Mathematics on Ice

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.

Cantor’s new theory had a mixed reception. His one-time teacher Leopold Kronecker reviled it as ‘humbug’ and ‘mathematical insanity’, whereas David Hilbert declared: ‘No one shall expel us from the paradise that Cantor has created for us.’ Bertrand Russell recalled in his autobiography that he ‘falsely supposed’ all Cantor’s ‘arguments to be fallacious’, only later realising that ‘all the fallacies were mine.’ In some cases, the reaction broke along national lines. French mathematicans, on the whole, were wary of its metaphysical aura. Henri Poincaré (who rivalled Hilbert as the greatest mathematician of the era) observed that higher infinities ‘have a whiff of form without matter, which is repugnant to the French spirit’. Russian mathematicians, by contrast, embraced the newly revealed hierarchy of infinities.

Why the contrary reactions? The authors of Naming Infinity see it as a matter of French rationalism versus Russian mysticism. And it was the mystics, they claim, who better served the cause of mathematical progress. Loren Graham, an American historian of science, and Jean-Michel Kantor, a French mathematician, argue that the intellectual milieu of the French mathematicians was dominated by Descartes, for whom clarity and distinctness were warrants of truth, and by Auguste Comte, who insisted that science should be purged of metaphysical speculation. Cantor’s theory seemed to offend against both.

The Russians, by contrast, warmed to it. In fact, the founding figures of the most influential school of 20th-century Russian mathematics were members of a heretical religious sect called the Name Worshippers, who believed that by repetitiously chanting God’s name they could achieve fusion with the divine. Name Worshipping, traceable to fourth-century Christian hermits in the deserts of Palestine, was revived in the modern era by a Russian monk called Ilarion. In 1907, Ilarion published On the Mountains of the Caucasus, a book that described the ecstatic experiences he induced in himself while chanting the names of Christ and God over and over again until his breathing and heartbeat were in tune with the words.

To the Russian Orthodox episcopacy, the Name Worshippers were guilty of the heresy of equating God with his name, and the tsarist regime acted to suppress the movement. (Naming Infinity opens in dramatic fashion in 1913 with Russian marines storming a monastery on the Aegean full of rebellious Name-Worshipping monks.) But for the sect’s mathematical adherents, Name Worshipping seemed to open up a special avenue to the infinite, and they were emboldened to make free use of higher infinities in their work.

Graham and Kantor are sure that Name Worshipping gave Russian mathematicians a leg up, but they are less certain that mysticism can play a genuine role in the attainment of mathematical knowledge. ‘We trust rational thought more than mystical inspiration,’ they write. The same, though, could be said of the French mathematicians who were supposedly surpassed by the Russians. It appears that mysticism in mathematics has a degree of pragmatic truth at least – that is, it works.

When Cantor began his work on infinity, calculus had long been the most important branch of mathematics for understanding the physical world, but its fundamental concepts were still in a muddle. Essentially, calculus deals with curves. Its two basic operations involve finding the direction of a curve at a given point (the ‘derivative’) and the area bounded by a curve (the ‘integral’). Curves are mathematically represented by ‘functions’. Some functions, like the sine wave, are smooth; they are ‘continuous’. But others are riddled with breaks and jumps: ‘discontinuities’. Cantor’s contemporaries were struggling with the question of how discontinuous a function could be before it was lost to the methods of calculus. The key to answering it is the idea of a set. Consider the collection of all the points where there is a discontinuity in a function. The larger and more complicated this set of discontinuities, the more ‘pathological’ the function. Cantor’s attention was drawn to sets of points. How could the size of such a set be measured? Well, he reasoned, if the members of one set can be paired off one-to-one with the members of another, the two sets must be of the same size. This is obviously true for finite sets: if my children can be paired off one-to-one with yours, then we have the same number of children. Cantor simply extended the method to infinite sets. First, he proved that the infinity of fractions was the same size as the infinity of whole numbers (1,2,3 . . . ). This is mildly astonishing, since between any two whole numbers there are infinitely many fractions. Could it be that all infinite sets were of the same size – that there was only one real infinity? Cantor at first suspected as much, but then he managed to prove that the ‘real numbers’, which correspond to the points on a line, constituted a larger infinity than that of the fractions. Finally, in 1891, he published his famous ‘diagonal argument’, which showed that for any given infinity, a greater one could always be generated. In other words, he showed that there are infinitely many orders of infinity.

Cantor’s theory of sets, and his distinction between ‘small’ and ‘large’ infinities, supplied what was needed to shore up the calculus and extend its basic concepts. Three French mathematicians led the way. Emile Borel and his students Henri Lebesgue and René Baire – whom the authors refer to as the ‘French trio’ – resolved some of the most vexing issues. Borel launched what became known as ‘measure theory’, today the basis of the study of probability; Baire deepened the understanding of continuity and its relation to the derivative; and Lebesgue produced a beautiful new theory of the integral that eliminated its most annoying flaws.

