by Ian Stewart
Schools—not just yours, Meg, but around the world—are so preoccupied with teaching sums that they do a poor job of preparing students to answer (or even ask) the far more interesting and difficult question of what mathematics is. And even though definitions are too limiting, we can still try to capture the flavor of our subject, using something that the human brain is unusually good at: metaphor. Our brains are not like computers, working systematically and logically. They are metaphor machines that leap to creative conclusions and belatedly shore them up with logical narratives. So, when I tell you that one of my favorite “definitions” of math is Lynn Arthur Steen’s phrase “the science of significant form,” you may feel that I’ve made a useful stab at the question, metaphorically speaking.
What I like about Steen’s metaphor is that it captures some crucial features. Above all, it is open-ended; it does not attempt to specify what kind of form should be considered significant, or what “form” or “significant” are even supposed to mean. I also like the word “science,” because math shares far more with the sciences than it does with the arts. It has the same reliance on stringent testing, except that in science this is done through experiments, whereas math employs proofs. It has the same character of operating within closely specified constraints: you can’t just make it up as you go along. Here I part company with the postmodernists, who assert that everything (except, apparently, postmodernism) is merely a social convention. Science, they tell us, consists only of opinions that happen to be held by a lot of scientists. Sometimes this is the case—the prevalent belief that the human sperm count is falling is probably an example— but mostly it is not. There is no question that science has a social side, but it also has the reality check of experiment. Even postmodernists must always enter a room through the door, not through the wall.
There is a famous book called What Is Mathematics? written by Richard Courant and Herbert Robbins. As with most books whose titles are questions, the question is never quite answered. Yet the authors say some very wise things. Their prologue begins, “Mathematics as an expression of the human mind reflects the active will, the contemplative reason, and the desire for aesthetic perfection.” It goes on to tell us that “All mathematical development has its psychological roots in more or less practical requirements. But once started under the pressure of necessary applications, it inevitably gains momentum in itself and transcends the confines of immediate utility.” And it ends like this: “Fortunately, creative minds forget dogmatic philosophical beliefs whenever adherence to them would impede constructive achievement. For scholars and layman alike it is not philosophy but active experience in mathematics itself that alone can answer the question: What is mathematics?” Or, as my friend David Tall often says, “Math is not a spectator sport.”
Some mathematicians are more interested in the philosophy of their subject than others, and among today’s prominent philosophers of mathematics we find Reuben Hersh. He observed that Courant and Robbins answered their question “by showing what mathematics is, not by telling what it is. After devouring the book with wonder and delight, I was still left asking, ‘But what is mathematics really?’” So Hersh wrote a book with that title, offering what he said was an unconventional answer.
Traditionally, there have been two main schools of mathematical philosophy: Platonism and formalism. Platonists believe that in some (slightly mystical) way, mathematical objects exist. They are “out there” in some abstract realm. This realm is not imaginary, however, because imagination is a human characteristic. It is real, in a nonphysical sense. The mathematician’s circle, with its infinitely thin circumference and a radius that remains constant to infinitely many decimal places, cannot take physical form. If you draw it in sand, as Archimedes did, its boundary is too thick and its radius too variable. Your drawing is only an approximation of the mathematical, Platonic circle. Inscribe it on a platinum slab with a diamond-tipped needle—the same difficulties still arise.
In what sense, then, does a mathematical circle exist? And if it doesn’t, how can it be useful? Platonists tell us that the mathematical circle is an ideal, not realized in this world but nevertheless having a reality that is independent of human minds.
Formalists find such statements fuzzy and meaningless. The first major formalist was David Hilbert, and he tried to put the whole of mathematics on a sound logical basis by effectively treating it as a meaningless game played with symbols. A statement like 2 + 2 = 4 was not, from this point of view, to be interpreted in terms of, say, putting two sheep in a pen with two others and thus having four sheep. It was the outcome of a game played with the symbols 2, 4, +, and =. But the game must be played according to an explicit list of absolutely rigid rules.
Philosophically, formalism died when Kurt Gödel proved, to Hilbert’s initial fury, that no formal theory can capture the whole of arithmetic and be proved logically consistent. There will always be mathematical statements that remain outside Hilbert’s game: neither provable nor disprovable. Any such statement can be added to the axioms for arithmetic without creating any inconsistency. The negation of such a statement has the same feature. So we can deem such a statement to be true, or we can deem it false, and Hilbert’s game can be played either way. In particular, the idea that arithmetic is so basic and natural that it has to be unique is wrong.
Most working mathematicians have ignored this, just as they have ignored the apparent mysticism of the Platonist view, probably because the interesting questions in math are those that can either be proved or disproved. When you are doing math, it feels as though what you are working on is real. You can almost pick things up and turn them around, squash them and stroke them and pull them to pieces. On the other hand, you often make progress by forgetting what it all means and focusing solely on how the symbols dance. So the working philosophy of most mathematicians is a mostly unexamined Platonist–Formalist hybrid.
