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At topological vector space, there's a spot where a uniform space is characterised by giving a base of entourages. Zoran thought that it would be a good way to make clear that ‘entourage’ is a technical term by making it into a link. So now there is a page entourage.
Most of the details are still to be found at uniform space, however.
Thanks. In the meantime I forgot the meaning of entourage so it is getting educating me again :)
The Idea section at entourage said that entourages are analogous to open subsets, but they aren't; they're analogous to neighbourhoods. So I fixed that.
I also put in a bit about the infinitesimal entourage in nonstandard analysis. (It is the adequality relation stating that two hyperpoints are infinitely close to each other.) I might put in the corresponding relation between [ultra]filters in standard analysis.
Even more confusing is that non-standard analysts call an entourage a “monad”.
This kind of language is even used on popular math videos on youtube: The Infinitesimal Monad, The Infintesimal Monad (Extra footage).
Not to say that an entourage can’t be described by a monad.
Do dual numbers also have a notion of entourage? What about their n-dimensional generalization Grassmann numbers?
You mean that they call an infinitesimal neighbourhood a ‘monad’. At least that's what I've seen before and what's described on Wikipedia; I didn't watch the videos. Fortunately, we have the alternate terminology ‘halo’ for this. (The infinitesimal entourage is the adequality relation.)
But it's not their fault; they have more right to that term than the category theorists do. As I understand it (not having read it myself), Leibniz connected his philosophical ideas about monads to his studies of infinitesimal calculus in a way that makes this nonstandard terminology reasonable. In contrast, the category theorists were pretty much just grabbing random words from the history of philosophy by that point, as far as I know.
Dual numbers, or Grassmann numbers, form a topological vector space, and every topological vector space has a uniform structure. An entourage in that case is as described at entourage for a topological abelian group, since only the topology and the additive group structure are relevant. (But note that the page writes the group multiplicatively, while the vector-space operation is written additively.)
Duh. What am I thinking, of course Grassmann numbers and dual number do. Entourages are defined using a simple distance metric. Edit: Whatever a “distance” maybe.
You mean that they call an infinitesimal neighbourhood a ‘monad’. At least that’s what I’ve seen before and what’s described on Wikipedia; I didn’t watch the videos. Fortunately, we have the alternate terminology ‘halo’ for this. (The infinitesimal entourage is the adequality relation.)
My worry is this is going to confuse people even more trying to understand what a monad is. Who first used the name “monad” in the sense of endofunctors with a monoid structure as 2-morphisms?
As I understand it (not having read it myself), Leibniz connected his philosophical ideas about monads to his studies of infinitesimal calculus in a way that makes this nonstandard terminology reasonable.
That’s interesting, in view of the discussion at Science of Logic. Do you have a reference for this?
Re #6: isn’t “monad” in category theory a kind of back-formation from “monoid”? That’s what I’ve always thought, and so (in contrast to other fundamental notions such as “category” and “functor”) this isn’t a case of purloining a word from the history of philosophy.
@Urs. Monads are discussed in Gottfried Wilhelm Leibniz, Monadologie.
Keith, sure, what I am asking for is a pointer to a place that specifically supports the claim that Leibniz’s was inspired to his monads by looking at just what the synthetic differential geometrs later called monads: infinitesimal neighbourhoods.
My understanding of Toby in #6 is that this is indeed the case, and I would be grateful if anyone could quote passages that would support this.
SEP: Continuity and Infinitesimals has
The philosopher-mathematician G. W. F. Leibniz (1646–1716) was greatly preoccupied with the problem of the composition of the continuum—the “labyrinth of the continuum”, as he called it. Indeed we have it on his own testimony that his philosophical system—monadism—grew from his struggle with the problem of just how, or whether, a continuum can be built from indivisible elements.
There’s a reference to Russell’s book on Leibniz.
Hegel certainly saw Leibniz’s philosophical conception of the monad as flawed:
§ 1186: …those absolute limitations in the being-in-self or the in-itself of the monads are not limitations in and for themselves, but vanish in the absolute. But in these determinations are to be seen only the usual conceptions which are not philosophically developed, nor raised into speculative Notions. Thus the principle of individuation does not receive its profounder realisation; the concepts concerning the distinction between the various finite monads and their relation to their absolute do not originate out of this being itself, or not in an absolute manner, but are the product of ratiocinative, dogmatic reflection and therefore have not achieved an inner coherence.
