The Princeton Companion to Mathematics.pdf

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The Princeton Companion to Mathematics

Part I Introduction

Part II The Origins of Modern Mathematics

Part III Mathematical Concepts

Part IV Branches of Mathematics

Part V Theorems and Problems

Part VI Mathematicians

Part VII The Influence of Mathematics

Part VIII Final Perspectives

Part I Introduction I.1 What Is Mathematics About? It is notoriously hard to give a satisfactory answer to the question, “What is mathematics?” The approach of this book is not to try. Rather than giving a deﬁnition of mathematics, the intention is to give a good idea of what mathematics is by describing many of its most impor- tant concepts, theorems, and applications. Nevertheless, to make sense of all this information it is useful to be able to classify it somehow. The most obvious way of classifying mathematics is by its subject matter, and that will be the approach of this brief introductory section and the longer section enti- tled some fundamental mathematical deﬁnitions [I.3]. However, it is not the only way, and not even obvi- ously the best way. Another approach is to try to clas- sify the kinds of questions that mathematicians like to think about. This gives a usefully diﬀerent view of the subject: it often happens that two areas of mathematics that appear very diﬀerent if you pay attention to their subject matter are much more similar if you look at the kinds of questions that are being asked. The last sec- tion of part I, entitled the general goals of mathe- matical research [I.4], looks at the subject from this point of view. At the end of that article there is a brief discussion of what one might regard as a third classi- ﬁcation, not so much of mathematics itself but of the content of a typical article in a mathematics journal. As well as theorems and proofs, such an article will contain deﬁnitions, examples, lemmas, formulas, conjectures, and so on. The point of that discussion will be to say what these words mean and why the diﬀerent kinds of mathematical output are important. 1 Algebra, Geometry, and Analysis Although any classiﬁcation of the subject matter of mathematics must immediately be hedged around with qualiﬁcations, there is a crude division that undoubtedly works well as a ﬁrst approximation, namely the division of mathematics into algebra, geometry, and analysis. So let us begin with this, and then qualify it later. 1.1 Algebra versus Geometry Most people who have done some high-school mathe- matics will think of algebra as the sort of mathemat- ics that results when you substitute letters for num- bers. Algebra will often be contrasted with arithmetic, which is a more direct study of the numbers themselves. So, for example, the question, “What is 3 × 7?” will be thought of as belonging to arithmetic, while the ques- tion, “If x + y = 10 and xy = 21, then what is the value of the larger of x and y?” will be regarded as a piece of algebra. This contrast is less apparent in more advanced mathematics for the simple reason that it is very rare for numbers to appear without letters to keep them company. There is, however, a diﬀerent contrast, between alge- bra and geometry, which is much more important at an advanced level. The high-school conception of geometry is that it is the study of shapes such as circles, trian- gles, cubes, and spheres together with concepts such as rotations, reﬂections, symmetries, and so on. Thus, the objects of geometry, and the processes that they undergo, have a much more visual character than the equations of algebra. This contrast persists right up to the frontiers of mod- ern mathematical research. Some parts of mathemat- ics involve manipulating symbols according to certain rules: for example, a true equation remains true if you “do the same to both sides.” These parts would typically be thought of as algebraic, whereas other parts are con- cerned with concepts that can be visualized, and these are typically thought of as geometrical. However, a distinction like this is never simple. If you look at a typical research paper in geometry, will it be full of pictures? Almost certainly not. In fact, the methods used to solve geometrical problems very often involve a great deal of symbolic manipulation, although good powers of visualization may be needed to ﬁnd and use

