/sci/ - Science & Math

this isn't reddit

>>11027863
>go to the lectures
>do the homework
>make friends

>>11027863
That for any theorem T that's classically provable, there is a theorem T' provable without use of the law of excluded middle (LEM), such that
LEM => (T' <=> T)
That is to say, LEM is convenient cope, which however disables you from looking at theories that are in contradiction to it (such as synthetic differential geometry).
They will tell you constructive logic is weaker than classical one. Don't fall for their lie. It's lazy cope.
gl

>>11027863 (OP)
That for any theorem that's classically provable, there is a theorem provable without use of the law of excluded middle (LEM), such that the latter is classically equivalent to the former.
That is to say, LEM is convenient cope which does not enable you to prove more and which, however, disables you from looking at theories that are in contradiction to it (such as synthetic differential geometry).
They will tell you constructive logic is weaker than classical one. Don't fall for their lie. It's lazy cope.
gl

>>11027898
woah. source?

>>11027906
For starters, the law of non-contradiction is easily seen to be classically provable.
For a stronger variant
https://en.wikipedia.org/wiki/Double-negation_translation

For a theory of geometry which beomes inconsistent as soon as you adopt LEM, see e.g.
https://en.wikipedia.org/wiki/Synthetic_differential_geometry

Don't listen to people who are into category theory and intuitionistic logic.
They are, without any exception, horrible human beings that make fundamentalist terrorists look like sane and nice people.

>>11027884
Unironically this.
You have to go to them, but be aware lectures should be considered closer to "reading time" than to "study time". Most of your work, and almost all of your deep learning happens in your own time.
Start all the homework as soon as it is assigned, and make sure you do every question.
The transition to university math can be rough depending on the quality of your highschool, but stay open minded and stick with it.
Higher math is more like a philosophy of pattern than whatever the abomination of (my)HS was. Try to ask "why" as much or more than you ask "how".

>>11027910
>combining diffgeo with category theory
Gay plus gay equals ubergay

>>11027863
>clueless math undergrad
Redundant.

>>11027913
I've noticed that a lot of people who are into category theory and intuitionistic logic aren't actually well versed in other fields of math - even those for which category theory was invented to deal with. In that sense some of them can get pretty annoying. But why do you say they're horrible human beings?

don't listen to
>>11027913
category theory is the promised land

>>11027927
>category theory is the promised land
Explain. To me it seems like it's just a tool to simplify certain parts of algebraic geometry and algebraic topology. What makes it "the promised land"?

>>11027925
Because they're the math version of continental philosophers. Utterly detached from everything and highly concerned with projecting an image of unapproachable genius.

>>11027931
Can you give some examples to elucidate why you think this way? It seems to me like a lot of areas of math, not just category theory, especially at high levels can be considered detached from everything. What makes them different from, say, algebraic geometers, an area that is also viewed as highly abstract with few applications?

>>11027930
it's like trying to explain colors to a blind person. you have to see it for yourself

>>11027936
Hahaha. Are you even aware of how arrogant you sound saying that? Posts like yours make me think this poster >>11027931
>>11027925
was right. Good job.

>>11027941
you sound like a heretic

>>11027945
You claim to love category theory, yet do you have a deep knowledge of any other field of mathematics? Or are you a stereotypical type of cat theorists we were talking about?

>>11027916
I posted the LEM fact and I like category theory, but I don't think the later can practically stand on its own. Categories (let alone topoi) don't seem to have a good way of being taught and so you'll always fall back to standard algebra first.
Nonetheless, knowing that e.g. the exterior derivative d is in fact a natural transformation may help remembering or viewing rules such as pic related, i.e.
d(f* ω) = f* dω

In any case, LEM is literally not a necessity and makes live harder if you want to implement things, see e.g.
https://en.wikipedia.org/wiki/Constructive_analysis#The_intermediate_value_theorem

And don't get me wrong, this is not to say one shouldn't study the nice theories that assume LEM. But it should be made explicit. It's a shortcut axiom that break a bunch of theories as well as symbol/computer implementation
https://en.wikipedia.org/wiki/Realizability

>>11027930
Most mathematicans live with the most naive pre 1900 Frege notion of extensional equality. There's a point to higher categories and not necessarily accepting artifacts of model theory.
E.g. if you take the standard definition of the ordered pair and natural numbers in the set theoreitcal foundation (Kuratowski pair and Neumann ordinals)
(a,b) := {{a},{a,b}}
0={}, 1={0}, 2={0,1},...
then you see that it's a theorem that the pair that holds the number 1 and the number 7 provably has the number 0 as an element. This accidental theorem shit doesn't happen with foundations that only use universal constructions, because things are set up relational, not material.
I'm not the promised land guy, but it's easy to argue that it's more than just a tool. At least if you're more philosophically/fundamentally inclined.

