- P R Halmos, The Foundations of Probability, Amer. Math. Monthly 51 (9) (1944), 493-510.
Probability is a branch of mathematics. It is not a branch of experimental science nor of armchair philosophy, it is neither physics nor logic. This is not to say that the experimenter and the philosopher should not discuss probability from their points of view. They should, and they do. The situation is analogous to that in geometry. No one denies that the physicist and the philosopher have made valuable contributions to our understanding of the space concept, nor, in spite of this, that geometry is a rigorous part of modern mathematics.
- P R Halmos, The Basic Concepts of Algebraic Logic, Amer. Math. Monthly 63 (6) (1956), 363-387.
It has often happened that a theory designed originally as a tool for the study of a physical problem came subsequently to have purely mathematical interest. When that happens, the theory is usually generalized way beyond the point needed for applications, the generalizations make contact with other theories (frequently in completely unexpected directions), and the subject becomes established as a new part of pure mathematics. The part of pure mathematics so created does not (and need not) pretend to solve the physical problem from which it arises; it must stand or fall on its own merits. Physics is not the only external source of mathematical theories; other disciplines (such as economics and biology) can play a similar role. A recent (and possibly somewhat surprising) addition to the collection of mathematical catalysts is formal logic; the branch of pure mathematics that it has precipitated will here be called algebraic logic.
- P Halmos, Mathematics as a creative art, American Scientist 56 (1968), 375-389.
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 in roofing and siding. If I tell him that I am a mathematician, his most likely reply will be that he himself could never balance his check book, and it must be fun to be a whiz at math. 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.
- P Halmos, How to write mathematics, Enseign. Math. (2) 16 (1970),
I think I can tell someone how to write, but I can't think who would want to listen. The ability to communicate effectively, the power to be intelligible, is congenital, I believe, or, in any event, it is so early acquired that by the time someone reads my wisdom on the subject he is likely to be invariant under it. To understand a syllogism is not something you can learn; you are either born with the ability or you are not. In the same way, effective exposition is not a teachable art; some can do it and some cannot. There is no usable recipe for good writing. Then why go on? A small reason is the hope that what I said isn't quite right; and, anyway, I'd like a chance to try to do what perhaps cannot be done. A more practical reason is that in the other arts that require innate talent, even the gifted ones who are born with it are not usually born with full knowledge of all the tricks of the trade. A few essays such as this may serve to "remind" (in the sense of Plato) the ones who want to be and are destined to be the expositors of the future of the techniques found useful by the expositors of the past. The basic problem in writing mathematics is the same as in writing biology, writing a novel, or writing directions for assembling a harpsichord: the problem is to communicate an idea. To do so, and to do it clearly, you must have something to say, and you must have someone to say it to, you must organize what you want to say, and you must arrange it in the order you want it said in, you must write it, rewrite it, and re-rewrite it several times, and you must be willing to think hard about and work hard on mechanical details such as diction, notation, and punctuation. That's all there is to it. ...
- P R Halmos, The Legend of John Von Neumann, Amer. Math. Monthly 80 (4) (1973), 382-394.
The heroes of humanity are of two kinds: the ones who are just like all of us, but very much more so, and the ones who, apparently, have an extra-human spark. We can all run, and some of us can run the mile in less than 4 minutes; but there is nothing that most of us can do that compares with the creation of the Great G-minor Fugue. Von Neumann's greatness was the human kind. We can all think clearly, more or less, some of the time, but von Neumann's clarity of thought was orders of magnitude greater than that of most of us, all the time. Both Norbert Wiener and John von Neumann were great men, and their names will live after them, but for different reasons. Wiener saw things deeply but intuitively; von Neumann saw things clearly and logically. What made von Neumann great? Was it the extraordinary rapidity with which he could understand and think and the unusual memory that retained everything he could understand and think and the unusual memory that retained everything he had once thought through? No. These qualities, however impressive they might have been, are ephemeral; they will have no more effect on the mathematics and the mathematicians of the future than the prowess of an athlete of a hundred years ago has on the sport of today. The "axiomatic method" is sometimes mentioned as the secret of von Neumann's success. In his hands it was not pedantry but perception; he got to the root of the matter by concentrating on the basic properties (axioms) from which all else follows. The method, at the same time, revealed to him the steps to follow to get from the foundations to the applications. He knew his own strengths and he admired, perhaps envied, people who had the complementary qualities, the flashes of irrational intuition that sometimes change the direction of scientific progress. For von Neumann it seemed to be impossible to be unclear in thought or in expression. His insights were illuminating and his statements were precise.
