ABSTRACT
The field of mathematics is unique in form as well as substance among the fields of intellectual endeavor. Fortunately, many mathematicians have written about their experience, explaining what it means and feels like to be a mathematician, allowing students and outsiders a glimpse of their world. This chapter will sketch the “culture of mathematics” by identifying some of the recurrent themes from the recommended reading list that follows: What do mathematicians feel is characteristic of and important about their field? THEMES
The first thing to mention is the self-deprecation many of these authors express in “writing about mathematics” rather than “doing mathematics.” This sets the tone, for example, of Hardy’s classic A Mathematician s Apology, despite the fact that even most mathematicians recognize the value of such reflective or popularizing writing-as long as someone else is doing it! Mordell commends the many eminent mathematicians who have given talks or who have written in this vein, saying that these endeavors “render a real service to mathematics and many have found great pleasure andinspiration in listening to or reading such expositions__ These havecontributed to the richness and vividness of mathematics and make it a living entity. Without them, mathematics would be much the poorer” (Mordell, 1970, p. 156). 1
The embarrassment of writing about mathematics is due to the great emphasis placed on creative mathematics; many writers discuss what they mean by this phrase, although they don’t necessarily use it in the same way: it connotes both new mathematics and truly innovative, important mathematics. Not all mathematicians do or are capable of “creative” work, much less the many more people who use existing, even if complex, mathematics for other purposes. Despite the apparent definiteness of what is or is not “creative mathematics,” there is something of a continuum, from new mathematics of primary significance down to rote calculational work. An extreme view is that mathematical ideas that are too thoroughly understood no longer count as mathematics (Halmos, 1968), since mathematicians aren’t interested in them.One meaning stressed is that doing mathematics is inherently creative as opposed to simply deductive, as the outsider or the student doing drills might think. According to de Morgan, “The moving power of mathematical invention is not reasoning, but imagination” (Gaither and Cavazos-Gaither, 1998, p. 131). Machines
This distinction is important in debating the mathematical potential of computers, which are good at performing calculations and following rules but not at generating new ideas. This is not to say that there is no relationship between these activities: calculating examples in pursuit of new insights has long been a standard research technique, and computer applications may be so efficient in this regard that they lead to results that would not otherwise have been possible. The “experimental mathematics” approach is increasingly utilized. Beyond calculation, more sophisticated capabilities appear in applications like automated theorem-provers, and such development is expected to progress (Gowers, 2002). Nevertheless, the majority view is that computers are and will remain tools and aids, not substitutes, for creative mathematicians. Youth
An offshoot of the preoccupation with truly significant innovations is the ubiquitous argument that the mathematician is most, or perhaps only, creative when young, to a degree greater than in other fields. The bar may be set very low: “Is it true that mathematicians are past it by the time they are 30?” (Gowers, 2002, p. 126). This often-quoted statement from the Apology
is the more common standard-bearer: “I do not know an instance of a major mathematical advance initiated by a man past 50” (Hardy, 1940, p. 72). For the truth of this judgment, “much depends on the definition of the advance” (Mordell, 1970, p. 157). Others point out that there are demonstrably many mathematicians who continue producing good, useful mathematics well past fifty (and frequent mention is made of Littlewood, who was still publishing original work while in his eighties). That mathematics is “a young man’s game” is a persistent belief, albeit little substantiated beyond anecdotes of mathematical prodigies, a prime example being Galois, who left a significant legacy although dying at 19. An actual application of this belief is that the French collective Bourbaki, from its beginnings in the mid-twentieth century, required a member to resign upon reaching the age of 50.The assumption that a mathematician is a man may be taken as a relic of the times (1940, in Hardy’s case). The apparently obvious issue of gender in mathematics is rarely mentioned in these general writings; an exception is the inclusion of “Why are there so few women mathematicians?” among Gowers’s frequently asked questions (Gowers, 2002, pp. 128-130). Beauty
