What is Mathematics I: The Question

Bonnie Gold

Monmouth University

bgold@monmouth.edu

Note: I wrote this several years ago, and hope to have a revised version of this paper available by a week prior to the meeting.

I. Introduction.

The question, "What is mathematics?", can have many interpretations. It can mean (and has most often been taken to mean), "What is the nature of mathematical objects?" It can mean, "What is special about how we reason in mathematics, or about the methods of mathematics?" What it has rarely meant in recent years, but what I would argue it should mean is, "What is the common nature of those subjects that are classified as mathematics which causes us to lump them together under the same name?" For example, biology can be described as the study of living organisms. This is the common property of all the subjects to which the name "biology" is applied. Political science is the study of the ways communities govern themselves. The question is, what is an equivalent description of mathematics?

I plan in this paper to examine some answers which have been given to this question in the past; why I believe these are not adequate; and by what criteria will we recognize a good answer to the question. However, before beginning to do this, I would like to address briefly the question of why it matters what mathematics is and why looking for the common thread to the various subjects of mathematics is both the most central form of the question as well as the one most promising in this era of success.

II. Why should we care?

Certainly, the value of mathematics as a form of human knowledge doesn't need justification by me here. Its value has been written on extensively. One can scarcely find a volume of the American Mathematical Monthly which does not have several articles exploring the value of mathematics in one aspect or another. Were mathematics not of value in its own right, its usefulness in other fields would adequately establish its value. However, despite its generally agreed usefulness, virtually no one with less than the equivalent of an undergraduate major in mathematics has any conception of what mathematics is. The man on the street, if asked what mathematics is, will reply to the effect that it's arithmetic (and perhaps geometry) and he doesn't much care for it anyway.

To try to help dispel this view of mathematics, a number of mathematicians have written books (see [Alexsandrov], [Courant], [Devlin], [Kac 79], [MacLane 86], [Steen], [Stewart]) trying to give the educated non-mathematician a feeling for what mathematics is by giving him a sampling of some mathematics which can be understood with no more technical background than that of a good high school education. This is certainly a necessary part of enabling non-mathematicians to learn what mathematics is. However, if we had some brief way of describing what mathematics is, it could help mathematicians to respond to some of the questions we are often asked when introduced to a non-mathematician: "What is it you folks do, anyway, compute big square roots?" "Why do we need mathematicians, now that we have computers which can do any arithmetic problem there is, and faster than any person?" It's all very well to have large books explaining mathematics, but one can't very well hand them to the interlocutor as a response to such queries. But if one could say, briefly, what mathematics is, and then illustrate with one or two favorite examples, the questioner might become motivated to learn more.

Having a reasonably brief criterion of what is and what isn't mathematics would enable the mathematician to decide whether new subjects which border on mathematics really are mathematics, and should be at least under the partial purview of the mathematics department, and even might be relevant for their work; and which are interesting applications of mathematics but not mathematics itself. For example, T. Fort [Fort, p. 606] writes of having given, at a meeting, reasons why mechanics is mathematics, upon which an electrical engineer asserted that by the same reasoning the science of electricity was also, and asks whether geometry as the science of actual physical space should be counted as mathematics. And if it isn't, why not? In what ways (if any), as Knuth asks [Knuth 74 and 85] is computer science different from mathematics (or is computer science simply a part of mathematics)? Furthermore, having a definition of what mathematics is in general helps to put one's own work in perspective, to consider how it is a part of the larger whole. In addition, it may be true, as MacLane asserts, that "Good understanding of the nature of mathematics helps us to realize when an apparent part of mathematics is in fact a dead end." [MacLane 92, p. 9]

As for its value to philosophers, they have wrangled long and hard and unsuccessfully trying to decide what the objects of mathematics are: physical, mental, spiritual or what. It doesn't look as if a head-on attack on this question is capable resolving it at the current time. So perhaps approaching the problem from a new perspective, of what is common to the subjects, may help answer questions of what mathematical objects are, and how we have knowledge of them: often, looking at an old question from a different perspective enables one to see a way out of what has seemed a hopeless morass.

