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March 2 - March 3, 2019
Or might the infinite be neither out there nor in here but only in language, a pretty conceit of poetry?
Mathematics promises certainty—but at the cost, it seems, of passion. Its initiates speak of playfulness and freedom, but all we come up against in school are boredom and fear, wedged between iron rules memorized without reason.
Gradually, then, the music of mathematics will grow more distinct. We will hear in it the endless tug between freedom and necessity as playful inventions turn into the only way things can be, and timeless laws are drafted—in a place, at a time, by a fallible fellow human. Just as in listening to music, our sense of self will widen out toward a more than personal vista, vivid and profound.
Whether we focus on the numbers we count with and their offspring or the shapes that evolve from triangles, ever richer structures will slide into view like beads on the wire of the infinite. And it is this wire, running throughout, that draws us on, until we stand at the edge of the universe and stretch out a hand.
Can we get behind numbers to find what it is they measure? Can we come to grips with the numbers themselves to know what they are and where they came from? Did we discover or invent them—or do they somehow lie in a profound crevice between the world and the mind?
With nothing more than the number one and the notion of adding, we are on the brink of a revelation and a mystery. Rubbing those two sticks together will strike the spark of a truth no doubting can ever extinguish, and put our finite minds in actual touch with the infinite.
We slip from the immensely concrete to the mind-bogglingly abstract with the slightest shift in point of view.
Does number measure time, or does time measure number? And in one case or both, have we just proven that ongoing time is infinite? Like those shifts from the concrete to the abstract, mathematics also alternates minute steps with gigantic leaps,
This was difficult in the case of zero, for it behaved badly in company. The sum of two numbers must be greater than either, but 3 + 0 is just 3 again.
What do you do when someone’s services are vital to your cause, for all his unconventionality? You do what the French did with Tom Paine and make him an honorary citizen. So zero joined the republic of numbers, where it has stirred up trouble ever since.
“A hidden connection is stronger than one we can see.”
How many directions now this insight may carry you off in: toward other polygonal shapes such as pentagons and hexagons, toward solid structures of pyramids and cubes, or to new ways of dividing up the arrays.
if it isn’t beautiful it isn’t mathematics.
We extend the franchise to them by calling the collection of natural numbers, their negatives and zero, the Integers: upright, forthright, intact. The letter Z, from the German word for number, Zahl, is their symbol, and −17 a typical member of their kind.
Fractions keep crowding whatever space you imagine between them, a claustrophobe’s nightmare. Thought of as ratios, however, they are a Pythagorean’s dearest dream: any two magnitudes, anywhere in the universe, would stand to one another as a ratio of two natural numbers.
Hippasus let the desire to simplify, and a craftsman’s feel for arithmetic, now take him where they would. It is this artistic motivation and reckless commitment to whatever consequences follow that is the mathematician’s real tetractys, the sign to kindred spirits across millennia; and it is what makes for the glories and despairs of mathematics.
You may find yourself now in the distracted state where mathematicians notoriously live.
The mathematician John von Neumann once said that in mathematics we never understand things but just get used to them. That can’t be quite right—yet our understanding must be stretched to the breaking point before it becomes flexible enough to adjust to the unthinkable.
It is at this point that a deep quality of the mathematical art emerges—let’s call it the Alcibiades Humor. For Alcibiades was the enfant terrible of ancient Athenian life at the time of Socrates: handsome and willful, outrageous and heroic, arrogant and playful, disrupter of discourse and envoy of passion to the feast of reason.
The Alcibiades Humor in mathematics is just this hubris, this refusal to stop playing when all seems lost. No square root of negative one? Then let’s make it up!
Give this new number a name and its habitation will follow. Call it i, for imaginary; let it be a number, a new sort of number whose only property is that its square is –1:
Now we have a little table of powers of this what-you-will:
We bring this alien slowly to earth by asking it to engage with the terrestrials. i + i is 2i, and 13i means i added to itself 13 times. 13 + i is just … 13 + i: the alien mixes with the natives on formal terms, keeping his distance. In that remoteness he generates further imaginaries, as I generated the natural numbers.
It might be a mistake to pause now and ask what these imaginaries really are. They had been described as “sophistic” by Italian mathematicians in the Renaissance; it was Descartes who dismissively first called them “imaginary.” Newton held them to be impossible, and Leibniz said that was an amphibian between being and not-being.
Do you feel we have been hustling you through inadequate justifications, like confidence-tricksters more eager to persuade than explain? We can’t after all just say that anything we choose is a number, or argue like lawyers from precedent, or like prophets, from revelation. We have to show that the franchise has been legitimately extended to these imaginaries, and that they can do work that none of the other citizens could manage.
