serious-discussion

[neat lintel]

LOL

Says cringe shit like this and asks if everyone ELSE is okay. XD

[charred mortar]

Unfortunately I am unable to do anything like this with respect to the second dimension

[neat lintel]

hey <@&268886789983436800> this is not nice!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

[neat lintel]

"DAAAADDDY! Guaappi is saying mean things to meeeeE~"

[cold needle]

what

[neat lintel]

snitching.

[meager sonnet]

???

you were rude out of the blue

[neat lintel]

Was being facetious and he like : "Are u okay? LOL"

ban plsss

[tall badge]

lets not backseat mod

[neat lintel]

....really?

[tall badge]

at the same time, @neat lintel grow the hell up

[neat lintel]

So, honest question : You have been through uni, right?

and Youtube?

Cuz his remarks about 4th dimension was. . .A bit "triggering" shall we say?

[tall badge]

pls dont deflect

[neat lintel]

Rokabe. . .

That was a vibe check to see how he would react.

[cold needle]

???

are youfr

[neat lintel]

one sec.

well anyways gn and thanks

[tall badge]

setting off other users intentionally is unwelcome here

[neat lintel]

You misunderstand. . . -_-.

When people hit 18, they typically go through their r/iamverysmart phase.

I am no exception to this.

I went through that phase.

That question he asked? Reminded me of my younger self and people I CONSTANTLY run into online.

[tall badge]

no matter what explanation u give for what u did, ur going to come off as an asshat

[neat lintel]

Nah. I reject that.

[tall badge]

i suggest just dropping this and moving on

[neat lintel]

K

[ancient flame]

lemme see whats going on in the math discord!!!

nvm this place sucks!!!

[sleek wing]

run

[grizzled grove]

profinite groups are strange

[charred mortar]

Powerwalk

[neat lintel]

i didn't

[neat lintel]

Then 18 year old you is MUCH better than 18 year old me.

But in all honesty. . . That's . . .Not a very high bar. . . . XD

[static loom]

I went through my iamverysmart phase at 5 years old

that's just how smart I am

[charred mortar]

I went through the r/iamverydumb phase

[neat lintel]

That's called being precocious, Merosity.

i was just horny when i was 18

[neat lintel]

Wait, you stopped being horny?

no

I'm literally horny all the fucking time.

I tried everything : The gym, meditation, no-fap, eating properly, that one meme. Nothing works.

[grizzled grove]

What do u want to work?

U want your hormones to stop working?

[sleek wing]

firstly, what the fuck

[neat lintel]

Going 5 minutes without thinking about butts.

[sleek wing]

secondly how the fuck is no-fap gonna make you less horny

[neat lintel]

Well it-

You see-

I. . .

[grizzled grove]

You lack a goal

[neat lintel]

this isn't really the right place for this kind of conversation

[grizzled grove]

That's the issue

[neat lintel]

Not true.

[sleek wing]

dw that's my only comment on da situ

[tall badge]

i go away for 2mins and this happens

[neat lintel]

My issue isn't a lack of goal, it's a lack of focus and consistency.

[charred mortar]

Anyways let’s move on to a different topic

[tall badge]

yes lets try to keep it family friendly

[grizzled grove]

I wish to talk about profinite groups

[neat lintel]

Sowwy fwiends.

[grizzled grove]

But no one is available

[neat lintel]

I would. . .But I'm not at that level yet.

What even is a profonite group?

[sleek wing]

glue the funny finite groups together

[grizzled grove]

Inverse limit of inverse system of finite groups

[neat lintel]

You can invert a limit?

[sleek wing]

inverse limits are so cool bro fr

[grizzled grove]

But is it just that?

[neat lintel]

Wait, I thought inversions were switching the input and output of functions.

And limits were something of a "status" of a function.

[grizzled grove]

We already defined inverse system and inverse limit

Then just giving special name

[neat lintel]

Yeesh, fine.

[grizzled grove]

For finite groups

[sleek wing]

it's also called the projective limit which is better imo

[grizzled grove]

To say profinite word

[neat lintel]

So inversions are projective limits?

[sleek wing]

no

ok this isn't like

a calc limit

[neat lintel]

Oh shit. . .

Hang on. . . ties headband

[sleek wing]

this is a limit defined in an arbitrary category, usually defined by a universal property

[static loom]

what interests you in profinite groups?

[grizzled grove]

Profinite groups is not really that necessary word needed

[neat lintel]

So, a limit set by an axiom?

Wait. . .

[grizzled grove]

Is it just to distinguish finite groups

[grizzled grove]

To understand profinite completions jn p adic stuff is my interest

[neat lintel]

join p-adic ???

[sleek wing]

wikipedia has a couple of alright examples

[neat lintel]

Ooooh. . . .

[static loom]

are you asking if every infinite group is a profinite group?

[sleek wing]

I'm not really sure how to describe it in a way you'll get

[neat lintel]

You're doing plenty by sharing this with me.

Thanks.

I'll find my way to it somehow.

[sleek wing]

yeah, in a grad course on algebra

[grizzled grove]

No I am asking what is the beauty of profinite groups

[neat lintel]

xD Yeah, that's gonna be FUN.

[sleek wing]

it is

[bronze pelican]

Profinite groups = galois groups

[vast surge]

A friend of mine told me that $\hat{\mathbb{Z}}\cong\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$. I'm not convinced that I believe him, this just feels weird.

[fathom swallow]

Zorn's Lemon

[vivid halo]

yeah that's not true at all

what is true is like

$\mathrm{Gal}(\mathbb{Q}^\mathrm{ab}/\mathbb{Q})=\widehat{\mathbb{Z}}^\times$

[fathom swallow]

nGroupoid

[vivid halo]

but the non-Abelian part $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}^\mathrm{ab})$ is extremely mysterious

[fathom swallow]

nGroupoid

[vivid halo]

there were some old conjectures that this non-Abelian part is free profinite on countably many generators with some prescription of how this interacts with the Abelian part but this is almost certainly false

[vivid halo]

to say a word about why this is true: by Kronecker-Weber every Abelian extension of Q is contained in some cyclotomic extension, so the maximal Abelian extension Q^ab is the direct limit of all cyclotomic extensions Q(\zeta_n)/Q, with Galois group the inverse limit of the Galois groups Gal(Q(\zeta_n)/Q)=(Z/nZ)*

[azure nymph]

My calc 2 prof has decided to skip Taylor series

[neat lintel]

that's depressing

[reef carbon]

your calc2 prof needs to be fired

[azure nymph]

I get that it's a summer course but we spent so much time on theory and proofs of other stuff just to skip Taylor series

[charred mortar]

Damn that sucks

[azure nymph]

I'll probably edit my course eval to complain even though it won't do anything most likely

[neat frost]

There’s only one person that I’d allow to be on me….

[deep mango]

I hope you are not an adult

That seems unrelated

It is not a good look to proposition randos on a discord server who could be minors

[neat lintel]

i think you should do more, and absolutely blast them for omitting such important material

pinging you because i think it's important to see

Cat wink

"arc"?

there you go

every TA and professor i get has an orbital laser focused on them at the start of the semester and if they screw up enough it fires

[sick kite]

every TA is TTerrafied

[neat lintel]

i'm literally paying them they should be fucking scared

[azure nymph]

We didn't get a TA cause it's a summer online. Honestly this class was not great though, I mean 30 people dropped it so

[sick kite]

what does it mean for the "orbital laser" to fire?

[neat lintel]

Taylor Series?

How can you skip taylor series in a calc 2 course

What

[neat lintel]

i write the most scathing and destructive course evaluation ever

i will put much more effort into blasting a professor's poor handling of a course than into their course itself

[azure nymph]

Idk I was pretty excited for calc 2 mainly cause of sequences and series but we did polar, parametric, and up to power series in a week which is very disappointing

[neat lintel]

Time for khan academy taylor series 🤣

[sick kite]

has this ever resulted to anything?

