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added earlier pointer for the use of “monad” for “infinitesimal neighbourhood”:
and added publication data for:
also this one:
Agreed; another case where “no relation” bites the dust. (But put me in the “happy coincidence” column!)
have dug out also Robinson’s original use of “monad”:
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”.
added pointer to:
and
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.
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.
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.)
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.
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).
In the definition
It is the intersection of all of the standard neighbourhoods of and is itself a hyper-neighbourhood of , the infinitesimal neighbourhood of .
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 is very pointedly not an internal set, and thus should not be a member of the transfer of the class of neighborhoods containing .
E.g. when applied to the real numbers, this remark seems to be implying that the set of infinitesimal hyperreals is an internal set.
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.
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.
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.
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.
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