Mathematics on Ice

Jim Holt

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A surprising number of mathematicians, even quite prominent ones, believe in a realm of perfect mathematical entities hovering over the empirical world – a sort of Platonic heaven. Alain Connes of the Collège de France once declared that ‘there exists, independently of the human mind, a raw and immutable mathematical reality,’ one that is ‘far more permanent than the physical reality that surrounds us’. Roger Penrose, another unabashed Platonist, holds that the natural world is only a ‘shadow’ of a realm of eternal mathematical forms.

The rationale for this otherwordly view appeared first in the Republic. Geometers, Plato observed, talk of perfectly round circles and perfectly straight lines, neither of which are to be found in the sensible world. The same is true of numbers, since they must be composed of perfectly equal units. The objects studied by mathematicians must therefore exist in another world, one that is changeless and transcendent. Seductive though this picture of mathematics might be, it doesn’t tell us how mathematicians are supposed to get in touch with this transcendent realm. How do we come to have knowledge of mathematical objects if they lie beyond the world of space and time? Contemporary Platonists tend to do a bit of hand-waving when confronted with this question. Connes invokes a special sense, ‘irreducible to sight, hearing or touch’, that allows him to perceive mathematical reality; Penrose believes that human consciousness somehow ‘breaks through’ to the Platonic world. Kurt Gödel, among the staunchest of 20th-century Platonists, wrote that ‘despite their remoteness from sense experience, we do have something like a perception’ of mathematical objects, adding, ‘I don’t see any reason why we should have less confidence in this kind of perception, i.e. in mathematical intuition, than in sense perception.’

But mathematicians, like the rest of us, think with their brains, and it’s hard to understand how the brain, a physical entity, could interact with a non-physical reality. ‘We cannot envisage any kind of neural process that could even correspond to the “perception of a mathematical object”,’ Hilary Putnam once observed. One way out of this dilemma is to throw over Plato for Aristotle. There may be no perfect mathematical entities in our world, but there are plenty of imperfect approximations. We can draw crude circles and lines on a chalkboard; we can add two apples to three apples, even if they are not identical, and end up with five. By abstracting from our experience of ordinary perceptible things, we arrive at basic mathematical intuitions, and logical deduction does the rest.

This Aristotelian view pretty much accords with common sense. But there is one putative mathematical object that it can’t handle: infinity. We have no experience of the infinite. We have no experience of anything like it. True, we do have a sense of numbers going on indefinitely – take the biggest number you can think of, and you’ll always be able to add one – and we think we can imagine space or time extending without limit. But an actual ‘completed’ infinity, as opposed to a merely potential one, is something we never encounter in the natural world. The idea of infinity was long regarded with suspicion, if not horror. Zeno’s paradoxes seemed to show that if space could be divided up into infinitesimal segments, then motion would be impossible. This absurd conclusion led Aristotle to ban infinity from Greek thought. But it eventually became apparent that mathematicians couldn’t do without it. Even ‘applied’ mathematics – the mathematical physics that grew out of Newton’s and Leibniz’s invention of calculus – had glitches that only a rigorous theory of sets, including infinite sets, could fix.

It was in the late 19th century that Georg Cantor, a Russian-born German mathematician, supplied the theory needed. Cantor did not set out to explore infinity for its own sake; rather, he claimed, the task ‘was logically forced upon me, almost against my will’. What he ended up with, after two decades of intellectual struggle, was a succession of higher and higher infinities – an infinite hierarchy of them, ascending towards an unknowable terminus that he called the Absolute. This seemed to him a divinely vouchsafed vision; in transmitting it to the world, he regarded himself (in the words of his biographer Joseph Dauben) as ‘God’s ambassador’. Cantor spent the rest of his life pondering the theological implications of infinity and, with equal enthusiasm, pursuing the hypothesis that the works of Shakespeare were written by Francis Bacon. He ended his life in an asylum.

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