PROFESSOR HARDY has defined mathematics partly by enumeration of shining examples of mathematicians, partly by reference to the whole body of mathetmatical knowledge with permanent aesthetic value. The criteria do not seem to be very clear. Does the permanent value depend on the results or the method? Many of the most important results of pure mathematics are now stated in ways that their original authors would find it difficult to understand, and the original proofs would often fail to pass muster by modern standards, being either non-rigorous or hopelessly long-winded. What, for instance, would Fourier make of his theorem as presented by Titchmarsh, and what would happen to a Tripos candidate who gave Fourier's original proof? Newton's Principia is a monumental work, but how much of it survives in anything like its original form?
It seems to me that "mathematician" is used in several different senses. To most people a mathematician is anybody that can solve a quadratic equation; in Cambridge he is perhaps anybody that has taken, or proposes to take, the Mathematical Tripos, though it becomes doubtful whether he retains the title if he takes any other Tripos too. But many people make use of mathematics and yet do not consider themselves mathematicians. To such people mathematics is a tool, and not a direct source of interest. Professor Dirac, for instance, considers himself a physicist, and says quite explicitly that he considers mathematics a tool. His work, also, makes far more appeal to experimental physicists than it does to most pure mathematicians. In spite of Professor Hardy's admission of him as a mathematician, Dirac has more in common with Rutherford than with Hardy. Now I think that this is a genuine distinction. If people must be classified as specialists, it seems to me much more important that they should be classified with regard to what they talk about than with regard to the technique they use. Dirac should be called a theoretical physicist, not an applied mathematician. The latter term would then be free to denote those who are interested in physical problems only as illustrations of mathematical methods. I mention no names, considering the description abusive.
Unfortunately the Mathematical Tripos cannot get on without applied mathematics. Theoretical physicists, whether they hope ever to make original advances of their own, or whether they would be satisfied to be able to take an intelligent interest in the work of others, must learn a certain amount of pure mathematics and the general principles and methods of theoretical physics. But problems of intrinsic physical interest usually turn out to be either bookwork or too hard to be done in the time available in an examination! The result is that ability to apply the principles can be tested in the examination only by reference to experiments invented for the purpose: rolling illustrated by a sphere inside a vertical cylinder, with no slipping at all at the point of contact, potential problems for boundaries that no experimenter could construct, and so on. That is the trouble about Parts I and II: either the questions apparently on theoretical physics will be too hard, or they will be artificial and unsatisfactory, thereby becoming applied mathematics. In Part III the difficulty is less serious, since a question that takes three hours to answer, if the candidate knows how, is possible.
In dynamics there is a special difficulty. All the technique needed for the enormous majority of actual problems was known to Laplace, with the possible exception of the modified Lagrangian function, due to Routh. For these problems these are still the best methods. The reason why dynamics to this stage sometimes appears stagnant is that it was the first branch of theoretical physics to be developed, and when Newton, d'Alembert, Lagrange, Euler and Laplace had all played their parts in improving the methods it is hardly surprising that little further remained to be done. Nevertheless gyroscopic motion and small oscillations remain interesting (though the associated mathematics is now called algebra). The more advanced development beginning with Hamilton's equations and the Hamilton-Jacobi theorem makes the harder problems easier, but it makes the easier ones harder.
De gustibus.... But I must dissent from Professor Hardy's remark that ballistics and aerodynamics are ugly and dull. It is interesting that for bodies of some shapes moving through a fluid the force is nearly perpendicular to the direction of motion; there is beauty in the circulation theorem of aerofoil lift and in Prandtl's theory of induced drag; also in G. I. Taylor's treatment of various problems of bodies moving at high velocity in compressible fluids. Incidentally the circulation theorem also explains why people catch crabs in rowing; and the ballistics problem is substantially the same as one that arises in the evolution of the solar system, a matter of interest though hardly of economic or military importance. The devil's possession of the good tunes need not be undisputed. Again, some of us enjoy the numerical solution of differential equations. I do myself in moderation; but where should we be without those, from Briggs and Napier to the British Association Tables Committee, who find their chief joy in numerical computation? E. W. Brown could write down the equations of the moon's motion in two minutes; it took him thirty years to solve them to the accuracy needed for comparison with observation. But when I met him his soul always seemed to be doing very well. And that is the ultimate difference between the mathematician and the theoretical physicist. The former is satisfied with a formula or even an existence theorem. The latter does not consider either an answer at all, until the work of numerical computation has been taken to a stage where comparison with observation is possible. No theoretical physicist is the worse for knowing some pure mathematics that he has never had occasion to use; but he is the worse if the characteristic outlook of the pure mathematician leads him to over-emphasize the importance of the mathematics at the expense of an understanding of why the problems are of interest.
Reproduced from Eureka 27 pages 16-18.
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Erratum: "mathetmatical" should read "mathematical".
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