1、a105Part IIntroductionI.1 What Is Mathematics About?It is notoriously hard to give a satisfactory answer tothe question, “What is mathematics?” The approach ofthis book is not to try. Rather than giving a definition ofmathematics, the intention is to give a good idea of whatmathematics is by describ
2、ing many of its most impor-tant concepts, theorems, and applications. Nevertheless,to make sense of all this information it is useful to beable to classify it somehow.The most obvious way of classifying mathematics is byits subject matter, and that will be the approach of thisbrief introductory sect
3、ion and the longer section enti-tled some fundamental mathematical definitionsI.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 tothink about. This gives a usefully dierent view of thesu
4、bject: it often happens that two areas of mathematicsthat appear very dierent if you pay attention to theirsubject matter are much more similar if you look at thekinds 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 t
5、he subject from thispoint of view. At the end of that article there is a briefdiscussion of what one might regard as a third classi-fication, not so much of mathematics itself but of thecontent of a typical article in a mathematics journal. Aswell as theorems and proofs, such an article will contain
6、definitions, examples, lemmas, formulas, conjectures,and so on. The point of that discussion will be to saywhat these words mean and why the dierent kinds ofmathematical output are important.1 Algebra, Geometry, and AnalysisAlthough any classification of the subject matter ofmathematics must immedia
7、tely be hedged around withqualifications, there is a crude division that undoubtedlyworks well as a first approximation, namely the divisionof mathematics into algebra, geometry, and analysis. Solet us begin with this, and then qualify it later.1.1 Algebra versus GeometryMost people who have done so
8、me 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 bethought of
9、 as belonging to arithmetic, while the ques-tion, “If x+y = 10 and xy = 21, then what is thevalue of the larger of x and y?” will be regarded as apiece of algebra. This contrast is less apparent in moreadvanced mathematics for the simple reason that it isvery rare for numbers to appear without lette
10、rs to keepthem company.There is, however, a dierent contrast, between alge-bra and geometry, which is much more important at anadvanced level. The high-school conception of geometryis that it is the study of shapes such as circles, trian-gles, cubes, and spheres together with concepts suchas rotatio
11、ns, reflections, symmetries, and so on. Thus,the objects of geometry, and the processes that theyundergo, have a much more visual character than theequations of algebra.This contrast persists right up to the frontiers of mod-ern mathematical research. Some parts of mathemat-ics involve manipulating
12、symbols according to certainrules: for example, a true equation remains true if you“do the same to both sides.” These parts would typicallybe thought of as algebraic, whereas other parts are con-cerned with concepts that can be visualized, and theseare typically thought of as geometrical.However, a
13、distinction like this is never simple. If youlook at a typical research paper in geometry, will it be fullof pictures? Almost certainly not. In fact, the methodsused to solve geometrical problems very often involvea great deal of symbolic manipulation, although goodpowers of visualization may be nee
14、ded to find and usea1052 I. Introductionthese methods and pictures will typically underlie whatis going on. As for algebra, is it “mere” symbolic manip-ulation? Not at all: very often one solves an algebraicproblem by finding a way to visualize it.As an example of visualizing an algebraic problem,co
15、nsider how one might justify the rule that if a andb are positive integers then ab = ba. It is possible toapproach 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 rectangulararray that consists ofaro
16、ws withb objects in each row.The total number of objects can be thought of as a lotsof b, if you count it row by row, or as b lots of a, if youcount it column by column. Therefore, ab=ba. Similarjustifications can be given for other basic rules such asa(b+c)=ab+ac and a(bc)=(ab)c.In the other direct
17、ion, it turns out that a good wayof solving many geometrical problems is to “convertthem into algebra.” The most famous way of doing thisis to use Cartesian coordinates. For example, supposethat you want to know what happens if you reflect acircle about a line L through its center, then rotate itthr