[neat lintel]

personal satisfaction

Do a rateyourprofessor and rate him as crap

Then no one will take his ocurse

[azure nymph]

I mean we went from 45 to 15 so no one is taking course

[neat lintel]

i had the chance to submit a student tenure review for a prof who got an extremely bad review, i could have pasted the course evaluation into it and everything. decided not to

[azure nymph]

But that's cause it's hard, like for instance one our homework questions tonight was just "Prove the ratio test"

[sick kite]

any reason why not?

[neat lintel]

i wasn't feeling spiteful anymore

i didn't even get an A+

something like a B+ or an A-

it's the lowest mark i've ever gotten on a math course after my first year

thankfully it's surrounded by a bunch of A+s and 100s so it looks fine

i wanted to forget the past

i would check what my grade was but i can't access the website

i also need to enrol in courses

I think my lowest grade in a math course was…

Combinatorics.

Don’t even ask how

counting is hard

it is okay

[reef carbon]

counting is so hard there's a running gag in my university that a 3rd year student can't do arithmetic

[azure nymph]

Yeah combinatorics is hard

[neat lintel]

i'm very excited for all of the courses i want to take to be full

if the department doesn't give me a wavier for each of the course i want to take expect to see my university on the news in a little while

[surreal sapphire]

[sick kite]

Is that a bomb threat?

[neat lintel]

don't ask me such questions

[sick kite]

TTerrarist

Jk

I know you wouldn't do such a thing... right?

[neat lintel]

6 hours of focused work >>>>>> any other feeling

[neat lintel]

i don't need violence to reach my goals but it is indeed an effective tool

[sick kite]

I guess if it works for you, then you do you bud 🙂

[neat lintel]

hey i never said id do anything bad to my university

who knows, i could be donating them a generous sum of money

[prisma swallow]

wavier?

[ancient flame]

I kinda agree with ryc here, especially since now im an adult and ur a minor (I think) so yeah we can mess around but uhhhh not too far

[neat frost]

😭

I can’t believe you’re a boomer now

[fervent pebble]

wat

is moddy old??

oh no

[bright hill]

if slurp won't do

I can be your daddy

[ancient flame]

LOL

PAIÑ

[deep mango]

[ancient flame]

HI RYC

[deep mango]

Yeah ok slurp was right about darq.

[ancient flame]

LMAOOO

[bright hill]

LMFAOO

[deep mango]

[ancient flame]

@deep mango ur so cute and fluffy

[deep mango]

What does that mean

[ancient flame]

I just wanna squeeze you

[bright hill]

you see ryc

that wasn't actually creepy

[ancient flame]

lol

[bright hill]

that's what you'd call being hot af

I'm sure you've never experienced that before

[deep mango]

[bright hill]

[deep mango]

I mean gmod did respond with "pain"

So idk if that's what's going on here

[bright hill]

masochism is a thing

[deep mango]

[bright hill]

you see ryc

I control how creepy the convo gets

[deep mango]

Do you?

[bright hill]

and I'm the only one that holds that power

I do

[deep mango]

I mean I could just send a battering ram through this rn

But I have cotton candy blue name so I wont do that

[bright hill]

exactly

unlike you, I'm not bound by mortal restraints

[deep mango]

That is my choice, motivated by a sense of integrity

I am not bound by any restraints but the ones of my own invention

[bright locust]

ryc can control the chat

[bright hill]

the illusion of choice is strong with this one

[bright locust]

He can lock it up

[median zinc]

Go to #cats if you want cute and fluffy

Objectively true

[ancient flame]

lol

[gleaming needle]

Hello! Have anyone heard of or studied Algebraic Fractals (aka structures with self-reference, a "formulaic" analog of "regular" geometric fractals) ? Any information about this topic would be appreciated.

[static loom]

I guess the geometric series S=1+x+x^2+... = 1+xS gets you S=1/(1-x) idk

[gleaming needle]

Are you giving an example ?

[static loom]

you tell me, I've never heard of algebraic fractals before

[gleaming needle]

It would look like something like this.

I googled few times to see if there a field of mathematics which speaks of such entities, but nothing interesting so far.

[surreal sapphire]

continued fractions?

but its hard to tell what precisely you want? just continue applying some algebra operation?

[turbid epoch]

Definitely not the right place to ask, but whats the dot product of the arclength element dL of a function f(x) and a vector? I'm going to go crazy.

[gleaming needle]

Ha interesting, I did not knew these exist.

[gleaming needle]

Well I read the definition of a continued fraction and it does not really fit the profile of what I am interested in. A continued fraction is something of this form,

Its definition doesn't conform to the criteria of being generated recursively from an expression with self-reference in it.

[neat lintel]

can someone explain a rough overview of how topological data analysis works

[surreal sapphire]

if they are periodic, they do

i still think you need to be more precise

i also think that there is no field of math that studies this in general, it seems very niche (and pointless most of the time)

[gleaming needle]

Ah alright

[gleaming needle]

True

[wet birch]

hello, is anyone here interested in negative progression systems and would like to chat about them?

[bright hill]

Check out my last message in #bots

[elder furnace]

Anyone is there who know what actually periodic function Laplace transform..

[bright hill]

I'm just too powerful

[elder furnace]

Can i find laplas of 4 different terms using shifting method

[bright hill]

do you guys have any recommendations for someone that wants to get into philosophy?

maybe something sorta useful like stoicism or smth

[surreal sapphire]

probably try reading philosophy

marcus aurelius is still very readible

[sick kite]

plato's republic is a good starting point

russell's "a history of western philosophy" or if you want to incorporate ideas from other places then maybe a.c. grayling's history of philosophy

if you want ideas

if you want a super intro then maybe try simon blackburn's think

or if you want an into into academic philosophy then the routledge textbooks on epistemology, philo of language and phenomenology are good places to go (especially for modern philo)

[bright hill]

I'm definitely not gonna approach philosophy at an academic level lol

at least, I'm not motivated rn

[surreal sapphire]

just start by reading meditations if you are interested in stoicism

[bright hill]

that's what I'm looking at rn

it seems to be his most popular book

although that could well be coz it's published by penguin classics

[sick kite]

i read a bit of the metaethics routledge textbook and it wasn't too bad, maybe took me like 5 or so minutes to understand each page, and it lets you engage with the ideas on a higher level which is appealing to me

so then, armed with the power of concepts and their pros/cons, arguments for/against you can critically analyse texts such as those of kant or plato or whatever other philosopher

also the SEP and the bibliographies attached to each article are super good resources

[bright hill]

I might be interested in that later on

but I don't want to overcommit straight away

ooooo

beyond good and evil also seems interesting

I've always heard nietzsche's work being praised

I think I'll start with meditations

[sick kite]

idk if nietzsche is the place to start

his work is super difficult to read imo

[bright hill]

I did hear about that too

[surreal sapphire]

the implication is the other way

its published by penguin classics because its the most popular one

its also quite good and surprisingly relevant in modern times

[bright hill]

[pearl moth]

i was never fond of nietzsche, but there are worse

i shold go reread tocqueville

[surreal sapphire]

nietzsche is a very good writer

i feel like he could write anything and i would believe it simply because he makes it sound so good lmao

maybe he did that

[neat lintel]

Picked up Thus Spoke Zarathustra the other week, been reading it slow and getting Nietszche's history and the context of the country he lived in when he wrote it.

Got 3 Dostoevsky books too : Crime and Punishment, Brothers Karamazov and Devils.

[neat lintel]

Beyond Good and Evil is the second book in the Thus Spoke Zarathustra trilogy, I recommend reading Thus Spoke Zarathustra for context as well as a basis for the ideas. After that there is. . . What was that third book again. . . ?

The third is on the Geneaology of Morality.

[pearl moth]

i tried to read crime and punishment years ago, but i couldn't force my way through it. it was verty depressing.

[maiden bear]

crime and punishment is a great book

still one of my very favorites

i also found like a really old version of it in an old bookstore

[ancient flame]

I've been involved in crime and punishment too

oh wait I think we're talking about different things

[maiden bear]

but the svetlana geier translation is superior to that one

[bright hill]

I wish to read crime and punishment too

[maiden bear]

do it

[bright hill]

yessir

probably after I check out meditations tho

[grizzled grove]

Can anyone help me understand why n dimension affine space over field k is same as spectrum of k[x_1,...x_n]?