- P Halmos, How to talk mathematics, Notices Amer. Math. Soc. 21 (1974), 155-158.
What is the purpose of a public lecture? Answer: to attract and to inform. We like what we do, and we should like for others to like it too; and we believe that the subject's intrinsic qualities are good enough so that anyone who knows what they are cannot help being attracted to them. Hence, better answer: the purpose of a public lecture is to inform, but to do so in a manner that makes it possible for the audience to absorb the information. An attractive presentation with no content is worthless, to be sure, but a lump of indigestible information is worth no more. ... Less is more, said the great architect Mies van der Rohe, and if all lecturers remember that adage, all audiences would be both wiser and happier. Have you ever disliked a lecture because it was too elementary? I am sure that there are people who would answer yes to that question, but not many. Every time I have asked the question, the person who answered said no, and then looked a little surprised at hearing the answer. A public lecture should be simple and elementary; it should not be complicated and technical. If you believe and can act on this injunction ("be simple"), you can stop reading here; the rest of what I have to say is, in comparison, just a matter of minor detail.
- P Halmos, The problem of learning to teach, Amer. Math. Monthly 82 (1975), 466-476.
The best way to learn is to do; the worst way to teach is to talk. About the latter: did you ever notice that some of the best teachers of the world are the worst lecturers? (I can prove that, but I'd rather not lose quite so many friends.) And, the other way around, did you ever notice that good lecturers are not necessarily good teachers? A good lecture is usually systematic, complete, precise - and dull; it is a bad teaching instrument. When given by such legendary outstanding speakers as Emil Artin and John von Neumann, even a lecture can be a useful tool - their charisma and enthusiasm come through enough to inspire the listener to go forth and do something - it looks like such fun. For most ordinary mortals, however, who are not so bad at lecturing as Wiener was - nor so stimulating! - and not so good as Artin - and not so dramatic! - the lecture is an instrument of last resort for good teaching. My test for what makes a good teacher is very simple: it is the pragmatic one of judging the performance by the product. If a teacher of graduate students consistently produces Ph.D.'s who are mathematicians and who create high-quality new mathematics, he is a good teacher. If a teacher of calculus consistently produces seniors who turn into outstanding graduate students of mathematics, or into leading engineers, biologists, or economists, he is a good teacher. If a teacher of third-grade "new math" (or old) consistently produces outstanding calculus students, or grocery store check-out clerks, or carpenters, or automobile mechanics, he is a good teacher.
- P Halmos, Four panel talks on publishing, Amer. Math. Monthly 82 (1975), 14-17.
Let me remind you that most laws (with the exception only of the regulatory statutes that govern traffic and taxes) are negative. Consider, as an example, the Ten Commandments. When Moses came back from Mount Sinai, he told us what to be by telling us, eight out of ten times, what not to do. It may therefore be considered appropriate to say what not to publish. I warn you in advance that all the principles that I was able to distill from interviews and from introspection, and that I'll now tell you about, are a little false. Counterexamples can be found to each one - but as directional guides the principles still serve a useful purpose. First, then, do not publish fruitless speculations: do not publish polemics and diatribes against a friend's error. Do not publish the detailed working out of a known principle. (Gauss discovered exactly which regular polygons are ruler-and-compass constructible, and he proved, in particular, that the one with 65537 sides - a Fermat prime - is constructible; please do not publish the details of the procedure. It's been tried.) Do not publish in 1975 the case of dimension 2 of an interesting conjecture in algebraic geometry, one that you don't know how to settle in general, and then follow it by dimension 3 in 1976, dimension 4 in 1977, and so on, with dimension k _ 3 in 197k. Do not, more generally, publish your failures: I tried to prove so-and-so; I couldn't; here it is - see?!
- P R Halmos, Progress Reports, Amer. Math. Monthly 84 (9) (1977), 714.