18、ough 40counterclockwise, and then reflect it oncemore about the same line L. One approach is to visualizethe 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 rotateit through 180about L (using the third dimension). Ther
19、esult will be upside down, but this does not matter ifyou simply ignore the thickness of the wood. Now if youlook up at the circle from below while it is rotated coun-terclockwise through 40, what you will see is a circlebeing rotated clockwise through 40. Therefore, if youthen turn it back the righ
20、t way up, by rotating about Lonce again, the total eect will have been a clockwiserotation through 40.Mathematicians vary widely in their ability and willing-ness to follow an argument like that one. If you cannotquite visualize it well enough to see that it is definitelycorrect, then you may prefer
21、 an algebraic approach,using the theory of linear algebra and matrices (whichwill be discussed in more detail in I.3 4.2). To beginPUP: firstnumeric-onlycross-referenceappears here. AllOK with you?with, one thinks of the circle as the set of all pairs ofnumbers (x,y) such that x2+y2lessorequalslant
22、1. The two trans-formations, reflection in a line through the center of thecircle and rotation through an angle , can both be rep-resented by 22 matrices, which are arrays of numbersof the form (abcd). There is a slightly complicated, butpurely algebraic, rule for multiplying matrices together,and i
23、t is designed to have the property that if matrix Arepresents a transformation R (such as a reflection) andmatrix B represents a transformation T, then the prod-uct AB represents the transformation that results whenyou first do T and then R. Therefore, one can solvethe problem above by writing down
24、the matrices thatcorrespond to the transformations, multiplying themtogether, and seeing what transformation correspondsto the product. In this way, the geometrical problem hasbeen converted into algebra and solved algebraically.Thus, while one can draw a useful distinction betweenalgebra and geomet
25、ry, one should not imagine that theboundary between the two is sharply defined. In fact,one of the major branches of mathematics is even calledalgebraic geometry IV.7. And as the above examplesillustrate, it is often possible to translate a piece of math-ematics from algebra into geometry or vice ve
26、rsa. Never-theless, there is a definite dierence between algebraicand geometric methods of thinkingone more symbolicand one more pictorialand this can have a profoundinfluence on the subjects that mathematicians chooseto pursue.1.2 Algebra versus AnalysisThe word “analysis,” used to denote a branch
27、of math-ematics, is not one that features at high-school level.However, the word “calculus” is much more familiar,and dierentiation and integration are good examples ofmathematics that would be classified as analysis ratherthan algebra or geometry. The reason for this is that theyinvolve limiting pr
28、ocesses. For example, the derivative ofa function f at a point x is the limit of the gradientsof a sequence of chords of the graph of f, and the areaof a shape with a curved boundary is defined to be thelimit of the areas of rectilinear regions that fill up moreand more of the shape. (These concepts
29、 are discussed inmuch more detail in I.3 5.)Thus, as a first approximation, one might say that abranch of mathematics belongs to analysis if it involveslimiting processes, whereas it belongs to algebra if youcan get to the answer after just a finite sequence of steps.However, here again the first ap
30、proximation is so crudeas to be misleading, and for a similar reason: if one looksmore closely one finds that it is not so much branchesof mathematics that should be classified into analysis oralgebra, but mathematical techniques.Given that we cannot write out infinitely long proofs,how can we hope
31、to prove anything about limiting pro-cesses? To answer this, let us look at the justification forthe simple statement that the derivative ofx3is 3x2. Theusual reasoning is that the gradient of the chord of theline joining the two points(x,x3)and(x+h),(x+h)3)a105I.1. What Is Mathematics About? 3is(x+
32、h)3x3x+hx,which works out as 3x2+3xh+h2.Ash“tends to zero,”this gradient “tends to 3x2,” so we say that the gradientat x is 3x2. But what if we wanted to be a bit more care-ful? For instance, ifx is very large, are we really justifiedin ignoring the term 3xh?To reassure ourselves on this point, we d
33、o a small cal-culation to show that, whatever x is, the error 3xh+h2can be made arbitrarily small, provided only that h issuciently small. Here is one way of going about it. Sup-pose we fix a small positive number epsilon1, which representsthe error we are prepared to tolerate. Then if|h|lessorequal
