in fact, difficult to give a concrete instance of a case in which the conditions for the application of Bernoulli’s Theorem are completely fulfilled.

It seldom happens, therefore, that we can apply Bernoulli’s Theorem with reference to a long series of natural events. For in such cases we seldom possess the exhaustive knowledge which is necessary. Even where the series is short, the perfectly rigorous application of the Theorem is not likely to be legitimate, and some degree of approximation will be involved in utilising its results.

Adherents of the Frequency Theory of Probability, who use the principal conclusion of Bernoulli’s Theorem as the defining property of all probabilities, sometimes seem to mean no more than that, relative to given evidence, every proposition belongs to some series, to the members of which Bernoulli’s Theorem is rigorously applicable. But the natural series, the series, for example, in which we are most often interested, … is not, as a rule, rigorously subject to the Theorem.

If, for instance, balls are drawn from a bag, which is one, but it is not certainly known which, out of a number of bags containing black and white balls in differing proportions, the knowledge of the colour of the first ball drawn affects the probabilities at the second drawing, because it throws some light upon the question as to which bag is being drawn from.

This last type is that to which most instances conform which are drawn from the real world. A knowledge of the characteristics of some members of a population may give us a clue to the general character of the population in question. Yet it is this type, where there is a change in knowledge but no change in the material conditions from one instance to the next, which is most frequently overlooked.

Keynes gives the following examples:

For consider the case of a coin of which it is given that the two faces are either both heads or both tails: at every toss, provided that the results of the other tosses are unknown, the probability of heads is and the probability of tails is 1/2; yet the probability of m heads and m tails in 2m tosses is zero, and it is certain à priori that there will be either 2m heads or none. Clearly Bernoulli’s Theorem is inapplicable to such a case. And this is but an extreme case of a normal condition.

If we are given a penny of which we have no reason to doubt the regularity, the probability of heads at the first toss is 1/2 ; but if heads fall at every one of the first 999 tosses, it becomes reasonable to estimate the probability of heads at the thousandth toss at much more than 1/2 . For the à priori probability of its being a conjurer’s penny, or otherwise biassed so as to fall heads almost invariably, is not usually so infinitesimally small as (1/2 )<sup>1000</sup>. We can only apply Bernoulli’s Theorem with rigour for a prediction as to the penny’s behaviour over a series of a thousand tosses, if we have à priori such exhaustive knowledge of the penny’s constitution and of the other conditions of the problem that 999 heads running would not cause us to modify in any respect our prediction à priori.

Dave Marsay

du Sautoy’s What we cannot know

Marcus du Sautoy What we Cannot Know: From consciousness to the cosmos, the cutting edge of science explained 4th Estate 2017

This is a very readable account of some interesting physics, leading to some speculations on consciousness, from the perspective of a mathematician with a UK professorship in the public understanding of science, written for a general audience. It is full of good examples and insights, and pithy remarks.

It limits itself to what most people would broadly consider physics, considering consciousness and free-will from the point of view of physics. It might nevertheless be useful for anyone reflecting on the topic more broadly.

The author seems to share a common view, that the world out to be somehow comprehensible, and that the role of science (in particular physics) is to make it so. Yet at the same time, like the scientists he asks about this, he wouldn’t want to know all the answers, for then what would he do? As a mathematician myself, it seems to me that what he would really like is for there to be a countable infinity of things to be known, such that he while nothing was unknowable, there would always be more to be known. This seems to be more or less where he ends up. But I found his journey of discovery very interesting.

One thing that I think is very important is that many of the questions that people may have explicitly call for straightforward answers (such as ’42’ or ‘No’)  where there is, in fact, no possible simple answer. But it is not that the answers are unknowable: we know that the questions as posed are some form of nonsense: often one can identify a genuine concern behind the question and answer that instead. He is concerned with meaningful unknowns, and asking if any of them are unknowable.

Marcus presents his thoughts in a series of ‘edges of knowledge’, each with a series of chapters. He starts from the familiar and accessible edge, ending with mathematics and logic. But I shall start commenting on his last, as this is fundamental.

Seventh Edge: The Christmas cracker

Marcus has made some crackers with logical ‘jokes’ based on mathematics. For most of us these are puzzles rather than jokes. His final section is ‘here be dragons’, and his account of mathematics and logic does much to point them out for the unwary. But early on in the final chapter (14) he quotes

a fantastic piece of logical trickery called the paradox of unknowability [credited to] Alonzo Church [via a paper by Fitch] promising a [logical] truth that will never be known by any means.

I comment on this here: Marcus seems to be quoting a philosopher’s interpretation rather than what the logic actually says, which doesn’t seem to go beyond the work of Gödel, as previously described by Marcus, to the effect that there is no algorithm for determining which statements about numbers are true.

Informally, if K is what we explicitly know, S is a ‘legal’ statement and M is a definite method (or ‘algorithm’) that applies to such statements to yield a binary ‘True’ or ‘False’ then either for some (a) false S M(S)=’True’ or (b) for some true S M(S)=’False’. Yet S may still be knowable in the sense that for some method M, we can be sure that (a) never happens and for every S there may be a valid construct C(S) such that M(S^C(S))=’True’. That is, recognizing that the construct is possible enables the truth of the statement to be determined. Marcus gives many examples of this, both in mathematics and physics. It implies that there will always be jobs for inventive logisticians, mathematicians and scientists.

Sometimes, as in Marcus’s discussion of a result by Cohen, there will be alternative possible constructs (or ‘conjectures’) that yield different answers, so only one of the constructs can be valid and we might never be able to known which one: each characterises a different subject. The possibility that is left open is that there may be an unknowable set of alternative subjects. It seems to me that an insight of Charles Dodgson applies here: if our subject is ‘essentially infinite dimensional’ then however much we try to model it mathematically there will always be infinite many dimensions left over. If one accepts the ‘axiom’ of choice, then such mathematical structures may be said to ‘exist’ even if they can’t be explicitly constructed. Marcus doesn’t consider this axiom. Some people limit themselves to mathematics which does not rely on this conjecture, but even so there seems no logical reason to suppose that their judgment in any ways constrains physical reality, whatever that is.

What I take from Marcus’ account of physics is that reality may be (and ‘probably’ is) essentially infinitely dimensional, or at least has way too many dimensions for us to cope with in any foreseeable future.

Edge Zero: The Known Unknowns

The desire to know is programmed [sic] into the human psyche. Those early humans with a thirst for knowledge are those who have survive, adapted, transformed their knowledge.

Just because the scientific community accepts a story as the current best fit, this doesn’t mean that it is true. … Mathematics perhaps has a slightly different quality, as I will discuss in the final two chapters [13 and 14].

The known unknowns outstrip the known knowns.

Rumsfeld is quoted:

[There] are also unknown unknowns, the ones we don’t know we don’t know.

Marcus adds:

He perhaps missed one interesting category: The unknown knowns. The things that you dare not admit to knowing.

If you are going to prove existence or otherwise in mathematics [or logic], you need a very clear definition of what it is that you are trying to prove exists.

First Edge: The Casino Dice


The unpredictable and the predetermined unfold together to make everything the way it is. …

Tom Stoppard, Arcadia


Today these probabilistic methods are our best weapon in trying to navigate everything from the behaviour of particles in a gas to the ups and downs of the stock market. Indeed, the very nature of matter itself seems to be at the mercy of the mathematics of probability, as we shall discover in the third edge … .


Any error … may render an acceptable prediction of the state in the distant future impossible.

“Like causes produce like effects” … is false.

The past even more than the future is probably something we can never truly know.

[The] idea of punctuated equilibria … captures the fact that species seem to remain stable for long periods and then undergo what appears to be quite rapid evolutionary change. This has also been shown to be a property of chaotic systems. The implications … are that many [questions] could well fall under the umbrella of things we cannot know because of their connections to the mathematics of chaos.

A dice that is fair when static may actually be biased when one adds in dynamics.

Note that the mathematics of probability theory is often introduced by reference to things like throwing dice and tossing coins, as if they are sure to be random. Thus mathematics teachers and some text books may reasonably be said to be wrong or at least misleading. But the error is scientific, not mathematical. It may be that mathematical probability theory is an accurate model of how some people think about dice and coins, but the validity of the mathematics does not depend on psychology: mathematics, as Marcus uses the term, has an internal validity that is separate from any empirical claims about it. That is why this section is about science, not mathematics. But, as with chaos theory, mathematics can be used to show when some common intuitions are wrong, which is not altogether useless or unimportant.

Second Edge: The Cello


Everyone takes the limits of his own vision for the limits of the world.

Arthur Shopenauer

The way science works is that you can hang on to your model of the universe until something pops up that doesn’t fit … .


Everything we call real is made of things that cannot be regarded as real.

Niels Bohr

Will we ever find ourselves at the point at which there are no new layers of reality to reveal? Can we ever know that the latest theory will be the last theory?

Third Edge: the pot of Uraniumb


It is absolutely necessary, for progress in science, to have uncertainty as a fundamental part of your inner nature.

Richard Feynman

[My] First Edge revealed that the randomness that is meant to describe the roll of the dice is just an expression of lack of knowledge, the world of the very small seems to have randomness at its heart … .

Repeat the experiment under precisely the same conditions and you may get a different answer each time.

It was the code-cracking mathematician Alan Turing who first realized that continually observing an unstable particle could somehow freeze it and stop it evolving. The phenomenon became known as the quantum Zeno effect … .

The probabilistic character and uncertainty occurs when I observe the particle and try to extract classical information.

[The] physicist David Mermin is reputed to have said to those, like me, who are unhappy with this unknown: “Shut up and calculate.” It is the same principle as the theory of probability applied to the throw of the dice.

I don’t demand that a theory correspond to reality because I don’t know what it is. Reality is not a quality you can test with litmus paper. All I’m concerned with is that the theory should predict the result of measurements.


Some would question if it makes sense to talk about setting up the experiment and running it again with exactly the same conditions – that in fact it is an impossibility.


How puzzling these changes are …

Lewis Carroll, Alice’s Adventures in Wonderland

[Marcus does not point out that Lewis Carroll is the pen-name of Charles Lutwidge Dodgson, who – as noted before – pointed out that a finite number of constraints applied to an infinite-dimension situation cannot produce a finite-dimensional situation, supporting the view of ‘some’ immediately above.]

Quantum physics isn’t about knowing answers to old questions, but about challenging the questions we are allowed to ask.

I’m happy with the maths – it’s trying top interpret where it’s got me that is tough. It’s almost as if we don’t have the language to reverse-translate what the maths is telling us about reality.

The uncertainty principle not only explains the unpredictability of my pot of uranium but also places limits on the knowledge that I can access as I try to zoom ever closer on the insides of my dice and see what is going on.

The uncertainty principle is perhaps more than just an expression of what we cannot know. Rather, it represents a limit of a definition of a concept.

Einstein … believed that there must be smaller cogs that control the outcomes of measurements.

What is proved by impossibility proofs is lack of imagination.

John Bell

The atoms or elementary particles themselves are not real; they form a world of potentialities or possibilities rather than one of things or facts.


Fourth Edge: The Cut-out Universe


For millennia it had been thought that Euclidean Geometry was ‘true’ of physical space. Eddington confirmed a theory of Einstein, that space is curved by mass. This raises a range of possibilities about the extent and form of space: infinite or finite? Curved in on itself, or outward?


Considering the dynamics of space raises even more questions.

Change the cosmological constant by something in the 123rd decimal place and suddenly it’s impossible to have habitable galaxies.

[We] call things beautiful because this is our body’s response to something that will be advantageous to our evolutionary survival.

Perhaps the real lesson is that ‘what we cannot know’ is something that we can never know because it is so hard to preclude the possibility of new ideas that might pull the unknowns into the known – just as Comte found when we discovered what stars are made from.

The Fifth Edge: The Wristwatch


When I was younger I thought it might be possible to know everything. I just needed enough time.

It is striking that often when a question arises to which it seems we cannot know the answer, it turns out that I need to acknowledge that the question is not well posed.


A singularity is a point at which our ability to model the scenario breaks down. A place where we throw up our hands and declare that we do not know.

We have to be careful about mathematical equations, because there may be some hidden piece that becomes significant only when we approach the singularity, and which will then play a large role in preventing any physical realisation of this infinity.


believes that th bumping together of black holes towards the closing stages of the last aeon [before the ‘big bang’] will have caused gravitational ripples that passed into our aeon. … [He doesn’t] like the term “unknowable”.’ It just means we’re not looking at the thing in the right way.’

[Quidditism is] the idea that there is more to the universe than just the relationship between objects – what they are (the quid is Latin for ‘what’) provides another level of distinction.

If there was no universe, no matter, no space, nothing, I think there would still be mathematics. [It] is a very strong candidate for the initial cause. It also explains the ‘unreasonable effectiveness of mathematics’.

Rovelli and Connes are able to demonstrate mathematically how this incomplete knowledge [of microscopic states] can give rise to a flow that has all the properties that  we associate with our sensation of time.

Sixth Edge: The Chatbot App


[The] sceptics believed that nothing could be known for certain.

There has been a growing trend for ’emergent phenomena’, a term coined to express how things arise from more fundamenetal entities and yet are themselves fundamental and irreducible.

I often wonder whether mathematics offers a good example of dualism, something which exists in a purely mental realm. Our own access to this world is certainly dependent on the physical realization of the mathematics.


Tononi … has developed a new theory of networks that he believes are conscious … integrated information theory [which] includes a mathematical formula [Φ] that measures the amount of integration and irreducibility in a network … . [When] Φ is high .. the network feeds back and forth .. .

It is [the] ability of the human mind to integrate and pick out what is significant that is at the heart of Tononi’s measure of consciousness Φ.

I can certainly define  as consciousness, but isn’t this the one thing that by its nature is beyond the ability of science to investigate empirically?

Consciousness is ultimately how much difference you make to yourself, the cause-effect power you have on yourself.

Wittgenstein explores how a sentence, or question, fools us into thinking it means something because it takes exactly the form of a real sentence, but when you examine it carefully you find that it doesn’t actually refer to anything.

Only when you can know there is a difference is there any point having a word to describe it.

Seventh Edge: The Christmas Cracker


In science the things we think we know … are things that match the data. … Eventually, we may well [sic] hit on the right model …, which won’t be rocked by further revelations. But we’ll never know for sure that we have got the right model.

Why is the process of attaining mathematical truths so different from that faced by the scientist who can never really know.


‘In mathematics the art of asking questions is more valuable than solving problems’. [Cantor]

Building on work done by Gödel, Paul Cohen, a logician at Stanford, demonstrated that you couldn’t prove from the axioms we currently use for mathematics whether or not there was a set of numbers whose size was strictly between the number of whole numbers and all infinite decimal numbers. In fact, he produced two different models of numbers that satisfied the axioms we use for mathematics: in one model the answer … was yes, in the other … the answer was no.

Any attempt to explain, for example, why induction is the right strategy for studying physical phenomena is going to rely on induction. The whole thing becomes very circular.

[There] are many different sorts of mathematics. … Sometimes … your choice may be based on your personal relationship to the consequences that follow from working within that system.
In mathematics we are freed up from this need to choose. As a mathematician I’m quite happy to move between different mathematical models that are individually self-consistent yet mutually contradictory.

[But you] can’t just assume that something you don’t know can be true or false.

[There is a] tension between mathematics and physics. Mathematics has for centuries been happy with the mathematical multiverse: different, mutually exclusive models of geometry or number. But even if the physicist is happy with the idea of the multiverse, there is still the desire to identify which of these possibilities describes the universe we are part of.

Science charts a single pathway through a tree of possible universes, mathematics maps every possible journey.

The limitations of language are at the heart of many of the limits of knowledge, and these could possibly evolve and change. … Try to translate … mathematics into the language of everyday experience and [sometimes] we create absurdities … . [Some ‘paradoxes’ are] a failure of translation from mathematics to natural language.
But we must always recognize that we are bound by the ways of thinking particular to our own moment in history. … I wonder if the safest bet is to say that we can never truly know for sure what it is we cannot know.

[A] state of humility is intellectually important, or we will live in a state of delusion and hubris. Yet … we cannot always know what it is that will forever transcend our understanding. That is essential for a scientist not to give in too early.

Marcus goes on to discuss some philosophers’ interpretation Fitch’s logic, as in my early section.

The fact [sic] that science works so well at making predictions of the way things appear is perhaps the best measure there is that we are close to explaining the truth. … Science may not really represent reality, but there isn’t anything that comes close as an alternative.

‘It is wrong to think that the task of physics is to find out how nature is. Physics concerns what we can say about nature’. [Nils Bohr]

[Science] flourishes when we share the unknowable with other disciplines.

‘Thoroughly conscious ignorance is the prelude to every real advance in science.’ [Maxwell]

‘The greatest enemy of knowledge is not ignorance but the illusion of knowledge’. [Stephen Hawking]

It is important to recognize that we must live with uncertainty, with the unknown, the unknowable.

My Conclusions

Marcus speculates that we might one day have a settled cosmology. If so, it might seem that  we must have a settled mathematical model. But it is only necessary that we have a set of models all of which have the same cosmological implications. These implications will follow from the interpretations of the associated logics.

A theory will seem settled when it has been maximally challenged. In science, there is always the possibility that an innovative new experiment, or just a ‘lucky’ observation, will falsify it, so – as Marcus says – we can never be sure that any such theory is settled come what may. All we can be sure of is the challenges that it has survived. But what about logic? It seems to me that we might reasonably hope that our logics will converge onto something final as we challenge them. At least, any test of a scientific theory is at least implicitly a test of our logics, and so I think it possible that we could derive a logic that is ‘more’ tested than any science. Mathematics should then be at least as reliable as its logic and our application of it, while the choice of which mathematics to use in a particular situation is only as reliable as the science. Thus a mathematical model has two aspects: as mathematics and as science. Euclidean Geometry, for example, seems to be good mathematics but bad cosmology.

My own view is that mathematical modelling is always potentially useful, but is potentially dangerous if its role in the science is misunderstood: the best a model can do is to represent an idea: it can show that the idea is possible, not that it is actually ‘true to reality’.

A good model will be consistent with both some notion of ‘how things work’ and with evidence, based on observation or experience. We can know that a model is good in this sense. But sometimes we seem to think that a model is uniquely good. How so? Different models may have different implications. For example, different precise models may make different predictions. Typically we consider a model to be almost final when we can’t imagine any alternative that is good but with different testable implications. But this depends both on our imagination and on our ability to test. And on the tendency of events to produce unlooked-for challenges. Mathematics ‘as such’ avoids these limitations by explicitly narrowing the scope of both the alternative models (using ‘axioms’) and of its implications (using pure logics), making no empirical claims.

The above quibbles apart, the book seems to provide a good overview of science. In particular, the conclusions of any science are only to be taken seriously to the extent to which they have survived serious challenges.

Dave Marsay





logic. But maybe

a set of logics that



Gerstein’s Flirting with Disaster

Marc Gerstein Flirting with Disaster: Why accidents are rarely accidental Union Square Press, 2008

With Michael Ellsberg, and a foreword and afterword by Daniel.


Learning from Past Disasters, Preventing Future Ones

I [Daniel Ellsberg] have participated in several major organizational catastrophes.

… The escalation in Vietnam was not the result of a universal failure of foresight among the president’s advisers, or to a lack of authoritative, precise, and urgently expressed warnings against his choice of policy.

Since 1961 … I have viewed the nuclear arms race as an ongoing catastrophe that has to be reveresed, and a situation that has to be understood. I assumed then, and still believe, that understanding the past and present realities is essential to changing them.

A major theme to be gained from this important book is that organizations do not routinely and systematically learn from past errors and disasters – in fact they rarely ever do.

One reaason for this folly is that many aspects of disasters in decision-making are known only within the organization, and not even by many insiders at that.

[The] deliberate decision within organizations not to try to learn internally what has gone wrong constitutes [an] anti-learning mechanism.

[There] is strong and succesful resistance within many organizations to studying or recording past actions leading to catastrophe – because doing so would reveal errors, lies, or even crimes.


This book is about disasters. From Chernobyl to Katrina, Challenger to Columbia, BP to Vioxx, and the Iraq War.  Were these – and practically every major catastophe that has befallen us in the past twenty-five years – unforeseen, unavoidable misfortunes that no-one could possibly have imagined? No. All of them, it turns out, were accidents waiting to happen, and many influential people inside saw them as such. The events were not really “accidental” at all.

[They] had long buildups and numerous warning signs. What’s more, they display a startling number of common causes, and the same triumph of misguided intuition over analysis … .

1. The Bystanders among us

Organizational bystanders are individuals who fail to take necessary action even when important threats or opportunities arise.

[This often happens in situations with the following characteristics]:

  • Ambiguous precipitating events.
  • A large number of people observe the event.
  • Failure of others to act.
  • Uncertainty regarding one’s ability to help.
  • Presence of formal authorities or “experts”.

Groupthink and other psychological, social and political phenomena may also be a problem.

Factors promoting bystander behaviour work in concert to maintain the status quo … . [Not] “rocking the boat” often becomes a way of life – even while that boat is going right over Niagra Falls.

2. Human Biases and Distortions

This summarises relevant work, mainly from the mainstream psychology literature. On evolutionary psychology it notes that:

[One] might expect humans to value direct experience, prefer hard facts to soft knowledge, and to put their trust in their own abilities rather than in abstractions while, at the same time, reserving respect for people they might encounter from whom they might acquire knowledge and new skills.

Rather than being generally cautious or risk-seeking, human beings appear to have been programmed by evolution to live with false positives to avoid a disadvantageous false negative, and sometimes the reverse. [We] seek to minimize the more costly error.

While some of the self-protective biases developed over countless evolutionary generations are not always functional in today’s society,  the principle of avoiding the greater mistake still works.

Unfortunately … the greater mistake appears to be the one that affects them personally rather than the one that one that produces the greater good (sic).

3. Understanding Uncertainty

Why did so many people bet against Katrina?

If ever there was “an accident waiting to happen” …

If we are to avoid such colossal mistakes in the future, we must learn to face the probabilities and likelihoods squarely, and be less sanguine that everything will work out okay if we merely follow our instincts.

4. Space Shuttle Challenger

Cold, Warm, and Hot causes of disasters.

5. Chernobyl

Faulty design, and the interplay of humans and technology

6. The Vioxx Disaster and BP

The seduction of profits

7. When all the Backups failed

How American F-15s accidentally shot down two U.S. Army Black Hawks.

8. Butterfly Wings and Stone Heads

How complexity influences catastrophe in policy decisions.

9. The Collapse of Arthur Andersen

The role of organizational culture.

10. When Countries go Bankrupt

The Prisoner’s Dilemma writ large.

11. What Have We Learned? What Can We Do?

   Despite its apparent simplicity, the advice offered here is often difficult to implement. [People] are tempted by short-term gains or coerced by social pressure, and then their risky behaviour is strongly reinforced when they repeatedly get away without incident.

Rules to Live By

  • Understanding the risks you face.
  • Avoid being in denial.
  • Pay attention to weak signals and early warnings.
  • Not to subordinate the chance to avoid catastrophe to other considerations.
  • Don’t delay by waiting for absolute proof or permission to act.

Avoiding the Bigger Mistake

   The principle of avoiding the bigger mistake underlies dealing sensibly with all types of risk.

The Disaster Time Line

Disasters can be partitioned into before, during, and after … .

[There] is usually ample time before any low-probability hazard breaks loose. The problem, of course, is getting people to pay attention so that the lead time can be produced productively.

Don’t squander your early warnings with delays or half measures.

 Real-Time Responses

The most important thing is to think through what’s possible and, most of all, what’s most likely, as well as what we can do and should do in each instance, including the worst-case scenario.

Believe the Facts, not Your Intuition

Some common examples are given, some of which may be surprising.

The most important lesson is that constant vigilance is required in all high-hazard categories.

Take Low-Probability Catastrophic Risks seriously

Understand each threat, its severity, and timing, decide in advance … .

Pay attention to Weak Signals and Near Misses

Perhaps the biggest fallacy associated with the signalling of risk is that certainty of an accident is required in order to take action. On the contrary, taking action on the basis of rational concern before an adverse event occurs is obviously better than preventing harm from the second such accident.

Always pay attention as if the worst had actually occurred, but develop efficient ways of confirming, or disconfirming, the actual danger to minimize your time and effort.

Everyday Life’s Low-Probability Risks

   When the consequences of an adverse event exceed our tolerance … we must fully acknowledge them, along with the costs of full protection. That is often an unpleasant recognition, and is prone to self-deception – and politics.

For very rare events, such as the outbreak of a devastating flu … preparations are a balancing act. [It] is wise to be as self-sufficient as possible [even though] there are som many potential scenarios.

Examine the cumulative risk of all low-probability threats and make your plans according to the rule of avoiding the greater mistake.

Endeavour to be “Risk-Neutral”

Be “Safe at any Speed”

Moving from Bystander to Witness to Whistle-Blower

  Sometimes just “active watching,” visibly taking notes, or writing a concerned e-mail is enough to change the course of a situation.

[Cautious] legal advisers suggest making one’s protests within the chain of command or other legitimate avenues, but then departing the organization on good terms if one’s complaint comes to naught.

12. Advice for Leaders

Virtually all the accidents described in this book occurred in organizational settings.

This book [is] not about … fraudulent enterprises deeply based on deceit, if not deliberate harm.

[Most] people are not harmed by malice but by risk blindness, the failure to see potential harm and imminent danger until it is too late.

We must all take a greater share in preventing unintended consequences.

[Conflicts] between espoused values and actual practice inevitably draw people into loyalty tests and cover-ups when an apparently sensible short-cut invites a catastrophic outcome.

Suggestions for Professionals and Managers

  1. We shouldn’t be bystanders and shouldn’t encourage bystander behaviour in those around us.
  2. We should all do what we can to ensure that dissent is encouraged, not repressed, and that the channels of complaint are open.
  3. We should do what we can to build viable information and reporting systems that widely disseminate risk-relate performance information.
  4. We should not collude in cover-ups, even minor ones.
  5. When there is likely and documentable unacknowledged risk, each of us should assemble our allies and pursue a complaint with the appropriate institutional body.

If all else fails, we should consider blowing the whistle (with documents).

Avoiding Catastrophe is Free

Or at least, cheaper than the alternative.

Suggestions for Leaders

The key to figuring out what to do is realizing that practicalities and shortcuts have costs that inevitably even out in time, and that one’s choice is to either pay now or pay later. (sic)

Unfortunately, unless a crisis erupts, in many organizations traditional organizational performance policies largely ignore safety and ethical risks by focusing on short-term backward-looking financial indicators. Such measures encourage imprudent risk-taking, although it takes exceptional candour to admit it.

… “What gets measured gets managed” is more true today than ever.

[Imposing] nonnegotiable performance objectives combined with severe sanctions for failure encourages the violation of safety rules, reporting distortions, and dangerous shortcuts.

  1. [Be] wary of excessive optimism. [Admitting] mistakes and accepting the need for radical solutions are essential.
  2. In organizational settings, accidents are never “accidental”: They are inevitably the result of faulty management, particularly the management of safety.
  3. Systematize paying attention to near-misses, weak signals and assessments of engineers and safety officials. … Create monitoring systems, systematic review procedures, and independent channels that do not report through the operational chain of command.
  4. [Recognize] that while every organization tolerates some dissent, on certain subjects it does not. Only leaders can eliminate these “undiscussables.” Without adequate protection for naysayers and internal whistle-blowers, widespread bystander behavior is inevitable.
  5. Create effective contingency plans for serious but low-probability risks. [The] combination of situational unfamiliarity, time pressure, and poor information is lethal – it increases the chances of an accident [hugely].
  6. Every organization requires robust, independent watchdogs. Actions that make watchdogs more “client-centre” and “efficient” run serious risks of reducing their effectiveness.
  7. Leadership must subject itself to relentless review and self-criticism.

Conclusion: An Opportunity, Not a Problem

  1. [Accept] the inevitability of accidents and catastrophes without giving in to them.
  2. [Appreciate] the difference between new ideas and unpracticed old ones. .. The later … is often where the bulk of the value actually lies.

Afterword: When the Leaders are The Problem

[Reasonable] people, who are not malicious, and whose intent is not to kill or injure other people, will nonetheless risk killing vast numbers of people. And they will do it predictably, with awareness.

What are the circumstances … ? [When] the potentially disasterous gamble offers the possibility of avoiding loss altogether, coming out even or a little ahead; and when the alternative to taking the gamble assures the certainty of loss in the short run – a loss that impacts the leader personally.

Dave Marsay

Essex’s Blunders of UK Government

Anthony King and Ivor Crewe The Blunders of Our Governments OneWorld 2014 (Revised and Updated)

This is a University of Essex view of UK Governments, focussing on the period 1979 (Thatcher, Tory) to 2010 (End of Brown, Labour), with an Epilogue from June 2014 (the Tory/Liberal coalition).


We believe that there have been far too many [blunders] and that most … of them could have been avoided. [Once, British government] was astonishingly competent. … Sadly, the British system is no longer held up as a model, and we suspect one reason is that today’s British governments screw up so often.

They screw up more often than most people realise. …

… Governments of both (sic) parties seem to blunder in much the same way … .

Part I: To begin with …

1 Blunders, judgement calls and institutions

We define a blunder as an episode in which a government adopts a specific course of action in order to achieve one or more objectives and, as a result largely or wholly of its own mistakes, either fails compleletlt=y to achieve those objectives , or does achieve some or all of them at a totally disproportionate cost, or else … contrives at the same time to cause a significant amount of “collateral damage” in the form of unintended and undesired consequences.

It is not enough that we don’t like the objectives.

2 An array of successes

The successes of governments are apt to pass unnoticed or else be taken for granted. They should not be.

The BBC, NHS, the Town and Country Planning Act 1947, house building 1950-1953, The Clean Air Act 1956, The Road Safety Act 1967, Decimalisation (1971), The Housing Act 1980, attracting Nissan to Sunderland, various Union legislation (1982-84), some early privatisations, the MMR vaccine, Citizens Charters, the minimum wage, smoking bans, the handling of the 2009 swine-flu pandemic and Gordon Brown’s response to the 2008 financial crisis and the 2012 London and Paralympic Games are all seen as successes, at least on their own terms. But …

Part II: Horror stories

3 Blunders past and present

The postwar Labour government … failed to draw up contingency plans to deal with the possibility, foreseen by some ministers, that severe fuel shortages might combine with an unusually cold winter to cause chaos.