Furthermore, an advantage to looking at the subjects which constitute mathematics, rather than at the objects of mathematics, is that it is not clear that the objects of mathematics are the most important or central aspect of? mathematics, that on which we should focus. From the perspective of some mathematicians, Halmos for example, the heart of mathematics is problems, not objects. [Halmos 87] For others, as we shall discuss below (section III-H) mathematics is the study of structure, with the objects on which those structures are based being arbitrary and inessential, or even nonexistent. For still others, it is the relationship between structures which is where the essence of mathematics lies.

I discuss in detail below (section III-E) why logic, the common method for mathematics, is not a good way to define mathematics. Principally, logic is not what distinguishes mathematics, for while the methods of the sciences do differ from mathematics, adding experiment to simple logic, the logic of mathematics is no different from that of science or philosophy, and the latter doesn't make use of the experimental methods of science either.

III. Answers which have been suggested.

A. The first category of answer, which I shall dismiss without much discussion, are the clever or bombastic answers. Among these are (allegedly Charles Darwin, quoted in [Fort], p. 606) "A mathematician is a blind man in a dark room looking for a black hat which isn't there," (Russell, quoted in [Fort], p. 608) "Mathematics is the subject in which one never knows what he is talking about nor if what he says is true," "Mathematics studies non-existing things and is able to find out the full truth about them." [Renyi, p. 11]. Each of these is a clever quip relevant to the debate on whether or not mathematical objects exist, and say, not what mathematics is, but rather what it perhaps isn't. The bombastic answers, given as part of after-dinner speeches or hortatory articles, such as "Mathematics is the science of saving thought" (a quote from Popular Science Monthly in Miller, p. 197] and "Mathematics is persistent intellectual honesty" [Richardson, p. 73] are reminiscent of Polus' reply (in Plato's Gorgias) to the question of what rhetoric is, with the answer that it is "the noblest of the arts": whether true or false, it does little to help understand its basic nature.

B. The next sort of answer, which I shall also dismiss rather briefly, is the attempt to get rid of the questioner by shifting the question. The first response of many mathematicians to the question of what mathematics is, is "mathematics is what mathematicians do." This answer appears to have the advantage over many other definitions of including all of mathematics, and not other things. However, on closer examination, even this is not so clear, for mathematicians do lots of things besides mathematics (for example, they eat, many play chess, many play musical instruments, etc.) and further, there are times when people who are not usually regarded as mathematicians do mathematics. One could modify it by saying that it is what mathematicians do when acting as mathematicians, but then it is not clear how to avoid circularity in specifying what it is to act as a mathematician. Further, it adds very little information, in that it gives us no practical method, when faced by a new piece of work which might or might not be mathematics, to decide whether it is or not, save by checking whether it was done by a mathematician acting in that role; and how do we know when he is acting in that role? Thus, we have merely postponed the question from "what is mathematics" to "what is acting as a mathematician."

C. If mathematicians tend to answer that mathematics is what mathematicians do, physicists tend to answer [Temple, p. 3] that "mathematics is the language of experimental physics." He goes on to dispute this: "Now there is no doubt that the language of physical science is mathematics, but the converse of this statement must be repudiated...pure mathematics is much more than an armory of tools and techniques for the applied mathematician. On the other hand, the pure mathematician has ever been grateful to the applied mathematician for stimulus and inspiration....there are great subjects in pure mathematics which have arisen in complete and serene independence of the necessities of physics." [ibid] Certainly a great deal of mathematics has been developed in the process of trying to answer questions in physics, but much has also arisen from other fields (more and more from computer science and the social sciences) and much more from its own internal relationships: there was no question from physics which motivated number theory, for example, even if the natural numbers themselves may have arisen trying to count sheep or enemies.

