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I have added some accompanying text to the list of links at monad (disambiguation).
One question: in the entry Gottfried Leibniz it is claimed that the term “monad” for a functor on a category with monoid structure also follows Leibniz’s notion of monads. Is this really so? What’s a reference for this claim?
I am asking because I don’t see how the notion of monoid in the endomorphisms of a category would be related to what Leibniz was talking about. What’s the idea, if there is one?
I can see how the nonstandard analysis sense of monad relates to Leibniz’s, but the other seems very unlikely. Wasn’t that monad chosen because of monoid, itself not the greatest of names? Isn’t it pure chance that the infinitesimal shape modality is a monad?
So that’s what I am thinking, too, but I don’t have a source. Why was “monoid” called “monoid”? Was it after realizing it as a one-object category (“mono-object category”)?
The etymological explanation that I’ve heard (probably from Albert Burroni, but I’m not sure) is that Bénabou coined the term not based on the point of view of monad as a monoid in endofunctors, but as a lax functor of type 1 -> Cat. The idea of monad as ‘indivisible unit’ is meant refer to the domain of the functor.
4 this is for a monad; for monoid, I think it is Bourbaki.
Jonas, thanks! That makes sense. I have added a remark to this extent to monad (disambiguation). But might you have a reference for this?
Benabou defines monads as lax functors in section 5.4 of “Introduction to Bicategories” (1967), this could be the earliest reference.
Urs, you call a lax functor of type 1 –> Cat a “point” of Cat viewed as 2-topos. Do you have any intuition behind that? It does not seem to be a categorification of “point of a Grothendieck topos” as a geometric morphism from Set.
We shouldn’t give the impression that Leibniz took a monad to be some smallest part of space. Rather for him it’s a mind-like simple substance. Space is something of an illusion. The sense of it arises in the monad which is our soul by this monad’s ability to reflect the functioning of the other monads. These monads are not in any kind of spatial contact with each other, and have no influence on each other.
It’s a very odd system, supremely confidently delivered in The Monadology, as though there’s nothing surprising about it. Yet,
Bertrand Russell, for example, famously remarked in the Preface to his book on Leibniz that he felt that “the Monadology was a kind of fantastic fairy tale, coherent perhaps, but wholly arbitrary.” And, in perhaps the wittiest and most biting rhetorical question asked of Leibniz, Voltaire gibes, “Can you really believe that a drop of urine is an infinity of monads, and that each of these has ideas, however obscure, of the universe as a whole?” (Oeuvres complètes, Vol. 22, p. 434)
I’ve slightly modified accordingly.
I suspect Urs was thinking of global elements as “points”, but I don’t think there is any 2-topos involving lax morphisms.
Mike, I just said “canonical 2-topos” to indicate that this point (in the sense of global element, yes) is not a point in an arbitrary random context, but in what might be called the canonical context for lax points to be in.
I should have written “the 2-category underlying the canonical 2-topos”. But I see you don’t like mentioning 2-toposes here, and I won’t insist.
David, thanks for making this more accurate. Of course we might add with the discussion at The monad of Leibniz that if following Hegel in the Lawverian perspective as discussed there, then monads are $\flat$-modal units (if we follow the tradition of nonstandard analysis) or maybe $\flat_{inf}$-modal units (if we follow Hegel’s complaint about monads just being “für sich” and not “für eins” (no halo!)) and these could in particular be atoms of space, but could also be something very different, depending on which differential cohesion we have realized.
So there could be non-spatial forms of differential cohesion? Is this related to the way modalities need not be taken as spatial, ’It is necessarily the case that…’ rather than ’It is locally the case that…’?
Something I mean to think about is Neel Krishnaswami’s comment on modalities as reflections of judgements in related categories.
I just mean that the axioms are rather general. Already in simple cases like super-cohesion people wouldn’t generally agree that the $\flat$-modal types are atoms of “space”. They might complain that this superspace is an imaginary, hypothetical “space”, only. And this is still very close to the standard model. More general notions of cohesion should encode more exotic notions of “space” even.
I don’t see how “the 2-category underlying the canonical 2-topos” is any better, since as I said, I don’t know of any 2-topos which contains lax functors.
This issue is maybe not worth heavy discussion, but just to clarify what I actually said: there is a 2-topos $Cat$ and it has an underlying 2-category $Cat$ and we can consider lax functors $\ast \to Cat$ and it’s not unreasonable to call them points of $Cat$.