These magnificent achievements all built on Cantor’s work, yet the French trio had reservations about it. Paradoxes discovered by Russell and others made them worry that the new set theory might be logically unsound. They were especially sceptical about a new assumption called the ‘axiom of choice’, which was introduced by the German mathematician Ernst Zermelo in 1904 as an extension to Cantor’s theory. The axiom of choice asserts the existence of certain sets even when there is no recipe for producing them. Suppose, for example, you start with a set consisting of an infinite number of pairs of socks, and you want to define a new set consisting of just one sock from each pair. Since the socks in a pair are identical, there is no rule for doing this. The axiom of choice nevertheless guarantees that such a set exists, even though it represents an infinite number of arbitrary choices.

The French ultimately rejected the axiom of choice – ‘such reasoning does not belong to mathematics,’ Borel declared – and, with it, the use of higher infinities. Graham and Kantor sense intellectual timidity: the French ‘lost their nerve’; they ‘confronted an intellectual abyss before which they came to a halt’. The price they paid for their qualms, we are given to believe, was psychological as well as mathematical. Borel retreated from the abstractions of set theory to the safer ground of probability: ‘Je vais pantoufler dans les probabilités,’ as he charmingly put it. Lebesgue in his ‘frustration’ became ‘somewhat sour’. Baire, whose physical and mental health were always delicate, finally killed himself.

In contrast to the French, the ‘Russian trio’ welcomed the metaphysical aspects of set theory. The senior figure, Dmitri Egorov, was a deeply religious man, and his student Pavel Florensky had trained as a priest. (Some years into the Bolshevik era, the sight of Father Florensky addressing a scientific conference in his clerical robes moved an incredulous Trotsky to exclaim: ‘Who is that?’) Florensky became the spiritual mentor of another of Egorov’s students, Nikolai Luzin. Both Egorov and Florensky were members of an underground cell of the Name Worshipping sect, which had spread from rural monasteries to the Moscow intelligentsia; Luzin, if not a member, was influenced by its philosophy. They carried their Name Worshipping over into mathematics, seeming to believe that the act of naming could put them in touch with infinite sets undefinable by ordinary mathematical means – indeed, that they could summon into existence new mathematical entities merely by naming them.

You can’t, however, point your finger at an infinite set and say, ‘I dub thee A,’ because such sets, if they exist, are not part of the spatio-temporal world. The only way to name an infinite set is by producing a mathematical description that it, and it alone, satisfies. Thus you might name a particular infinite set by saying, ‘Let A be the set of all rational numbers whose squares are less than 2.’ Here the name is mere shorthand; what actually does the referential work is the definition, without which there is no way to assert the set’s existence.

That is what the French trio realised. ‘To define always means naming a characteristic property of what is being defined,’ Lebesgue wrote. In quoting this, Graham and Kantor add italics to the word naming; here, they claim, ‘we catch a hint of the importance of the concept later to the Russian Name Worshippers.’ This is misleading. Lebesgue was not talking about dubbing a mathematical object with a name, like calling an infinite set ‘Bob’. Rather, he was using the French verb nommer in its sense of ‘cite’ or ‘indicate’ (the way we might say in English, ‘Name me one thing they have in common!’). His point was logically sound: defining a thing means citing a property that distinguishes it from other things. And this was just the sort of definition with which the axiom of choice allowed one to dispense – dangerously, in the French view.

In claiming that the freewheeling use of infinity by the Russians enabled them to make breakthroughs denied to the more cautious French, Graham and Kantor exaggerate. It was the French trio who wrote the dramatic final chapter in the logical development of calculus. Every working mathematician is intimately acquainted with the ‘Borel algebra’, the ‘Baire category theorem’, and above all the ‘Lebesgue integral’. To this the Russians merely added a few footnotes. (Egorov’s most famous theorem, which concerns an infinite sequence of functions, was essentially a rediscovery of one of Borel and Lebesgue’s results.) It’s true that Luzin helped found ‘descriptive set theory’, a sub-branch of set theory that uses Cantor’s higher infinities to describe complicated subsets of the points making up the real-number line. But to call this ‘a new field of modern mathematics’, as the authors do, is to inflate its importance.

The real achievement of Egorov and Luzin was to put Moscow on the mathematical map. A circle of young mathematicians at Moscow University formed around them in the early 1920s, taking the name ‘Lusitania’ in Luzin’s honour. Mathematical creativity flowered amid famine and civil war. Seminars were held in sub-freezing temperatures because of fuel shortages, but the students created an ice rink inside the mathematics building, ‘singing while they glided on the ice around the central staircase under the skylight’.

