serious-discussion

[compact tartan]

now we just have to come up with a way to compute $$e^? = (e^w)^2 + e^c$$

[fathom swallow]

mniip

[compact tartan]

the squaring part is obviously easy, the fun part is in the addition

$$ae^{ib} + ce^{id} = \sqrt{a^2 + c^2 + 2ac\cos(d-b)}e^{i<atan2>(c \sin(d-b), a + c\cos(d-b))}$$

[fathom swallow]

mniip

[compact tartan]

so we get
$$e^{p+ib} + e^{q+id} = e^{\log\left(\sqrt{e^{2p} + e^{2q} + 2e^{p+q}\cos(d-b)}\right) + i<atan2>(e^q \sin(d-b), e^p + e^q\cos(d-b))}$$

[fathom swallow]

mniip

[compact tartan]

now here we get to branch on p < q or p > q

[pale orchid]

how come

[compact tartan]

well if p >> q these expressions are going to be inaccurate

$$\log\left(\sqrt{e^{2p} + e^{2q} + 2e^{p+q}\cos(d-b)\right) =$$ $$=\begin{cases} p + \frac{1}{2}\log\left(1 + e^{2q - 2p} + 2e^{q-p}\cos(d-b)\right) & ,p \ge q \ q + \frac{1}{2}\log\left(1 + e^{2p - 2q} + 2e^{p-q}\cos(d-b)\right) & ,p \le q \end{cases}$$

[fathom swallow]

mniip
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[compact tartan]

idk where the error is

but yea, this is really good because the e is always taken to a negative power

serving as a correction to 1

[pale orchid]

that's clever

[compact tartan]

you can do that same for the atan2 part

[vivid halo]

god I remember trying to do optimizations like this to my holomorphic dynamics plotter I wrote years ago

[compact tartan]

just use mpf

[turbid epoch]

I feel intellectually outpaced in this discord chat.

[vivid halo]

I mean it was a programming class project so I did everything from scratch in C++

[pale orchid]

[compact tartan]

yea bignum arithmetic is one thing I know not to do by hand

libgmp is just too good

[turbid epoch]

Anyone took analysis here?

[sick burrow]

scratch

[vague lion]

does this remind anyone else of a fruit lolly

[compact tartan]

sigh

I understand type 4 functions but I don't understand the coordinate system it uses

[pale orchid]

i tried googling type 4 function and all the results are about type 4 collagen, so i can't help you

[toxic schooner]

[turbid epoch]

Too late to flaunt Calc 1 prowess, too early to understand Analysis.

Despair

[sick burrow]

[compact tartan]

ok it's pretty stupid

[wicked ore]

Literally what is this

I’m trying to think of a set that could possibly satisfy these conditions

for all x in A, x and A are disjoint
for all x, y where x ≠ y, then x is in y or vice versa
if x is in y and y is in A, then x is in A

[vivid halo]

the natural numbers are a good example of this, suitably constructed

[zinc basin]

@vivid halo where I can get 10th class solutions?

[wicked ore]

What suitable construction

[compact tartan]

ok I made a better and more compact mandelbrot in latex

[vivid halo]

like the Von Neumann construction of the natural numbers

where it's like

[wicked ore]

Von neumann naturals didn’t work at all

Oh what

[compact tartan]

all von neumann naturals satisfy this condition

so does the set of all von neuman naturals

[wicked ore]

But 2 intersect 1 = {Ø, {Ø}} intersect {Ø} = Ø right? I feel like I’m being like super dumb rn

[vivid halo]

{Ø, {Ø}} intersect {Ø} is {Ø}

[compact tartan]

2 intersect 1 = 1

[wicked ore]

Oh shoot

[leaden torrent]

🧠

[wicked ore]

Yeah intersect is the set of common ones

[compact tartan]

not sure what this has to do with anything though

[wicked ore]

Nah i was just testing them out a bit

[vivid halo]

yea so the naturals form the first infinite ordinal, \omega_0

each Von Neumann natural forms a finite ordinal contained in \omega_0

[wicked ore]

Contained in meaning subset right

That makes sense

[vivid halo]

yea

[wicked ore]

Would it be weird to write that N = omega_0 ?

[vivid halo]

it depends on what structure you're considering

usually N refers to just the underlying set

[wicked ore]

I’m trying to understand the difference between ordinals and cardinals

Ok

[vivid halo]

\omega_0 refers to the underlying set along with the order relation

cardinals are sets considered up to bijection

whereas ordinals are sets along with a well order relation considered up to order preserving bijection

[turbid epoch]

Are there any functions f:N -------> R that are one-to-one?

[vivid halo]

sure, the obvious inclusion

[turbid epoch]

Hmm, but isn't the condition for one-to-oneness the fact that both sets have the same cardinalities?

[vivid halo]

no not at all

any inclusion is one to one

just not onto

one to one and onto means bijection and then yea in that case you need same cardinality

[wicked ore]

I think he has a definition that one to one is bijective

[turbid epoch]

Both a surjection and a bijection.

[compact tartan]

I hate the terms "one to one" and "onto"

[wicked ore]

Me too

I like how they sound a little closer to their meaning, but after using the -jective words enough times, it becomes intuitive anyway

[compact tartan]

how is "onto" closer to its meaning?

[deep mango]

You're probably thinking of "one to one correspondence". This is not the same as "one to one". I know, it's stupid.

[wicked ore]

Like you’re putting something big onto something small

[pale orchid]

this one always fucks me up. i always have to google it

[wicked ore]

When you learn that stuff for the first time, it’s just a bit easier

But it’s stupid because you confuse injection and bijection

[devout nacelle]

I'm okay with both but I prefer injective/surjective/bijective over the alternatives

[compact tartan]

mono/epi/iso

[sick burrow]

All injections are bijections into their image

[wicked ore]

Ok so cardinals and ordinals are the same except ordinals have WO on them and the bijections preserve the WO

[compact tartan]

no?

[sick burrow]

Kinda?

[compact tartan]

cardinals are sets modulo bijection

[wicked ore]

ok wait then what is it

[compact tartan]

ordinals set well-ordered sets modulo order-preserving bijection

[wicked ore]

Oh

[turbid epoch]

what in the fever dream is all this?

[sick burrow]

Yeah that's a good way of putting it mniip

[turbid epoch]

I feel intellectually outpaced in this chat.

