Yet doing mathematics requires

*symbols,*which are necessarily concrete in their origin and use. The use of symbols in mathematics arises naturally even at its very beginning: counting. There is nothing more fundamental to mathematics than a collection of symbols representing different quantities. What often goes overlooked by those who don't study mathematics is that embedded in those symbols is a theory of numbers. For instance, many people hardly notice that our counting system is a "base 10" or "decimal" system. The number 10 itself is not so significant. What

*is*significant is that somehow everyone knows that when I put a "1" directly to the left of a "0" I actually mean ten. This principle allows us to express remarkably large numbers with relatively few digits; indeed, the value of a number grows

*exponentially*with the length of the number: 10 is ten times 1, 100 is ten times 10, 1000 is ten times 100, and so on. Thus a deep fundamental principle is embedded into the very symbols we use to count things: that all whole numbers are uniquely expressible as a sum of powers of ten, where the coefficients of each power is something between zero and nine.

To realize the significance of this, it is worth comparing this with Roman numerals. To make ten, thankfully I don't have to write down ten I's; I can just write X. But if I want to make eleven, I have to write down XI, and then to make twelve and thirteen I write down XII and XIII. Not nearly as efficient as 12 and 13. The problem only gets worse. If I want to make a hundred, I write down some new symbol, C, which has nothing to do with anything that's come before it. And then to write a thousand, I write M, which again has nothing to do with anything I started with. And what if I want to write a million, a billion, or a trillion? The Romans didn't have such numerals, and if they wanted to write those numbers, they would simply have to make up new symbols. Their number system didn't have the same kind of power as our current number system does, because it wasn't embedded with the same theory as ours.

In general, when mathematicians discover new abstract principles, they encode those principles in symbols. Thus the abstract becomes tied to the concrete. The physical nature of these symbols is not insignificant; in fact, it is essential. It is first worth noting that often the inspiration for a proof of a very abstract result comes from looking at (or mentally picturing) a very concrete picture. (For instance, the proof of the projection theorem, for me, always begins with drawing a line and then a point away from that line, and looking at how the distance between the line and the point can be minimized by choosing the right line segment; this even leads to the idea for the Riesz representation theorem in Hilbert spaces.) But even more significant is that a large portion of a typical mathematician's work is devoted to scribbling down symbols on paper and attempting to logically manipulate them in such a way as to discover something new. (I have often wondered what this would be like for a blind mathematician. But it can be done!) This suggests that without some physical connection to the symbols which we have inherited, most of our work simply wouldn't happen.

Because mathematics deals with the abstract, and because all of its results are ultimately provable by logical deduction alone, it is tempting to think of mathematics as something which could be derived in its entirety by a single mind working from basic principles--a mind so powerful, of course, that it would immediately see all necessary logical relationships. However, the idea of such a mind does not do justice to the slow evolution of symbols and definitions which have accumulated over centuries of study. It is quite fascinating, for instance, that the numbers π and

*e*did not get their own symbols until the 1700s (both popularized by Euler). Negative numbers, imaginary numbers, and even the number zero were all subjects of controversy up until relatively recently in the history of mathematics. Yet simply having symbols for each of these ideas makes them difficult to erase from use. Once certain rules of symbolic manipulation have been established, and it can be clearly seen that these rules are consistent with previously known identities, new symbolic connections gradually become accepted as legitimate or "true." Mathematics, then, evolves in much the way culture does. It not only creates artifacts; it is also in turn shaped by those artifacts in ways that cannot necessarily be predicted

*a priori.*

Once we accept the use of a symbol in mathematics, we then turn around and justify its place in mathematics using some formal construction. Most mathematics can be done without ever formally constructing the natural numbers or the real number line. Moreover, it should be seen that such formal constructions really are an abstraction from the symbols which we have largely inherited. For some, I suppose, this is perhaps necessary to justify the symbols we use. I think it's better to view these formal constructions as explanations of the underlying theory which is embedded in the symbols we use, even if those symbols came about through a slow evolution over time rather than formally.

What a single mind seeking to construct all mathematics from scratch would be missing is the need for certain concepts and definitions. It is never clear

*a priori*which concepts and definitions we need to make progress in mathematics. For instance, one can derive matrix algebra by first trying to solve linear systems of equations, but it is not immediately obvious that one would like to have the more abstract idea of a linear operator on vector spaces. In my area of mathematics, the need for studying Hilbert spaces and Banach spaces becomes quite evident when one is trying to view a partial differential equation as a dynamical system. The state space of such a system must be some function space, and it then becomes necessary to define some metric topology on that space. If you are not a mathematician (or even if you are, but don't study my particular field) then it should not be surprising if (some of) what I am saying makes no sense to you. In fact, that is precisely my point; these terms and concepts don't make any sense outside the study of particular questions.

