Paul Halmos: Mathematics as a Creative Art
By P R Halmos, Department of Mathematics, Indiana University, Bloomington.
Do you know any mathematicians - and, if you do, do you know anything about what they do with their time? Most people don't. When I get into conversation with the man next to me in a plane, and he tells me that he is something respectable like a doctor, lawyer, merchant, or dean, I am tempted to say that I am an odd job man. If I tell him that I am a mathematician, his most likely reply will be that he himself could never balance his chequebook, and it must be fun to be an expert at mathematics. If my neighbour is an astronomer, a biologist, a chemist, or any other kind of natural or social scientist, I am, if anything, worse off - this man thinks he knows what a mathematician is, and he is probably wrong. He thinks that I spend my time (or should) converting different orders of magnitude, comparing binomial coefficients and powers of 2, or solving equations involving rates of reactions.
C P Snow points to and deplores the existence of two cultures; he worries about the physicist whose idea of modern literature is Dickens, and he chides the poet who cannot state the second law of thermodynamics. Mathematicians, in converse with well-meaning, intelligent, and educated laymen (do you mind if I refer to all non-mathematicians as laymen?) are much worse off than physicists in converse with poets. It saddens me that educated people don't even know that my subject exists. There is something that they call mathematics, but they neither know how the professionals use that word, nor can they conceive why anybody should do it. It is, to be sure, possible that an intelligent and otherwise educated person doesn't know that egyptology exists, or haematology, but all you have to tell him is that it does, and he will immediately understand in a rough general way why it should and he will have some empathy with the scholar of the subject who finds it interesting.
Usually when a mathematician lectures, he is a missionary. Whether he is talking over a cup of coffee with a collaborator, lecturing to a graduate class of specialists, teaching a reluctant group of freshman engineers, or addressing a general audience of laymen - he is still preaching and seeking to make converts. He will state theorems and he will discuss proofs and he will hope that when he is done his audience will know more mathematics than they did before. My aim today is different - I am not here to proselyte but to enlighten - I seek not converts but friends. I do not want to teach you what mathematics is, but only that it is.
I call my subject mathematics - that's what all my colleagues call it, all over the world-and there, quite possibly, is the beginning of confusion. The word covers two disciplines - many more, in reality, but two, at least two, in the same sense in which Snow speaks of two cultures. In order to have some words with which to refer to the ideas I want to discuss, I offer two temporary and ad hoc neologisms. Mathematics, as the word is customarily used, consists of at least two distinct subjects, and I propose to call them mathology and mathophysics. Roughly speaking, mathology is what is usually called pure mathematics, and mathophysics is called applied mathematics, but the qualifiers are not emotionally strong enough to disguise that they qualify the same noun. If the concatenation of syllables I choose here reminds you of other words, no great harm will be done; the rhymes alluded to are not completely accidental. I originally planned to entitle this lecture something like "Mathematics is an art," or 'Mathematics is not a science', or 'Mathematics is useless', but the more I thought about it the more I realised that I mean that 'Mathology is an art', 'Mathology is not a science', and 'Mathology is useless'. When I am finished, I hope you will recognise, that most of you have known about mathophysics before, only you were probably calling it mathematics; I hope that all of you will recognise the distinction between mathology and mathophysics; and I hope that some of you will be ready to embrace, or at least applaud, or at the very least, recognise mathology as a respectable human endeavour.
In the course of the lecture I'll have to use many analogies (literature, chess, painting), each imperfect by itself, but I hope that in their totality they will serve to delineate, what I want delineated. Sometimes in the interest of economy of time, and sometimes doubtless unintentionally, I'll exaggerate; when I'm done, I'll be glad to rescind anything that was inaccurate or that gave offence in any other way.
