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In the category:people-entry “William Lawvere” I have created a subsection “Motivation from foundations of physics” where I want to collect pointers to where and how Lawvere was/is motivated from finding foundations for (classical continuum) physics.
Explicit evidence for this that I am aware of includes notably the texts Toposes of laws of motion and the introduction to the book Categories in Continuum Physics.
The Wikipedia entry has this about motivation from physics:
Lawvere studied continuum mechanics as an undergraduate with Clifford Truesdell. He learned of category theory $[...]$ found it a promising framework for simple rigorous axioms for the physical ideas of Truesdell and Walter Noll. $[...]$ meeting on “Categories in Continuum Physics” in 1982. Clifford Truesdell participated in that meeting, as did several other researchers in the rational foundations of continuum physics and in the synthetic differential geometry which had evolved from the spatial part of Lawvere’s categorical dynamics program). Lawvere continues to work on his 50-year quest for a rigorous flexible base for physical ideas, free of unnecessary analytic complications.
Question: Can anyone point me to more on this early phase of the story (graduate student is supposed to start to look into continuum mechanics, starts to wonder “What is a vector field, really?, what a differential equation?” and ends up revolutionizing the foundations of differential calculus)?
Found something concerning my question above: p. 8 of
The link seems to be down.
Sorry, which link seems to be down?
(The interview pdf works for me. Maybe the nLab was down, is that what you mean?)
Have you seen ’Comments on the development of topos theory’, in Development of Mathematics 1950-2000, Volume 1, Google Books? Especially p. 726. But there’s not much detail.
Ah, thanks for the pointer. I must have seen this before, but forgot about it.
Okay, let’s just be careful with that happens on the top of p. 726: that’s about “distribution on a topos” and that’s the notion where I still don’t really see what the impact is. There is the other notion of “distribuion on an object in a topos” which is the one I refer to above, and which is the one that has a clear impact, in that it subsumes and unifies traditional distributions, traditional generalized homology etc.
Then further below on that p. 726 I find exactly what I was looking for, another explicit statement of how he was all motivated by formalizing physics. I’ll extract this to the nLab entry now, too. Thanks.
I meant the pdf with the interview, but it’s working now.
Regarding distributions on toposes, I think Bunge and Funk talk about this in their monograph, I think. I didn’t spend time trying to understand it, but they gave a very down to earth example early on (in fact it’s mind-blowing how they get from a weather balloon to what they discuss next - but this is all from memory so caveat emptor).
Thanks, David, I know about Bunge-Funk. What i maybe still need is somebody to point out to me what is mind-blowing about it. I am still not sure if I see it.
I have further edited the entry William Lawvere. Added at the beginning more lines on the formalization of unity of opposites. Then I expanded the list of writings a bit more and tried to make sure that it is entirely in correct chronological order.
Finally I added the following further quote in the section on motivation from physics:
In 2000 in Comments on the development of topos theory Lawvere writes in the closing section 7 titled “From and to continuum physics”:
What was the impetus which demanded the simplification and generalization of Grothendieck’s concept of topos, if indeed the application to logic and set theory were not decisive. $[...]$ My own motivation came from my earlier study of physics. The foundation of the continuum physics of general materials, in the spirit of Truesdell, Noll and others, involves powerful and clear physical ideas, which unfortunately have been submerged under a mathematical appartus $[...]$. But, as Fichera $[25]$ lamented, all this apparatus gives often a very uncertain fit to the phenomena. The apparatus may well be helpful in the solution of certain problems, but can the problems themselves and the needed axioms be stated in a direct and clear manner? And might this not lead to a simpler, equally rigorous account? These were the questions to which I began to apply the topos method in my 1967 Chicago lectures $[$ Categorical dynamics $]$. It was clear that work on the notion of topos itself would be needed to achieve the goal. I had spent 1961-62 with the Berkeley logicians, believing that listening to experts on foundations might be the road to clarifying foundational questions.