2 I. Introduction these methods and pictures will typically underlie what is going on. As for algebra, is it “mere” symbolic manip- ulation? Not at all: very often one solves an algebraic problem by ﬁnding a way to visualize it. As an example of visualizing an algebraic problem, consider how one might justify the rule that if a and b are positive integers then ab = ba. It is possible to approach the problem as a pure piece of algebra (per- haps proving it by induction), but the easiest way to con- vince yourself that it is true is to imagine a rectangular array that consists of a rows with b objects in each row. The total number of objects can be thought of as a lots of b, if you count it row by row, or as b lots of a, if you count it column by column. Therefore, ab = ba. Similar justiﬁcations can be given for other basic rules such as a(b + c) = ab + ac and a(bc) = (ab)c. In the other direction, it turns out that a good way of solving many geometrical problems is to “convert them into algebra.” The most famous way of doing this is to use Cartesian coordinates. For example, suppose that you want to know what happens if you reﬂect a circle about a line L through its center, then rotate it through 40◦ counterclockwise, and then reﬂect it once more about the same line L. One approach is to visualize the situation as follows. Imagine that the circle is made of a thin piece of wood. Then instead of reﬂecting it about the line you can rotate ◦ it through 180 about L (using the third dimension). The result will be upside down, but this does not matter if you simply ignore the thickness of the wood. Now if you look up at the circle from below while it is rotated coun- terclockwise through 40 , what you will see is a circle being rotated clockwise through 40 . Therefore, if you then turn it back the right way up, by rotating about L once again, the total eﬀect will have been a clockwise ◦ rotation through 40 ◦ ◦ . PUP: ﬁrst numeric-only cross-reference appears here. All OK with you? Mathematicians vary widely in their ability and willing- ness to follow an argument like that one. If you cannot quite visualize it well enough to see that it is deﬁnitely correct, then you may prefer an algebraic approach, using the theory of linear algebra and matrices (which will be discussed in more detail in [I.3 §4.2]). To begin with, one thinks of the circle as the set of all pairs of numbers (x, y) such that x2 + y 2 1. The two trans- formations, reﬂection in a line through the center of the circle and rotation through an angle θ, can both be rep- resented by 2× 2 matrices, which are arrays of numbers of the form ( a b c d ). There is a slightly complicated, but purely algebraic, rule for multiplying matrices together, and it is designed to have the property that if matrix A represents a transformation R (such as a reﬂection) and matrix B represents a transformation T , then the prod- uct AB represents the transformation that results when you ﬁrst do T and then R. Therefore, one can solve the problem above by writing down the matrices that correspond to the transformations, multiplying them together, and seeing what transformation corresponds to the product. In this way, the geometrical problem has been converted into algebra and solved algebraically. Thus, while one can draw a useful distinction between algebra and geometry, one should not imagine that the boundary between the two is sharply deﬁned. In fact, one of the major branches of mathematics is even called algebraic geometry [IV.7]. And as the above examples illustrate, it is often possible to translate a piece of math- ematics from algebra into geometry or vice versa. Never- theless, there is a deﬁnite diﬀerence between algebraic and geometric methods of thinking—one more symbolic and one more pictorial—and this can have a profound inﬂuence on the subjects that mathematicians choose to pursue. 1.2 Algebra versus Analysis The word “analysis,” used to denote a branch of math- ematics, is not one that features at high-school level. However, the word “calculus” is much more familiar, and diﬀerentiation and integration are good examples of mathematics that would be classiﬁed as analysis rather than algebra or geometry. The reason for this is that they involve limiting processes. For example, the derivative of a function f at a point x is the limit of the gradients of a sequence of chords of the graph of f , and the area of a shape with a curved boundary is deﬁned to be the limit of the areas of rectilinear regions that ﬁll up more and more of the shape. (These concepts are discussed in much more detail in [I.3 §5].) Thus, as a ﬁrst approximation, one might say that a branch of mathematics belongs to analysis if it involves limiting processes, whereas it belongs to algebra if you can get to the answer after just a ﬁnite sequence of steps. However, here again the ﬁrst approximation is so crude as to be misleading, and for a similar reason: if one looks more closely one ﬁnds that it is not so much branches of mathematics that should be classiﬁed into analysis or algebra, but mathematical techniques. Given that we cannot write out inﬁnitely long proofs, how can we hope to prove anything about limiting pro- cesses? To answer this, let us look at the justiﬁcation for the simple statement that the derivative of x3 is 3x2. The usual reasoning is that the gradient of the chord of the line joining the two points (x, x3) and ((x+h), (x+h)3)