What Mathematicians Do
As a first step toward telling you what mathematicians do, let me tell you some of the things they do not do. To begin with, mathematicians have very little to do with numbers. You can no more expect a mathematician to be able to add a column of figures rapidly and correctly than you can expect a painter to draw a straight line or a surgeon to carve a turkey-popular legend attributes such skills to these professions, but popular legend is wrong. There is, to be sure, a part of mathematics called number theory, but even that doesn't deal with numbers in the legendary sense - a number theorist and an adding machine would find very little to talk about. A machine might enjoy proving that 13 + 53 + 33 = 153, and it might even go on to discover that there are only five positive integers with the property that the equation indicates (1, 370, 371, 407), but most mathematicians couldn't care less; many mathematicians enjoy and respect the theorem that every positive integer is the sum of not more than four squares, whereas the infinity involved in the word 'every' would frighten and paralyse any ordinary office machine, and, in any case, that's probably not the sort of thing, that the person who relegates mathematicians to numbers had in mind.
Not even those romantic objects of latter day science fiction, the giant brains, the computing machines that run our lives these days-not even they are of interest to the mathematician as such. Some mathematicians are interested in the logical problems involved in the reduction of difficult questions to the sort of moronic baby talk that machines understand: the logical design of computing machines is definitely mathematics. Their construction is not, that's engineering, and their product, be it a payroll, a batch of sorted mail, or a supersonic plane, is of no mathematical interest or value.
Mathematics is not numbers or machines; it is also not the determination of the. heights of mountains by trigonometry, or compound interest by algebra, or moments. of inertia by calculus. Not today it isn't. At one point in history each of those things, and others like them, might have been an important and non-trivial research problem, but once the problem is solved its repetitive application has as much to do with mathematics as the work of a Post Office telegraph boy has to do with Marconi's genius.
There are at least two other things that mathematics isn't; one of them is something it never was, and the other is something it once included and by now has sloughed off. The first is physics. Some laymen confuse mathematics and theoretical physics and speak, for instance, of Einstein as a great mathematician. There is no doubt that Einstein was a great man, but he was no more a great mathematician than he was a great violinist. He used mathematics to find out facts about the universe, and that he successfully used certain parts of differential geometry for that purpose adds a certain piquancy to the appeal of differential geometry. Withal, relativity theory and differential geometry are not the same thing. Einstein, Schrödinger, Heisenberg, Fermi, Wigner, Feynman - great men all, but not mathematicians; some of them, in fact, strongly anti-mathematical, preach against mathematics, and would regard it as an insult to be called a mathematician.
What once was mathematics remains mathematics always, but it can become so thoroughly worked out, so completely understood, and, in the light of millennia of contributions, with hindsight, so trivial, that mathematicians never again need to or want to spend time on it. The celebrated Greek problems (trisect the angle, square the circle, duplicate the cube) are of this kind, and the irrepressible mathematical amateur to the contrary notwithstanding, mathematicians are no longer trying to solve them. Please understand it isn't that they have given up. Perhaps you have heard that, according to mathematicians, it is impossible to square a circle, or trisect an angle, and perhaps you have heard or read that, therefore, mathematicians are a pusillanimous chicken-hearted lot, who give up easily, and use their ex-cathedra pronouncements to justify their ignorance. The conclusion may be true, and you may believe it if you like, but the proof is inadequate. The point is a small one but a famous one and one of historical interest: let me digress to discuss it for a moment.
A Short Digression
The problem of trisecting the angle is this: given an angle, construct another one that is just one third as large. The problem is perfectly easy, and several methods for solving it are known. The catch is that the original Greek formulation of the problem is more stringent: it requires a construction that uses ruler and compasses only. Even that can be done, and I could show you a perfectly simple method in one minute and convince you that it works in two more minutes. The real difficulty is that the precise formulation of the problem is more stringent still. The precise formulation demands a construction that uses a ruler and compasses only and, moreover, severely restricts how they are to be used; it prohibits, for instance, marking two points on the ruler and using the marked points in further constructions. It takes some careful legalism (or some moderately pedantic mathematics) to formulate really precisely just what was and what wasn't allowed by the Greek rules. The modern angle trisector either doesn't know those rules, or he knows them but thinks that the idea is to get a close approximation, or he knows the rules and knows that an exact solution is required but lets wish be father to the deed and simply makes a mistake. Frequently his attitude is that of the visitor from outer space to golf. (If all you want is to get that little white ball in that little green hole, why don't you just go and put it there?)