That’s fine if all you want is to do mathematics. As Hersh says, “Mathematics comes first, then philosophizing about it, not the other way round.” But if, like Hersh, you still wonder whether there might be a better way to describe that working philosophy, it all comes back to that same basic question of what mathematics is.
Hersh’s answer is what he calls the humanist philosophy. Mathematics is “A human activity, a social phenomenon, part of human culture, historically evolved, and intelligible only in a social context.” This is a description, not a definition, since it does not specify the content of that activity. The description may sound a bit postmodern, but it is made more intelligent than postmodernism by Hersh’s awareness that the social conventions that govern the activities of those human minds are subject to stringent nonsocial constraints, namely, that everything must fit together logically. Even if mathematicians got together and agreed thatπ equals 3, it wouldn’t. Nothing would make sense.
A mathematical circle, then, is something more than a shared delusion. It is a concept endowed with extremely specific features; it “exists” in the sense that human minds can deduce other properties from those features, with the crucial caveat that if two minds investigate the same question, they cannot, by correct reasoning, come up with contradictory answers.
That’s why it feels as if math is “out there.” Finding the answer to an open question feels like discovery, not invention. Math is a product of human minds but not bendable to human will. Exploring it is like exploring a new tract of country; you may not know what is around the next bend in the river, but you don’t get to choose. You can only wait and find out. But the mathematical countryside does not come into existence until you explore it.
When two members of the Arts Faculty argue, they may find it impossible to reach a resolution. When two mathematicians argue—and they do, often in a highly emotional and aggressive way—suddenly one will stop, and say, “I’m sorry, you’re quite right, now I see my mistake.” And they will go off and have lunch together, the best of friends.
I agree with Hersh, pretty much. I
f you feel that the humanist description of math is a bit woolly, that this type of “shared social construct” is a rarity, Hersh offers some examples that might change your mind. One is money. The entire world runs on money, but what is it? It is not pieces of paper or disks of metal; those can be printed or minted anew, or handed into a bank and destroyed. It is not numbers in a computer: if the computer blew up, you would still be entitled to your money. Money is a shared social construct. It has value because we all agree it has value.
Again, there are strong constraints. If you tell your bank manager that your account contains more than his computer says it does, he does not respond, “No problem, it’s just a social construct, here’s an extra ten million dollars. Have a nice day.”
It is tempting to think that even if we consider math to be a shared social construct, it has a kind of logical inevitability, that any intelligent mind would come up with the same math. When the Pioneer and Voyager spacecraft were sent off into space, they carried coded messages from humanity to any alien race that might one day encounter them. Pioneer bore a plaque with a diagram of the hydrogen atom, a map of nearby pulsars to show where our sun is located, line drawings of a naked man and woman standing in front of a sketch of the spacecraft, for scale, and a schematic picture of the solar system to show which planet we inhabit. The two Voyager craft carried records with sounds, music, and scientific images.
Would an alien recipient be able to decode those messages? Would a picture like o–o, two circles joined by a line, really look like the hydrogen atom to them? What if their version of atomic theory relied on quantum wave functions instead of primitive “particle” images, which even our own physicists tell us are wildly inaccurate? Would the aliens understand line drawings, given that humans from tribes that have never encountered such things fail to do so? Would they consider pulsars significant?
In most discussions about such questions, one eventually hears it argued that even if they grasped nothing else, any intelligent alien would be able to comprehend simple mathematical patterns, and the rest can be built from there. The unstated assumption is that math is somehow universal. Aliens would count 1, 2, 3, . . . just as we do. They would surely see the implied pattern in diagrams like * ** *** **** .
I’m not convinced. I’ve been reading Diamond Dogs, by Alistair Reynolds, a novella about an alien construct, a bizarre and terrifying tower, through whose rooms you progress by solving puzzles. If you get the answer wrong, you die, horribly. Reynolds’s story is powerful, but there is an underlying assumption that aliens would set mathematical puzzles akin to those that a human would set. Indeed, the alien math is too close to human; it includes topology and an area of mathematical physics known as Kaluza–Klein theory. You are as likely to arrive on the fifth planet of Proxima Centauri and find a Wal-Mart. I know that narrative constraints demand that the math should look like math to the reader, but even so, it doesn’t work for me.
I think human math is more closely linked to our particular physiology, experiences, and psychological preferences than we imagine. It is parochial, not universal. Geometry’s points and lines may seem the natural basis for a theory of shape, but they are also the features into which our visual system happens to dissect the world. An alien visual system might find light and shade primary, or motion and stasis, or frequency of vibration. An alien brain might find smell, or embarrassment, but not shape, to be fundamental to its perception of the world. And while discrete numbers like 1, 2, 3, seem universal to us, they trace back to our tendency to assemble similar things, such as sheep, and consider them property: has one of my sheep been stolen? Arithmetic seems to have originated through two things: the timing of the seasons and commerce. But what of the blimp creatures of distant Poseidon, a hypothetical gas giant like Jupiter, whose world is a constant flux of turbulent winds, and who have no sense of individual ownership? Before they could count up to three, whatever they were counting would have blown away on the ammonia breeze. They would, however, have a far better understanding than we do of the math of turbulent fluid flow.