There is much attention paid to the notion of beauty in mathematics. One aspect is the aesthetic motivation for developing mathematics: the mathematician, in expanding or synthesizing the structure of mathematics, does so in order to purify its form and prefers methods that promise the greatest simplicity with the greatest depth. A major insight or discovery gives the creator a great aesthetic and emotional reward, often described as joy. Another facet is the mathematician’s great appreciation of an “elegant” idea and proof; the elegant proof exhibits conciseness of development, originality, intuitive penetration, and appropriateness of means to ends, as well as a result that is unexpected, yet inevitable, significant, general, and deep. According to Scott Buchanan, “The best proofs in mathematics are short and crisp like epigrams, and the longest have swings and rhythms that are like music” (Henderson, 1945, p. 250). Art and Science
This concern for aesthetics is so essential that another active debate is whether mathematics is an art rather than a science. Those who feel it is an art point to its independence of physical reality, so that it is pursued for its own sake, and to the intuition and creativity employed by the mathematician. Mathematics is variously compared to music, poetry, and the visual arts. One analogy likens the argument that rhyme and meter are necessary
for producing meaningful poetry to the formal constraints of mathematics that ensure the truth and beauty of the result. The beauty of mathematics is because of, not in spite of, the formal rules that must be followed (contrary to the view of the nonmathematician who feels that the formality of proofs stultifies the subject). Nevertheless, the absolute nature of mathematical results militates against it being considered solely an art. “If mathematics is the most intellectual of the arts, however, it is also strikingly like a science, particularly in its insistence that there is only one version of truth” (Hammond, 1980, p. 23). The distinction has also been made based on process: according to Knuth, “Science is what we understand well enough to explain to a computer. Art is everything else we do” (Petkovsek et al., 1996, p. ix). Pure Versus Applied
Most of those on the “art” side qualify their arguments as applying to “pure mathematics”—the distinction between pure and applied mathematics is perhaps the most central theme in discussions of the nature of mathematics. There is some debate about how meaningful the dichotomy is, given that abstract ideas can later be, and often are, found to have an area of application, and that ideas developed in working on an applied problem can then be, and frequently are, pursued abstractly. It is noted that some mathematicians do both pure and applied work (although the feeling is that any individual is fundamentally either a pure or an applied mathematician) and that some pure and applied papers look very much alike. However, pure and applied mathematics are held to differ “in motivation, in purpose, frequently in method, and almost always in taste” (Halmos, 1968, p. 26). The distinction is relatively recent and is held by some to be counterproductive. In its exaggerated form, “to the applied mathematician the antonym of ‘applied’ is ‘worthless,’ and to the pure mathematician the antonym o f ‘pure’ is ‘dirty’” (Halmos, 1968, p. 25). Emphasis on one area or the other changes with time and place and can affect such things as curriculum design, departmental character, funding policy, and hiring practices. Absolute and Enduring Truth
The aforementioned contention about the absolute nature of mathematical truth may require some qualification. In general, mathematics is considered to be a solid structure built from incontrovertible building blocks, all the mathematical facts ever established-once a theorem is proved, it stays proved, although it may be considered more or less important at different
times. This endurance of knowledge can be an inspirational and comforting feature to mathematicians. For those motivated by a desire for fame, it is a heady thought that their contributions will last forever, in contrast to other fields in which changing interpretation and new data may completely overturn a once-prevailing view. Mathematicians also speak of their field as pure because a given piece of work is demonstrably true or false, so that no faking is possible. It is obligatory to mention, however, that Godel’s incompleteness theorem caused some revision of the notion of absolute truth in mathematics (Kline, von Neumann, etc.), and that some scholars assert that mathematics is not immune from charges of being a social construct (De Millo). The basic conviction of mathematics’ absolute and enduring truth remains remarkably intact despite these caveats. Halmos (1968) suggests that the cumulative effect remains objective: “The criterion for quality is beauty, intricacy, neatness, elegance, satisfaction, appropriateness-all subjective, but all somehow mysteriously shared by all” (p. 28). Certainly old research literature remains very relevant in mathematics, much more so than in most sciences (at the other extreme is computer science, in which articles more than a very few years old are useless). The mathematician makes frequent and intensive use of books and journal articles, which, combined with the absence of experimental equipment, leads to the adage that the library is the mathematician’s laboratory.