>>11027925
Basically this >>11027931
They are obsessed with technical details and some vague notion of beauty. I have to deal with their shit all the time, one makes some nice system to verify some property or model something and they start rambling about how it is not elegant and how it would be more beautiful if it would be expressed in a categorical framework and how the world would be a better place if we would only think in terms of categories. Sadly they can never explain why elegance is something to strive for besides mental masturbation or why they produce almost no results that are applicable.

>>11027946
I know some commutative algebra, algebraic geometry, algebraic topology

>>11027951
What's the last thing you worked on in those fields? Just naming them doesn't tell me much.

>>11027935
Detached as in detached from other fields of math, as noted here >>11027925
>people who are into category theory and intuitionistic logic aren't actually well versed in other fields of math
I didn't mean "muh engineering applications".

>>11027947
I flipped it around, 1 is an element of (0,7).

Also to qualify the material statement, most mathematicians e.g. work with the set of natural number or the uncountbale set of reals and for their work in geometry, say, the already view e.g. the real numbers as things in their own. And know how to map N into R in this way or another. This is the info that is highlighted by the category theoretical approach.
But the category machinery is also very bulky, so it's not exactly nice. Dependent type theories to save the day?

>>11027863
sorry op, this is now a cat theory thread

>>11027863
Take some time to study, in your free time, things that might be a semester away from you. If you do this for 3 hours a week, you'll already have questions and an access point to when you actually need to learn stuff.

Also, I second the make friends statement - and I'd also try to make friends in the crowd that is not the people who will leave after 1 year. There will be enough of those guys and you don't want to stand alone in year 2 because the plebs exited.

>>11027951
>>11027954
Why have you stopped responding? lol

>>11027863
more abstract doesn't mean better/harder/easier. it just means more abstract, and for some people it works, for some people it doesn't.
category theory is NOT the peak of mathematics, it's just a discipline like any other. if you happen to like it, do it, I'm not trying to discourage you. but don't get the impression that you're more patrician than people doing differential equations and don't turn into that undergrad who knows what a topos is, but can't compute a double integral.

>>11027959
Typical category autist

>>11027884
Literally all you need to succeed is in those tips. Engineer senior here, almost all my professors have taken around 1/2 of the questions from hw/lectures, do the one and attend the other.

>>11027980
lol good job anon. Made me laugh.

>>11027980
>he actually bothered to produce some OC
You're a good guy, anon. Whether I agree with you or not.

>>11027980
Kek, good stuff

>>11027980
Top kek
essential /sci/

>>11027980
kek'd hard

>>11027863
Here, OP. I haven't read through it carefully yet, but it might offer some of the reassurance you're after.

>>11027947
>calls other people naive
>acts like it's a big problem that if you choose to treat everything as a set, then everything is a set
Like the patient who said: "doctor, it hurts when I do THIS"

>>11027863 (OP)
Here, OP. I haven't read through it carefully yet, but it might offer some of the reassurance you're after.

>>11027994
>Lara al Cock

>>11027980
lol

>>11027916
They cancel by the remainder theorem

>>11027980
I was inspired by your post and made my own OC. :)

>>11028003
>not even interested in the fields that category theory was invented to deal with
this one triggers me hard

>>11028012
That's literally the least crucial one on the list.
Topology doesn't have to interest you just because you find natural transformations natural.

>>11027863
look at the median salary of graduates
look at the employment rate of graduates
ask myself is it worth it

>>11027863
Nice blog post

bump

>>11027884
These. Don't neglect making friends too.

Also make sure that you enjoy your major and that you/your parents have enough money to make sure you complete your degree, and not abandon it half way through wasting all of your early and mid-20s in a fucking spiral of death leading to being an unemployed NEET with no future. Hypothetically speaking of course.

>>11028899
Too me me irl for comfort

>>11027863
skip class whenever possible and just read straight from the textbook

>t. CS & Math double major junior with 3.7 GPA

Drop STEM
Go Business