I.1. What Is Mathematics About? 3 is (x + h)3 − x3 x + h − x , which works out as 3x2+3xh+h2. Ash “tends to zero,” this gradient “tends to 3x2,” so we say that the gradient at x is 3x2. But what if we wanted to be a bit more care- ful? For instance, if x is very large, are we really justiﬁed in ignoring the term 3xh? To reassure ourselves on this point, we do a small cal- culation to show that, whatever x is, the error 3xh+ h2 can be made arbitrarily small, provided only that h is suﬃciently small. Here is one way of going about it. Sup- pose we ﬁx a small positive number , which represents the error we are prepared to tolerate. Then if |h| /6x, we know that |3xh| is at most /2. If in addition we know that |h| /2, then we also know that h2 /2. So, provided that |h| is smaller than the minimum of /2, the diﬀerence between the two numbers /6x and 3x2 + 3xh + h2 and 3x2 will be at most . There are two features of the above argument that are typical of analysis. First, although the statement we wished to prove was about a limiting process, and was therefore “inﬁnitary,” the actual work that we needed to do to prove it was entirely ﬁnite. Second, the nature of that work was to ﬁnd suﬃcient conditions for a certain fairly simple inequality (the inequality |3xh + h2| ) to be true. Let us illustrate this second feature with another example: a proof that x4 − x2 − 6x + 10 is positive for every real number x. Here is an “analyst’s argument.” Note ﬁrst that if x −1 then x4 x2 and 10 − 6x 0, so the result is certainly true in this case. If −1 x 1, then |x4−x2−6x| cannot be greater than x4+x2+6|x|, which is at most 8, so x4 − x2 − 6x −8, which implies that x4−x2−6x+10 2. If 1 x 3 2 , then x4 x2 and 6x 9, so x4 − x2 − 6x + 10 1. If 3 x 2, then 2, so x4 − x2 = x2(x2 − 1) 2. Also, 6x 12, x2 9 so 10 − 6x −2. Therefore, x4 − x2 − 6x + 10 0. Finally, if x 2, then x4 − x2 = x2(x2 − 1) 3x2 6x, from which it follows that x4 − x2 − 6x + 10 10. 2 4 The above argument is somewhat long, but each step consists in proving a rather simple inequality—this is the sense in which the proof is typical of analysis. Here, for contrast, is an “algebraist’s proof.” One simply points out that x4−x2−6x+10 is equal to (x2−1)2+(x−3)2, and is therefore always positive. This may make it seem as though, given the choice between analysis and algebra, one should go for alge- bra. After all, the algebraic proof was much shorter, and makes it obvious that the function is always positive. However, although there were several steps to the ana- lyst’s proof, they were all easy, and the brevity of the algebraic proof is misleading since no clue has been given about how the equivalent expression for x4 −x2 − 6x + 10 was found. And in fact, the general question of when a polynomial can be written as a sum of squares of other polynomials turns out to be an interesting and dif- ﬁcult one (particularly when the polynomials have more than one variable). There is also a third, hybrid approach to the problem, which is to use calculus to ﬁnd the points where x4−x2− 6x+ 10 is minimized. The idea would be to calculate the derivative 4x3−2x−6 (an algebraic process, justiﬁed by an analytic argument), ﬁnd its roots (algebra), and check that the values of x4 − x2 − 6x + 10 at the roots of the derivative are positive. However, though the method is a good one for many problems, in this case it is tricky because the cubic 4x3 − 2x − 6 does not have integer roots. But one could use an analytic argument to ﬁnd small intervals inside which the minimum must occur, and that would then reduce the number of cases that had to be considered in the ﬁrst, purely analytic, argument. As this example suggests, although analysis often involves limiting processes and algebra usually does not, a more signiﬁcant distinction is that algebraists like to work with exact formulas and analysts use estimates. Or, to put it even more succinctly, algebraists like equalities and analysts like inequalities. 2 The Main Branches of Mathematics Now that we have discussed the diﬀerences between algebraic, geometrical, and analytical thinking, we are ready for a crude classiﬁcation of the subject matter of mathematics. We face a potential confusion, because the words “algebra,” “geometry,” and “analysis” refer both to speciﬁc branches of mathematics and to ways of think- ing that cut across many diﬀerent branches. Thus, it makes sense to say (and it is true) that some branches of analysis are more algebraic (or geometrical) than oth- ers; similarly, there is no paradox in the fact that alge- braic topology is almost entirely algebraic and geometri- cal in character, even though the objects it studies, topo- logical spaces, are part of analysis. In this section, we shall think primarily in terms of subject matter, but it is important to keep in mind the distinctions of the pre- vious section and be aware that they are in some ways more fundamental. Our descriptions will be very brief: further reading about the main branches of mathemat- ics can be found in parts II and IV, and more speciﬁc points are discussed in parts III and V.