D. One very common answer to the question "What is Mathematics?" is to list the subjects of mathematics: "The mathematician studies numbers and geometric forms" [Renyi, p. 7], "mathematics is arithmetic, music, geometry and astronomy" (for the ancient Greeks), "There are now at least five distinct sources of mathematical ideas. They are number, shape, arrangement, movement, and chance." [Stewart, p. 7] As my asking what is common to the subjects grouped under the name mathematics indicates, I find this simple listing of subjects to be unsatisfactory, and for a number of reasons. First, as Kuntzman [Kuntzman, p. 12] warns, such definitions risk becoming dated by the evolution of mathematics; even if we make our list include all the current Mathematics Reviews subject classifications, new subjects are being added all the time. Second, they emphasize the separateness of the different branches of mathematics, whereas if there has been any lesson from the development of mathematics in the last 50 years, it is the unity of mathematics, the complex web of interconnections between the supposedly different fields, even those which seem to have very different flavors (more on this in section IV). Third, they give no assistance in recognizing a new kind of mathematics when it appears.

One could try to force all of mathematics into a few categories, such as "number and geometry," but even today there are subjects such as category theory which fit under neither category, and many which fit under both, thus indicating there are not two disjoint categories which simply happen to be lumped together for convenience even though they have nothing in common. This presumably prompted Stewart's somewhat longer list (and his qualifier "at least"), but what reason do we have to believe, even if indeed all the current mathematical subjects could be fit tolerably well under one or another of these

rubrics, that this would continue to be true for even the next 20 years: certainly, it's not clear that his last three would have been included by mathematicians of the last century, and new subjects are developing much more rapidly than ever before.

E. This difficulty with finding a common subject has caused people to turn to the methodology of mathematics to find its unifying theme, mathematics being unique among the sciences in making deductions from axioms the cornerstone of its reasoning. One of the most common answers in the first 3/4 of this century, to the question of what mathematics is, comes from the foundationalist schools of formalism and logicism, which relate mathematics in one form or another to logic and formal systems. "Mathematics as a pure formal science is indeed identical with logic." [Cohen, p. 173] "Mathematics ... is usually recognized to be properly a branch of logic." [Bridegman, p. 47] "Pure mathematics is the class of all propositions of the form 'p implies q' where p and q are propositions containing one or more variables, the same in the two propositions, and neither p nor q contains any constants except logical constants." [Russell, p. 3] "Mathematics is only a part of logic, and is the class of all propositions of the

form: A(x,y,z,...) implies, for all values of the variables, A(x,y,z,...)." [Jourdain, p. 70] "Mathematics is the science which draws necessary conclusions." [Peirce, p. 97]

Then there is the formalist or combinatorial point of view. (Both view mathematics as a process of starting with certain givens and doing some computations on these which give mathematical truth. The principle difference is that formalists view the axioms as arbitrary or at least, not necessarily true, while the combinatorial point of view sometimes starts from things taken as known, such as the integers.) "Mathematics includes computation with natural numbers and everything that can be founded upon it, but nothing else." [Study, quoted in Fort]. "Hilbert holds that mathematics is essentially the manipulation of symbols." [Fort, p. 608, speaking of Courant and Hilbert, Grundlagen der Mathematik]. "A mathematical system is any set of strings of recognizable marks in which some of the string are taken initially and the remainder derived from these by operations performed according to rules which are independent of any meaning assigned to the marks." [Lewis, p. 355-356] Mathematics is "the science which studies all possible types of pairings" with the restriction that "a study is not to be called mathematics unless it is conducted with the aim of analyzing the logical structure and implications of the pairing operations." [Denbow, p. 235]

A variant of the "mathematics is logic" type of response attempts to locate mathematical objects within logic. "Mathematics is the theory of universals," according to Bigelow [Bigelow, p. 13], which he later modifies (p. 16) to "perhaps mathematics only discusses a small range of unusually neat and tidy interrelationships among universals." This in a sense echoes similar sentiments by mathematicians and other philosophers: "The point of mathematics is that in it we have always got rid of the particular instance, and even of any particular sorts of entities." [Whitehead 25, p. 21] "But if mathematics could be reduced to a series of such verifications (as that of 2 + 2 = 4) it would not be a science....There is no science but the science of the general." [Poincare 83, p. 396] However, abstraction to universals is not an adequate defining criterion of mathematics. "But abstraction is not the exclusive property of mathematics; it is characteristic of every science, even of all mental activity in general. Consequently the abstractness of mathematical concepts does not in itself give a complete description of the peculiar character of mathematics." [Aleksandrov, p. 2] Abstraction is perhaps more properly viewed as an important tool of mathematics: "A powerful tool here is the process of 'abstraction'... identifying (i.e., regarding as identical) those entities that differ only in nonessential properties. A set of such entities thus becomes a single unit and in this way a new entity is created." [Hermes, p. 5]