[To] address a serious postwar shortage of vegatable oils … Ministers and their advisers chose to plant [peanuts] in thousands of acres that lacked both proper soil and adequate rainfall. [The scheme] imported more nuts for seed than it managed to harvest.

[The] disasterous 1956 Suez expedition [was] easily Britain’s most egregious postwar foreign-policy blunder.

Another … was … Blue Streak.

Concorde … quickly became known as “the flying white elephant”.

Possibly the biggest blunder committed by Harolf Wilson’s postwar 1964 Labour government was its decision not to devalue the pound … .

Two of the most spectacular government blunders of the postwar period … concerned industrial relations.

And so on …

More recent blunders (chapters 4-15)

The poll tax, pension mis-selling, the so-called ‘Child Support Agency’, exiting the ERM, the millenium dome, individual learning accounts, tax credits, the (criminal) Assets Recovery Agency, the single payment scheme for farming subsidies, many IT projects and London undeground renewal are all exposed at length as blunders. But why do they keep happening?

Part III: Human failures

16 Cultural Disconnect

Everyone projects onto others his or her lifestyles, preferences and attitudes. Some people do it all the time; most of us do it some of the time. … We call these kinds of assumptions “cultural disconnect”.

[In the case of the poll tax, a] minister who arrived later on the scene was blunt: “It needed exceptionally clever people to produce anything so stupid.” {Later, someone] said he still could not decide whether the ministers … had been guilty of “woolly thinking or no thinking.”

One [way to counteract cultural disconnect] is actively to seek out the views and draw on the experience of people on the other side of whichever cultural divide it is thought may exist – and also to draw on the experience of those who have direct dealings with people on the other side of that divide.

If piloting is not practicable, dummy runs and siumulations often are.

[Activists] … behaved as though they imagined that the subculture in which they were embedded was somehow represerntative of the culture of the majority of the whole nation.

17 Group-think

[Cultural ] disconnect … has a near neighbour … “group-think” [, and] they are frequently found cohabiting.

[Group-think] is liable to occur when the members of any face-to-face group feel under pressure to maintain the group’s cohesion or are anyway inclined to want to do that. [It] often does manifest itself in various pathological ways [such as an undue] “Yes, we can” attitude.

“When you are totally focussed on something, you forget to ask yourself whether it is the right thing to do.” [Chanelling Patton, one should realise that] “Everyone agrees. So we must be wrong.”

[Of The London underground fiasco:] Courtiers may have disagreed and argued among themselves; courtiers often do. But they seldom, perhaps never, reached out to others.

[The original group-think concept is] a purely psychological concept. But [it] can be extended … to refer to a variety of situations in which there exists such widespread agreement amoing the members of a group about the desirability of a given course of action that no threats to the groups’ internal cohesion ever arise.

[It] would have been a good idea “to have someone … to put up the case for the other side”.

[There] needs always to be “grit in the oyster”, at least one person present in all group discussions who has been assigned the task of arguing the case on the other side and of ferreting out potential defects in what otherwise seems an unassailable proposition. The danger, of course, is that if the individual in question is sufficiently persuasive … .

An alternative approach is .. to assign the same policymaking task to two or more groups … .

[Or] “after reaching a preliminary consensus about what seems to be the best policy alternative, the policymaking group should hold a ‘second-chance’ meeting at which the members are expected to express as vividly as they can all their residual doubts … ” [Perhaps lubricated by alcohol.]

[An excessively good presentation] priviledges the presenter in an unhealthy way and has the effect of discouraging critical discussion and analysis. It may also encourage group-think. PowerPoint is  a potentially dangerous instrument of persuasion.

Group-think … renders blundering more probable.

18 Prejudice and pragmatism

[Cultural] disconnect .. group-think [and] “intellectual prejudice” … can easily co-exist.

[Intellectual] prejudices are simplifying devices, presumptions, mental short-cuts, hunches even [which] typically embody informal theories concerning how the world works now and how it could be made to work in the future. [They] are likely to be influential because they often go unspecified and unstated [and so] they remain unchallenged. [Those] who hold them are likely to regard them as obvious truths, undeniable facts, things to be taken for granted.

Ronald Reagan’s first director of the Bureau of the Budget [opined] “The world was less manageable than he had imagined; this machine has too many crazy moving parts to incorporate into a single lucid theory.”

[Strong] prejudices seems to preclude … pragmatism – that is, a careful, dispassionate approach to problem solving, one that evaluates beliefs or theories in terms of the probable success of their practical application.

One probable consequence of intellectual prejudice, especially if it is combined with either cultural disconnect (or both) is that policymakers never get around to doing any contingency planning. There are no Plans B.

[They] neglected … to contemplate worst-case scenarios and to be clear in their own minds what they would do if one of their putative worst-case scenarios turned out to be what actually happened.

19 Operational disconnect

[Anyone] planning [an] operation should ideally be put in charge of it – and should know in advance that he is going to be put in charge of it.

No feature of the blunders … stands out more promninently – or more frequently – than the divorce between policymaking and implementation and, in human terms, between those who made policies and those charged with implementing them.

[The] frontline worker does not need to know a great deal about the organisation in which he or she works, but the top manager needs to have at least some grasp of what actually happens on or close to the front line.

[On the poll tax:] Everyone operated on the basis of false assuptions about the thinking of others without realsing that that was what they were doing.

Experience … suggests that mapping backwards, instead of relying solely on mapping forwards, concentrates the mind. It alerts planners to pitfalls (as well as, possibly, to previously undetected opportunities.)

The authors [of a US report on a cabinet blunder noted:] ” “Implementation … is not only something to be done after a decision, it is as much or more a thing to think about before decision, right along with substance.”

The report is critical of Next Steps agencies, which it blamed for many blunders.

[The] easy bit … is deciding what ought to be done: the hard bit is the doing of it, and the hard bit is likely to be very hard.

20 Panic, symbols and spin

Frequently the cry goes up, “Something must be done!”

The Dangerous Dogs Act, the two Firearms acts of 1997, the New Millenium Experience, the Assets Recovery Agency, attempts to introduce ID cards and The Fiscal Responsibility Act (2010) are given as examples of panic, symbols and spin.

All of the phenomena described in the last five chapters – cultural disconnect, group-think, intellectual prejudice, operational disconnect and symbolism and spin – can be found everywhere and in all walks of life. … But all of them are more pervasive in British government than is often realised.

Part IV: System failures

21 The centre cannot hold

British government is not a single, unified entity. [No] prime minister can command and control everything that goes on inside his or her administration.

By international standards, the British prime minister also has limited staff resources. There are not many people, and certainly not many policy experts and administrators, who are his people.

Commentators on British government frequently describe Whitehall departments as living their lives within “silos”. One can see what they mean … .

Neither the prime minister nor any other powerful institution at or near the centre of government is capable in practice of checking and balancing, let alone controlling and directing, much of what goes on elsewhere.

The dangers inherent in the routines of cabinet government is that matters will appear to have been thoroughly discussed and thrashed out when in fact they have not been. Proper form will conceal defective practice.

[Both] decision making and policy development have inevitably been devolved outwards, downwards and side ways to smaller, often ad hoc clusters of people .. individuals whom the prime minister or other senior ministers believe do actually have something to contribute. … “Decision are well made if the right people are in the room and they have all the available facts before them, on paper or orally, if those in the room feel free to challenge propositions and argue, and if the decisions are properly recorded and disseminated.”

A striking feature of the policymaking process that culminated in the blunders described in this book is that “the right people” were often not in the room and that neither the prime minisater or anyone else at the centre possessed both the knowledge and the clout “to challenge propositions and to argue”.

The prime minister’s Policy Unit … has been very small [with] neither the time nor the remit to act effectively, or at all, as an early-warning system, alerting the boss to potential blunders lying in wait and capable of briefing him, or organising a brief to him, on what might be done to avoid them.

22 Musical chairs

In Britian, holders of important portfolios come and go; in most other countries, they come and stay, at least for a while. {New] ministers have little or no idea what is required if their wonderfully brioght ideas are to be given practical effect.

[Ministers] have have every incentive to focus on the short term … .

23 Ministers as activists

[Officials], at least in many government departments and in many policy areas, have become remarkably reluctant to sopeak truth to power.

“No one had the balls to say, ‘Look, let’s stand back and reassess.'” [Civil] servants had failed to perform their historic function.

[Tony Blair’s former chief of staff] regards it as part of [civil servants’] job “to warn their ministers of the elephant traps into which they are about to walk … and they know where the traps are, unlike their minister”.

24 Accountability, lack of

The almost to total lack of accountability in the strong sense – of ministers being held to account for their actions and being penalised for their more egregious misjudgements and errors – is one of the most striking features of the British system of government. Or rather, it would be striking if only more people were aware of it.

The chances of anyone being held to account are also likely to be reduced if the blunder, however large, manifests itself … as a sequence of lesser blunders.

[The] Public Accoiunts Committee is specifically precluded by its remit from enquiring into either the formulation or the substance of government policy .. .

The National Audit Office … concerns itself with how efficiently and effectively government money is spent, not with how policy is formulated or with the wisdom or otherwise of the policy itself.

[Important] policy decisions … typically take years, sometimes even decades, … to play themselves out.

[Politicians lack any incentive] to think a long way ahead, to contemplate seriously the probable consequences for future generations of whatever it is that they do today,

Moral hazard is evidently a feature of policymaking in government as well as in economic life.

There would be a lot to be said for encouraging [relevant bodies] to assess how well government initiatives were continuing to achieve their declared objectives after, say, five, ten or twenty years.

25 A peripheral parliament

As a legislative assembly, the parliament of the United Kingdom is, much of the time, either peripheral or totally irrelevant. It might as well not exist.

[It] is defects in parliament’s institutions, practices and mindsets that have led government in Britain to be more blunder-prone than it needs to be or should be.

26 Asymetries of expertise

[Skills] shortages increase markedly the chances that gross blunders of the kind committed so frequentrly in the past will continue to be committed well into the future. [The] UK government appears destined to go on buying lemons by the basketful, always at other’s expense.

27 A deficit of deliberation

The activity of deliberation has three distinct but interconnected components:. The first is one of careful consideration, of weighing up. … The second .. is that of not being over hasty … . The third … is that of conferring and taking counsel, of reaching out. British politicians in general have a curious habit of functioning in crisis mode – at high speed and in an agitated state – even when no crisis exists. They seem to enjoy it. It seems to give many of them a high.

… Britain’s political system is a power-hoarding system … .

Ministers who take time to make up their minds are said to be dithering, and the same is likely to be said of the few minsitsers who publicly acknowledge that they are weighing up finely balanced arguments.

Understandably, ministers’ elemental fear of losing the next election is a great inducer of caution. It tempts them to put off tackling … “wicked issues”.

[Really serious deliberation is often best conducted in private, behind closed doors, in settings where people find it easier to change their minds and work out compromises.

Epilogue (June 2014)

[The] political process that led to the passage of the Health and Social Care Act 2012 was remarkably shambolic.

Whether civil servants did as they should have to warn Lansley of the practical and political elephants traps ahead is an open question … there was no-one in Number 10 they could talk to and, if need be, warn.

One of their principla aims was to take the NHS effectively out of politics, to create … “a machine tht would go of itself”. But any such effort was doomed from the start.

The handling of the bidding for the West Coast Mainline franchise during the 2010s undoubtedly was [a blunder]. [The] department had “yet again failed to learn from previous disasters”.

There were also blunders in the ‘bonfire of the quanqos’, the introduction of police commissioners, the increase in university tuition fees, attempts to limit net migration to below 100,000, the provision of staff for the 2012 Olympics, disability assessements and the introduction of NHS 111 to replace NHS Direct.

The book also puts seven initiatives on ‘negative watch’, including Universal Credit.

The errors of previous governments … have been been replicated almost uncannily. Cultural disconnect and operational disconnect … have been constantly in evidence. So have intellectual prejudice … and also … proness to to spinning and indulging in symbolic politics, … .

[People] throughout the UK … have … begun to regard constant blundering as the new normal, as something that just happens, something unavoidable. We disagree.


This book will make you gasp in disbelief and stamp your feet in rage … [The Guardian, quoted on the cover]

If this were true, it would be a severe indictment of British news coverage, let alone its investigative reporting. The book itself simply claims that blunders are more common, endemic even, than most people realise. It is in the examples and the analysis that the value of the book lies. Its diagnosis seems plausible, as far as it goes, and gains credence from having been reviewed in 2014. Its prognoses also seem sensible, but – as is clear from the review – had no impact.

Part of the problem, it seems to me, is that the diagnosis is one that is largely familiar to generations of academics and civil servants, and not a few ministers. But what to do about it.  Blunders seem to be due to poor decision-making. Every generation accepts this of its predecessors and many take on board recommendations very similar to those of this book and improve. While this book provides a more compelling case for change than any I can think of, it begs the question: How do we know when w have improved enough? Given that not all are adept at good decisionj making (whatever we that would be), how do we identify particular instances where better decision making is required? What else, apart from this book, is needed to help improve?

I have been involved in too many attempts to improve decision making of many kinds, with partial success. The last UK governmment project was in 2010. It seemed to me that the policymakers were well clued up on all the above issues – perhaps they had been interviewed for this book – and all seemed set fair at last. But, it got immediately messed up right at the start of implementation – unusually soon. In my view a large part of the problem was that the project management policy was changed in ways that the operational policy makers didn’t seem to have anticipated, so that the management structure was compeletely inapprop[riate to the task at hand. Having read this book, I wonder if the management policy makers hadn’t also been involved in it, and had made some changes intended to reduce the ‘chance’ of blunders? If so, talking to the authors may have been a gross blunder.

All this is speculation. But previous failings, it seems to me, have often come about when different areas seek to respond positively to well-intention initiatives or insights, but there – at leat with hindsight – too little co-ordination between them. How could we facilitate appropriate foresight on this issue?

My own long-standing view is that where there are challenging issues, different teams with different perspewtives are going to take time to come to a mutual understanding that is good enough to even begin to comprehend the particular issues, not just those of ‘implementation’. One can’t turn on a sixpence and ‘reform’. One needs a longer bedding-in, based on a sound foundation. But what foundation? How can we tell, with foresight, what will be good enough unless we are clear abouit what will be built on it?

It seems to me that the foundation needs to come before the policy, and that one needs to be clear about the range of possible policies it is fit to support. But how to tell? Lacking any guidance, it seems to me inevitable that policymakers will sometimes go to far. I also incline to the view that many, perhaps most, of our most pressing chronic issues are simply beyond our current foundations, which is why they are chronic.

So, a worthwhile and stimulating book.

Dave Marsay


Having read this book, I wonder if the mangers of the implementation


More … TBC

Eddington’s New Pathways

A.S. Eddington New Pathways in Science: The Messenger Lectures, 1934, Cambridge University Press, 1935


“Does the harmony which human intelligence thinks it discovers in Nature exist apart from such intelligence ? Assuredly no. …What we call “objective reality” is, strictly speaking, that which is common to several thinking beings and might be common to all; this common part, we shall see, can only be the harmony expressed by mathematical laws.”

POINCARE, The Value of Science


As a conscious being I am involved in a story. The perceiving part of my mind tells me a story of a world around me. The story tells of familiar objects. It tells of colours, sounds, scents belonging to these objects; of boundless space in which they have their existence, and of an ever-rolling stream of time bringing change and incident. It tells of other life than mine K.tcy [?] about its own purposes.

As a scientist I have become mistrustful of this story. … One of our first tasks must be to try to understand the relation between die familiar story and the scientific story of what is happening around us.

Scientific thinkers generally agree that the channel of communication between the external world and man’s consciousness is severely limited in this way; but, whilst giving intellectual assent, they do not always adjust their scientific outlook to correspond. They are strangely reluctant to doubt the assertions of the familiar story teller even when it is evident that he is talking through his hat.


Broadly speaking the task of physical science is to infer knowledge of external objects from a set of signals passing along our nerves. But that rather underrates the difficulty of the problem. The material from which we have to make our inferences is not the signals themselves, but a fanciful story which has been in some way based on them. It is as though we were asked to decode a cipher message and were given, not the cipher itself, but a mistranslation of it made by a clumsy amateur.

It is the inexorable law of our acquaintance with the external world that that which is presented for knowing becomes transformed in the process of knowing.

The solution of a cryptogram is found by studying the recurrency of the various signs and indications. I do not think we should ever have made progress with the problem of inference from our sensory experience, and theoretical physics would never have originated, if it were not that certain regularities and recurrencies are noticeable in sensory experience. We call these regularities of experience laws of Nature. When such a law has been established it becomes also a rule of inference, so that it helps us in further decipherment just as in solving an ordinary cryptogram.

I do not know how a logician would classify the process of solving a cryptogram. The decoded message is inferred from the cryptogram, but the method of inference can scarcely be described as logical deduction. In saying that the scientific description of the external world is inferred from our sensory experience, and that the entities of the physical world are inferences, I use the word inference in this broad sense.


But, it may be suggested, if all observation is reduced to coincidences and pointer-readings, can we ever infer from it anything but a system of relationship of coincidences and pointer-readings? In one sense the answer is No. But if the question is put in the form ‘ * Can we by manipulating pointer- readings ever arrive at a knowledge which does not smell of pointer-readings?” I suggest that it might equally well be asked “Can an artist by manipulating paint ever achieve a creation which does not smell of paint ? ” But I do not wish to set this question aside lightly, for it goes to the very heart of the difference between the new and the old scientific outlook. We shall see later that a scheme of relationship, or a structure, has a significance which can be abstracted from the intrinsic nature of that which is the subject of the relationship. The structure is the object of our search, and when we have reached knowledge of the structure we can disregard the scaffolding by which we reached it. It does not lessen the dignity of the structure that its elements are pointer- readings which after all is only the story teller’s name for them.


The rejection of determinism is in no sense an abdication of scientific method. It is rather the fruition of a scientific method which had grown up under the shelter of the old causal method and has now been found to have a wider range. It has greatly increased the power and precision of the mathematical theory of Observed phenomena. On the other hand I cannot agree with those who belittle the philosophical significance of the change.


[The] measurement of probability employed in mathematics and physics has an altogether different basis, as we shall see. …

[There] is no reason to think that these probabilities can be graded systematically in order of magnitude. It has been maintained by some writers that probability always has a numerical measure even when the word is used in this elementary way; and that the beliefs of a right-thinking person could ideally be arranged in a unique sequence in order of intensity. I rate this on a level with the view that to a person with a right sense of humour all jokes can be arranged in a unique sequence in order of funniness.

The probability is p that an event a has an outcome e has to be translated:

The event a is a member of a certain class of events A, and the proportion of events in the class A which have an outcome e is p.

[It] is often assumed that certain events will in the future occur with the same frequency as they have been observed to do in the past. The study of probability is often distracted by a discussion as to whether we have any proof of these assumptions. But the function of probability theory is to utilise such information, not to supply it. When once it is realised that there is nothing illogical in a numerical probability being itself only probable, we can utilise any reasonable belief as to the frequency of events and so determine “a reasonably probable probability”; just as we may use a reasonable belief as to the cause of the recession of die spiral nebulae and so determine a reasonably probable cosmical constant. …

If we maintained, as some have done, that scientific (numerical) probability is the basis of all rational belief other than strict logical deduction, thereby annexing the whole subject of inference to the mathematical theory of probability, it would be necessary to go into the matter further. But that is not the position here adopted.


According to the principle of relativity we can only observe and have knowledge of the relations of things. So when we refer to the properties of any object we must always have a comparison object in mind. …


“There has been a great deal of speculation in traditional philosophy which might have been avoided if the importance of structure, and the difficulty of getting behind it, had been realised. For example, it is often said that space and time are subjective, but they have objective counterparts; or that phenomena are subjective, but are caused by things in themselves, which must have differences inter se corresponding with the differences in the phenomena to which they give rise. Where such hypotheses are made, it is generally supposed that we can know very little about the objective counterparts. In actual fact, however, if the hypotheses , as stated were correct, the objective counterparts would form a world having the same structure as the phenomenal world In short, every proposition having a communicable significance must be true of both worlds or of neither: the only difference must lie in just that essence of individuality which always eludes words and baffles description, but which, for that very reason, is irrelevant to science.”

BERTRAND RUSSELL, Introduction to Mathematical Philosophy, p. 61. I

Let us suppose that a thousand years hence archaeologists are digging over the sites of the forgotten civilisation of Great Britain. They have come across the following literary fragment, which somehow escaped destruction when the abolition of libraries was decreed.

‘Twas brillig, and the slithy toves
Did gyre and gimble in the wabe,

All mimsy were the borogoves
And the mome raths outgrabe.

This is acclaimed as an important addition to the scanty remains of an interesting historical period. But even the experts are not sure what it means. It has been ascertained that the author was an Oxford mathematician; but that does not seem wholly to account for its obscurity. It is certainly descriptive of some kind of activity; but what the actors are, and what kind of actions they are performing, remain an inscrutable mystery. It would therefore seem a plausible suggestion that Mr Dodgson was expounding a theory of the physical universe.

Our account of the external world (when purged of the inventions of the story teller in consciousness) must necessarily be a “Jabberwocky” of unknowable actors executing unknowable actions. How in these conditions can we arrive at any knowledge at all? We must seek a knowledge which is neither of actors nor of actions, but of which the actors and actions are a vehicle. The knowledge we can acquire is knowledge of a structure or pattern contained in the actions. [ think that the artist may partly understand what I mean. (Perhaps that is the explanation of the Jabberwockies that we see hung on the walls of Art exhibitions.) In mathematics we describe such knowledge as knowledge of group structure.

It does not trouble the mathematician that he has to deal with unknown things. At the outset in algebra he handles unknown quantities x and y. His quantities are unknown, but he subjects them to known operations addition, multiplication, etc. Recalling Bertrand Russell’s famous definition, the mathematician never knows what he is talking about, nor whether what he is saying is true; but, we are tempted to add, at least he does know what he is doing. The last limitation would almost seem to disqualify him for treating a universe which is the theatre of unknowable actions and operations. We need a super-mathematics in which the operations are as unknown as the quantities they operate on, and a super-mathematician who does not know what he is doing when he performs these operations. Such a super- mathematics is the Theory of Groups.

It is sufficient to say that what physics ultimately finds in the atom, or indeed in any other entity studied by physical methods, is the structure of a set of operations. We can describe a structure without specifying the materials used; thus the operations that compose the structure can remain unknown. Individually each operation might be anything; it is the way they interlock that concerns us. The equation P 6 P a =P c [I need to fix this!] is an example of a very simple kind of interlocking.The mode of interlocking of the operations, not their nature, is responsible for those manifestations of the external universe which ultimately reach our senses. According to our present outlook this is the basal principle in the philosophy of science.

This mathematical way of describing everything with which we deal emphasises, perhaps inadvertently, an important physical truth. Usually when we wish to consider a problem about a hydrogen atom, we take a blank sheet of paper and mark in first the proton and then the electron. That is all there is in the problem unless or until we draw something else that we suppose to be present. The atom thus presents itself as a work of creation a creation which can be stopped at any stage. When we have created our hydrogen atom, we may or may not go on to create a universe for it to be part of. But the real hydrogen atoms that we experiment on are something selected from an always present universe, often selected or segregated experimentally, and in any case selected in our thoughts. And we are learning to recognise that a hydrogen atom would not be what it is, were it not the result of a selective operation performed on that maze of interrelatedness which we call the universe.


His difficulty rather suggests that a cyclic scheme of knowledge with which science has familiarised us is not yet appreciated in philosophy. I have formerly [ The Nature of the Physical World, p. 262.] illustrated the nature of a cyclic scheme by a revised version of “The House that Jack Built” which instead of coining to an end repeats itself indefinitely ” . . .that worried the cat, that killed the rat, that ate the malt, that lay in the house, that was built by the priest all shaven and shorn, that married . . .”. Wherever we start in the cycle we presuppose something that we reach again by following round the cycle. The scheme of physics constitutes such a cycle; and equally we may contemplate a wider cycle embracing that which is beyond physics. Starting at the point of the cycle which corresponds to our individual perceptions, we reach other entities which are constructs from our perceptions. From these we reach other entities, and so on for a number of steps. When we seem to have travelled a long way from our starting point, we find that our perceptions (or more strictly the recurrencies in our perceptions) reappear as constructs from the last-reached entities. The fact that we return by a circuit and not by retracing our steps secures that our adventure is an extension of knowledge and not an excursion in tautology. By the method of Chapter xii we can extract the group structure from the cycle and so express the same truth symbolically without a formal presupposition if we prefer.


Modern science, in so far as I am familiar with it through my own scientific work, mathematics and physics make the world appear more and more as an open one, as a world not closed but pointing beyond itself. . . . Science finds itself compelled, at once by the epistemological, the physical and the constructive-mathematical aspect of its own methods and results, to recognise this situation. It remains to be added that science can do no more than show us this open horizon; we must not by including the transcendental sphere attempt to establish anew a closed (though more comprehensive) world.


[I Intend to read this!]

I think it is insufficiently recognised that modern theoretical physics is very much concerned with the study of organisation; and from organisation to organism does not seem an impossible stride. But equally it would be foolish to deny the magnitude of the gulf between our understanding of the most complex form of inorganic matter and the simplest form of life. Let us suppose, however, that some day this gulf is bridged, and science is able to show how from the entities of physics creatures might be formed which are counterparts of ourselves even to the point of being endowed with life. The scientist will perhaps point out the nervous mechanism of the creature, its powers of motion, of growth, of reproduction, and end by saying “That’s you”. But it has yet to satisfy the inescapable test. Is it concerned with truth as I am ? Then I will acknowledge that it is indeed myself. The scientist might point to motions in the brain and say that these really mean sensations, emotions, thoughts; and perhaps supply a code to translate the motions into the corresponding thoughts. Even if we could accept this inadequate substitute for consciousness as we intimately know it, we must still protest: “You have shown us a creature which thinks and believes; you have not shown us a creature to whom it matters that what it thinks and believes should be true”. The inmost ego, possessing what I have called the inescapable attribute, can never be part of the physical world unless we alter the meaning of the word ‘physical” o as to be synonymous with ” spiritual” a change scarcely to the advantage of clear thinking. But having disowned our supposed double, we can say to the scientist: “If you will hand over this Robot who pretends to be me, and let it be filled with the attribute at present lacking and perhaps other spiritual attributes which I claim as equally self-evident, we may arrive at something that is indeed myself”.

Our present conception of the physical world is hollow enough to hold almost anything. I think the reader will agree. There may indeed be a hint of ribaldry in his hearty assent. What we are dragging to light as the basis of all phenomena is a scheme of symbols connected by mathematical equations. That is what physical reality boils down to when probed by the methods which a physicist can apply. A skeleton scheme of symbols proclaims its own hollowness.

It can be nay it cries out to be filled with something that shall transform it from skeleton into substance, from plan into execution, from symbols into an interpretation of the symbols. And if ever the physicist solves the problem of the living body, he should no longer be tempted to point to his result and say “That’s you”. He should say rather “That is the aggregation of symbols which stands for you in my description and explanation of those of your properties which I can observe and measure. If you claim a deeper insight into your own nature by which you can interpret these symbols a more intimate knowledge of the reality which I can only deal with by symbolism you can rest assured that I have no rival interpretation to propose”. The skeleton is the contribution of physics to the solution of the Problem of Experience; from the clothing of the skeleton it stands aloof.


The scientific conception of the world has come to differ more and more from the commonplace conception until we have been forced to ask ourselves what really is the aim of this scientific transmutation. The doctrine that things are not what they seem is all very well in moderation; but it has proceeded so far that we have to remind ourselves that the world of appearances is the one to which we have actually to adjust our outward lives.

So long as physics in tinkering with the familiar world was able to retain those aspects which appeal to the aesthetic side of our nature, it might with some show of reason make claim to cover the whole of experience; and those who claimed that there was another, religious aspect of our existence had to fight for their claim. But now that its picture omits so much that is obviously significant, there is no suggestion that it is the whole truth about experience, To make such a claim would bring protest not only from the religiously minded but from all who recognise that Man is not merely a scientific measuring machine.

Physics provides a highly perfected answer to one specialised problem which confronts us in experience. I do not wish to minimise the importance of the problem and the value of the solution. We have seen (p. n) how in order to focus the problem the various faculties of the observer have been discarded, and even his sensory equipment simplified, until the problem becomes such as our methods are adequate to solve. For the physicist the observer has become a symbol dwelling in a world of symbols. But before ever we handed over the problem to the physicist we had a glimpse of Man as a spirit in an environment akin to his own spirit.

Why should anyone suppose that all that matters to human nature can be assessed with a measuring rod or expressed in terms of the intersections of world-lines?

If our method consists in codifying, what can we possibly obtain but a code?

Interpreting the term material (or more strictly, physical); in the broadest sense as that with which we can become! acquainted through sensory experience of the external world, we recognise now that it corresponds to the waves not to the water of the ocean of reality. My answer does not deny the existence of the physical world, any more than the answer that the ocean is made of water denies the existence of ocean waves; only we do not get down to the intrinsic nature of things that way. Like the symbolic world of physics, a wave is a conception which is hollow enough to hold almost anything; we can have waves of water, of air, of aether, and (in quantum theory) waves of probability. So after physics has shown us the waves, we have still to determine the content of the waves by some other avenue of knowledge. If you will understand that the spiritual aspect of experience is to the physical aspect in the same kind of relation as the water to the wave form, I can leave you to draw up your own answer to the question propounded at the beginning of this section and so avoid any verbal misunderstanding. What is more important you will see how easily the two aspects of experience now dovetail together, not contesting each other’s place. It is almost as though the modern conception of the physical world had deliberately left room for the reality of spirit and consciousness.

It is a commonplace reflection that we understand very little about our own minds, but it is here if anywhere that all knowledge begins. As for the external objects, remorselessly dissected by science, they are studied and measured, but they are never known. Our pursuit of them has led from solid matter to molecules, from molecules to sparsely scattered electric charges, from electric charges to waves of probability. Whither next?

… I think there can be no doubt that the scientist has a much more mystic conception of the external world than he had in the last century when every scientific ” explanation” of phenomena proceeded on the assumption that nothing could be true unless an engineer could make a model of it. The cruder kind of materialism which sought to reduce everything in the universe, inorganic and organic, to a mechanism of fly-wheels or vortices or similar devices has disappeared altogether. Mechanical explanations of gravitation or electricity are laughed at nowadays. You could now safely hand over the human intellect to the tender mercies of the physicist without fear that he would discover in its workings a grinding of cog-wheels. But we must not make too much of these signs of grace in modern physical science. The tyranny of the engineer has been superseded by the tyranny of the mathematician. At least that is a view very widely taken. But alongside this there is a growing realisation that the mathematician is less oppressive a master than the engineer, for he does not claim any insight deeper than his own symbols.

