serious-discussion

[static loom]

if every rub is painful you'll never be polished

[neat lintel]

then why is math so hard?

my experience with group theory has been a torture so far

[charred mortar]

new concepts are often difficult

[neat lintel]

yea

difficult af

[charred mortar]

for me I just kind of sat on them for a while and it magically made sense eventually

do not ask me how my brain did the work

[neat lintel]

hope this will be the case for me tbh

also hope it won't take long

[charred mortar]

I think it's pretty common

for me it usually happens about a week before finals, so I'm always on time lol

[neat lintel]

yea, it happened to me with calculus

[charred mortar]

like throughout the whole semester I'm just like "wtf is this shit, none of this gibberish makes sense"

and then in the final week it's "ooooohhhh that's the thingy"

[neat lintel]

I mean

groups do make sense

but it's just difficult

it's easy to solve some group theory problems with algebraic play without understanding the intuition

but I'm no man of blind rigor

[charred mortar]

yeah that's just unfortunately a thing early on

getting a good intuition for this stuff can be a lot harder than the algebraic manipulations you'll need to do in your course

I had a similar experience in analysis

[neat lintel]

my problem with analysis was the rigor part

[charred mortar]

I could manipulate epsilon-delta proofs in my sleep, but the intuition for everything only came after a while

[neat lintel]

can relate

[cobalt star]

what undergraduate level math topics are there that are more visual

ive been trying to learn real analysis but its like

Very symbol dense

i mean i'll eventually learn it ofc

[leaden torrent]

graph theory?

[cobalt star]

ok

[halcyon mist]

How long did it take to get that level of skill with epsilon-delta proofs? just had it thrown on us 1st week of calc and I'm struggling as of yet to figure out what we're doing with them

[charred mortar]

for me I think it was about 2 months

and that was mostly with active practice, i.e. doing as many of those sort of problems/going through proofs as I could

[neat lintel]

one variable real analysis is visual af actually

[cobalt star]

really

[halcyon mist]

true - I'm guessing exploring the other facets of limits helped a lot? was it totally mind boggling at first too? I think after a few days it's gotten a bit clearer

Ahh definitely - gonna try and do a few every day

[cobalt star]

random question how come the way you construct real numbers is so different to the way you construct Q and Z

like with the dedekind cuts

[charred mortar]

well going from Q to R is a pretty different process compared to going from Z to Q

[devout nacelle]

You're moving to a construction that builds on limits of sequences of rationals

[cobalt star]

just feels a bit weird to me

[neat lintel]

you poor poor thing

[cobalt star]

Like the continuum of the reals is a pretty intuitive concept

[neat lintel]

is it

[cobalt star]

isnt t

like a continuum is pretty intuitive to me

[neat lintel]

i think the usual ways to construct reals is pretty intuitive too

you just want sequences that get small to converge

can u learn ring theory without learning group theory

no

ima do fields -> rings -> groups

why

that's how it works for me

I didn't do integrals till it got to triples

then it started to interest me

you are a Benjamin Button

lmao

field theory uses group theory

I hope very little

[tall badge]

here we're using dedekind cuts not sequences in Q

[neat lintel]

dedeking cuts are kinda same idea

I'll do groups -> rings -> fields but not so thoroughly, then I'll come back

just think of it as sequences converging from below

[neat lintel]

do it

[cobalt star]

right

[neat lintel]

you are amazing, Amber Li

[tall badge]

if u think about how Z & Q are constructed by taking equivalence classes under properties we want members of those sets to have, the construction of R actually follows quite a similar pattern

[jovial ember]

Yes

I think a lot of departments are switching to a rings first approach, because students tend to be more familiar with rings

[neat lintel]

wow

[charred mortar]

Yeah I learned rings first too

It kinda makes sense, since Z is a ring that everyone is familiar with, and so is Q as an example of a field

[neat lintel]

and GL(M) where M is a module are examples of rings

[charred mortar]

That too yeah

[neat lintel]

theres a motivation thats lost tho cuz

a great reason to study rings in general

is that Aut(G) for abelian group G is a ring

[fluid rapids]

what

i mean, some helpers are offline

when you delete a ping it gets removed from the notif centre

so that’s nice

[uncut socket]

Did you get the ping but was confused because it got deleted?

But wait that's a thing?

