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Auguste Comte is often described (e.g., on Wikipedia) as a “mathematician” besides being a philosopher of science. I am aware that he taught mathematics (he was at various times a répétiteur and/or an examinateur at the École polytechnique), and he was acquainted or even friends with prominent French mathematicians of the time such as Henri Navier (of “Navier-Stokes” fame) and Joseph Fourier. Also, of course, he placed mathematical sciences at one end of his hierarchy of sciences. But I am not aware of any original mathematical result by Comte himself, or whether he did any mathematical research. Nor could I find any publication of his, but I imagine “publication” worked very differently in the early 1800's than now, and I didn't really know how to search, so this doesn't mean anything.

So anyway, question: did Auguste Comte do any original work in mathematics himself? If so, what did it involve?

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    $\begingroup$ In the narrow sense, none, by Comte's own admission, see Pickering, Comte and the Académie des sciences:"Admitting that he had not made any special contribution to mathematics, he asserted that an eminent specialist was often a mediocre teacher... he explained that his concentration on philosophy had precluded him from making any special contributions to the sciences... members noted that Comte had not studied mathematics since being expelled from the École polytechnique in 1816. He had not even received a degree." $\endgroup$
    – Conifold
    Feb 22 at 1:02
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    $\begingroup$ In the broader sense, Comte's book length Philosophy of Mathematics (book I of his Course in Positive Philosophy) was one of the first comprehensive studies of the subject. $\endgroup$
    – Conifold
    Feb 22 at 1:05

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Comte's main contribution to mathematics is the careful set of notes he kept of Cauchy's course in analysis in the 1820s. This is mentioned in Schubring's book

Schubring, Gert. Conflicts between generalization, rigor, and intuition. Number concepts underlying the development of analysis in 17–19th Century France and Germany. Sources and Studies in the History of Mathematics and Physical Sciences. Springer-Verlag, New York, 2005.

As I recall, the first few pages of Comte's notes are unfortunately missing.

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