Windowless monads reflecting to a greater or lesser extent the activities of other monads require an absolute monad to maintain this coherence.
Hegel’s Critique of the Infinitesimal Calculus and Analytical Practice might be interesting then.
Thanks. It’s precisely the thought that thinking of Leibniz’s “monads” not as being like “closed points” but as being like “points with (infinitesimal) halos” around them would both address Hegel’s criticism and at the same time make the formalization via differential cohesion go along neatly with the philosophical chat.
That’s why I am asking: What is support of the claim (suggested by Toby in #6) that when Leibniz came up with his concept of monad, that he was inspired by the concept in differential calculus that today we call an “infinitesimal neighbourhood”. Or in other words: is the use of “monad” as used in synthetic differential geometry actually the correct mathematical reflection of the concept that Leibniz was really thinking of, or that at least inspired him in the beginning.
That’s what I guess Toby is saying in #7. What would be a passage supporting this?
From English wikipedia:
(I) As far as Leibniz allows just one type of element in the building of the universe his system is monistic. The unique element has been ’given the general name monad or entelechy’ and described as ’a simple substance’ (§§1, 19). When Leibniz says that monads are ’simple,’ he means that “which is one, has no parts and is therefore indivisible”.[5] Relying on the Greek etymology of the word entelechie (§18),[6] Leibniz posits quantitative differences in perfection between monads which leads to a hierarchical ordering. The basic order is three-tiered: (1) entelechies or created monads (§48), (2) souls or entelechies with perception and memory (§19), and (3) spirits or rational souls (§82). Whatever is said about the lower ones (entelechies) is valid for the higher (souls and spirits) but not vice versa. As none of them is without a body (§72), there is a corresponding hierarchy of (1) living beings and animals (2), the latter being either (2) non-reasonable or (3) reasonable. The degree of perfection in each case corresponds to cognitive abilities and only spirits or reasonable animals are able to grasp the ideas of both the world and its creator. Some monads have power over others because they can perceive with greater clarity, but primarily, one monad is said to dominate another if it contains the reasons for the actions of other(s). Leibniz believed that any body, such as the body of an animal or man, has one dominant monad which controls the others within it. This dominant monad is often referred to as the soul.
and then in:
(III) Composite substances or matter are “actually sub-divided without end” and have the properties of their infinitesimal parts (§65). A notorious passage (§67) explains that “each portion of matter can be conceived as like a garden full of plants, or like a pond full of fish. But each branch of a plant, each organ of an animal, each drop of its bodily fluids is also a similar garden or a similar pond”. There are no interactions between different monads nor between entelechies and their bodies but everything is regulated by the pre-established harmony (§§78-9). Much like how one clock may be in sync with another, but the first clock is not caused by the second (or vice versa), rather they are only keeping the same time because the last person to wind them set them to the same time. So it is with monads; they may seem to cause each other, but rather they are, in a sense, “wound” by God’s pre-established harmony, and thus appear in sync. Leibniz concludes that “if we could understand the order of the universe well enough, we would find that it surpasses all the wishes of the wisest people, and that it is impossible to make it better than it is — not merely in respect of the whole in general, but also in respect of ourselves in particular” (§90).
@Todd #10:
Do you mean that ‘monad’ was derived from ‘monoid’ by editing the Greek suffix without regard for any previous uses that the new word may have had? As opposed to being chosen as a term from philosophy that resembles ‘monoid’, ignoring the philosophical term's meaning but not its existence (which is what I thought). In other words, you are saying that it's just a coincidence that the words that they came up with was a word that had already existed?
Some time spent with Google has found no justification for the use of ‘monad’ in nonstandard analysis. I have found a couple of things that seem to take the connection for granted (including some things in or linked from the nLab), but nothing that explains it.
I don't understand Leibniz's philosophy, but I would have thought that his monads, which are said to have no internal structure, would be points without their infinitesimal neighbourhoods, which seems to be exactly what Hegel is complaining about in the bit about Leibniz at Science of Logic.
@Toby: Foundations of Infinitesimal Calculus , Page 12, Def. 1.2.
Don’t forget the linear algebra version: https://en.m.wikipedia.org/wiki/Monad_(linear_algebra). This is often used in the context of vector bundles or sheaves of modules.