Paul Halmos was the editor of 'Progress Reports' in the American Mathematical Monthly. Here is his description of what he was seeking to publish:
It is easy to be too busy to pay attention to what anyone else is doing, but not good. All of us should know, and want to know, what has been discovered since our formal education ended, but new words, and relations between them, are growing too fast to keep up. It is possible for a person to learn of the title of a recent work and of the key words used in it and still not have the faintest idea of what the subject is. Progress Reports is to be an almost periodic column intended to increase everyone's mathematical information about what others have been up to. Each column will report one step forward in the mathematics of our time. The purpose is to inform, more than to instruct: what is the name of the subject, what are some of the words it uses, what is a typical question, what is the answer, who found it. The emphasis will be on concrete questions and answers (theorems), and not on general contexts and techniques (theories). References will be kept minimal: usually they will include only one of the earliest papers in which the answer appears and a more recent exposition of the discovery, whenever one is easily available. Everyone is invited to nominate subjects to be reported on and authors to prepare the reports. The ground rules are that the principal theorem should be old enough to have been published in the usual sense of that word (and not just circulated by word of mouth or in preprints); it should be of interest to more than just a few specialists; and it should be new enough to have an effect on the mathematical life of the present and near future. In practice most reports will probably be on progress achieved somewhere between 5 and 15 years ago.
- P R Halmos, Logic from A to G, Mathematics Magazine 50 (1) (1977), 5-11.
Originally "logic" meant the same as "the laws of thought" and logicians studied the subject in the hope that they could discover better ways of thinking and surer ways of avoiding error than their forefathers knew, and in the hope that they could teach these arts to all mankind. Experience has shown, however, that this is a wild-goose chase. A normal healthy human being has built in him all the "laws of thought" anybody has ever invented, and there is nothing that logicians can teach him about thinking and avoiding error. This is not to say that he knows how he thinks and it is not to say that he never makes errors. The situation is analogous to the walking equipment all normal healthy human beings are born with. I don't know how I walk, but I do it. Sometimes I stumble. The laws of walking might be of interest to physiologists and physicists; all I want to do is to keep on walking. The subject of mathematical logic, which is the subject of this paper, makes no pretence about discovering and teaching the laws of thought. It is called mathematical logic for two reasons. One reason is that it is concerned with the kind of activity that mathematicians engage in when they prove things. Mathematical logic studies the nature of a proof and tries to forecast in a general way all possible types of things that mathematicians ever will prove, and all that they never can. Another reason for calling the subject mathematical logic is that it itself is a part of mathematics. It attacks its subject in a mathematical way and proves things exactly the same way as do the other parts of mathematics whose methods it is concerned with.
- P R Halmos, The Heart of Mathematics, Amer. Math. Monthly 87 (7) (1980), 519-524.
What does mathematics really consist of? Axioms (such as the parallel postulate)? Theorems (such as the fundamental theorem of algebra)? Proofs (such as Godel's proof of undecidability)? Concepts (such as sets and classes)? Definitions (such as the Menger definition of dimension)? Theories (such as category theory)? Formulas (such as Cauchy's integral formula)? Methods (such as the method of successive approximations)? Mathematics could surely not exist without these ingredients; they are all essential. It is nevertheless a tenable point of view that none of them is at the heart of the subject, that the mathematician's main reason for existence is to solve problems, and that, therefore, what mathematics really consists of is problems and solutions. "Theorem" is a respected word in the vocabulary of most mathematicians, but "problem" is not always so. "Problems," as the professionals sometimes use the word, are lowly exercises that are assigned to students who will later learn how to prove theorems. These emotional overtones are, however, not always the right ones. The commutativity of addition for natural numbers and the solvability of polynomial equations over the complex field are both theorems, but one of them is regarded as trivial (near the basic definitions, easy to understand, easy to prove), and the other as deep (the statement is not obvious, the proof comes via seemingly distant concepts, the result has many surprising applications). To find an unbeatable strategy for tic-tac-toe and to locate all the zeroes of the Riemann zeta function are both problems, but one of them is trivial (anybody who can understand the definitions can find the answer quickly, with almost no intellectual effort and no feeling of accomplishment, and the answer has no consequences of interest), and the other is deep (no one has found the answer although many have sought it, the known partial solutions require great effort and provide great insight, and an affirmative answer would imply many non-trivial corollaries). Moral: theorems can be trivial and problems can be profound. Those who believe that the heart of mathematics consists of problems are not necessarily wrong.
- P R Halmos, The Thrills of Abstraction, The Two-Year College Mathematics Journal 13 (4) (1982), 243-251.