The advantage of these sorts of definition, as Kuntzman [Kuntzman, p. 13] says, is that, unlike definition by subject matter, definition by method is much more stable, not having changed since the ancient Greeks; and if there is no unity of subject, a unity of method is better than nothing. Certainly, logic is important to mathematics: "If you attempt to organize any body of subject matter logically, you will ultimately cast it in the form of an abstract mathematical science." [Richardson, p. 74] However, none of these will do as a definition of mathematics, for mathematics is at once both much larger than logic and much smaller, and these definitions have been repeatedly repudiated in the last part of this century. Mathematics is smaller than logic in that, although the style of argument is the style of mathematical discourse, the deduction "All men are mortal, Socrates is a man, hence Socrates is mortal" is not mathematics, nor am I doing mathematics when I read a murder mystery and, logically combining the clues, deduce who the murderer is. Logic is the reasoning method of mathematics, but it is also the reasoning method of much else (one might say, of all rational discourse), and so is much larger than mathematics.

On the other hand, mathematics is much larger than logic in a number of senses. First, logic is simply one of many branches of mathematics. Secondly, mathematics is not simply a style of reasoning, but also what is being reasoned about. "We often hear that mathematics consists mainly in 'proving theorems.' Is a writer's job mainly that of 'writing sentences?'" [Kac 86, p. 154] Logic is "an unassailable stronghold, inside which we could scarcely confine ourselves without risk of famine, but to which we are always free to retire in case of uncertainty or of external danger." [Weil, p. 297] Logic is "the external form which the mathematician gives to his thought, the vehicle which makes it accessible to others (in footnote: Indeed, every mathematician knows that a proof has not really been understood if one has done nothing more than verifying step by step the correctness of the deductions of which it is composed and has not tried to gain a clear insight into the ideas which have led to the construction of this particular chain of deductions in preference to every other one), in short, the language suited to mathematics; this is all, no further significance should be attached to it." [Bourbaki, p. 223] Even if, for example, one can pick out the integers inside set theory, or even if one could find them inside logic itself, logic gives no reason to look at that particular selection. It is only an understanding of the deep ideas of mathematics that enables mathematicians to make a significant subject inside the tautologies of logic.

Thirdly, it is not clear that all of mathematics can be embedded in logic, even if the integers and real numbers can: category theory, for example, seems just too big to fit into set theory and in any case, the experience of the last 100 years in foundations seems to be that set theory is not a part of logic anyway. Fourthly, as the incompleteness theorems show, no one system of logic can ever suffice for doing all of mathematics: one has to keep going outside the system to answer questions of the system. As Fraissé says [Fraissé], with games one fixes rules before starting, but in mathematics, there is both the theory (the axioms) and the metatheory, and with the latter one can go outside of the axioms to draw conclusions about the theory.

Furthermore, the whole idea of identifying mathematics with logic or formal systems is misguided. It does nothing to help tell which sets of axioms are interesting mathematically and which are merely spinning one's wheels. "Mathematics is, after all, not a collection of theorems, but a collection of ideas." [Halmos 90, p. 562] Indeed, a good definition of mathematics should be such that good mathematics partakes more of it than bad mathematics, and yet all mathematics partakes equally of logic.

F. Davis and Hersh propose "the study of mental objects with reproducible properties is called mathematics." [Davis 81, p. 399] It is a moot point (indeed, the major debate in the last 25 years in the philosophy of mathematics has been) whether the objects of mathematics are mental or otherwise; but in any case, if they are, so are lots of other objects, such as the game of chess, which is an equally mental object (the fact that we have physical representations of it is no different from mathematical objects) with reproducible properties. So which of these objects are mathematical ones? The definition is too broad.