Oh, I misunderstood. I can see “lax point of the 2-category Cat”.
I have expanded further (thanks to prodding by Alexander Campbell in another thread here) adding references that point out Euclid’s and Sextus Empiricus’s use of “monad”:
Semën S. Kutateladze, Leibniz’s Definition of Monad, NeuroQuantology 4 3 (2006) 249-241 [arXiv:math/0608298, pdf]
E. I. Gordon, A. G. Kusraev, Semën S. Kutateladze, Infinitesimal Analysis, Mathematics and its Applications 544, Springer (2002) [doi:10.1007/978-94-017-0063-4]
and expanding the text a little further, accoringly.
Where Euclid’s points are atoms in the sense of geometry, so his monads are units in the sense of arithmetic.
Yes, I think that’s exactly right. I recall someone arguing that we should take Euclid as treating geometry and arithmetic quite separately.
Interesting the link from Leibniz to Bruno. I wonder what that earlier tradition was. The infamous charge that the Scholastics would waste their time in arguing about how many angels fit on the head of a pin is probably relevant. I believe it was used in the 16th Century to indicate how degenerate that style of thought was. But if you believe in angels it was natural enough to wonder about their space-filling properties. And I believe Aquinas was interested in whether multiple angels could be in the same place.
I see this wikipedia page is saying all this.
By the way, historians typically don’t use ’Dark Ages’ anymore, and if they did it would be long before Bruno, perhaps now ’Early Middle Ages’ (5th or 6th to 10th C). Then High Middle ages is about 1000-1350, and Late Middle Ages about 1350-1500. Bruno is then in the Early Modern Period. I’ll change the section name.
Ah, I see ’monad’ (or it’s latin equivalent) comes up in Aquinas when pondering the Trinity here. Makes sense that it should appear when a counting problem appears, three in one and one in three, etc. I’ll add a word on that.
Changed section heading to ’Middle Ages and Early Modern Period: Monads as metaphysical units’.
I was going to mention Aquinas on the Trinity, but to leave it at that would be to misrepresent things, as though the Ancients were rational, non-metaphysical enquirers, and the Middle Ages brings religion into it.
But then to tell the story properly we’d have to go back long before Euclid to the Pythagoreans who already mix arithmetic and religion. Then there’s Plato, Aristotle, the neo-Platonists, …
This wikipedia page indicates what we’d need.
Glad that we agree, for a change. :-)
Incidentally, I wrote “dark ages” on purpose. That whole affair gets quite in the way of monads as a mathematical term. But I guess we can trust the inclined reader to figure it out by themselves.
Glad that we agree, for a change. :-)
Typically about things that don’t matter too much :)
If I were to raise another issue here, I wonder a little why you want to cast things according to a kind of rational-mathematical/dark ages-religious split. That Pythagorean monad-God thing keeps reappearing through the Platonic then neo-Platonic/neo-Pythagorean line that I thought you liked. That feeds into the mystic element of Hegel we once spoke about - mysticism.
I have added more definitions from Euclid, Elements VII which make it crystal clear that by “monad” he just means the natural number 1:
Book VII, Definition 7:
περισσὸς δὲ ὁ μὴ διαιρούμενος δίχα ἢ ὁ μονάδι διαφέρων ἀρτίου ἀριθμοῦ.
Book VII, Definition 15:
In these definitions, “monad” is clearly the element “1” in the natural numbers, seen with the additive structure.
Euclid also talks about this as the multiplicative unit number. For this one has to note that by “one number measures” another he means “one number divides” another:
Book VII, Definition 5:
πολλαπλάσιος δὲ ὁ μείζων τοῦ ἐλάσσονος, ὅταν καταμετρῆται ὑπὸ τοῦ ἐλάσσονος.
The greater number is a multiple of the less when it is measured by the less.
With that understood, Euclid gives a crystal clear definition of prime number as one whose only lesser number dividing it is a monad:
Book VII, Definition 11:
I have dug out (now here) sources for the text by “Trismegistus” that Aquinus cites: This is the Book of 24 philosophers from ~1200 AD
which has this one line (as the first of a list of such statemenets):
Deus est monas monadem gignens is se unum reflectens ardorem.
The earliest attribution of the use of “monad” in this “gnostic” sense that I have found is to some Monoimus (150-210 AD) who the Wikipedia page claims “is known for coining the usage of the word Monad in a Gnostic context.”
I have briefly recorded that (here). Would be good to find a referenced source for this claim, but I am out of time for such investigations for today.
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