[compact tartan]

ordinals have more structure as well as a smaller quoetienting relation

[wicked ore]

So it’s like an equivalence class of all sets with the same cardinality

[sick burrow]

Though don't you need AC to get a cardinality for every set

[compact tartan]

equivalence class

[wicked ore]

Ye ye

[compact tartan]

yea that's a cardinal

vaguely

[sick burrow]

Yeah aren't cardinals well-orderable sets modulo bijection?

[compact tartan]

all sets are well-orderable

[sick burrow]

I mean yeah

[wicked ore]

So I understand the purpose of cardinals, but why do ordinals exist

[compact tartan]

well-orderable != well ordered

[sick burrow]

Join the club

[compact tartan]

to describe transfinite processes

[sick burrow]

I mean I'm keeping up right now

[wicked ore]

Transfinite induction?

[compact tartan]

a cardinal number describes how many there is of something, an ordinal number describes when something comes up in "sequence"

[sick burrow]

But like half the time this chat talks about serious math it's just gibberish

Though it's fun to see myself gradually understanding more and more

[wicked ore]

i used to be in ur position, it’s not actually as weird as it sounds

[wicked ore]

Oh that makes sense

Because it’s got the order, right

[deep mango]

If you spend a lot of time here you will pick up on what the big words mean until you become another one of us, people who use big words without really knowing what they mean

[compact tartan]

whenever nG posts pictures I don't understand a single symbol

[sick burrow]

That's because he just makes things up

[turbid epoch]

I swear when I start learning this stuff, I'm stunting on all of you so hard.

[deep mango]

Hmm yes. Etale cohomology. Ah yes, bundle epimorphism.

[wicked ore]

So ordinals seem like a generalization of the natural numbers (since N describes when something comes up in sequence)

[neat lintel]

how do i do this?

[turbid epoch]

Ahhh yes, Aether.

[compact tartan]

yes

[wicked ore]

That’s pretty cool

Ok, thanks

Wait one more question

[compact tartan]

ordinals are for like, "longer" sequences

[sick burrow]

Category theorists just scribble arrows everywhere and call it math

[wicked ore]

Modulo requires an equivalence relation on the “set of all sets”, no?

[compact tartan]

yeah hence the actual definition is trickier

[sick burrow]

Wait can you not have equivalence relations on a class?

[wicked ore]

So it gets finicky with the axioms you’re using, I’m guessing

[compact tartan]

with choice we can actually provide a "canonical" representative for every cardinality

[sick burrow]

I don't see why you couldn't

[wicked ore]

Cause set of all sets isn’t a thing, right

[compact tartan]

you want to take the quotient

it's too large

[sick burrow]

You just define it with formulas rather than a set

Right you can't actually take modulo on V

Hence the canonical cardinals

God I find it so cool that I actually know (basic) set theory now

[compact tartan]

one way to do it is take a set X, come up with a well-ordering, which will give you an order-preserving isomorphism with some ordinal alpha, and then in alpha+1 take the least ordinal with the same cardinality as X, which is well-defined because alpha+1 is well-ordered and contains at least alpha

you can show this is independent of the choice of the well-ordering (and resp. alpha)

in this definition all cardinals are ordinals, but only very particular ones

[sick burrow]

I mean you just take the smallest cardinal with an injection into the set as its cardinality

*largest

[compact tartan]

we're defining what a cardinal is

that's like very circular

[sick burrow]

Cardinals are just ordinals k s.t. alpha<k implies there is no injection from k into alpha

[compact tartan]

ok

that's isomorphic to my definition

[sick burrow]

Yeah I just didn't read what you said :chad:

[compact tartan]

but also in your definition you can't take the "largest cardinal"

not well defined

[sick burrow]

Just reverse the well ordering

Right you just take the smallest cardinal that the set can inject into

[compact tartan]

sure

[wicked ore]

wait what’s an ordinal defined as here

[compact tartan]

.

[proper tide]

the real no. number line seems in 1st dimension only back and forth but to reach 4 th dimansion we need a 3d dimesion, my thought

[wicked ore]

So what did you mean by this then

That definition I pasted doesn’t have modulo

[compact tartan]

abstract idea vs concrete implementation

[sick burrow]

The way I saw oridnals defined is a transitive set well-ordered by set inclusion

[wicked ore]

Ok so like

The modulo is the idea

But since that’s impossible

You have to define it kinda carefully

[compact tartan]

well

there's a relation of isomorphism on the category of well orders

and the von-neumann ordinals (defined in your image) form a skeleton

meaning there's exactly one von neuman ordinal in each isomorphism class of the category of well orders

i.e. we can use the class of von neumann ordinals as an "implementation" of V/~

[wicked ore]

Ok

Do the vn ordinals act as a sort of representative for the “equivalence class” or does that not apply here

[compact tartan]

exactly

[sick burrow]

This is your brain on computer science

[wicked ore]

Ok that sort of makes sense yeah

[compact tartan]

no this is cat theory

[wicked ore]

Yeah I was gonna ask

[compact tartan]

there's a universal property you want to satisfy (quotient)

[wicked ore]

Doesn’t this have some things outside ZFC

[compact tartan]

and this is an object that satisfies this

yes, this is outside ZFC

[sick burrow]

Isn't everything you do cat theory mniip?

[compact tartan]

large categories and all that

[compact tartan]

I find a way to relate everything to cat theory

[sick burrow]

Since you're a cat and you do theory

[wicked ore]

Small cats are like, each object is a set right

[compact tartan]

no

[wicked ore]

Small cats are also cute

[compact tartan]

the collection of objects is small enough to be a set

[wicked ore]

Oh ok

[compact tartan]

you'd think basic arithmetic is safe

but alas

[sick burrow]

What is a universal property mniip

[wicked ore]

What’s V

[sick burrow]

L

[wicked ore]

No way

[compact tartan]

initiality in some category, usually category of cones

could be a slice category of a profunctor

[sick burrow]

You say this like you think I have any idea what those words mean

The extent of my category theory knowledge is the definitions of category and functor

[wicked ore]

Lol yeah

[wicked ore]

Is there a nice, formal implementation of ordinals and cardinals in ZFC? Or do they all rely on some weird properties to sidestep things like “set of all sets”

[sick burrow]

Yes

[compact tartan]

in vague terms, we're looking for some object with a certain pattern of arrows, such that for any other object with its own pattern of arrows, there exists a unique "mediating arrow" such that the second pattern factors through that arrow and the first pattern

[sick burrow]

Ordinals and cardinals can be formalized within ZFC. If they couldn't, ZFC would be kinda lame

[wicked ore]

Sounds like a morphism of morphisms in a sense

Or like, a morphism between entire categories

[compact tartan]

no

[sick burrow]

So... a functor

[compact tartan]

yes there is

the "set of all sets" is mostly there for intuition

[sick burrow]

mniip how should I learn cat theory

[compact tartan]

you can do the rigorous details

[sick burrow]

Like how much should I be learning for alg top

[compact tartan]

up to you

for "normal maths" it's useful to learn at least the "categorical language", i.e. the definitions of initial objects, products, equalizers, pullbacks, their duals; as well as the notion of functoriality and naturality

[wicked ore]

Category theory is completely formalized right? Like ZFC?