4 2.1 Algebra The word “algebra,” when it denotes a branch of math- ematics, means something more speciﬁc than manipu- lation of symbols and a preference for equalities over inequalities. Algebraists are concerned with number sys- tems, polynomials, and more abstract structures such as groups, ﬁelds, vector spaces, and rings (discussed in some detail in some fundamental mathematical deﬁnitions [I.3]). Historically, the abstract structures emerged as generalizations from concrete instances. For instance, there are important analogies between the set of all integers and the set of all polynomials with rational (for example) coeﬃcients, which are brought out by the fact that they are both examples of algebraic struc- tures known as Euclidean domains. If one has a good understanding of Euclidean domains, one can apply this understanding to integers and polynomials. This highlights a contrast that appears in many branches of mathematics, namely the distinction between general, abstract statements and particular, concrete ones. One algebraist might be thinking about groups, say, in order to understand a particular rather complicated group of symmetries, while another might be interested in the general theory of groups on the grounds that they are a fundamental class of mathemat- ical objects. The development of abstract algebra from its concrete beginnings is discussed in the origins of modern algebra [II.3]. A supreme example of a theorem of the ﬁrst kind is the insolubility of the quintic [V.24]—the result that there is no formula for the roots of a quintic poly- nomial in terms of its coeﬃcients. One proves this theo- rem by analyzing symmetries associated with the roots of a polynomial, and understanding the group that is formed by them. This concrete example of a group (or rather, class of groups, one for each polynomial) played a very important part in the development of the abstract theory of groups. As for the second kind of theorem, a good example is the classiﬁcation of ﬁnite simple groups [V.8], which describes the basic building blocks out of which any ﬁnite group can be built. Algebraic structures appear throughout mathematics, and there are many applications of algebra to other areas, such as number theory, geometry, and even math- ematical physics. 2.2 Number Theory Number theory is largely concerned with properties of the set of positive integers, and as such has a consid- I. Introduction erable overlap with algebra. But a simple example that illustrates the diﬀerence between a typical question in algebra and a typical question in number theory is pro- vided by the equation 13x − 7y = 1. An algebraist would simply note that there is a one-parameter fam- ily of solutions: if y = λ then x = (1 + 7λ)/13, so the general solution is (x, y) = ((1 + 7λ)/13, λ). A num- ber theorist would be interested in integer solutions, and would therefore work out for which integers λ the number 1 + 7λ is a multiple of 13. (The answer is that 1 + 7λ is a multiple of 13 if and only if λ has the form 13m + 11 for some integer m.) Other topics studied by number theorists are properties of special numbers such as primes. However, this description does not do full justice to modern number theory, which has developed into a highly sophisticated subject. Most number theorists are not directly trying to solve equations in integers; instead they are trying to understand structures that were origi- nally developed to study such equations but which then took on a life of their own and became objects of study in their own right. In some cases, this process has hap- pened several times, so the phrase “number theory” gives a very misleading picture of what some number theorists do. Nevertheless, even the most abstract parts of the subject can have down-to-earth applications: a notable example is Andrew Wiles’s famous proof of fermat’s last theorem [V.12]. Interestingly, in view of the discussion earlier, num- ber theory has two fairly distinct subbranches, known as algebraic number theory [IV.3] and analytic number theory [IV.4]. As a rough rule of thumb, the study of equations in integers leads to algebraic number theory and the study of prime numbers leads to analytic number theory, but the true picture is of course more complicated. 2.3 Geometry A central object of study is the manifold, which is dis- cussed in [I.3 §6.9]. Manifolds are higher-dimensional generalizations of shapes like the surface of a sphere, which have the property that any small portion of them looks fairly ﬂat but the whole may be curved in compli- cated ways. Most people who call themselves geometers are studying manifolds in one way or another. As with algebra, some will be interested in particular manifolds and others in the more general theory. Within the study of manifolds, one can attempt a fur- ther classiﬁcation, according to when two manifolds are regarded as “genuinely distinct.” A topologist regards