In an earlier book [The Nature of the Physical World, pp. 104, 209] I have referred to the unconscious habit of the modern physicist of looking on the Creation as though it were the work of a mathematician. Perhaps the irony of these passages is not so evident now as it was at the time. I could not foresee that a few years later a colleague would seriously put forward the view that “from the intrinsic evidence of his creation, the Great Architect of the Universe now begins to appear as a pure mathematician” [Sir James Jeans, The Mysterious Universe, p. 134]. Jeans has previously considered but rejected another explanation. ” So it may be suggested, the mathematician only sees nature through the mathematical blinkers he has fashioned for himself.”

In rejecting what seems to me to be the right explanation, Jeans dwells on the failure of anthropomorphic theories and later the devices of the engineer to explain the universe, and he contrasts them with the success of the mathematical conception. There are two factors which, it seems to me, explain the comparative success of the mathematician. In the first place the mathematician is the professional wielder of symbols; he can deal with unknown quantities and even unknown operations. Clearly he is the man to help us to sift a little knowledge from a vast unknown. But the main reason why the mathematician has beaten his rivals is that we have allowed him to dictate the terms of die competition. The fate of every theory of the universe is decided by a numerical test. Does the sum come out right ? I am not sure that the mathematician understands this world of ours better than the poet and the mystic. Perhaps it is only that he is better at sums.


The stress here laid on the limitations of physical science will, I hope, not be misunderstood by the reader. There is no suggestion that science has become a declining force; rather we obtain a clearer appreciation of the contribution which it is able to make, both now and in the future, to human development and culture. Within its own limitations physical science has become greatly strengthened by the changes. It has become more sure of its aims and perhaps less sure of its achievements. Since the last most bewildering revolution of physical theory (wave mechanics) there has been an interval of some years during which it has been possible to settle down to steady progress. Recently the most striking developments have been on the experimental side. In quick succession the artificial transmutation of the elements, the discovery of the neutron and the discovery of the positive electron have startled the scientific world and opened up new realms for exploration. But I count this as normal prosperity rather than revolution.

In contemplating the gradually developing scheme of
scientific knowledge which never seems to reach finality in any direction, there are times when we are tempted to doubt the substantiality of our gains. Questions, which seem to have been settled, become unsettled.

… I sometimes think that the progress of knowledge is to be measured not by the questions that it has answered but by the questions that it provokes us to ask.


All this new growth of science has its roots in the past. If we see farther than our predecessors it is because we stand on their shoulders and it is not surprising if they receive a few kicks as we scramble up. A new generation is climbing on to the shoulders of the generation to which I belong; and so it will go on. Each phase of the scientific advance has contributed something that is preserved in the succeeding phase. That, indeed, is our ground for hope that the coming generation will find something worth preserving something that is not wholly illusory in the scientific thought of the Universe as it stands to-day.


This all seems very reasonable. I shall try to reduce my quotes above to make them an easier read without – hopefully – mangling their meaning too much. Please let me know if you think I have already gone too far, or if you have any good arguments against the views expressed above.

Dave Marsay

Eddington’s Physical World

A.S. Eddington The Nature of the Physical World: The Gifford Lectures 1927, Macmillan, 1929

(I intend to develop this, but have distracted by later work:)


… The aim is to make clear the scientific view of the world as it stands at the present day, and, where it is incomplete, to judge the direction in which modern ideas appear to be tending.

… I would like to recall that the idealistic tinge in my conception of the physical world arose out of mathematical researches on the relativity theory. In so far as I had any earlier philosophical views, they were of an entirely different complexion.


I have settled down to the task of writing these lectures and have drawn up my chairs to my two tables. Two tables! Yes; there are duplicates of every object about me — two tables, two chairs, two pens.

One of them has been familiar to me from earliest years. It is a commonplace object of that environment which I call the world. How shall I describe it? It has extension; it is comparatively permanent; it is coloured; above all it is substantial. … I do not think substantiality can be described better than by saying that it is the kind of nature exemplified by an ordinary table. And so we go round in circles.^ After all if you are a plain commonsense man, not too much worried with scientific scruples, you will be confident that you understand the nature of an ordinary table. I have even heard of plain men who had the idea that they could better understand the mystery of their own nature if scientists would discover a way of explaining it in terms of the easily comprehensible nature of a table.

Table No. 2 is my scientific table. It is a more recent acquaintance and I do not feel so familiar with it. It does not belong to the world previously mentioned — that world which spontaneously appears around me when I open my eyes, though how much of it is objective andhow much subjective I do not here consider. It is part of a world which in more devious ways has forced itself on my attention.

… Reviewing their properties one by one, there seems to be nothing to choose between the two tables for ordinary purposes; but when abnormal circumstances befall, then my scientific table shows to advantage. If the house catches fire my scientific table will dissolve quite naturally into scientific smoke, whereas my familiar table undergoes a metamorphosis of its substantial nature which I can only regard as miraculous.

The whole trend of modern scientific views is to break down the separate categories of “things”, “influences”, “forms”, etc., and to substitute a common background of all experience. Whether we are studying a material object, a magnetic field, a geometrical figure, or a duration of time, our scientific information is summed up in measures ; neither the apparatus of measurement nor the mode of using it suggests that there is anything essentially different in these problems. The measures themselves afford no ground for a classification by categories. We feel it necessary to concede some background to the measures — an external world; but the attributes of this world, except in so far as they are reflected in the measures, are outside scientific scrutiny. Science has at last revolted against attaching the exact knowledge contained in these measurements to a traditional picture-gallery of conceptions which convey no authentic information of the background and obtrude irrelevancies into the scheme of knowledge.

… It makes all the difference in the world whether the paper before me is poised as it were on a swarm of flies and sustained in shuttlecock fashion by a series of tiny blows from the swarm underneath, or whether it is supported because there is substance below it, it being the intrinsic nature of substance to occupy space to the exclusion of other substance; all the difference in conception at least, but no difference to my practical task of writing on the paper.

It is true that the whole scientific inquiry starts from the familiar world and in the end it must return to the familiar world; but the part of the journey over which the physicist has charge is in foreign territory.

After the physicist has quite finished his world-building a linkage or identification is allowed; but premature attempts at linkage have been found to be entirely mischievous.

Science aims at constructing a world which shall be symbolic of the world of commonplace experience. It is not at all necessary that every individual symbol that is used should represent something in common experience or even something explicable in terms of common experience. The man in the street is always making this demand for concrete explanation of the things referred to in science; but of necessity he must be disappointed. It is like our experience in learning to read. That which is written in a book is symbolic of a story in real life. The whole intention of the book is that ultimately a reader will identify some symbol, say BREAD, with one of the conceptions of familiar life. But it is mischievous to attempt such identifications prematurely, before the letters are strung into words and the words into sentences.

The frank realisation that physical science is concerned with a world of shadows is one of the most significant of recent advances. I do not mean that physicists are to any extent preoccupied with the philosophical implications of this. From their point of view it is not so much a withdrawal of untenable claims as an assertion of freedom for autonomous development. At the moment I am not insisting on the shadowy and symbolic character of the world of physics because of its bearing on philosophy, but because the aloofness from familiar conceptions will be apparent in the scientific theories I have to describe. If you are not prepared for this aloofness you are likely to be out of sympathy with modern scientific theories, and may even think them ridiculous — as, I daresay, many people do.

It is difficult to school ourselves to treat the physical world as purely symbolic. We are always relapsing and mixing with the symbols incongruous conceptions taken from the world of consciousness. Untaught by long experience we stretch a hand to grasp the shadow, instead of accepting its shadowy nature. Indeed, unless we confine ourselves altogether to mathematical symbolism it is hard to avoid dressing our symbols in deceitful clothing. … I can well understand that the younger minds are finding these pictures too concrete and are striving to construct the world out of Hamiltonian functions and symbols so far removed from human preconception that they do not even obey the laws of orthodox arithmetic. For myself I find some difficulty in rising to that plane of thought; but I am convinced that it has got to come.

In these lectures I propose to discuss some of the results of modern study of the physical world which give most food for philosophic thought. This will include new conceptions in science and also new knowledge. In both respects we are led to think of the material universe in a way very different from that prevailing at the end of the [19th] century. … I am convinced that a just appreciation of the physical world as it is understood today carries with it a feeling of open-mindedness towards a wider significance transcending scientific measurement, which might have seemed illogical a generation ago; and in the later lectures I shall try to focus that feeling and make inexpert efforts to find where it leads. …


Frames of space

An exceptionally modest observer might take some other planet than his own as the standard of rest. Then he would have to correct all his measurements for the FitzGerald contraction due to his own motion with respect to the standard, and the corrected measures would give the space-frame belonging to the standard planet as the original measures gave the space-frame of his own planet. For him the dilemma is even more pressing, for there is nothing to guide him as to the planet to be selected for the standard of rest. Once he gives up the naive assumption that his own frame is the one and only right frame the question arises, Which then of the innumerable other frames is right? There is no answer, and so far as we can see no possibility of  an answer. Meanwhile all his experimental measurements are waiting unreduced, because the corrections to be applied to them depend on the answer. I am afraid our modest observer will get rather left behind by his less humble colleagues.

The trouble that arises is not that we have found anything necessarily wrong with the frame of location that has been employed in our system of physics; it has not led to experimental contradictions. The only thing known to be “wrong” with it is that it is not unique. If we had found that our frame was unsatisfactory and another frame was preferable, that would not have caused a great revolution of thought; but to discover that ours is one of many frames, all of which are equally satisfactory, leads to a change of interpretation of the significance of a frame of location.


few indeed are the experiments contributing to our scientific knowledge which would not be invalidated if our methods of measuring lengths were fundamentally unsound. We now find that there is no guarantee that they are not subject to a systematic kind of error. Worse still we do not know if the error occurs or not, and there is every reason to presume that it is impossible to know.


Einstein’s Principle

The modest observer mentioned in the first chapter was faced with the task of choosing between a number of frames of space with nothing to guide his choice. They are different in the sense that they frame the material objects of the world, including the observer himself, differently; but they are indistinguishable in the sense that the world as framed in one space conducts itself according to precisely the same laws as the world framed in another space. Owing to the accident of having been born on a particular planet our observer has hitherto unthinkingly adopted one of the frames; but he realises that this is no ground for obstinately asserting that it must be the right frame. Which is the right frame?

At this juncture Einstein comes forward with a suggestion —

“You are seeking a frame of space which you call the right frame. In what does its rightness consist?”

You are standing with a label in your hand before a row of packages all precisely similar. You are worried because there is nothing to help you decide which of the packages it should be attached to. Look at the label and see what is written on it. Nothing.

“Right” as applied to frames of space is a blank label. It implies that there is something distinguishing a right frame from a wrong frame; but when we ask what is this distinguishing property, the only answer we receive is “Rightness”, which does not make the meaning clearer or convince us that there is a meaning.

I am prepared to admit that frames of space in spite of their present resemblance may in the future turn out to be not entirely indistinguishable. …

But it does not seem a profitable procedure to make odd noises on the off-chance that posterity will find a significance to attribute to them. To those who now harp on a right frame of space we may reply in the words of Bottom the weaver —

“Who would set his wit to so foolish a bird? Who would give a bird the lie, though he cry ‘cuckoo’ never so?”

The next point to notice is that the other quantities of physics go along with the frame of space, so that they also are relative.

… [But:] It is a common mistake to suppose that Einstein’s theory of relativity asserts that everything is relative. Actually it says, “There are absolute things in the world but you must look deeply for them. The things that first present themselves to your notice are for the most part relative.”

A very homely illustration of a relative quantity is afforded by the pound sterling. Whatever may have been the correct theoretical view, the man in the street until very recently regarded a pound as an absolute amount of wealth. But dire experience has now convinced us all of its relativity. At first we used to cling to the idea that there ought to be an absolute pound and struggle to express the situation in paradoxical statements — the pound had really become seven-and-six-pence. But we have grown accustomed to the situation and continue to reckon wealth in pounds as before, merely recognising that the pound is relative and therefore must not be expected to have those properties that we had attributed to it in the belief that it was absolute.

Nature’s Plan of Structure

… I do not think Nature has been particularly subtle in concealing which frame she prefers. It is just that she is not enthusiastic about frames of space. They are a method of partition which we have found useful for reckoning, but they play no part in the architecture of the universe. Surely it is absurd to suppose that the universe is planned in such a way as to conceal its plan. It is like the schemes of the White Knight —

But I was thinking of a plan
To dye one’s whiskers green,
And always use so large a fan
That they could not be seen.

If this is so we shall have to sweep away the frames of space before we can see Nature’s plan in its real significance. She herself has paid no attention to them, and they can only obscure the simplicity of her scheme. I do not mean to suggest that we should entirely rewrite physics, eliminating all reference to frames of space or any quantities referred to them; science has many tasks to perform, besides that of apprehending the ultimate plan of structure of the world. But if we do wish to have insight on this latter point, then the first step is to make an escape from the irrelevant space-frames.

No aethereal frame has been found. We can only discover motion relative to the material landmarks scattered casually about the world; motion with respect to the universal ocean of aether eludes us. We say, “Let V be the velocity of a body through the aether”, and form the various electromagnetic equations in which V is scattered liberally. Then we insert the observed values, and try to eliminate everything that is unknown except V. The solution goes on famously; but just as we have got rid of the other unknowns, behold! V disappears as well, and we are left with the indisputable
but irritating conclusion —

This is a favourite device that mathematical equations resort to, when we propound stupid questions. If we tried to find the latitude and longitude of a point north-east from the north pole we should probably receive the same mathematical answer. “Velocity through aether” is as meaningless as “north-east from the north pole”.

This does not mean that the aether is abolished. We need an aether. The physical world is not to be analysed into isolated particles of matter or electricity with featureless interspace. We have to attribute as much character to the interspace as to the particles, and in present-day physics quite an army of symbols is required to describe what is going on in the interspace. We postulate aether to bear the characters of the interspace as we postulate matter or electricity to bear the characters of the particles. Perhaps a philosopher might question whether it is not possible to admit the characters alone without picturing anything to support them — thus doing away with aether and matter at one stroke. But that is rather beside the point.

You receive a balance-sheet from a public company and observe that the assets amount to such and such a figure. Is this true? Certainly; it is certified by a chartered accountant. But is it really true? Many questions arise; the real values of items are often very different from those which figure in the balance-sheet. I am not especially referring to fraudulent companies. There is a blessed phrase “hidden reserves”; and generally speaking the more respectable the company the more widely does its balance-sheet deviate from reality. This is called sound finance. But apart from deliberate use of the balance-sheet to conceal the actual situation, it is not well adapted for exhibiting realities, because the main function of a balance-sheet is to balance and everything else has to be subordinated to that end

Perhaps you will think we ought to alter our method of keeping the accounts of space so as to make them directly represent the realities. That would be going to a lot of trouble to provide for what are after all rather rare transactions. But as a matter of fact we have managed to meet your desire. Thanks to Minkowski a way of keeping accounts has been found which exhibits realities (absolute things) and balances. There has been no great rush to adopt it for ordinary purposes because it is a four-dimensional balance-sheet.

We have travelled far from the old standpoint which demanded mechanical models of everything in Nature, seeing that we do not now admit even a definite unique distance between two points. The relativity of the current scheme of physics invites us to search deeper and find the absolute scheme underlying it, so that we may see the world in a truer perspective.

Chapter III TIME

Simultaneity (Now) is seen to be relative. The denial of absolute simultaneity is intimately con- nected with the denial of absolute velocity; knowledge of absolute velocity would enable us to assert that certain events in the past or future occur Here but not Now; knowledge of absolute simultaneity would tell us that certain events occur Now but not Here. Removing these artificial sections, we have had a glimpse of the absolute world-structure with its grain diverging and interlacing after the plan of the hour-glass figures. By reference to this structure we discern an absolute distinction between space-like and time-like separation of events — a distinction which justifies and explains our instinctive feeling that space and time are fundamentally different.



If you take a pack of cards as it comes from the maker and shuffle it for a few minutes, all trace of the original systematic order disappears. The order wiil never come back however long you shuffle. Something has been done which cannot be undone, namely, the introduction of a random element in place of arrangement

We shall put forward the contention that —

Whenever anything happens which cannot be undone, it is always reducible to the introduction of a random element analogous to that introduced by shuffling.

he possibility of the shuffling becoming complete is significant. If after shuffling the pack you tear each card in two, a further shuffling of the half-cards becomes possible. Tear the cards again and again; each time there is further scope for the random element to increase. With infinite divisibility there can be no end to the shuffling. The experimental fact that a definite state of equilibrium is rapidly reached indicates that energy is not infinitely divisible, or at least that it is not infinitely divided in the natural processes of shuffling. Historically this is the result from which the quantum theory first arose. We shall return to it in a later chapter.

Chapter V: “BECOMING”

In sorting out the confused data of our experience it has generally been assumed that the object of the quest is to find out all that really exists. There is another quest not less appropriate to the nature of our experience — to find out all that really becomes.


I cannot but think that Newton himself would rejoice that after 200 years the “ocean of undiscovered truth” has rolled back another stage. I do not think of him as censorious because we will not blindly apply his formula regardless of the knowledge that has since accumulated and in circumstances that he never had the opportunity of considering.


I have too long delayed dealing with the criticism of the pure mathematician who is under the impression that geometry is a subject that belongs entirely to him. Each branch of experimental knowledge tends to have associated with it a specialised body of mathematical investigations. The pure mathematician, at first called in as servant, presently likes to assert himself as master; the connexus of mathematical propositions becomes for him the main subject, and he does not ask permission from Nature when he wishes to vary or generalise the original premises. Thus he can arrive at a geometry unhampered by any restriction from actual space measures; a potential theory unhampered by any question as to how gravitational and electrical potentials really behave; a hydrodynamics of perfect fluids doing things which it would be contrary to the nature of any material fluid to do. But it seems to be only in geometry that he has forgotten that there ever was a physical subject of the same name, and even resents the application of the name to anything but his network of abstract mathematics. I do not think it can be disputed that, both etymologically and traditionally, geometry is the science of measurement of the space around us; and however much the mathematical superstructure may now overweigh the observational basis, it is properly speaking an experimental science. This is fully recognised in the “reformed” teaching of geometry in schools; boys are taught to verify by measurement that certain of the geometrical propositions are true or nearly true. No one questions the advantage of an unfettered development of geometry as a pure mathematical subject; but only in so far as this subject is linked to the quantities arising out of observation and measurement, will it find mention in a discussion of the Nature of the Physical World.


The Origin of the Trouble. Nowadays whenever enthusiasts meet together to discuss theoretical physics the talk sooner or later turns in a certain direction. You leave them conversing on their special problems or the latest discoveries; but return after an hour and it is any odds that they will have reached an all-engrossing topic — the desperate state of their ignorance. This is not a pose. It is not even scientific modesty, because the attitude is often one of naive surprise that Nature should have hidden her fundamental secret successfully from such powerful intellects as ours. It is simply that we have turned a corner in the path of progress and our ignorance stands revealed before us, appalling and insistent. There is something radically wrong with the present fundamental conceptions of physics and we do not see how to set it right.

The cause of all this trouble is a little thing called h which crops up continually in a wide range of experiments. In one sense we know just what h is, because there are a variety of ways of measuring it; h is

.0000000000000000000000000065 5 erg-seconds.

That will (rightly) suggest to you that h is something very small; but the most important information is contained in the concluding phrase erg-seconds. The erg is the unit of energy and the second is the unit of time; so that we learn that h is of the nature of energy multiplied by time.

… Erg-seconds or action belongs to Minkowski’s world which is common to all observers, and so it is absolute. It is one of the very few absolute quantities noticed in pre-relativity physics. Except for action and entropy (which belongs to an entirely different class of physical conceptions) all the quantities prominent in pre-relativity physics refer to the three-dimensional sections which are different for different observers.

… It is remarkable that just as Einstein found ready prepared by the mathematicians the Tensor Calculus which he needed for developing his great theory of gravitation, so the quantum physicists found ready for them an extensive action-theory of dynamics without which they could not have made headway.

… It would seem that the atom carelessly throws overboard a lump of energy which, as it glides into the aether, moulds itself into a quantum of action by taking on the period required to make the product of energy and period equal to h. If this unmechanical process of emission seems contrary to our preconceptions, the exactly converse process of absorption is even more so. Here the atom has to look out for a lump of energy of the exact amount required to raise an electron to the higher orbit. It can only extract such a lump from aetherwaves of particular period — not a period which has resonance with the structure of the atom, but the period which makes the energy into an exact quantum.

We must, of course, look forward to an ultimate reconstruction of our conceptions of the physical world which will embrace both the classical laws and the quantum laws in harmonious association. There are still some who think that the reconciliation will be effected by a development of classical conceptions. But the physicists of what I may call “the Copenhagen school” believe that the reconstruction has to start at the other end, and that in the quantum phenomena we are getting down to a more intimate contact with Nature’s way of working than in the coarse-grained experience which has furnished the classical laws. …

The classical laws are the limit to which the quantum laws tend when states of very high quantum number are concerned.

This is the famous Correspondence Principle enunciated by Bohr.

For an example I will borrow a quantum conception from the next chapter. It may not be destined to survive in the present rapid evolution of ideas, but at any rate it will illustrate my point. In Bohr’s semi-classical model of the hydrogen atom there is an electron describing a circular or elliptic orbit. This is only a model; the real atom contains nothing of the sort. The real atom contains something which it has not entered into the mind of man to conceive, which has, however, been described symbolically by Schrodinger. This “something” is spread about in a manner by no means comparable to an electron describing an orbit. Now excite the atom into successively higher and higher quantum states. In the Bohr model the electron leaps into higher and higher orbits. In the real atom Schrodinger’s “something” begins to draw itself more and more together until it begins sketchily to outline the Bohr orbit and even imitates a condensation running round. Go on to still higher quantum numbers, and Schrodinger’s symbol now represents a compact body moving round in the same orbit and the same period as the electron in Bohr’s model, and moreover radiating according to the classical laws of an electron. And so when the quantum number reaches infinity, and the atom bursts, a genuine classical electron flies out. The electron, as it leaves the atom, crystallises out of Schrodinger’s mist like a genie emerging from his bottle.


[The] theory has already gone through three distinct phases associated with the names of Born and Jordan, Dirac, Schrodinger. … As regards philosophical ideas the three theories are poles apart; as regards mathematical content they are one and the same.

All authorities seem to be agreed that at, or nearly at, the root of everything in the physical world lies the mystic formula

qppq = ih/2π

… The righthand side contains nothing unusual, but the left-hand side baffles imagination. We call q and p co-ordinates and momenta, borrowing our vocabulary from the world of space and time and other coarse-grained experience; but that gives no real light on their nature, nor does it explain why qp is so ill-behaved as to be unequal to pq.

For Dirac p is a symbol without any kind of numerical interpretation; he calls it a ^-number, which is a way of saying that it is not a number at all.

I venture to think that there is an idea implied in Dirac’s treatment which may have great philosophical significance, independently of any question of success in this particular application. The idea is that in digging deeper and deeper into that which lies at the base of physical phenomena we must be prepared to come to entities which, like many things in our conscious experience, are not measurable by numbers in any way; and further it suggests how exact science, that is to say the science of phenomena correlated to measure-numbers, can be founded on such a basis.

One of the greatest changes in physics between the nineteenth century and the present day has been the change in our ideal of scientific explanation. It was the boast of the Victorian physicist that he would not claim to understand a thing until he could make a model of it; and by a model he meant something constructed of levers, geared wheels, squirts, or other appliances familiar to an engineer. Nature in building the universe was supposed to be dependent on just the same kind of resources as any human mechanic; and when the physicist sought an explanation of phenomena his ear was straining to catch the hum of machinery. The man who could make gravitation out of cog-wheels would have been a hero in the Victorian age.

Nowadays we do not encourage the engineer to build the world for us out of his material, but we turn to the mathematician to build it out of his material. Doubtless the mathematician is a loftier being than the engineer, but perhaps even he ought not to be entrusted with the Creation unreservedly. We are dealing in physics with a symbolic world, and we can scarcely avoid employing the mathematician who is the professional wielder of symbols; but he must rise to the full opportunities of the responsible task entrusted to him and not indulge too freely his own bias for symbols with an arithmetical interpretation. If we are to discern controlling laws of Nature not dictated by the mind it would seem necessary to escape as far as possible from the cut-and-dried framework into which the mind is so ready to force everything that it experiences.

I think that in principle Dirac’s method asserts this kind of emancipation. He starts with basal entities inexpressible by numbers or number-systems and his basal laws are symbolic expressions unconnected with arithmetical operations. The fascinating point is that as the development proceeds actual numbers are exuded from the symbols. Thus although p and q individually have no arithmetical interpretation, the combination qppq has the arithmetical interpretation expressed by the formula above quoted. By furnishing numbers, though itself non-numerical, such a theory can well be the basis for the measure-numbers studied in exact science. The measure-numbers, which are all that we glean from a physical survey of the world, cannot be the whole world; they may not even be so much of it as to constitute a self-governing unit. This seems the natural interpretation of Dirac’s procedure in seeking the governing laws of exact science in a non-arithmetical calculus

Schrodinger’s theory is now enjoying the full tide of popularity, partly because of intrinsic merit, but also, I suspect, partly because it is the only one of the three that is simple enough to be misunderstood. Rather against my better judgment I will try to give a rough impression of the theory. It would probably be wiser to nail up over the door of the new quantum theory a notice, “Structural alterations in progress — No admittance except on business”, and particularly to warn the doorkeeper to keep out prying philosophers. I will, however, content myself with the protest that, whilst Schrodinger’s theory is guiding us to sound and rapid progress in many of the mathematical problems confronting us and is indispensable in its practical utility, I do not see the least likelihood that his ideas will survive long in their present form.

Principle of Indeterminacy.

My apprehension lest a fourth version of the new quantum theory should appear before the lectures were delivered was not fulfilled; but a few months later the theory definitely entered on a new phase. It was Heisenberg again who set in motion the new development in the summer of 1927, and the consequences were further elucidated by Bohr. The outcome of it is a fundamental general principle which seems to rank in importance with the principle of relativity. I shall here call it the “principle of indeterminacy”.

[We] assert that the description of the position and velocity of an electron beyond a limited number of places of decimals is an attempt to describe something that does not exist; although curiously enough the description of position or of velocity if it had stood alone might have been allowable

Ever since Einstein’s theory showed the importance of securing that the physical quantities which we talk about are actually connected to our experience, we have been on our guard to some extent against meaningless terms.

A general kind of reason for this can be seen without much difficulty. Suppose it is a question of knowing the position and momentum of an electron. So long as the electron is not interacting with the rest of the universe we cannot be aware of it. We must take our chance of obtaining knowledge of it at moments when it is interacting with something and thereby producing effects that can be observed. But in any such interaction a complete quantum is involved; and the passage of this quantum, altering to an important extent the conditions at the moment of our observation, makes the information out of date even as we obtain it.

This is not a casual difficulty; it is a cunningly arranged plot — a plot to prevent you from seeing something that does not exist, viz. the locality of the electron within the atom.

Other examples of the reciprocal uncertainty have been given, and there seems to be no doubt that it is entirely general. The suggestion is that an association of exact position with exact momentum can never be discovered by us because there is no such thing in Nature.

A New Epistemology

The principle of indeterminacy is epistemological. It reminds us once again that the world of physics is a world contemplated from within surveyed by appliances which are part of it and subject to its laws. What the world might be deemed like if probed in some supernatural manner by appliances not furnished by itself we do not profess to know.

There is a doctrine well known to philosophers that the moon ceases to exist when no one is looking at it. I will not discuss the doctrine since I have not the least idea what is the meaning of the word existence when used in this connection. At any rate the science of astronomy has not been based on this spasmodic kind of

What should we regard as a complete description of this scientific world? We must not introduce anything like velocity through aether, which is meaningless since it is not assigned any causal connection with our experience. On the other hand we cannot limit the description to the immediate data of our own spasmodic observations. The description should include nothing that is unobservable but a great deal that is actually unobserved. Virtually we postulate an infinite army of watchers and measurers. From moment to moment they survey everything that can be surveyed and measure everything that can be measured by methods which we ourselves might conceivably employ. Everything they measure goes down as part of the complete description of the scientific world. We can, of course, introduce derivative descriptions, words expressing mathematical combinations of the immediate measures which may give greater point to the description — so that we may not miss seeing the wood for the trees.

This theory of knowledge is primarily intended to apply to our macroscopic or large-scale survey of the physical world, but it has usually been taken for granted that it is equally applicable to a microscopic study. We have at last realised the disconcerting fact that though it applies to the moon it does not apply to the electron.

I expect that at first this will sound to you like a merely dialectical difficulty. But there is much more in it than that. The deliberate frustration of our efforts to bring knowledge of the microscopic world into orderly plan, is a strong hint to alter the plan.

It means that we have been aiming at a false ideal of a complete description of the world. There has not yet been time to make serious search for a new epistemology adapted to these conditions. It has become doubtful whether it will ever be possible to construct a physical world solely out of the knowable — the guiding principle in our macroscopic theories. If it is possible, it involves a great upheaval of the present foundations. It seems more likely that we must be content to admit a mixture of the knowable and unknowable. This means a denial of determinism, because the data required for a prediction of the future will include the unknowable elements of the past. I think it was Heisenberg who said,

The question whether from a complete knowledge of the past we can predict the future, does not arise because a complete knowledge of the past involves a self-contradiction.

It is only through a quantum action that the outside world can interact with ourselves and knowledge of it can reach our minds. A quantum action may be the means of revealing to us some fact about Nature, but simultaneously a fresh unknown is implanted in the womb of Time. An addition to knowledge is won at the expense of an addition to ignorance. It is hard to empty the well of Truth with a leaky bucket.


We have an intricate task before us. We are going to build a World — a physical world which will give a shadow performance of the drama enacted in the world of experience. We are not very expert builders as yet; and you must not expect the performance to go off without a hitch or to have the richness of detail which a critical audience might require. But the method about to be described seems to give the bold outlines; doubtless we have yet to learn other secrets of the craft of world building before we can complete the design.

