The Fabric of Reality: Towards a Theory of Everything (Penguin Science)

by David Deutsch

However, as with solipsism generally, the truly fatal flaw of intuitionism is revealed not when it is attacked, but when it is taken seriously in its own terms, as an explanation of its own, arbitrarily truncated world.

In 1900, to mark the turn of the century, Hilbert published a list of problems that he hoped mathematicians might be able to solve during the course of the twentieth century. The tenth problem was to find a set of rules of inference with the above properties, and, by their own standards, to prove them consistent. Hilbert was to be definitively disappointed. Thirty-one years later, Kurt Gödel revolutionized proof theory with a root-and-branch refutation from which the mathematical and philosophical worlds are still reeling: he proved that Hilbert’s tenth problem is insoluble. Gödel proved first that any set of rules of inference that is capable of correctly validating even the proofs of ordinary arithmetic could never validate a proof of its own consistency. Therefore there is no hope of finding the provably consistent set of rules that Hilbert envisaged. Second, Gödel proved that if a set of rules of inference in some (sufficiently rich) branch of mathematics is consistent (whether provably so or not), then within that branch of mathematics there must exist valid methods of proof that those rules fail to designate as valid. This is called Gödel’s incompleteness theorem.

Thanks to Gödel, we know that there will never be a fixed method of determining whether a mathematical proposition is true, any more than there is a fixed way of determining whether a scientific theory is true. Nor will there ever be a fixed way of generating new mathematical knowledge. Therefore progress in mathematics will always depend on the exercise of creativity. It will always be possible, and necessary, for mathematicians to invent new types of proof. They will validate them by new arguments and by new modes of explanation depending on their ever improving understanding of the abstract entities involved.

So explanation does, after all, play the same paramount role in pure mathematics as it does in science. Explaining and understanding the world – the physical world and the world of mathematical abstractions – is in both cases the object of the exercise. Proof and observation are merely means by which we check our explanations.

However, even if it is conceded that Penrose’s Gödelian argument fails to prove what it sets out to prove, and his proposed new physical theory seems unlikely to explain what it sets out to explain, Penrose is nevertheless right that any world-view based on the existing conception of scientific rationality creates a problem for the accepted foundations of mathematics (or, as Penrose would have it, vice versa). This is the ancient problem that Plato raised, a problem which, as Penrose points out, becomes more acute in the light of both Gödel’s theorem and the Turing principle. It is this: in a reality composed of physics and understood by the methods of science, where does mathematical certainty come from? While most mathematicians and computer scientists take the certainty of mathematical intuition for granted, they do not take seriously the problem of reconciling this with a scientific world-view. Penrose does take it seriously, and he proposes a solution. His proposal envisages a comprehensible world, rejects the supernatural, recognizes creativity as being central to mathematics, ascribes objective reality both to the physical world and to abstract entities, and involves an integration of the foundations of mathematics and physics. In all those respects I am on his side.

The reliability of the knowledge of a perfect circle that one can gain from a diagram of a circle depends entirely on the accuracy of the hypothesis that the two resemble each other in the relevant ways. Such a hypothesis, referring to a physical object (the diagram), amounts to a physical theory and can never be known with certainty. But that does not, as Plato would have it, preclude the possibility of learning about perfect circles from experience; it just precludes the possibility of certainty. That should not worry anyone who is looking not for certainty but for explanations.

Euclidean geometry can be abstractly formulated entirely without diagrams. But the way in which numerals, letters and mathematical symbols are used in a symbolic proof can generate no more certainty than a diagram can, and for the same reason. The symbols too are physical objects – patterns of ink on paper, say – which denote abstract objects. And again, we are relying entirely upon the hypothesis that the physical behaviour of the symbols corresponds to the behaviour of the abstractions they denote. Therefore the reliability of what we learn by manipulating those symbols depends entirely on the accuracy of our theories of their physical behaviour, and of the behaviour of our hands, eyes, and so on with which we manipulate and observe the symbols.

Incidentally, Plato’s idea that physical reality consists of imperfect imitations of abstractions seems an unnecessarily asymmetrical stance nowadays. Like Plato, we still study abstractions for their own sake. But in post-Galilean science, and in the theory of virtual reality, we also regard abstractions as means of understanding real or artificial physical entities, and in that context we take it for granted that the abstractions are nearly always approximations to the true physical situation. So, whereas Plato thought of Earthly circles in the sand as approximations to true, mathematical circles, a modern physicist would regard a mathematical circle as a bad approximation to the real shapes of planetary orbits, atoms and other physical things.

How do we verify it? We prove it. A proof is traditionally defined as a sequence of statements satisfying self-evident rules of inference, but what does the ‘proving’ process amount to physically? To prove a statement about infinitely many triangles at once, we examine certain physical objects – in this case symbols – which have properties in common with whole classes of triangles.