To be honest, I'm not altogether familiar with the evolution of the symbols currently used in my field, but I am not for that reason any less able to use them. Because of this, I am able to contribute without the need to build the whole field from the ground up. The symbols I use direct my thinking toward certain fundamental relationships that are needed to obtain results in my field. For instance, it is surely no accident (wherever it comes from) that the Sobolev spaces W^{s,2} are labeled H^s, with "H" a constant reminder that these are Hilbert spaces. That "s" should be written as an exponent is natural for those who know the meaning of these spaces. Even more valuable is that this "s" can be any real number, and that the spaces H_0^s and H^{-s} form a duality pairing which behave in the same way we would expect if we were merely integrating L^2 functions against one another (thus the s and the -s "cancel out"). To those not familiar with these spaces, this is all gibberish; but the point is that a lot of information is encoded in these symbols, and it is by these symbols that we hand down progress which has been made in function space theory.

Mathematics, then, can be seen just like any other forms of human endeavor as a cultural practice that evolves over time through concrete processes. Perhaps for that very reason it is all the more significant that it is so universal. Yet I would suggest that this universality is more a result of our time, rather than an inevitable feature of mathematics. True, there have always been certain mathematical ideas which naturally arise in all cultures; yet certain other concepts seem to have been developed by some cultures and not by others (for instance, the number system we use comes not from the West, but from India, Arabia, and Persia). Other concepts turn out not to be based so much on fixed principles as on human intuition, which turns out to be at least incomplete (I don't know if I would say "wrong"). Euclid gave us five axioms of geometry; the fifth axiom, it turns out, is neither necessary nor inevitable. It matters, then, what are the symbols, and what are the concepts embedded in those symbols, which we pass down to others.

It is true that mathematics deals with abstract truths, and for that reason it is probably more likely for a wide variety of cultures to converge in their pursuit of mathematical knowledge, or at least for mathematicians from a variety of cultures to quickly be able to learn new concepts from one another. But it does not follow that it is meaningful to talk about mathematics as a fixed set of abstract truths which exist in the realm of ideas, to which our minds ascend as we discover new theorems. Every theorem is a statement about the relationship between concepts which are necessarily incarnated in certain symbols. The particular incarnation is not so important is the fact that they are thus incarnated. One can define

*e*to mean whatever he wants, but once he has decided, that symbol is now essential for whatever truth he is then going to discover.

Thus we find an inseparable link between the concrete and the abstract. It is not fair to say that mathematics is separated from time and space, since it relies on the exchange of physical symbols created by human beings. It is interesting that in recent times collaboration has become more and more common among mathematicians. As I see it, the reason is not merely that we can "combine brainpower" to solve problems, but that we can gain insights by seeing different uses of symbols. As a graduate student, probably the best thing I have learned from my advisor is how to treat symbols in my equations as "good guys" and "bad guys" (the good guys can "eat up" the bad guys, allowing me to prove an essential estimate that gives me a new theorem). It seems to me that a large part of progress in mathematics comes from learning how we ought to relate to the symbols we use on a concrete, personal level.

All of this is not to say that mathematics is simply about knowing how to

*manipulate*symbols. We want to understand their purpose and meaning. The proper manipulations of mathematical symbols will not become evident until those symbols are related to other, more concrete concepts. It seems, then, that understanding the abstract principles is really a matter of having a general approach to treating particular instances. I know that whenever I look at a linear system of equations, I can write it as a matrix equation. My theoretical understanding of matrices and linear algebra is then embodied in my approach to solving particular problems. Really understanding mathematical theorems, then, is not that different from a politician learning how to talk to a variety of people, or a teacher learning how to deal with a variety of students, or a mechanic learning how to work on a variety of cars. Understanding the abstract is not so much a matter of separating from the concrete as it is being faithful to a particular approach in all matters concrete.

What this amounts to is my current understanding of the epistemology of mathematics, based on my consideration of other epistemological theories I've come across. I think there are some important applications of these ideas, especially to teaching mathematics, but I won't go into them here. I simply think these ideas are fun to think about, and I've always enjoyed reflecting on what it is that I'm actually doing as a mathematician.

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