The first problem is the building material. I remember that as an impecunious schoolboy I used to read attractive articles on how to construct wonderful contrivances out of mere odds and ends. Unfortunately these generally included the works of an old clock, a few superfluous telephones, the quicksilver from a broken barometer, and other oddments which happened not to be forthcoming in my lumber room. I will try not to let you down like that. I cannot make the world out of nothing, but I will demand as little specialised material as possible. Success in the game of World Building consists in the greatness of the contrast between the specialised properties of the completed structure and the unspecialised nature of the basal material.

Relation Structure. We take as building material relations and relata. The relations unite the relata; the relata are the meeting points of the relations. The one is unthinkable apart from the other. I do not think that a more general starting-point of structure could be conceived.

To distinguish the relata from one another we assign to them monomarks. …

The relation between two human individuals in its broadest sense comprises every kind of connection or comparison between them — consanguinity, business transactions, comparative stature, skill at golf — any kind of description in which both are involved. For generality we shall suppose that the relations in our world-material are likewise composite and in no way expressible in numerical measure. Nevertheless there must be some kind of comparability or likeness of relations, as there is in the relations of human individuals; otherwise there would be nothing more to be said about the world than that everything in it was utterly unlike everything else. To put it another way, we must postulate not only relations between the relata but some kind of relation of likeness between some of the relations. The slightest concession in this direction will enable us to link the whole into a structure.

We assume then that, considering a relation between two relata, it will in general be possible to pick out two other relata close at hand which stand to one another in a “like” relation. By “like” I do not mean “like in every respect”, but like in respect to one of the aspects of the composite relation. How is the particular aspect selected? If our relata were human individuals different judgments of likeness would be made by the genealogist, the economist, the psychologist, the sportsman, etc.; and the building of structure would here diverge along a number of different lines. Each could build his own world-structure from the common basal material of humanity. There is no reason to deny that a similar diversity of worlds could be built out of our postulated material. But all except one of these worlds will be stillborn. Our labour will be thrown away unless the world we have built is the one which the mind chooses to vivify into a world of experience. The only definition we can give of the aspect of the relations chosen for the criterion of likeness, is that it is the aspect which will ultimately be concerned in the getting into touch of mind w T ith the physical world. But that is beyond the province of physics.

This one-to-one correspondence of “likeness” is only supposed to be definite in the limit when the relations are very close together in the structure. Thus we avoid any kind of comparison at a distance which is as objectionable as action at a distance. Let me confess at once that I do not know what I mean here by “very close together”. As yet space and time have not been built. Perhaps we might say that only a few of the relata possess relations whose comparability to the first is definite, and take the definiteness of the comparability as the criterion of contiguity. I hardly know. The uilding at this point shows some cracks, but I think it should not be beyond the resources of the mathematical logician to cement them up. We should also arrange at this stage that the monomarks are so assigned as to give an indication of contiguity.

Let us start with a relatum A and a relation AP radiating from it. Now step to a contiguous relatum B and pick out the “like” relation BQ. Go on to another contiguous relatum C and pick out the relation CR which is like BQ. (Note that since C is farther from A than from B } the relation at C which is like AP is not so definite as the relation which is like BQ.) Step by step we may make the comparison round a route AEFA which returns to the starting-point. There is nothing to ensure that the final relation AP’ which has, so to speak, been carried round the circuit will be the relation AP with which we originally started.

We have now two relations AP, AP’ radiating from the first relatum, their difference being connected with a certain circuit in the world AEFA. The loose ends of the relations P and P have their monomarks, and we can take the difference of the monomarks (i.e. the difference of the identification numbers comprised in them) as the code expression for the change introduced by carrying AP round the circuit. As we vary the circuit and the original relation, so the change PP’ varies; and the next step is to find a mathematical formula expressing this dependence. There are virtually four things to connect, the circuit counting double since, for example, a rectangular circuit would be described by specifying two sides. Each of them has to be specified by four identification numbers (either monomarks or derived from monomarks) ; consequently, to allow for all combinations, the required mathematical formula contains 4^4 or 256 numerical coefficients. These coefficients give a numerical measure of the structure surrounding the initial relatum.

The axiom of comparability of contiguous relations only discriminates between like and unlike, and does not initially afford any means of classifying various decrees and kinds of unlikeness; but we have found a means of specifying the kind of unlikeness of AP and AP’ by reference to a circuit which “transforms” one into the other. Thus we have built a quantitative study of diversity on a definition of similarity.

The numerical measures of structure will be dependent on, and vary according to, the arbitrary code of monomarks used for the identification of relata. This, however, renders them especially suitable for building the ordinary quantities of physics. … Physical quantities in general have no absolute value, but values relative to chosen frames of reference or codes of monomarks.

We have now fashioned our bricks from the primitive clay and the next job is to build with them. The 256 measures of structure varying from point to point of the world are somewhat reduced in number when duplicates are omitted; but even so they include a great deal of useless lumber which we do not require for the building. That seems to have worried a number of the most eminent physicists; but I do not quite see why. Ultimately it is the mind that decides what is lumber — which part of our building will shadow the things of common experience, and which has no such counterpart. It is no part of our function as purveyors of building material to anticipate what will be chosen for the palace of the mind. The lumber will now be dropped as irrelevant in the further operations, but I do not agree with those who think it a blemish on the theory that the lumber should ever have appeared in it.

I have said that violation of these laws of conservation is unthinkable. Have we then found physical laws which will endure for all time unshaken by any future revolution? But the proviso must be remembered, “granting that the identification [of their subject matter] is correct”. The law itself will endure as long as two and two make four; but its practical importance depends on our knowing that which obeys it. We think we have this knowledge, but do not claim infallibility in this respect. From a practical point of view the law would be upset, if it turned out that the thing conserved was not that which we are accustomed to measure with the above-mentioned instruments but something slightly different.

Selective Influence of the Mind.

By following this particular plan of building we construct things which satisfy the law of conservation, that is to say things which are permanent. The law of conservation is a truism for the things which satisfy it; but its prominence in the scheme of law of the physical world is due to the mind having demanded permanence. We might have built things which do not satisfy this law. In fact we do build one very important thing “action” which is not permanent; in respect to “action” physics has taken the bit in her teeth, and has insisted on recognising this as the most fundamental thing of all, although the mind has not thought it worthy of a place in the familiar world and has not vivified it by any mental image or conception.

he world which we have built from the relationstructure is no doubt doomed to be pulled about a good deal as our knowledge progresses. The quantum theory shows that some radical change is impending. But I think that our building exercise has at any rate widened our minds to the possibilities and has given us a different orientation towards the idea of physical law. The points which I stress are:

Firstly, a strictly quantitative science can arise from a basis which is purely qualitative. The comparability that has to be assumed axiomatically is a merely qualitative discrimination of likeness and unlikeness.

Secondly, the laws which we have hitherto regarded as the most typical natural laws are of the nature of truisms, and the ultimate controlling laws of the basal structure (if there are any) are likely to be of a different type from any yet conceived.

Thirdly, the mind has by its selective power fitted the processes of Nature into a frame of law of a pattern largely of its own choosing; and in the discovery of this system of law the mind may be regarded as regaining from Nature that which the mind has put into Nature.

Three Types of Law. So far as we are able to judge, the laws of Nature divide themselves into three classes:

(1) identical laws,
(2) statistical laws,
(3) transcendental laws.

We have just been considering the identical laws, i.e. the laws obeyed as mathematical identities in virtue of the way in which the quantities obeying them are built. They cannot be regarded as genuine laws of control of the basal material of the world. Statistical laws relate to the behaviour of crowds, and depend on the fact that although the behaviour of each individual may be extremely uncertain average results can be predicted with confidence. Much of the apparent uniformity of Nature is a uniformity of averages. Our gross senses only take cognisance of the average effect of vast numbers of individual particles and processes; and the regularity of the average might well be compatible with a great degree of lawlessness of the individual. I do not think it is possible to dismiss statistical laws (such as the second law of thermodynamics) as merely mathematical adaptations of the other classes of law to certain practical problems. They involve a peculiar element of their own connected with the notion of a priori probability; but we do not yet seem able to find a place for this in any of the current conceptions of the world substratum.

If there are any genuine laws of control of the physical world they must be sought in the third group — the transcendental laws. The transcendental laws comprise all those which have not become obvious identities implied in the scheme of world-building. … It is a natural suggestion that the greater difficulty in elucidating the transcendental laws is due to the fact that we are no longer engaged in recovering from Nature what we have ourselves put into Nature, but are at last confronted with its own intrinsic system of government. But I scarcely know what to think. We must not assume that the possible developments of the new attitude towards natural law have been exhausted in a few short years. It may be that the laws of atomicity, like the laws of conservation, arise only in the presentation of the world to us and can be recognised as identities by some extension of the argument we have followed. But it is perhaps as likely that after we have cleared away all the superadded laws which arise solely in our mode of apprehension of the world about us, there will be left an external world developing under genuine laws of control.

At present we can notice the contrast that the laws which we now recognise as man-made are characterised by continuity, whereas the laws to which the mind as yet lays no claim are characterised by atomicity. The quantum theory with its avoidance of fractions and insistence on integral units seems foreign to any scheme which we should be likely subconsciously to have imposed as a frame for natural phenomena. Perhaps our final conclusion as to the world of physics will resemble Kronecker’s view of pure mathematics.

“God made the integers, all else is the work of man.”


Familiar Conceptions and Scientific Symbols. We have said in the Introduction that the raw material of the scientific world is not borrowed from the familiar world. It is only recently that the physicist has deliberately cut himself adrift from familiar conceptions. He did not set out to discover a new world but to tinker with the old. Like everyone else he started with the idea that things are more or less what they seem, and that our vivid impression of our environment may be taken as a basis to work from. Gradually it has been found that some of its most obvious features must be rejected. … But this new knowledge can still be grasped by a rearrangement of familiar conceptions. I can picture to myself quite vividly the state of affairs just described; if there is any strain, it is on my credulity, not on my powers of conception. Other advances of knowledge can be accommodated by that very useful aid to comprehension — “like this only more so”. For example, if you think of something like a speck of dust only more so you have the atom as it was conceived up to a fairly recent date.

In addition to the familiar entities the physicist had to reckon with mysterious agencies such as gravitation or electric force; but this did not disturb his general outlook. … It was taken to be one of the main aims of research to discover how to reduce these agencies to something describable in terms of familiar conceptions — in short to “explain” them. …

Then at last it was seen that the linkage to familiar concepts should be through the advanced constructs of physics and not at the beginning of the alphabet. We have suffered, and we still suffer, from expectations that electrons and quanta must be in some fundamental respects like materials or forces familiar in the workshop — that all we have got to do is to imagine the usual kind of thing on an infinitely smaller scale. It must be our aim to avoid such prejudgments, which are surely illogical; and since we must cease to employ familiar concepts, symbols have become the only possible alternative.

Although this book may in most respects seem diametrically opposed to Dr.
Whitehead‘s widely read philosophy of Nature, I think it would be truer to regard him as an ally who from the opposite side of the mountain is tunnelling to meet his less philosophically minded colleagues. The important thing is not to confuse the two entrances.

Nature of Exact Science. One of the characteristics of physics is that it is an exact science, and I have generally identified the domain of physics with the domain of exact science.

… exact science invokes, or has seemed to invoke, a type of law inevitable and soulless against which the human spirit rebels. …

[The] whole subject-matter of exact science consists of pointer readings and similar indications.  … The essential point is that, although we seem to have very definite conceptions of objects in the external world, those conceptions do not enter into exact science and are not in any way confirmed by it. Before exact science can begin to handle the problem they must be replaced by quantities representing the results of physical measurement.

There is always the triple correspondence

(a) a mental image, which is in our minds and not in the external world;
(b) some kind of counterpart in the external world, which is of inscrutable nature;
(c) a set of pointer readings, which exact science can study and connect with other pointer readings.

And so we have our schedule of pointer readings ready to make the descent. And if you still think that this substitution has taken away all reality from the problem, I am not sorry that you should have a foretaste of the difficulty in store for those who hold that exact science is all-sufficient for the description of the universe and that there is nothing in our experience which cannot be brought within its scope.

I should like to make it clear that the limitation of the scope of physics to pointer readings and the like is not a philosophical craze of my own but is essentially the current scientific doctrine. It is the outcome of a tendency discernible far back in the last century but only formulated comprehensively with the advent of the relativity theory. The vocabulary of the physicist comprises a number of words such as length, angle, velocity, force, potential, current, etc., which we call “physical quantities”. It is now recognised as essential that these should be defined according to the way in which we actually recognise them when confronted with them, and not according to the metaphysical significance which we may have anticipated for them.

Limitations of Physical Knowledge. Whenever we state the properties of a body in terms of physical quantities we are imparting knowledge as to the response of various metrical indicators to its presence, and nothing more. After all, knowledge of this kind is fairly comprehensive. A knowledge of the response of all kinds of objects — weighing-machines and other indicators — would determine completely its relation to its environment, leaving only its inner un-get-atable nature undetermined.

Mathematics is the model of exact inference; and in physics we have endeavoured to replace all cruder inference by this rigorous type. Where we cannot complete the mathematical chain we confess that we are wandering in the dark and are unable to assert real knowledge. Small wonder then that physical science should have evolved a conception of the world consisting of entities rigorously bound to one another by mathematical equations forming a deterministic scheme. This knowledge has all been inferred and it was bound therefore to conform to the system of inference that was used. … But making all allowance for future progress in developing the scheme, it seems to be flying in the face of obvious facts to pretend that it is all comprehensive, Mr. X is one of the recalcitrants. When sound-waves impinge on his ear he moves, not in accordance with a mathematical equation involving the physical measure numbers of the waves, but in accordance with the meaning that those sound-waves are used to convey. To know what there is about Mr. X which makes him behave in this strange way, we must look not to a physical system of inference, but to that insight beneath the symbols which in our own minds we possess.



In the scientific world the conception of substance is wholly lacking, and that which most nearly replaces it, viz. electric charge, is not exalted as star-performer above the other entities of physics. For this reason the scientific world often shocks us by its appearance of unreality. It offers nothing to satisfy our demand for the concrete.

… The modern scientific theories have broken away from the common standpoint which identifies the real with the concrete. …

he cleavage between the scientific and the extrascientific domain of experience is, I believe, not a cleavage between the concrete and the transcendental but between the metrical and the non-metrical. I am at one with the materialist in feeling a repugnance towards any kind of pseudo-science of the extrascientific territory. Science is not to be condemned as narrow because it refuses to deal with elements of experience which are unadapted to its own highly organised method ; nor can it be blamed for looking superciliously on the comparative disorganisation of our knowledge and methods of reasoning about the non-metrical part of experience. But I think we have not been guilty of pseudo-science in our attempt to show in the last two chapters how it comes about that within the whole domain of experience a selected portion is capable of that exact metrical representation which is requisite for development by the scientific method.


To put the conclusion crudely — the stuff of the world is mind-stuff. As is often the way with crude statements, I shall have to explain that by “mind” I do not here exactly mean mind and by “stuff” I do not at all mean stuff. Still this is about as near as we can get to the idea in a simple phrase. The mind-stuff of the world is, of course, something more general than our individual conscious minds; but we may think of its nature as not altogether foreign to the feelings in our consciousness. The realistic matter and fields of force of former physical theory are altogether irrelevant — except in so far as the mind-stuff has itself spun these imaginings. The symbolic matter and fields of force of present-day theory are more relevant, but they bear to it the same relation that the bursar’s accounts bear to the activity of the college. Having granted this, the mental activity of the part of the world constituting ourselves occasions no surprise; it is known to us by direct self-knowledge, and we do not explain it away as something other than we know it to be — or, rather, it knows itself to be. It is the physical aspects of the world that we have to explain, presumably by some such method as that set forth in our discussion on world-building. Our bodies are more mysterious than our minds — at least they would be, only that we can set the mystery on one side by the device of the cyclic scheme of physics, which enables us to study their phenomenal behaviour without ever coming to grips with the underlying mystery.

The mind-stuff is not spread in space and time; these are part of the cyclic scheme ultimately derived out of it. But we must presume that in some other way or aspect it can be differentiated into parts. … When messages relating to a table are travelling in the nerves, the nerve-disturbance does not in the least resemble either the external table that originates the mental impression or the conception of the table that arises in consciousness. … We are acquainted with an external world because its fibres run into our consciousness; it is only our own ends of the fibres that we actually know; from those ends we more or less successfully reconstruct the rest, as a palaeontologist reconstructs an extinct monster from its footprint.

Again Bertrand Russell writes —

What the physiologist sees when he examines a brain is in the physiologist, not in the brain he is examining. What is in the brain by the time the physiologist examines it if it is dead, I do not profess to know; but while its owner was alive, part, at least, of the contents of his brain consisted of his percepts, thoughts, and feelings. [Analysis of Matter, p. 320.]

I assume that we have left the illusion of substance so far behind that the word “stuff” will not cause any misapprehension. I certainly do not intend to materialise or substantialise mind. Mind is — but you know what mind is like, so why should I say more about its nature? The word “stuff” has reference to the function it has to perform as a basis of world-building and does not imply any modified view of its nature.

It is true that I have a strong impression of an external world apart from any communication with other conscious beings. But apart from such communication I should have no reason to trust the impression.

This domestic definition of existence for scientific purposes follows the principle now adopted for all other definitions in science, namely, that a thing must be defined according to the way in which it is in practice recognised and not according to some ulterior significance that we imagine it to possess.

No familiar conceptions can be woven round the electron; it belongs to the waiting list. Similarly the description of the processes must be taken with a grain of salt. The tossing up of the electron is a conventional way of depicting a particular change of state of the atom which cannot really be associated with movements in space as macroscopically conceived. Something unknown is doing we don’t know what — that is what our theory amounts to. It does not sound a particularly illuminating theory. I have read something like it elsewhere —

The slithy toves
Did gyre and gimble in the wabe.

There is the same suggestion of activity. There is the same indefiniteness as to the nature of the activity and of what it is that is acting. And yet from so unpromising a beginning we really do get somewhere. We bring into order a host of apparently unrelated phenomena; we make predictions, and our predictions come off. The reason — the sole reason — for this progress is that our description is not limited to unknown agents executing unknown activities, but numbers are scattered freely in the description.

It would not be a bad reminder of the essential unknownness of the fundamental entities of physics to translate it into “Jabberwocky”; provided all numbers — all metrical attributes — are unchanged, it does not suffer in the least. Out of the numbers proceeds that harmony of natural law which it is the aim of science to disclose. We can grasp the tune but not the player. Trinculo might have been referring to modern physics in the words, “This is the tune of our catch, played by the picture of Nobody”.


In the old conflict between freewill and predestination it has seemed hitherto that physics comes down heavily on the side of predestination. Without making extravagant claims for the scope of natural law, its moral sympathy has been with the view that whatever the future may bring forth is already foretold in the configurations of the past … .

… It
seems contrary to our feeling of the dignity of the mind to suppose that it merely registers a dictated sequence of thoughts and emotions; but it seems equally contrary to its dignity to put it at the mercy of impulses with no causal antecedents. I shall not deal with this dilemma. Here I have to set forth the position of physical science on this matter so far as it comes into her territory. …

The foregoing paragraph is from the manuscript of the original lecture delivered in Edinburgh. …

In rewriting this chapter a year later I have had to mingle with this attitude of indifference an attitude more definitely hostile to determinism which has arisen from the acceptance of the Principle of Indeterminacy (p. 220). There has been no time for more than a hurried examination of the far-reaching consequences of this principle; and I should have been reluctant to include “stop-press” ideas were it not that they appear to clinch the conception towards which the earlier developments were leading. The future is a combination of the causal influences of the past together with unpredictable elements — unpredictable not merely because it is impracticable to obtain the data of prediction, but because no data connected causally with our experience exist. It will be necessary to defend so remarkable a change of opinion at some length. Meanwhile we may note that science thereby withdraws its moral opposition to free-will. Those who maintain a deterministic theory of mental activity must do so as the outcome of their study of the mind itself and not with the idea that they are thereby making it more conformable with our experimental knowledge of the laws of inorganic nature.

[It] is a healthy attitude to assume that nothing is beyond the scope of scientific prediction until the limits of prediction actually declare themselves. (sic) …

[ We] must recall that knowledge of the physical world has to be inferred from the nerve-messages which reach our brains, and the current epistemology assumes that there exists a determinate scheme of inference (lying before us as an ideal and gradually being unravelled). But, as has already been pointed out, the chains of inference are simply the converse of the chains of physical causality by which distant events are connected to the nerve-messages. If the scheme of transmission of these messages through the external world is not deterministic then the scheme of inference as to their source cannot be deterministic, and our epistemology has been based on an impossible ideal. In that case our attitude to the whole scheme of natural knowledge must be profoundly modified.

Whether or not there is a causal scheme at the base of atomic phenomena, modern atomic theory is not now attempting to find it; and it is making rapid progress because it no longer sets this up as a practical aim.

[When] we ask what is the characteristic of the phenomena that have been successfully predicted, the answer is that they are effects depending on the average configurations of vast numbers of individual entities. But averages are predictable because they are averages, irrespective of the type of government of the phenomena underlying them.

The New Epistemological Outlook. Scientific investigation does not lead to knowledge of the intrinsic nature of things. “Whenever we state the properties of a body in terms of physical quantities we are imparting knowledge of the response of various metrical indicators to its presence and nothing more”

But if a bodyis not acting according to strict causality, if there is an element of uncertainty as to the response of the indicators, we seem to have cut away the ground for this kind of knowledge. It is not predetermined what will be the reading of the weighing-machine if the body is placed on it, therefore the body has no definite mass; nor where it will be found an instant hence, therefore it has no definite velocity; nor where the rays now being reflected from it will converge in the microscope, therefore it has no definite position; and so on. It is no use answering that the body really has a definite mass, velocity, position, etc., which we are unaware of; that statement, if it means anything, refers to an intrinsic nature of things outside the scope of scientific knowledge. We cannot infer these properties with precision from anything that we can be aware of, because the breach of causality has broken the chain of inference. Thus our knowledge of the response of indicators to the presence of the body is non-existent; therefore we cannot assert knowledge of it at all. So what is the use of talking about it? The body which was to be the abstraction of all these (as yet unsettled) pointer readings has become superfluous in the physical world. That is the dilemma into which the old epistemology leads us as soon as we begin to doubt strict causality.

n phenomena on a gross scale this difficulty can be got round. A body may have no definite position but yet have within close limits an extremely probable position. When the probabilities are large the substitution of probability for certainty makes little difference; it adds only a negligible haziness to the world. But though the practical change is unimportant there are fundamental theoretical consequences. All probabilities rest on a basis of a priori probability, and we cannot say whether probabilities are large or small without having assumed such a basis. In agreeing to accept those of our calculated probabilities which are very high as virtually equivalent to certainties on the old scheme, we are as it were making our adopted basis of a priori probability a constituent of the world-structure — adding to the world a kind of symbolic texture that cannot be expressed on the old scheme.

The Principle of Indeterminacy. Thus far we have shown that modern physics is drifting away from the postulate that the future is predetermined, ignoring it rather than deliberately rejecting it. With the discovery of the Principle of Indeterminacy (p. 220) its attitude has become more definitely hostile.


In assessing whether the symbols which the physicist has scattered through the external world are adequate to predetermine the future, we must be on our guard against retrospective symbols. It is easy to prophesy after the event.

Volition. From the philosophic point of view it is of deep interest to consider how this affects the freedom of the human mind and spirit. A complete determinism of the material universe cannot be divorced from determinism of the mind. Take, for example, the prediction of the weather this time next year. The prediction is not likely ever to become practicable, but “orthodox” physicists are not yet convinced that it is theoretically impossible; they hold that next year’s weather is already predetermined. We should require extremely detailed knowledge of present conditions, since a small local deviation can exert an ever-expanding influence. We must examine the state of the sun so as to predict the fluctuations in the heat and corpuscular radiation which it sends us. We must dive into the bowels of the earth to be forewarned of volcanic eruptions which may spread a dust screen over the atmosphere as Mt. Katmai did some years ago. But further we must penetrate into the recesses of the human mind. A coal strike, a great war, may directly change the conditions of the atmosphere; a lighted match idly thrown away may cause deforestation which will change the rainfall and climate. There can be no fully deterministic control of inorganic phenomena unless the determinism governs mind itself. Conversely if we wish to emancipate mind we must to some extent emancipate the material world also. There appears to be no longer any obstacle to this emancipation.

… It seems that we must attribute to the mind power not only to decide the behaviour of atoms individually but to affect systematically large groups — in fact to tamper with the odds on atomic behaviour. This has always been one of the most dubious points in the theory of the interaction of mind and matter.

… To use an analogy — we have admitted an uncertainty which may take or spare human lives; but we have yet to find an uncertainty which may upset the expectations of a life-insurance company. Theoretically the one uncertainty might lead to the other, as when the fate of millions turned on the murders at Sarajevo. But the hypothesis that the mind operates through two or three key-atoms in the brain is too desperate a way of escape for us, and I reject it for the reasons already stated.

… There can be no unique probability attached to any event or behaviour; we can only speak of “probability in the light of certain given information”, and the probability alters according to the extent of the information. It is, I think, one of the most unsatisfactory features of the new quantum theory in its present stage that it scarcely seems to recognise this fact, and leaves us to guess at the basis of information to which its probability theorems are supposed to refer.

XV. Science and Mysticism

Possibly TBC.


…Starting from aether, electrons and other physical machinery we cannot reach conscious man and render count of what is apprehended in his consciousness. Conceivably we might reach a human machine interacting by reflexes with its environment; but we cannot reach rational man morally responsible to pursue the truth as to aether and electrons or to religion. …

… The physicist now regards his own external world in a way which I can only describe as more mystical, though not less exact and practical, than that which prevailed some years ago, when it was taken for granted that nothing could be true unless an engineer could make a model of it. There was a time when the whole combination of self and environment which makes up experience seemed likely to pass under the dominion of a physics much more iron-bound than it is now. That overweening phase, when it was almost necessary to ask the permission of physics to call one’s soul one’s own, is past. The change gives rise to thoughts which ought to be developed. Even if we cannot attain to much clarity of constructive thought we can discern that certain assumptions, expectations or fears are no longer applicable.

I think that the “success” theory of reasoning will not be much appreciated by the pure mathematician. For him reasoning is a heaven-sent faculty to be enjoyed remote from the fuss of external Nature. It is heresy to suggest that the status of his demonstrations depends on the fact that a physicist now and then succeeds in predicting results which accord with observation. Let the external world behave as irrationally as it will, there will remain undisturbed a corner of knowledge where he may happily hunt for the roots of the Riemann-Zeta function. …

… The accusation is often made that, by its neglect of aspects of human experience evident to a wider culture, physical science has been overtaken by a kind of madness leading it sadly astray. It is part of our contention that there exists a wide field of research for which the methods of physics suffice, into which the introduction of these other aspects would be entirely mischievous.

It will perhaps be said that the conclusion to be drawn from these arguments from modern science, is that religion first became possible for a reasonable scientific man about the year 1927. If we must consider that tiresome person, the consistently reasonable man, we may point out that not merely religion but most of the ordinary aspects of life first became possible for him in that year. Certain common activities (e.g. falling in love) are, I fancy, still forbidden him. If our expectation should prove well founded that 1927 has seen the final overthrow of strict causality by Heisenberg, Bohr, Born and others, the year will certainly rank as one of the greatest epochs in the development of scientific philosophy. But seeing that before this enlightened era men managed to persuade themselves that they had to mould their own material future notwithstanding the yoke of strict causality, they might well use the same modus vivendi in religion.

Scientific discovery is like the fitting together of the pieces of a great jig-saw puzzle; a revolution of science does not mean that the pieces already arranged and interlocked have to be dispersed; it means that in fitting on fresh pieces we have had to revise our impression of what the puzzle-picture is going to be like. One day you ask the scientist how he is getting on; he replies, “Finely. I have very nearly finished this piece of blue sky.” Another day you ask how the sky is progressing and are told, “I have added a lot more, but it was sea, not sky; there’s a boat floating on the top of it”. Perhaps next time it will have turned out to be a parasol upside down ; but our friend is still enthusiastically delighted with the progress he is making. The scientist has his guesses as to how the finished picture will work out; he depends largely on these in his search for other pieces to fit; but his guesses are modified from time to time by unexpected developments as the fitting proceeds. These revolutions of thought as to the final picture do not cause the scientist to lose faith in his handiwork, for he is aware that the completed portion is growing steadily. Those who look over his shoulder and use the present partially developed picture for purposes outside science, do so at their own risk.

The lack of finality of scientific theories would be a very serious limitation of our argument, if we had staked much on their permanence. …

If the scheme of philosophy which we now rear on the scientific advances of Einstein, Bohr, Rutherford and others is doomed to fall in the next thirty years, it is not to be laid to their charge that we have gone astray. Like the systems of Euclid, of Ptolemy, of Newton, which have served their turn, so the systems of Einstein and Heisenberg may give way to some fuller realisation of the world. But in each revolution of scientific thought new words are set to the old music, and that which has gone before is not destroyed T^ut refocussed. Amid all our faulty attempts at expression the kernel of scientific truth steadily grows; and of this truth it may be said — The more it changes, the more it remains the same thing.


This all seems very reasonable. I shall try to reduce my quotes above to make them an easier read without – hopefully – mangling their meaning too much. Please let me know if you think I have already gone too far, or if you have any good arguments against the views expressed above.

See Also

Eddington’s New Pathways in Science for an updated view.

Dave Marsay

Bateson’s Mind and Nature

Gregory Bateson Mind and Nature: A necessary unity Wildwood House 1979

Bateson was an anthropologist with some interesting and influential views on biological evolution, cybernetics and information theory, not least that it is difficult to communicate ideas outside of a broad range characteristic of a particular culture. He is known as the ‘father’ of family therapy. As a mathematician I can only try to interpret him as best I can.

This is Bateson’s last book, summarising his ideas.

I Introduction

[While making] an attempt to reexamine the theories of biological evolution in the light of cybernetics and infromation theory …. [it] became monstrously evident that schooling in this country [USA] and England was so careful to avoid all crucial issues that I would have to write a second book to explain what seemed to me elementary ideas relevant to evolution and top almost any other biological or social thinking – to daily life and the eating of breakfast. … Even grown-up persons with children of their own cannot give a reasonable account of concepts such as entropy, sacrement, syntax, number, quantity, pattern, linear relation, name, class, relevance, energy, redundancy, force, probability, parts, whole, information, tautology, homology, mass (either Newtonian or Christian), explanation, description, rule of dimension, logical type, metaphor, topology, and so on. …

It seemed to me that the writing out of some of these very elementary ideas could be entitled, with a  little irony, “Every Schoolboy (sic) Knows.”

