The best mathematical biography on the market is, arguably, Constance Reid’s Hilbert. To be sure, tastes can differ on this matter, but a strong case can be made for the thesis that, together with Weil’s Apprenticeship of a Mathematician and Hardy’s poignant if not tragic A Mathematician’s Apology, the distinctive characteristics of mathematicians’ lives are best conveyed here. The latter two autobiographical works are products of the pens of actual master practitioners of the art, and Reid was the sister of none other than Julia Robinson, whence also an insider in mathematical circles: guilty by association if not participation. Another property these three books have in common is the high quality of the prose in their pages. With Reid’s book additionally providing an unsurpassed account of life at Göttingen’s mathematics and physics departments in the first half of the twentieth century, her descriptions of sundry prominent members of Hilbert’s circle (or ellipse, with Klein at the other focus) are particularly evocative. The galaxy of Göttingen’s stars was phenomenal, of course, including Minkowski, Landau, Hecke, Artin, and Weyl, but none shone brighter than Emmy Noether.
Hilbert and Klein famously imported her, with her expertise in the theory of invariants, to aid in their researches in physics. She rather quickly established what is now one of the mainstays of modern physics, the physicists’ Noether’s theorem, to the effect that symmetries and invariants and conservation laws are ultimately part and parcel of the same thing (with apologies for putting it so profanely).
Noether, generously crediting Dedekind, was soon captivated by rather more abstract “pure” questions, leading to nothing less than what was not so long ago still called “modern algebra.” Specifically, Noether took abstraction to a different level, unrelentingly championing the new axiomatic approach. And as she was moving down these untrodden roads she proved theorem after theorem — all of great beauty and depth. We have, for instance, “our” Noether’s theorem, whose manifestations include the fundamental isomorphisms of group theory and (e. g.) the rank-nullity theorem of linear algebra. But there is so much more: division algebras, crossed products, proto-cohomology, and of course modern invariant theory. Regarding this latter, there is an amusing anecdote to be recounted (cf. p. 166 of Reid’s book). Noether’s doctorate was earned under the direction of Paul Gordan, the once “king of invariants,” but as she matured as a mathematician and embraced a far more abstract methodology in her algebraic pursuits, she was wont to refer dismissively to her doctoral work, which contained a huge amount of formula-manipulation, as Formelgestrüpp, which translates to “a jungle of formulas.”
Regarding invariant theory as such, it is the case that one of Hilbert’s earliest breakthroughs was a solution of Gordan’s Problem addressing the question of finite bases for invariant systems. Hilbert caused something of a stir by proving existence without constructing anything: the Hilbert basis theorem for polynomial rings over a suitably well-behaved integral domain like the integers (more generally any Noetherian ring) conveys the same spirit. Gordan’s famous reaction to this revolution was the remark, Aber dass ist nicht Mathematik, dass ist Theologie! Says Reid, again on p. 166, that in the decade following her appearance in Göttingen “she was destined … to make Hilbert’s ‘theology’ look like mathematics.”
A propos, one more irresistible tale: Hilbert had an impossible time getting Emmy Noether appointed to the Göttingen faculty because of her sex, and counterarguments included that men coming home from the fighting in World War I should not be asked to return to their studies at the feet of a woman. Exasperated and furious, Hilbert replied to his faculty senate something to the effect of “but gentlemen, we are a mathematics faculty, not a bathhouse!”
All this having been said, I am ecstatic to have the book under review, Emmy Noether’s Gesammelte Abhandlungen — Collected Papers, in my hands. I have idolized her since my undergraduate days, many years ago, ever since I came to connect the magical character of abstract algebra, as such, with this fabulous scholar. One of my favorite professors, the late Indian algebraic geometer and algebraist, C. Musili, teaching a terrific undergraduate linear algebra course, introduced the first isomorphism theorem for vector spaces as “Noether’s Theorem” — and I was completely hooked. Thus, to have Noether’s Werke before me is a pure pleasure.
The forty three papers in this compendium are in and of themselves a study in the evolution of a major mathematical movement and methodological approach, even as they contain marvelous individual results that broke new ground at the time of their publication. Beyond this, the first page of the book contains a facsimile of Noether’s handwritten Lebenslauf, which is to say, her curriculum vitae, penned at the time of her writing of her doctoral dissertation under Gordan; it is accompanied by a translation — unlike the aforementioned forty three papers: the reader should be comfortable with German. The next entry is the Editor’s Preface, the editor being none other than Nathan Jacobson (it’s an algebraic bonanza!), and after that we’re off to the races — in German, as indicated.
So there it is. Emmy Noether, after escaping the madness of the Third Reich, died in 1935 in Pennsylvania, having been recently appointed to a professorship at Bryn Mawr. She was only in her early fifties. In her short life she completely revolutionized higher algebra, and her Gesammelte Abhandlungen are a treasure.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.