$[...]$ Several books treating the simplified topos theory (MacLane-Moerdijk being the most recent and readable text), together with the three excellent books on synthetic differential geometry $[...]$ provide a solid basis on which further treatment of functional analysis and the needed development of continuum physics can be based.
In the course of this I have created further category:reference entries such as for
Speaking of Bill Lawvere: somewhere I read an interview of him where he was describing a debate he had attended between Dana Scott and someone else, maybe Montague, on the semantics of higher-order logic, involving at some point a discussion about induction along iterated membership chains, and how his (Lawvere’s) negative reaction to this was partly influential in his development of categorical set theory in the early 60’s. (This is to the best of my recollection, so possibly garbled.) But my past few attempts to track down this interview have failed. Does anyone know about this?
Not sure, I suppose you did look through the interview linked to above (pdf).
He does talk a bit there about his point of rejecting the material membership relation etc. But maybe it’s not exactly what you had seen.
But generally I second the reference request: if anyone has further pointers to documents by/about Lawvere, please share. It’s time to collect them usefully in one place.
added one more quote to the section on motivation from physics to the entry William Lawvere:
In the talk Toposes of laws of motion in 1997, Lawvere starts with the following remark
I read somewhere recently that the basic program of infinitesimal calculus, continuum mechanics, and differential geometry is that all the world can be reconstructed from the infinitely small. One may think this is not possible, but nonetheless it’s certainly a program that has been very fruitful over the last 300 years. I think we are now finally in a position to actually make more explicit what that program amounts to.
$[...]$ I think that on the basis of these developments we can focus on this question of making very explicit how continuum physics etc. can be built up mathematically from very simple ingredients.
In the same talk, a few lines later after discussion of infinitesimally thickened points $T$, it says:
The basic spaces which are needed for functional analysis and theories of physical fields are thus in some sense available in any topos with a suitable object $T$.
Urs, yes, I’ve read that interview. Not the one I was asking about.
Edit: There’s also this.
The website titled “Archive for Mathematical Sciences and Philosophy” claims here that it has archived several interviews with Lawvere over the years.
But I have trouble finding any link to anything archived on that site (?)
Just for the record, I found the source of that other interview (which was well hidden..)
It is
I have added it to the entry.
“Archive for Mathematical Sciences and Philosophy”
This is Michael Wright’s collection of hours of audio and visual recordings over decades. 35000 recordings! It includes a several day interview of Lawvere by some top people. I think one was Pierre Cartier.
The trouble has been one of finding the money to set up the archive.
So none of these recordings are available for download anywhere?
Not as far as I am aware, but I could ask if you like. Surely you know him too, photo. He attends many category theoretic events, and films them.
their website says: archmathsci.org is unavailable at the moment.
Do ask Michael Wright he is usually very helpful. The contact e-mail is at http://www.archmathsci.org/contact/
Thanks, maybe I’ll try. But concretely I would have needed material for tomorrow. That’s a bit late now.
@Urs #8
Only mindblowing in the sense analogous to moving, in one step, from the physical system of a mass falling freely under gravity to symplectic geometry, as if the former were sufficient motivation for the latter. As I said, I didn’t try to understand it either, so I’m in the same boat as you.
(continued from previous message)
The title of Goldblatt’s book (and not only his!) is in itself misleading. The purpose of topos theory and category theory is not primarily to provide an analysis of logic, but to permit the development of algebraic topology, algebraic geometry, differential topology and geometry, dynamical systems, combinatorics, etc. It emerged in the 1960’s that logic and set theory can and should be viewed as a special distillation of this geometry. In that way the actual achievements of logic and set theory are, reciprocally, enjoying much wider mathematical application.
Bill Lawvere
(The original discussion is here.)
Thank you so much! That is very interesting. Do you have any idea what the paper of Mitchell about constructibility he mentions is? I would very much like to read it.
@Mike
It appears to be this one. In MR, Blass wrote
In a final section he proves the consistency of the axiom of choice and the generalized continuum hypothesis relative to the theory of Boolean topoi with natural-numbers-object, by combining the tree technique with Gödel’s theory of constructible sets.
