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1 | That science works – it creates what we mean by progress – is proof and evidence enough for most of us that there is a world out there, separate from us and full of things that have separate existences. And then there is mathematics, the strongest evidence of all.

2 | For the philosopher René Descartes the only certainties were mathematics and theology.

The mathematics appear to be there in the behaviour of physical things and not merely imposed by us.

Roger Penrose, mathematical physicist and philosopher

3 | Roger Penrose believes that mathematics has real existence in a kind of Platonic world parallel – but somehow connected (by what mechanism is not known) – to our world of experience.

I believe that mathematical reality lies outside us, that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our ‘creations’, are simply our notes of our observations.

G.H. Hardy, A Mathematician’s Apology

Pure mathematics … seems to me a rock on which all idealism founders: 317 is a prime, not because we think it is, or because our minds are shaped in one way rather than another, but because it is, because mathematical reality is built that way.

Ibid.

4 | If we can believe with G.H. Hardy, Roger Penrose and others that mathematics is out there, and not intertwined with our own perspective on the world, then materialists may claim, as some do, that one day we will be able to write down, in the language of mathematics, laws that fully describe the physical world. Mathematics appears to be proof that the world can be transcended. Processes are real because they can be described by mathematics, which is itself real. Mathematics looks like proof that there is an external world, and that science is an investigation of its nature and substance.

Mathematics is the only religion that has proved itself a religion.¹

F. de Sua, mathematician

The unreasonable effectiveness of mathematics.

Unattributed

The equation is smarter than I am.

The theoretical physicist Paul Dirac (1902–84), on his equation that predicted the existence of antiparticles

5 | Deeper descriptions of the universe require more and more sophisticated mathematical formalisms. Einstein took ten years to find the mathematical language in which to write his general theory of relativity. Unusually, the mathematical formalism that quantum mechanics is written in came first, and its interpretation – still argued over – came afterwards.

It is only the unsophisticated outsider who imagines mathematicians make discoveries by turning the handle of some miraculous machine.

G.H. Hardy, reminding us that mathematics is carried out by human beings, not machines

There is no sort of agreement about the nature of mathematical reality among either mathematicians or philosophers. Some hold that it is ‘mental’ and that in some sense we construct it, others that it is outside and independent of us. A man who could give a convincing account of mathematical reality would have solved very many of the most difficult problems of metaphysics. If he could include physical reality in his account, he would have solved them all.

G.H. Hardy

6 | Mathematical truths seem to have been out there waiting out the ages, to be discovered or not, by whoever or whatever.² It looks as if idealism founders on the rocks of mathematics. Not quite. There are loopholes. The philosopher Immanuel Kant (1724–1804) argues that science is an exploration of the co-evolution of humans and the universe; the doll and the dolls’ house are inextricably entangled. Science is an exploration of that entanglement. Even mathematics, Kant believed, is not outside us: it is as much in our brains as in the outside world; yet it is less a mental construct than a product of the co-evolution of everything together, there being no meaningful separation between inside and outside. For Kant science is access to one kind of truth, another path being our sense of morality.

Two things fill the heart with renewed and increasing awe and reverence the more often and the more steadily that they are meditated on: the starry skies above me and the moral law inside me.

Immanuel Kant, Critique of Practical Reason

7 | Roger Penrose writes that mathematical models describe reality with ‘a precision enormously exceeding that of any description free of mathematics’. Clearly this is true but somewhat circular. Physics does precision, poetry does metaphor. They are incommensurate. Biology cares for decimal points only somewhat; poetry not at all.

Neither physicists nor philosophers have ever given any convincing account of what ‘physical reality’ is, or of how the physicist passes, from the confused mass of fact or sensation with which he starts, to the construction of the objects which he calls ‘real’. Thus we cannot be said to know what the subject matter of physics is; but this need not prevent us from understanding roughly what a physicist is trying to do. It is plain that he is trying to correlate the incoherent body of crude fact confronting him with some definite and orderly scheme of abstract relations, the kind of scheme he can borrow only from mathematics.

A mathematician, on the other hand, is working with his own mathematical reality.

G.H. Hardy

8 | Einstein called mathematics the poetry of logical ideas.