[storm sage]

I love brothers karamazov

thus spake zarathustra is so poetic :)

I don't love crime and punishment tbh, at least compared to the brothers karamazov

it's pretty repetitive in the first part but after that it picks up and has a lot of interesting ideas

[full isle]

Do you know definition of affine space over k?

You can think of it as n dimensional vector space

Over k

[grizzled grove]

But the forgetting 0 part is sometimes confusing

Because you have to have a feeling of difference between vector space and affine space

[full isle]

Its not that you forget 0 but more that affine transformations dont need to preserve 0

More like basepoint is free

Do you know what spec R is?

[grizzled grove]

Yes I know

Prime ideals collection

Of R

[full isle]

Yes!

Now what are prime ideals of a field

[grizzled grove]

Only 0 and itself

[full isle]

Now did you know that the polynomial ring of a field is a PID?

[grizzled grove]

Yes I know

[full isle]

So what does it mean to be a PID

[grizzled grove]

All ideals are principle

Generated by single element

[full isle]

Also do you know that irreducibles correspond to prime ideals?

[grizzled grove]

Yes

But also they are maximal

Right?

[full isle]

Yes

I think we can show that quickly

We know irreducible means gh=1 implies g or h is degree 0, constant

This is analogous to if gh in p then g in or h in p for a prime ideal p

[frail lagoon]

i love the word play and stuff in also sprach zarathustra

[full isle]

@grizzled grove Do you see it?

[grizzled grove]

I am kinda remembering how I learnt that

[full isle]

The last step to reasoning is seeing that since k is algebraically closed the only irreducibles are linear

So this means that polynomials such as sum x_i-a for a in k are prime

[grizzled grove]

I don't think that is what irreducible means. I mean it says if f=gh then either g is unit or h is unit

[full isle]

Yes

if either one is a unit

you can multiply by the inverse

[grizzled grove]

Are you saying that the bijective map between these two sets is that a goes to x_i-a?

[full isle]

Yes or more like

xi-ai

[neat lintel]

by definition

[full isle]

where ai are coordinates

[grizzled grove]

so (a_1,a_2) corresponds to which prime ideal in Spec k x1,x2

[neat lintel]

which one do you think?

[grizzled grove]

(X1-a1,x2-a2)?

[neat lintel]

seems right but i haven't read the whole conversation

[full isle]

I said some false things by accident

I gave wrong definition for irreducible

I meant to say gh=f like you said

[neat lintel]

the non-maximal prime ideals in k[x_1, ..., x_n] are going to cause problems

[grizzled grove]

But just the ideal x1-a1 is not maximal and hence prime or is not prime and hence not maximal because it is sitting inside (X1-a1,x2-a2)?

[full isle]

Indeed

[grizzled grove]

Mary but you can still help me clear out distinction between affine and vector space. You used the word affine transformation

What's that

[full isle]

Its a linear transformation along with a map that preserves a choice of basepoint is the way I think of it

So if T is your transformation

and you want to have a basepoint of a

[neat lintel]

you're thinking of polynomial rings over a field in a single variable, but they're asking about several variables.

k[x, y] is not a principal ideal domain

[grizzled grove]

Oops

[full isle]

Yeah I made a mistake in that

Is it a UFD?

[grizzled grove]

Yes

I think it is notherian

[neat lintel]

let me clarify my previous comment. a non-maximal prime ideal in k[x_1, ..., x_n] is not going to correspond to a single point in A^n

maybe i should not have crossed out my "by definition" reply

[grizzled grove]

I am now confused. Let's rewrite why n dimension affine space over field k is same as spectrum of k[x_1,...x_n]?

I think it is also important that k is algebraically closed

[full isle]

The way I would phrase affine transformations is that it is a transformation F that is composed of a linear transformation T and a translation x-a to basepoint a. So if you want to see what happens to a vector v and changing the basepoint to a then F(v)=Tv+(v-a)

There is probably a better way to phrase it

[grizzled grove]

No this makes sense. So vector space is different from affine space because affine transformation may preserve any other fixed pt other than 0 but in vector space it has to fix 0

[full isle]

Yessir

[neat lintel]

if "n dimensional affine space over field k" means A^n = {(a_1, ..., a_n) : a_i \in k}, then the maximal ideals of k[x_1, ..., x_n] are in one-to-one correspondence with the points of A^n. this is because they are precisely of the form (x_1 - a_1, ..., x_n - a_n) (by nullstellensatz; algebraic closure is used here). the moment you consider non-maximal prime ideals (which are still elements of Spec(k[x_1, ..., x_n])) this correspondence breaks down

[grizzled grove]

But do non maximal prime ideals even exist in this case?

[neat lintel]

(x_1) is prime, but not maximal since it's contained properly in (x_1, x_2)

[grizzled grove]

Do you think affine space has any advantage over vector space. Like an example of result?

[full isle]

They are used in different contexts. I guess mainly when we work with objects where translations are allowed

They come up in tropical geometry

Well these objects called affine integral manifolds

[grizzled grove]

Then how come A_n= Spec k[x_1,...x_n]?

[full isle]

Thats just the definition after working out that Spec of that corresponds to a n dim vectspace over k

[grizzled grove]

Well in algebraic geometry why we didn't do all stuff on vector space. Why we did on affine space.

[full isle]

Im not sure how affine transformations come into it though

Im planning on learning soon

[neat lintel]

let me make my concerns clear. how is "n dimensional affine space over k" defined for you?

i would like the definition

[full isle]

Personally I see it as the same as a vectorspace along with a translation map sending a vector v to x-v

[grizzled grove]

(a_1,....a_n) tuples collection. Each tuple corresponds to one unique maximal prime ideal. But Spec does not consists of those only...so it feels like Spec is bigger than A_n

[full isle]

But this is just a personal intuition

[neat lintel]

which is exactly what i wrote down

so they're not really the same

something is going on here...

[grizzled grove]

https://en.wikipedia.org/wiki/Spectrum_of_a_ring

See the second example

[neat lintel]

this looks more like a definition of n-dimensional affine space in the scheme theoretic setting, than a statement of equality

[grizzled grove]

No it's not definition...i also have seen this in poonen notes

Without any scheme etc

[neat lintel]

post notes

[pure sun]

the "traditional" model of algebraic geometry only looked at maximal ideals

whereas now when people talk about spectra of rings they include all prime ideals

often times when discussing a spectrum, though, people only refer to the maximal ideals, since those give the closed points of the space

the rest of the prime ideals are only really useful in giving some topological information.

[neat lintel]

i am really glad that i do not study algebraic geometry

[full isle]

Why is that

That is a topic I think I would like to go into

[grizzled grove]

https://www.google.com/url?sa=t&source=web&rct=j&url=https://math.mit.edu/~poonen/782/782notes.pdf&ved=2ahUKEwiCif_j8_H4AhW-hGMGHVKRCfoQFnoECAoQAQ&usg=AOvVaw1KxG1-kT_LehGXRARdMrBc

[full isle]

but im on the edge between it and algebraic topology

I took a difftop class but it isnt my to my liking

[grizzled grove]

See definition 22.1

[pure sun]

yes, affine space is spec(k[x1, ..., xn])

[full isle]

Speaking of

I dont ser how adic numbers are that useful

Maybe I need to spend more time reading

[pure sun]

adic numbers? do you mean p-adic numbers?