[compact tartan]

uhh

category theory is more of a vibe in that it extends beyond ZFC

but if you insist you can formalize category theory inside ZFC

but there will be caveats

[sick burrow]

is Riehl's book good for intro cat theory

[compact tartan]

I guess

[fast ivy]

[wicked ore]

Well take the “category of sets”, for instance, how is that formalized

Like are categories undefined objects sorta like sets and you define their properties

[compact tartan]

you take a universum U and take the category of elements of U

a category whose set of objects is in U is called U-small

if C is U-small then [C, C] might not be U-small anymore

[wicked ore]

Hold up, what’s a universum, and you said take the category of elements of U

But what is a category (formally)

[compact tartan]

if we're restricting our attention to ZFC, it's a set Ob, together with a family of sets Hom(X, Y) where X, Y range over Ob

together with a family of elements id_X in Hom(X, X) for every X in Ob, and a family of functions \circ : Hom(Y, Z) x Hom(X, Y) -> Hom(X, Z)

that satisfy some identity and associativity conditions

[wicked ore]

Yep

[compact tartan]

the "categorical vibe" implies that Ob doesn't have to be a set and can be something more vague, like a class, or something without a notion of equality at all

[vivid halo]

people talking about size issues
ng has joined the chat

[compact tartan]

for Hom we actually care about equality

[wicked ore]

I love this, categorical vibe

[compact tartan]

but it can still happen to be a class

[wicked ore]

If it’s not a set, then what is it? Lol

That’s what I’m asking

[vivid halo]

it can be taken to be a proper class

[compact tartan]

no that's not what you're asking

[wicked ore]

What

[compact tartan]

"formally, in ZFC" and "if it's not a set then what is it" are incompatible requirements

[vivid halo]

okay yea that too

[sick burrow]

wait so like with the cat theory definition of a group

the morphisms are group elements right?

[vivid halo]

in NBG you can talk about classes I guess

[wicked ore]

Ok I’m dumb

I thought everything was a set in ZFC

[compact tartan]

which one?

you could be referring to at least 2 different things here

[sick burrow]

groupoid with one element

[vivid halo]

sure, you can define groups as one object categories where all morphisms are isomorphisms, and then the group elements correspond to the morphisms

[compact tartan]

sure

[compact tartan]

exactly

but cat theory goes beyond ZFC

[wicked ore]

Yes

[compact tartan]

ergo when you're asking for a ZFC formalization we're only exploring a part of it

[late skiff]

hello people!

[wicked ore]

Ohhhh ok

[sick burrow]

oh right because every group is a subgroup of a symmetric group

[wicked ore]

So then how does the formalization work outisde ZFC?

[compact tartan]

magic

[vivid halo]

so there are a few ways to do this

NBG is a bad way to do it but it works for really crude purposes

[wicked ore]

vibes

[vivid halo]

since NBG gives you a way to talk about sets and proper classes

so you can talk about the proper class of all sets being the class of objects in the category of sets for instance

there are some issues with taking this approach in general though, since you can't talk about a "metaclass" of all classes and repeat constructions (e.g. for the Yoneda embedding)

so you need something more subtle

the two standard ways to do this are either through ZFC + TG (Tarski-Grothendieck axiom) or ZFC + exists a proper class of inaccessible cardinals

[compact tartan]

1. ZFC and copium
2. ZFC+large cardinals as universes
3. NBG with class of all sets as your universe
4. Tarski-Grothendieck with as many universes as you want
5. A ramified type theory

[vivid halo]

these are equivalent statements, just depends on whether you like the setup of Grothendieck universes or the setup of inaccessible cardinals

1. ZFC and copium
   

[compact tartan]

6. vibes and magic

most normal mathematicians I've seen go for ZFC+TG I think

[vivid halo]

yea

[compact tartan]

I'm a type theory junkie though

U_n : U_n+1

there's your TG

[vivid halo]

yea exactly

either of these approaches is more common among category theorists/type theorists

though some people still like inaccessibles

there's also some reasons why the inaccessible approach is a little more expressive but either is fine

[compact tartan]

I still don't understand how hTop is not concrete

I've read the proof it checks out but like

how

[vivid halo]

oh yea it's a weird proof

I never understood it

like it's one of those proofs you can read each line and still not understand what just happened

[compact tartan]

ok so like reality check

if we have an ess. surjective functor between small categories, does it always have a section?

hmm

ess. surjective and full

I guess

sounds like this should be a form of choice

which means non-concreteness of hTop is a failure of "large" choice?

choomers btfo

[turbid epoch]

Help, this chat is a fever dream.

Also, can someone enlighten me about the Axiom of Choice?

More specifically, what it is and why it amounts to the statement "The cartesian product of a non-empty set is non-empty."

[compact tartan]

The Axiom of Choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma?

[turbid epoch]

Bruh, I'm a lvl 0 goon.

[sick kite]

what does it mean for an axiom to be true?

[turbid epoch]

Wtf is the axiom of choice, I need information.

[sick kite]

wait is this a troll

[compact tartan]

the formal statement is:
$$\forall A, \varnothing \not\in A \implies \exists F : A \to \bigcup A, \forall X \in A, F(X) \in X$$

[fathom swallow]

mniip

[compact tartan]

this is the "C" part of ZFC

[turbid epoch]

Yeah, more like the Axiom of Incomprehension.

[compact tartan]

you can prefer to include or exclude this axiom from your assumptions

[sick kite]

oh its a stackexchange copypasta

wait nvm

im so fucking stupid

[turbid epoch]

@compact tartan I'm baby, explain it to me like Im 5.

