/sci/ - Science & Math

>— I want to make the strong claim that in the foundations of mathematics, one should replace topological spaces with condensed sets (except when they are meant to be topoi — topoi form a separate variant of topological spaces that is useful, and somewhat incomparable to condensed sets). This claim is only tenable if condensed sets can also serve their purpose within real functional analysis.

>— if it stands, the theorem gives a powerful framework for real functional analysis, making it into an essentially algebraic theory. For example, in the Masterclass, Clausen sketched how to prove basic results on compact Riemann surfaces or general compact complex manifolds (finiteness of cohomology, Serre duality), and one can black box all the functional analysis into this theorem. Generally, whenever one is trying to mix real functional analysis with the formalism of derived categories, this would be a powerful black box. As it will be used as a black box, a mistake in this proof could remain uncaught.

>— while I was very happy to see many study groups on condensed mathematics throughout the world, to my knowledge all of them have stopped short of this proof. (Yes, this proof is not much fun…)

>— I have occasionally been able to be very persuasive even with wrong arguments. (Fun fact: In the selection exams for the international math olympiad, twice I got full points for a wrong solution. Later, I once had a full proof of the weight-monodromy conjecture that passed the judgment of some top mathematicians, but then it turned out to contain a fatal mistake.)

>— the Lean community has already showed some interest in formalizing parts of condensed mathematics, so the theorem seems like a good goalpost.

>— If achieved, it would be a strong signal that a computer verification of current research in very abstract mathematics has become possible. I’ll certainly be excited to watch any progress.

>>10420747
https://math.stackexchange.com/questions/583609/p-adic-numbers-and-group-characters
Same question here

The thread shows that it's a bit hard to find a mathematician that fits your criteria. You want to rule out anybody before Riemann but at the same time have a "profound effect". You can always argue that Poincare and Hilbert were sort of universalists and that they actively tried to build physics theories, and at the same time after 1930 there was only stuff like laser physics that's really profound in terms of consequences and not engineering. I don't know how you'd rate the pureness of Lie or Kolmogorov. That leaves about 50 good years you can pick from, doesn't it.

But to at least mention potentially valid candidates, let's go with Minkowski.

>>9303164
While I'm not sure if your stance can hold, I didn't want to defend Kuhn. I think he tells a too simple narrative, but there are surely changes in mindsets that come with switching of generations of academics. I think it's valid to speak of a Maxwell/Planck generation, or a Einstein/Bohr generation, and then Feynman/Dyson and so on. Those folks determine the mindset and accept things that have previously not been accepted, and there you have a sort of "revolution".

>>9304040
I'd argue that most scientists don't care, or are not very aware of the various philosophical streams of thought. Especially in India/China, there doesn't seem to be much discussion at all. I might be wrong about this.
Feyerabend is "too edgy" or too extreme for any larger part of the scientific community to agree with him. Popper is 100 years old by now, no? I've read some books by folks after those, but whenever I have the feel I'm dealing with theorists who don't properly had the goal to advance the scientific subjects themselves, I tend to quickly drop those books.

>>9304046
I think positivism has been watered down and we mostly just have pragmatists who don't actually care or larp for positivists once they have to answer hard questions they don't know the answer to.
ad. Ah, yes I finally read your last sentence and I agree

In case you are actually up to a challenge an a project of a few months, I want to write a compiler from a dependently typed language to a smart contract virtual machine (NEO blockchain). The target platform already has a compiler from a subset of C# and some Java and Pythons in development too.
There are already a few projects that use pure functional languages for this purpose, and one of them even does the whole node implementation in Haskell.

>Mathematics is the language of the realm of mind; it is the very fabric of mental activity.
doubt it.

>The image I have in my head is one of those Japanese paper fans, where the basepoint is the hinge, the algebra is the handle, and the paper folds are copies of the algebra held together by algebraic rules.
Interesting.
>copies of the algebra
It's representations?

>held together
do you speak of maps among those copies, or the rules of the algebra that the copies obey?

>duality
>R^n and Z^{\oplus n}
If we were to start from Z^{\oplus n} (which appears inherently countable), and work through to R^n from there, where do we introduce a seed of accountability here?

I read the other posts when I come back from the dentist. Add dates to the posts.

>>8570565
I've never encountered a "book" made of a sequence of articles by different authors like that, which I liked.
And especially if it's about branes, you need not feel bad for being lost as each author will assume a different background. And the supersymmetry signs will always be wrong.

>>8570564
Why is that? Apart from my never being happy with the treatment of more philosophical aspects of GR, like the equivalence principle, I think Wald is a good read.

On that topic: I'm interested in the expression of the derivative of a vector in curvelinear coordinates. That is...
In Euclidean space, if

[math] {\bf x} (t) = \sum_{ i=1 }^n x^i(t) {\bf e}_i (t) [/math]

you have

[math] \dfrac {\partial } {\partial t} {\bf x} (t) = \sum_{i=1}^n \left( \left ( \dfrac {\partial } {\partial t} x^i (t) \right){ \bf e}_i (t) + x^i (t) \dfrac {\partial } {\partial t} {\bf e}_i (t) \right) [/math]

And I'm looking for that general expression (and in fact the second derivative too) for a setting where the inner product is a general Riemannian metric g, with all em Christoffels and whatnot. Is it maybe even in that book?

>>8416345
Since I'm interested in what you do, you speaking about your stuff would be great for a conversation - assuming you try and explain the basic stuff too. I think I remember you doing an elaboration thread and a series of posts, but it wasn't really basic at all.

>>8416317
Demo the idea? I'm not at my new place yet, but you can do videos and rant quickly, like so
https://youtu.be/eAffVHGjeR8

Well who are you and what can you do?
I'd eventually do "teachy" videos, but to gain some critical mass an interest, one would first want to do very informal discussion or accessible topic stuff, then one can start getting to the roots of things and go towards topics that are more popular in research math.