Wikipedia:Reference desk/Archives/Mathematics/2023 November 14

Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is a transcluded archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.

November 14

Spinoza's ethics, mathematically analyzed

Is there an attempt to examine Spinoza's ethics mathematically, using the tools of modern mathematics? 2A02:8071:60A0:92E0:785D:C373:7F90:3E99 (talk) 12:34, 14 November 2023 (UTC)

There is a six-page paper on the Academia.edu website by Florian Marion, entitled "A Mathematical Interpretation of Spinoza's Ethics: Short preliminary remarks".^[1] If you don't have access to Academia, you can ask the author to email you the paper; his email address is on his university webpage. Not necessarily related to ethics, but possibly of interest:

• J. Friedman, "Some Set-Theoretical Partition Theorems Suggest by the Structure of Spinoza's God", Synthese, 27–1, 1974, p. 199–209.
• R. Ariew, "The Infinite in Spinoza's Philosophy", in P.-F. Moreau (ed.), Spinoza. Issues and Directions. The Proceedings of the Chicago Spinoza Conference, Brill, 1990, p. 16–31.
• P. Bussotti, Ch. Tapp, "The influence of Spinoza's concept of infinity on Cantor's set theory", in Studies in History and Philosophy of Science, Part A 40 (1), 2009, p. 25–35.

 --Lambiam 18:10, 14 November 2023 (UTC)

Wow, mathematisized ethics! I wonder if it can calculate how many people need to be on the other track to switch in the trolley problem ;-) NadVolum (talk) 19:47, 15 November 2023 (UTC)

Notation of representations

In the article on the Schur-Weyl duality, why are $S^{2}$ and $\Lambda ^{2}$ used for, I think, the sign representation and the trivial representation respectively. I'm not so familiar with the topic and I don't find these symbols used this way in related articles. I suppose $\Lambda ^{2}$ just "multiplies away" the sign in this case, but wouldn't something like $1$ be the standard notation for the trivial representation? Icek~enwiki (talk) 21:12, 14 November 2023 (UTC)

I'm pretty sure they're referring to the symmetric algebra and exterior algebra, with the superscript 2 meaning the second component of these graded algebras. The space of 2-tensors has dimension n². This breaks up into the symmetric part with dimension (n²+n)/2 and the antisymmetric part with dimension (n²-n)/2. Here, the symmetric part is generated by tensors of the form u⊗v+v⊗u and the antisymmetric part is generated by tensors of the form u⊗v-v⊗u. Note that when k=n the representation given by $\Lambda ^{k}$ is simply the determinant. --RDBury (talk) 22:40, 14 November 2023 (UTC)