The attempt to write down his ideas was a transformative experience.

It began to seem that the old-fashioned and still-established ideas about epistemology … were a reflection of an obsolete physics and contrasted in a curious way with the little we seem to know about living things.

… What is the difference between the physical world of pleroma [the non-living world that is undifferentiated by subjectivity], where forces and impacts provide sufficient basis of explanation, and the creatura, [the living world], where nothing can be understood until difference and distinctions are invoked?

I offer you the phrase the pattern which connects as a synonym, another possible title for this book.

… Why do schools teach almost nothing of the pattern which conncets? Is it that teachers know that they carry the kiss of death which will turn to tastelessness whatever they toiuch and therefore they are wisely unwilling to touch or teach anything of real-life importnavce? Or is it that they carry the kiss of death because they dare not teach anything of real-life importance? What’s wrong with them?

We have bene trained to think of patterns, with the exception of those of music, as fixed affairs. It is easier and lazier that way but, of course, nonsense. In truth, the right way to think about the pattern whgich connects is to think of it as primarily (whatever that means) a dance of interacting parts and only secondarily pegged down by various sors of physical limits and by those limits which organisms characteristically impose.

[If] the world be connected, if I am at all fundamentally right in what I am saying, then thinking in terms of stories must be shared by all mind or minds, whether ours or those of redwood forests and sea anemones.
Context and relevance must be characteristic not only of all so-called behaviour (those stories which are projected into “action”), but also those of all internal stories … .

What is a story that it may connect the As and Bs, its parts? And is it true that the general fact that parts are connected in this way is at the very root of what it is to be alive? I offer you the notion of context, of pattern through time.

… I am asserting that whatever the word context means, it is an appropriate word, the necessary word, in the description of all these differently related processes [that he gives as examples].

There is a parallel confusion in the teaching of language that has never been straightened out. Professional linguists may know what’s what, but children in schoold are still taught nonsense. They are told that a “noun” is the “name of a person, place or thing,” that a “verb” is “an action word,” and so on. That is, they are taught … to define something by what it supposely is in itself, not by its relation to other things.

I what is offered in this book, the heirarchical structure of thought, which Bertrand Russell called logical typing, will take the place of heirarchical strcture of [Lamarck’s] Great Chain of Being … What is importnat is that, right or wrong, the epistemology shall be explicit.

So the immediate task of this book is to construct a picture of how the wwrold is joined together in its mental aspects. How do ideas, information, steps of logical or pragmatic consistency, and the like fit together? How is logic … related to an outside world of things and creatuyres, parts and wholes? … How is the world of logic, which eschews “circular argument,” related to a world in which circular trains of causation are hte rule rather than the exception?

[We] shall see as every schoolboy ought to know that [classical] logic is precisely unable to deal with recursive circuits without generating paradox and that quantities are precisely not the stuff of complex communicating systems.
In other words, logic and quantity turn out to be inappropriate devices for describing organisms and their intercations and internal organization.

At present, there is no existing science whose special interest is the combining of pieces of information.

Throughout, the thesis will be tht it is possibel and worthwhile to think about many problems of order and disorder in the biological universe and that we have a considerable supply of tools which we do not use … partly because we are unwilling to accept the necessities that follow from a clear view of the human dilemma.

II Every Schoolboy Knows

Science, like art, religion, commerce, warfare, an even sleep, is based on presuppositions. It differs, however from most other branches of human activity in that not only are the pathways of scientific thought determnine by the presupposiitons of the scientists but their goals are the testing and revision of old presuppositions and the creation of new.
[It] is clearty desirable … for the scientist to know conciously and be able to state his own presuppositions. It is also conventient and necessary for scientific judgment to know the presuppositions of colleaugues working in the same field. Above all, it is necessry for the reader of scientific matter to know the presupposition of the writer.

Those who lack all idea that it is possible to be wrong can learn nothing except know-how.

It is worthwhile to attempt a tentative recognition of certain basic presuppositions which all minds must share or, conversely, to define mind by listing a number of such basic communicational characteristics.

1. Science never proves anything

2. The map is not the territory, and the name is not the thing named

3. There is no objective experience

4. The processes of image formation are unconcious

5. The division of the perceived universe into parts and wholes is convenient and may be necessary, but no necessity determines how it shall be done

6. Divergent sequences are unpredictable

7. Convergent sequences are predictable

8. “Nothing will come of nothing”

9. Number is different from quantity

10. Quantity does not determine pattern

11. There are no monotone “values” in biology

12. Sometimes small is beautiful

13. Logic is a poor model of cause and effect

14. Causality does not work backwords

15. Language commonly stresses only one side of any interaction

16. “Stability” and “Change” describe parts of our descriptions

III Multiple Versions of the World

In this chapter … I ask … “What bonus or increment of knowing follows from combining information from two or more sources? [But] my ultiumate goal is an inquiry into the learger pattern which connects.

1. The case of Difference

[It] take sat least two somethings to create a difference.

The stuff of sensation … is a pair of values of some variable, presented over time to a sense organ whose response depends upon the ratio between the members of the pair.

2. The case of Binocular Vision

   The binocular image, which appears undivided, is in fact a complex synthesis of informatioin from the left fron tin the right brain and a corresponding synthesis of m,aterail from th right front in the left brain.

[The] difference between the information provided by [the two retinas] is itself information of a different logical type.

9. The case of “description,” “toutology,” and “explanation”

   A pure description would include all the facts (i.e., all the effective differences) immanent in the phenomena to be described but would indicate no kind of connection among these phenomena that might make them more understandable. …

In science … descriptiopn and explanation … are connected by what is technically a tautology. … Al that is claimed is that if the axioms be such and such and the postulate such and such, then the theorems will be so and so. In otherwords, all that tautology affords is connections between propositions. …
Tautology contains no information whatever, and explanation (the mapping of description onto tautology) contains only the information tat was presetn in the description. … Description, on the other hand, contains information but no logic and no explanation. For some reason, human beings enormously value this combining of ways of organizing information or material.

It is necessary to understand that right and left cannot be defined … .

[An] explanation is a mapping of the pieces of a description onyto a tautology, and an explanation becomes accepytable to the degree that you arev willing to accept the links of the tautology. … That is all. It is always a matter of natural history, a matter of the faith, trust, rigidity and so on of the organism … .

The manner of search … might be called the method of double or multiple comparison.

….  It is the Platonic thesis of this book that epistemology is an indivisiblem integraeted meta-sceince whose subject matter is the world of evolution, thought, adaptation, embryology, and genetics – the science of mind in its widest sense.

IV Criteria of Mental Process

This chapter is an attempt to maker a list of criteria such that if any aggregate of phenomena, and system, satisfies all the criteria listed, I shall unhestitatingly say that the aggregate is a mind and shall expect that, if I am to understand that aggregate, I shall need sorts of explanation different from those which would suffice to explain the charateristics of its smaller parts.
This list is the cornerstone of the whole book. … This book must stand or fall … by the validity of the idea that some such structuring of pistemology, evolution, and epigenesis is possible.

  1. A mind is an aggregate of interacting parts or components.
  2. The interaction between parts of mind is triggered by difference, and … difference is related to negentropy and entropy rather than to energy.
  3. Mental process requires collateral energy.
  4. Mental process requires circular (or more complex) chains of determination.
  5. In mental process the effects of difference are to be regarded as transforms (i.e., coded versions) of the difference which preceded them. The rules of such transformation must be comparatively stable … but are not themselves subject to transfromation.
  6. The description and classification of these processes of transformation discloses a hierarchy of logical types immanent in the phenomena.

I shall argue that the phenomena which we call thought, evolution, ecology, life, learning, and the like occur only in systems that sastisfy these criteria.

V Multiple versions of relationship

In this chapter, in addition to talking about double description, I want to examine the subject of boundaries. What limits the units, what limits “things”, and above all, what, if anything, limits the self?
… “Inside” and “outside” are not appropriate metaphors for inclusion and exclusison when we are speaking of the self.
The mind contains no things, no pigs, no people, no midwife toads, or what have you, only ideas (i.e. news of difference), information about “things” in quotes, always in quotes. Similarly, the mind contains no time and no space, only ideas of “time” and “space”. It follows that the boundaries of the individual, if real at all, will be … something … like the sacks that represent stes in set theoretical diagrams or bubbles that come out of the mouths of the characters in comic strips.

Relationship is not internal to the single person. It is nonsense to tal;k about “dependency” or “aggressiveness” or “pride”, and so on. … All characterological adjectives are to be reduced or expanded to derive their definitions from patterns of interchange, i.e., from combinations of double description. [The] understanding (concious and unconcious) of behaviour through relationship gives a new logical type of learning.

The whole matter is a little difficult to grasp because we have been taught to think of learning as atwo-unti affair: The teacher “taught,” and the stuent (or experimental animal) “learned.” But the lineal model becomes obsolete when we learned bout cybernetic cicuits of intercation. The minimum unit of interction conatins three components. … stimulus, response and reinforcement.

Similalrly, we can expect self-validftion in other examples of the same logical tying. {These are] categories of contextual organization of behaviour.

1. Know thyself

2. Totemism

3. Abduction

VI The Great Stochastic Processes

It is a general assumption of this book that both genetic change and the process called learning (including the somatic chnges induced by habit and environment) are stochastic processes. In each case there is, I believe, a stream of events that is random in certain aspects and in each case there is a nonrandom selective process which causes certain of the random components to “survive” longer than others. Without the random, there ca be no new thing.
I assume that in evolution the production of mutatnt forms is either random within  whatver set of alternatives the status quo ante will permit or that, if mutation be oredred, the criteria of that rdering are irrelevant to the stresses of the organism.

We face, then, two great stochastic systems that are partly interaction and partly isolated from each other. One system … is .. learning; the other is … evolution.

The task of this chapter is to show how these two stochastic systems, working at different levels of logical typing, fit together into a single ongoing biosphere that could not endure if neitehr somatic or genetic change were fundmentally different from what it is.
The unity of the combined system is necessary.

Finally, it is necessary to put together the two stoichastic processes which I have separated for the sake of analysis. What formal relationship exists between the two?
[When] we admit naming as a phenomenon occuring in and organizing the phenomena we study, we acknowledge ipso facto that in these phenomena, we expect hierarchies of logical typing.
So far we can go with Russell and Principia. But we are not in Russell’s world of abstarct logic or mathematics and cannot acept an empty heirarchy of names or classes. For the mathematician, it is all very well to speak of names of names of names or of classes of classes of classes. But for the scientits, this empty world is insufficient. We are trying to deal with an interlocking or interaction of digital (i.e., naiming) and anologic steps. The process of naming is itself nameable, and this fact comples us to substitute an alternation for the simple ladder of logical types that Principia would suppose.

… There must always be a generative process whereby the classes are created efore they can be named.

VII From Classification to Process

The thrust of my argument is that the very process of perception is an act of logical typing. Every image is a complex of many-level coding and mapping.

It is necessary to expand on the relationship between form  and process, treating the notion of form as an analogue of what I have been calling tautology and process as an analogue of the aggregate of phenomena to be explained. As form is to process, so tautology is to description.

{There are] three dichotomies: form-process, calibration-feedback, and higher-lower logical type. … I shall argue that [the first two] are … mutually synonymous but that the relationship between higher and lower logical type is more complex. [Structure] may determine process and … process may determine structure. I believe that this is the analogue in the real world of Rusell’s abstract step from class to class of classes.

The effect of any … jumping of levels, upward or downward, is that infromation appropriate as a basis for decision at level will be used as a basis for decision at some other level, a common variety of error in logical typing.
In legal and administrative systems, such jumping of logical levels is called ex post facto legislation. In families, the anologous errors are called double blinds.

[It] appears that the idea of “logical typing,” when transplanted from the abstarct realms inhabited by mathematical philosphers to the hurly-burly of organisms, takes on a very different appearance. Instead of heirarchy of classes, we face a hierarchy of orders of recursiveness.

A world of sense, organization, and communication is not conceiveable without discontinuuity, without threshold. If sense organs can recieve news only of difference, and if neurons either fire or do not fire, then threshold becomes necessarily a feature of how the living and mental world is put together.
Chairoscuro is all very well, but William Blake tells us firmly that wise men see outlines and therefore they draw them.

My Comments

If not for the introduction, what ‘Every schoolboy knows’ might seem a reasonable summary of what British workers used to make of ‘workers education’, when it was a ‘thing’, of the ideas that used to be reflected in ‘worthy’ British TV  programmes when they were aimed at educating the masses, and what some seemed to think students should be taking from their ‘progressive’ education. Since then, these things seemed largely to have become accepted. It follows from Bateson’s (reasonable) views on ‘differences that make a difference’ that the promulgation of such insights became both less important and more difficult. But what is the situation now?

At least Bateson provides a useful check-list, backed up by a reasonable discursion. Usefully, this makes repeated reference to Russell’s notion of logical typing, and to Russell & Whitehead’s Principia. In essence, seems to be proposing a dual inter-connected ‘ladders’, both of which resemble Russel”s hierarchies. This is reminiscent of Whitead’s process view. As ‘every schoolboy knows’, it would be a mistake to assume (as some seem to) that reality had a structure of the same general ‘type’ as mathematical ‘models’, but maybe something like Whitehead’s notions of processes and epochs could be used to reason logically about Bateson’s process and form? In any case, Bateson’s account is calls to mind category theory, albeit with more complicated structures than the usual ‘linear’ applications. Maybe something like category theory could be used to reason logically about what can be known? That is, maybe one can reason about life, mind and nature logically, just so long as one uses a logic that takes account of what ‘every schoolboy knows’?






Abduction. Used by Bateson to refer to a third scientific methodology (along with induction and deduction) which was central to his own holistic and qualitative approach. Refers to a method of comparing patterns of relationship, and their symmetry or asymmetry (as in, for example, comparative anatomy), especially in complex organic (or mental) systems. The term was originally coined by American Philosopher/Logician Charles Sanders Peirce, who used it to refer to the process by which scientific hypotheses are generated.
Criteria of Mind (from Mind and Nature A Necessary Unity):[31]

Creatura and Pleroma. Borrowed from Carl Jung who applied these gnostic terms in his “Seven Sermons To the Dead”.[32] Like the Hindu term maya, the basic idea captured in this distinction is that meaning and organisation are projected onto the world. Pleroma refers to the non-living world that is undifferentiated by subjectivity; Creatura for the living world, subject to perceptual difference, distinction, and information.
Deuterolearning. A term he coined in the 1940s referring to the organisation of learning, or learning to learn:[33]
Schismogenesis – the emergence of divisions within social groups.
Information – Bateson defined information as “a difference which makes a difference.” For Bateson, information in fact mediated Alfred Korzybski’s map–territory relation, and thereby resolved, according to Bateson, the mind-body problem.[34][35][36]

UK Defence Applied Research 1945-1990

Eds. Robert Bud and Philip Gummett (1999), Cold War Hot Science: Applied Research in Britain’s Defence Laboratories 1945-1990, Harwood ISBN 90-5702-481-0


The history described in these pages is … sometimes cheering, sometimes saddening, but always worth recounting.

SirHermann Bondi FRS

Introduction: Don’t You Know There’s a War ON?

The Cold War is central to the understanding of science in the second half of the twentieth century. …

… At a central moment of the confrontation, 1967, the US National Academy of Sciences reported to the US House of Representatives on ‘applied science and technological progress’. … Among its key conclusion were that ‘studies of the history and sociology of applied sceince are important‘.

Unlike its larger US counterpart, British defence research … was conducted largely within dedicated civil service establishments, employing at its height over 30,000 scientists and administrators. … In 1959 … about 3,000 technicians and 2,000 professional scientists were employed.

[The] establishments were not simply  centres of research: they were (and are) also a reservoir of technical advice available to government, whether on the feasibility of proposals from would-be contractors, on the seriousness with technical dimensions of intelligence should be viewed, or the future of technology as a factor in strategic planning. In times of crisis they also proved to be a valuable asset, able to mobilise effort rapidly to address novel problems, or to work out how to adapt existing equipment, which has so often been found itself being used in circumstances different from those for which it had been intended. Finally, civilian spinoff has also been an important aspect of their work, even though … this has been only a secondary concern.

… Scientists came to play new roles as advisers at the highest levels: Tizard and Lindeman, .. Blackett and Jones, Zuckerman and Bernal … . Their impact was felt not only on equipment … but also in the development of tactical and strategic ideas, as the ‘Sunday Soviets’, in which the radar scientists freely brainstorm with military officers … . In short, scientists came to play a new role as colleagues and advisers of those men in Whitehall and in the field.

The postwar position

Overall, Britain remained a world leader in many fields of defence technology, and was well placed in many others. [But] Industry was concerned that an excessively military focus would detract from civilain competance. … As a 1958 … memorandum explained: ‘the aim must be also to ensure .. a fund of scientific knowledge and resources is built up which will be adequate for the development of future generations of weapons, even though their precise nature cannot yet be foreseen

The history of defence policy … until the late 1980s could be summed up as reflecting, on the one hand, the withdrawal from Empire and the definition of a new world role in the face of growing superpower strength, and on the other the search for a defence policy that would be economically sustainable. … [The] main focus of policy was a potential war against the Soviet Union, which would be fought on the north German plain. … Not until the 1982 Falklands conflict was a a major conventional campaign waged.

[The] life-expectancy of every defence review between 1945 and the mid-1960s had been a mere two and a half years. As the usual gestation period for new weapon systems was from two to four times longer than that figure, this was clearly too short. Things were to get worse in the mid-1960s, ‘with doubts and dissension developing at almost every level of British defence policy, with strategy, fundamental weapons and administrative methods all being thrown into the melting pot’. …

There developed … growing unease at managerial control of the equipment programme. Much of the blame was attributed to inter-service rivalry and the incapicity of the central Ministry of Defence to control the Service ministries and the supply agencies. A review … led to the creation of a new, more unified, Ministry of Defence from April 1964. …

By the mid 1970s … the need for further defence cuts had become imperative. … Britain’s Armed Forces were still performing a greater range of missions than any other in the world, except those of the superpowers and, possibly, France. …

An attempt to deal with these problems radically was made in 1981 … .

The Falklands and after

[British] industry and the defence research establishments played an important behind-the-scenes part. …

By 1985, all [the old] items remained on the agenda and some new ones had been added. The MInistry of Defence … underwent a major reorganisation [, partly to exploit] new opportunities for international collaboration … seing here a route to economies. [Sources] increasingly pressed the case for a greater emphasis to be placed on conventional weapons.

The [US Strategic Defense Initiative] also raised issues of cooperation within Europe. … [France] proposed … Eureka … .

By means of enormous exertion the country had retained the capacity for (relatively) autonomous development and production of a wide range of weapons systems .. . In some fields (notably VTOL, communications equipment, night vision, display systems, short-range antiaircraft missules, and computer software, to name but some), Britain’s capability could be seen to be in the front rank by world standards.

At the same time, it must be acknowledged that British weapons have rarely been tested in the kind of confrontation for which they were designed … . [Hence] the jury remains out on just how effective the products of the Cold War defence equipment programme might have been.

Overview of R&D effort

The UK has … traditionally been a high spender .. . [Britain] remained, at least until the late 1980s, the clear leader of the second division, although an order of magnitude behind the superpowers … half of government expenditure went on R&D went on defence. Thre concerns would be repeated … : How could the research establishments best focus on priorities, select key affordable programmes that would benefit the military most efficiently, and maximise spinoff to the civilian economy? [A committee under] the chairmanship of Sir Solly Zuckerman [made an] urgent recommendatrion that there should be a seemless web between the definition of an operational requirement and the monitoring of a final manufacturing process.

The role of the establishments in the 1980s ands 1990s … have been restricted to those that fall most naturally to goivernment, and without which government would find it hard to act as an informed customer of the defence industries and to stay abreast of international trends. Thus, the establishments assist with the setting of specifications, perform long-range research, run certain expensive central facilities, support industrial contractors, engage in evaluation and acceptance work, and generally act as a reservoir of advice for government.

[But] in 1991 the main non-nuclear defence research establishments were moved into a more contractural relationship with the Ministry of Defence … .

In 1995 DRA underwent a 60% expansion, with the addition of test and operational analysis facilities, pluse CBDE, and was renamed the Defence Evaluation and Research Agency (DERA). {this] was the largest science and technology organistaion in Europe, with a turnover of more than £1 billion. Staff (excluding contract staff), numbered 11,700, compared with 8600 in 1995, though it was projected to fall to about 10,000 by 1997-8.


Time and time again we see the leapfrogging of offence and defence, but often all on the side of the West. [The] high level of investment in western military research and development undoubtedly stimulated research to nullify its consequences. …

The second theme in this book is the nature of applied science and technology. The focus of research … was upon entire systems, hence the importance of having access within the establishments to a wide range of cognate technologies and – critically – the capacity to integrate them … for a particular military task. [If] there really has been an information-led revolution in military affairs, together with movement towards the much-vaunted ‘system of systems’, then its foundations were laid half a century ago in the UK by the admiralty research establishments .

[Zuckerman] concluded that efficient financial management of military R&D mattered far less than choosing the right projects on which to work, and the right people to do the job.

The function of the establishments within ‘UK plc’ has … been ambivalent: pressed to contribute, yet in the nature of the innovation process not best placed to do so.


[The] essential dilemmas remain: how to maintain technological capabilities against uncertain and shifting requirements: how to organise projects so as to achieve success; how to recruit and retain the necessary staff; how to cope with the budget set; and how to contribute effectively to the wider science, technology and innovation base. These … are, it seems, inherent to the activity itself.

Some more Chapters

  • The Royal Aircraft Establishment from 1945 to Concorde
  • Rotary-Wing Aircraft
  • Ground-Based Air Defence and ABM Systems
  • Armoured Fighting Vehicles
  • Aircraft Carriers and Submarines: Naval R&D in Britain in the Mid-Cold War
  • Thermal Radiation and its Applications

Open Systems in a Closed World: Ground and Airborne Radar in the UK, 1945-90

Jon Agar and Jeff Hughes

… Through the iaspora of radar scientists and engineers who returned to and reoriented civil science after the war, and those who remained in government employment and continued the development of electronic systems for miltary purposes, and those who helped create the new electronic industries servicing both civil and military sectors, radar technology and the skills that produced it became pervasive elements of the postwar sociotechnical ensemble, part of the reason why the academic-military-industrial complex existed, and simuklateneously the ‘glue’ that held it together.

Yet … Radar technology does not exist in and of itself: it is always part of some larger system, be it the Chain Home reporting system established during the Second World War and entrenched today as air traffic control and early warning systems … . At the same time, however, it si another characteristic feature of radar that particular assemblies or subsystems might well have multiple uses. … A particular piece of radar technology could thus be deployed and made useful in various ways which depended on complex negotiations between producers (scientists and engineers at the research establishments) and operational users. It is this systemic yet flexible aspect of radar that makes it difficult to abstract the development of radar per se from the development of the wider networks in which it is designed to play a part.

In the postwar context … however, with a radically different institutional framework and a very different set of relationships between the producers and the users of radar technology, the development of these component technologies was subject to different demands and an entirely different institutional philosophy of innovation. …

[Paul Edwards] characterises the closed world as ‘a dome of technological oversight … within which every event was interpreted as part of a titanic struggle between the superpowers’. Its key themes, he argues, were ‘global surveillance and control through high-technology military power’, and the cyborg-like integration of human and machine in ‘weapon systems and strategies whose human and machine componets could function as a seamless web, even on the global scales and in the vastly compressed timescale of global nuclear war’. In this regime computers – and, we argue radars and othyer electronic devices which were linked to and through them – ‘made the closed world work simultaneously as technology, as political system, and as ideological mirage’.

… Our main conclusions centre on the relations between research establishments and service users, the trend towards integration of systems, but also thier suprising vulnerability – ‘open’, then, in several senses of the word.

A world of their own? Radar establishements and their work 1945-49

At … TRE a very particular institutional style of innovation had developed during the war, in which very close communictaion between entrepreneurial innovating scientists and potential users was a key part of the process by which new technologies cam einto beingand were put into active service. Typically, informal contacts between Service staff (whose visits to the research establishements were promoted by events such as A.P. Rowe’s celebrated ‘Sunday Soveiets’ at TRE) and scientists would lead to a two-way discussion of operational problems and suggested solutions, and of new devices and possible applications. Confluences of interest could be negotiated in this informal setting, and scientists ‘selling’ a new idea wopuld usually identify closely with the needs of a particualr user or group of users, wgho wopuld become their ‘product champion’. Authority to produce a ‘mock up’ of proposed new equipment was easily obtained, and only succesful and very promising results would usually be drawn to the attention of higher levels of management Only after many proitotypes and trials, when an innovation reached the stage of engineering for prouction, would the management of a development project become formalised. Crucially, TRE would continuen to be involved with applications as they were introduced into operational service, an importnat factor in the success of TRE in developing new systems to help counter the Nazi threat.

The best staff were not ‘back-room boys’ (e.g Frank Jones, Bernard Lovell).

   Responsive and efficient, the informality of innovation at TRE worked admirably well during the war. The close liaison between providers and users essential to the success of TRE all but ceased with the end of the wewr, howveer, when the reeqarchers in the establishments were not required to reacxt quickly to unfolding events but to work systematically within a long-term research programme. This change of approach fostered a very different philosophy of innovation, which must have been quite alien to those who had come to scientific maturity during TYRE’s war years.

Remobilisationand Rearmament: Radar develpment 1948-53

[Following the start of the Cold War, the Korean war and the establishment of NATO there was a reorganisation] stemming from concern at ‘higher levels of management that … there might be duplication and wasted effort’. [However] the organisational change … raised fundamental issues concerned with the relationships between establishemnts and ‘users’, and with the trend …  towards comprehensive integreated radar systems.

Consolidation but not unification: The Radar Research Establishment 1953-61

[The amalgamation of TRE and RRDE] was a decision [not] ‘to abolish the split of work by “user” – and the close relationship between service and establishment was and is an often-cited factor behind the succes of TRE.

[In 1953 the links between RRE and industry] were very strong. Academic links were weaker.

The Limits of Integration: Data handling, Airborne Radar and TSR-2

The expansion of research at RRE in the wake of Sputnik was first accommodated within the organsiational structure imposed in the 1953 merger. {in 1962 a further reorganisation took place … .

A significant feature of the RRE programme in the 1950s was a group under A.M. Uttley investigating the role of the huamn operator as a link in the control chain. This grouop originated in wartime TRE, and explored problems of reaction times, response characteristics and patternm recognition in radar users, all with the aim of understanding errors in control operations. Although the group was wound up after Uttley’s resignation to take up a senior post at NPL in 1957, RRE neverhteless retained a marked sensitivity to the ole of the human operator, and it may be that the work of Uttley’s group fundamentally shaped the philosophy of the establishment. … Two radically different solutions [to radar integration] were possible. Both RRE and ASWE (ASRE) succeeded in developing data processing and display systems meeting the operational requirement, one (RRE) employing analogue and the other (ASWE) digital techniques. [DRPC’s Cockburn] noted that ‘the important issue was that two experienced establishments faced with meeting similar operational requirements in a similar time sclae were depending on different technological solutions’.

Intimately tied to this difference in approach – what Cockburn called a ‘transition from well-tried analogue techniques to more promising but risky digital techniques – were the attitudes of the two establishements to automation: ASWE used automatic data extraction as part of their precocious ADA system [a language for embedded, real-time, safety-critical computing], developed since 1953 and described elsewhere in this volume, whereas RRE had chosen manual data extraction and processing … . [A working party ] found that RRE had not pursued manual systems out of ignorance … but felt that automatic systems could not cope with jamming, insisting that instead the discrimination of a human was needed. [Behind] the difference in technical solutions were radically different attitudes t human operators, jamming, digital techniques, and a natural preference for intramural development.

Ground Radar and Radars for other users, 1960-80

We have argued that the relationship between establishment and user was an importnat factor in the development of radar. The establishments were confronted with a variety of users whose needs might diverge widely, of course, and again it is eay to see how the flexible and pervasive nature of radar allowed the technology to be creatively adapted to new situations.

Conclusion: Reorganisation, Rationalisation and RSRE into the 1980s

 … In 1948 … a TRE report had [noted] that ‘In peace as in war there must always be the closest possible co-operation between the scientist and the user to ensure that the user gets as nearly as possible what he wants and that the scientist is not allowed to design an equipment so complicate dthat it cannot be used’. This could fairly be said to characterise the attitude of the radar establishemnsts for the follwoing 50 years, up to and including RSRE becoming one of the founders of the DRA in 1990. Indeed, the continuyity ad stability of the relationships between establishments and users against a rapidly (and occasionally radically) changing organisational and strategic background is one of the most marked features of our account. [The] establishments have always displayed a sensitivity towards users, at first in their traditional Army and Air Force constituencies, and increasingly in the 1960s in industry … .

‘In almost every field of Defence interests, electronic equipment forms the basis of modern weapon systems’. … This widespread applicability of the [radar] technology, and its openness … was important in protecting basic radar R&D, and again allowed for maximum creativity in user-establishment relations.

… It would be easy to conclude that increasing integartion and centralisation implied increasing automation. Yet, as we have seen, there were significant differences of opinion between establishments with respect to the role of the human operator in the command and control chain – and therefore to the kinds of technologies and, ultimately, the kinds of global strategies that might be appropriate to manage the closed world.

An integrated system, embodying the fruits of new research from the establishemnts and in which automation promised minimisation ofm human erro, should not, howver,be equated easily with success. … The vulnerability of the closed world is perhaps the most significant theme to emerge from a study of postwar radar.

Even more Chapters

  • Naval Command and Control Equipment: The Birth of the Late Twentieth Century ‘Revolution in Military Affairs’
  • Laser Research and Development
  • Chemical and Biological Warfare and Defence, 1945-90
  • Defence Physiology
  • Civil Spinoff from the Defence Research Establishments

Government Management of Defence Research since the Second World War

Stephen Robinson

Opportunities for exploiting science are normally identified by key individuals working at the leading edge, and many defence establishments grew up ad hoc during and just afetr the war as technology, with obvious application in key military fields, was identified. …

Public affairs are run largely by civil servants in the administrative and executive classes, and as government reserach expanded between the wars, more and more scientists were recruited. In 1945 arrangements were formalised and the science class was created, with separate career streams for scientists, scientific assistants and experimental staff.