I.1. What Is Mathematics About? 5 two objects as the same if one can be continuously deformed, or “morphed,” into the other; thus, for exam- ple, an apple and a pear would count as the same for a topologist. This means that relative distances are not important to topologists, since one can change them by suitable continuous stretches. A diﬀerential topologist asks for the deformations to be “smooth” (which means “suﬃciently diﬀerentiable”). This results in a ﬁner classi- ﬁcation of manifolds and a diﬀerent set of problems. At the other, more “geometrical,” end of the spectrum are mathematicians who are much more concerned by the precise nature of the distances between points on a man- ifold (a concept that would not make sense to a topolo- gist) and in auxiliary structures that one can associate with a manifold. See riemannian metrics [I.3 §6.10] and ricci ﬂow [III.80] for some indication of what the more geometrical side of geometry is like. 2.4 Algebraic Geometry As its name suggests, algebraic geometry does not have an obvious place in the above classiﬁcation, so it is eas- ier to discuss it separately. Algebraic geometers also study manifolds, but with the important diﬀerence that their manifolds are deﬁned using polynomials. (A simple example of this is the surface of a sphere, which can be deﬁned as the set of all (x, y, z) such that x2+y 2+z2 = 1.) This means that algebraic geometry is algebraic in the sense that it is “all about polynomials” but geometric in the sense that the set of solutions of a polynomial in several variables is a geometric object. An important part of algebraic geometry is the study of singularities. Often the set of solutions to a system of polynomial equations is similar to a manifold, but has a few exceptional, singular points. For example, the equa- tion x2 = y 2 + z2 deﬁnes a (double) cone, which has its vertex at the origin (0, 0, 0). If you look at a small enough neighborhood of a point x on the cone, then, provided x is not (0, 0, 0), the neighborhood will resemble a ﬂat plane. However, if x is (0, 0, 0), then no matter how small the neighborhood is, you will still see the vertex of the cone. Thus, (0, 0, 0) is a singularity. (This means that the cone is not actually a manifold, but a “manifold with a singularity.”) The interplay between algebra and geometry is part of what gives algebraic geometry its fascination. A fur- ther impetus to the subject comes from its connections to other branches of mathematics. There is a particu- larly close connection with number theory, explained in arithmetic geometry [IV.6]. More surprisingly, there are important connections between algebraic geom- etry and mathematical physics. See mirror symmetry [IV.14] for an account of some of these. 2.5 Analysis Analysis comes in many diﬀerent ﬂavors. A major topic is the study of partial differential equations [IV.16]. This began because partial diﬀerential equa- tions were found to govern many physical processes, such as motion in a gravitational ﬁeld, for example. But they arise in purely mathematical contexts as well— particularly in geometry—so partial diﬀerential equa- tions give rise to a big branch of mathematics with many subbranches and links to many other areas. Like algebra, analysis has some abstract structures that are central objects of study, such as banach spaces [III.64], hilbert spaces [III.37], C∗-algebras [IV.19 §3], and von neumann algebras [IV.19 §2]. These are all inﬁnite-dimensional vector spaces [I.3 §2.3], and the last two are “algebras,” which means that one can multi- ply their elements together as well as adding them and multiplying them by scalars. Because these structures are inﬁnite dimensional, studying them involves limit- ing arguments, which is why they belong to analysis. However, the extra algebraic structure of C∗ -algebras and von Neumann algebras means that in those areas substantial use is made of algebraic tools as well. And as the word “space” suggests, geometry also has a very important role. 0+2, and then let z2 = z2 dynamics [IV.15] is another signiﬁcant branch of analysis. It is concerned with what happens when you take a simple process and do it over and over again. For example, if you take a complex number z0, then let z1 = z2 1+2, and so on, then what is the limiting behavior of the sequence z0, z1, z2, . . .? Does it head oﬀ to inﬁnity or stay in some bounded region? The answer turns out to depend in a compli- cated way on the original number z0. The study of how it depends on z0 is a question in dynamics. Sometimes the process to be repeated is an “inﬁnites- imal” one. For example, if you are told the positions, velocities, and masses of all the planets in the solar sys- tem at a particular moment (as well as the mass of the Sun), then there is a simple rule that tells you how the positions and velocities will be diﬀerent an instant later. Later, the positions and velocities have changed, so the calculation changes; but the basic rule is the same, so one can regard the whole process as applying the same simple inﬁnitesimal process inﬁnitely many times. The correct way to formulate this is by means of partial dif- ferential equations and therefore much of dynamics is

