David Deutch - The Fabric of Reality

As I have said, there is no such method of justification as ‘induction’. The idea of reasoning one’s way to ‘near-certainty’ in science is myth. How could I prove with ‘near-certainty’ that a wonderful new theory of physics, overturning my most unquestioned assumptions about reality, will not be published tomorrow? Or that I am in not inside a virtual-reality generator? But all this is not to say that scientific knowledge is indeed ‘second-class’. For the idea that mathematics yields certainties is a myth too.

Since ancient times, the idea that mathematical knowledge has a privileged status has often been associated with the idea that some abstract entities, at least, are not merely part of the fabric of reality but are even more real than the physical world. Pythagoras believed that regularities in nature are the expression of mathematical relationships between natural numbers. ‘All things are numbers’ was the slogan. This was not meant quite literally, but Plato went further and effectively denied that the physical world is real at all. He regarded our apparent experiences of it as worthless or misleading, and argued that the physical objects and phenomena we perceive are merely ‘shadows’ or imperfect imitations of their ideal essences (‘Forms’, or ‘Ideas’) which exist in a separate realm that is the true reality. In that realm there exist, among other things, the Forms of pure numbers such as 1, z, 3, …, and the Forms of mathematical operations such as addition and multiplication. We may perceive some shadows of these Forms, as when we place an apple on the table, and then another apple, and then see that there are two apples. But the apples exhibit ‘one-ness’ and ‘two-ness’ (and, for that matter, ‘apple-ness’) only imperfectly. They are not perfectly identical, so there are never really two of anything on the table. It might be objected that the number two could also be represented by there being two different things on the table. But such a representation is still imperfect because we must then admit that there are cells that have fallen from the apples, and dust, and air, on the table as well. Unlike Pythagoras, Plato had no particular axe to grind about the natural numbers. His reality contained the Forms of all concepts. For example, it contained the Form of a perfect circle. The ‘circles’ we experience are never really circles. They are not perfectly round, nor perfectly planar; they have a finite thickness; and so on. All of them are imperfect.

Plato then pointed out a problem. Given all this Earthly imperfection (and, he could have added, given our imperfect sensory access even to Earthly circles), how can we possibly know what we know about real, perfect circles? Evidently we do know about them, but how? Where did Euclid obtain the knowledge of geometry which he expressed in his famous axioms, when no genuine circles, points or straight lines were available to him? Where does the certainty of a mathematical proof come from, if no one can perceive the abstract entities that the proof refers to? Plato’s answer was that we do not obtain our knowledge of such things from this world of shadow and illusion. Instead, we obtain it directly from the real world of Forms itself. We have perfect inborn knowledge of that world which is, he suggests, forgotten at birth, and then obscured by layers of errors caused by trusting our senses. But reality can be remembered through the diligent application of ‘reason’, which then yields the absolute certainty that experience can never provide.

I wonder whether anyone has ever believed this rather rickety fantasy (including Plato himself, who was after all a very competent philosopher who believed in telling ennobling lies to the public). However, the problem he posed — of how we can possibly have knowledge, let alone certainty, of abstract entities — is real enough, and some elements of his proposed solution have been part of the prevailing theory of knowledge ever since. In particular, the core idea that mathematical knowledge and scientific knowledge come from different sources, and that the ‘special’ source of mathematics confers absolute certainty upon it, is to this day accepted uncritically by virtually all mathematicians. Nowadays they call this source mathematical intuition, but it plays exactly the same role as Plato’s ‘memories’ of the realm of Forms.

There have been many bitter controversies about precisely which types of perfectly reliable knowledge our mathematical intuition can be expected to reveal. In other words, mathematicians agree that mathematical intuition is a source of absolute certainty, but they cannot agree about what mathematical intuition tells them! Obviously this is a recipe for infinite, unresolvable controversy.

Inevitably, most such controversies have centred on the validity or otherwise of various methods of proof. One controversy concerned so-called ‘imaginary’ numbers. Imaginary numbers are the square roots of negative numbers. New theorems about ordinary, ‘real’ numbers were proved by appealing, at intermediate stages of a proof, to the properties of imaginary numbers. For example, the first theorems about the distribution of prime numbers were proved in this way. But some mathematicians objected to imaginary numbers on the grounds that they were not real. (Current terminology still reflects the old controversy, even though we now think that imaginary numbers are just as real as ‘real’ numbers.) I expect that their schoolteachers had told them that they were not allowed to take the square root of minus one, and consequently they did not see why anyone else should be allowed to. No doubt they called this uncharitable impulse ‘mathematical intuition’. But other mathematicians had different intuitions. They understood what the imaginary numbers were and how they fitted in with the real numbers. Why, they thought, should one not define new abstract entities to have any properties one likes? Surely the only legitimate grounds for forbidding this would be that the required properties were logically inconsistent. (That is essentially the modern consensus which the mathematician John Horton Conway has robustly referred to as the ‘Mathematicians’ Liberation Movement’.) Admittedly, no one had proved that the system of imaginary numbers was self-consistent. But then, no one had proved that the ordinary arithmetic of the natural numbers was self-consistent either.

There were similar controversies over the validity of the use of infinite numbers, and of sets containing infinitely many elements, and of the infinitesimal quantities that were used in calculus. David Hilbert, the great German mathematician who provided much of the mathematical infrastructure of both the general theory of relativity and quantum theory, remarked that ‘the literature of mathematics is glutted with inanities and absurdities which have hail their source in the infinite’. Some mathematicians, as we shall see, denied the validity of reasoning about infinite entities at all. The runaway success of pure mathematics during the nineteenth century had done little to resolve these controversies. On the contrary, it tended to intensify them and raise new ones. As mathematical reasoning became more sophisticated, it inevitably moved ever further away from everyday intuition, and this had two important, opposing effects. First, mathematicians became more meticulous about proofs, which were subjected to ever increasing standards or rigour before they were accepted. But second, more powerful methods of proof were invented which could not always be validated by existing methods. And that often raised doubts as to whether a particular method of proof, however self-evident, was completely infallible.