The answer is that we must use rules of inference which, to the best of our understanding, will cause our symbols to behave, in the relevant ways, like the abstract entities they denote. How can we be sure that they will? We cannot. Suppose that some critics object to our rules of inference because they think that our symbols will behave differently from the abstract entities. We cannot appeal to the authority of Aristotle or Plato, nor can we prove that our rules of inference are infallible (quite apart from Gödel’s theorem, this would lead to an infinite regress, for we should first have to prove that the method of proof that we used was itself valid). Nor can we haughtily tell the critics that there must be something wrong with their intuition, because our intuition says that the symbols will mimic the abstract entities perfectly. All we can do is explain. We must explain why we think that, under the circumstances, the symbols will behave in the desired way under our proposed rules. And the critics can explain why they favour a rival theory. A disagreement over two such theories is, in part, a disagreement about the observable behaviour of physical objects. Such disagreements can be addressed by the normal methods of science. Sometimes they can be readily resolved; sometimes not.

A conventional symbolic proof seems at first sight to have quite a different character from the ‘hands-on’ virtual-reality sort of proof. But we see now that they are related in the way that computations are to physical experiments. Any physical experiment can be regarded as a computation, and any computation is a physical experiment. In both sorts of proof, physical entities (whether in virtual reality or not) are manipulated according to rules. In both cases the physical entities represent the abstract entities of interest. And in both cases the reliability of the proof depends on the truth of the theory that physical and abstract entities do indeed share the appropriate properties.

We can also see from the above discussion that proof is a physical process. In fact, a proof is a type of computation. ‘Proving’ a proposition means performing a computation which, if one has done it correctly, establishes that the proposition is true. When we use the word ‘proof to denote an object, such as an ink-on-paper text, we mean that the object can be used as a program for recreating a computation of the appropriate kind.

It follows that neither the theorems of mathematics, nor the process of mathematical proof, nor the experience of mathematical intuition, confers any certainty. Nothing does. Our mathematical knowledge may, just like our scientific knowledge, be deep and broad, it may be subtle and wonderfully explanatory, it may be uncontroversially accepted; but it cannot be certain. No one can guarantee that a proof that was previously thought to be valid will not one day turn out to contain a profound misconception, made to seem natural by a previously unquestioned ‘self-evident’...

Proofs are not abstract. There is no such thing as abstractly proving something, just as there is no such thing as abstractly calculating or computing something. One can of course define a class of abstract entities and call them ‘proofs’, but those ‘proofs’ cannot verify mathematical statements because no one can see them.

Computations, including the special computations that qualify as proofs, are physical processes. Proof theory is about how to ensure that those processes correctly mimic the abstract entities they are intended to mimic.

one cannot prove, independently of any knowledge of physics, that a proof must consist of finitely many steps. The same considerations apply to the intuition that there must be finitely many rules of inference, and that these must be ‘straightforwardly applicable’. None of these requirements is meaningful in the abstract: they are physical requirements.

That mathematicians throughout the ages should have made various mistakes about matters of proof and certainty is only natural. The present discussion should lead us to expect that the current view will not last for ever, either. But the confidence with which mathematicians have blundered into these mistakes and their inability to acknowledge even the possibility of error in these matters are, I think, connected with an ancient and widespread confusion between the methods of mathematics and its subject-matter.

Unlike the relationships between physical entities, relationships between abstract entities are independent of any contingent facts and of any laws of physics. They are determined absolutely and objectively by the autonomous properties of the abstract entities themselves. Mathematics, the study of these relationships and properties, is therefore the study of absolutely necessary truths. In other words, the truths that mathematics studies are absolutely certain. But that does not mean that our knowledge of those necessary truths is itself certain, nor does it mean that the methods of mathematics confer necessary truth on their conclusions. After all, mathematics also studies falsehoods and paradoxes. And that does not mean that the conclusions of such a study are necessarily false or paradoxical.

Necessary truth is merely the subject-matter of mathematics, not the reward we get for doing mathematics. The objective of mathematics is not, and cannot be, mathematical certainty. It is not even mathematical truth, certain...

Why, then, does mathematics work as well as it does? Why does it lead to conclusions which, though not certain, can be accepted and applied unproblematically for millennia at least? Ultimately the reason is that some of our knowledge of the physical world is also that reliable and uncontroversial. And when we understand the physical world sufficiently well, we also understand which physical objects have properties in common with which abstract ones. But in principle the reliability of our knowledge of mathematics remains subsidiary to our knowledge of physical reality. Every mathematical proof depends absolutely for its val...