[The] individual merit (IM) promotion scheme, introduced in 1946 … has served research well. … [It] has allowed scientists to remain closely associated with important programmes, and to provide the core of excellence so importnat to national defence and to international knowledge exchange. Nevertheless, the scheme has suffered from two flaws. The panel of independent assessors has included world-class scientists more familiar with pure than applied work, and the five-year review, stipulated to retain the new rank, has sometimes enhanced the predisposition to remain in a particular field against the run of changing defence priorities. Apocryphal stories relating to IM promotion abound, including that of candidates who failed for being involved in research ‘too closely associated with defence matters’!

In practice, research, more than most activities, is programmed in detail by those doing the work, within constarints imposed from above. … [Two] aspects have remained in common: the need for military staff to interpret user requirements and set programme constraints with the help of scientific advice, and the attentuation of unwanted projects by prioritising candidate programmes within fixed resource allocations. [MoD] methods to some extent inspired he customer-contractor principle made famous by Lord Rothschild in his 1971 report on the organisation and management of government R&D. His proposdals were aimed mainly at applied research in the civil sector … . Lord Rothschild was well aware that overzealous customer control would stultify research progress, and he proposed a surcharge of 10 per cent should be available to scientists for discretionary work. It is a cause of concern that although the MoD allows such cost recoveries in industry, it has not seen them as appropriate in its own laboratories.

The key to good applied research is that high-calibre research staff should be able to steer their work by intimate knowledge of users and their potenbtial needs, and transfer technology effectively to those who will apply it to meet such needs. The challenge to management is to lead, and to deploy capital, human resources and advice in a way that sharpens rather than blunts this process. It is a challenge that is rarely met as well as might be hoped, but it is fair to claim that performance within the MoD has compared favourably over the years to similar organisations overseas.

… Thirty years ago a fortunate scientist could apply the wisdom he gained inn research and design of one generation of equipment to the next. Now he is lucky if he sees one design through to completion, and even if he should survive to its successor, there is a likelihood that rapid advances in technology will have passed him by. … Growing difficulties in the maintenance of appropriate industrial capacity for the deep understanding of ever more complex military systems have long been recognized. Increasingly, these are the overiding considerations in formulating policy for research programmes and in the career development of staff.

Transfer to Industry of Design, System Integration and Project Management

None of these subjects is easy. It is difficult to recruit scientists while laying off engineers, and much of industry was only too happy to accept the low-risk option of manufacturing equipment to government drawings. …

… Lack of clarity concerning the proper development and project management responsibility of industry may also have contributed to the incredible concept of the 1980s that, by more rigorous contracts, a  monopoly customer could increase competition and reduce technical risk at the same time. Sadly for managers, it is all too easy to fail in the implementation of a good policy, but virtually impossible to succeed when it is bad.

… Wiht budgets to hand, research staff have been able to lead military requirements by u technical demonstration, and dictate the direction of industrial research through extramural contracts. Consequently, well-meaning, excellent and articulate scientific staff have sometimes presented management with problems they have not been able to recognise, let alone solve, and valuablem industrial resources have bene misused. As might be expected, this phenomenon has been most evident in areas such as data processing, where civil technology has beenn overtaking that more familiar within the defence community. …

Although this problem may, on occasion, have led to suboptimal defence procurement, the exploitation record of defnece research has generally been good. Queen’s Awards for Tehnology are given when commeriocal success in industry has derived from highly originalk scientifi wor, and bearing in mind the small fraction of defence establishment activity which is appropriate, it is highly commendable that, between 1976 and 996, government defence laboratoroes (20 awards) have shared in awards three times as often as Research Council laboratories (7 awrads), and ten times as often as university departments (2 awards). On balance, British industry has certainly gained considerable commercial advantage fromm knowledge discovere by government scientists for defence purposes.

Failure of Inter-Service Prgramme Coordination

[In] time ofm peace, with budgets on a tri-Service basis and the work moving towards genberic research, there is little doubtthat research establishments have become too parochial. … [Scientists] hardly communicate at all. Moreover ‘focus’ and ‘viable levels of resource’ are watchwords of applied research, and dispersal is almost always a recipe for fragmentation and failure.

As developments moved to industry after the war the inefficiencies of uncoordinated rseearch were exposed … .Electronics, being generic and of increasing importance to all three Services, was the main problem area, and matters improved somehwat when the principle electronice laboratories were merged at Malvern.

Electronic Component Development

… In future conflicts a protagonist with access to superior components, for example for target acquisition, intelligence gathering or communications, is likely to prevail.


Although management theories and fashionable styles developed to address particualr problems abound, management remains a largely pragmatic activity, a game with uncertain rules, played on an unevn field, in an unfair world. ..


… Applied research workers in the defence community should feel involved and strongly committed to succesful equipment procurement, for it is written: ‘the Lord said unto Cain, Where is Abel they brother? and he said, I know not: Am I my brother’s keeper?’ After all their endevours, it would be sad if the Defence Evaluation and Research Agency survived, only to carry he mark of Cain into the next millenium. … [Government] and industry should not forget that the [DERA] is a jewel in their crown. In a world of increasing technical sophistication, where industry will be less and less able to carry out training and applied research at the level demanded by the market, it may well prove to be a model of the type of integrated staff development and applied research institute that society requires.





Aesop’s Boy who Cried Wolf

Aesop The Boy Who Cried Wolf

The Fable

Wikipedia’s summary is:

The tale concerns a shepherd boy who repeatedly tricks nearby villagers into thinking a wolf is attacking his town’s flock. When a wolf actually does appear and the boy again calls for help, the villagers believe that it is another false alarm and the sheep are eaten by the wolf.


Teachers have used the fable as a cautionary tale about telling the truth, but an educational experiment in the first decade of the 21st century suggested that reading “The Boy Who Cried Wolf” increased children’s likelihood of lying.

My Comments

A Variation

As commonly told the tale concerns ‘villagers’. Its seems odd that they do not find a more trusted person to watch over their flock, and so even if the boy had been lying the villagers would seem to deserve some blame for the sheep’s deaths. In the original, though, the people who respond are actually ‘labourers’, which makes more sense. So in what follows I do not suppose that the villagers were responsible for the selection and tasking of the boy to watch the flock.


This is intended as a moral tale, to discourage children from lying. But there are two problems:

  1. It makes no sense, logically.
  2. It appears not to have the intended effect.

There is a puzzle here, depending on what you make of the tale. Either the teachers or their students are mistaken: why? I find this interesting as an application of mathematics that does not involve numbers.

If we try to model the situation logically, then how do we know that the boy was originally lying? Maybe he later confessed, but can we trust a confession that he may have felt was coerced? A reasonable interpretation is:

  • On many occasions:
    • The boy raised the alarm.
    • The villagers responded.
    • The villagers found no wolf.
  • Later:
    • The boy raised the alarm.
    • The villagers did not respond.
    • The wolf ate the sheep.
    • The villagers blamed the boy.

More than this, we do not know. It may be that the earlier alarms were false. From a probability theory point of view if this is the only hypothesis we consider then we should assign a probability of ‘1’, and the conventional moralist interpretation would seem justified. But what if the earlier alarms were genuine, and the response of the villagers had scared off the wolf? Once we recognize this possibility then the conventional interpretation no longer seems justified.

From a pedagogic point of view, what lesson might we expect others to take from the tale? If we suppose that the student has never been falsely accused of lying by those in authority (parents, teachers, ‘adults’) then possibly it might not occur to them that the villagers were mistaken, and hence it might be thought that they would draw the intended conclusion. But how could we ever now that someone else doesn’t think that they have been falsely accused? If they have, what lesson might they draw?

It may seem that the boy was given an unreasonable job to do. By being diligent and truthful he ended up in disgrace, seeming worse than if he had simply failed to raise the alarm. Maybe after the first time, he would have done better to ‘raised the alarm’ in such a way that the villagers would arrive too late to save the sheep. (Maybe he could rush and trip or otherwise invent a plausible excuse for delay). Surely everyone would be better off?

The boy would have been to lie (about the reason for the delay) than to act as instructed to the best of his ability.

This seems a reasonable conclusion about lying for a student to draw. (He may also draw conclusions about the insightfulness of his teachers and – once he discovers that the conventional interpretation is widely accepted – about the supposed ‘wisdom’ of those supposedly ‘in authority’.

Induction and probability

Inductive reasoning  is where a prediction is made based upon experience. In this case, the villagers have never found a wolf after many alarms, so – inductively – they may reasonably come to expect not to find a wolf if they responded to the next alarm. But this is not to say that the alarm was necessarily false.

A ‘mathematical’ interpretation is given by Bayesian inference: If at some point a villager thinks that the boy might be false alarming (with non-zero probability), and that if he were not then with some non-zero probability a wolf would be noticed, then with each failure to find a wolf the probability that the boy is falsely alarming should be increased by the ‘Bayes factor’, and so tend to 1, virtual certainty. Thus this fable provides an example of the failure of Bayesian inference. (In this case, because the alternative to the boy false-alarming is not a suitable hypothesis to which Bayesian inference can be properly applied.)

A logically minded student might suppose that the villagers were not very good at inference, and that their teachers had failed to appreciate this.


From the point of view of an individual villager, if there is a strong enough majority in favour of not responding, there may seem no point in suggesting that there may be a wolf, and every incentive not to. There is not even any incentive to consider any alternatives to the majority belief that no wolf was found because there was no wolf. Once the sheep have been killed there may seem little point in defending the boy by ‘speaking truth to power‘ and some  strong incentives not to challenge the common view. Thus the story suggests a difference between saying what is actually true and going along with a common view.

Social truth

It is clearly sensible for villagers and children not to challenge a common view once sufficiently strongly held. But how do such odd views form?

If we suppose that the villagers are intolerant of psychological (as against logical) uncertainty, then in particular the recognition that there may have been a wolf is uncomfortable, and the only obvious way to remove the uncertainty is to deny it, and to suppose that there was no wolf. Thus all those with such intolerance have an incentive to suppose the alarms false. Once the sheep have been savaged there is an additional incentive to blame the boy: the alternative would be to blame the common view of the villagers, when there is no obvious way to remedy the discomfort that this would cause.

Thus the fable can be read as suggesting that the tendency to reduce ‘felt’ uncertainty and establish some ‘common-sense truth’ can not only lead to disaster (the loss of the sheep) but also to necessary ‘white lies’ and abuse.


The fable also speaks to a dilemma of adaptability: continuing to respond when there may be no wolf would seem maladaptive, yet the obvious adaptation leads to harm. But the story can also be read as providing a socially acceptable solution.

Suppose that the villagers differ in their tendency to form various beliefs and to consider alternatives, and do not feel obliged to have a common reaction. Then one might expect the response to decline with each alarm. Eventually the wolf might attack when there has still been some response, and even a single villager could call for reinforcements and so save some sheep.


There are many circumstances in which activity needs to be concerted to be effective, yet in the face of uncertainty it may, as above, be better for a minority to behave usefully than for everyone to be useless. The ability to act in concert is something that needs developing and maintaining, so having a general concern for social cohesion seems reasonable, and so a default to act in concert might seem indicated. That is, cooperation might seem a better default than collaboration. Such an attitude might well promote a tendency to deny uncertainty as far as seems reasonably possible.

The key, here, seems to be to recognize that collaboration, in which social cohesion is developed and maintained while at the same time at least tolerating any tendencies of the collaborators that aren’t certainly harmful, and even encouraging the development of reasonable differences (e.g., of viewpoint, experience, skills, affiliations). Possibly the ‘villagers’, suspecting that the child was raising false alarms, might have tried engaging with him to resolve the situation, rather than simply ignoring him or telling him off.


It seems to me that ‘the boy who cried wolf’ can be used as an example of what is, in fact, a very common problem: if one views a situation in the common way, one may well find the common interpretation plausible, even compelling. But outside of the common view, the ‘correct’ interpretation is much less clear. Sometimes, it pays to ‘think outside the box‘, even when the box is not so obvious. Mathematical modelling may help to clarify the box and challenge ‘common sense’ as well as have its more common use, of solving problems when taken ‘at face value’.

Dave Marsay

Roux’s Forms of Mathematization

Sophie Roux. Forms of Mathematization (14th-17th Centuries). Early Science and Medicine, Brill Academic Publishers, 2010, 15, pp.319-337

As a mathematician I am intrigued by notions of ‘mathematization’. This fascicle presents a point of view that seems to me similar to that of many social scientists.

[Mathematization] was founded on the metaphysical conviction that the world was created pondere, numero et mensura, or that the ultimate components of natural things are triangles, circles, and other geometrical objects. This metaphysical conviction had two immediate consequences: that all the phenomena of nature can be in principle submitted to mathematics and that mathematical language is transparent; it is the language of nature itself and has simply to be picked up at the surface of phenomena.

The assumption that reality is measurable or consists of geometrical objects is no longer tenable, yet the consequences seem to still hold: ‘mathematization in British. or mathematisation (ˌmæθɪmətaɪˈzeɪʃən)’ is variously described as being:

  • Reduction to mathematical form (Mirriam-Webster)
  • The act of interpreting or expressing mathematically, or the state of being considered or explained mathematically. (Collins)
  • The characterization of objects and events in terms of mathematical relationships. (Various papers on sciencedirect.)


Language as an empirical phenomenon (just like many other empirical phenomena) is described in mathematical terms to obtain a ‘model’, which is investigated using mathematical means and the results are then projected back on the phenomenon. (We can also understand this mathematization as a matter of extracting the structure of the phenomenon in the form of a mathematical object.)  [Linguistics and Philosophy Jaroslav Peregrin, in Philosophy of Linguistics, 2012]

That is, mathematization seems to presuppose that the real phenomena of interest has a structure that is capable of being represented mathematically. This contrasts with that of a blogger:

the mathematization of a problem or area of study consists of applying mathematical ideas to that problem or field so as to think more precisely or clearly about things. …. Whether any human can follow the argument and understand the subject matter is another question

The main point of difference being to what extent the representation of a problem or field assists in thinking more precisely or clearly. The fascicle opines:

Grand narratives such as this [the effectiveness of mathematization] are perhaps simply fictions doomed to ruin as soon as they are clearly expressed.

Perhaps. But really?

The fascicle makes a non-controversial observation:

In general, the term “mathematization” refers to the application of concepts, procedures and methods developed in mathematics to the objects of other disciplines or at least of other fields of knowledge. A definition of this kind seems to assume that there is an agreement, first, on what is mathematics, second, on the profits that various disciplines can make out of its application and, third, on the relevance of the very notion of application. But there are many good reasons to think that such an agreement might be difficult to achieve.

It goes on:

There was never a working definition of mathematics in general; even at the time when the traditional definition of mathematics as the “science of quantities” or “magnitudes in general” emerged and was commonly accepted, there were different conceptions of quantities, and consequently different ways of conceiving of the unity of mathematics. Now, if the second-order question of how to define mathematics was ever raised, it is because it is a fundamentally complex field, that included various domains from its very beginning and that kept developing new domains throughout history.

Wikipedia notes that the term ‘mathematics’ still has no ‘generally accepted definition’. The fascicle characterises mathematics thus:

[With] the emergence of calculus and of infinite series, new domains began to be explored. In these circumstances, [to social scientists] it seems inevitable to admit that we should neither look for a definition of mathematics in general, nor think of mathematics as a unified field of knowledge, but, rather, submit to an historically situated and empirical determination of mathematics: what should be called “mathematics” is the activities of those who called themselves or were called by others “mathematicians”. As tautological and circular as it may appear, such a determination is not without consequence on how we should conceive of mathematization.

No reason for this baleful conclusion is offered, other than its seeming inevitability, so any other view is at least as credible. (I shall offer one, later.)

Arithmetic, in as far as it is the practice of numbers, generates a first form of mathematization: what we call “quantification” consists in capturing in numerical form certain aspects of material things. Such a capture requires indeed measurements, concrete apparatus and a growing concern for precise and standardized data, but also graphical techniques to present numerical results and intellectual techniques of approximation and averaging. Of course, quantification may be only peripherically related to the disinterested search of laws in natural philosophy: the alledged benefits of quantification are sometimes practical.

Even today, mathematization is often conceived of as mere quantification, and hence mathematization is often thought to imply various beliefs (such as measurability) that are, in fact, not mathematically credible. Such concepts are easy to refute. The current fascicle needs to be taken more seriously. For example:

Practices may refer to the non-verbal commitments shared by mathematicians that help them defining a scientific style and constituting an intellectual community; practices in this sense are opposed to explicit beliefs and assumed to be invisible to the mathematicians themselves.

The fascicle regards all kinds of mathematicians as mathematicians, so that most mathematicians are part of some broader community of scientists, technologists, engineers and ‘practical people’ rather than forming some relatively closed community of mathematicians. Hence their ‘non-verbal commitments’ are not just or even mainly mathematical: they are at least heavily influenced by the culture of the community in which they work. Contrary to what the above paragraph seems to suppose, there are often (as with other professions) some very clear issues that arise due to the ‘opposition’ (or at least differences and seeming contradictions) between the commitments of mathematicians as mathematicians and their commitments to their wider communities, for example between their professional integrity and their role as employees. These issues are far from ‘invisible’: even in academia they are a frequent topic of discussion.

[Mathematical] practices can be identified with practical mathematics, as contrasted with pure mathematics and which refers to the real world, with its economic interests, practical concerns, material instrumentation, local settings and complex social networks.

Quite so.


It seems to me that the social sciences generally have some very important things to say about ‘mathematical practices’. In particular, however highly one regards mathematics ‘as such’, mathematicians who are a part of a community (e.g., a work-force) are subject to much the same influences as other professionals, and one should not simply assume that they are as open and honest about their mathematics and its implications as one might wish. If ‘mathematization’ is what mathematicians do, and if we can’t trust mathematicians then clearly we can’t trust mathematization. In this case, mathematization might even seem to be a bad thing, in so far as creates an artefact that can only be understood and used by a relative few, and largely excludes people who otherwise seem to have essential insights into real problems.

If, as a mathematician, I try to adopt the viewpoint of social scientists, then it seems ‘inevitable’ that mathematization of ‘wicked problems‘ is going to make things worse.

But could one devise or develop communities in which mathematicians could be more open and honest, and what might the implications for ‘mathematization’ be? Could it not be a ‘force for good’?

It seems to me that there is a key point to be made:

Working mathematicians often seek to mathematize a problem or an area of interest, whereas – as mathematicians – all they can properly do is to mathematize a conception of a problem or a theory about an area. It seems to me that the difference matters.

For example, the role of mathematicians in economics and finance has been criticised, not unreasonably. But mathematizing a wrong theory isn’t going to correct it. It may make it clearer, less ambiguous. It may enable more efficient methods, or even totally new instruments. But if the theory is wrong, more efficient or more effective misguided methods may simply lead one to realise the inadequacies of the theory sooner.

This leads to another key point:

  1. Mathematizing a theory considered as if it were a straightforward description of some ‘real system’ is essentially using mathematics as a tool for developing the theory. It may accidentally reveal inconsistencies in a theory, but not other mistakes.
  2. An alternative is to consider a theory together with a description of how the theory is justified and tested. Mathematizing the theory in this broader context (i.e., ‘as a theory’ not just ‘as a description’) can then reveal the extent to which the theory is justified and has been tested.

My own experience is that whenever I try to mathematize an existing theory (which I do in the second sense) I have always found it wanting, even when the theory has already been mathematized in the first sense. Practitioners often initially disagree about whether this is ‘a difference that makes a difference’, and even those who think that it could make a difference (including me) recognize that there are other issues that could prevent a properly mathematized theory making a difference. But it seems to me that in considering a whole range of areas that have been mathematized in the first sense, there could be substantial payoffs in at least some of them in mathematizing in the second sense. But such mathematization alone is not enough: all it really does is to reveal ambiguities, imprecisions and potential or actual mistakes before brute experience does: one still needs appropriate subject-matter ‘experts’ to resolve the areas for investigation that have been revealed.

My experience is that for sufficiently complex problems one never ends up with a ‘reduction to mathematical form’, and of course such a reduction is not even possible in principle for ‘wicked problems’. But one can use insights from mathematization (in the second sense) to reduce the wickedness, which often seems to me to be a ‘good thing’.

Dave Marsay


I see that I offered to give a characterisation of mathematics. I think the core concept is ‘logical proof’. There are a few different schools of thought within mathematics as to what types of proof are appropropriate for which subjects. If we pick one, say x-proof, we can define an x-claim as one which has an x-proof. An x-method is one which x-provably can only produce x-claims. An x-logic is one which is used in x-proofs. An x-conjecture is one which for which no x-proof of falsehood is known and which is amenable to x-logic, and which we have some reason to think may be x-provable. An x-heuristic is a method which we x-conjecture often produces reasonable x-conjectures. An x-mathematics is the collection of the above x-types. I have never found the particular notion of x-proof to be that important, as long as it is genuinely ‘logical’ and not simply an appeal to some supposed authority.

P.P.S. What applies to mathematics seems to have some truth when applied to other disciplines. All disciplines seems to interpret situations in their own terms and make recommendations. To me, the the key question is always: have they cosnidered whether there discipline is adequate, or the extent to which its recommendations might usefully be complemented by those from other disciplines? How should the recommendations be caveated? More generally, a common practice is to treat different disciplines as ‘viewpoints’ and to meld their conclusions togther to reach some sort of ‘common sense’ resolution. But when is this adequate? Could logic play a special role?

Locke’s Essay: Book IV

John Locke Essay Concerning Human Understanding, Final Edition, 1699 (Original 1690).

Book IV: The Reality of Knowledge

Locke points out some implications of Book III. The key point is to distinguish between, for example, the idea of a tree and ideas about how an idea of a tree might relate to any supposed external reality. (When we reason about a map, we can’t just assume that the territory will be constrained by our conclusions, any more than we can expect a SATNAV to behave ‘intelligently’ in unusual circumstances.)


1. Our Knowledge conversant about our Ideas only.

2. Knowledge is the Perception of the Agreement or Disagreement of two Ideas.

3. This Agreement or Disagreement may be any of four sorts.

But to understand a little more distinctly wherein this agreement or disagreement consists, I think we may reduce it all to these four sorts:



1. Of the degrees, or differences in clearness, of our Knowledge: I. Intuitive

All our knowledge consisting, as I have said, in the view the mind has of its own ideas, which is the utmost light and greatest certainty we, with our faculties, and in our way of knowledge, are capable of, it may not be amiss to consider a little the degrees of its evidence. The different clearness of our knowledge seems to me to lie in the different way of perception the mind has of the agreement or disagreement of any of its ideas. For if we will reflect on our own ways of thinking, we will find, that sometimes the mind perceives the agreement or disagreement of two ideas IMMEDIATELY BY THEMSELVES, without the intervention of any other: and this I think we may call INTUITIVE KNOWLEDGE. For in this the mind is at no pains of proving or examining, but perceives the truth as the eye doth light, only by being directed towards it. Thus the mind perceives that WHITE is not BLACK, that a CIRCLE is not a TRIANGLE, that THREE are more than TWO and equal to ONE AND TWO. Such kinds of truths the mind perceives at the first sight of the ideas together, by bare intuition; without the intervention of any other idea: and this kind of knowledge is the clearest and most certain that human frailty is capable of. This part of knowledge is irresistible, and, like bright sunshine, forces itself immediately to be perceived, as soon as ever the mind turns its view that way; and leaves no room for hesitation, doubt, or examination, but the mind is presently filled with the clear light of it. IT IS ON THIS INTUITION THAT DEPENDS ALL THE CERTAINTY AND EVIDENCE OF ALL OUR KNOWLEDGE; which certainty every one finds to be so great, that he cannot imagine, and therefore not require a greater: for a man cannot conceive himself capable of a greater certainty than to know that any idea in his mind is such as he perceives it to be; and that two ideas, wherein he perceives a difference, are different and not precisely the same. He that demands a greater certainty than this, demands he knows not what, and shows only that he has a mind to be a sceptic, without being able to be so. Certainty depends so wholly on this intuition, that, in the next degree of knowledge which I call demonstrative, this intuition is necessary in all the connexions of the intermediate ideas, without which we cannot attain knowledge and certainty.

II. Demonstrative.

The next degree of knowledge is, where the mind perceives the agreement or disagreement of any ideas, but not immediately. Though wherever the mind perceives the agreement or disagreement of any of its ideas, there be certain knowledge; yet it does not always happen, that the mind sees that agreement or disagreement, which there is between them, even where it is discoverable; and in that case remains in ignorance, and at most gets no further than a probable conjecture. The reason why the mind cannot always perceive presently the agreement or disagreement of two ideas, is, because those ideas, concerning whose agreement or disagreement the inquiry is made, cannot by the mind be so put together as to show it. In this case then, when the mind cannot so bring its ideas together as by their immediate comparison, and as it were juxta-position or application one to another, to perceive their agreement or disagreement, it is fain, BY THE INTERVENTION OF OTHER IDEAS, (one or more, as it happens) to discover the agreement or disagreement which it searches; and this is that which we call REASONING. Thus, the mind being willing to know the agreement or disagreement in bigness between the three angles of a triangle and two right ones, cannot by an immediate view and comparing them do it: because the three angles of a triangle cannot be brought at once, and be compared with any other one, or two, angles; and so of this the mind has no immediate, no intuitive knowledge. In this case the mind is fain to find out some other angles, to which the three angles of a triangle have an equality; and, finding those equal to two right ones, comes to know their equality to two right ones.

3. Demonstration depends on clearly perceived proofs.

Those intervening ideas, which serve to show the agreement of any two others, are called PROOFS; and where the agreement and disagreement is by this means plainly and clearly perceived, it is called DEMONSTRATION; it being SHOWN to the understanding, and the mind made to see that it is so. A quickness in the mind to find out these intermediate ideas, (that shall discover the agreement or disagreement of any other,) and to apply them right, is, I suppose, that which is called SAGACITY.

9. Demonstration not limited to ideas of mathematical Quantity.

[It has been generally taken for granted, that mathematics alone are capable of demonstrative certainty: but to have such an agreement or disagreement as may intuitively be perceived, being, as I imagine, not the privilege of the ideas of number, extension, and figure alone, it may possibly be the want of due method and application in us, and not of sufficient evidence in things, that demonstration has been thought to have so little to do in other parts of knowledge, and been scarce so much as aimed at by any but mathematicians.] For whatever ideas we have wherein the mind can perceive the immediate agreement or disagreement that is between them, there the mind is capable of intuitive knowledge; and where it can perceive the agreement or disagreement of any two ideas, by an intuitive perception of the agreement or disagreement they have with any intermediate ideas, there the mind is capable of demonstration: which is not limited to ideas of extension, figure, number, and their modes.

My dictionary characterises mathematics as ‘an abstract science’, in which case using mathematical demonstrations to set the standard may not seem all that ambitious.

The notion of a mathematical demonstration has been considerably refined since Locke’s day. Thus demonstrations that Locke thought ‘as good as’ a mathematical one may now be considered not so reliable.

III. Sensitive Knowledge of the particular Existence of finite beings without us.

These two, viz. intuition and demonstration, are the degrees of our KNOWLEDGE; whatever comes short of one of these, with what assurance soever embraced, is but FAITH or OPINION, but not knowledge, at least in all general truths.


4. As, First All Simple Ideas are really conformed to Things.

FIRST, The first are simple ideas, which since the mind, as has been showed, can by no means make to itself, must necessarily be the product of things operating on the mind, in a natural way, and producing therein those perceptions which by the Wisdom and Will of our Maker they are ordained and adapted to. From whence it follows, that simple ideas are not fictions of our fancies, but the natural and regular productions of things without us, really operating upon us; and so carry with them all the conformity which is intended; or which our state requires: for they represent to us things under those appearances which they are fitted to produce in us: whereby we are enabled to distinguish the sorts of particular substances, to discern the states they are in, and so to take them for our necessities, and apply them to our uses. Thus the idea of whiteness, or bitterness, as it is in the mind, exactly answering that power which is in any body to produce it there, has all the real conformity it can or ought to have, with things without us. And this conformity between our simple ideas and the existence of things, is sufficient for real knowledge.

Locke is sometimes taken to mean that however we divide our idea of totality into parts, there must be separate real ‘things’ that correspond to those parts. He doesn’t actually seem to argue this, though, so it might be better to think of a real totality that is comprehended as separate things but may not actually be so divided. (As in physics, chemistry and biology.)

5. Secondly, All Complex Ideas, except ideas of Substances, are their own archetypes.

Secondly, All our complex ideas, EXCEPT THOSE OF SUBSTANCES, being archetypes of the mind’s own making, not intended to be the copies of anything, nor referred to the existence of anything, as to their originals, cannot want any conformity necessary to real knowledge.

6. Hence the reality of Mathematical Knowledge

I doubt not but it will be easily granted, that the knowledge we have of mathematical truths is not only certain, but real knowledge; and not the bare empty vision of vain, insignificant chimeras of the brain: and yet, if we will consider, we shall find that it is only of our own ideas. The mathematician considers the truth and properties belonging to a rectangle or circle only as they are in idea in his own mind. For it is possible he never found either of them existing mathematically, i.e. precisely true, in his life. But yet the knowledge he has of any truths or properties belonging to a circle, or any other mathematical figure, are nevertheless true and certain, even of real things existing: because real things are no further concerned, nor intended to be meant by any such propositions, than as things really agree to those archetypes in his mind. Is it true of the IDEA of a triangle, that its three angles are equal to two right ones? It is true also of a triangle, wherever it REALLY EXISTS. Whatever other figure exists, that it is not exactly answerable to that idea of a triangle in his mind, is not at all concerned in that proposition. And therefore he is certain all his knowledge concerning such ideas is real knowledge: because, intending things no further than they agree with those his ideas, he is sure what he knows concerning those figures, when they have BARELY AN IDEAL EXISTENCE in his mind, will hold true of them also when they have A REAL EXISTANCE in matter: his consideration being barely of those figures, which are the same wherever or however they exist.

Berkeley criticises this exposition, as it could be interpreted as meaning that abstract reasoning can be ‘real’. Locke, who was not a mathematician, may have been trying to convey his interlocutor, Newton’s, views. Exactly what these ‘really’ were is unclear, but never mind.

18. Recapitulation.

Wherever we perceive the agreement or disagreement of any of our ideas, there is certain knowledge: and wherever we are sure those ideas agree with the reality of things, there is certain real knowledge. Of which agreement of our ideas with the reality of things, having here given the marks, I think, I have shown WHEREIN IT IS THAT CERTAINTY, REAL CERTAINTY, CONSISTS. Which, whatever it was to others, was, I confess, to me heretofore, one of those desiderata which I found great want of.

