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Among the universal attributes of homo sapiens, several have become established as special fields of study—language, art and music, religion, political economy. But mathematics, another universal attribute of our species, is still modeled separately by logicians, historians, neuroscientists, and others. Could it be integrated into “mathematics studies,” a coherent, many-faceted branch of empirical science? Could philosophers facilitate such a unification? Some philosophers of mathematics identify themselves with “positions” on the nature of mathematics. Those “positions” could more productively serve as models of mathematics.

This collection of fifteen articles on mathematics and/or narrative originated in a conversation between the two editors, during a vacation on a Greek island owned by one of them. The “circles dist...

This paper has four objectives: (1) to address some psychological sources that motivate creative mathematicians to do sustained research, (2) to use case studies and self-reports to identify cognitive and mathematical strategies, (3) to give inspiring examples of creative breakthroughs in the teaching of mathematics, (4) to report on a startling recent discovery in artificial intelligence, with thought-provoking implications for the management of human intelligence: the pursuit of novelty, unrestricted by any other prescribed goal or objective, radically speeds up evolutionary adaptation.

rhile working on our new book, Loving and Hating Mathematics.. The Emotional Side of MathematicalLife, we were surpr ised at how many wellknown mathematicians have created mathematics while in prison. We found five prisoners of war from three different wars, plus two political prisoners, and one pr isoner convicted of evading military service. Mongs ide these, there was one mathematician who used mathematics to escape an excruciating toothache, one who was revived to life by a mathematical p rob lem while bedr idden and almost 90 years old, a novelist who was distracted by mathematics from his decadeslong writer 's block, and an idealistic youngster who was he lped to endure the agony of part icipating in a senseless, brutal bombing war. Of all escapes from reality, mathematics is the most successful ever. It is a fantasy that becomes all the more addict ive because it works back to improve the same reality we are trying to evade. All other e s c a pe s s e x , drugs, hobbies , w h a t e v e r a r e ephemera l by compari son . . . . The mathemat ic ian becomes totally committed, a monster, like Nabokov 's chess p layer who eventual ly sees all life as subordina te to the game of chess [13].

There is a widely held assumption that the appeal of novelty, the rewards of discovery and the beauty of artistic expression provide their own rewards and constitute a claim for the human capacity for goodness. In celebrating these contributions to our lives, we seldom examine the possibility that such a creative contribution can also pose a serious threat to our very survival. In this chapter, we explore the ethics of choices that accompanied creative contributions in three different historical settings.

Practical calculation of the limit of a sequence often violates the defini- tion of convergence to a limit as taught in calculus. Together with examples from Euler, Polya and Poincare, this fact shows that in mathematics, as in science and in everyday life, we are often obligated to use knowledge that is derived, not rigor- ously or deductively, but simply by making the best use of available information- plausible reasoning. The "philosophy of mathematical practice" fits into the general framework of "warranted assertibility", the pragmatist view of the logic of inquiry developed by John Dewey.

ANIVE DEFINITION, adequate for the dictiuonary and for an initial understanding, is that mathematics is the science of quality and space. Expanding this definition a bit, one might add that mathematics also deals with the symbolism relating to quantity and to space.

IF YOU DO mathematics every day, it seems the most natural thing in the world. if you stop to think about what you are doing and what you means, it seems one of the most mysterious. How are we able to tell about things no one has ever seen, and understand them better than the solid objects of daily life? Why is Euclidean geometry still correct, while Aristotelian physics is dead long since? What do we know in mathematics, and how do we know it?

1. WHY SZEGED? The year 1933 was dreary and dismal for most people. This was precisely at the end of my studies for the doctorate. By some miracle, I was spared unemployment. I was awarded a National Research Council Fellowship, for a year of postdoctoral study at Harvard University under the guidance of Marshall Stone. There were four N.R.C. fellows at Harvard: Magnus Hestenes, David Nathan, Deane Montgomery, and me. Montgomery and I formed a seminar, to lecture each other on the material in Oswald Veblen's Analysis Situs. We were joined by Norman Steenrod, who shone by his determination to find out what was really going on. These fellowships had been severely cut, both in number and in stipend. Nevertheless, even with the reduced stipend of $1,600 for twelve months, I managed to live in Boston like a Bohemian, dividing my activities between wooing the recalcitrant muse of mathematics and indulging in the follies of youth: drinking beer, going to symphony concerts, and jogging in the park. This extra year at Harvard was supposed to give us a "coat of varnish," as one of my friends put it. Whether it turned us into gentlemen or scholars is a moot question. It did provide a line in my curriculum vitae. Future employers were impressed. Stone told me that the two mathematicians in all the world who could be most helpful to my development were John von Neumann and Frederick Riesz. (His Hungarian name, Frigyes, became Frederick when anglicized). von Neumann's name was well known to me, of course. Of Riesz, situated at the other end of the world, I was only vaguely aware. But Stone's remark was prophetic. Stone and I agreed that my energies were best spent digging deeper into Hilbert and Banach spaces. That meant a frontal attack on Stone's Colloquium Series Publication, Linear Transformations in Hilbert Space, a remarkable 622-page book more often quoted than read. Stone's style has sometimes been subject to comment. It is infinitely correct but, like many of his other qualities, carried out "a outrance," with some bizarre results. For example, one of his theorems (p. 590) takes two complete pages-for the statement. The last chapter of the book fills 218 pages. Stone had a special sense of humor. At one of our infrequent meetings I mentioned some problems as possible candidates for research topics. About the best one of my problems, Stone said, "Oh, I don't know. Somebody must have