G. Before I turn to several sorts of definitions which are worthy, it seems? to me, of more serious consideration, I should mention several descriptions of mathematics which, although not attempts at definitions, give some useful information which a correct definition should in some way incorporate. First a mathematical haiku: "Fire and Ice, Strange anomaly, The flame of intuition, Frozen in rigor." [O'Brian] Nevanlinna expresses a similar sentiment: "Mathematics combines two opposites, exactitude and freedom." [Nevanlinna, p. 456] These quotations capture an aspect of mathematics which I (and many mathematicians, as quoted in section E above in response to logicism or formalism) view as essential to mathematics, and which therefore a definition must include to have a chance of being adequate, the intuitive, non-formal side of mathematics.

"But the supreme value of mathematics, insofar as understanding the world about us is concerned, is that it reveals order and law where mere observation shows chaos." [Kline, p. 1] "Mathematics provides the dies by which science is formed, and mathematics is the essence of our best scientific theories." [ibid] While these quotations are not definitions of mathematics, they express eloquently one of the central properties of mathematics, its importance to, indeed, its presence at the heart of, science.

Knuth, in his discussions of the differences between mathematics and computer science, points out several: mathematics is interested in infinite systems, while computer science is not; mathematics tends to look for some sort of uniformity or homogeneity, while computer science is willing to tolerate a multitude of different cases [Knuth 85, p. 181]. Furthermore, mathematics deals with theorems and static relationships while computer science deals with algorithms and dynamic relationships. [Knuth 74, p. 326]

H. A view of mathematics which is currently enjoying popularity, and which is certainly an improvement on formalism or logicism is structuralism, the view that there are no mathematical objects, per se, but rather that mathematics studies pattern or form. This is not so terribly new: Poincaré wrote "Mathematicians do not study objects but the relations between objects.... Matter does not engage their attention, they are interested in form alone." [Poincaré 52, p. 20] Whitehead [Whitehead 51, p. 678] wrote that "Mathematics is the most powerful technique for the understanding of pattern, and for the analysis of the relationships of patterns."

One aspect of this view is not so different from formalism. According to Marshall Stone, Mathematics is "the study of general abstract systems, each one of which is an edifice built of specified abstract elements and structured by the presence of arbitrary but unambiguously specified relations among them." [Stone, p. 717] However, he goes on to qualify this (p. 719): "Nevertheless it [this characterization] cannot adequately suggest the intimate structural connections which have actually been found among the different branches of? mathematics, as a result of modern researches." Bourbaki in the middle of the century saw mathematics as a collection of interrelated structures, with a few

fundamental "mother structures" and then derivative structures which were combinations of these. [Bourbaki, p. 228-230] However, the Bourbaki group is since then moving away from this definition: "this notion has since been superseded by that of category and functor, which includes it under a more general and convenient form." [Dieudonné, p. 138] As Shapiro sees it [Shapiro, p. 538], "Mathematics is to reality as universal is to instantiated particular," or "Mathematics is to reality as pattern is to patterned." Later (p. 543) he says, "Mathematics per se is the 'pure' study of patterns." Felix Browder writes that Leibnitz and Descartes gave us "a vision of mathematics as the total science of

intellectual order, as the science of pattern and structure." [Browder, p. 252] Lang, however, when asked about the idea that mathematics is the manipulation of structures, objects "Yes but which ones?...When you do physics you also manipulate structures." Mathematics is no more the manipulation of structures than music is the manipulation of notes. [Lang, p. 3]

Thus, one of the problems with mathematics as the study of structures or patterns is that it doesn't really account for the important and unexpected connections which are found between various parts and structures of mathematics. Another is that it doesn't help us distinguish the structures found in mathematics from other structures. However, making it the science of pattern helps with this problem: we're not studying each individual pattern, and thus, for example, are not interested in the patterns of atoms or molecules. Rather, mathematics is concerned with the properties of patterns, the general relationships between patterns, how they behave, and so on. This definition looks good also because it may help us understand the usefulness of mathematics. Since some patterns occur in the world, understanding how patterns work should help us understand some aspects of the world. However, the most important quality of a definition of mathematics is that it should be true of mathematics, and this definition is not. We do, in mathematics, study certain patterns and how they relate to certain other patterns, but we are, for the most part (with the possible exception of certain subdisciplines, such as category theory and logic) not interested in pattern in general, and certainly that is not our sole concern. Furthermore, we are much more interested in some patterns - the integers, the complex numbers, for example - than in others. Again, the definition misses the problem of the varying significance of various parts of mathematics.

