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



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