The Bactra Review: Occasional and eclectic book reviews by Cosma Shalizi   22

Emmy Noether


by Auguste Dick

translated from the German by Heidi I. Blocher

Boston, Basel, Stuttgart: Birkhäuser, 1981.
Emmy Noether is universally acknowledged to be one of the great mathematicians of modern times, responsible for not only one of the most important principles in mathematical physics, but fundamental innovations in abstract algebra as well. Her importance to the history of science and mathematics is only enhanced by the fact that, while in many ways typical of the great mathematicians of the first half of this century (not just either German or Jewish but both, child of another mathematician, passionate about abstraction and rigor, extremely unworldly, an exile, and emotionally the sort of person who'd drive a guidance counselor to distraction), she was of course a woman, and the only female mathematician to have made it into the pantheon. The book under review seems to be the closest thing to a proper, full-length study of her life and work, at least in English, but it is hardly satisfactory on any count.

Modern mathematics (especially algebra, Noether's special field) states highly abstract ideas in a jargon where non-technical words (``class'', ``group'', ``field'', ``ring'', ``ideal'', ``kernel'', even ``axiom'') assume very technical meanings. There are thus essentially two ways of writing about the work of a mathematician. The easiest is to assume that the reader knows mathematics, and use the specialized terms and abstractions, discuss the relevant theorems and their proofs, etc. This is the proper course when writing a true scientific biography, which Noether has apparently yet to receive. The other way of writing about mathematics is to try to convey, through simple examples, analogies, and concrete instances, something of the flavor of the math, of what the abstractions are about and why they are important. Dick synthesizes the worst of both approaches, mentioning the technical terms but neither using them fully nor explaining or analogizing them, which frustrates initiated readers and can only perplex others. We are told there are such things as Noetherian rings, but not what a ring is, nor are they given a proper definition. We are told that Noether developed a highly abstract and axiomatic approach to algebra, which contrasts strongly with the ``symbolic computation'' approach of her teachers, but the non-mathematical reader will gain no clue what this means from this book. (Things are made worse by Dick's way of repeating titles and descriptions which use now-obsolete jargon, instead of translating it into the modern terms.)

Outside of pure mathematics, Noether is most famous for her theorem about invariants in variational problems, commonly known as just ``Noether's Theorem.'' While stated with a high degree of generality, it is most usually applied to physics, where its meaning can be made somewhat intuitive, provided some ground-clearing work is done first. One has to begin with the notion of a transformation, which is a change (perhaps purely imaginary) to either the things we're interested in, or the way we measure them. A typical transformation is to rotate one's experimental apparatus, or (equivalently) to rotate our coordinate grid, or to start the experiment at a different time (which is equivalent to zeroing our clock at a different time, and is called time translation), or to mirror-reverse either the apparatus or our coordinate grid (parity reversal, more or less). We are particularly interested in transformations which don't change anything, or at least don't change the equations which govern the behavior of our system. The equations, and the system, are then said to be invariant under those transformations, or to have one or another sort of symmetry (rotational symmetry, time-translation symmetry, mirror-symmetry, etc.). Now, some transformations are continuous, and some are, at least conceptually, reversible (i.e. there's another transformation of the same sort which returns you to your original set-up). Those which are both continuous and reversible, like rotation and time-translation, are said to form continuous groups. (Parity reversal doesn't qualify, since it is a discrete and not a continuous transformation. You can't be just a little mirror-reversed.) Noether's Theorem essentially states that, whenever a physical system is invariant under a continuous group of transformations, there is a conserved quantity, one which is a function of the system and whose value does not change with time. More, the theorem even shows how to go about computing the conserved quantity. The conserved quantity which goes with rotation, for instance, is angular momentum, and that with time translation, energy. This is a very impressive result, which, in the seventy-odd years since Noether first published it, has gradually been elevated to one of the first principles of physics. We nowadays believe that all the conservation laws come from continuous symmetries of the fundamental interactions, and so symmetry studies are essential to the understanding of all physical forces. (Naturally, I've left out a lot of subtleties and technicalities, some of which have been discussed before in these pages.)

Dick mentions Noether's Theorem on three pages, without ever describing it, even as poorly as I just did, or giving a hint of its importance. This is of a piece with his general inability to see, not the forest for the trees, but the trees for the twigs. (For instance, we get three full pages on various obscure ancestors and relatives, of no apparent influence on Noether.) Not only are her mathematical and scientific work mis-presented, we get only disjointed, barely anecdotal statements about her as a person, and little more than a list of titles (her own and her papers') about her career. It would be of great interest to know just what Noether had to do to become a mathematician in Wilhelmine Germany, and to learn just who objected to her becoming a Privatdozent and latter a full professor at Göttingen, and who her allies were, and why they sided with her. Instead we are merely told that she was opposed, and get treated to the well-known remark Hilbert is supposed to have made on her behalf: ``I do not see that the sex of the candidate is an argument against her admission as Privatdozent. After all, we are a university and not a bathing establishment.'' It's a good line, but not exactly deep. And so it goes.

Does this book explain Noether's work for her fellow mathematicians and heirs? No. Does it convey a sense of what she accomplished to non-mathematicians? No; it is distinctly inferior to what (for instance) Edna Kramer's The Nature and Growth of Modern Mathematics has on her. Does it shed any light on the history of modern science and mathematics? No. Would it serve even as a juvenile biography, to inspire the young? No. Shall we consign it then to the flames? Not altogether: Dick does have the sense to reprint, in their entirety, various obituaries of Noether by mathematicians with whom she had collaborated, which are far better than his text (when it is not a paraphrase or direct quotation of those obituaries). The memorial address by Hermann Weyl, in particular, does manage to convey a sense of Noether's personality, and to explain, somewhat, just what she accomplished. We are still waiting for someone to carry on from where Weyl left us, and give Noether's memory the permanent form it deserves.

193 pp., chronology of principal dates in Noether's life, bibliography of Noether's publications and doctoral dissertations prepared under her supervision, appendix of obituaries, appendix of German academic titles, index of names.
Biography / Mathematics
Officially in print as a paperback, ISBN 0817605193, US$32, and even as a hardback, ISBN 0817630198, no listed price, but frankly I'm skeptical; LoC QA29 N6 D513
27 September 1997