For Grothendieck it’s particularly ironic, because he had the right ingredients in his hands, too: The true microscopic nature of spacetime appears not from fiddling with the site for the classical geometry topos, but from replacing the phase space topos by its *motive*.

This works wonders already in the simplest instances: Consider a first approximation to a phase space on $n$ point particles roaming around in Euclidean space to be $\Omega \underset{{}^{\{1,\cdots, n\}}}{Conf}( \mathbb{R}^3 )$.

The corresponding motive is $\Sigma^\infty \Omega \underset{{}^{\{1,\cdots, n\}}}{Conf}( \mathbb{R}^3 )$. A decent first approximation to that is $H \mathbb{Z} \wedge \Sigma^\infty \Omega \underset{{}^{\{1,\cdots, n\}}}{Conf}( \mathbb{R}^3 )$. And this contains precisely the objects that Polyakov is talking about in that quote above.

(I am just running a tweet-seminar on this here. See there fore more details.)

]]>I have seen this happening a few times, that pure mathematicians, established and even famous for their deep work in pure mathematics, eventually become interested in physics… take the first step and immediately fall into this tarpit of naive discrete spacetime – set up by the gods there really to train young physics student to understand that they must trust the Dao of maths, not their wetware intuition.

(The same happened more recently to the founder of modern type theory, he spoke to me from out of this tarpit when we met in Paris some years back. I don’t know what became of him, maybe he managed to climb out again.)

Let it be known to all the pure mathematicians out there who feel interdisciplinary ambition arise within them: While spacetime is indeed not the Riemannian continuum at the Planck scale, its true nature is rooted in subtle deep modern mathematics, not in the naive discrete point set dust of the engineers.

Just in the other thread, we were quoting Polyakov on this topic, here.

]]>I should add too that in a July 9 1986 letter to Jun-Ichi Yamashita, Grothendieck said he is working:

to justify (within the framework of the “geometry of forms” or “analysis situs” I am developing) in terms of extant “general topology” a certain definition of “regular figure” (the combinatorial substitute for “variety”) I had in mind….

Apparently that idea is contained in the manuscript that this thread is about. (I learned this via John Alexander Cruz Morales via Zalamea’s Grothendieck book). Winfred Scharlau has a copy of this letter, but there are only a couple very brief snippets online.

It might interest people to hear that Grothendieck also says in that letter:

I have been intensely busy for about a month now, with writing down some altogether different foundations of “topology”, starting with the “geometrical objects” or “figures”, rather than with a set of “points” and some kind of notion of “limit” or (equivalently) “neighborhoods”. Like the language of topoi (and unlike the so-called “moderate space” theory forshadowed in the Esquisse, still waiting for someone to take hold of the work in store…), it is a kind of topology “without points” - a direct approach to “shape”. I do hope the language I have started developing will be appropriate for dealing with finite spaces, which come off very poorly in “general topology” (even when working with nonHausdorff spaces). After all, presumably the space-time space we are living in is finite - at any rate there is no philosophical evidence whatever that it isn’t, and still less, that it is adequately represented as a mathematical “continuum” (more specifically, as a topological or differentiable or Riemannian or pseudo-Riemannian “manifold”) - and as for physical evidence; it is clear there cannot be any by the very nature of things, as measurements never yield anything else but approximate locations of the would-be (ideal) “points”. These “points” however do not have any empirical existence whatever. As Riemann pointed out, I believe, the mathematical continuum is a convenient fiction for dealing with physical phenomena, and the mathematics of infinity are just a way of approximating (by simplification through “idealization”) an understanding of finite aggregates, whose structures seem too elusive or too hopelessly intricate for a more direct understanding (at least it has been so up to now).

(so, whatever this theory is, it is - while synthetic - very different than sdg)

]]>Did anything ever come of this? The github link now returns “404 not found”.

]]>@Shulman @Roberts, thanks for your suggestions the project is still in its embryonic stage and needs feedback. ]]>

Git(hub) repositories should *not* contain compiled files like PDFs, just as source code repositories should not contain executable files. I agree that it would be nice to share a PDF of the current state of a project for the benefit of folks who don’t want to go through the trouble of cloning/forking/compiling, but that should be done elsewhere, e.g. on a github.io site, github wiki, personal web space, nLab page, etc.

I guess some people only supply TeX, and let others compile at home after downloading/forking. Mike does (did?) the same in his GH repository. At this stage all the files look blank to me.

]]>Sure, but why not provide a pdf of the current state of the transcription.

Anyway. What I see there regarding founding geometry on “shapes” reminds me a lot of how Lawvere has been speaking, say at “Space ad Quantity” and elsewhere. Possibly the transcription and processing of these ideas already exists and is known as SDG. :-)

]]>@Urs the idea is that one should fork the GitHub repository, go to a file in the Grothendieck archives that is listed in the Readme (say 156-1), and attempt to transcribe it into TeX. The efforts are then merged in the main repo.

]]>It would be fantastic if this project does indeed succeed in typing up the notes. Looking at the originals, it is some job! But it is highly likely that there are interesting ideas in there.

]]>When I go to github.com/gcarmonamateo/GeomFormes i only see a list of TeX source files and a readme-file. The content of the readme file doesn’t seem to be for public consumption.

Could you provide a link to a pdf?

]]>I would like to know the great nForum community of "categorical physics" would be interested in contribute. Give it a chance and take a look into the project https://github.com/gcarmonamateo/GeomFormes and hopefully caught your interest in it.

(Sorry for the imprecisions in the English language). ]]>