/sci/ - Science & Math

>>9409905
I've seen monads in places you people wouldn't believe

I write dapps on NEO and have this channel

https://youtu.be/cnK1E3yZGx0

desu I also consider it largely a meme, but it's hard not to make tens of thousands of dollars and I'm pretty sure it will go on like that for a few years and so I'll play along. Might well be that the platform that convinces me in all facets is coming along and then things just go right as well.

>>9322735
This isn't actually that bad of an idea. Can you elaborate on your interest?

>>9322766
Topic of interest?

>>9067113
For a system of digital money" transactions, in 2008 someone made the proposal for a practical use of what's now called blockchain. Roughly, it's a protocol (like http, if you will) for writing data on a servers (or rather a data set which copied to thousands of peoples computers) in a way that's immutable. You can effectively only remove stuff from a blockchain if you destroy everyones machines at once. That was implemented as bitcoin.
I actually made a video about the blockchain principle here
https://youtu.be/w3sI8WVX-cc
(otherwise the channel is for making videos about the dependently typed language Idris)
In 2013 people started to create protocols that let you deploy computer programs ("smart contracts"), to be executed in a decentralized fashion. And so that's the hype now.

>>9067129
Only vaguely. Next week I'll, for example, interview a startup which will make a "sidechain" to bitcoin that enables one to do financial contracts on this platform, and they are going to use F* (that's one of the two Microsoft projects, the other being LEAN)
https://en.wikipedia.org/wiki/F*_(programming_language)
I am interested in providing decentralized services in that regard, but mostly I just want to test the waters if I can make this a supported project, whether it be bitcoin or attention that make the time investment worth it.

Why don't you like it and what are the other books you have?

The standard, 'Categories for the Working Mathematician', can be a tough intro.
There is a book by Adwodey, but it's for CS people, mostly. Then there is one by Simmons, which does drawing diagrams as a tool very right, but lacks examples.

For the most part, knowing examples is the most important aspect, otherwise it's stale. Understanding the sense in which isomorphisms replace equality in the theory (as a theory itself I mean, written down in logic) is also quite relevant.

I don't know why you have a problem with natural transformations. Then /maybe/ look at functors as a homomorphism for functions an [math] \circ [/math]
[math] F(f\circ_C g) = F(f) \circ_D F(g) [/math]
akin to
[math] \exp(x+y) = \exp(x) \cdot \exp(y) [/math]
and a natural transformation as a homotopy of such homomorphism.
Whatever is the range of the one homomorphism (functor), you can direct it through the homotopy (natural transfomration) to get the image of another.

>Yoneda lemma
Then /maybe/ Try understanding the Yoneda embedding first

Depending on how quickly you need to learn it, I'll discuss category theory in a series of youtube videos this year.

To me, interesting science is either
- science news of any subject (items you consume in 2 minutes and then, often, there isn't much to discuss)
- 3rd year or above physics problems (which mostly translate to a problem in math immediately)
And even the math miss-understandings (if they are about a more advanced subject) often come down to an miss-understanding on a lower formal level, and thus people do more formal stuff here. My guess.

PS I'd add philosophy of math and science to your list of stuff worth discussing.

>>8183602
Going to higher equivalences for gauge theories was part of his PhD and so I assume someone else brought it to him.
I'm not sure to what extend you can claim he's reformalizing quantization. He's more extending the classical gauge theory to all n, with some tweaks so he can call it pre-quantization, but he doesn't really have proper results that are looked for in QFT. And who needs those anyway, tbqh.
The promise of the n-stuff to have cohomology theories pop out as cases of a general clean perspective on math itself is very appealing, I agree.

or
[math] b \mapsto (a \mapsto \sin(a)+3b) [/math]

The axioms of a topos have more than the sets as model, for example it provides an (intuitionistic logic), where the iso above means
„A and B being true implies C“
is equivalent to
„A being true implies B being true implies C“

Or you can take the classical identifications of the set theoretic operations as numerical operations. Forgetting what your finite sets are about, the above relation is the basic identity of numbers
[math] c^{a·b} = {c^a}^b [/math]
In the other direction, the relation
[math] c^{a+b} = c^a·c^b [/math]
provable in a topic, at the same time tells you for the categories of sets (where + is the disjoint/tagged union) that the function space
[math] (A+B) \to C [/math]
is iso (i.e. in bijection to) to
[math] (A\to C) \times (B\to C) [/math]
In logic,
„one of A or B being true already implies C“
is equivalent to
„A implies C and also B implies C“

In the other direction, a derivation rule like modus p.
„From A and A implies B follows B“
being true guarantees you that the function space
[math](A\times B^A)\to B[/math]
isn’t empty
Indeed, the evaluation function
[math] eval (a,f) := f(a) [/math]
works no matter which sets A and B are.

What I want to say is this: Formulating analysis synthetically in a basic object like a topos (as opposed to something hard to construct like R), makes theorems of analysis more than theorems of just analysis. And secondly, if you know that analysis works if a topos has this and that axioms, then you can further build your theory (like quantum field theory, kek) in such a setting without restricting you to muh reals and muh epsilons. Although the reals are nice of course.