[compact tartan]

I don't think I can

[sick kite]

but what if im 10 mniip

[compact tartan]

if you have a bag of bags, and you know none of the bags is empty, sounds like you should be able to pick an item from each bag

and there's no dispute about this if the number of bags you have is finite

but turns out it gets complicated in the infinite case

[turbid epoch]

But what does this choice of ours have to do with cartesian products?

wait a minute, I think I just got it.

[compact tartan]

functions are related to cartesian products

[turbid epoch]

yeah thanks

although let me be frank @compact tartan

[compact tartan]

the AoC is surprising because of the statements that are equivalent to it and statements that follow from it

existence of non-measurable sets or more famously the banach-tarski paradox is a consequence of AoC

[cinder zephyr]

TFW your prof gives a HW answer in class

[ancient flame]

politics in a math server smh

[mint patio]

partial fraction decomposition is beautiful and useful.

[turbid epoch]

Until you have 5 terms in the denominator and have to solve a system of 5 equations.

Yikes.

[bronze pelican]

uh oh

new veritasium math video

will this break math discord 😬

[sick kite]

this is just math history

[turbid epoch]

Yeah, it's just the history of i.

[bronze pelican]

good

[turbid epoch]

And depressed cubes and the cubic formula.

[bronze pelican]

there cant be any posssible way for cranks to misinterpret that right ?

[sick kite]

surely

[bronze pelican]

right?

[turbid epoch]

cranks?

[compact tartan]

[bronze pelican]

amazing emote

[deep mango]

[sick kite]

😬

[turbid epoch]

yeah, but what does crank refer to?

[deep mango]

New veritasium just dropped

[bronze pelican]

crank is someone who is delusional in their claims about math

[compact tartan]

is this a video about cubics and complex numbers?

[sick kite]

yes

[compact tartan]

(I haven't watched it)

[turbid epoch]

yeh

[sick kite]

math history of i

[compact tartan]

"led to quantum physics"

[sick kite]

mostly talks about 16th century italians

[compact tartan]

bold claim

[deep mango]

Wow... The cubic equation... Has square roots of negative numbers??? Wtf???

[compact tartan]

I'm sure physicists would have invented complex numbers if they needed to

[sick kite]

not only is it bold, it is unsubstantiated

all it says is

"oh look multiplication by i is like rotation"

therefore physics

[bronze pelican]

that's also the basis of conformal geometry

[compact tartan]

"it's like a number but if you square it you get -1" - shit physicists say all the time

[velvet dagger]

I know you're being sarcastic but there's always that slight "what if" when it comes to you lol

[turbid epoch]

I think this is build up to another video.

[sick kite]

BRO LITERALLY

[turbid epoch]

idk, just a feeling.

[sick kite]

the same happened yesterday aha

[pale orchid]

prepare for "there is a hole at the base of physics"

[crystal stone]

Then why didn't they

Huh

[compact tartan]

mathematicians beat them to it

[crystal stone]

#excuses

[sick kite]

mathematicians in shambles

[bronze pelican]

complex geometry is the most based thing #Riemann #Poincare

[velvet dagger]

Try again lol

[turbid epoch]

Mathematicians make math, physicists find uses for it a hundred years later.

All is as it should be.

[crystal stone]

Eh

I dunno

[bronze pelican]

We need a Poincare fan club

[crystal stone]

Witten is better than most mathematicians

[deep mango]

[compact tartan]

sometimes physicists shit something out and mathematicians are tasked with making sense of it

[deep mango]

What if what?

[compact tartan]

sometimes it's as simple as "wait this is just group theory"

sometimes it's an open problem for 50 years

[deep mango]

I know what if. You don't need to say it.

[crystal stone]

Dirac did some crazy stuff

I'm still skeptical about dirac's delta function

[bronze pelican]

Poicare is really my hero

[crystal stone]

I have a suspicion that Green beat him to it

[velvet dagger]

All math is but a prelude to p-adic hodge theory

[bronze pelican]

and Jacobi

[compact tartan]

"it's like a smooth function but it's zero everywhere except origin"

[bronze pelican]

and Abel

and Riemann

[turbid epoch]

It's integral is 1.

[crystal stone]

I think Green really beat him to it

[sick kite]

• c (probably)

[crystal stone]

But I don't know enough about history and haven't read Green's original paper where he came up with his functions

[compact tartan]

ah yes, the green's function for the constant PDE

[crystal stone]

Well

Hey, beating someone to it 100 years early

Is a big deal

I'd have to sit down and sort through it

I dunno if I have the patience for that though

[pale orchid]

moonbears like

"ah, yes, c1 + c2"

[compact tartan]

how do you rationalize learning math history if there's so much interesting math to learn instead

couldn't care less who invented it and when

[crystal stone]

I like the development of ideas

[turbid epoch]

Why do people write QED at the end of every proof? What does Quantum ElectroDynamics have to do with anything?

[bronze pelican]

it motivates where the math your are using / learning came from

[crystal stone]

I enjoy history and I enjoy math

and I enjoy settling misconceptions about history of math

[bronze pelican]

you see math is not just abstract ideas that exists with out human input. Every definition you use has a historical context

[turbid epoch]

Some math stories are amazing.

[compact tartan]

I mean

[crystal stone]

le meme dies in duel lmao haha dude you could imagine just dude

genius

[compact tartan]

that sounds like applications

yes you can apply math to other things

but why do I care who did it though?

[turbid epoch]

Like when they gutted a ####ing guy because he drew the square root of 2 outside Pythagoras' house.

[bronze pelican]

what do you mean applications

[compact tartan]

applicataions and or motivation

[bronze pelican]

I find learning about the history of math and mathematical ideas very rewarding

[crystal stone]

I guess in the longer term I think I'm more interested in curriculum, education, and development of ideas

[compact tartan]

in my mind those things are not related to specific persons at all

[crystal stone]

Rather than doing pure math research for the rest of my life

but we'll see how gripping I find more research

It is interesting and I do like solving problems

[bronze pelican]

its very interesting to see the historical development of ideas

it gives you a newfound respect from people like Riemann

[crystal stone]

I guess the research I'd do is interesting

but I don't jump for joy over it

[velvet dagger]

I find history of math to be cool more from a history pov than a math pov

[crystal stone]

I disagree

You don't find it interesting from that point of view

[velvet dagger]

Ballsy claim

What's your alternative suggestion?

[crystal stone]

You find history of math to be cool from a math point of view

and you think the historians are full of shit

So the history point of view is bad

According to you, of course.