[full isle]

Yes sorry

[neat lintel]

it feels like way too much of it is grappling with increasingly verbose abstraction, definition after definition. that's kind of the vibe i got from my algebraic geometry class

[full isle]

are adic numbers different

[pure sun]

"adic numbers" aren't really a thing

[grizzled grove]

But how? We could only correspond maximal prime ideals with points in An

[neat lintel]

in the "geometry" part of the algebraic geometry class we spent too much time defining things and too little time actually proving cool facts about anything

the "algebra" part of the class was fantastic, though

at least i know what a sheaf is now, i guess

[pure sun]

yes? what's the problem? the closed points of affine space are exactly the n-tuples of points as you want

the remaining non-maximal prime ideals are only giving extra topological information

when you are visualizing a spectrum of a ring, you should only visualize the closed points, i.e., the maximal ideals

here's how it was described to me once: it's like a bowl of cereal. the pieces of cereal are the clsoed points, i.e. the maximal ideals. the non-maximal prime ideals are the milk which holds everything together

(actually it was described to me in terms of raisin pudding but i dont like raisins)

[neat lintel]

based

[pure sun]

it's also been described to me by an australian professor as a family of koalas all on a big eucalyptus tree. the koalas are the closed points that you can actually see, and the branches of the tree are the non-maxmial ideals which are providing "structure"

you don't "see" the non-maximal ideals when you look at spec, they are just there in the background providing the structure/glue to hold it all together

so you should visualize spec(k[x, y]) as the plane consisting of points (a1, a2) in the traditional sense

[grizzled grove]

Ok, so i get it that you send (x1-a1,x2-a2) to (a1,a2) but where do you send the prime ideal (x1-a1)

[pure sun]

it's the line x1 = a1

and similarly the ideal (y-x^2) corresponds to the line y = x^2

the ideal (y-x^2) is a "point" in spec(k[x,y]), but it doesn't live in any aprticular place. it is the "branch" of the tree which goes along the parabola y = x^2

and which holds all of the points on it together

[grizzled grove]

I see so these are equal as sets. But no bijection exists between them

[pure sun]

i dont understand what you are saying

strictly speaking, as a set, spec(k[x,y]) is larger than just the set {(a,b) | a, b in k}

in algebraic geometry, "affine space" is used to refer to spec(k[x,y])

[neat lintel]

i was right all along

[pure sun]

there might be other fields which use the phrase "affine space" to mean something different, for example, maybe just to mean the set {(a,b) | a, b in k}.

i am also telling you that even in an algebraic geometry setting, you should visualize affine space as just the traditional {(a,b) | a, b in k}

because the other "points" don't really look like "points" in the traditional sense

[grizzled grove]

Let B=Spec kx1,x2...,xn
A=An affine space of n dimension
Now in the notes it is written A=B

But the way we are sending a point in A to maximal ideal in B, it seems that this way of mapping is not bijection because
Both prime ideals (x1-a1,x2-a2) and (x1-a1) have image containing (a1,a2).

[pure sun]

i'm sorry but did you read anything i just wrote?

[neat lintel]

[pure sun]

A is equal to B by definition

that is the definition of A

[grizzled grove]

I see. But For defining A you need the word Spectrum? I do understand that as sets they are equal. I am just trying to identify each single element of spectrum with each single element of Affine space.

[pure sun]

I'm not sure what you're having trouble with here. affine space is, by definition, the spectrum of k[x,y]. (only 2 variables here for convenience, of course this is also true for more variable)

"elements of affine space" are literally the same thing as "prime ideals in k[x,y]"

there's no need to identify them with each other

if we look just at the subset of affine space consisting of closed points, i.e. those which correspond to maximal ideals

we can identify that subset with {(a,b) | a, b in k}

via (a,b) --> (x-a, y-b)

affine space contains additional points

which are not of that form

corresponding to the non-maximal prime ideals

[neat lintel]

buncho look at latest ivory message

[zenith forge]

So the geometry was great?

[odd narwhal]

Ah yes, the raisin pudding model of the atom affine space

[pure sun]

hahaha exactly

[full isle]

(x) is a prime ideal

What element does this correspond to?

[neat lintel]

please

[pure sun]

i dont know what more you want me to say

[neat lintel]

read the entire conversation you missed

[vivid halo]

[pure sun]

elements of affine space are literally the same thing as prime ideals in k[x,y]

the ideal (x) corresponds to itself

[full isle]

This

[pure sun]

when you visualize spec(k[x,y])

you should imagine the "plane" {(a,b) | a, b in k}

the ideal (x), which is the same as (x-0), should be thought of as the vertical line x = 0

sitting in the coordinate plane

[full isle]

Ok yes thats what I thought

[full isle]

Sorry if I was being annoying

[tender tulip]

Is this "technically" a differential equation if P(x) is some function but x' and y' are unknown $\\frac{\dot{x}}{P(x)} + \frac{\dot{y}}{P(y)} = \frac{\frac{d}{dt}[T(x,y)]}{T(x,y)} \
\text{Where:} T(x,y) = T(y,x)$

[full isle]

I already see this

And Im scared that it doesnt stop

[neat lintel]

you are now by ping spamming me

[full isle]

But algebraic topology seems tonhave the same feeling for me

Oh sorry I forgot that replies spam

[neat lintel]

it happens

[vivid halo]

yeah both AG and AT involve a lot of machinery

[fathom swallow]

Mizalign

[pure sun]

lots of math gets very abstract, of course different fields get more abstract more quickly

algebraic geometry is really fucking hard

[vivid halo]

yeah

[pure sun]

people didn't just make up definitiosn for shits and giggles

[vivid halo]

it does get to a point where you have the machinery to do more conceptual things, but it takes a while to get that machinery going

[pure sun]

yeah

[neat lintel]

maybe my gripe is just with the class itself, maybe it's with how long it takes for it to get going

i liked when we were just doing stuff with affines

[vivid halo]

yeah AG is a hard topic to teach

[full isle]

Its not necessarily that I think the definitions are redundant. Im just scared that there wont be a point where im not learning definitions

[vivid halo]

it does get better eventually

[full isle]

Seems like every problem has its own tailored definitions

[pure sun]

ok actually dinner time nw

[neat lintel]

the class itself was enjoyable, but i came out of it feeling like i just learned a bunch of definitions. more like a "okay, now you can go take another AG class" instead of learning anything super substantial

[pure sun]

glhf everyone

[blazing pawn]

The solution is to try to learn anabelian geometry while not knowing anything beyond the definition of a scheme and get absolutely obliterated

[neat lintel]

have a good meal buncho

[vivid halo]

I mean if you take something like Hartshorne as an example, chapters 2 and 3 are all machinery and that can take a really substantial amount of time to get through

[pure sun]

thanks :^)

[blazing pawn]

have a nice dinner buncho

[vivid halo]

by chapter 4 you have enough to develop the theory of algebraic curves

[full isle]

Isnt a scheme a ringed spaced covered by spectra of commutative rings

[blazing pawn]

Unironically i dont think this is a horrible idea

[vivid halo]

so by then things are more geometric again and you can actually compute things that relate to classical things again

[vivid halo]

essentially yes

[blazing pawn]

locally ringed but yes

[vivid halo]

locally ringed space

[full isle]

I dont see what locally adds here

[neat lintel]

when the AG prof writes "C-ringed space"

[full isle]

Usually it would be clear

[vivid halo]

yeah so the standard example is like

if you look at Spec(R) where R is a discrete valuation ring

then Spec(R) consists of two points

one closed point s, and a generic point \eta

If R has residue field k, then you get a map Spec(k)->Spec(R)

[blazing pawn]

it doesnt really matter for the definition of a scheme itself because you're already demanding that its locally Spec(R) which is clearly locally ringed

but for morphisms it does matter

[vivid halo]

Spec(k) consists of only one point, namely (0)

but if you think about the map Spec(k)->Spec(R)

there's two maps of ringed spaces like this: one that sends (0) to s, and one that sends (0) to \eta

only one of these is a map of locally ringed spaces though!

and for schemes, you should only have one map Spec(k)->Spec(R)

so the locally ringed part is what enforces this

[blazing pawn]

I think maybe a good way to think about it intuitively is that the "locally ringed" condition is saying that given f: X -> Y, the maximal ideal of the local ring at f(x) is mapped to the maximal ideal of the local ring at x. If you think about the classical situation where the maximal ideal consists of regular maps that vanish at f(x) this is saying that they pull back to regular maps that vanish at x

[vivid halo]

and yeah what moth said is more conceptual

the example I gave is sort of the minimal example where the difference becomes visible

[full isle]

Ok thanks for the elaboration

[neat lintel]

Yeah, a youtuber made a video on the Flame of Frenzy as a concept and basically tricked me into watching a 25+ minute lecture on the brothers Karamazov.

