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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeAug 12th 2023

    added earlier pointer for the use of “monad” for “infinitesimal neighbourhood”:

    • Jerome Keisler, Def. 1.2 in: Foundations of Infinitesimal Calculus, Prindle Weber & Schmidt (1976, 2022) [pdf]

    and added publication data for:

    diff, v12, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeAug 12th 2023

    also this one:

    • Wilhemus A. J. Luxemburg, A General Theory of Monads, in: Applications of Model Theory to Algebra, Analysis and Probability, Holt, Rinehart and Minston (1966) 18–86 [ark]

    diff, v12, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeAug 12th 2023
    • (edited Aug 12th 2023)

    I have deleted the following comment from this page

    It is best to avoid the term ’monad’ for this concept on this wiki, since it has more or less nothing to with the categorial monads that are all over the place here (including elsewhere on this very page).

    and instead added another paragraph here.

    diff, v13, current

    • CommentRowNumber4.
    • CommentAuthorTodd_Trimble
    • CommentTimeAug 12th 2023

    Agreed; another case where “no relation” bites the dust. (But put me in the “happy coincidence” column!)

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeAug 12th 2023

    have dug out also Robinson’s original use of “monad”:

    • Abraham Robinson, p. 57 of: Non-standard analysis, Studies in Logic and the Foundations of Mathematics 42, North-Holland (1966), Princeton University Press (1996) [ISBN:9780691044903]

    While Robinson’s books refers back to Leibniz a lot (in the Introduction and then in chapter X) this all concerns Leibniz’s ideas of infinitesimals and calculus. I don’t see Robinson ever explicitly justifying the use of the word “monad”.

    diff, v14, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeAug 13th 2023

    added pointer to:

    and

    diff, v15, current

    • CommentRowNumber7.
    • CommentAuthorzskoda
    • CommentTimeAug 13th 2023
    • (edited Aug 15th 2023)

    Monads (edit: in categorical sense) play role also when talking the categories of quasicoherent sheaves on the infinitesimal neighborhoods of subschemes. In particular, the filtration of infinitesimal neighborhoods induces the filtration which in the endomorphism ring singles out regular differential operators along with a filtration on that subring. You may compare the situation in the theory of noncommutative schemes: differential monad, neighborhood of a topologizing subcategory, where the resolution of the diagonal, in the spirit of Grothendieck’s costratification leading to crystals and regular differential operators, leads to regular differential operators in the noncommutative setup.

    • CommentRowNumber8.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 15th 2023

    Re #6

    Only 3 pages of text and really very strange.

    I don’t think I’m moved from where we all seemed to agree (#17, #24, #27-29) in the discussion of entourage that Hegel was right to be critical of Leibniz’s notion of a monad.

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeAug 15th 2023

    Which is why it’s so useful that Kutateladze et al. recall that a sane notion of “monad” sensibly applicable to infinitesimal neighbourhoods is due to Euclid. (I am grateful to Alexander Campbell for pointing this out here.)

    • CommentRowNumber10.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 15th 2023
    • (edited Aug 15th 2023)

    OK, but to talk about Leibniz in this context is to misunderstand the weird thing he’s trying to do. We’re not in space and time. These are appearances generated by the reflections of other monads within the monad that is our soul. Monads don’t make up anything. They’re units unto themselves, with internal structure that is set up so as to reflect better or worse the structures of other monads, not through interaction but because they’re preordained to work that way.

    May seem whacky, but something of this is still in Kant. Space and time are merely forms of sensible intuition for the creatures that we are. They have no application to the thing-in-itself, theyre just the means we have to organise its effects upon us.

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeAug 15th 2023

    Probably these comments don’t pertain to the entry “infinitesimal neighbourhood”, where my reply from #9 seems to still apply to them?

    If you feel like expanding on Leibniz’s Monadology, the most pertinent entry would be either the entry Leibniz itself or monad (disambiguation), which currently has — after a more extensive paragraph on Euclid’s sane notion of monads — one sentence on Leibniz, here: “Leibniz’s Monadology (1714) … is now more famous than Euclid’s usage, but also more mysterious, if not puzzling.”

    Probably one could add after “…if not puzzling” that the irrational aspect of the discussion of monads may originate somewhere in the dark ages — Giordano Bruno already insisted to include gods and souls in his monadology, but even he does not seem to be the first: “the striking similarities between aspects of Leibniz’s monadology and Bruno’s doctrine of minims are probably attributable to the sources and philosophical interests that they shared in common” (from here).

    • CommentRowNumber12.
    • CommentAuthorHurkyl
    • CommentTimeAug 15th 2023
    • (edited Aug 15th 2023)

    In the definition

    It is the intersection of all of the standard neighbourhoods of pp and is itself a hyper-neighbourhood of pp, the infinitesimal neighbourhood of pp.

    Unless I’m misunderstanding what’s meant (’hyper-neighborhood’ isn’t given a definition, so I’m making what I think to be the natural assumption), this is wrong: generically, the monad at pp is very pointedly not an internal set, and thus should not be a member of the transfer of the class of neighborhoods containing pp.

    E.g. when applied to the real numbers, this remark seems to be implying that the set of infinitesimal hyperreals is an internal set.

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeAug 16th 2023

    Just to say that this passage was written in 2013 by Toby Bartels (revision 2), I haven’t looked into this. If you can improve on this, please do.

    • CommentRowNumber14.
    • CommentAuthorHurkyl
    • CommentTimeAug 16th 2023
    • (edited Aug 16th 2023)

    I’m mainly just checking to see if there’s something I’m missing (e.g. if “hyper-neighborhood” means something different that I expect) before making the change (I would just remove the clause calling it a hyper-neighborhood), since I’m sufficiently less than certain I’m not missing something.

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeAug 16th 2023

    No need to dance around a vague line from a stub entry as if it were gospel. If you have the expertise and energy, add a decent paragraph actually explaining infinitesimal neighbourhoods in nonstandard analysis! Would be a great service to the community.

    • CommentRowNumber16.
    • CommentAuthorHurkyl
    • CommentTimeAug 16th 2023
    • (edited Aug 16th 2023)

    I’ve only picked up a little NSA and in a rather piecemeal fashion, which is why my confidence is less than certain and wanted to take a moment to make sure nobody objects before making a change.

    I hadn’t really grokked the impact edit summaries have – I don’t have to worry about silently introducing an error, because the summaries mean I would be noisily introducing an error. I’ll keep that in mind going forward with small changes like this.

    • CommentRowNumber17.
    • CommentAuthorHurkyl
    • CommentTimeAug 16th 2023

    Removed the clause suggesting the monad of a point is a hyperneighborhood, since it will usually not be an internal set.

    diff, v19, current

    • CommentRowNumber18.
    • CommentAuthorHurkyl
    • CommentTimeAug 16th 2023

    Added a remark that it is constructed an external set.

    diff, v19, current

    • CommentRowNumber19.
    • CommentAuthorUrs
    • CommentTimeAug 24th 2023

    I was looking for concrete references in nonstandard analysis that say “halo” for “infinitesimal neighbourhood”.

    Found this one:

    • Diego Rayo, Def. 4.11 in: Introduction to non-standard analysis (2015) [pdf]

    but there must be more canonical ones.

    diff, v20, current