[velvet dagger]

Lolol

[sick kite]

Dami, you DO NOT THINK WHAT YOU THINK

[compact tartan]

math history from the history point of view
math history in the 1940s

[crystal stone]

Teichmuller reax only

[turbid epoch]

Are there any new fields in math that have just emerged in the past decade or so?

[compact tartan]

homotopy type theory came about in 2013 ish

[turbid epoch]

just the one?

[crystal stone]

What do you mean by "field"

[compact tartan]

probably plenty of others

[velvet dagger]

But yeah I guess my thing with math history is, it's cool for general stories/it's cool to sometimes contrast how we think about things today vs how less enlightened people from 1700s/1800s thought about stuff

[turbid epoch]

I guess, field to me means that it is, to a degree, separate from other math?

[crystal stone]

"separate from other math"

Ain't gonna happen buddy

[compact tartan]

how old is condensed mathematics?

[turbid epoch]

That's why I added "to a degree"

[compact tartan]

3 years?

[crystal stone]

what is condesned math

[compact tartan]

I have no idea

[crystal stone]

oh it's that shitsee guy

[compact tartan]

yet another "the most ambitious crossover in the history of math"

[velvet dagger]

But I think it doesn't often inform how I look at current math, and generically I just prefer modern arts

[turbid epoch]

Condensed mathematics is the (potential) unification of various mathematical subfields, including topology, geometry, and number theory. Ripped from Wiki.

[velvet dagger]

Oh yeah condensed math is some new age stuff

But yeah I'll just say don't expect entirely new fields of math to pop up that frequently

[compact tartan]

shit like this is good leverage for constructivist propaganda

I should be all over this

but no time

still stuck on HoTT memes

[velvet dagger]

Since it turns out a lot of stuff is done by more incremental work or adapting existing math, which turns out to be still be very robust

[turbid epoch]

idk this is some fields medalist guy saying this. Is this truly bs?

[compact tartan]

no one said it's bs

[velvet dagger]

Condensed math?

[crystal stone]

It's done by shitzee

So it's probably abstract nonsense

[velvet dagger]

Yeah nobody thinks it's Bs. I don't yet know what exactly it does but

[crystal stone]

I have to take an algebra qual sloth

That includes sylow and galois

[turbid epoch]

lol Why do you call Scholze shitzee?

[crystal stone]

I'll be face to face with the enemy

[crystal stone]

It's how I pronounce scholze in my accent

[sick kite]

'accent'

[velvet dagger]

Scholze's passive observations probably destroy my entire math career lol

[surreal sapphire]

there was a recent talk about (also) condensed math that was quite accessible btw: https://www.youtube.com/watch?v=rec9uHzrDM8

[crystal stone]

I watched one video by laurie or lurie

and they got the fourier transform definition wrong

and I stopped watching

[surreal sapphire]

his camera is super annoying though

it keeps trying to focus

[velvet dagger]

What'd he say?

Also I'm p sure you mean the categorification one lol

[crystal stone]

they missed a negative on the fourier transform

Yeah

I was like I'm out

[velvet dagger]

Oh that's an alternative convention lol

[crystal stone]

Nope

Why

I don't wanna know why

All I know

Is no

[velvet dagger]

I'm not sure why but I remember that conventions disagree

So he's not quite wrong here lol

[crystal stone]

It's non-canon

[compact tartan]

how many of 2pi do you write in your forward and backward fourier transform

[crystal stone]

It ain't a part of my gospel

[velvet dagger]

Whatever makes the stuff unitary

[crystal stone]

I got a chuckle out of that

[velvet dagger]

So iirc 1/sqrt(2π)

[compact tartan]

understandable

in CS it's common to use non-unitary fourier transforms

no factor on the forward

[crystal stone]

Yeah who would ever study

Fourier transforms of boolean functions

[compact tartan]

excuse me it's called Z_2

[crystal stone]

{-1,1}^n

Deal

[velvet dagger]

2-adics?

[crystal stone]

W/f

it

[compact tartan]

dyadic integers

[crystal stone]

I can't believe I'm gonna have to learn all that shit

[pale orchid]

to be fair, a sequence of alternating -1s and 1s is and transforms into useful stuff

[crystal stone]

yuh, it's what my research advisor does

And what I will probably do

[compact tartan]

does there exist a hadamard matrix of order 668

tell me, signals guy

[pale orchid]

let me google hadamard matrix

yeah sure

they use that sort of stuff for coded excitation with large arrays

[compact tartan]

it's the shit that most spread spectrum transmission formats use

[pale orchid]

sounds about right

(pseudo) orthogonal sequences make nice coded signals

both from the spectrum/interference standpoint and from the SNR standpoint

"sIgNaL ComPrESsIoN"

[crystal stone]

Do you work as a signal processing engineer

or do you just study signals

[compact tartan]

I think his undergrad was in antennas or something

[pale orchid]

lmao

it wasn't

and both, technically

[neat lintel]

antenna engineering

[pale orchid]

since i'm hired both at the uni and at a company that collabs with it

[crystal stone]

Oh nice

Signal processing is like my back up plan

I almost got a job as an RF engineer

but they found out I didn't wanna live in sacramento

So then I didn't get the offer

e.g. I would've worked for like 1.5 years

and then asked for a transfer

[cold needle]

hadamard matrices my algebra prof might give a talk on it next semestr

[bronze pelican]

I feel like the part about shrodinger's equation was unnecessary in the veritasium video

He could've just left it at the history of the cubic equation

I don't think the discussion about the shrodinger equation actually added anything of substance

[sick burrow]

what does orientation look like on an 3-simplex

[crystal stone]

I only got to the cubic stuff

I liked his explanation of the cubic formula

[bronze pelican]

Yeah

[crystal stone]

I've heard the story so many times, it's one of my favorites in math

[bronze pelican]

It is good

[crystal stone]

Cardano, Tartaglia

[bronze pelican]

And deserved to be on YouTube

[crystal stone]

I think it's way better than "Galois in a duel" meme

[bronze pelican]

They're both memes

[crystal stone]

I dunno

That's a meme

[bronze pelican]

What really needs to be showcased

To a wide audience

Is Poincare and Klein's race to prove the Uniformization theorem in complex geometry

Now thats an exciting story

And I don't know the full story

But I wish I did

[crystal stone]

Oh hey fourier was the first to speculate that the Earth's atmosphere traps heat coming from solar radiation

That's pretty neat

Well at least first to speculate it in a real, academic sense with mathematicalisms at the foundation

It's fun to read old books on science

"Scientists speculate there could be other galaxies"

[neat lintel]

Am I the only one who thinks that Nasa does fake shit?

