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

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