Added

A video of the commemorative special session in honour of Bill Lawvere from CT2023 can be found here: recording

that was brought up in the other link. Not quite sure if I chose a place in the entry though.

]]>Thanks, I see.

Not sure what caused that error. I have tried again now and it worked. The file is now at :

]]>Thanks for fixing this. Actually, I meant to upload the file to nlab, to have a mirror. However, when I tried to do this, I got an HTTP 500 error when trying to upload, and later forgot to retry/edit.

It’s not a big deal either way. However, perhaps the server error needs to be looked into?

]]>Thanks. I have made it point (here) to that github page instead

and removed the request for a stand-along nLab page (which probably was not the intention)

]]>Matt Earnshaw is the guy who maintains a github for Lawvere’s works, and it seems he’s uploaded there a 94 page pdf on these lectures. Did he mean to upload it here, or just point to his page (here)?

]]>just to note that earlier today (revision 91) somebody added (here) a couple of broken links for “Introduction to Linear Categories and Applications”

Let’s either fix these links if we have them or else remove them

(GoogleSearch doesn’t know which file could be meant)

]]>As I’m sure most have seen now, Lawvere is no longer with us. Someone edited the page accordingly, but didn’t make a comment, so the change didn’t end up here.

]]>Added a recent reprint

*Toposes generated by codiscrete objects in combinatorial topology and functional analysis*, Reprints in Theory and Applications of Categories, No. 27 (2021) pp. 1-11, pdf.

[deleted]

]]>Changed dead link.

]]>Replaced a broken link by a link to Web Archive.

]]>Corrected a date.

]]>have added journal-links to

]]>Added the reference: *An elementary theory of the category of sets*, 1964

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.

]]>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 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.

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 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.

]]>@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.

]]>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.

]]>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.

“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).

]]>@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.