6 I. Introduction concerned with the long-term behavior of solutions to these. 2.6 Logic The word “logic” is sometimes used as a shorthand for all branches of mathematics that are concerned with fundamental questions about mathematics itself, notably set theory [IV.1], category theory [III.8], model theory [IV.2], and logic in the narrower sense of “rules of deduction.” Among the triumphs of set theory are gödel’s incompleteness theorems [V.18]and Paul Cohen’s proof of the independence of the contin- uum hypothesis [V.21]. Gödel’s theorems in particular had a dramatic eﬀect on philosophical perceptions of mathematics, though now that it is understood that not every mathematical statement has a proof or disproof most mathematicians carry on much as before, since most statements they encounter do tend to be decid- able. However, set theorists are a diﬀerent breed. Since Gödel and Cohen, many further statements have been shown to be undecidable, and many new axioms have been proposed that would make them decidable. Thus, decidability is now studied for mathematical rather than philosophical reasons. Category theory is another subject that began as a study of the processes of mathematics and then became a mathematical subject in its own right. It diﬀers from set theory in that its focus is less on mathemati- cal objects themselves than on what is done to those objects—in particular, the maps that transform one to another. A model for a collection of axioms is a mathematical structure for which those axioms, suitably interpreted, are true. For example, any concrete example of a group is a model for the axioms of group theory. Set theo- rists study models of set-theoretic axioms, and these are essential to the proofs of the famous theorems men- tioned above, but the notion of model is more widely applicable and has led to important discoveries in ﬁelds well outside set theory. 2.7 Combinatorics There are various ways in which one can try to deﬁne combinatorics. None is satisfactory on its own, but together they give some idea of what the subject is like. A ﬁrst deﬁnition is that combinatorics is about counting things. For example, how many ways are there of ﬁlling an n×n square grid with 0s and 1s if you are allowed at most two 1s in each row and at most two 1s in each col- umn? Because this problem asks us to count something, it is, in a rather simple sense, combinatorial. Combinatorics is sometimes called “discrete math- ematics” because it is concerned with “discrete” as opposed to “continuous” structures. Roughly speaking, an object is discrete if it consists of points that are isolated from each other and continuous if you can move from one point to another without making sud- den jumps. (A good example of a discrete structure is the integer lattice Z2, which is the grid consisting of all points in the plane with integer coordinates, and a good example of a continuous one is the surface of a sphere.) There is a close aﬃnity between combinatorics and theoretical computer science (which deals with the quintessentially discrete structure of sequences of 0s and 1s), and combinatorics is sometimes contrasted with analysis, though in fact there are several connections between the two. A third deﬁnition is that combinatorics is con- cerned with mathematical structures that have “few con- straints.” This idea helps to explain why number theory, despite the fact that it studies (among other things) the distinctly discrete set of all positive integers, is not considered a branch of combinatorics. In order to illustrate this last contrast, here are two somewhat similar problems, both about positive inte- gers. (i) Is there a positive integer that can be written in a thousand diﬀerent ways as a sum of two squares? (ii) Let a1, a2, a3, . . . be a sequence of positive integers, and suppose that each an lies between n2 and (n+ 1)2. Will there always be a positive integer that can be written in a thousand diﬀerent ways as a sum of two numbers from the sequence? The ﬁrst question counts as number theory, since it concerns a very speciﬁc sequence—the sequence of squares—and one would expect to use properties of this special set of numbers in order to determine the answer, which turns out to be yes.1 The second question concerns a far less structured sequence. All we know about an is its rough size—it is fairly close to n2—but we know nothing about its more detailed properties, such as whether it is a prime, or a 1. Here is a quick hint at a proof. At the beginning of analytic number theory [IV.4] you will ﬁnd a condition that tells you pre- cisely which numbers can be written as sums of two squares. From this criterion it follows that “most” numbers cannot. A careful count shows that if N is a large integer, then there are many more expres- sions of the form m2+n2 with both m2 and n2 less than N than there are numbers less than 2N that can be written as a sum of two squares. Therefore there is a lot of duplication.