Thing is : It perfectly elaborates on the Flame of Frenzy so much that you have to wonder : Is it REALLY a trick?

[storm sage]

LOL

what is the flame of frenzy

[velvet dagger]

Oh whoops

[neat lintel]

Ts....What is the Flame of Frenzy. . .?

Remember that monologue in the Brothers Karamazov where they are talking in the bar?

[storm sage]

yes

[neat lintel]

One guy mentions the children. . . . .

[storm sage]

yes

[neat lintel]

Oh the children. . .

[storm sage]

oh it's in a video game

[neat lintel]

And then he ultimately says : "Maybe god has a plan such that the man who tore the woman's child apart, the child and the mother can unite together in tears. But it's not worth it unless I can see it. And unless I can see it, I must have justice"

Yes.

Now imagine after he says that, he basically burns every living thing in the world down with a Yellow flame that drives people mad.

Which after the work is done : Somehow everyone in that game will all return to one primordial life form again?

[neat lintel]

A damn good one at that.

[storm sage]

Well.. that makes the novel a lot less interesting

[tawdry mountain]

I'm absolutely obsessed with this curve. I know exactly how to make it using some python code, but I have no equation for it

it loops forever like a sine wave

[neat lintel]

Eh, well to be fair some faustian bodyjacking cult-leader who goes from 0-100 real quick and his (potential) collaborator convinced you.

[neat lintel]

That. is an eldritch truth that will slowly drive you insane the more you look at it.

Said Eldritch truth is called math.

[tawdry mountain]

its driving me insane already

the fact that i created this curve

it exists nowhere on the vast internet

and if i can't find some sort of meaning

it will be lost

no one else will

[static loom]

dare I ask what code you use to generate it with

[tawdry mountain]

from turtle import *

speed(0)
left(90)

while True:
    forward(1)
    right(ycor() / 100) #The 100 is for scaling

basically to start points straight up
moves forward by 1, then turns clockwise by the y position, and repeat those two steps

simple but annoying

it creates a curve with no rhyme or reason

and it doesn't matter if it's degrees or radians

it's the same

f(x) is dependent upon f(x-1)

wait

f_n(x) is dependant upon f_n-1(x)

however that works

since it creates two sequences at the same time (x and y)

i guess it takes 3 inputs

f(x,y,a) where a is the angle

and sure i could probably write an equation for that

that wouldn't help generate a general function

or generalize it to f(c,theta) and use complex inputs

[frozen merlin]

here's what I'm thinking

I think the angle is arctan(df/dx)

[tawdry mountain]

x -> cos(a)+x
y -> sin(a)+y
a -> y

That's what the function does (i think)

[frozen merlin]

and the relation thing is $\dv{x} \arctan \dv{f}{x} = -f(x)$ up to scaling

[fathom swallow]

Average J∘du=du∘j enjoyer

[frozen merlin]

this seems like an ugly DE

bingo

looks similar

[tawdry mountain]

similar

[foggy finch]

studying geometry is cool, I just feel like I understand reality in general better

[azure nymph]

18 page mocktest for a 7 question test

[charred mortar]

Urgh

What are those 18 pages

[neat lintel]

Can someone tell me if I got this right because my mind is kind of blown. Like people usually say that probability has no memory, so that if you flip a coin and get heads three times, then you aren’t any more likely to get heads the forth time. (Which is called the gamblers fallacy I think). But I think this is wrong now. Or to be more precise, this presumes a fair coin. But in the real world, and when we do statistical analysis of actual coin flips, it turns out that all coins are biased. So doesn’t this mean that the gamblers fallacy isn’t a fallacy?

It’s only a fallacy in an idealized world?

This probably isn’t true for slot machines, but is it true for coin flips?

[velvet dagger]

Probability represents both "actually random" things and things for which you don't have info

In reality a coin is deterministic

If you know initial position and velocity and physical properties of the coin you know everything

If you treat everything but the coin properties as random, and you know how the coin is biased, now it's not 50/50

But if you don't know the coin is biased, or in what direction

Then it's by some metric "functionally 50/50"

[neat lintel]

Ohhh okay, so 3 flips heads isn’t enough to tell me if the coin is biased heads?

So let’s just say you win a million dollars if you guess what the forth flip is gonna be. Would it be rational to bet heads?

[velvet dagger]

You could probably do some kinda statistics to figure that out

Almost like

"If you have 3 heads in a row what's the probability that this is random chance vs something's up with the coin?"

Something something p value something something

[neat lintel]

Oh okay then I’ll try and figure that out then

[charred mortar]

Gamblers fallacy assumes that the events are basically independent

So like if you’ve done a million flips and the distribution of heads/tails is “reasonably random” (I.e. p-value is fairly high and stuff) then there’s no good reason to believe that the next flip is one of heads/tails

But if you’ve done a million flips and they’re all heads then it’s extremely likely something’s wonky

[velvet dagger]

In particular ab probably biased the coin somehow

[shadow hare]

How much of a bad habit is it to study a mathematical topic by reading what the theorems state and then read their proofs almost immediately? I usually try to think about theorems and prove them myself before proceeding, but I think sometimes that takes quite a good deal of my time and concentration, so sometimes I just read the proof almost immediately.

[leaden torrent]

not really a big deal

like it's better to try the proofs yourself where feasible

not even to succeed at them, just trying and failing is useful

but it's not too big of a deal to skip that if you're doing all the exercises and stuff

(by "all the" i don't mean literally all the exercises, i mean enough so that you're comfortable with the material)

[sick kite]

when can one say they're "comfortable with the material"

[leaden torrent]

that's an intuition thing, and kind of hard to answer

my general metric is "can look at a problem and, with only a bit of thought, have a good idea of how to approach it and what special considerations might arise"

[sick kite]

that seems like a good heuristic

[shadow hare]

I see, thank you.

I don't know how comfortable I am with problems on invariants, to be honest.

Like, after detecting a hypothetical invariant, it's generally not difficult for me to prove it is an invariant using induction. But detecting an invariant can sometimes be hard for me.

[neat lintel]

Yo, you go mad yet?

[neat lintel]

Really really really bad, but then again….I’m not on y’all’s level yet so….

[neat lintel]

This honestly sounds more like the “prep” phase than the study phase tbh though.

[shadow hare]

hmm I arrived at a contradiction 😄

what do you call a prep phase? (and a study phase?)

[leaden torrent]

have you even read a definition-theorem-proof textbook before @neat lintel

[neat lintel]

Wtf??? No!!!! Those exist?!? Like books with JUST definitions AND theorems and no problems?

[leaden torrent]

well they have problems as well (usually), the point is that it omits all of the exposition that you might see in, say, a calc textbook

that's spent on nonrigorous considerations

[neat lintel]

Dayum…..

[shadow hare]

The textbook I'm reading currently isn't too dense. It's your average discrete math textbook taught at the undergraduate level. But well, it has an emphasis on proofs (as a discrete math textbook should) and that's just what my current level is. (btw, it's the Mathematics for Computer Science textbook from MIT.)

[leaden torrent]

at that level, i wouldnt expect you to even be familiar with enough proof techniques/common patterns that you'd be able to get all the proofs without reading them

it's still good practice to make an attempt

but dont feel too bad following the textbook's intended progression

just make sure you actually UNDERSTAND the proof as written

[shadow hare]

I mean, I'd say I actually do get most of them myself (as I have discussed their correctness on this server too), but sometimes that takes much of my time.

[leaden torrent]

not just "I can see why this manipulation works", things like "I can see why this manipulation was necessary in the proof strategy"

[neat lintel]

So, I haven’t touched the books namington told me about….But I’ve been reading how to read and remember.

Everything points to having a prep phase where you read all of the content in a book loosely to see where you are and where you are going.

