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