I.2. The Language and Grammar of Mathematics 7 perfect cube, or a power of 2, etc. For this reason, the second problem belongs to combinatorics. The answer is not known. If the answer turns out to be yes, then it will show that, in a sense, the number theory in the ﬁrst problem was an illusion and that all that really mattered was the rough rate of growth of the sequence of squares. 2.8 Theoretical Computer Science This branch of mathematics is described at considerable length in part IV, so we shall be brief here. Broadly speak- ing, theoretical computer science is concerned with eﬃ- ciency of computation, meaning the amounts of various resources, such as time and computer memory, needed to perform given computational tasks. There are math- ematical models of computation that allow one to study questions about computational eﬃciency in great gen- erality without having to worry about precise details of how algorithms are implemented. Thus, theoretical computer science is a genuine branch of pure mathe- matics: in theory, one could be an excellent theoretical computer scientist and be unable to program a com- puter. However, it has had many notable applications as well, especially to cryptography (see mathematics and cryptography [VII.7] for more on this). 2.9 Probability There are many phenomena, from biology and eco- nomics to computer science and physics, that are so complicated that instead of trying to understand them in complete detail one tries to make probabilistic state- ments instead. For example, if you wish to analyze how a disease is likely to spread, you cannot hope to take account of all the relevant information (such as who will come into contact with whom) but you can build a math- ematical model and analyze it. Such models can have unexpectedly interesting behavior with direct practical relevance. For example, it may happen that there is a “critical probability” p with the following property: if the probability of infection after contact of a certain kind is above p then an epidemic may very well result, whereas if it is below p then the disease will almost certainly die out. A dramatic diﬀerence in behavior like this is called a phase transition. (See probabilistic models of critical phenomena [IV.26] for further discussion.) Setting up an appropriate mathematical model can be surprisingly diﬃcult. For example, there are physical cir- cumstances where particles travel in what appears to be a completely random manner. Can one make sense of the notion of a random continuous path? It turns out that one can—the result is the elegant theory of brown- ian motion [IV.25]—but the proof that one can is highly sophisticated, roughly speaking because the set of all possible paths is so complex. 2.10 Mathematical Physics The relationship between mathematics and physics has changed profoundly over the centuries. Up to the eigh- teenth century there was no sharp distinction drawn between mathematics and physics, and many famous mathematicians could also be regarded as physicists, at least some of the time. During the nineteenth cen- tury and the beginning of the twentieth century this situation gradually changed, until by the middle of the twentieth century the two disciplines were very sepa- rate. And then, toward the end of the twentieth cen- tury, mathematicians started to ﬁnd that ideas that had been discovered by physicists had huge mathematical signiﬁcance. There is still a big cultural diﬀerence between the two subjects: mathematicians are far more interested in ﬁnd- ing rigorous proofs, whereas physicists, who use math- ematics as a tool, are usually happy with a convincing argument for the truth of a mathematical statement, even if that argument is not actually a proof. The result is that physicists, operating under less stringent con- straints, often discover fascinating mathematical phe- nomena long before mathematicians do. Finding rigorous proofs to back up these discoveries is often extremely hard: it is far more than a pedantic exer- cise in certifying the truth of statements that no physi- cist seriously doubted. Indeed, it often leads to further mathematical discoveries. The articles vertex opera- tor algebras [IV.13], mirror symmetry [IV.14], gen- eral relativity and the einstein equations [IV.17], and operator algebras [IV.19] describe some fasci- nating examples of how mathematics and physics have enriched each other. I.2 The Language and Grammar of Mathematics 1 Introduction It is a remarkable phenomenon that children can learn to speak without ever being consciously aware of the sophisticated grammar they are using. Indeed, adults too can live a perfectly satisfactory life without ever thinking about ideas such as parts of speech, subjects, predicates, or subordinate clauses. Both children and

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