Then you read the chapters and before delving into a chapter or a section, one looks at the problems in the back. (I just write down the hard ones)

And looks at the explanation to see what the sections teachings are for.

The goal of this is to sort of build sustainable understanding so that you don’t have to memorize.

Like when you learn enough about words and language, if you forget a definition , you look at its root words and come to the conclusion as to what the word means.

[shadow hare]

ah right

I don't know if this is good or not, but I rarely study math at different levels of focus. Generally, I start by studying at more or less full attention, even though that usually results in slow progress.

[leaden torrent]

that approach seems like it might work for retention-based material but not really for mathematics

perhaps i'm just a nonbeliever

but it seems like it prioritizes learning to like, memorize and contextualize things

which is just... not the hard part of mathematics

[neat lintel]

We’ll see that’s the funny part, the point is to NOT memorize

[wicked ore]

I don’t understand, what is unique about this approach exactly

[neat lintel]

It’s like….remember linear algebra and how every matrix has a basis?

The point is to engage with the material in a way so that you understand it, and memorization in a way IS a part of this yes: but you memorize CERTAIN things by committing them to memory which acts like a net of sorts, and this net protects the understanding you have of the subject.

Take me memorizing how willpower works.

I have a mnemonic, Roy baumeister the brewmeister drinking a beer he crafted on Sunday.

Roy Baumeister did the experiment with the marshmallows.

This marshmallow experiment involved kids waiting.

Ultimately, the experiment proved that saying “no” to something is a drain on willpower, the more you say no the more your willpower reduces.

But willpower is like a muscle. One must expend it, or it will atrophy.

Another fun fact : those who learn to master their willpower are more likely to be successful. (Baumeister kept tabs on the kids who he did the experiment with. Those who waited and got the second marshmallow ended up becoming very successful people.)

So with this context, If I were to say…..read a book on willpower and the parts of the brain that make it happen, it would not only be easier to understand but to remember. Because I have a basis.

[leaden torrent]

...

okay cool

[wicked ore]

I really don’t understand lol

[leaden torrent]

i dont see how any of that relates to what i said

[neat lintel]

……

[leaden torrent]

if you want to use a different word than "memorize" then sure

my point is that "definitional/theorem/algorithmic recall" is literally the easiest part of mathematics

most of my exams in undergrad were straight-up open book

[sleek wing]

Word salad

[leaden torrent]

or at most "bring your own cheat sheet"

because they simply didnt test on that

[neat lintel]

An analogy is word salad….?

[leaden torrent]

i do not understand the point of the analogy

[sleek wing]

That message is word salad

[neat lintel]

I’m sorry you feel that way.

[wicked ore]

I concretely do not understand what is different about this approach

[sleek wing]

“Mnemonic to memorise how will power works” what the fuck

[neat lintel]

That is what we call building a straw man.

The mnemonic was the beginning of how I remember how willpower works….

I never said the mnemonic was memorizing how willpower works by itself.

[sleek wing]

I really don’t care about your logical fallacy fuckery, a mnemonic is completely unneeded to remember how willpower works

As nami said the message was completely unrelated

[leaden torrent]

i dont see how the mnemonic connects to the conclusion either

i feel like "willpower is like a muscle" is like

a far easier and more direct analogy to memorize

than whatever the fuck the beer thing is

[neat lintel]

Okay….?

[wicked ore]

I just literally don’t understand what this beer thing is

[neat lintel]

Roy baumeisters name rhymes with brewmeister…..

[sleek wing]

Fractal

[wicked ore]

Like… he crafted a beer? Marshmallows? What?

[leaden torrent]

my point is that i dont see how the brewmeister thing connects to the rest of the message

also who tf uses "brewmeister"

[bright hill]

TIL: surd is a thing

[frail lagoon]

Tfw mnemonics are meant to be memorable

[sleek wing]

Boss this is like finding where the chicken nuggets in the fridge are by considering their gravitational affects on proxima centuri like wtf

[leaden torrent]

like dont get me wrong

if this works for you

then thats good

i wont object

[neat lintel]

Okay look….y’all can disagree…..? You don’t have to be assholes about it.

Like holy shit.

[leaden torrent]

but it seems weird to be making authoritative statements on how other peoples' study methods are terrible

[sleek wing]

Nah I do

[wicked ore]

CONCEDE

No I’m kidding but

I just don’t really get it

[leaden torrent]

when you havent even taken a course of a similar level to them before

or of a similar style

[shadow hare]

is this discussion a corollary of the original message I sent?

[last oxide]

honestly, I have been personally struggling quite a bit with contextualizing more "advanced" material

it seems like the standard classica dont make that much effort for it

areas that started historically concrete and got too abstract feel like they present everything kinda "backwards" if that makes sense. a specific complaint tho

[neat lintel]

Well, here is the thing….if you can’t be bothered to have this thing called manners? I don’t have to engage with you.

[sleek wing]

Ok then why are you engaging with me

[leaden torrent]

this is fair

and this is a case where i'd genuinely recommend skimming the problems first

and then reading the text

i just dont like their "read it twice" approach

seems really weird

[bright hill]

wew having manners?

[leaden torrent]

but yeah

[neat lintel]

You don’t really “read it twice “ you skim first and then read.

[bright hill]

impossible

[last oxide]

yeah
I dont even recommend reading big books
and maybe Im projecting here
they are fun to go around, but too hard to finish

[wicked ore]

Yeah I feel like sometimes seeing the “original” problem is a good motivator for material

[neat lintel]

Good thing I blocked him, then. Can’t stand rude people.

[sleek wing]

I can’t stand bullshitters

[wicked ore]

Otherwise sometimes it’s like “why in the world would someone invent this thing… oh the application is on the next page… why didn’t they show the original problrm first”

[neat lintel]

Yup.

[shadow hare]

||first, you learn to read. then, you read to learn||
I'm sorry

[last oxide]

this is happenning more and more tbh

[neat lintel]

Also, Google Roy Baumeister.

There was clearly some confusion by what I wrote and that’s cool.

I guess I didn’t provide enough context for you to understand where I was coming from.

[shadow hare]

I think I have actually come to appreciate math for its own sake regardless if it has applications in a particular area or not. like, it is enough of a fun game for me on its own to be honest, even though that probably sounds badly pretentious.

[neat lintel]

The mad cycle of being human….when anijfri….? When will the madness end and when will we be set free from this Samsara??

[neat lintel]

Pretentious? That sounds charmingly childish in all the right ways and I fucking love it.

[leaden torrent]

on the bright side, youre in good company

When I first saw [Lang's Diophantine geometry], about a year ago, I was disgusted with the way in which my own contributions to the subject had been disfigured and made unintelligible. My feeling is very well expressed when you mention Rip van Winkle!

The whole style of the author contradicts the sense for simplicity and honesty which we admire in the works of the masters in number theory - Lagrange, Gauss, or on a smaller scale, Hardy, Landau. Just now Lang has published another book on algebraic numbers which, in my opinion, is still worse than the former one. I see a pig broken into a beautiful garden and rooting up all flowers and trees.

Unfortunately there are many "fellow-travellers" who have already disgraced a large part of algebra and function theory; however, until now, number theory had not been touched. These people remind me of the impudent behaviour of the national socialists who sang: "Wir werden weiter marschieren, bis alles in Scherben zerfällt!"

I am afraid that mathematics will perish before the end of this century if the present trend for senseless abstraction - as I call it: theory of the empty set - cannot be blocked up.

• letter from Siegel to Mordell

he literally compared Bourbakists to Nazis lmao

[wicked ore]

Lmao wow

Wait what is bourbakism

I’ve heard of the group, but what is the ideology

[leaden torrent]

a certain pedagogical-slash-philosophical approach to mathematics textbook writing that emphasizes abstract, hyper-formal ground-up definitions and constructions

the modern axiomatic method traces its centrality back to this school of thought

obviously axioms were used before Bourbaki, but they werent so central

a ring was only axiomatized in the 1920s by noether!

[neat lintel]

So math that is built with definitions instead of theorems?

The justification being : “we have axioms and they are self evident, so back off!”

[leaden torrent]

???

no???