That is, while empirical knowledge is always liable to Cromwell’s rule, formal, idealised, knowledge need not be. For example, we can know a lot about triangles, but we can’t know for certain that any real thing actually corresponds or ‘is’ a triangle as we conceive it.


1. Maxims or Axioms are Self-evident Propositions.

THERE are a sort of propositions, which, under the name of MAXIMS and AXIOMS, have passed for principles of science: and because they are SELF-EVIDENT, have been supposed innate, without that anybody (that I know) ever went about to show the reason and foundation of their clearness or cogency. It may, however, be worth while to inquire into the reason of their evidence, and see whether it be peculiar to them alone; and also to examine how far they influence and govern our other knowledge.

20. Their Use dangerous where our Ideas are not determined

And as these maxims are of little use where we have determined ideas, so they are, as I have showed, of dangerous use where [our ideas are not determined; and where] we use words that are not annexed to determined ideas, but such as are of a loose and wandering signification, sometimes standing for one, and sometimes for another idea: from which follow mistake and error, which these maxims (brought as proofs to establish propositions, wherein the terms stand for undetermined ideas) do by their authority confirm and rivet.


1. Knowledge is not got from Maxims.

But from comparing clear and distinct Ideas.

7. The true Method of advancing Knowledge is by considering our abstract Ideas.

9. Our Knowledge of Substances is to be improved, not by contemplation of abstract ideas, but only by Experience.

EXPERIENCE HERE MUST TEACH ME WHAT REASON CANNOT: and it is by TRYING alone, that I can CERTAINLY KNOW, what other qualities co-exist with those of my complex idea, v.g. whether that yellow heavy, fusible body I call gold, be malleable, or no; which experience (which way ever it prove in that particular body I examine) makes me not certain, that it is so in all, or any other yellow, heavy, fusible bodies, but that which I have tried. … For example, I cannot be certain, from this complex idea, whether gold be fixed or no; because, as before, there is no NECESSARY connexion or inconsistence to be discovered betwixt a COMPLEX IDEA OF A BODY YELLOW, HEAVY, FUSIBLE, MALLEABLE; betwixt these, I say, and FIXEDNESS; so that I may certainly know, that in whatsoever body these are found, there fixedness is sure to be. Here, again, for assurance, I must apply myself to experience; as far as that reaches, I may have certain knowledge, but no further.


  • That for Locke ‘experience’ is not passive.  It relies on active ‘trying’ and ‘application’.
  • Also, knowledge is just knowledge of what has been found by experience: If one extrapolates beyond experience (as in ‘induction’) one is going beyond what can be known.

Locke continues:

10. Experience may procure is Convenience, not Science.

I deny not but a man, accustomed to rational and regular experiments, shall be able to see further into the nature of bodies, and guess righter at their yet unknown properties, than one that is a stranger to them: but yet, as I have said, this is but judgment and opinion, not knowledge and certainty. This way of GETTING AND IMPROVING OUR KNOWLEDGE IN SUBSTANCES ONLY BY EXPERIENCE AND HISTORY, which is all that the weakness of our faculties in this state of mediocrity which we are in this world can attain to, makes me suspect that NATURAL PHILOSOPHY IS NOT CAPABLE IS BEING MADE A SCIENCE. We are able, I imagine, to reach very little general knowledge concerning the species of bodies, and their several properties. Experiments and historical observations we may have, from which we may draw advantages of ease and health, and thereby increase our stock of conveniences for this life; but beyond this I fear our talents reach not, nor are our faculties, as I guess, able to advance.

13. The true Use of Hypotheses.

Not that we may not, to explain any phenomena of nature, make use of any probable hypothesis whatsoever: hypotheses, if they are well made, are at least great helps to the memory, and often direct us to new discoveries. But my meaning is, that we should not take up any one too hastily (which the mind, that would always penetrate into the causes of things, and have principles to rest on, is very apt to do) till we have very well examined particulars, and made several experiments, in that thing which we would explain by our hypothesis, and see whether it will agree to them all; whether our principles will carry us quite through, and not be as inconsistent with one phenomenon of nature, as they seem to accommodate and explain another. And at least that we take care that the name of PRINCIPLES deceive us not, nor impose on us, by making us receive that for an unquestionable truth, which is really at best but a very doubtful conjecture; such as are most (I had almost said all) of the hypotheses in natural philosophy.

14. Clear and distinct Ideas with settled Names, and the finding of those intermediate ideas which show their Agreement or Disagreement, are the Ways to enlarge our Knowledge.

15. Mathematics an instance of this.

Chapter xv: Probability

This is not just about classical, numeric, probability. It is not even considered as a part of mathematics.

1.•Demonstration is showing the agreement or disagreement of two ideas by the intervention of one or more proofs,·theseparate links of·which have a constant, unchangeable,and visible connection with one another; and•probability is nothing but the appearance of such an agreement or disagreement, by the intervention of proofs whose connection isn’t perceived to be constant and unchangeable, but is or appears for the most part to be so, sufficiently to induce the mind to judge the proposition to be true or to be false.

2…. most of the propositions that we think with, reason with, use in discourse, and indeed act on,are ones of whose truth we can’t have undoubted knowledge. … But here there are degrees·of•confidence·from the very neighbourhood of certainty and demonstration right down to improbability and unlikelihood of truth, and downfurther to the brink of impossibility; and also degrees•ofassent from full assurance and confidence right down to conjecture, doubt, and distrust.

4.. . . .The grounds of probability are the two following. First, the conformity of something with our own knowledge, observation, and experience….

This may involve the dreaded ‘abstract reasoning’.

Chapter xvi: The degrees of assent

1.The grounds of probability laid down in the preceding chapter serve not only as the basis on which to decide whether to assent·to a proposition·but also as the measure of how strongly we should assent. Bear in mind, though, that whatever grounds of probability there maybe, they will operate on the truth-seeking mind only to the extent that they appear to it in its first judgment or its first look into the matter. … It suffices that they did once carefully and fairly sift the matter as far as they could, and that they have searched into everything that they can imagine might throw light on the question, and done their best to evaluate the evidence as a whole; and having thus once found on which side the probability appeared tot hem, after as full and exact an enquiry as they can make, they store the conclusion in their memories as a truth they have learned; and for the future they remain satisfied with the testimony of their memories that they have seen evidence for this opinion that entitles it to the degree of their assent that they are now giving to it.

3.I have to admit that men’s sticking to their past judgments and adhering firmly to conclusions formerly made often leads them to be obstinate in maintaining errors and mistakes. But their fault is not that they rely on their memories for what they previously judged well, but that they judged before they had examined well…. But in matters of probability we can’t always be sure that we have taken account of everything that might be relevant to the question, and that there is no evidence still to be found which could turn the probability-scales the other way, and outweigh everything that now seems to us to carry the most weight.

  1. …I the person you want to win over to your opinions is•one who examine sbefore he assents, you must allow him time to go over the account again, to recall points favouring his own side—ones he has currently forgotten—and to see on which side the advantage lies. And if he doesn’t think your arguments are good enough to indicate that he should take all that trouble reconsidering the matter, this is only what you often do insimilar cases; and you wouldn’t like it if others told to you what points you should study. … hose whohavefairly and truly examined·the grounds for their beliefs·, and have been brought by this beyond doubt about the doctrines they profess and live by, would have a fairer claim to requireothers to follow them. But there are so few of these, andthey find so little reason to be dogmatic in their opinions,that nothing insolent and bullying is to be expected from them; and there is reason to think that·in general·if men were better instructed themselves they wouldn’t push others around so much.

  2. Concerning … particular matters of fact,·I distinguish three kinds of case, to which I give a section each·. First, when something that fits with theconstant observation of ourselves and others in similarcases is supported by reports of all who mention it, weaccept it as easily and build on it as firmly as if it were certain knowledge; and we reason and act on it with as little doubt as if we had a perfect demonstration of it. Thus, if all Englishmen who have occasion to mention it were to affirm that it froze in England last winter, or that there were swallows seen there in the summer, I think one could hardly doubt this more than one does that seven and four are eleven. Thus, the first and highest degree of probability occurs when the general consent of all men in all ages, as faras it can be known, fits one’s own constant and never-failing experience in similar cases.

This is a kind of comparative probability, forming a partial order. Roughly speaking, something is regarded as more probable the more severely it has been tested. (Compare Keynes.) Note also that ‘swallows’ are species and hence abstractions, but despite that Locke argues that ceratin seeming ‘matters of fact’, if supported well enough, may reasonably be regarded as almost as certain as mathematical knowledge. (But his examples are very straightforward observations, with no deductions, inductions or abductions.)

Chapter xx: Wrong assent, or error

  1. Knowledge can be had only of visible and certain truth.So error isn’t a fault of our knowledge, but a mistake of our judgment when it gives assent to something that isn’t true.But if assent is based on likelihood, if what assent especially aims at is probability, and if probability is what I said it isin chapters xv and xvi, you will want to know how it comes about that men sometimes accept propositions that are not probable. For there’s nothing more common than contrariety of opinions; nothing more obvious than that one man wholly disbelieves what another only doubts of and a third firmly believes. The reasons for this may be very various, but Ithink they all come down to these four:
  1. Lack of proofs,·to be discussed in sections 2–4·.
  2. Lack of ability to use them,·section 5·
  3. Lack of will to use them.·section 6·.
  4. Wrong measures of probability,·sections 7–17·.

.Fourthly, there remains the last sort·of belief contrary to probability·, which occurs when people who have the real probabilities plainly laid before them nevertheless don’t accept the conclusion, and instead either suspend their assent or give it to the less probable opinion. This is the danger that threatens those who adopt wrong measures of probability. These wrong measures are

  1. Propositions that are not in themselves certain andevident, but doubtful and false, accepted as principles;·discussed in sections 8–10·.
  2. Received hypotheses;·section 1 .
  3. Predominant passions or inclinations;·sections 12–16·.
  4. Authority;·section 17·

8.The first and firmest ground of probability is the conformity something has to our own knowledge, especially the part of our knowledge that we have made our own and continueto regard as principles. These have so much influence on our opinions that it is usually by them that we judge concerning truth, and we measure probability in terms of them so strictly that if something is inconsistent with them—that is, with our ‘principles’—we count it not merely as improbable but as impossible. The reverence we give to these principles is so great, and their authority so supreme, that the testimony ofother men and even the evidence of our own senses are often rejected when they threaten to testify to something contrary to these established rules. …anyone who swallows wrong principles, blindly giving himself up to the authorityof some opinion that isn’t in itself evidently true, puts into his understanding a strong bias that will inevitably lead his assent astray.

17.The fourth and last wrong measure of probability thatI shall discuss keeps more people in ignorance or errorthan do the other three combined. I mentioned it in the foregoing chapter: it is the practice of giving our assent tothe common received opinions of our friends, our party, our neighbourhood, or our country.

18.Despite the great noise that is made about errors andopinions, I must be fair to mankind and say: There aren’t so many men with errors and wrong opinions as is commonly supposed.  ….For if we were to interrogate most partisans of most sects, so far from finding evidence that they acquired their opinions on the basis of examining arguments and the appearance of probability, we wouldn’t even find that they have any opinions of their own on the matters they are so zealous about!  … It is enough for him to obey his leaders, to have his hand and his tongue ready for the support of the common cause, in this way winning the approval of those who can give him credit, promotion, or protection in that society. Thus men become supporters of, and combatants for, opinions that they were never convinced of—indeed, ones that they never even had floating in their heads! I’m not playing down how many improbable or erroneous opinions there are out therein the world; but I am saying that there are fewer people that actually assent to them, and mistake them for truths, than there are generally thought to be.

Chapter xxi: The division of the sciences

1.All that can fall within the range of human understanding is in three categories.

  1. The nature of things as they are in themselves, their relations, and their manner of operation.
  2. What man himself ought to do, as a thinking and willing agent, for the attainment of any end, especially happiness.
  3. The ways and means by which the knowledge of each of those two is attained and communicated.

I think that science[= ‘high-level disciplined knowledge’] can properly be divided into these three sorts.

5. This is the first and most general Division of the Objects of our Understanding.

This seems to me the first and most general, as well as natural division of the objects of our understanding. For a man can employ his thoughts about nothing, but either, the contemplation of THINGS themselves, for the discovery of truth; or about the things in his own power, which are his own ACTIONS, for the attainment of his own ends; or the SIGNS the mind makes use of both in the one and the other, and the right ordering of them, for its clearer information. All which three, viz. THINGS, as they are in themselves knowable; ACTIONS as they depend on us, in order to happiness; and the right use of SIGNS in order to knowledge, being TOTO COELO different, they seemed to me to be the three great provinces of the intellectual world, wholly separate and distinct one from another.

The End

Dave Marsay

Locke’s Essay: Books I, II

John Locke Essay Concerning Human Understanding, Final Edition, 1699 (Original 1690).


Locke argues against the view that all decent humans share ceratin principles or ideas, not even enough to inform ‘right action’. This was strongly advertised in the prologue, where I comment further.



9. Abstraction.

The use of words then being to stand as outward mark of our internal ideas, and those ideas being taken from particular things, if every particular idea that we take up should have a distinct name, names must be endless. To prevent this, the mind makes the particular ideas received from particular objects to become general; which is done by considering them as they are in the mind such appearances,—separate from all other existences, and the circumstances of real existence, as time, place, or any other concomitant ideas. This is called ABSTRACTION, whereby ideas taken from particular beings become general representatives of all of the same kind; and their names general names, applicable to whatever exists conformable to such abstract ideas. Such precise, naked appearances in the mind, without considering how, whence, or with what others they came there, the understanding lays up (with names commonly annexed to them) as the standards to rank real existences into sorts, as they agree with these patterns, and to denominate them accordingly. Thus the same colour being observed to-day in chalk or snow, which the mind yesterday received from milk, it considers that appearance alone, makes it a representative of all of that kind; and having given it the name WHITENESS, it by that sound signifies the same quality wheresoever to be imagined or met with; and thus universals, whether ideas or terms, are made.

Compare Berkeley.

13. Difference between Idiots and Madmen.

In fine, the defect in naturals seems to proceed from want of quickness, activity, and motion in the intellectual faculties, whereby they are deprived of reason; whereas madmen, on the other side, seem to suffer by the other extreme. For they do not appear to me to have lost the faculty of reasoning, but having joined together some ideas very wrongly, they mistake them for truths; and they err as men do that argue right from wrong principles. For, by the violence of their imaginations, having taken their fancies for realities, they make right deductions from them. Thus you shall find a distracted man fancying himself a king, with a right inference require suitable attendance, respect, and obedience: others who have thought themselves made of glass, have used the caution necessary to preserve such brittle bodies. Hence it comes to pass that a man who is very sober, and of a right understanding in all other things, may in one particular be as frantic as any in Bedlam; if either by any sudden very strong impression, or long fixing his fancy upon one sort of thoughts, incoherent ideas have been cemented together so powerfully, as to remain united. But there are degrees of madness, as of folly; the disorderly jumbling ideas together is in some more, and some less. In short, herein seems to lie the difference between idiots and madmen: that madmen put wrong ideas together, and so make wrong propositions, but argue and reason right from them; but idiots make very few or no propositions, and reason scarce at all.


1. Number the simplest and most universal Idea.

Amongst all the ideas we have, as there is none suggested to the mind by more ways, so there is none more simple, than that of UNITY, or one: it has no shadow of variety or composition in it: every object our senses are employed about; every idea in our understandings; every thought of our minds, brings this idea along with it. And therefore it is the most intimate to our thoughts, as well as it is, in its agreement to all other things, the most universal idea we have. For number applies itself to men, angels, actions, thoughts; everything that either doth exist or can be imagined.

2. Its Modes made by Addition.

By repeating this idea in our minds, and adding the repetitions together, we come by the COMPLEX ideas of the MODES of it. Thus, by adding one to one, we have the complex idea of a couple; by putting twelve units together we have the complex idea of a dozen; and so of a score or a million, or any other number.

3. Each Mode distinct.

The SIMPLE MODES of NUMBER are of all other the most distinct; every the least variation, which is an unit, making each combination as clearly different from that which approacheth nearest to it, as the most remote; two being as distinct from one, as two hundred; and the idea of two as distinct from the idea of three, as the magnitude of the whole earth is from that of a mite. This is not so in other simple modes, in which it is not so easy, nor perhaps possible for us to distinguish betwixt two approaching ideas, which yet are really different. For who will undertake to find a difference between the white of this paper and that of the next degree to it: or can form distinct ideas of every the least excess in extension?

4. Therefore Demonstrations in Numbers the most precise.

The contemporary view is much the same, with some logical refinements.

6. Another reason for the necessity of names to numbers.

This I think to be the reason why some Americans I have spoken with, (who were otherwise of quick and rational parts enough,) could not, as we do, by any means count to 1000; nor had any distinct idea of that number, though they could reckon very well to 20. Because their language being scanty, and accommodated only to the few necessaries of a needy, simple life, unacquainted either with trade or mathematics, had no words in it to stand for 1000; so that when they were discoursed with of those greater numbers, they would show the hairs of their head, to express a great multitude, which they could not number; which inability, I suppose, proceeded from their want of names.


1. Sensation, Remembrance, Contemplation, &c., modes of thinking.

When the mind turns its view inwards upon itself, and contemplates its own actions, THINKING is the first that occurs


31. Uneasiness determines the Will.

To return, then, to the inquiry, what is it that determines the will in regard to our actions? And that, upon second thoughts, I am apt to imagine is not, as is generally supposed, the greater good in view; but some (and for the most part the most pressing) UNEASINESS a man is at present under.

35. The greatest positive Good determines not the Will, but present Uneasiness alone.

It seems so established and settled a maxim, by the general consent of all mankind, that good, the greater good, determines the will, that I do not at all wonder that, when I first published my thoughts on this subject I took it for granted; and I imagine that, by a great many, I shall be thought more excusable for having then done so, than that now I have ventured to recede from so received an opinion. But yet, upon a stricter inquiry, I am forced to conclude that GOOD, the GREATER GOOD, though apprehended and acknowledged to be so, does not determine the will, until our desire, raised proportionably to it, makes us uneasy in the want of it. Convince a man never so much, that plenty has its advantages over poverty; make him see and own, that the handsome conveniences of life are better than nasty penury: yet, as long as he is content with the latter, and finds no uneasiness in it, he moves not; his will never is determined to any action that shall bring him out of it. Let a man be ever so well persuaded of the advantages of virtue, that it is as necessary to a man who has any great aims in this world, or hopes in the next, as food to life: yet, till he hungers or thirsts after righteousness, till he FEELS AN UNEASINESS in the want of it, his WILL will not be determined to any action in pursuit of this confessed greater good; but any other uneasiness he feels in himself shall take place, and carry his will to other actions.

This is in contrast to mainstream utility theory, which by contrast seems more magical.

48. The Power to suspend the Prosecution of any Desire makes way for consideration.

49. To be determined by our own Judgment, is no Restraint to Liberty.

50. The freest Agents are so determined

53. Power to Suspend.

This is the hinge on which turns the LIBERTY of intellectual beings, in their constant endeavours after, and a steady prosecution of true felicity,—That they CAN SUSPEND this prosecution in particular cases, till they have looked before them, and informed themselves whether that particular thing which is then proposed or desired lie in the way to their main end,

55. How Men come to pursue different, and often evil Courses.

From what has been said, it is easy to give an account how it comes to pass, that, though all men desire happiness, yet their wills carry them so contrarily; and consequently, some of them to what is evil. And to this I say, that the various and contrary choices that men make in the world do not argue that they do not all pursue good; but that the same thing is not good to every man alike.

61. Our wrong judgments have regard to future good and evil only.

68. Wrong judgment in considering Consequences of Actions.

(II). As to THINGS GOOD OR BAD IN THEIR CONSEQUENCES, and by the aptness that is in them to procure us good or evil in the future, we judge amiss several ways.

  1. When we judge that so much evil does not really depend on them as in truth there does.

  2. When we judge that, though the consequence be of that moment, yet it is not of that certainty, but that it may otherwise fall out, or else by some means be avoided; as by industry, address, change, repentance, &c.

That these are wrong ways of judging, were easy to show in every particular, if I would examine them at large singly: but I shall only mention this in general, viz. that it is a very wrong and irrational way of proceeding, to venture a greater good for a less, upon uncertain guesses; and before a due examination be made, proportionable to the weightiness of the matter, and the concernment it is to us not to mistake. This I think every one must confess, especially if he considers the usual cause of this wrong judgment, whereof these following are some:—

69. Causes of this.

(i) IGNORANCE: He that judges without informing himself to the utmost that he is capable, cannot acquit himself of judging amiss.

(ii) INADVERTENCY: When a man overlooks even that which he does know. This is an affected and present ignorance, which misleads our judgments as much as the other. Judging is, as it were, balancing an account, and determining on which side the odds lie. If therefore either side be huddled up in haste, and several of the sums that should have gone into the reckoning be overlooked and left out, this precipitancy causes as wrong a judgment as if it were a perfect ignorance. That which most commonly causes this is, the prevalency of some present pleasure or pain, heightened by our feeble passionate nature, most strongly wrought on by what is present. To check this precipitancy, our understanding and reason were given us, if we will make a right use of them, to search and see, and then judge thereupon. How much sloth and negligence, heat and passion, the prevalency of fashion or acquired indispositions do severally contribute, on occasion, to these wrong judgments, I shall not here further inquire. I shall only add one other false judgment, which I think necessary to mention, because perhaps it is little taken notice of, though of great influence.

70. Wrong judgment of what is necessary to our Happiness.

73. Recapitulation—Liberty of indifferency.

To conclude this inquiry into human liberty, which, as it stood before, I myself from the beginning fearing, and a very judicious friend of mine, since the publication, suspecting to have some mistake in it, though he could not particularly show it me, I was put upon a stricter review of this chapter. Wherein lighting upon a very easy and scarce observable slip I had made, in putting one seemingly indifferent word for another that discovery opened to me this present view, which here, in this second edition, I submit to the learned world, and which, in short, is this: LIBERTY is a power to act or not to act, according as the mind directs.

75. Summary of our Original ideas.

And thus I have, in a short draught, given a view of OUR ORIGINAL IDEAS, from whence all the rest are derived, and of which they are made up; which, if I would consider as a philosopher, and examine on what causes they depend, and of what they are made, I believe they all might be reduced to these very few primary and original ones, viz. EXTENSION, SOLIDITY, MOBILITY, or the power of being moved; which by our senses we receive from body: PERCEPTIVITY, or the power of perception, or thinking; MOTIVITY, or the power of moving: which by reflection we receive from OUR MINDS.

I crave leave to make use of these two new words, to avoid the danger of being mistaken in the use of those which are equivocal.

To which if we add EXISTENCE, DURATION, NUMBER, which belong both to the one and the other, we have, perhaps, all the original ideas on which the rest depend. For by these, I imagine, might be EXPLAINED the nature of colours, sounds, tastes, smells, and ALL OTHER IDEAS WE HAVE, if we had but faculties acute enough to perceive the severally modified extensions and motions of these minute bodies, which produce those several sensations in us. But my present purpose being only to inquire into the knowledge the mind has of things, by those ideas and appearances which God has fitted it to receive from them, and how the mind comes by that knowledge; rather than into their causes or manner of Production, I shall not, contrary to the design of this Essay, see myself to inquire philosophically into the peculiar constitution of BODIES, and the configuration of parts, whereby THEY have the power to produce in us the ideas of their sensible qualities. I shall not enter any further into that disquisition; it sufficing to my purpose to observe, that gold or saffron has power to produce in us the idea of yellow, and snow or milk the idea of white, which we can only have by our sight without examining the texture of the parts of those bodies or the particular figures or motion of the particles which rebound from them, to cause in us that particular sensation, though, when we go beyond the bare ideas in our minds and would inquire into their causes, we cannot conceive anything else to be in any sensible object, whereby it produces different ideas in us, but the different bulk, figure, number, texture, and motion of its insensible parts.


8. Our Ideas of Relations often clearer than of the Subjects related


This is along the lines of Russell’s process theory, but more readable.


1. Whence the Ideas of cause and effect got.

In the notice that our senses take of the constant vicissitude of things, we cannot but observe that several particular, both qualities and substances, begin to exist; and that they receive this their existence from the due application and operation of some other being. From this observation we get our ideas of CAUSE and EFFECT. THAT WHICH PRODUCES ANY SIMPLE OR COMPLEX IDEA we denote by the general name, CAUSE, and THAT WHICH IS PRODUCED, EFFECT. Thus, finding that in that substance which we call wax, fluidity, which is a simple idea that was not in it before, is constantly produced by the application of a certain degree of heat we call the simple idea of heat, in relation to fluidity in wax, the cause of it, and fluidity the effect. So also, finding that the substance, wood, which is a certain collection of simple ideas so called, by the application of fire, is turned into another substance, called ashes; i. e., another complex idea, consisting of a collection of simple ideas, quite different from that complex idea which we call wood; we consider fire, in relation to ashes, as cause, and the ashes, as effect. So that whatever is considered by us to conduce or operate to the producing any particular simple idea, or collection of simple ideas, whether substance or mode, which did not before exist, hath thereby in our minds the relation of a cause, and so is denominated by us.



2. Simple Ideas are all real appearances of things.

First, Our SIMPLE IDEAS are all real, all agree to the reality of things: not that they are all of them the images or representations of what does exist; the contrary whereof, in all but the primary qualities of bodies, hath been already shown. But, though whiteness and coldness are no more in snow than pain is; yet those ideas of whiteness and coldness, pain, &c., being in us the effects of powers in things without us, ordained by our Maker to produce in us such sensations; they are real ideas in us, whereby we distinguish the qualities that are really in things themselves.

12. Simple Ideas, [word in Greek], and adequate.

Thus the mind has three sorts of abstract ideas or nominal essences:

First, SIMPLE ideas, which are [word in Greek] or copies; but yet certainly adequate. Because, being intended to express nothing but the power in things to produce in the mind such a sensation, that sensation, when it is produced, cannot but be the effect of that power. So the paper I write on, having the power in the light (I speak according to the common notion of light) to produce in men the sensation which I call white, it cannot but be the effect of such a power in something without the mind; since the mind has not the power to produce any such idea in itself: and being meant for nothing else but the effect of such a power that simple idea is [* words missing] the sensation of white, in my mind, being the effect of that power which is in the paper to produce it, is perfectly adequate to that power; or else that power would produce a different idea.

13. Ideas of Substances are Echthypa, and inadequate.

Secondly, the COMPLEX ideas of SUBSTANCES are ectypes, copies too; but not perfect ones, not adequate: which is very evident to the mind, in that it plainly perceives, that whatever collection of simple ideas it makes of any substance that exists, it cannot be sure that it exactly answers all that are in that substance. Since, not having tried all the operations of all other substances upon it, and found all the alterations it would receive from, or cause in, other substances, it cannot have an exact adequate collection of all its active and passive capacities; and so not have an adequate complex idea of the powers of any substance existing, and its relations; which is that sort of complex idea of substances we have. And, after all, if we would have, and actually had, in our complex idea, an exact collection of all the secondary qualities or powers of any substance, we should not yet thereby have an idea of the ESSENCE of that thing. For, since the powers or qualities that are observable by us are not the real essence of that substance, but depend on it, and flow from it, any collection whatsoever of these qualities cannot be the real essence of that thing. Whereby it is plain, that our ideas of substances are not adequate; are not what the mind intends them to be. Besides, a man has no idea of substance in general, nor knows what substance is in itself.

14. Ideas of Modes and Relations are Archetypes, and cannot be adequate.

Thirdly, COMPLEX ideas of MODES AND RELATIONS are originals, and archetypes; are not copies, nor made after the pattern of any real existence, to which the mind intends them to be conformable, and exactly to answer. These being such collections of simple ideas that the mind itself puts together, and such collections that each of them contains in it precisely all that the mind intends that it should, they are archetypes and essences of modes that may exist; and so are designed only for, and belong only to such modes as, when they do exist, have an exact conformity with those complex ideas The ideas, therefore, of modes and relations cannot but be adequate.

In particular, ‘abstract thought’, as distinct from reasoning about simple ideas, is never adequate. (Locke seems to regard mathematical reasoning, which many people consider to be abstract, the ‘gold standard’, in which case there is an apparent paradox. It disapears if we regard the mathematical reasoning as being about ‘simple ideas’, as distinct from the application of mathematics, which entails abstraction.)


1. Truth and Falsehood properly belong to Propositions, not to Ideas.


1. Something unreasonable in most Men.

There is scarce any one that does not observe something that seems odd to him, and is in itself really extravagant, in the opinions, reasonings, and actions of other men. The least flaw of this kind, if at all different from his own, every one is quick-sighted enough to espy in another, and will by the authority of reason forwardly condemn; though he be guilty of much greater unreasonableness in his own tenets and conduct, which he never perceives, and will very hardly, if at all, be convinced of.

4. A Degree of Madness found in most Men.

I shall be pardoned for calling it by so harsh a name as madness, when it is considered that opposition to reason deserves that name, and is really madness; and there is scarce a man so free from it, but that if he should always, on all occasions, argue or do as in some cases he constantly does, would not be thought fitter for Bedlam than civil conversation. I do not here mean when he is under the power of an unruly passion, but in the steady calm course of his life. That which will yet more apologize for this harsh name, and ungrateful imputation on the greatest part of mankind, is, that, inquiring a little by the bye into the nature of madness, (b. ii. ch. xi., Section 13,) I found it to spring from the very same root, and to depend on the very same cause we are here speaking of. This consideration of the thing itself, at a time when I thought not I the least on the subject which I am now treating of, suggested it to me. And if this be a weakness to which all men are so liable, if this be a taint which so universally infects mankind, the greater care should be taken to lay it open under its due name, thereby to excite the greater care in its prevention and cure.

5. From a wrong Connexion of Ideas.

19. Conclusion.

Having thus given an account of the original, sorts, and extent of our IDEAS, with several other considerations about these (I know not whether I may say) instruments, or materials of our knowledge, the method I at first proposed to myself would now require that I should immediately proceed to show, what use the understanding makes of them, and what KNOWLEDGE we have by them. This was that which, in the first general view I had of this subject, was all that I thought I should have to do: but, upon a nearer approach, I find that there is so close a connexion between ideas and WORDS, and our abstract ideas and general words have so constant a relation one to another, that it is impossible to speak clearly and distinctly of our knowledge, which all consists in propositions, without considering, first, the nature, use, and signification of Language; which, therefore, must be the business of the next Book.