The concept of number is, at least in mathematical circles, very well known. We all use words such as "five" every day, but do many people ask themselves what "five" is? And, by the way, shouldn't they be ashamed of themselves? We wouldn't use words such as "grandfather", or "tax", or "lawnmower" without being able to define them - without, to be specific, being able to tell a ten-year old child exactly what a grandfather, or a tax, or a lawnmower is, but the challenge is to tell him exactly what a number is. I don't mean what a number does, or how a number can be used - I mean what it is. All right: what is "5"? We may not know that, but we know that if it's the answer to "How many fingers are there on your right hand", then it's also the answer to "How many players are there on a basketball team?" In other words, while we may not know what "number" is, we do know when two sets of objects (be they fingers, or whatever) are "equinumerous". They are that just when we can establish a correspondence between them (for example, by pointing to each basketball player on the team with a different finger) that is a one-to-one correspondence - each object in each set corresponds to a unique object in the other set. What then is a number? Or, better asked, what is the number of objects in a set? Answer: the collection of all sets equinumerous with it.
- P R Halmos, Pure Thought Is Better Yet ..., The College Mathematics Journal 16 (1) (1985), 14-16.
"The algorithmic way of life" is, we are told, the right way. Since I disagree, I fear I'll be viewed as being against motherhood, on the side of the devil, and at war with the angels, but I'd like at least to try to explain what I mean before I am boiled in oil. Which is necessary for salvation: good works or faith? What makes a house: ten-penny nails or blueprints? Is an opera the libretto or the music? Is an essay an exercise in syntax or an exposition of a subject? Is real physics done in a laboratory or on paper? The answer is the same in every case (namely, both), and even to try to decide which component is more important is not much more meaningful than to decide whether for walking you need your right foot more or your left. Does real mathematics consist of algorithms or abstractions, and when they are both present, which is more important? The answer is that every mathematician must be both an effective calculator and an abstract thinker, and the relative importance of the two kinds of activities depends on the task at hand.
- P R Halmos, Has Progress in Mathematics Slowed Down?, Amer. Math. Monthly 97 (7) (1990), 561-588.
Do we know anything that Dedekind didn't know? We should. Dedekind died in February, 1916. Six weeks before that, late Friday afternoon on New Year's Eve 1915, the MAA was born in Columbus, Ohio. In connection with the celebration of its diamond anniversary, in August 1990, in Columbus, Ohio, it became my mission to report on whether and how mathematics has changed during the 75 years of the Mathematical Association of America's existence, and what follows is an attempt at such a report. I am not trying to teach any mathematics in this report, nor even any history of mathematics - all I am trying to do is share an interesting look at the growth of mathematics in the last 75 years. Everybody could find out everything I found out by spending a few months looking at all extant volumes of Mathematical Reviews and a few dozen other journals, but everybody hasn't done it, and I have, and I am ready to tell what I have found. The question that I set out to answer might be phrased this way: if you had a time machine to take you back to Dedekind, what could you teach him about progress in mathematics since his day? In an attempt to organize the possible answers, I propose to put them into three classes: concepts, explosions, and developments.
- P R Halmos, What is Teaching?, Amer. Math. Monthly 101 (9) (1994), 848-854.
There are three types of knowledge that we commonly speak of as subjects for teaching or learning; they can be most effectively identified as what, how, and why. To be educated means to remember something, to be able to use it, and to understand it. Frequently these three kinds of education are thought of as belonging to altogether different kinds of human activity, but ideally they are all present every time. ... Many students confuse education with memorization. They tend to think that if we know the boiling point of beer, the gestation period of elephants, the conjugation of French irregular verbs, and the population of Burma, together with many other such goodies about the moon, whales, protons, synapses, schizophrenia, and interest rates, then we are educated. A walking encyclopaedia is, however, rarely an educated person. To a historian, history is not just a collection of facts but an organized understanding of how we got to be what we are; Waterloo is not just a fact, but, possibly, a tool to be used to avoid catastrophes in the future. To a chemist, chemistry is not just purple liquids in test tubes, but a scheme for prediction and a way of understanding the world and the same sort of thing is true of the physicist, the astronomer, the psychologist, and the economist. ... How do we teach logic and mathematics, how do we teach abstract concepts and the relations among them, how do we teach intuition, recognition, understanding? How do we teach these things so that when we are done our ex-student can not only pass an examination by naming the concepts and listing the relations, but he can also get pleasure from his insight, and, if he is talented and lucky, be vouchsafed the discovery of a new one? The only possible answer that I can see is: nohow. Don't do nuttin'; just wait. The only way I know of for an individual to share in humanity's slowly acquired understanding is to retrace the steps. Some old ideas were in error, of course, and some might have become irrelevant to the world of today, and therefore no longer fashionable, but on balance every student must repeat all the steps ontogeny must recapitulate philogeny every time.