The Art of the Infinite: The Pleasures of Mathematics

by Robert M. Kaplan

If they were consistent with one another, so that no paradox could follow from their workings, a rounded body of connections would grow musically from them.

“I call axioms propositions that are true but are not proved because our knowledge of them flows from a source very different from the logical source, a source which might be called spatial intuition. From the truth of the axioms it follows that they do not contradict each other.”

But the growing variety of geometries meant that Kant had to be wrong in some of the details. Gauss was the first to spot this: the variety must show, he said, that our knowledge of space turns out to depend on experience; but there are no competing systems of number because it alone comes from an intuition prior to experience.

Hilbert agreed. As part of his doctoral examination in 1885 he defended the a priori nature of arithmetical judgments (those, that is, about the natural numbers). Forty-five years later, in the farewell address he gave to his native city of Königsberg, he explained that after the dross had been removed from Kant’s theory, “only that a priori will remain which also is the foundation of pure mathematical knowledge”—a foundation, he said, of intuitive insight.

Kurt Gödel conclusively showed in 1931 that no proof of consistency and completeness could ever be made within the system to which it referred, as Hilbert’s was meant to—for any system which was strong enough to deal with the mathematics of the natural numbers.

There was no hope, then, of securing the axioms by Hilbert’s clever outflanking maneuver. He and his rival had moved through time, intent on its expression, to have their paths ultimately coincide in failure.

“I can more easily imagine cutting something up zero times than I can a million times!”

The product of our two negatives is positive, no matter how happy and undeserving a may appear, or how wretched and meritorious b.

The very finite proof that in fact the number of primes is infinite stands framed in Euclid—so it is at least 2,300 years old. It needs only the briefest introduction.

Euclid wants to prove that there is no last prime. He does this by showing that no matter how many primes you have, you are forced to produce another.

The twentieth-century mathematician Paul Erdos often spoke of “The Book”: the book, he meant, in which God keeps all the most beautiful proofs. “You don’t have to believe in God,” said Erdos, “but you do have to believe in The Book.” Everyone has his own edition of this book, but Euclid’s proof of the infinitude of primes is likely to be in them all.

Curiouser and curiouser.

The startling news is that there are stretches of numbers a thousand long with not a single prime among them. More: there are primeless stretches a million long! Since the primes never end, you will come on one eventually after such a span—which begins to give a horrifying sense of how big very big numbers are, and how immeasurably bigger than big the infinity of the natural numbers is.

The story of how many primes there are in such sequences was finally told in 1837 by a remarkable man named Johann Peter Gustav Lejeune-Dirichlet (an ancestor, a young man from Richelet—“le jeune de Richelet”—moved from Belgium to the Rhineland.

Gauss—whom we saw as a schoolboy triumphantly writing on his slate—used to contemplate tables of primes for sheer amusement, the way Russians always and the English on country house weekends love browsing through railway timetables.

The amazing fifteen-year-old Gauss came up with such a function. He looked at the data we saw on page 79–80 and realized that the ratio of x to π(x) increased by roughly 2.3 from one power of 10 to the next. 2.3?

There is a number —an irrational close to 2.7—which lies at the heart of biology and economics, because it expresses organic growth. When is raised to about 2.3 you get 10.

Gauss therefore leapt to the conjecture—on the basis of how the primes were distributed among the first 3,000,000 integers!

Gauss was unable to prove his conjecture, which did not become a theorem until 1896, when it was proved independently in two very different ways by two very different men, who were born a year apart and died a year apart, almost a century later; their lives were to diverge radically from their intersection at the proof of this conjecture.

One, who gloried in the name Charles Jean Gustave Nicolas de la Vallée Poussin, was born, lived, and died in Louvain, in Belgium: a professor, like his father before him, at the university there; survivor of two world wars and fifty years of teaching.

Somewhere very far out, π(x) becomes larger than Li(x). We don’t yet know where this happens, but it has been shown to be past 1020, and is likely to be around 1.39822 · 10316. This number, far greater than the number of particles in the universe (a mere 1075 or so), is no more than a peak in the mountainous landscape where number theorists stride.

To what, then, should we compare our present condition—as icebound as was Shackleton at the Pole? Are we like those brilliant, autistic people who understand that there must be something which facial expressions reflect and can with avid intelligence catch clues to correlate some with others, yet have no idea what the cause of such effects might be?

Mathematics is the art of the infinite because whatever it focuses on with its infinite means discloses limitless depth, structure, and extent.