[neat lintel]

Damn…..

[leaden torrent]

the theorems are proved from the definitions

[neat lintel]

But….

But …..

[wicked ore]

I hate this, a lot

It’s terrible and lazy pedagogy

[surreal sapphire]

do you have a source for this

[neat lintel]

Yesterday you told me definitions can’t be proven…..

[sleek wing]

I actually quite like it lol

[surreal sapphire]

Siegel is such an interesting person

[sleek wing]

It’s quite easy to follow

[neat lintel]

How do you use something that can’t be proven as proof…? That sounds absurd.

[wicked ore]

It’s only easy to follow logically

[surreal sapphire]

i am 100% convinced that Siegel was one of the smartest people to ever live

[leaden torrent]

https://www.ams.org/notices/199503/199503FullIssue.pdf

[last oxide]

unironically, riemannian geo (which is part of what I had in mind) had huge contributions of Gauss and looks "nothing" like what he did

[sleek wing]

How so?

[leaden torrent]

page 21 in the PDF

[surreal sapphire]

but nobody knows about him and his type of mathematics died out, save in niche fields of analytic NT

thanks

[shadow hare]

keep up with this and we'll reach The Goedel in 50 messages

[neat lintel]

I’m mad…..

[leaden torrent]

you can prove that something satisfies a definition

the point is that we need some "starting ground" for our reasoning

[wicked ore]

I mean understanding the motivation and problem-context and all of that stuff. The hyper formalism takes all of that away. It feels like you’re in an empty void

Maybe that sounds weird idk

[leaden torrent]

the Bourbakist approach has a VERY low-to-the-ground, concrete starting ground

starting from low-level set theory and all

super granular and nitpicky

[surreal sapphire]

also you can just assume things if you want

[frail lagoon]

Do you want an infinite regress lol

[surreal sapphire]

nobody will arrest you

[neat lintel]

Namington, I want to take your word but when i read that my mind immediately jumped to fallacy of begging the question. Which raises the question: how is this different?

[leaden torrent]

???

[surreal sapphire]

i can assume GRH if i want to

and i will

[sleek wing]

I agree with you on motivation, if that’s important to you I can see why this style would suck. I personally don’t really care about motivation that much and don’t mind if I understand something before knowing the motivation

[frail lagoon]

lol do you like just naming fallacies

[leaden torrent]

like

here's an example where we prove a definition

[wicked ore]

I see

[sleek wing]

Moving the goal posts fallacy

[leaden torrent]

Definition. A matrix A is called "invertible" if there exists another matrix B such that AB = BA = I (the identity matrix).

[neat lintel]

shivering in fear uhhh…. N-no??

[neat lintel]

I follow.

[leaden torrent]

Fact. The matrix I is invertible.
Proof. We prove it satisfies the definition of invertible. Let A = I, B = I. Then II = II = I. QED

[frail lagoon]

So why do you have a problem with people proving stuff from stuff that can't be proven

[leaden torrent]

obviously this is a very simple case

[frail lagoon]

Or rather what kinda framework would you prefer

[neat lintel]

I think I’m starting to understand, but holy shit I’m gonna need to read more to fully get it.

[last oxide]

hmm
hmmmm
seems like a big cope

[sleek wing]

I would never cope

It simply isn’t in my nature

[neat lintel]

Because I actually remember that from linear algebra, thing is….I remembered it being a theorem.

I didn’t remember it being a definition.

[leaden torrent]

Fact 2. The matrix (6 -7; 0 3) is invertible.
Proof. The inverse of this matrix is (1/6 7/18; 0 1/3). [insert computations showing that they multiply to I here] QED

[surreal sapphire]

you can copy paste the handwritten printed letter in the pdf

[leaden torrent]

what?

[surreal sapphire]

thats interesting

[leaden torrent]

the definition of "invertible" is a theorem to you?

i dont even see how that makes sense

[neat lintel]

No dude…here is the thing.

[wicked ore]

I suppose it feels less like you’re an active participant/spectator, and more like they’ve taken all the mathematics and cleaned it up to the point where it’s impossible to understand who did what when and for what reason. Like you’re reading a hyper-filtered version of the original math

[neat lintel]

I didn’t have proper context for definitions vs theorems until y’all educated me yesterday.

[leaden torrent]

i feel like you seriously misunderstood some facets of the terminology at play here

[neat lintel]

I don’t doubt it.

[leaden torrent]

a definition is where we explain what a word or phrase means

[sleek wing]

Yeah gimme that streamlined shitttt, as long as there’s exercises I honestly don’t mind

[leaden torrent]

a theorem is a proven fact about something

typically we take definitions and prove things about them

[neat lintel]

Question.

[leaden torrent]

for example, we take the definition of a field and prove that finite fields have prime power cardinality

[sleek wing]

The clutch edit

[neat lintel]

Definitions can’t be proven, but by taking things about them and proving things about them, are we affirming the definitions existence?

[leaden torrent]

or we prove that some particular construct satisfies the definition of a field

[neat lintel]

I follow.

[leaden torrent]

no, we're just exploring the consequences of that definition

mathematics does not deal with "existence" in a philosophical sense

that's in the domain of physics, engineering, etc

mathematics is just reasoning.

[last oxide]

precisely
sometimes it gives you the opposite of thay nice concrete feeling of doing something "hands on"
by the end, it feels like youve learned nothing about the essence of whatever it is and that you cannot hang on with the people that have

[neat lintel]

I’m not sure I get it. If we’re not dealing with existence in a philosophical sense, why even reason at all?

[leaden torrent]

(to be clear, there's an entire philosophical rabbit hole one can go down with, to what extent, mathematical objects "exist" or "represent things that exist")

[leaden torrent]

because we can apply the reasoning to things that do exist

[neat lintel]

@.@

[leaden torrent]

this is what, say, a model in physics is

our understanding of physics is necessarily incomplete and imperfect

our understanding of mathematics is pretty good

so we create a "model" of some physical behaviour that lets us use mathematical conclusions to say things about physics

technically we're only saying things about the MODEL

but the idea is that, if our model is good, it should match the physics

(or at least be close enough for our purposes)

[neat lintel]

Just to be clear :

[leaden torrent]

newtonian mechanics is known to be "wrong" in an abstract sense, but for most motion on earth, it's a perfectly suitable model

with like 99.999% precision

[neat lintel]

We need not delve into questions like “is this real or is it a simulation?”

We have a reality in front of us, and we’re using math to understand how it works, is that correct?

[leaden torrent]

the point is that mathematics itself, in the abstract, is about how to work with abstract mathematical objects, properties, and relationships

and other fields take that math and try to apply it to something "real" by constructing models

with mixed success, usually; models in chemistry tend to be a lot better than in, say, psychology

"better" in the sense of more predictive

[neat lintel]

I think I understand.

So some mathematics work well for things that are “real” better than others, while some are completely off the mark?

[leaden torrent]

mathematics in a pure sense is wholly unconcerned with the "real"

naturally a lot of mathematical development is motivated by "hey, investigating this stuff might be useful for physics" or whatever

a lot of fields trace back their birth to statistics, for example

[neat lintel]

So it’s a utility device rather than a philosophical one?

[leaden torrent]

but it is not the job of (pure) mathematics to answer questions about reality

[neat lintel]

Ahhh….So THATS why you guys didn’t like my analogy…..okay, that makes sense now.

[leaden torrent]

it is the job of mathematics to answer questions about mathematics

[wicked ore]

Yep exactly. I’ve played around with (very basic) stuff on my own, but when I finally write up my results, it feels like I’ve grappled in the mud and finally gotten out and discovered a nugget of gold. Whereas with these textbooks, everything is a clean, perfectly shaven cube of gold, and when everything is gold, nothing is gold, so you have no real appreciation for the results

Lol not sure if that made sense

[neat lintel]

Because the analogy admittedly didn’t make sense now that you’ve contextualized math for me.

