/sci/ - Science & Math

https://www.forbes.com/sites/cognitiveworld/2019/07/29/the-future-will-be-formulated-using-category-theory/#58ce9d4d625e

>>11111796

When talking about monads etc, I will now lead with the fact that the universe's consciousness is written in category theory

bump

Black ops now.

>>11111796
Isn't category theory just arrow autism? I get it's usefulness as a foundation but how is it meant to explain conciousness?

Does anyone else find this recent obsession of /sci/ with category theory fascinating? From close to zero mentions of it to two active threads as of right now plus some memes around it, within just a year, and it's not like it didn't exist last year either. I think that /sci/ is simply ahead of the curve of trends because the culture here has a high "velocity" so to speak, i.e we get through things quite quickly, and that we might see a similar shift in the mainstream soon enough. I wouldn't be surprised at all if popsci youtubers start making videos about it within a few months. However the fact that this shithole was somehow ahead of the trend is still quite fascinating.

>>11112096
>obsession with category theory
>recent trend
dude are you in the 60's or what ?

>>11111796

>>11112096
dude walk around the undergrads in a department sometime, the category theory meme is not a meme at all, it's real
obviously not everyone is like that but there are definitely quite a few people who think it's hilarious and epic to dive headfirst into categorical autism (reading all of nlab, taking everything anyone says and putting it in categorical context, etc)
perhaps it's not new, though i've only noticed it more recently. but it's definitely not something /sci/ is ahead on.
i'm surprised that there don't seem to be popmath youtube channels going through all the categorical definitions and examples. that actually sounds like something people would like since it's just arrow following usually.

>>11112098
I'm talking about mainstream culture, not academia

>>11112220
>i'm surprised that there don't seem to be popmath youtube channels going through all the categorical definitions and examples. that actually sounds like something people would like since it's just arrow following usually.
This is exactly what I'm trying to say, that a lot of people here (and undergrads in general) are all obsessed with it but the most you can see on youtube are some lectures with 20k views. What my post was around is the fact that /sci/ is the first place on the internet to create memes and culture around that topic, before people like 3b1b start creating videos and the rest of the mainstream becomes category theorists. There is literally not a single other site that talks about category theory as much as this place which sometimes feels like an isolated room of the same 10 people flinging shit at each other, yet somehow managed to get ahead of a pop culture trend and perhaps even define it with its stereotyping memes. If any of the popsci math channels start creating content about it tomorrow, it won't be because the undergrads suddenly started finding it interesting, it will be because stuff like this >>11112205 and this >>11111930 will reach the science subreddits as imported shitposts, which will then influence culture there, which will then make one youtuber interested enough to transfer that culture on to youtube as well to pander to the subreddit. I'm saying this because this has already happened before but no one seems to see it, with shit like https://www.youtube.com/watch?v=N0IGrSjcBZs . Yes, it's a problem that has existed before the /sci/ shitposts but it was specifically /sci/ which defined it as mainstream culture, which is what my post was around - a dissection of the influence of this irrelevant small place on mainstream culture.

>>11111796

>>11112096
>>11112237
I think the increasing popularity of category theory is coming from people getting to know it from programming.
Haskell and fp is getting more and more popular

Holy shit you dense faggots. Stop calling everything a meme, this isn't 9fag.
Meme isn't synonymous to joke. Category theory became a subject of interest among many people, so it is literally a meme.

name 1(one) practical use of category theory

>>11112439
It makes you feel smarter.

>>11112439
i made my phd on it

>>11112439
>name 1(one) practical use of category theory
listen to the fucking video

Every useful application of CT to generalising higher mathematics can be done quicker and easier with HTT. The only reason CT gets all the fame is because it's slightly more approachable for undergrads.

Baez is a cunt and I can't stand him. I am the Lord.

>>11112439
Keeping the aspies/OCD occupied

>>11112096
its not new, its been a meme for quite some time here

what are pre requisites for category theory? is it just extension of set theory?

>>11113590
good knowledge of abstract algebra, general topology, and algebraic topology.

Mathlet here (the highest level stuff I know is basic analysis, integrals, differentials). What exactly is category theory and why is it such a meme here?

starter pack

>>11112439
It gives you language to talk about commonly occuring things succinctly in algebraic topology and algebraic geometry.
For example instead of saying "the maps in the LES of a pair commute with the maps induced by post-composition as in the following diagram...", we just say "the maps in the LES of a pair are natural" and it is clear/obvious what we mean.
Another example would be instead of "there is a bijective correspondence between affine varieties and fin gen reduced k algebras. Regular maps of affine varieties induce maps of fin gen reduced k algebras and vice versa (so the maps are also in bijective correspondence). The map induce by a composite of two maps is the composite of the two induced maps, blah blah blah..." we can just say "there is an equivalence between the category of affine varieties and the category of fin gen reduced k algebras."

