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Noether as mathematical structuralist. (English) Zbl 1506.01006

Reck, Erich H. (ed.) et al., The prehistory of mathematical structuralism. Oxford: Oxford University Press. Log. Comput. Philos., 166-186 (2020).
The author discusses the notion of structures in mathematics in connection with the work of Emmy Noether. In the reviewer’s opinion, the conclusion that “Noether’s methodological inclinations […] were to generalize given results to more abstract settings” shortchanges her achievements.
The discussion of the mathematical achievements of Kummer, Dedekind and Noether is somewhat superficial. Several problematic points are related to the author’s reliance on secondary literature.
In her discussion of Kummer’s ideal numbers, the author seems to believe that one has to “conjecture the existence” of these ideal primes before we can “provide rules for calculating divisibility by them”. This misses the point completely. Kummer’s ideal numbers may be interpreted as an algorithm that sends a cyclotomic integer to a residue class modulo this ideal number (see the reviewer’s [Abh. Math. Semin. Univ. Hamb. 79, No. 2, 165–187 (2009; Zbl 1178.01020)]).
The author writes that Kummer’s methods “were limited in their application”, but fails to mention what this means. As a matter of fact, one of Dedekind’s main contributions was recognizing the fundamental role of algebraic integers in his ideal theory. Similarly, the author regards Noether’s axiomatic characterization of what are now called Dedekind rings as a generalization of Dedekind’s work from algebraic integers to more general domains, but this not at all the whole story. The idea of characterizing the objects you want to study by a few axioms has profoundly influenced Hasse, Artin, Lang and other mathematicians of the 20th century; E. Artin and G. Whaples, for example, characterized global fields using the product formula in [Bull. Am. Math. Soc. 51, 469–492 (1945; Zbl 0060.08302)], and Lang’s books contain many more examples of this kind. In addition, Noether’s axiom that the domain in question be integrally closed means that you cannot prove unique factorization into ideals without using this concept; since Kummer did not know about algebraic integers, this implies that his proof of unique factorization into ideal numbers must have a gap.
Dedekind is quoted in English translation, which is credited to his [Theory of algebraic integers. Translated from the French by John Stillwell. Cambridge: Cambridge Univ. Press (1996; Zbl 0863.11068)]; the fact that this is a translation, that it was done by J. Stillwell and that it was published in 1996 is not mentioned at all.
The author claims that “we find little in the way of autobiographical reflection on her approach to mathematics”, and instead of providing the reader with what little she has found she quotes Weyl’s and Van der Waerden’s assessment of Noether’s methods. As a matter of fact, Noether did indeed reflect on her methods in her correspondence; see e.g. [F. Lemmermeyer (ed.) and P. Roquette (ed.), Helmut Hasse und Emmy Noether. Die Korrespondenz 1925–1935. Göttingen: Universitätsverlag Göttingen (2006; Zbl 1101.01010); P. Roquette and F. Lemmermeyer, The Hasse-Noether correspondence 1925–1935. English translation with extensive commentary (to appear). Cham: Springer (2022; Zbl 1514.01005)].
For the entire collection see [Zbl 1440.03007].

MSC:

01A60 History of mathematics in the 20th century
00A30 Philosophy of mathematics

Biographic References:

Noether, Emmy; Dedekind, Richard; Kummer, Ernst
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