Trends in mathematics today
The three principal broad trends in mathematics today I would characterize as (i) variety of applications, (ii) a new unity in the mathematical sciences, and (iii) the ubiquitous presence of the computer. Of course, these are not independent phenomena, indeed they are strongly interrelated, but it is easiest to discuss them individually.
The increased variety of application shows itself in two ways. On the one hand, areas of science, hitherto remote from or even immune to mathematics, have become "infected". This is conspicuously true of the social sciences, but is also a feature of present-day theoretical biology. It is noteworthy that it is not only statistics and probability which are now applied to the social sciences and biology; we are seeing the application of fairly sophisticated areas of real analysis, linear algebra and combinatorics, to name but three parts of mathematics involved in this process.
But another contributing factor to the increased variety of applications is the conspicuous fact that areas of mathematics, hitherto regarded as impregnably pure, are now being applied. Algebraic geometry is being applied to control theory and the study of large-scale systems; combinatorics and graph theory are applied to economics; the theory of fibre bundles is applied to physics; algebraic invariant theory is applied to the study of error-correcting codes. Thus the distinction between pure and applied mathematics is seen now not to be based on content but on the attitude and motivation of the mathematician. No longer can it be argued that certain mathematical topics can safely be neglected by the student contemplating a career applying mathematics. I would go further and argue that there should not be a sharp distinction between the methods of pure and applied mathematics. Certainly such a distinction should not consist of a greater attention to rigour in the pure community, for the applied mathematician needs to understand very well the domain of validity of the methods being employed, and to be able to analyse how stable the results are and the extent to which the methods may be modified to suit new situations.
These last points gain further significance if one looks more carefully at what one means by "applying mathematics". Nobody would seriously suggest that a piece of mathematics be stigmatized as inapplicable just because it happens not yet to have been applied. Thus a fairer distinction than that between "pure" and "applied" mathematics, would seem to be one between "inapplicable" and "applicable" mathematics, and our earlier remarks suggest we should take the experimental view that the intersection of inapplicable mathematics and good mathematics is probably empty. However, this view comes close to being a subjective certainty if one understands that applying mathematics is very often not a single-stage process. We wish to study a "real world" problem; we form a scientific model of the problem and then construct a mathematical model to reason about the scientific or conceptual model . However, to reason within the mathematical model, we may well feel compelled to construct a new mathematical model which embeds our original model in a more abstract conceptual context; for example, we may study a particular partial differential equation by bringing to bear a general theory of elliptic differential operators. Now the process of modeling a mathematical situation is a "purely" mathematical process, but it is apparently not confined to pure mathematics! Indeed, it may well be empirically true that it is more often found in the study of applied problems than in research in pure mathematics.
Thus we see, first, that the concept of applicable mathematics needs to be broad enough to include parts of mathematics applicable to some area of mathematics which has already been applied; and, second, that the methods of pure and applied mathematics have much more in common than would be supposed by anyone listening to some of their more vociferous advocates. For our purposes now, the lessons for mathematics education to be drawn from looking at this trend in mathematics are twofold; first, the distinction between pure and applied mathematics should not be emphasized in the teaching of mathematics, and, second, opportunities to present applications should be taken wherever appropriate within the mathematics curriculum.
The second trend we have identified is that of a new unification of mathematics. We would only wish to add to the discussion the remark that this new unification is clearly discernible within mathematical research itself. Up to ten years ago the most characteristic feature of this research was the "vertical" development of autonomous disciplines, some of which were of very recent origin. Thus the community of mathematicians was partitioned into subcommunities united by a common and rather exclusive interest in a fairly narrow area of mathematics (algebraic geometry, algebraic topology, homological algebra, category theory, commutative ring theory, real analysis, complex analysis, summability theory, set theory, etc., etc.). Indeed, some would argue that no real community of mathematicians existed, since specialists in distinct fields were barely able to communicate with each other. I do not impute any fault to the system which prevailed in this period of remarkably vigorous mathematical growth - indeed, I believe it was historically inevitable and thus "correct" - but it does appear that these autonomous disciplines are now being linked together in such a way that mathematics is being reunified. We may think of this development as "horizontal", as opposed to "vertical" growth. Examples are the use of commutative ring theory in combinatorics, the use of cohomology theory in abstract algebra, algebraic geometry, fuctional analysis and partial differential equations, and the use of Lie group theory in many mathematical disciplines, in relativity theory and in invariant gauge theory.
I believe that the appropriate education of a contemporary mathematician must be broad as well as deep, and that the lesson to be drawn from the trend toward a new unification of mathematics must involve a similar principle. We may so formulate it: we must break down artificial barriers between mathematical topics throughout the student's mathematical education. The third trend to which I have drawn attention is that of the general availability of the computer and its role in actually changing the face of mathematics. The computer may eventually take over our lives; this would be a disaster. Let us assume this disaster can be avoided; in fact, let us assume further, for the purposes of this discussion at any rate, that the computer plays an entirely constructive role in our lives and in the evolution of our mathematics. What will then be the effects? The computer is changing mathematics by bringing certain topics into greater prominence - it is even causing mathematicians to create new areas of mathematics (the theory of computational complexity, the theory of automata, mathematical cryptology). At the same time it is relieving us of certain tedious aspects of traditional mathematical activity which it executes faster and more accurately than we can. It makes it possible rapidly and painlessly to carry out numerical work, so that we may accompany our analysis of a given problem with the actual calculation of numerical examples.
However, when we use the computer, we must be aware of certain risks to the validity of the solution obtained due to such features as structural instability of round-off error. The computer is especially adept at solving problems involving iterated procedures, so that the method of successive approximation (iteration theory) takes on a new prominence. On the other hand, the computer renders obsolete certain mathematical techniques which have hitherto been prominent in the curriculum - a sufficient example is furnished by the study of techniques of integration. There is a great debate raging as to the impact which the computer should have on the curriculum . Without taking sides in this debate, it is plain that there should be a noticeable impact, and that every topic must be examined to determine its likely usefulness in a computer age. It is also plain that no curriculum today can be regarded as complete unless it prepares the student to use the computer and to understand its mode of operation. We should include in this understanding a realization of its scope and its limitations; and we should abandon the fatuous idea, today so prevalent in educational theory and practice, that the principal purpose of mathematical education is to enable the child to become an effective computer even if deprived of all mechanical aids!
Let me elaborate this point with the following table of comparisons. On the left I list human attributes and on the right I list the contrasting attributes of a computer when used as a calculating engine. I stress this point because I must emphasize that I am not here thinking of the computer as a research tool in the study of artificial intelligence. I should also add that I am talking of contemporary human beings and contemporary computers. Computers evolve very much faster than human beings so that their characteristics may well undergo dramatic change in the span of a human lifetime.
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