Late in his life, Newton said: “I seem to have been only like a boy playing on the seashore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me.” Although mathematics has grown exponentially since his time, we still find ourselves children standing on the edge of the limitless unexplored.

There are some, like the distinguished twentieth-century number theorist André Weil, for whom a conjecture once proven, like a mountain climbed, becomes no more than a trophy: another name on Don Giovanni’s list. In contrast, a piece of mathematics heard as music is inexhaustibly filled with promises for the future and houses as well an inexhaustible presence, like a fugue from the Well-Tempered Clavier.

Let’s act like mathematicians, with faith in design and confidence in our powers to find it. Above all, let’s use the art of the infinite: going, that is, for all the patterns at once.

This kind of naming and these sorts of symbols drive more people away from mathematics than teachers who tell you you’re wrong because they say so. We are perfectly happy to think of someone as James Smith or even James Topaz Smith, and if his son is James Topaz Smith Junior we take that easily in stride. Should the son become a Doctor or even the Right Reverend Doctor James Topaz Smith Junior we may smile, but can handle it. Yet attach a pair of numbers to a letter and we beg for mercy.

Lest you think that nothing could be more boring than proving the obvious, it would be enough to remember that mathematics is the one skyscraper of thought which rises above mere opinion to utter certainty.

There it is: seen all at once and so naturally, so convincingly, that you look back in wonder at all the sour wrangle over foundations and formal proofs.

Yet such pictures as this have their critics, who would caution us to speak of them instead as “more or less proofs.” Pictures can lie; at the very least they can persuade the eye to take for granted what the mind should examine in detail.

When the number theorist J. E. Littlewood said of a drawing that it was all the proof needed for a professional, he was suggesting that a professional would know where and how to grow the connective tissue.

Lost in the tangle of tactics we agree with Descartes that thought should move through a demonstration continuously and smoothly.

I consulted Browne, the Professor of Psychology, about it the other day, but he assured me that it isn’t a dangerous habit. He said: ‘When you find yourself getting into multiples of seven, come to me again.’”

The aim is always to aid intuition, not to fossilize insight into formalism.

With triangles, what you see is what you get.

This circle is called the triangle’s circumcircle, since it is circumscribed about it; and O is therefore called the circumcenter.

For the sake of our intuition, let’s do something Euclid would never have done and imagine our triangle actually cut out of a thin sheet of metal, with its mass spread out uniformly; and then picture balancing this triangular gusset on a knife-edge.

So the nimble mind coaxes new insights from old with that economy that marks the noblest arts.

“Arrogance” is what Alcibiades’s enemies called what his friends saw as insouciant confidence. Intuition urges straight lines on a mind in pursuit of least distance, and reason has to contrive how to form intuition to fit the circumstances.

The Formalist seems to have the last, cautionary word: if anything is infinite, it is the subtlety of the world.

The Eagle of Algebra

Mathematics seems ever to teach us two lessons: there is no limit to our mind’s ingenuity; and there is even less of a limit to the intransigence of the world.

It was Gauss—once again Gauss, whose name runs through the last two centuries of mathematics like Louis Armstrong’s through the evolution of jazz—who on March 30, 1796, when he was still eighteen, discovered how to construct the 17-gon. No one had seen a way, or was even sure that it could be done, in the two thousand years of thinking about it before him.

By concentrated analysis I succeeded, during a vacation in Braunschweig, in the morning of the day, before I got up, to see [the general idea so] clearly that I was able to make the specific application to the 17-gon and to confirm it numerically right away.

A proof, if one is needed, that adolescents should be allowed to get up ...

What are we doing here? Many a climber has found that the little chaos of life grows ordered and makes a new sense when seen from afar, just as writers like James Joyce discover in exile the vivid structure of home, concealed by its cluttered presence.

Complex events in simple contexts become simple when the context grows sophisticated. So on this complex plane, exceptions and peculiarities, such as those we recently met, will all at once be seen as outcroppings of deeper symmetries.

Simplicity and symmetry: how often the impulse toward understanding takes its bearings from these two markers, in the belief that ultimate answers lie just beyond them (we lust after the subtle and ...

But if we allow complex roots, quadratics always have two, cubics three, quartics four—and nth degree polynomials always have n complex roots.15 This truth (once again, proved by Gauss) is so important that it is called the Fundamental Theorem of Algebra. Roots are buried all over the complex plane, there for our extracting.

Algebra helped geometry in the last chapter: here geometry will repay the debt.