>>11114010
why are you on this board?

>>11111796
shouldn't undergrad category theorists be told to study model theory
it seems pretty natural to tell them to stop studying a meme subject and study a real subject instead

>>11114010
the concept behind category theory is to amp up the abstraction level to 11
https://www.youtube.com/watch?v=uMSV4OteqBE
it's based on the idea that instead of being a set of pairs, a homomorphism (now called just a morphism) is just an opaque object that can be composed with other homomorphisms
this is a fine concept, but category theory tends to study proper classes instead of sets, so it is really a subject of meta-mathematics
as motivation for studying meta-mathematics, category theory is fine, but I'm no fan of the idea of treating maps as opaque objects, nor of the practice of considering proper classes before developing the theory as it relates to sets

>>11112240
old meme, keep up with the times

>>11111830
1. woman
2. not white or Russian
3. in industry, not academia
4. publishing in Forbes
5. CEO, not engineer
6. cites Scientific American
7. the only formula is in an unrelated image, and the formula isn't even from category theory
8. irrelevant futurist theme
9. slathered in philosophy, metaphysics, and hippie-talk
https://www.youtube.com/watch?v=GmhcZqI-4Qg

>>11114701
To say category theory speaks of classes is to presuppose a set theoretic foundation of mathematical objects.
You don't have to put all mathematically conceivable and classical-logic-consistent objects in one framework to start with.
Fair enough to consider it meta-mathematics

>>11114759
Saunders MacLane is simply lying when he says, in Chapter 1,
>First we describe categories directly by means of axioms, without using any set theory,...
and then goes on to say
>A *metagraph* consists of *objects* a,b,c, ...
first of all, he doesn't know what set theory is for, which is why he wants to get rid of it
second, any mathematician will naturally want to use set theory to study the objects he has defined, so saying
>don't use set theory to study categories
is perfectly ridiculous, he's essentially demanding special treatment from his peer group (other mathematicians)
frankly he is peer pressuring mathematicians in a way that is totally unethical and inappropriate
people like this who want special treatment are trouble
they aren't honest and they aren't willing to share their work at the same level as other mathematicians
if you are a real mathematician, you won't present your findings while begging the reader
>please don't use set theory to study this subject

>>11114793
from Categories for the Working Mathematician, 2/e

>>11111830
>schizoposting in Forbes Magazine

>>11114793
>>11114798
>doesn't use set theory
isn't the same as
>doesn't use sets or anything like a set
The idea is presumably to use only naive finitary operations and avoid the higher-powered axioms such as foundation and replacement. Though on closer inspection they sometimes end up using those anyway, e.g. famously the "grothendieck universes" behaving like inaccessible cardinals

>>11114793
Not sure why you cite MacLane, though.
It might well be that he is not doing foundations without set theory, but you can indeed do it.
Moreover, using collections isn't equal to using set theory. Counting things and considering their interaction isn't set theory as such either. Set theory is a theory of the membership relation - it's axiom do a lot that in any given situation where you handle some notion of collection you will probably not need.
If you do universal algebra and study the theory of groups, then the domain of discourse (all groups) is also a collection, but you're not doing or sneaking in set theory.

Btw. I personally don't think one should drop either framework.

>>11111796
>>11111830
Absolutely based

>>11114840
>The idea is presumably to use only naive finitary operations and avoid the higher-powered axioms
>>11114855
>foundations without set theory
this is why I can't stand category theory: it has become wedded to this idea that you can get rid of set theory without understand why set theory is useful
furthermore, category theory seems to be hostile to the idea that set theory could shed some light on category theory
I think I could argue that category theory is anti-mathematical in nature and is actually a part of sophistry.
For example: what other math book has the word "mathematician" in its title?
https://en.wikipedia.org/wiki/Categories_for_the_Working_Mathematician
I think the most damning thing I can say about this book is that it makes anti-mathematical demands on the reader by launching an unjustified, prejudiced attack on set theory.
Math books simply don't have the authority to say
>don't use subject X to study the material in this book
yet that is what this book has done when X=set theory

>>11115392
>For example: what other math book has the word "mathematician" in its title?
It's some kind of meme title based on other book titles that were popular at the time. Sort of like how "The Joy of Sets" is patterned after "The Joy of Cooking" which was an enormously popular cookbook.
>I think the most damning thing I can say about this book is that it makes anti-mathematical demands on the reader by launching an unjustified, prejudiced attack on set theory.
No it doesn't. It just says you don't need to be working in any particular system of axiomatic set theory to look at these objects, at least at the level considered in the book. Just like you don't need to worry about axiomatic set theory to prove the infinitude of the primes -- even though you would normally form "the set of all primes" as part of the proof. Any other interpetation is sophomoric, and Mac Lane was no sophomore.

has literally anything important been done with category theory? and no, intentionally choosing to solve a tiny problem with category theory instead of using something else that would have worked as well doesn't count

>>11115572
It is the language of the consciousness and of Cyberspace, Aquaspace, Geospace, and Space (CAGS). It is the language of the universe's internal monologue.