Allow me to add a short digression to the digression. I'd like to remind you that when a mathematician says that something is impossible, he doesn't mean that it is very very difficult, beyond his powers, and probably beyond the powers of all humanity for the foreseeable future. That's what is often meant when someone says it's impossible to travel at the speed of sound five miles above the surface of the earth, or instantaneously to communicate with someone a thousand miles away, or to tamper with the genetic code so as to produce a race of citizens who are simultaneously intelligent and peace-loving. That's what is belittled by the classic business braggadocio (the impossible takes a little longer). The mathematical impossible is different: it is more modest and more secure. The mathematical impossible is the logical impossible. When the mathematician says that it is impossible to find a positive number whose sum with 10 is less than 10, he merely reminds us that that's what the words mean (positive, sum, 10, less); when he says that it is impossible to trisect every angle by ruler and compasses, he means exactly the same sort of thing, only the number of technical words involved is large enough and the argument that strings them together is long enough that they fill a book, not just a line.
The Start of Mathematics
No one knows when and where mathematics got started, or how, but it seems reasonable to guess that it emerged from the same primitive physical observations. (counting, measuring) with which we all begin our own mathematical insight (ontogeny recapitulates phylogeny). It was probably so in the beginning, and it is true still, that many mathematical ideas originate not from pure thought but from material necessity; many, but probably not all. Almost as soon as a human being finds it necessary to count his sheep (or sooner?) he begins to wonder about numbers and shapes and motions and arrangements-curiosity about such things seems to be as necessary to the human spirit as curiosity about earth, water, fire, and air, and curiosity - sheer pure intellectual curiosity - about stars and about life. Numbers and shapes and motions and arrangements, and also thoughts and their order, and concepts such as 'property' and 'relation' - all such things are the raw material of mathematics. The technical but basic mathematical concept of 'group' is the best humanity can do to understand the intuitive concept of 'symmetry' and the people who study topological spaces, and ergodic paths, and oriented graphs are making precise our crude and vague feelings about shapes, and motions, and arrangements.
Why do mathematicians study such things, and why should they? What, in other words, motivates the individual mathematician, and why does society encourage his efforts, at least to the extent of providing him with the training and subsequently the livelihood that, in turn, give him the time he needs to think? There are two answers to each of the two questions: because mathematics is practical and because mathematics is an art. The already existing mathematics has more and more new applications each day, and the rapid growth of desired applications suggests more and more new practical mathematics. At the same time, as the quantity of mathematics grows and the number of people who think about it keeps doubling over and over again, more new concepts need explication, more new logical interrelations cry out for study, and understanding, and simplification, and more and more the tree of mathematics bears elaborate and gaudy flowers that are, to many beholders, worth more than the roots from which it all comes and the causes that brought it all into existence.
Mathematics is very much alive today. There are more than a thousand journals that publish mathematical articles; about 15,000 to 20,000 mathematical articles are printed every year. The mathematical achievements of the last hundred years are greater in quantity and in quality than those of all previous history. Difficult mathematical problems, which stumped Hilbert, Cantor, or Poincaré, are being solved, explained, and generalized by beardless (and bearded) youths in Berkeley and in Odessa.
Mathematicians sometimes classify themselves and each other as either problem-solvers or theory-creators. The problem-solvers answer yes-or-no questions and discuss the vital special cases and concrete examples that are the flesh and blood of mathematics; the theory creators fit the results into a framework, illuminate it all, and point it in a definite direction - they provide the skeleton and the soul of mathematics. One and the same human being can be both a problem-solver and a theory-creator, but, usually, he is mainly one or the other. The problem-solvers make geometric constructions, the theory-creators discuss the foundations of Euclidean geometry; the problem-solvers find out what makes switching diagrams tick, the theory-creators prove representation theorems for Boolean algebras. In both kinds of mathematics and in all fields of mathematics the progress in one generation is breath-taking. No one can call himself a mathematician nowadays who doesn't have at least a vague idea of homological algebra, differential topology, and functional analysis, and every mathematician is probably somewhat of an expert on at least one of these subjects and yet when I studied mathematics in the 1930's none of those phrases had been invented, and the subjects they describe existed in seminal forms only.
