Even if any physical or mathematical intuition were inborn, that would not confer any special authority upon it. Inborn intuition cannot be taken as a surrogate for Plato’s ‘memories’ of the world of Forms. For it is a commonplace observation that many of the intuitions built into human beings by accidents of evolution are simply false. 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. It is just possible (though I do not believe it) that Euclidean geometry or Aristotelian logic are somehow built into the structure of our brains, as the philosopher Immanuel Kant believed. But that would not logically imply that they were true. Even in the still more implausible event that we have inborn intuitions that we are constitutionally unable to shake off, such intuitions would still not be necessary truths.
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. It is a very narrow window and gives us only a limited range of perspectives. Some of the structures that we see out there, such as the natural numbers or the rules of inference of classical logic, seem to be important or ‘fundamental’ to the abstract world, in the same way as deep laws of nature are fundamental to the physical world. But that could be a misleading appearance. For what we are really seeing is only that some abstract structures are fundamental to our understanding of abstractions. We have no reason to suppose that those structures are objectively significant in the abstract world. It is merely that some abstract entities are nearer and more easily visible from our window than others.
mathematics The study of absolutely necessary truths.
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.
mathematical intuition (Traditionally:) An ultimate, self-evident source of justification for mathematical reasoning.
(Actually:) A set of theories (conscious and unconscious) about the behaviour of certain physical objects whose behaviour models that of interesting abstract entities.
intuitionism The doctrine that all reasoning about abstract entities is untrustworthy except where it is based on direct, self-evident intuition. This is the mathematical version of solipsism.
Hilbert’s tenth problem To ‘establish once and for all the certitude of mathematical methods’ by finding a set of rules of inference sufficient for all valid proofs, and then proving those rules consistent by their own standards.
Gödel’s incompleteness theorem A proof that Hilbert’s tenth problem cannot be solved. For any set of rules of inference, there are valid proofs not designated as valid by those rules.
Abstract entities that are complex and autonomous exist objectively and are part of the fabric of reality. There exist logically necessary truths about these entities, and these comprise the subject-matter of mathematics. However, such truths cannot be known with certainty. Proofs do not confer certainty upon their conclusions. The validity of a particular form of proof depends on the truth of our theories of the behaviour of the objects with which we perform the proof. Therefore mathematical knowledge is inherently derivative, depending entirely on our knowledge of physics. The comprehensible mathematical truths are precisely the infinitesimal minority which can be rendered in virtual reality. But the incomprehensible mathematical entities (e.g. Cantgotu environments) exist too, because they appear inextricably in our explanations of the comprehensible ones.
I have said that computation always was a quantum concept, because classical physics was incompatible with the intuitions that formed the basis of the classical theory of computation. The same thing is true of time. Millennia before the discovery of quantum theory, time was the first quantum concept.
11
Time: The First Quantum Concept
Like as the waves make towards the pebbled shore,
So do our minutes hasten to their end;
Each changing place with that which goes before,
In sequent toil all forwards do contend.
Even though it is one of the most familiar attributes of the physical world, time has a reputation for being deeply mysterious. Mystery is part of the very concept of time that we grow up with. St Augustine, for example, said:
What then is time? If no one asks me, I know; if I wish to explain it to one who asks, I know not. (Confessions)
Few people think that distance is mysterious, but everyone knows that time is. And all the mysteries of time stem from its basic, common-sense attribute, namely that the present moment, which we call ‘now’, is not fixed but moves continuously in the future direction. This motion is called the flow of time.