A possible response to this comes again from Browder [Browder, p. 252]: mathematics is "the science of significant form" (emphasis added). This definition seems to me to be one of the best which I have found, but it has the problem that we have now transferred the problem from defining mathematics to defining "significant." What makes one form more significant than another: that it is more important for mathematics? Certainly one doesn't want to go this circular route, but it is not clear how else to proceed?

Saunders MacLane is perhaps currently the most significant mathematician working on this structuralist view of mathematics. He started with formalism, but a formalism which didn't mean so much that we just study formal systems as that we are concerned with forms: mathematics "has developed to be a deductive analysis of a large number of very different but interlocking formal structures." [MacLane 81, p. 463] "At each stage of development in mathematics, the structure at issue can be recorded as a formal deductive system." [ibid, p. 465] "Mathematics is formal, but not simply 'formalistic' - since the forms studied in mathematics are derived from human activities and used to understand these activities." [ibid, p. 464] Mathematics consists in the "discovery of successive stages of the formal structures underlying the world and human activities in that world, with emphasis on those structures of broad applicability and those reflecting deeper aspects of the world." [ibid, p. 471] This is a very good statement for the emphasis on the different depths of different parts of mathematics, and the connection between mathematics and the world, but it is not really a definition; rather it is a description of the development of mathematics. More recently, he has refined this to make the broad applicability the essential characteristic of mathematics: "Mathematics is 'protean.' This means that one and the same mathematical structure has many different realizations." [MacLane 92, p. 3] This then leads to the definition: "Mathematics is that part of science

which applies in more than one empirical context." [ibid] However, this definition tries to replace the vagueness of "deeper aspects" in the earlier statement with a formal definition, and in doing so, misses the mark. "2 + 2 = 4" applies in at least as many empirical contexts as any mathematical observation, and yet, while in one sense that makes it very deep, in the sense that mathematicians use this word, it is hardly a deep result, or even a deep application of mathematics.

I. Similar to MacLane's assertion that mathematics is concerned with formal structures underlying the world is James Franklin's "Mathematics investigates the necessary interconnections between the parts of the global structure." [Franklin, p. 293] This definition is vague in many places: what does he mean by "necessary" interconnections, and what is the "global structure." Both MacLane's and Franklin's definitions are not concerned solely with the logic underlying mathematics, but also with its connection with the world at a deep level, and this is an important aspect of a correct definition.

IV. If none of these definitions work, maybe it's hopeless?

Of course, it is possible that there is no thing which is common to all those subjects called by the name "mathematics." The current belief among mathematicians seems support this attitude. For example, in their book [Kac 79, p. 9], Kac and Ulam say "We shall not undertake to define mathematics, because to do so would be to circumscribe its domain....The structure, however, changes continually and sometimes radically and fundamentally. In view of this, an attempt to define mathematics with any hope of completeness and finality is, in our opinion, doomed to failure." "At each stage in the advance of mathematical thought the outstanding characteristics are novelty and originality. This is why mathematics is such a delight to study, such a challenge to practice, and such a puzzle to define." [Temple, p. 2] Priestly [Priestly, p. 517] suggests that a satisfactory answer to our question "appears to be nearly as far off as the settling of the seemingly parallel questions of aesthetics, viz., 'What is art?' and 'What is significant art?'" However, people are at least working seriously on these latter questions, and it is certainly much more satisfying when one can give such a definition, and so it seems to me it is at least worth the search. Furthermore, this seems to me an ideal time in the history of mathematics to attempt an answer to the question, for many new fields of mathematics have developed in the last century, as well as new fields bordering mathematics, providing us thus with a great deal of raw material to test our theories on. As Howard Stein proposes, in the 19th century there was essentially a second birth of mathematics (the first being in the time of ancient Greece) - "of the very same subject", unlike physics which was different in Greek times. The expansion occurred "by the very same capacity of thought that the Greeks discovered; but in the process, something new was learned about the nature of that capacity - what it is, and what it is not." [Stein, p. 238]. And the unity which has also emerged in the last 50 years gives one strong reason to believe that this common thread must be there. "Today, we believe however that the internal evolution of mathematical science has, in spite of appearance, brought about a closer unity among its different parts, so as to create something like a central nucleus that is more coherent than it has ever been." [Bourbaki, p. 222] "Suddenly we seem to be able again to look at mathematics as a whole and to see that its various parts are much more closely united with one another than could have been suspected before." [Köthe, p. 506] If there is this unity, which was not foreseen but which has forced itself upon us, it

