Math Coursework Sample
The main difficulty in the teaching of the history of modern mathematics lies not in the richness and exuberance of the subject, but in its technicality. The students attending a general course could not be expected to be sufficiently prepared for the understanding of each lecture. It would be necessary to explain the subject as well as the history of the subject, and this would soon prove impossible.
On the other hand, it seems to be the duty of instructors teaching special subjects to explain their history. For example, consider the teaching of the theory of elliptic functions. Would not the most natural way of introducing that teaching be to explain by what series of circumstances those functions were recognized, how their study forced itself gradually upon the attention of geometers, how it grew by leaps and bounds within the analytic domain, how it branched off and connected itself, little by little, with other branches of mathematics? Thus the course would naturally begin with one or two historical lectures, and if the teacher had the historical sense, he would easily contrive, as the theory unfolded itself, to introduce the main creators: Fagnano, Euler, Lagrange, Legendre, Gauss, Abel, Jacobi, Clebsch, Kronecker, Halphen, Hermite. Please note that if he did that part of his task well enough, if he succeeded in evoking those great personalities, his audience would become familiar with some of the leading mathematicians of the last century. It would be merely a cross-section of nineteenth-century mathematics, but it would be a magnificent cross-section. Every subject taught in our universities to advanced students, theory of surfaces, theory of probabilities, projective geometry, general theory of functions, theory of potential, analysis situs, and so on, would lead to historical cross-sections of a similar kind, and the student guided in such courses by imaginative and historicallyminded teachers would begin to have a fair idea of the achievements of the last century. Laplace would not be for him simply the name of a function or an equation, but a man, a man of flesh and blood, a man of towering genius yet as full of vanities as himself; and the same applies to all the names which have become indissolubly attached to mathematical ideas.
In short, the history of modern mathematics should be taught by mathematical teachers in the course of their ordinary teaching, while the history of older mathematics can be properly taught only by a specialist, who must be as much of a historian as of a mathematician, if not more.
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