I think it is still credible that where blimp math and ours made contact, they would be logically consistent with each other. They could be distant regions of the same landscape. But even that might depend on which type of logic you use.
The belief that there is one mathematics—ours—is a Platonist belief. It’s possible that “the” ideal forms are “out there,” but also that “out there” might comprise more than one abstract realm, and that ideal forms need not be unique. Hersh’s humanism becomes Poseidonian blimpism: their math would be a social construct shared by their society. If they had a society. If they didn’t—if different blimps did not communicate—could they have any conception of mathematics at all? Just as we can’t imagine a mathematics not founded on the counting numbers, we can’t imagine an “intelligent” species whose members don’t communicate with each other. But the fact that we can’t imagine something is no proof that it doesn’t exist.
But I am drifting off the topic. What is mathematics? In despair, some have proposed the definition “Mathematics is what mathematicians do.” And what are mathematicians? “People who do mathematics.” This argument is almost Platonic in its perfect circularity. But let me ask a similar question. What is a businessman? Someone who does business? Not quite. It is someone who sees opportunities for doing business when others might miss them.
A mathematician is someone who sees opportunities for doing mathematics.
I’m pretty sure that’s right, and it pins down an important difference between mathematicians and everyone else. What is mathematics? It is the shared social construct created by people who are aware of certain opportunities, and we call those people mathematicians. The logic is still slightly circular, but mathematicians can always recognize a fellow spirit. Find out what that fellow spirit does; it will be one more aspect of our shared social construct.
Welcome to the club.
4
Hasn’t It All Been Done ?
Dear Meg,
In your last letter you asked me about the extent to which mathematics at university can go beyond what you have already done at school. No one wants to spend three or four years going over the same ideas, even if they are studied in greater depth. Now, looking ahead, you are also right to worry about the scope that exists for creating new mathematics. If others have already explored such a huge territory, how can you ever find your way to the frontier? Is there even any frontier left?
For once, my task is simple. I can set you at ease on both counts. If anything, you should worry about the exact opposite: that people are creating too much new mathematics, and that the scope for new research is so gigantic that it will be difficult to decide where to start or in which direction to proceed. Math is not a robotic way of replacing thought by rigid ritual. It is the most creative activity on the planet.
These statements will be news to many people, possibly including some of your teachers. It always astonishes me that so many people seem to believe that mathematics is limited to what they were taught at school, so that basically “it’s all been done.” Even more astonishing is the assumption that because “the answers are all in the back of the book,” there is no scope for creativity, and no questions remain unanswered. Why do so many people think that their school textbook contains every possible question?
This failure of imagination would amount to deplorable ignorance, were it not for two factors that together go a long way to explain it.
The first is that many students quickly come to dislike mathematics as they pass through the school system. They find it rigid, boring, repetitive, and, worst of all, difficult. Answers are either right or wrong, and no amount of clever verbal jousting with the teacher can convert a wrong answer into a correct one. Mathematics is a very unforgiving subject. Having developed this negative attitude, the last thing the student wants to hear is that there is more mathematics, going beyond the already daunting contents of the set text. Most p
eople want all the answers to be at the back of the book, because otherwise they can’t look them up.
Dame Kathleen Ollerenshaw, one of Britain’s most distinguished mathematicians and educators, who continues to do research at the age of ninety, makes exactly this point in her autobiography To Talk of Many Things. (Do read it, Meg; it’s inspirational, and very wise.) “When I told a teenage friend that I was doing mathematical research, her reply was, ‘Why do that? We have enough mathematics to cope with already—we don’t want any more.’”
The assumptions behind that statement bear examination, but I content myself with just one. Why did Kathleen’s friend assume that any newly invented mathematics would automatically appear in school texts? Again we encounter the same belief, that the math you are taught at school is the entire universe of mathematics. But no one thinks that about physics, or chemistry, or biology, or even French or economics. We all know that what we are taught at school is just a tiny part of what is currently known.
I sometimes wish schools would go back to using words like “arithmetic” to describe the content of “math” courses. Calling them “mathematics” debases the currency of mathematical thought; it’s a bit like using the term “composing” to describe routine exercises in playing musical scales. However, I lack the power to change the name, and if in fact the name were changed, the main effect would be to decrease public recognition of mathematics. For most people, the only time they are aware of their life and mathematics intersecting is at school.
As I wrote in my first letter, this does not imply that mathematics has no relevance to our daily lives. But the profound influence of our subject on human existence takes place behind the scenes and therefore passes unnoticed.