Development of Mathematics in the 19th Century (Lie Groups) (English and German Edition) 1st Edition by Felix Klein (PDF)

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Text: English, German (translation)

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Reviews from Amazon users which were colected at the time this book was published on the website:

⭐Klein had apparently meant to expand this work, so one should not take omissions and relative weights of the topics too seriously. Nevertheless what stands must mean something, and the message is clear on what is the main accomplishment of 19th mathematics: complex function theory, comprising almost half the book. The heart and soul of this theory is the theory of elliptic functions and its generalisations (abelian functions, elliptic modular functions, automorphic functions). Those of us who did not absorb the theory of elliptic functions with our mothers’ milk feel that we are missing out on something big. Klein is worried as well: “When I was a student, abelian functions were, as an effect of the Jacobian tradition, considered the uncontested summit of mathematics, and each of us was ambitious to make progress in this field. And now? The younger generation hardly knows abelian functions.” (p. 294). On the whole, Klein is very fond of grand theories rather than individual results. Gauss took a hint from number theory for his work on elliptic functions, but the working out of the general theory is what we love most. Or take the second largest topic of the book: projective-algebraic geometry. Sure, at first it is a way of dealing with matters of classical geometry, but we are of course overjoyed when it points the way to invariant theory, a grand theory par excellence. There is only one thing we love more than a big theory, and that is when two big theories can be brought together. Thus we rejoice immensely when Riemann’s geometrisation of complex function theory enables us to establish deep connections with both algebraic geometry and algebraic number theory. Hooray for the unity of mathematics. “[Narrow specialisation] would bring about the death of the science if it should gain a one-sided acceptance. We ourselves have always striven for the opposite. In our generation we have more or less kept up the contacts between 1) invariant theory, 2) the theory of equations, 3) function theory, 4) geometry, and 5) number theory; and this has been our particular pride.” (p. 308). But although function theory has been a success story also in this respect, the potential pompousness of a too grandiose theory construction causes even Klein to have minor reservations about its future development: “Here an enormous prospect opens out into a purely theoretical domain, which, through its general regularities, exerts the greatest aesthetic charm, but, as we cannot neglect to remark, is utterly distant from all applications” (p. 314).The German edition of this work has a second part which is not included here. It is not a harmonious continuation of the first part (in fact, it’s not so much a historical narrative as a monograph aimed at exposing an intellectual path from the beginnings of invariant-theoretic ideas in geometry to their role in Einstein’s general theory of relativity, which is by no means the actual historical path, as Klein admits). Still, a translation of this second part would have been infinitely more valuable than Hermann’s hundreds of pages of appendixes on modern differential geometry, unkleinian in style and unrelated to Klein’s text.

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