9 | In the 1920s an attempt was made to make mathematical logic the sole means of advancing philosophical knowledge, and so rid the scientific method of metaphysics once and for all. The methodology was called logical positivism, and assumed experience as the sole source of knowledge. Logical positivists wanted to admit as meaningful only those sentences that can be independently verified, ultimately by mathematics.

Logical positivism is the idea that a sentence or another fragment – something you can put in a computer file – means something in a freestanding way that doesn’t require invoking the subjectivity of a human reader. Or, to put it in nerd-speak: ‘The meaning of a sentence is the instruction to verify it.’

Jaron Lanier, computer scientist and virtual-reality pioneer

Logical positivism is a form of solipsism. If you say physics is only about predicting the outcomes of experiments, you can only really say it’s about experiments that you personally do, because to you any other person is just another thing you’re observing. But solipsism is a dead-end philosophy and when it comes to science it’s a poison.

David Deutsch

10 | At the laboratory in Lagado, the capital city of the nation Balnibarbi, there is a great machine that manipulates all the words of the language of the people. Scientists hope to extract knowledge out of the random generation of words. Where three or four words are found together that might make part of a sentence, they are dictated and transcribed into a ‘large Folio already collated, of broken sentences’. Out of this process it is ‘intended to piece together, and out of [these] rich materials to give the World a complete Body of all Arts and Sciences …’ At the Mathematical School in the same city, formulae are written in ink on wafers and eaten, in the belief that the ink with its message will eventually reach the brain, where the information will be processed. Other scientists have attempted to reduce the nation’s language entirely to nouns, since in reality ‘all things imaginable are but Nouns’. Instead of speaking, language is reduced to pointing at things. The disadvantage is that it means carrying around a large number of things to point at. The women of the country rebelled and sought ‘the Liberty to speak with their Tongues, after the manner of their Ancestors; Such constant irreconcilable Enemies to Science are the Common People.’

The quotes are from Gulliver’s Travels by Jonathan Swift (1667–1745).

11 | Some materialists believe that out of enough data, meaning is self-generated. And so logical positivism creeps back into fashion. Larry Page, one of the founders of Google, believes that the internet will come alive at some point. The futurist George Dyson believes that it already has. Other futurists, Ray Kurzweil notably, predict that a moment will come when machines will outsmart humans. It is predicted that this singular event will occur in the twenty-first century.

12 | Kurt Gödel’s incompleteness theorem of 1931 (published when he was only twenty-five years old) presented the logical positivists’ programme in a new light. Gödel’s Theorem is often misstated. It does not show that mathematics is incomplete, but that mathematics is incomplete within any particular mathematical formalism, which is crucially different. It means, for example, that there are mathematical problems that can be written in the language of, for instance, arithmetic (an example of one kind of mathematical formalism), but cannot be proved in arithmetic. But arithmetic can be made complete within a more encompassing formalism, say, geometry; yet geometry is itself incomplete, and so on. Mathematics is a series of nesting formalisms, one inside another like a Russian doll. Our physical understanding of the universe – written as it is in mathematics – may be like this too.³

Gödel’s theory is useful when we think about computers or any mechanical device that works in some formal way. It tells us that there are mathematical truths that cannot be proved by any computer or mechanical device, no matter how sophisticated that device may be. If our minds are formal in the way computers are, then there will always be some mathematics that is beyond our reach, and so also some understanding of the physical world that is beyond our reach. Alternatively, as Gödel pointed out, if humans can always delve deeper into mathematical reality, then they cannot be machines. We are left with two possibilities: humans are machines and their understanding of the world has a limit, or they are not machines and are free to explore the physical world forever. Either way the world is more mysterious than we are.

If we accept that mathematics is the strongest evidence we have of an externally existing world, Gödel’s theorem throws a spanner in the works when we move on to consider the manufacture of our human doll. If human beings are not machines, then how can we make one? On the other hand, if they are machines, Gödel’s theorem casts doubt on the possibility that humans themselves could ever construct a machine of equal complexity. We must hope that out of the scientific method we can create machines that are intelligent enough to evolve, and in that way eventually transcend the intelligence of their makers.