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Neither wikipedia nor the first few pages of Google are showing me much about the history of the development of equivalence classes. When was this notion first formalized?

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    $\begingroup$ Kaplansky told me that Gauss equivalence for binary quadratic forms was the first time this happened, other than given examples such as equality of fractions en.wikipedia.org/wiki/Binary_quadratic_form#Equivalence $\endgroup$
    – Will Jagy
    Jul 22, 2021 at 23:51
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    $\begingroup$ You might want to try this on the history of science and math stackexchange. hsm.stackexchange.com $\endgroup$
    – Alan
    Jul 23, 2021 at 0:20
  • $\begingroup$ Using Google ngram, the usage in English of “equivalence relation” and “equivalence class” seems to start climbing in the late 1920s.The concept could certainly be used under other names before, of course, but that gives a clear idea of when perhaps modern usage started. books.google.com/ngrams/… $\endgroup$ Jul 23, 2021 at 0:38
  • $\begingroup$ (I’d expect it was formalized initially in German, just because so much formalization started there. I just don’t know the correct German terms to search.) $\endgroup$ Jul 23, 2021 at 1:02
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    $\begingroup$ Of interest: link.springer.com/article/10.1007/s11229-018-1674-2 $\endgroup$ Jul 23, 2021 at 1:07

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As recommended by Alan, I asked this question over on hsm.SE.

Conifold captured some of the historical messiness in their answer. I recommend you read his answer, but the gist is that the notion was being developed near the start of the 20th century. Reference together of the modern name and formal definition of equivalence classes existed as early as the 1930's.

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