Dave Marsay

Locke’s Essay: Prologue

John Locke Essay Concerning Human Understanding, Final Edition, 1699 (Original 1690).

The prologue is:



. … The imputation of Novelty is a terrible charge amongst those who judge of men’s heads, as they do of their perukes, by the fashion, and can allow none to be right but the received doctrines.


… Every step the mind takes in its progress towards Knowledge makes some discovery, which is not only new, but the best too, for the time at least.

… If thou judgest for thyself I know thou wilt judge candidly, and then I shall not be harmed or offended, whatever be thy censure. For though it be certain that there is nothing in this Treatise of the truth whereof I am not fully persuaded, yet I consider myself as liable to mistakes as I can think thee, and know that this book must stand or fall with thee, not by any opinion I have of it, but thy own

This discontinued way of writing may have occasioned, besides others, two contrary faults, viz., that too little and too much may be said in it. … But to confess the truth, I am now too lazy, or too busy, to make it shorter.  … There are few, I believe, who have not observed in themselves or others, that what in one way of proposing was very obscure, another way of expressing it has made very clear and intelligible; though afterwards the mind found little difference in the phrases, and wondered why one failed to be understood more than the other. But everything does not hit alike upon every man’s imagination. We have our understandings no less different than our palates; and he that thinks the same truth shall be equally relished by every one in the same dress, may as well hope to feast every one with the same sort of cookery: the meat may be the same, and the nourishment good, yet every one not be able to receive it with that seasoning; and it must be dressed another way, if you will have it go down with some, even of strong constitutions.

… Men’s principles, notions, and relishes are so different, that it is hard to find a book which pleases or displeases all men. … The commonwealth of learning is not at this time without master-builders, whose mighty designs, in advancing the sciences, will leave lasting monuments to the admiration of posterity: but every one must not hope to be a Boyle or a Sydenham; and in an age that produces such masters as the great Huygenius and the incomparable Mr. Newton, with some others of that strain, it is ambition enough to be employed as an under-labourer in clearing the ground a little, and removing some of the rubbish that lies in the way to knowledge;—which certainly had been very much more advanced in the world, if the endeavours of ingenious and industrious men had not been much cumbered with the learned but frivolous use of uncouth, affected, or unintelligible terms, introduced into the sciences, and there made an art of, to that degree that Philosophy, which is nothing but the true knowledge of things, was thought unfit or incapable to be brought into well-bred company and polite conversation. Vague and insignificant forms of speech, and abuse of language, have so long passed for mysteries of science; and hard and misapplied words, with little or no meaning, have, by prescription, such a right to be mistaken for deep learning and height of speculation, that it will not be easy to persuade either those who speak or those who hear them, that they are but the covers of ignorance, and hindrance of true knowledge. To break in upon the sanctuary of vanity and ignorance will be, I suppose, some service to human understanding; though so few are apt to think they deceive or are deceived in the use of words; or that the language of the sect they are of has any faults in it which ought to be examined or corrected, that I hope I shall be pardoned if I have in the Third Book dwelt long on this subject, and endeavoured to make it so plain, that neither the inveterateness of the mischief, nor the prevalency of the fashion, shall be any excuse for those who will not take care about the meaning of their own words, and will not suffer the significancy of their expressions to be inquired into.

The bookseller will not forgive me if I say nothing of this New Edition, which he has promised, by the correctness of it, shall make amends for the many faults committed in the former. He desires too, that it should be known that it has one whole new chapter concerning Identity, and many additions and amendments in other places. These I must inform my reader are not all new matter, but most of them either further confirmation of what I had said, or explications, to prevent others being mistaken in the sense of what was formerly printed, and not any variation in me from it.

I must only except the alterations I have made in Book II. chap. xxi.

Of this the ingenious author of the Discourse Concerning the Nature of Man has given me a late instance, to mention no other. For the civility of his expressions, and the candour that belongs to his order, forbid me to think that he would have closed his Preface with an insinuation, as if in what I had said, Book II. ch. xxvii, concerning the third rule which men refer their actions to, I went about to make virtue vice and vice virtue, unless he had mistaken my meaning; which he could not have done if he had given himself the trouble to consider what the argument was I was then upon, and what was the chief design of that chapter, plainly enough set down in the fourth section and those following…. whatever authority the learned Mr. Lowde places in his Old English Dictionary, I daresay it nowhere tells him (if I should appeal to it) that the same action is not in credit, called and counted a virtue, in one place, which, being in disrepute, passes for and under the name of vice in another.

‘Tis to this zeal, allowable in his function, that I forgive his citing as he does these words of mine (ch. xxviii. sect. II): “Even the exhortations of inspired teachers have not feared to appeal to common repute, Philip, iv. 8;” without taking notice of those immediately preceding, which introduce them, and run thus: “Whereby even in the corruption of manners, the true boundaries of the law of nature, which ought to be the rule of virtue and vice, were pretty well preserved. So that even the exhortations of inspired teachers,” &c. By which words, and the rest of that section, it is plain that I brought that passage of St. Paul, not to prove that the general measure of what men called virtue and vice throughout the world was the reputation and fashion of each particular society within itself; but to show that, though it were so, yet, for reasons I there give, men, in that way of denominating their actions, did not for the most part much stray from the Law of Nature; which is that standing and unalterable rule by which they ought to judge of the moral rectitude and gravity of their actions, and accordingly denominate them virtues or vices. Had Mr. Lowde considered this, he would have found it little to his purpose to have quoted this passage in a sense I used it not; and would I imagine have spared the application he subjoins to it, as not very necessary. But I hope this Second Edition will give him satisfaction on the point, and that this matter is now so expressed as to show him there was no cause for scruple.

The booksellers preparing for the Fourth Edition of my Essay, gave me notice of it, that I might, if I had leisure, make any additions or alterations I should think fit. Whereupon I thought it convenient to advertise the reader, that besides several corrections I had made here and there, there was one alteration which it was necessary to mention, because it ran through the whole book, and is of consequence to be rightly understood. What I thereupon said was this:—

CLEAR and DISTINCT ideas are terms which, though familiar and frequent in men’s mouths, I have reason to think every one who uses does not perfectly understand. And possibly ‘tis but here and there one who gives himself the trouble to consider them so far as to know what he himself or others precisely mean by them. I have therefore in most places chose to put DETERMINATE or DETERMINED, instead of CLEAR and DISTINCT, as more likely to direct men’s thoughts to my meaning in this matter. By those denominations, I mean some object in the mind, and consequently determined, i. e. such as it is there seen and perceived to be. This, I think, may fitly be called a determinate or determined idea, when such as it is at any time objectively in the mind, and so determined there, it is annexed, and without variation determined, to a name or articulate sound, which is to be steadily the sign of that very same object of the mind, or determinate idea.

To explain this a little more particularly. By DETERMINATE, when applied to a simple idea, I mean that simple appearance which the mind has in its view, or perceives in itself, when that idea is said to be in it: by DETERMINED, when applied to a complex idea, I mean such an one as consists of a determinate number of certain simple or less complex ideas, joined in such a proportion and situation as the mind has before its view, and sees in itself, when that idea is present in it, or should be present in it, when a man gives a name to it. I say SHOULD be, because it is not every one, nor perhaps any one, who is so careful of his language as to use no word till he views in his mind the precise determined idea which he resolves to make it the sign of. The want of this is the cause of no small obscurity and confusion in men’s thoughts and discourses.

I know there are not words enough in any language to answer all the variety of ideas that enter into men’s discourses and reasonings. But this hinders not but that when any one uses any term, he may have in his mind a determined idea, which he makes it the sign of, and to which he should keep it steadily annexed during that present discourse. Where he does not, or cannot do this, he in vain pretends to clear or distinct ideas: it is plain his are not so; and therefore there can be expected nothing but obscurity and confusion, where such terms are made use of which have not such a precise determination.

Upon this ground I have thought determined ideas a way of speaking less liable to mistakes, than clear and distinct: and where men have got such determined ideas of all that they reason, inquire, or argue about, they will find a great part of their doubts and disputes at an end; the greatest part of the questions and controversies that perplex mankind depending on the doubtful and uncertain use of words, or (which is the same) indetermined ideas, which they are made to stand for. I have made choice of these terms to signify, (1) Some immediate object of the mind, which it perceives and has before it, distinct from the sound it uses as a sign of it. (2) That this idea, thus determined, i.e. which the mind has in itself, and knows, and sees there, be determined without any change to that name, and that name determined to that precise idea. If men had such determined ideas in their inquiries and discourses, they would both discern how far their own inquiries and discourses went, and avoid the greatest part of the disputes and wranglings they have with others.

My Comments

Locke may have had in mind something like Cromwell’s rule. Even if not, he ends up by taking mathematics as a ‘gold standard’ in reasoning, so it would be reasonable for us to think in terms of this as a vital principle. (It is perhaps unfortunate that it does not seem to be innate.)

In any case, Locke doubts the then commonplace logics and common sense, and starts by doubting the notion that ideas and principles are innate. I’m not sure if he really belives there are no such innate ideas or principles, and even if he did we might apply Cromwell’s rule to obtain:

We are never justified in supposing that any particular idea or principle is innate, or even shared completely.

Locke’s aim, I think, is to refine key ideas and principles to the point where they may be reasonably understood and accepted, as the basis for ongoing refinement, at least to be developed and maintained as fit for constitutional, legal and democratic purposes.

For example, Locke seems to regard the then mathematics as a ‘gold standard’, and specifically refers to Newton. This has led some to think that Locke was admitting the principles in Newton’s Principia. But, firstly, the Principia was published after Locke’s death and, secondly, while Newton’s principles are ‘mathematical’, they are principles of mathematical physics, not mathematics as such. Thus we might do well to follow his advice in critiqueing our ideas about mathematics and his example in further developing and refining them, thus creating an improved standard. (Compare Berkeley.)

We might also sympathize with Locke’s ‘laziness’ and with his response to his critics.

Main Contents

The main contents, as I see them are:






Book IV: The Reality of Knowledge

  • CHAPTER XV: Probability
  • CHAPTER XVI: The degrees of assent
  • CHAPTER XX: Wrong assent, or error
  • CHAPTER XXI: The division of the sciences

The End

Dave Marsay

Berkeley’s Human Knowledge

George Berkeley  A Treatise Concerning the Principles of Human Knowledge Part 1, London 1710

This has become infamous for an extreme version of ‘the observer effect’, but this may be a misunderstanding of what Berkeley meant. I have yet to read the full text, but his introduction seems to me to give some sensible advice on how to read his text, from which I agree with wikipedia that the ‘trees in forest’ view attributed to him may be misleading.


[When] we lay the blame for our paradoxes and difficulties on our faculties rather than on our wrong use of them, perhaps we are letting ourselves down too lightly.

It is hard to believe that right deductions from true principle should ever lead to conclusions that can’t be maintained or made consistent.

I deny that I can perform ‘abstraction’ in the standard meaning of that word, which covers two kinds of mental performance: (1) conceiving abstractly and in isolation a quality that couldn’t exist in isolation as we are said to do with colour and motion·; and (2) forming a general notion by abstracting from particulars in the way I have described, as we are said to do with man and animal·. There is reason to think that most people are like me in this respect. The majority of people, who are simple and illiterate, never claim to have abstract notions.

Suppose for example that a geometrician, proving the validity of a procedure for cutting a line in two equal parts, draws a black line one inch long. As used in this geometrical proof, this particular line is general in its significance because it is used to represent all particular lines, so that what is proved regarding it is proved to hold for all lines. And just as that particular line becomes general by being used as a sign, so the word ‘line’—which in itself is particular—is used as a sign with a general meaning. The line is general because it is the sign not of an abstract or general line but of all particular straight lines that could exist, and the word is general for the same reason—namely that it stands equally well for each and every particular line.

Who could believe that a couple of children cannot chatter about sugar-plums and toys until they have first tacked together numberless inconsistencies and so formed abstract general ideas in their minds, attaching them to every common name they make use of?

Abstract ideas are no more needed, in my opinion, for the growth of knowledge than they are for communication. I entirely agree with the widespread belief that all knowledge and demonstration concerns universal notions; but I can’t see that those are formed by abstraction. The only kind of universality that I can grasp doesn’t belong to anything’s intrinsic nature; a thing’s universality consists how it relates to the particulars that it signifies or represents.

[Although] the idea I have in view while I make the demonstration may be (for instance) that of an isosceles right-angled triangle whose sides are of a determinate length, I can still be certain that it applies also to all other triangles, no matter what their sort or size. I can be sure of this because neither the right angle nor the equality of sides nor length of the sides has any role in the demonstration.

So let us examine how words have helped to give rise to the mistaken view that there are abstract ideas.·… ·.(1) People assume that every name does or should have just one precise and settled signification.

It is one thing to make a name always obey the same definition, and another to make it always stand for the same idea: one is necessary, the other useless and impracticable.

(2) Words helped in another way to produce the doctrine of abstract ideas, namely through the widespread opinion that language is for the communicating of our ideas ..

It can’t be denied that words are extremely useful: they make it possible for all the knowledge that has been gained by the enquiries of men at many times and in all nations to be pulled together and surveyed by a single person. But at the same time it must be admitted that most branches of knowledge have been made enormously much darker and more difficult by the misuse of words and turns of phrase. Therefore, since words are so apt to influence our thoughts, when I want to consider any ideas I shall try to take them bare and naked, keeping out of my thoughts—as much as I can—the names that those ideas have been given through long and constant use. From this I expect to get the following·three·advantages:-

22 First, I shall be sure to keep clear of all purely verbal controversies. Secondly, this seems to be a sure way to extricate myself from that fine and delicate net of abstract ideas, which has so miserably perplexed and entangled the minds of men (with this special feature: the more sharp-witted and exploratory any man’s mind is, the more completely he is likely to be trapped and held by the net!). Thirdly, so long as I confine my thoughts to my own ideas with the words peeled off, I don’t see how I can easily be mistaken.

23 But I can’t get all these advantages unless I free myself entirely from the deception of words. I hardly dare promise myself that, because the union between words and ideas began early and has been strengthened by many years of habit·in thought and speech·, making it very difficult to dissolve. This difficulty seems to have been very much increased by the doctrine of abstraction. For so long as men thought their words have abstract ideas tied to them, it isn’t surprising that they used words in place of ideas: they found that they couldn’t set aside the word and retain the abstract idea in the mind, because abstract ideas are perfectly inconceivable.

24 But when you know that these are mistakes, you can more easily prevent your thoughts from being influenced by words. Someone who knows that he has only particular ideas won’t waste his time trying to conceive the abstract idea that goes with any name. And someone who knows that names don’t always stand for ideas will spare himself the labour of looking for ideas where there are none to be had.

25 Unless we take care to clear the first principlesof knowledge from being burdened and deluded by words,we can reason from them for ever without achieving any-thing; we can draw consequences from consequences and be never the wiser. The further we go, the more deeply and irrecoverably we shall be lost and entangled in difficulties and mistakes. To anyone who plans to read the following pages, therefore, I say: Make my words the occasion of your own thinking, and try to have the same sequence of thoughts in reading that I had in writing. This will make it easy foryou to discover the truth or falsity of what I say. You will run no risk of being deceived by my words, and I don’t see howyou can be led into an error by considering your own naked, undisguised ideas.

Main Body

First 50 sections

3. Everyone will agree that our thoughts, emotions, and ideas of the imagination exist only in the mind. It seems to me equally obvious that the various sensations or ideas that are imprinted on our senses cannot exist except in a mind that perceives them—no matter how they are blended or combined together (that is, no matter what objects they constitute). You can know this intuitively [= ‘you can see this as immediately self-evident’] by attending to what is meant by the term ‘exist’ when it is applied to perceptible things. The table that I am writing on exists, that is, I see and feel it; and if I were out of my study I would still say that it existed, meaning that if I were in my study I would perceive it, or that some other spirit actually does perceive it. …

4. It is indeed widely believed that all perceptible objects—houses, mountains, rivers, and so on—really exist independently of being perceived by the understanding. But however widely and confidently this belief may be held, anyone who has the courage to challenge it will—if I’m not mistaken—see that it involves an obvious contradiction. For what are houses, mountains, rivers etc. but things we perceive by sense? And what do we perceive besides our own ideas or sensations? And isn’t it plainly contradictory that these, either singly or in combination, should exist unperceived?

5. If we thoroughly examine this belief in things existing independently of the mind it will, perhaps, be found to depend basically on the doctrine of abstract ideas.

… To be convinced of this, you need only to reflect and try to separate in your own thoughts the existence of a perceptible thing from its being perceived—·you’ll find that you can’t·

8. ‘But’, you say, ‘though the ideas don’t exist outside the mind, still there may be things like them of which they are copies or resemblances, and these things may exist outside the mind in an unthinking substance.’ I answer that the only thing an idea can resemble is another idea; a colour or shape can’t be like anything but another colour or shape.

20. In short, if there were external bodies, we couldn’t possibly come to know this; and if there weren’t, we might havethe very same reasons to think there were that we have now. No-one can deny the following to be possible: A thinking being might, without the help of external bodies, be affected with the same series of sensations or ideas that you have,imprinted in the same order and with similar vividness in his mind. If that happened, wouldn’t that thinking being have all the reason to believe ‘There are corporeal substances thatare represented by my ideas and cause them in my mind’ that you can possibly have for believing the same thing? Of course he would; and that consideration is enough, all on its own, to make any reasonable person suspect the strength of whatever arguments he may think he has for the existence of bodies outside the mind.

23. ‘But’, you say, ‘surely there is nothing easier than to imagine trees in a park, for instance, or books on a shelf, with nobody there to perceive them.’ I reply that this is indeed easy to imagine; but let us look into what happens when you imagine it. You form in your mind certain ideas that you call ‘books’ and ‘trees’, and at the same time you omit to form the idea of anyone who might perceive them. But while you are doing this, you perceive or think of them! So your thought- experiment misses the point; it shows only that you have the power of imagining or forming ideas in your mind; but it doesn’t show that you can conceive it possible for the objects of your thought to exist outside the mind. To show that, you would have to conceive them existing unconceived or unthought-of, which is an obvious contradiction. However hard we try to conceive the existence of external bodies, all we achieve is to contemplate our own ideas.

35. I don’t argue against the existence of any one thing that we can take in, either by sense or reflection. I don’t in the least question that the things I see with my eyes and touch with my hands do exist, really exist. The only thing whose existence I deny is what philosophers call ‘matter’ or ‘corpo-real substance’. And in denying this I do no harm to the rest of mankind—·that is, to people other than philosophers·—because they will never miss it. The atheist indeed will lose the rhetorical help he gets from an empty name, ‘matter’, which he uses to support his impiety; and the philosophers may find that they have lost a great opportunity for word-spinning and disputation.

36. If you think that this detracts from the existence or reality of things, you are very far from understanding what I have said in the plainest way I could think of.

Much of the above seems uncontroversial and logical in the modern sense, and we can easily follow Locke’s advice in making some sense of the rest for ourselves, without necessarily claiming to have much insight not what Berkeley actually thought. He is critiquing a narrowly mechanistic and deterministic view of life and seems to be ‘creating space’ for the possible existence of ‘spirits’. Fair enough.

Second 50 sections

I haven’t quoted from any of this. Sometimes Berkeley seems to use the term ‘spirit’ to simply mean anthing that can affect an idea that is not itself an idea, such as a ‘thinking being’. At other times he seems to be showing signs of his journey to becoming a Bishop. So I take Locke’s advice and for now only comment that I have failed to make much sense of it.

Final Sections

Berkeley considers the implications of his views, which (perhaps fortunately) dont much depend on how we (or he) conceive of ‘spirits’. In particular, while the previous sections seem (on a quick scan) to suggest something like ‘As long as we follow God’s will we can develop sciences in ways which will meet with his approval and hence be beneficial’, we can ignore this suggestion and still make sense of his conclusions.

107. … by diligently observing the phenomena within our view, we can discover the general laws of nature, and from them deduce further phenomena. I don’t say demonstrate [= ‘prove in a rigorously valid manner’]; for all deductions of this kind depend on supposing that the author of nature always operates uniformly, constantly keeping to those rules that we regard as principles—though we can’t know for sure that they are.

(This distinction between ‘deduce’ and ‘demonstrate’ seems vital.)

108. Those men who make general rules from phenomena, and afterwards derive phenomena from those rules, seem to be considering signs rather than causes. A man may understand natural signs well without being able to say bywhat rule a one .event is a sign of another. And just as it ispossible to write improperly through too strictly observing general rules of grammar, so also in arguing from general rules of nature we may extend the analogy too far and thus run into mistakes.

110. The best key to natural science is widely agreed to be a certain celebrated treatise of mechanics—·Newton’s Principia·. At the start of that justly admired treatise, time, space, and motion are each distinguished into absolute and relative,·or, giving the same distinction in different words·,true and apparent, or·in yet other words·mathematical and vulgar [= ‘that of the plain uneducated ordinary person’]. According to the author’s extensive account of it, this distinction does presuppose that time, space and motion exist outside the mind, and that they are ordinarily•conceived as relating to perceptible things; but really in their own nature they have no relation to them at all.

112. Despite all this, it doesn’t appear to me that there can be any motion except relative motion. To conceive motion,·it seems to me·, one must conceive at least two bodies that alter in their distance from, or position in relation to, each other. Hence if there was one only body in existence, it couldn’t possibly be moved. This seems obvious, because the idea that I have of motion necessarily includes relation.

116. From what has been said, it follows that the scientific consideration of motion doesn’t imply the existence of an absolute space, distinct from the space that is perceived by the senses, is related to bodies, and cannot exist outside the mind, as is clear from the principles that prove the samething of all other objects of sense. If we look into it closely we shall perhaps find that we can’t even form an idea of pure space without bodies. This, I must confess, seems impossible, as being a most abstract idea.

118. Up to here I have written about natural science. Now letus enquire into that other great branch of speculative knowledge, namely mathematics. See the start of 101·.Celebrated though it is for its clearness and certainty of demonstration, which is matched hardly anywhere else, mathematics cannot be supposed altogether free from mistakes if in its principles there lurks some secret error that mathematicians share with the rest of mankind. Mathematicians deduce their theorems from premises that are highly certain; but their first principles are confined to the concept of quantity; and they don’t ascend into any enquiry concerning those higher maxims that influence all the particular sciences including ones that aren’t quantitative·. Any errors involved in those higher maxims will infect every branch of knowledge, including mathematics. I don’t deny that the principles laid down by mathematicians are true, or that their methods of deduction from those principles are clear and beyond dispute. But I hold that there are certain erroneous maxims that spread wider than mathematics, and for that reason are not explicitly mentioned there, though they are tacitly assumed throughout the whole progress of that science; and that the bad effects of those secret, unexamined errors are diffused through all the branches of mathematics. To be plain, I suspect that mathematicians as well as other men are caught in the errors arising from the doctrines of abstract general ideas and of the existence of objects outside the mind.

119. Arithmetic has been thought to have for its object abstract ideas of number. A considerable part of speculative knowledge is supposed to consist in understanding the properties and mutual relations of numbers. The belief in the pure and intellectual nature of numbers in the abstract has won for them the esteem of those thinkers who put on a show of having an uncommon subtlety and elevation of thought. It has put a price on the most trifling numerical theorems that are of no practical use and serve only to pass the time; and it has infected the minds of some people so much that they have dreamed of mighty mysteries involved in numbers, and tried to explain natural things by means of them. But if we look into our own thoughts, and consider the doctrines I have laid down, we may come to have a low opinion of those high flights and abstractions, and to look on all researches into numbers as mere earnest trivialities insofar as they aren’t practically useful in improving our lives.

122. In arithmetic therefore we have to do not with the things but with the signs, though these concern us not for their own sake but because they direct us how to act in relation to things, and how to manage them correctly. Just as I have remarked concerning language in general (19 intro), so here oo abstract ideas are thought to be signified by numerals or number-words at times when they don’t suggest ideas of particular things to our minds. I shan’t go further into this subject now, except to remark that what I have said shows clearly that the things that are taken to be abstract truths and theorems concerning numbers are really about nothing but particular countable things—or about names and numerals, which were first attended to only because they are signs that can aptly represent whatever particular things men needed to calculate about. To study these names or numerals for their own sake, therefore, would be just as wise and pointful as to neglect the true use or original intention and purpose of language, and to spend one’s time on irrelevant criticisms of words, or on purely verbal reasonings and controversies.

123. From numbers we move on to discuss extension, which (considered as relative) is the object of geometry. The infinite divisibility of finite extension, though it isn’t explicitly asserted either as an axiom or as a theorem in the elements of geometry, is assumed throughout it, and is thought to have so inseparable and essential a connection with the principles and proofs in geometry that mathematicians never call it into question. This notion is the source of all those deceitful geometrical paradoxes that so directly contradict the plain common sense of mankind, and are found hard to swallow by anyone whose mind is not yet perverted by learning. It is also the principal source of all the fine-grained and exaggerated subtlety that makes the study of mathematics so difficult and tedious. So if I can make it appear that nothing whose extent is finite contains innumerable parts, or is infinitely divisible, that will immediately free the science of geometry from a great number of difficulties and contradictions that have always been thought a reproach to human reason, and also make the learning of geometry a much less lengthy and difficult business than it has been until now.

126. I have pointed out that the theorems and demonstrations of geometry are about universal ideas (15 intro). And I explained in what sense this ought to be understood, namely that the particular lines and figures included in the diagram are supposed to stand for innumerable others of different sizes. In other words, when the geometer thinks about them he abstracts from their size; this doesn’t imply that he forms an abstract idea, only that he doesn’t care what the particular size is, regarding that as irrelevant to the demonstration.

132. It may be said that various undoubtedly true theorems have been discovered by methods in which infinitesimals were used, which couldn’t have happened if their existence included a contradiction in it. I answer that when you look into this thoroughly you won’t find any case where you need to conceive infinitesimal parts of finite lines, or even quantities smaller than the smallest you can perceive. You’ll find that this is never done, because it is impossible. This completes my discussion of infinite divisibility.


Locke’s Ideas


Berkeley is critiquing John Locke’s An Essay Concerning Human Understanding, in particular the somewhat quaint view that our perceptions ‘directly’ correspond to some real ‘thing’. Given Locke’s “Substances are “nothing but the assumption of an unknown support for a group of qualities that produce simple ideas in us”, I’m not sure if Berkeley applied the idea in his last paragraph above to his reading of Locke, but maybe that doesn’t matter?


Berkeley makes a critical distinction between a deduction from one’s ideas about a topic, and a demonstration of the validity of the deduction. (Economists take note!) I would have to re-read Locke to see if he gets this. (Newton did?)


Berkeley’s first publication was in mathematics, and he went on to write his ‘Analyst’, that led to a process of reform that can be traced until at least 1966, and which may not be complete. (I hope it is!) It might therefore be reasonable to read that to clarify any issues.

[Wikipedia] opines:

“The logical criticism is that of a fallacia suppositionis, which means gaining points in an argument by means of one assumption and, while keeping those points, concluding the argument with a contradictory assumption.” “Berkeley, however, found it paradoxical that “Mathematicians should deduce true Propositions from false Principles, be right in Conclusion, and yet err in the Premises.”

Berkeley was himself a mathematician, whereas Locke was not, and it is not clear to me that Berkeley was criticising mathematicians as such, rather than Locke’s understanding of mathematics. In any case, mathematics has moved on, and there seems no reason to think that the above comment would apply today.

(Berkeley is particularly critical of the characterisation of arithmetic, geometry and calculus as being about abstarctions from physical ‘reality’ as then conceived. He advocated  a formal approach to number, and a relativistic and finitistic approach to space and time.)


Locke writes about abstraction by analogy with his view of mathematical reasoning. Berkeley makes a similar analogy, but this results in a more refined view of methods that had been thought of in terms of ‘abstraction’.


I am not quite clear that contemporary psychologists quite get Berkeley’s point, but I haven’t come across any evidence to the contrary, centuries later. Kahneman, for example, opines:

System 1 represents categories by a prototype or a set of typical examples.

I’m not sure this actually true, but it is at least consistent with Berkeley: no abstraction as such.


Some psychologists use computers as a metaphor for human cognition. Computers can go beyond what Kahneman supposes, above, in the sophistication of their representations (and typically do), but many contemporary psychologist’s views would still seem consistent with Berkeley.

The main essay

Wikipedia opines:

This theory denies the existence of material substance  .. His arguments were a precursor to the views of Mach and Einstein. …Interest in Berkeley’s work increased after World War II because he tackled many of the issues of paramount interest to philosophy in the 20th century.

George Berkeley’s theory that matter does not exist comes from the belief that “sensible things are those only which are immediately perceived by sense”. The only causes that exist in Berkeley’s worldview are those that are a result of the use of the will.

But an alternative reading seems to be that there may be a reality that remains that would be perceived as a tree if we looked: it is simply that the reality could be beyond our powers of perception, and so nothing like ‘a tree’ as we perceive it.

Russell ( A History of Western Philosophy and Its Connection with Political and Social Circumstances from the Earliest Times to the Present Day) has some important technical criticisms of Berkeley’s logic, showing that logicians had made progress in the centuries since Berkeley.

One could argue that Berkeley still seems ‘essentially correct’ and he seems much more accessible than anything from the last 100 years. (But maybe that’s not saying much!)

An interpretation

The following scholastic beliefs were once widespread:

  • That words stand for ideas, particularly abstract ideas, that can be communicated.
  • That every name does or should have just one precise and settled signification.
  • That science and mathematics, for example, depends on abstract ideas, such as ‘point’ and ‘line’.

If true, they would seem to suggest that it would, at least in principle, be possible for science to establish empirical ‘facts’. That would be convenient.

From a contemporary viewpoint, Berkeley’s objections seem reasonable. For example, current SATNAV’s are very good at reasoning about their maps, which is often adequate for our purposes, but sometimes they would do better for us if they somehow could reason about actual roads, or at least roads as we perceive them. ‘The map is not the territory‘.

No-one has been able to explain what possible real processes might bring about such fortunate circumstances, or how we could know that they had, even if they did. They seem to need either inherited ‘innate ideas’, ESP or at least one ‘god’. For example, it is not at all clear how computers might become ‘intelligent’ to take account of the limitations of their internal representations and demonstrate what Keates called ‘negative capabilties‘. Yet controversy seems to linger. Perhaps we would could simply agree that even if these beliefs might be true, we could never be absolutely sure about them in particular circumstances. There is always some irreducible uncertainty, even if we don’t realise it, and no matter how ‘pragmatic‘ we think we are or how authoritative we wish to appear.

Dave Marsay