For example, the human eye and its controlling software implicitly embody the false theory that yellow light consists of a mixture of red and green light (in the sense that yellow light gives us exactly the same sensation as a mixture of red light and green light does). In reality, all three types of light have different frequencies and cannot be created by mixing light of other frequencies. The fact that a mixture of red and green light appears to us to be yellow light has nothing whatever to do with the properties of light, but is a property of our eyes. It is the result of a design compromise that occurred at some time during our distant ancestors’ evolution.

The fabric of reality, then, does have a more unified structure than would have been possible if mathematical knowledge had been verifiable with certainty, and hence hierarchical, as has traditionally been assumed. Mathematical entities are part of the fabric of reality because they are complex and autonomous. The sort of reality they form is in some ways like the realm of abstractions envisaged by Plato or Penrose: although they are by definition intangible, they exist objectively and have properties that are independent of the laws of physics. However, it is physics that allows us to gain knowledge of this realm. And it imposes stringent constraints. Whereas everything in physical reality is comprehensible, the comprehensible mathematical truths are precisely the infinitesimal minority which happen to correspond exactly to some physical truth – like the fact that if certain symbols made of ink on paper are manipulated in certain ways, certain other symbols appear. That is, they are the truths that can be rendered in virtual reality. We have no choice but to assume that the incomprehensible mathematical entities are real too, because they appear inextricably in our explanations of the comprehensible ones.

There are physical objects – such as fingers, computers and brains – whose behaviour can model that of certain abstract objects. In this way the fabric of physical reality provides us with a window on the world of abstractions.

proof A way of establishing the truth of mathematical propositions. (Traditional definition:) A sequence of statements, starting with some premises and ending with the desired conclusion, and satisfying certain ‘rules of inference’. (Better definition:) A computation that models the properties of some abstract entity, and whose outcome establishes that the abstract entity has a given property.

Unfortunately it is not a good theory. To put it bluntly, the reason why the common-sense theory of time is inherently mysterious is that it is inherently nonsensical. It is not just that it is factually inaccurate. We shall see that, even in its own terms, it does not make sense.

Admittedly, different snapshots of the observer perceive different moments as ‘now’. But that does not mean that the observer’s consciousness – or any other moving or changing entity – moves through time as the present moment is supposed to. The various snapshots of the observer do not take it in turns to be in the present. They do not take it in turns to be conscious of their present. They are all conscious, and subjectively they are all in the present. Objectively, there is no present.

We do not experience time flowing, or passing. What we experience are differences between our present perceptions and our present memories of past perceptions. We interpret those differences, correctly, as evidence that the universe changes with time. We also interpret them, incorrectly, as evidence that our consciousness, or the present, or something, moves through time.

Everything that exists in time is supposed to take the form of unchanging snapshots arrayed along the time-line. That includes the conscious experiences of all observers, including their mistaken intuition that time is ‘flowing’. They may imagine a ‘moving present’ travelling along the line, stopping and starting, or even going backwards or ceasing to exist altogether. But imagining it does not make it happen. Nothing can move along the line. Time cannot flow.

The idea of the flow of time really presupposes the existence of a second sort of time, outside the common-sense sequence-of-moments time. If ‘now’ really moved from one of the moments to another, it would have to be with respect to this exterior time. But taking that seriously leads to an infinite regress, for we should then have to imagine the exterior time itself as a succession of moments, with its own ‘present moment’ that was moving with respect to a still more exterior time – and so on.

For example, in saying that Faraday discovered electromagnetic induction ‘in 1831’ we are assigning that event to a certain range of moments. That is, we are specifying on which set of snapshots, in the long sheaf of snapshots of world history, that discovery is to be found. No flow of time is involved when we say when something happened, any more than a ‘flow of distance’ is involved if we say where it happened. But as soon as we say why something happened, we invoke the flow of time.

Our theories of physics are, unlike common sense, coherent, and they first achieved this by dropping the idea of the flow of time.

Spacetime is sometimes referred to as the ‘block universe’, because within it the whole of physical reality – past, present and future – is laid out once and for all, frozen in a single four-dimensional block.

Relative to spacetime, nothing ever moves. What we call ‘moments’ are certain slices through spacetime, and when the contents of such slices are different from one another, we call it change or motion through space.

In spacetime physics (which is, effectively, all pre-quantum physics, starting with Newton) the future is not open. It is there, with definite, fixed contents, just like the past and present. If a particular moment in spacetime were ‘open’ (in any sense) it would necessarily remain open when it became the present and the past, for moments cannot change.

It so happens that all fundamental theories in the era of spacetime physics had the property that given everything that happens before a given moment, the laws of physics determine what happens at all subsequent moments. The property of snapshots being determined by other snapshots is called determinism.