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I worked on synthetic differential geometry:
I rearranged slightly and then expanded the "Idea" section, trying to give a more comprehensive discussion and more links to related entries. Also added more (and briefly commented) references. Much more about references can probably be said, I have only a vague idea of the "prehistory" of the subject, before it became enshrined in the textbooks by Kock, Lavendhomme and Moerdijk-Reyes.
Also, does anyone have an electronic copy of that famous 1967 lecture by Lawvere on "categorical dynamics"? It would be nice to have an entry on that, as it seems to be a most visionary and influential text. If I understand right it gave birth to topos theory, to synthetic differential geometry and all that just as a spin-off of a more ambitious program to formalize physics. If I am not mistaken, we are currently at a point where finally also that last bit is finding a full implmenetation as a research program.
added DOI-s to:
Anders Kock, Synthetic Differential Geometry, Cambridge University Press 1981, 2006 (pdf, doi:10.1017/CBO9780511550812)
Anders Kock, Synthetic geometry of manifolds, Cambridge Tracts in Mathematics 180 (2010) (pdf, doi:10.1017/CBO9780511691690)
Re remark 2.1 (Peirce’s quote that the “idea of an infinitesimal involves no contradiciton” supposedly foreshadowing non-classical logic): doesn’t he rather mean that the concept of infinitesimal is not inconsistent? The immediately preceding paragraph in the article (Peirce 1892: 537) is this (emphasis mine):
Most of the mathematicians who during the last two generations have treated the differential calculus have been of the opinion that an infinitesimal quantity is an absurdity ; although, with their habitual caution, they have often added “or, at any rate, the conception of an infinitesimal is so difficult, that we practically cannot reason about it with confidence and security.” Accordingly, the doctrine of limits has been invented to evade the difficulty, or, as some say, to explain the signification of the word “infinitesimal.” This doctrine, in one form or another, is taught in all the text-books, though in some of them only as an alternative view of the matter ; it answers well enough the purposes of calculation, though even in that application it has its difficulties.
Sorry for being unclear: the Peirce quote is intended to exhibit only his advocacy of infinitesimals. I added a further quote (from John Bell) as a pointer to his views of the role of LEM in “this” context. In fact, the passage on LEM that Bell goes on quoting concerns points on a continuous line for which LEM is supposed to be invalid and is taken from a 1903 note of Peirce published in vol.III of “The New Elements of Mathematics”, p.xvi. Feel free to improve the passage!
Wikipedia has this article called Abstract differential geometry
Added reference
Anonymous
I couldn’t fix it, but (p. 9 ) is for some reason linked to the nonexisting page https://users-math.au.dk//kock/sdg99.pdf#page=9 and not https://users-math.au.dk/kock/sdg99.pdf#page=9 Seems like a bug in markdown? Anyway, in order to signal the error I removed the bracket, so at least the reader can see the link. Sorry, if that is not the accepted way, but I couldn’t save the page unchanged.
Dan Synek
Re #8: This was caused by an incorrect redirect at the Aarhus University website, not an nLab problem. Fixed now.
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