What Mitchell does is build a model of Z+GCH (check the paper for the precise axioms this means) from a boolean topos (not assuming AC) using constructible versions of the trees in the usual construction of a model of material set theory. He then takes the model of ETCS that comes from this.
“Archive for Mathematical Sciences and Philosophy”
I overheard Michael Wright talking to one of the Cambridge University Press people at a shindig we had at their shop on Monday. Wright said that they are working on transcribing a lot of material, and were looking for a publisher. Apparently Oxford UP said they weren’t interested, but the CUP guy gave Wright his business card (and then I moved on).
Oh, that’s disappointing. Lawvere’s comments made it sound to me as though Mitchell had given a categorical interpretation of constructibility, when actually all he does there is mimic the usual $\in$-theoretic construction inside the tree model.
McLarty mentions Mitchell’s result as circa 1975, whereas the paper David cited was 1972. So maybe there’s hope. Investigating…
Edit: Yeah, I have a strong suspicion that it’s actually this paper that Lawvere and McLarty were referring to:
W. Mitchell. Sets constructible from sequences of ultrafilters. J. Symbolic Logic, 39:57{66}, 1974.
I don’t have institutional access to say more on this, though.
@Todd that paper looks like pure set theory to me (I don’t have access, but I can read the MR review). He shows that from a collection $U$ of filters satisfying some conditions, the model $L[U]$ satisfies GCH plus $\diamond$ and the existence of a well-ordering of $\mathbb{R}$ of a certain restricted form. This hardly looks like something Lawvere would hold up as a good example.
Unless there was some unpublished work, I would claim that the best bet is the 1972 paper.
I assist David on this: the topic came up again on the list (link) and there Lawvere gives explicit reference with brief comment to
W. Mitchell Boolean topoi and the theory of sets (the membership-free content of Goedels constructible sets still needs to be clarified further) Journal of Pure and Applied Algebra, vol. 2, 1972, pp 261-274
It is also the only Mitchell paper that Lawvere refers to in ’development of topos theory’ (tac reprint), a paper that exposes his views on universes and set theory (funnily, a picture of Specker appears there although his name is mentioned only once in passing!). The man really likes to play paper-chase.
Hm, okay. I’m all out of ideas then. (Lawvere says this work of Mitchell is a “tour de force” and that his result is “startling”; Mike seems to think it doesn’t really merit such descriptors. I haven’t read it myself.)
I have added another reference to the entry on William Lawvere, which I also gave its own category:reference-entry:
In the course of this I tried to streamline the paragraph on Relation to philosophy a little and added the following quote from Law92:
It is my belief that in the next decade and in the next century the technical advances forged by category theorists will be of value to dialectical philosophy, lending precise form with disputable mathematical models to ancient philosophical distinctions such as general vs. particular, objective vs. subjective, being vs. becoming, space vs. quantity, equality vs. difference, quantitative vs. qualitative etc. In turn the explicit attention by mathematicians to such philosophical questions is necessary to achieve the goal of making mathematics (and hence other sciences) more widely learnable and useable. Of course this will require that philosophers learn mathematics and that mathematicians learn philosophy.
I should say that in this vein I have been collecting material on the nLab as of late, all of which however is in rough form, just collecting notes. For instance at objective and subjective logic, Baruch Spinoza, Spinoza’s system, idealism, speculation, mysticism, Meister Eckhart and maybe other. And, yes, I keep working on the entry Science of Logic, even though I am trying to stop.
I added a couple more items to the list of Lawvere’s papers, and tried to get the list closer to being in chronological order. (It was close, except for a bunch of items at the bottom.)
I also removed the word “pure” from the sentence
F. William Lawvere is an influential pure category theorist.
As far as I can tell he was always interested in applying his ideas, and - unless someone convinces me otherwise - I think he’d reject being characterized as a “pure category theorist”.
I updated links to a couple of IMA preprints that had changed, and links to Numdam from pdf-direct links to the article page (so people can get djvu if they want).
Note also that the wordpress blog conceptual mathematics is not accessible any more, without a password, so links to there are kind of useless to most people.
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