Look at how these nuts are fliying, it looks so fake.

https://www.youtube.com/watch?v=EtaWWCXbtbY&ab_channel=CanadianSpaceAgency

The lighting of them looks like the CGI of a highschool project.

[leaden torrent]

nasa
canadian space agency

[neat lintel]

It's made on the ISS.

[leaden torrent]

anyway, looks fine to me

im sure the video is touched up (obviously, the label is blurred)

but that would be very impressive cgi considering the reflections on the aluminium foil

[neat lintel]

The way it is fliying looks so fake, as if they were putting a 2d clip on a 3d.

[leaden torrent]

...i dont see it

[neat lintel]

They even admited to me that it was made the same way Star Trek was.

[leaden torrent]

thats... a different video

are you just trying to spread conspiratorial bs in a random big server

screw off

[neat lintel]

Nah, bro I am actually triying to discuss it with people.

Cause if it is really all fake I would be sad.

[wild lantern]

What else is there to discuss other than that it looks fake to you?

[leaden torrent]

not sure that bringing up a comment from a totally unrelated video is a good faith discussion

you havent tried to explain why it seems fake other than "it looks 2d", but it doesnt look 2d to me

and the reflections on the foil and the can make the depth fairly clear

at least the people who find reptilians in video compression artifacts have an image they can point to

[wild lantern]

Nasa's videos look fake: Space is fake:

Jk

[leaden torrent]

nah space is real, NASA is fake

[wild lantern]

Lol

[leaden torrent]

NASA is a big money laundering scheme

they say every space shuttle launch costed billions

where did that billions come from, huh?

clearly illegal black market activity

[sick kite]

pablo escobar didn't die and is just head of money laundering space agency

faked his death

[leaden torrent]

actually space stuff would probably be a surprisingly effective way to launder money rn lmao

eh i guess its too public facing/sensationalized

but like

no one knows how much shit should cost

[wild lantern]

I would support a nasa money laundring scheme if it somehow managed to further space travel and junk.

[leaden torrent]

if spacex says a rocket costs $800 million

who are we to argue

maybe its only $400 million

meanwhile, if you claim that car wash costed $60, im gonna get suspicious

[wild lantern]

Markup for government purchases is pretty crazy in general.

Seems plausible for space stuff too

[sick kite]

yeah but it did bro, trust me

[leaden torrent]

reporting income tax on your car wash

sales: 7
total earnings: $522 000

[sick kite]

who's saying I'm reporting income tax?

[neat lintel]

I actually really hope that stuff is real.

Cause espacilly stuff like LISA is big shit (Even though that's from ESA)

[leaden torrent]

why focus on such out there conspiracy theories instead of ones that actually make sense for the government to pursue

like mass selling stocks in early march 2020

[eager reef]

[sage stag]

What shape is this lol

[leaden torrent]

what shape is what

[sage stag]

these beads

like two cones

[leaden torrent]

the object itself is an abacus

[sage stag]

I get that but the shape it is used, I know there is a name just cant remember?

[leaden torrent]

dicones?

[sage stag]

yes but inverted

[leaden torrent]

(dicone is a synonym)

[sage stag]

I see

[leaden torrent]

but tbh id call those closer to "saucer shape"

[neat lintel]

Anyone actually able to quote stuff like like EGA or Harthshorne. Looked up books saying intro to AG and a lot of people mentioned it, reddit called finishing them a rite of passage. Im doubtful becsuse both look long and tedious to go through in full.

[leaden torrent]

i read hartshorne in undergrad

i would not recommend it

ega is pedagogically terrible and only really valuable as a reference

hartshorne is fine but not great

[bronze pelican]

I feel like I'm ready for Hartshorne

Now that I've done some complex geometry

[neat lintel]

oof

[light needle]

based yamin

yeah like, i read ch1 so far, and all of it feels like a continuition of complex geo

I should get back to hartshorne soon

[hollow ginkgo]

Stack of frusta

[jovial ember]

Ch1 of Hartshorne kekw

Do you also enjoy dragging ur balls through broken glass?

[cold needle]

huh

[frozen venture]

Are there professors in this discord?

[neat lintel]

every professor is busy, every professor is in every uni

[frozen venture]

I'm wondering who all the smart yellow namers are

are you guys grad students? advanced undergrads?

[jovial ember]

We are legion

ReLisrixallt we are all different ages lol

There’s hsers, undergrads, grad students. There is one professor

He’s also a cat

Actually maybe two now that buncho graduated

And there’s zeta

Lol

[neat lintel]

[cold needle]

i am a second year undergraduate

[frozen venture]

So how does one achieve the coveted yellow name?

[cold needle]

unknown

you must be a good role model and accepted in the community or something

idk

i should know

[jovial ember]

Be friends with the other ones

Idfk

[toxic schooner]

chmonkey will u be my friend

[jovial ember]

No

[toxic schooner]

[jovial ember]

Chmonkey is everyone’s friend

Except yours

[toxic schooner]

why

[jovial ember]

😨

Cuz ur my buddy

[toxic schooner]

[jovial ember]

Can’t believe my buddy hasn’t wished me happy bday

[neat lintel]

happy belated birthday

[jovial ember]

It’s still today

:3

Where I live

[toxic schooner]

[neat lintel]

tomorrow's my birthday

[toxic schooner]

its your birthday!?

[jovial ember]

Oh yes I remember

[toxic schooner]

Happy birthday!

