@HollisWilliams Hardly specific enough for the purposes of the question, which is about how mechanics in particular informs geometry; to me it's along the very general lines of Andy Putman's "And as the answers we have gotten show, what this question is really going to attract are the usual “inspirational quotes” about how amazing physics is." (Also I find your assertion of specificity here at odds with your comment below Carlo Beenakker's answer.)

@Vincent Is it? The quote from Fourier sounds pretty generic and non-specific; cf. Andy Putman's comments under the question, and also the comment by Sam Hopkins under Trunk's answer (now converted to a comment).

I read your post at MathSE. A few commenters said this looks hopeless (I agree), and one said that people here at MO might be able to explain why this is so hard. I don't think anyone will be able to answer your question as posed. Nor am I confident that they can convincingly explain why it should be so hard (why is the Collatz problem hard?), but maybe you'd get better replies to the latter question. But is this an idle question? Or do you have some good motivation for asking?

@SidharthGhoshal Oh, absolutely. I recommend Joyal's theory of species to your attention. (A species is a functor from the groupoid of finite sets and bijections to a suitable receiving category like the category of sets. The derivative $F'$ of a species $F$ is defined by $F'[S] = F[S+1]$. The free commutative monoid construction is the "analytic functor" induced from the terminal species; it is manifestly true that the derivative of the terminal species is again the terminal species.) But this comment box is too small to give such a hurried introduction.

The line between AM and PM is quite nebulous, as I'm sure you realize, and that first bullet point will read to many as quite offensive. In any case, it's too general and too strongly worded (as potentially impugning an entire community), and I would get rid of it. In any case, since it's clear that OP has issues with the reasoning of Riesel and Gohl, you're better off if you drill down on any specific weaknesses, and leave the generalities behind. The more precise you are, the better.

I don't understand what the first bullet point is driving at, but it sounds very much like it's saying "applied mathematics is not real mathematics". Otherwise, what is the connection between the first sentence and the second?

An awesome story. Only in Canada.

Anyway, @TheMathemagician [sic] is right -- it is a ridiculous sentiment.

@SylvainJULIEN No disrespect, but I think the question has become unwieldy; or perhaps more precisely, as can be seen from the most recent edit, the question(s) continue to be updated, and this is really not good MO practice. Could I ask you to please discontinue further editions of this question, now more than 10 years old? MO works best if the question statement can be quickly comprehended by an expert, without having to plow through large walls of text.

@D.R. Well, I agree it should be looked into. A "toy example", where I might begin, centers on the classical Stone duality, which from an algebraic geometry perspective concerns functors not on all commutative rings but just on Boolean rings. If the interplay between these various notions of duality could be fleshed out satisfactorily in that case, then this could reveal hints about a more general story.

You should wait at least a week before cross-posting to another site. Duplication of effort across multiple platforms = waste of people's time.

@ronno You're right. So "compactly generated" in the sense of this post is much stronger than "compactly generated" in the topology sense (which as you say is automatic for Banach spaces).

See ncatlab.org/nlab/show/colimits+in+categories+of+algebras Specifically, here: ncatlab.org/nlab/show/…

A left $R$-module $A$ is flat if $- \otimes_R A$ preserves monos, and similarly we can say an $R$-bimodule is flat if tensoring on the right and left preserves monos. So if the ring maps $R \to E_i$ are all injective and all the $E_i$ are flat as $R$-bimodules, this would seem to do the trick (prove first for finitely many tensor factors, then take the filtered colimit or directed union to extend to infinitely many tensor factors).

zeraoulia, as a general matter of policy and guidelines, multiple very open-ended questions are much frowned upon at MO; the term of art is "fishing expeditions". Also, the perception that you are posting this question mostly in order to advertise your paper will inevitably arise in the minds of community members, and in view of other issues raised by David Roberts, this may ultimately do you harm. My strong advice: close this question. Ask a proper MO question: a single question that is technically precise and requires no guesswork from the reader, and avoids the look of self-advertisement.