In the early years of the Soviet era, mathematicians were largely left alone by the authorities because of the abstract nature of their work. Egorov and Luzin kept religion out of their lectures, hinting only at the ‘mystical beauty’ of the mathematical world and the importance of bestowing names on its objects. But when Stalin came to power, Egorov was denounced as ‘a reactionary supporter of religious beliefs, a dangerous influence on students, and a person who mixes mathematics and mysticism’. His accuser was Ernst Kol’man, an impish and sinister Marxist mathematician nicknamed the ‘dark angel’. Egorov and Florensky, along with other Name Worshippers, were eventually arrested. Egorov starved to death in prison in 1931 – his last words, reputedly, were ‘Save me, O God, by Thy name!’ – and Florensky was tortured and sent to a camp in the Arctic, and then executed in Leningrad in 1937. Luzin too was targeted by Kol’man, who deployed esoteric mathematical arguments to show that Luzin’s theorising clashed with Marxist materialism. But Luzin had powerful defenders, one of whom, appealing to Stalin, observed that Newton himself had been a ‘religious maniac’. After undergoing a humiliating trial – at which among other charges, he was accused of publishing his results in foreign journals – he was spared.

Several of Luzin’s former students took part in the campaign against him, among them Pavel Alexandrov and Andrei Kolmogorov. A surprising (to me, at least) revelation of this book is that these two legendary mathematicians – both of them outshone their mentor, and Kolmogorov was one of the half-dozen greatest mathematicians of the 20th century – were long-time lovers. Their favourite activity was swimming vast distances and then doing mathematics together in the nude. Graham and Kantor speculate that the hostility they showed Luzin was probably more a matter of professional rivalry and personal friction than ideology.

The Moscow school flourished long after the eclipse of its founders; in the postwar era, only Paris rivalled the Russian capital. But the higher infinities revered by the Name Worshippers ceased to be of great importance, and the elaborate set theory favoured by Luzin was displaced by the more mainstream methods of Kolmogorov and Alexandrov. As for the once controversial axiom of choice, Gödel removed any need for a mystical rationale when he proved in 1938 that it was logically consistent with the other, generally accepted axioms of set theory. Since no harmful contradictions could result from its use, mathematicians were now free to employ the axiom as they saw fit – no need to worry about whether it truly described a Platonic world of infinite sets.

Herein lies a clue to the demystification of mathematics. Suppose there are no transcendent objects: does mathematics then become like theology without God? Is it (as philosophical nominalists insist) mere make-believe? In a sense, yes. If there is no actual mathematical reality to be described, mathematicians are free to make up stories – that is, to explore whatever hypothetical realities they can imagine. As Cantor once declared: ‘The essence of mathematics is freedom.’ By these lights, their work consists of if-then assertions: if such-and-such a structure satisfies certain axioms, then that structure must also satisfy certain further conditions. (This if-then view has been held at times by Russell and Putnam, among others.) Some of these axioms might describe hypothetical structures that have analogues in the physical world, making them useful for ‘applied’ mathematics. Others may be of no help in understanding the physical world but still have utility within mathematics. The axiom of choice, for example, is not needed for applied mathematics, but most mathematicians use it because it streamlines the more make-believe areas like topology. There is only one constraint on the storytelling of mathematicians (other than the need to get tenure), and that is consistency. As long as a collection of axioms is consistent, then it describes some possible structure. But if the axioms turn out to be inconsistent – that is, if they harbour a contradiction – they can describe no possible structure and are hence a waste of the mathematician’s time.

As evidence that the belief in a Platonic mathematical reality is still very much alive, Graham and Kantor cite the baroque case of Alexander Grothendieck. Working in Paris in the 1960s, Grothendieck (the son of a Russian anarchist who died at Auschwitz) created a new abstract framework that revolutionised mathematics, enabling the expression of hitherto inexpressible ideas. Grothendieck’s approach has a strong mystical bent. In his voluminous autobiographical writings, he describes a creative process involving ‘visions’ and ‘messenger dreams’. Like the Name Worshippers, he sees naming ‘as a way to grasp objects even before they have been understood’. Grothendieck thus looks like an advertisement for the pragmatic power of mysticism: Graham and Kantor even suggest that he may be ‘a prelude of the future development of mathematics’. Grothendieck himself long ago abandoned the Parisian mathematical world, and now lives in seclusion in the Pyrenees, where, according to visitors, ‘he is obsessed with the Devil, which he sees at work everywhere in the world, destroying the divine harmony.’