34、slantepsilon1/6x,we know that |3xh| is at most epsilon1/2. If in addition weknow that |h|lessorequalslantradicalbigepsilon1/2, then we also know that h2lessorequalslantepsilon1/2.So, provided that |h| is smaller than the minimum ofthe two numbersepsilon1/6x andradicalbigepsilon1/2, the dierence betw
35、een3x2+3xh+h2and 3x2will be at most epsilon1.There are two features of the above argument thatare typical of analysis. First, although the statement wewished to prove was about a limiting process, and wastherefore “infinitary,” the actual work that we needed todo to prove it was entirely finite. Sec
36、ond, the nature ofthat work was to find sucient conditions for a certainfairly simple inequality (the inequality |3xh+h2| lessorequalslant epsilon1)to be true.Let us illustrate this second feature with anotherexample: a proof that x4x26x+10 is positive forevery real number x. Here is an “analysts ar
37、gument.”Note first that if xlessorequalslant1 then x4greaterorequalslantx2and 106xgreaterorequalslant 0,so the result is certainly true in this case. If 1 lessorequalslantxlessorequalslant 1,then|x4x26x|cannot be greater thanx4+x2+6|x|,which is at most 8, so x4x26xgreaterorequalslant8, which implies
38、thatx4x26x+10 greaterorequalslant 2. If 1 lessorequalslantxlessorequalslant32, thenx4greaterorequalslantx2and6x lessorequalslant 9, so x4x26x+10 greaterorequalslant 1. If32lessorequalslant x lessorequalslant 2, thenx2greaterorequalslant94greaterorequalslant 2, so x4x2=x2(x21)greaterorequalslant 2. A
39、lso, 6xlessorequalslant 12,so 10 6x greaterorequalslant 2. Therefore, x4x2 6x+ 10 greaterorequalslant 0.Finally, if xgreaterorequalslant 2, then x4x2=x2(x21)greaterorequalslant 3x2greaterorequalslant 6x,from which it follows that x4x26x+10 greaterorequalslant 10.The above argument is somewhat long,
40、but each stepconsists in proving a rather simple inequalitythis isthe sense in which the proof is typical of analysis. Here,for contrast, is an “algebraists proof.” One simply pointsout thatx4x26x+10 is equal to(x21)2+(x3)2,and is therefore always positive.This may make it seem as though, given the
41、choicebetween analysis and algebra, one should go for alge-bra. After all, the algebraic proof was much shorter, andmakes it obvious that the function is always positive.However, although there were several steps to the ana-lysts proof, they were all easy, and the brevity of thealgebraic proof is mi
42、sleading since no clue has beengiven about how the equivalent expression forx4x26x+10 was found. And in fact, the general question ofwhen a polynomial can be written as a sum of squares ofother polynomials turns out to be an interesting and dif-ficult one (particularly when the polynomials have more
43、than one variable).There is also a third, hybrid approach to the problem,which is to use calculus to find the points wherex4x26x+10 is minimized. The idea would be to calculate thederivative 4x32x6 (an algebraic process, justified byan analytic argument), find its roots (algebra), and checkthat the
44、values of x4x26x+10 at the roots of thederivative are positive. However, though the method isa good one for many problems, in this case it is trickybecause the cubic 4x3 2x 6 does not have integerroots. But one could use an analytic argument to findsmall intervals inside which the minimum must occur
45、,and that would then reduce the number of cases that hadto be considered in the first, purely analytic, argument.As this example suggests, although analysis ofteninvolves limiting processes and algebra usually does not,a more significant distinction is that algebraists like towork with exact formula
46、s and analysts use estimates. Or,to put it even more succinctly, algebraists like equalitiesand analysts like inequalities.2 The Main Branches of MathematicsNow that we have discussed the dierences betweenalgebraic, geometrical, and analytical thinking, we areready for a crude classification of the
47、subject matter ofmathematics. We face a potential confusion, because thewords “algebra,” “geometry,” and “analysis” refer both tospecific branches of mathematics and to ways of think-ing that cut across many dierent branches. Thus, itmakes sense to say (and it is true) that some branchesof analysis
48、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, weshall think primarily in
49、 terms of subject matter, but itis important to keep in mind the distinctions of the pre-vious section and be aware that they are in some waysmore 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 specificpoints are discussed in parts III and V.a1054 I. Introduction2.1 AlgebraThe word “algebra,” when it denotes a branch of math-ematics, means something more specific than manipu-lation of symbols and a preference for eq