Mathematics is abstract thought, mathematics is pure logic, mathematics is creative art. All these statements are wrong, but they are all a little right, and they are all nearer the mark than 'mathematics is numbers' or 'mathematics is geometric shapes'. For the professional pure mathematician, mathematics is the logical dovetailing of a carefully selected sparse set of assumptions with their surprising conclusions via a conceptually elegant proof. Simplicity, intricacy, and above all, logical analysis are the hallmark of mathematics.
The mathematician is interested in extreme cases - in this respect he is like the industrial experimenter who breaks light-bulbs, tears shirts, and bounces cars on ruts. How widely does a reasoning apply, he wants to know, and what happens when it doesn't? What happens when you weaken one of the assumptions, or under what conditions can you strengthen one of the conclusions? It is the perpetual asking of such questions that makes for broader understanding, better technique, and greater elasticity for future problems.
Mathematics - this may surprise or even shock you - is never deductive in its creation. The mathematician at work makes vague guesses, visualises broad generalisations, and jumps to unwarranted conclusions. He arranges and rearranges his ideas, and he becomes convinced of their truth long before he can write down a logical proof. The conviction is not likely to come early-it usually comes after many attempts, many failures, many discouragements and many false starts. It often happens that months of work result in the proof that the method of attack they were based on cannot possibly work, and the process of guessing, visualising, and conclusion-jumping begins again. A reformulation is needed-and this too may surprise you-more experimental work is needed. To be sure, by 'experimental work' I do not mean test tubes and cyclotrons. I mean thought-experiments. When a mathematician wants to prove a theorem about an infinite-dimensional Hilbert space, he examines its finite-dimensional analogue, he looks in detail at the 2- and 3-dimensional cases, he often tries out a particular numerical case, and he hopes that he will gain thereby an insight that pure definition-juggling has not yielded. The deductive stage, writing the result down, and writing down its rigorous proof are relatively trivial once the real insight arrives; it is more like the draughtsman's work, not the architect's.
The Mathematical Fraternity
The mathematical fraternity is a little like a self-perpetuating priesthood. The mathematicians of today train the mathematicians of tomorrow and, in effect, decide whom to admit to the priesthood. Most people do not find it easy to join-mathematical talent and genius are apparently exactly as rare as talent and genius in painting and music - but anyone can join, everyone is welcome. The rules are nowhere explicitly formulated, but they are intuitively felt by everyone in the profession. Mistakes are forgiven and so is obscure exposition-the indispensable requisite is mathematical insight. Sloppy thinking, verbosity without content, and polemic have no role, and this to me is one of the most wonderful aspects of mathematics-they are much easier to spot than in the non-mathematical fields of human endeavour (much easier than, for instance, in literature among the arts, in art criticism among the humanities, and in your favourite abomination among the social sciences).
Although most of mathematical creation is done by one man at a desk, at a blackboard, or taking a walk, or, sometimes, by two men in conversation, mathematics is nevertheless a sociable science. The creator needs stimulation while he is creating and he needs an audience after he has created. Mathematics is a sociable science in the sense that I don't think it can be done by one man on a desert island (except for a very short time), but it is not a mob science, it is not a team science. A theorem is not a pyramid; inspiration has never been known to descend on a committee. A great theorem can no more be obtained by a 'project' approach than a great painting; I don't think a team of little Gausses could have obtained the theorem about regular polygons under the leadership of a rear admiral anymore than a team of little Shakespeares could have written Hamlet under such conditions.
JOC/EFR November 2007
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