[leaden torrent]

(applied mathematics is a slightly different beast; it mostly deals with building models and stuff)

(so it is actively taking the math and applying it)

[last oxide]

its about the relationship of abstract objects
you have a concrete problem and you abstract it away by extracting the parts you care about and/or generalizing

for instance, you could try asking if there is a nice algebraic way of representing roots of integer polynomials and the relationships between them and end up with the whole of the complex numbers

[leaden torrent]

(but the point is that it's bridging the gap)

[sleek wing]

See the thing is, when I read a text book I don’t want a struggle, - I want a quick, condensed, exposition about whatever I want to learn about

[last oxide]

(algebraic integers, but you get the point)

[leaden torrent]

the complex numbers are an interesting case since

i am wholly unconvinced they "exist" in any real philosophical sense

(admittedly i think the same thing about the reals, but i'm less sure on that since some things do seem to be real continua)

[last oxide]

but what are you really learning

[leaden torrent]

but they are incredibly incredibly useful in a whole lot of physical models of everything

thanks in large part to fourier bullshit

[last oxide]

what about the algebraic integers then

[leaden torrent]

that regardless of whether they have a platonistic "existence", they are inseparable from our understanding of reality

[wicked ore]

Yeah, I respect that. I guess it just comes down to what you want as a reader

[leaden torrent]

honestly most cases where ℂ comes up in applications feel like contrivances

like "you could've modelled this with just ℝ²"

[sleek wing]

Properties of an object that I will later use in my own work. The motivation is my own, I feel like that’s where it differs for me

[leaden torrent]

but ℂ adds some computational machinery that is incidentally useful for, at least, expressing certain formulas in a convenient way for manipulations

[last oxide]

but wouldnt it be R^2 with multiplication?

then you just get C, right?

[leaden torrent]

well my point is that a lot of applications dont even need to multiply them

[sleek wing]

I don’t care for your motivation as much because I have my own reasons to learn about stuff

[leaden torrent]

indeed, as soon as you embed ℝ² with ℂ's multiplication, you just have ℂ

but a lot of cases where complex numbers come up dont even do that

or dont need to do that, rather

but like, theres no reason NOT to either

[last oxide]

the instances where I feel it makes sense for C to come up are either some kind of completion or bringing in algebraic tools to the plane

[leaden torrent]

and maybe being able to write shit in polar form is incidentally useful

yeah im talking about physical applications

like okay, signal/wave stuff does need ℂ for the fourier-type stuff to work

but basically every other engineering application of ℂ does not need its multiplication

and that's fine

[last oxide]

but the object is a construction somebody else made, right?
I mean, you have to at least believe in the value of that thing to some extent to even want to use it

[sleek wing]

Idk most times I can just kind of look at the definition and go “oh ok yeah that seems like it would be useful”

[last oxide]

what are some good ones?
I actually cant think of many

[sleek wing]

I accept that my way of learning is quite non-standard though

[last oxide]

I dont think it is

otherwise mathbooks wouldnt be like this anymore

[leaden torrent]

rotations in computing are an obvious example

[sleek wing]

Hmmmm

[leaden torrent]

yeah rotating is multiplying blahblah but

youre converting from polar form to a+bi form to do that

usually

at which point youre just... applying a rotation matrix with extra steps

[wicked ore]

Well my example is prob the determinant, it’s very easy to see why it would be useful, but I really have no idea why someone came up with the idea in the first place, what problem it was assisting, etc. I feel like I would gain a much deeper understanding if I knew that. But who knows

[leaden torrent]

why go through the ℂ formalism at all?

[last oxide]

I mean
in some sense thats the same thing, right?

[leaden torrent]

yeah, my point is that viewing this as ℂ is completely unnecessary

[last oxide]

its not like youre bringing the baggage of the lebesgue measure or some extra topology to do that

[leaden torrent]

its ℝ² and you apply some rotation matrices

these rotation matrices ARE complex numbers, in a sense

ℂ is isomorphic to [a b; -b a] or some shit

[last oxide]

sure but
is there really much distinction
I think the actual extra baggage only comes up if you bring it in

[leaden torrent]

okay, here's another common example

in electromagnetism you'll very commonly seen the magnetic and electrical parts of a system represented as complex numbers

but multiplying them makes absolutely no sense

these are 2 disjoint quantities

[last oxide]

hell
most people thing of C as just a+bi and nothing else
a real part and an imaginary part

[leaden torrent]

a norm on them doesnt even make sense

even calling it ℝ² is being generous

since theyre 2 very different parameters with no real relation

yet its written in a+bi form for convenience

[last oxide]

okay
this one is fair

no multiplication really doesnt make sense

it would be like counting balls using rationals or some meme like that

[sick kite]

hence proving that applied mathematics makes no sense qed

[leaden torrent]

sim. https://physics.stackexchange.com/questions/566605/electromagnetic-waves-complex-numbers

[wicked ore]

Ford circles?

[leaden torrent]

even in the case of fourier analysis, i am not sure that any of the transformations we do actually represent any "real" thing

they are very very useful for computing stuff obviously

breaking up shit into parts is useful

but it's also a pretty heavy layer of abstraction

i would say the complex numbers in this stuff only arise to make the computations work

rather than somehow being fundamentally inherent to how waves work

admittedly thats a more nebulous idea

[last oxide]

is it really?

I mean
they are a big set

[leaden torrent]

i mean, even being able to "map out a waveform" is a pretty heavily abstracted idea lmao

[last oxide]

but I think they can be described quite simply

[leaden torrent]

waves are weird

[sleek wing]

For you

[leaden torrent]

like, as they actually manifest in nature

if you follow one "particle" or "point" in a wave, its behaviours are essentially unpredictable without knowing the entire wave

"waves" are an abstraction of a lot of minute mechanical interactions

(at least things like sound waves and fluid waves)

now it's a sensible enough abstraction that i wouldnt say waves "dont exist"

or anything like that

but when you now say "we can also break up this wave into a bunch of nicer waves that, when we add them together, produce this resultant wave"

[last oxide]

I think of them more like these little numbers that are roots of polynomials tbf

[leaden torrent]

like... what do those nicer waves actually represent physically?

they dont really represent anything physically, at least afaik

maybe im missing my intuition

and this should be intuitive if you've, say, ever heard different timbres before

[last oxide]

you mean the fourier thingies?

[leaden torrent]

yes

[sleek wing]

Oh nice we’re talking about music again

[last oxide]

couldnt that be thought of as like waves of a certain frequency verbatim?

[leaden torrent]

my point is that complex numbers only really enter the picture when you apply this weird, un-physically-motivated process to something thats already kind of an abstraction

and they only arise for computational purposes

its not even as direct a connection as, say, finding roots of a polynomial

i am unconvinced that they are somehow "real"

[last oxide]

well, it should be, right?

theyre gonna show up because of a polynomial somewhere

[sick kite]

nami do you believe that any number is "real"?

[leaden torrent]

i understand why complex numbers come up in fourier stuff

i know basic math

[last oxide]

[leaden torrent]

my point is that we're on like

the 3rd or 4th layer of abstraction

by the time we get to that point

motion of individual things in a system → waves → breaking up waves into simpler waveforms

perhaps im just stupid/my physical intuition is shot

but like

i cant point to anything physically where its like

"yeah it's obvious why complex numbers are necessary here"

it's clear why they're a useful tool to have for this stuff

[last oxide]

are we?
I just dont see essential abstraction being added on
there is no necessity for taking much extra care
I think of it like
you now have a+bi and i^2=-1

[leaden torrent]

the abstraction is VERY handy

[sleek wing]

I just naturally associated any oscillation with complex numbers ngl

[last oxide]

in the same way we dont really construct real numbers
we just pretend we do and we just get the algebraics and some meme transcendentals

[leaden torrent]

fractal my point is that, yet again, youre talking about complex numbers by saying mathematical constructs they must satisfy

rather than physical ones

i cant qualitatively explain why complex numbers arise in these physical systems

i can explain it if you let me explain the mathematical reasoning

but i cant point to something that gives birth to complex numbers directly

[last oxide]

at some point, its hard to see the distinction for me
would you consider sqrt 2 a physicsl thing that shows up for instance?