Toby, if what you are saying is that whoever it was (Mac Lane?) wanted a term (for ’triple’ or standard construction’) that had to be from philosophy, then I pause to wonder. So I think your first sentence is slightly closer to what I speculate, although the fact that it happened also to be a term from philosophy would have been known to Mac Lane right away, with a smile. It seems we agree that the meaning in philosophy was being ignored however.
that the words that they came up with
I was only talking about the one word, ’monad’. Of course Mac Lane says in Categories for the Working Mathematician that ’category’ and ’functor’ (back in 1945, or a little earlier before their first paper was published) were happily purloined from philosophy, ’monad’ came much later, around 1965 or so, I think.
Maybe someone on the Categories list knows for sure…
@Keith # 18: What do you want me/us to get from that reference? (By the way, for anybody else looking, it's page 12 of the PDF but page 2 of the document's page numbering system.)
@Toby, you were asking for justification. Unless I’m misunderstanding you.
Is it possible to talk about linguistic entourages? I could see something like this helping in computer science, but working on terms instead of topologies.
I don't see anything in Keisler to justify using Leibniz’s term ‘monad’ for an infinitesimal neighbourhood. He just uses it; he doesn't explain it, at least not on that page.
To clarify: I'm not looking for evidence that this term is in fact used in nonstandard analysis; I can find many examples where it is. I'm looking for a reason that doing so is in accordance with Leibniz's ideas on metaphysics (monadology) and analysis (the infinitesimal calculus).
Also, here is an English StackExchange question about the origin of the term in category theory with a thorough answer.
I don't understand Leibniz's philosophy, but I would have thought that his monads, which are said to have no internal structure, would be points without their infinitesimal neighbourhoods, which seems to be exactly what Hegel is complaining about in the bit about Leibniz at Science of Logic.
That’s exactly my impression. I was asking because it seemed to me that above in #6 you were saying that there is evidence that in fact really Leibniz thought of points with their infinitesimal neighbourhood included. But apparently not.
Aren’t Leibniz’s monads what nlab would consider truth values, his God being the value true?
@Urs #24:
The evidence is the rumour that the use of the term ‘monad’ in nonstandard analysis is related to Leibniz's use of that term. The rumour exists, so there is some evidence, but we've found no more than that.
Right, my module swings round to consider Leibniz, so may at last be able to answer #24.
One claim already in
The disappearance of the infinitesimal from Leibniz’s serious philosophical thinking is a bit mysterious. After only the briefest flirtation with the idea of “real infinitesimals” in metaphysics (the “mathematical points”), he consistently avoids speculating about the composition of matter from infinitesimals, and repeatedly points out that the existence of infinitely small portions of matter does not follow from anything he says (GM 3:524, 535f.) I suspect that while Leibniz feels pressure not to commit himself to the existence of infinitesimal quantities, he is not fully satisfied that they have been proved to be impossible. The most popular grounds for denying the existence of infinitesimals, besides a lingering sense of their simply being conceptually “repugnant,” are (1) the lack of any useful mathematical application, and (2) their incompatibility with the so-called axiom of Archimedes.
This continues:
By Leibniz’s lights, of course, “infinitesimals can at least be used in the calculus and in reasoning” (GM 3:535), so (1) lacks force for him. As regards (2), the axiom of Archimedes is so directly a denial of the existence of infinitesimals, it is hard to see that it supports any substantive argument for rejecting them. If infinitesimals are to be rejected, and the axiom upheld, some other, logically prior considerations ought to be brought to bear. At any rate, it is perfectly clear that by the spring of 1676 Leibniz takes the infinitesimals of his mathematics to be “fictions.” And in writings of many years later, he occasionally suggests that he has a positive argument against the very possibility of infinitesimal quantities (GM 3:524, 551). I have so far been unable to find such an argument. I suspect that the argument, if there is one, must be somewhere in his (extensive) mathematical writings of 1676, probably occurring in mid-March.
And for those following, the monad/infinitesimal relation is broken apart by
The determinacy of discrete objects owes to the presence of “real substances” that, as he says, “are in” actuals, by which he means that actuals are constructed out of, or founded on, “the multitude of monads or simple substances”(G 2:282). The world of matter decomposes in a particular way into parts, because there is a real underlay of substances that it is constructed out of in a particular way. The indeterminacy characteristic of continuous objects, on the other hand, is due precisely to the absence of any monadic substructure. There are “no limitations at all on how one might wish to assign parts” in continua because there is nothing back there constraining how those parts are to be assigned. Continua, unlike material objects, are pure constructs of the mind-ideals, entia rationis-and in no way actually incorporate substantial reality.
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