Our investigation focused on historical, pedagogical, and social-political aspects of Hungarian mathematical life. We did not attempt to survey Hungarian mathematical research of the present. Even so, our time proved too short for our ambitions. The important Hungarian mathematicians whom we missed are certainly more numerous than those we interviewed. We spoke in depth to a dozen people, and carried out formal interviews with eight: in Hungary, Belaszokefalvi-Nagy, Pal Erdos, Tibor Gallai (recently deceased), Istvan Vincze, and Lajos Posa; in the United States, Agnes Berger, John Horvath, and Peter Lax. (While we were in Budapest, two of the leading newspapers carried major articles honoring Szokefalvi - Nagy's 75th birthday.) We asked all our interviewees the question “What is so special about Hungarian mathematics? What made possible the production of so many famous mathematicians in such a small, poor country, in the period between the two Wars?” In our interviews, and also in our reading, we got two quite distinct kinds of answers. Type 1 was internal. It related to institutions and practices within the world of mathematics. The other kind, type 2, was external. It related to trends and conditions in Hungarian history and social life at large. Perhaps one contribution of this article is to point out the importance of both types of answer. One could conjecture that favorable conditions of both types---within mathematical life and within socio-politico-economic life at large--- are necessary to produce a brilliant result such as the Hungarian mathematics of the 1920s and 1930s. In the terminology used by Mihaly Csikszentmihalyi and Rick Robinson [5] in their study of creativity, perhaps conditions have to be right both in the "domain"---the area of creative work and in the "field"---the ambient culture.

THERE IS A LIMITIED amount of knowledge, practice, and aspiration which is currently manifested in the thoughts and activites of contemporary mathematicians. The mathematics that is frequently used or is in the process of emerging is part of the current consciousness. This is the material which—to use a metaphor from computer science—is in the high speed memory or storage cells. What is done, created, practiced, at any given moment of time can be viewed in two distinct ways: as part of the larger cultural and intellectual consciousness and milieu, frozen in time, or as part of a changing flow of consciousness.

Once upon a time, when I was a teaching assistant, teaching a class of the kind mockingly called "Math for Poets," an obnoxious freshman said to me, "Zero isn't a number." I have forgotten my answer, but I remember finding her remark a shocking expression of profound ignorance. Years later, it dawned on me she was right! If I say "I own a number of calculus books" or "I have a number of friends at the Courant Institute," I don't mean zero books or zero friends. I don't even mean one book or one friend. I mean two or more. That's what "number" means in plain English. I read recently that the famous phenomenologist Edmund Husserl meant by "number" something greater or equal to 2. So did Plato. In mathematical talk, "number"has several meanings. None is the plain English meaning. The ordinary math teacher, like me back then, is so deeply embedded in math lingo that he/she doesn't notice the inconsistency. But the inconsistency can confuse students. I say "math lingo,"not language. It's a jargon, a semidialect of English (or some other natural language), not a complete language. You can't say " I have a headache" or "You bore me" in math lingo. In math lingo, a straight line is the simplest example of a curve. In plain English, quite otherwise: a straight line isn't a curve, and a curve isn't a straight line. In English, what we call a " line segment" is just a " line." What we call a " line" is "an infinite line." "Difference," "product," "factor," "prime" all have different meanings in plain English and in math lingo. I may ask a student, "If you subtract zero from zero, what's the difference?" While answering math-linguistically, "zero," she may be thinking, plain-Englishly, "That's right! Who cares? What's the difference?" In English, "adding" increases what you've got. In math lingo, it may increase it, decrease it or neither, depending on whether you happen to be adding something positive, negative or zero. Correspondingly, subtracting decreases. In math lingo, it may decrease or increase or neither. In English, "adding" and "subtracting" are opposite. In math lingo, they're opposite, and yet they're the same! For adding a number is the same as subtracting some other number (its negative). In English, "multiplying" means repeated adding. It makes things bigger. In math lingo, multiplying makes them bigger, smaller, or neither, depending on what you multiply with. Correspondingly, "divide" means cut into pieces, possibly equal pieces. In math lingo, "divide" is the same as "multiply," in the sense that dividing by a number other than zero is the same as multiplying by some other number (its reciprocal).