It is easy to ask how we know that a statement is always true, but very hard to answer. A twelfth-century Indian proof of the Pythagorean Theorem consists of no more than two puzzle-like diagrams with the single explanatory word: “Look!”
Proofs—those minimalist structures that end up on display in glass cases—come from people mulling things over in strikingly different ways, with different leapings and lingerings.
Gauss may have had better access to his intuition than most of us do, but isn’t it clear that what is common to us all is this very intuition? Yet ever since the earth turned out not to be flat, our trust in the obvious has been weakened.
This meant deducing results by pure logic from as trim and tight a foundation as he could find. These foundations were “axioms,” like the familiar “equals added to equals make equals”—statements so weighty and worthy of belief that we don’t even know how to doubt them. Their evolution is curious, because we are such inveterate doubters.
On November 10, 1619, the young Descartes had a dream in the midst of the Lowland Wars, where he served with Prince Maurice of Orange. In it he saw that authority counted for nothing in mathematics, whose methods were able to find unimpeachable truths.
When he wrote up the principles of this method nine years later, in Règles pour la direction de l’esprit, he said that in order to gain knowledge we must begin with what we can intuit clearly and immediately, pass one by one through all the relevant stages in a continuous and uninterrupted movement of thought, to see in the end the truth directly and transparently.
You hear it in 1810, when the French geometer Gergonne wrote that axioms were theorems whose mere statement sufficed for recognizing their truth.
“Whatever form is algebraically equivalent to another form expressed in general symbols, must continue to be equivalent, whatever those symbols denote.”
Formalism—where relations hold among symbols that need have no further referents—became an ideal shelter in the revolution that was sweeping through mathematics itself in the nineteenth century. Everyone had taken for granted, over the past two millennia, that Euclid’s geometry described this precious only endless world in which we say we live—or in Kant’s terms, the way mind must spatially conceive.
Questions about how to consider mathematical existence became the special concern of David Hilbert, a German mathematician whose outlook dominated much of the twentieth century. His was an existence haunted by existence.
Had you supposed that adding two numbers would produce a caterpillar, or multiplying them, a butterfly? Such axioms verge on mere definition—almost beneath the dignity of self-evidence.
So numbers gradually came to be thought of as secondary phenomena and sets emerged as fundamental.
But the Garden of Eden is famous for its snake, and the snake is the desire for more precise knowledge.
Not even the sort of induction used in science, which concludes from a lot of test cases that a hypothesis probably holds. Certainty is the outcome here, and from many more than three or even a lot of instances: in fact, from all of them.
An example will help. If you like, you may then adopt medical school practice in mastering an operation: watch one, do one, teach one
What induction does, in effect, is show that the insight spreads contagiously from a first number to the rest of the naturals, by making the insight clamp a number and its successor (n and n + 1) together: “the empty form,” as it was called by a troubling figure of twentieth-century
The Mind and Truth
Riemann said “Just give me the insights. I can always come up with the proofs!” But his work is strung with diamonds of dazzling insights. The prolific eighteenth-century Swiss mathematician Leonhard Euler was renowned for his insights—but his proofs are gems. Both sorts live like two souls in the Faustian breast of each, not at war but in conversation with one another.
Caricatures only, because of the immense variety of people who have prospered in mathematics—the reticent and the contentious, the companionable and the morose—and because this art is the birthright of us all, so deep in the structure of our thought that it is no respecter of origin or upbringing, of morality, age, even of sanity or madness.
Each had profound insights, each produced stunning proofs—but Hilbert longed to establish mathematics on unshakable, impersonal foundations, Brouwer to free it from logic and even language, asserting the supremacy of private intuition.
For language, Brouwer wrote, “only touches the outside of an automaton.”
Mathematicians generally believe that for all they invent their paths, the landscape through which those paths wander is out there, independent of them, its granite truths indifferent to their climbing.
These truths, he said, are “fascinating by their immovability but horrifying by their lifelessness, like stones from barren mountains of disconsolate infinity.”
the foundations themselves. The rot was everywhere. Doubts increased about Euclidean geometry’s claim to the throne, with the pretenders from France, Russia, Hungary, and Germany brandishing their credentials. For each pictured space differently (Euclid’s surfaces were flat, the others’ differently curved), and nothing about them revealed which was the correct portrait of actual space. And there was the infinite.
It would have been folly to beg these questions by shifting the burden onto some other subject, as if mathematics were descriptive of the world so that physics, say, or chemistry or the physiology of the brain would be ultimately responsible for its axioms—and in need therefore of axioms in its turn.