[jovial ember]

I’m 2 days older than you

[neat lintel]

😊

[toxic schooner]

happy bday in advance

[jovial ember]

Also I forget did u used to have like

[neat lintel]

happy birthday

[jovial ember]

Jantheman

As ur name

Or was that some1 else

Or Jan something

[neat lintel]

janman

[jovial ember]

Oh okay

Cool cool

I wasn’t sure if that was u

Or not

[toxic schooner]

🐒

🪑

chmonkey in a nutshell^

[jovial ember]

I think I remember that the bios were the same or something

But I guess ppl didn’t read ur bio

Hence the new name

Actually

👑

Me in a nutshell

[toxic schooner]

ofc

👑

🐒

🪑

[jovial ember]

Chmonkey

https://gizmodo.com/new-squid-game-cryptocurrency-launches-as-obvious-scam-1847961584

Lmaooooooo

[neat lintel]

lmfao

dalgona with mandelbrot on it

[jovial ember]

https://gizmodo.com/squid-game-cryptocurrency-scammers-make-off-with-2-1-m-1847972824

Follow up btw

[neat lintel]

oof

[pale orchid]

wait, read my about me is janman?

wtf

i see a random person with generic roles and a generic nickname lol

[neat lintel]

or either nJanmoid

[pale orchid]

i thought it was someone new

[fast ivy]

https://ncatlab.org/nlab/show/Gelfand-Fuks+cohomology

Fuks

[jovial ember]

Fuks

[wicked ore]

Maybe this is a dumb question but can you have a sequence of functions (f, f’, f’’, …), where f’ means derivative? Can such a sequence “converge”?

I’m thinking about how polynomials “converge” to zero, some functions “converge” to e^x, and some functions diverge

[devout nacelle]

Polynomials converging to 0

But I think you can certainly have the sequence of derivatives

Provided f is infinitely differentiable of course

[wicked ore]

Yeah

[devout nacelle]

And it's meaningful to interpret its convergence as any other sequence of functions

[wicked ore]

Any other sequence of functions? How do those work?

[devout nacelle]

The one I'm familiar with is pointwise convergence

F_n converges to f if for any x in the domain of f and any e>0, there exists some n such that |F_m(x)-f(x)|<e for all m>n

Basically just the usual convergence but for images of your functions

[last oxide]

it depends on the norm you choose

[devout nacelle]

I'm just assuming R here

With the usual norm

[last oxide]

you are doing pointwise, ye

but the usual norm is sup

[devout nacelle]

Ah okay

[last oxide]

for some suitable sense of usual

[devout nacelle]

sup|F_m(x)-f(x)|->0 as m->\infty?

[wicked ore]

This sounds like normal convergence just saying fn converges to f if the sequence (f1(x), f2(x), …) converges to f(x) for all x in the domain

[devout nacelle]

x ranging over the domain

[devout nacelle]

Yep!

[last oxide]

the for any x quantifier goes after the property

[wicked ore]

Lit lit lit

[last oxide]

if you want sup norm

but ye

thats the same

[rocky mulch]

hey guys, i'm currently in grade 12 and working on a math investigation (for those of you who know what the IB is, i'm working on my math IA) and was planning to do an engineering optimization problem, do you think it's a good idea to do something simple like the most efficient rocket launch trajectory, my requisites are it can't have complicated physics but the mathematics have to be like simple calculus(as a reference it's my first time studying calc but i could learn a bit of more sophisticated calc if neccesary)

[gloomy marten]

most efficient rocket launch trajectory
Requisits can't have complicated physics

I'm guessing air resistance is negligible

[pale orchid]

you can do a much more simplified version of that problem

if you throw an object with, say, a catapult, at which angle should it leave the ground so that it reaches the largest horizontal distance possible?

[gloomy marten]

Oh you mean that?

That's rather easy

If there are no walls and air resistance is negligible, just calculate de derivative

I made an animation of that for physics in my senior year

[pale orchid]

(that's just my suggestion, cuz the other problem they gave is out of their reach)

but yeah, it's like beginner level physics and you can use calc to optimize it, so

[gloomy marten]

@pale orchid I think you learn about Kepler's laws in 12th grade, so it might be possible to solve using elementary physics and some calculus, the best trajectory for a rocket. Assuming boundary restrictions on the gravitational pull of a planet

[pale orchid]

knock yourself out then

[gloomy marten]

I mean they said it was an investigation thing

the most efficient rocket launch trajetory is also a weird question, what type of optimizations are you looking for

least fuel consumption is probably like close to enough for escape velocity, and just use planets as catapults

[inner finch]

solve triple body problem

[rocky mulch]

@gloomy marten @pale orchid I've just read your discussion and i think I'm gonna use the catapult idea considering air resistance and that i think I should get the difficulty required, thanks for your help :D

[gloomy marten]

Ah yes

gaming

[turbid epoch]

Is this actually valid? lol

[toxic schooner]

but 2^(1/2) is also irrational,
is the theorem trying to say otherwise

[gloomy marten]

it's not though

the proof by fermat's last theorem is simply not valid for n = 2

[toxic schooner]

yes, its not, ik

[turbid epoch]

but is this proof actually valid tho?

[toxic schooner]

aahh
ig maybe? i mean its a theorem proven to be true, so i dont see a problem

[gloomy marten]

the normal proof is really simple though

[turbid epoch]

Some people were saying "circularity" something something but I see no way the usage of this is circular.

[toxic schooner]

[gloomy marten]

it might be, I don't know the proof for fermat's last theorem

[toxic schooner]

how is this circular?

[gloomy marten]

anyway

if 2^(1/n) is rational, exists p,q coprime such that it is equal to p/q.
2q^n = p^n => 2|p => (by sub) 2|q. contradiction (for n > 1) qed

[turbid epoch]

It's a basic 10th grade proof.

yeh

[pulsar ruin]

fun

[inner finch]

fermat's last theorem might use this fact within the known proof

[toxic schooner]

oh

[neat lintel]

Proof is valid, doesnt look like Fermat’s last theorem uses this fact

Or you can just stop before last line and say a^n is even and b^n even which is contradiction

a^n even because divisible by 2, b^n even because n>=2 on both sides implies such.

[nimble shuttle]

andrew wiles is most famous for his proof of the modularity conjecture, which finally resolved the famous, long-standing problem of whether the cube root of 2 is rational

[inner finch]

lmao

[steep mountain]

too bad it cant prove square root of 2 is irrational

[turbid epoch]

We need a stronger version of Fermat's Last Theorem for such sorcery.

[neat lintel]

Is it possible to write like this? r ∧ n ∈ ℕ, x ∈ R?

[pale orchid]

you just did

what do you want ^ to represent tho

[neat lintel]

That r and n are natural and x is a positive real number

[pale orchid]

you can just use a comma

[neat lintel]

Where?

[pale orchid]

$r,n \in \mathbb{N}, ,, x \in \mathbb{R}$

[fathom swallow]

Edd

[neat lintel]

Ohh thanks!

Just another quick question

If I have a three statements equation

Where do I say this about r,n and x?