should be possible to find out what drives it.

V. Criteria for a good definition.

While I have suggested problems with all of the definitions which I have found in the literature, each has some important aspect of the truth (otherwise they would not have been proposed) which gives us a criterion on which to judge any proposed definition. From these unsatisfactory answers, let's try to extract some criteria a correct answer should satisfy. What properties should a good definition of mathematics have?

Three are essential.

1. It must be true of all of mathematics and of nothing which is not mathematics. It must allow new subjects to enter mathematics without having to change the definition.

2. The terms in which mathematics is defined must themselves be well-defined and with clear meanings so that one hasn't just reduced the problem to an equivalently difficult one.

3. It should be capable of being effectively applied. It must be possible to tell if an object fits or not. It must enable us to separate things which are not mathematics from things which are: chess, physics, computer science, etc. With subjects which are on the borderline, it must help us see in what ways they are part of mathematics and what ways not. It must help us decide if a new subject is part of mathematics.

Then there are a number of other things a good definition could help us do; a good definition should help us do many of them.

4. It should explain the "unreasonable effectiveness" of mathematics [Wigner], why mathematics is so useful in describing the physical world, why often mathematics, which has arisen out of purely mathematical considerations or a particular application, finds unexpected applications unrelated to its origins, despite the apparent independence of mathematics from real-world objects.

5. It should explain the unity of parts of mathematics which start out separate and apparently unrelated, and then turn out to be different aspects of the same fundamental ideas, such as number theory, algebraic geometry and several complex variables; the phenomenon of structures reappearing in various unexpected guises.

6. It should give the basis for a system of values for mathematics, which would explain why 12345 + 54321 = 66666 is inherently (independent of applications) not as valuable as Fermat's last theorem. It should explain the near unanimity of judgment of mathematical merit. This value system would include an appreciation for value of constructive results, but should emphasize the importance of ideas over calculations.

7. It should account for the rarity of contradictions in mathematics, which allow for the development of mathematics without having to change the theorems, even on those rare occasions where contradictions have appeared. Although in some important ways the picture changes, the collection of acceptable methods and objects, much more than in any other science, grows without having to throw any away as outdated.

8. While mathematics is neither identical to, nor a subset of, logic, a good definition should account for the importance of logic and deductive reasoning in mathematics. On the other hand, it should also account for the importance of intuition in mathematics.

9. The idea (mentioned in IIIA) that mathematics is persistent intellectual honesty does express the observation that one cannot fool oneself or others long in mathematics; it would be good if a definition would help us understand why this is true. Is mathematics more certain than other sciences? This has been debated in the philosophy of mathematics and a correct definition of mathematics should help in this debate.

10. It should help resolve the tension between the feeling that one is totally free, when doing mathematics, to define new objects as one pleases, and the feeling, often, when the object has been defined, that the definition was "necessary" or "natural" and wouldn't have been as effective if defined any other way.

11. It should account for the preference on the part of mathematicians to find a common idea behind a theorem rather than let a proof or solution rest with a lot of special cases, and the fact that this is so often eventually successful, even though initially such a unifying explanation appeared impossible.

12. Mathematics is objective, not just one mathematician's pipe dream nor a mass psychosis of mathematicians. A good definition should account for at least this gentle form of mathematical platonism. At the same time, it should account in some form for man's role in the development of mathematics.

Bibliography

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This paper was begun while on sabbatical at the Université Catholique de Louvain in Belgium, and was partially supported by a Lilly Faculty Open Fellowship grant number 910043.



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