>>11115392
>this is why I can't stand category theory: it has become wedded to this idea that you can get rid of set theory without understand why set theory is useful
You could not be more wrong. Category theory is one of the few branches of math where set-theoretic foundations are present. In arithmetic, algebra, geometry, analysis, and most of topology and discrete math, set theory is ignored at best and dismissed as irrelevant at worst.
Outside of set theory itself, you won't find anything like Freyd's adjoint functor theorem that gives as much weight to set-theoretic considerations (and no, the continuum hypothesis isn't set theory, it's second-order arithmetic), short of maybe some results on the axiom of choice and its variants. And even then, it'd have to measure up against the categorical formulation in terms of projective objects, and all its attendant generalizations and refinements.
Instead of obsessing over a single chapter in a single book by a single author, consider looking at the results and achievements of category theory in its entirety.

>>11115586
>the continuum hypothesis is second-order arithmetic
>functions on the power set of the power set
>second order
Good luck with that and good luck with your third year of undergrad faggot

>>11115572
Anything in algebraic geometry since the 60s used the language.
The collections of all projections of some sort onto a topological space is a sheaf. Category theory novelties amounts to that and not much more.
The other part is universal constructions, which you can barely do without whenever you have a notion of tensor. As such, it's barely avoidable.
>>11115392
You're projecting - few users of category theory are hostile towards set theory.

There are not many topos theorists overall, you'd be the ones to make such a point. And those guys usually have a deep understanding of set theory axioms

>>11114752
1. incel

>>11115572

>>11115586
>set theory is ignored in analysis
>CH isn't set theory it is second order arithmetic

Holy shit, you people really have no idea what you're talking about.

we need some kind of shootout to determine whether
- category theory
- set theory
- model theory / meta-mathematics
is the one, true undisputed supreme master

>>11117549
no, we don't. why not use all?

>>11117581
because categories are the future

>>11117581
because we need to turn math into a high-velocity adrenaline-charged sport

Now what are those fags arguing about?

>>11111796
Here is my main gripe with category theory: this book
> Category Theory for the Working Mathematician
should be more of an ad for model theory than it is
in fact, there should be some correspondence between logic and category theory since category theory is just meta-mathematics
the book is fundamentally naive, and that is what makes it so infuriating
it should have been sophisticated and beatific, but instead it was slow and disabled
the reason for this, I think, is this obsession with diagrams as if that somehow were sufficient argument to dispense with questions of meta-mathematical rigor
furthermore, category theory injects pure arrogance into mathematics with insistence that objects that exist are
> small
in the sense of small cateogries
whereas objects that don't exist are either regular sized or large, depending on your point of view
the problem here is simply adding a new non-standard name for a concept that already has two names:
- condition
- class
and instead of introducing the necessary meta-mathematical equipment and saying
- a (large) category is defined to be an n-tuple of conditions of the following form
where the n-tuple is
- condition: is this an object in the category
- condition: is this a morphism in the category
- ...
we are just supposed to assume that all of that meta-mathematics and logic is trivial
the level of rigor simply isn't up to that of the rest of mathematics
not only rigor, but finding a fit with existing fields of mathematics and meta-mathematics

>>11114175
Why are you a pompous asshole?

>>11112096
If you want to do practically anything nontrivial with software without writing a ball of mud, you'll end up leveraging category theory even if you don't know it.
We've known this for literally several decades.

>>11118119
>homotopy type theory
what if im not the homo top type? asking for a friend

Category theory is a language for faggots

>>11119433
you can pick one of
>homobottomy type theory
and if that isn't to your liking then just get a
>homolobotomy

today is a great day
it's a day of learning math
in this happy place

Category theory = type theory + religion.

>>11120701
Stop diddling little kids in the name of categories.

All it needs is a blockchain

>>11120739
I sense opportunity

>>11111796 reminds me when i do a self solve electronic control sistem in uni with an open antena for the static to enter the sistem and add the signal to the retroalimentation to gradualy solve the circuit. and see if it solve faster.