[pale orchid]

wdym

write it out in words

[neat lintel]

So I have an equation with three statements. I want to say that r,n is natural and x is a positive real for every three statements

I've never done this before

Right now it looks like this

And I don't know if it's right

[pale orchid]

idk why there are ifs

[pale orchid]

but you can just separate the stuff with commas as you did

[neat lintel]

There are three different equations

[pale orchid]

for example

and you can separate the conditions with commas

but honestly you should just describe r,n, and x BEFORE you define P

cuz you're writing the same thing over and over

[neat lintel]

yeah I was wondering that. How do I do that? just put it infront of P in brackets?

like this

[pale orchid]

you can write it above and use words

nothing wrong with words

given r,n \in \mathbb{R}, etc,. the function P is defined as

[neat lintel]

Thank you so much!!!

I'm in my second year in highschool so you can image that this is WAY a head of my knowledge

Or thrid maybe if you translate it from I live.

This all started with 3b1b video, and then I got obsessed

[pale orchid]

that's aight

there's no need to use symbols for everything, especially if it makes things more difficult

[neat lintel]

Yeah but if I don't state that I will get like 100 solutions.

[pale orchid]

what i mean is you can use words and that doesn't make it any less correct

[neat lintel]

Riight

But this kind of looks cooler. Like if I show it to my teacher

But she won't really care

[pale orchid]

e.g. "consider the numbers $n,r \in \mathbb{N}$, $x \in \mathbb{R}$. We define the function $P: \mathbb{N} \times \mathbb{N} \times \mathbb{R} \to \text{(idk what your codomain is)}$ as:
$$P = 0$$

[fathom swallow]

Edd

[pale orchid]

just as an example lol

[neat lintel]

Riiight

It's kind of hard to understant the P: N x N x R ->, part

never done that lol

[sick burrow]

It's just a function that takes ordered pairs of natural numbers as input

[gloomy marten]

what did you use for the spacing?

[sick burrow]

Wait NxNxR so I guess ordered pairs of natural numbers and one real number as input

[gloomy marten]

\,,?

[pale orchid]

\,\,

[gloomy marten]

I see

thanks

[sick burrow]

So ordered triples where the first two entries are natural numbers and the last is any real number

[neat lintel]

@pale orchid sorry if this is bothering lol. Did I do it right now?

n and r are both known values

we want to solve for x

I should write that P is also known

[pale orchid]

sure i guess

[neat lintel]

So maybe it's less than a function and more than an equation???

[pale orchid]

i mean, you have a binary relation that gives one output in R+ to any input in N x N x R+, so

i guess what i would change is that i'd write P(n, r, x) instead of P

[neat lintel]

yeah I didn't really know what that part meant. P could be anything but I always say that P is equal to erf/sqrt(2)

Or P could be anything between 0 and 1

[pale orchid]

i dont understand what you mean

[neat lintel]

P is what the range of the probability density should be.

so (r+1)/(n+2) -+ x give the range. And the area under that range times (n+3) should equal P

[pale orchid]

idk what you mean by range here

[neat lintel]

So I am calculating probabilities of probabilities

[pale orchid]

the range is a set

[neat lintel]

Oh

[pale orchid]

(and the word is also ambiguous)

when you say range, it's all the possible outputs

[neat lintel]

Right so between two values then?

Is better to say

[pale orchid]

i assume you mean to fix a value of n and r, and then vary x

to yield a curve P(x) for a fixed n and r

and then varying n and r yields a family of functions

and the definite (probably improper) integral of P(x) yields the probability of x being within some range of values

[neat lintel]

Yes. I believe so(?)

Like lets say for instance a factory produces 100 cars where 2 cars have defects. What is their true success?

In %

[pale orchid]

sounds like a probability distribution that already exists and you should google how to write properly 😛

[neat lintel]

Have not found anything.

I used laplace rule of succession

Because I want to find the two values that are most likely is the car producers succession

I would love to find a formula for it, that isn't this complicated and hard to calculate as mine is.

[pale orchid]

i would really suggest asking in one of the questions channels tbh

and regarding notation, a quick read on what functions are and their notation on wikipedia should kickstart you

for specifics on latex, the latex help channel

[neat lintel]

Thanks!

[mint patio]

yooooooooooooooooo

im actually enjoying doing engineering for once

analyzing trusses is so fun

it's like solving a puzzle

[pale orchid]

sounds kinda truss

what's a truss btw?

[sick kite]

truss more like sus

[pale orchid]

trussy baka

[steep mountain]

kinda truss

[sick kite]

truss 🥴

[mint patio]

hey edd

im analyzing dnusses now

[pale orchid]

stun seed analysis

[sick kite]

dnussy baka

🥴

[mint patio]

edd don't you want to know what dnusses are : (

[steep mountain]

dnusses deez nutz idk

[mint patio]

thanks for ruining my joke james

[pale orchid]

that i answered you with stun seed analysis should've been clear enough

[mint patio]

I don't get it

[pale orchid]

the fundamental theorem of stun seed analysis goes as follows: "read it backwards"

[sick kite]

i will crucify you

[pale orchid]

😌

got'em

[sick kite]

[mint patio]

you are an animal

I've lost all respect for you Edd

[pale orchid]

https://tenor.com/view/soobkitty-otter-dance-gif-19501900

[sick burrow]

anyone know anything about the Churchill scholarship?

[bronze pelican]

yes

[sick burrow]

what

what do you know nYaminoid

[sick kite]

what do you know nYaminoid

[sick burrow]

why bully @sick kite

[sick kite]

i want to know too & its just funny

[sick burrow]

@bronze pelican

[bronze pelican]

I applied to it

i didnt get it tho

this webpage has info about applying https://www.churchillscholarship.org/apply.html

o fuq the deadline was yesterday

[sick burrow]

I was just curious because a research program in math straight out of undergrad seems strange

[bronze pelican]

its sort of like a masters actually

[sick burrow]

is it not a research thing?

[bronze pelican]

https://www.churchillscholarship.org/documents/Part3MathematicsforChurchillScholars.pdf

no, you take courses

[sick burrow]

yeah I read through that

sounds like you get material vomited at you and then kinda die

[bronze pelican]

https://www.maths.cam.ac.uk/postgrad/part-iii/part-iii-guide-courses

[sick burrow]

but then if you survive you might actually learn some math

[bronze pelican]

look at how many number theory courses they have

[sick kite]

too many

[bronze pelican]

I wish i had that at my school 😔