[fluid rapids]

yea

[uncut socket]

Thought it still ghost pings

[fluid rapids]

doesn’t

lol

[compact tartan]

it does

[dawn bridge]

i need ideas
what topic good

[signal kestrel]

So I'm in a major that basically needs every math class In college

But I hate math

Should I just swap majors

Not sure if I wanna do this shit the rest of my life

[leaden skiff]

I mean, if you hate math you probably should go into a math hevay field

[rancid meadow]

is it CS or some engineering discipline

this is like

hilariously common among CS majors

[signal kestrel]

Oh u know it's CS

This major is just kinda balls honestly

[rancid meadow]

yes this is tremendously common

[rose dock]

Why'd you pick it then

[rancid meadow]

shut up lmao

[rose dock]

If he has a goal then that might motivate him to get through the math

[rancid meadow]

honestly the math in cs sort of falls off after ur second year depending on your specific program

like the skills are still needed but

you srota do calc 1 - 3 and then LA 1 & 2

and then ur DSA and ToC course

and after that its free for all usually

[signal kestrel]

I'm barely into precalc 2

Doesn't help that my prof is a doofus and ive had him for the past 2 sems

[rancid meadow]

and honestly if you just want a nice job you can sorta scrape by in these courses

oh hm

precalc?

thats odd

[tiny marten]

the math takes time but it's worth it

[rancid meadow]

honestly it's not

for CS majors

lol

I've done the CS math track

[tiny marten]

okay if you just want the bachelor's and a programming job then yeah

[rancid meadow]

yes

I can't imagine someone who "hates math" is trying to get a phd

[tiny marten]

then it def isnt worth it

yeah

you would prolly be better off trying to do coding camps and stuff

for 3-4 years

[rancid meadow]

my understanding of most cs programs is that they expect you to have done precalc/calc in hs

yeah you can always do bootcamps

[tiny marten]

kinda yeah, i am a nontraditional student and got calc 1 done at a community college

then took on pure math at a uni

[rancid meadow]

i'm personally turned off by bootcamps but I tihnk they are fitting for certain ppl

[tiny marten]

it would have been trickier without having had calc 1

well not really, but if i was cs it woulda been since you petition to get into the program

yeah i would rather just do it myself

[rancid meadow]

anyways @signal kestrel I guess alphyte asked a fair question

if you just want a nice cushy 80k/yr programming job

you can sorta just

deal with it

as long as you are passing it's whatever

[signal kestrel]

I probably won't pass this sem

[rancid meadow]

oh well

then you might want to reconsider

[signal kestrel]

And its all due to the teacher, the guy is a complete joke

He doesn't teach

[rancid meadow]

¯_(ツ)_/¯

happens

[signal kestrel]

He has a socially awkward problem

[rancid meadow]

part of uni is dealing with shitty profs like this honestly

and in CS it is especially important to learn to self teach

[rose dock]

Did he assign a textbook to read?

[signal kestrel]

I guess, math would be a lot more bearable if I had anyone with half a brain

[rancid meadow]

it sucks but you can't just sit around and stew about it

[signal kestrel]

Looking it up all on my own is N O T how I learn

[rancid meadow]

not that im accusing you of doing this lol

[signal kestrel]

Never has been

[rancid meadow]

well, CS might not be for you

[signal kestrel]

With me its face to face or fuck all

[rancid meadow]

the fact of the matter is that every CS major I know has self taught most of their skills

like "job skills", I guess

like I'm doing internships and stuff as a pure math major, I learned everything myself

so idk if this is like a general issue for you I would actually reconsider the majro

[signal kestrel]

Its just something I've always had, I need someone to talk with shit about and learn from that. I'm not googling shit to learn

[rancid meadow]

yeah I get that

I'm just saying that if you want a job in software specifically, the biggest skill you can have is googling & self-teaching

[signal kestrel]

Which i just don't have

[rancid meadow]

yeah

[signal kestrel]

Just wasent born that way

[rancid meadow]

this is something you can work on of course, a pre-calc course is a good time to try to learn

there a bajillion youtube videos out there

[dense belfry]

Do you have friends to do work with lije?

[signal kestrel]

No

I dont have friends in general

[dense belfry]

Maybe you should try finding some study buddies

[rancid meadow]

well, you have that part of being a CS major down

[dense belfry]

[rancid meadow]

yeah anyways uh

[neat lintel]

yubel!

[rancid meadow]

If you want to do CS you need to work on this skill, absolutely

you really do not have a choice, especially once you're actually employed

I don't think you are incapable of learning it

[leaden skiff]

what skill

[rancid meadow]

but if you think you are then you should consider a new major

googling/self teaching

it is weird to call googling a skill but if you've talked to software engineers they will often say it is one of the best skills to develop

knowing what to click on

[leaden skiff]

googling and self teaching is definitely a learned skill

no one is innately born being good at googling

[stable ginkgo]

https://youtu.be/cEBkvm0-rg0

[rose dock]

I'd say being able to read documentation is a valid skill

[rancid meadow]

Yes

[rose dock]

But honestly as long as it's not like geeksforgeeks or quora and it's the first link I click on it

[rancid meadow]

it is

[leaden skiff]

I mean, what's there to know about reading docs

[rancid meadow]

eh

[leaden skiff]

you just look shit up as you need it

[rancid meadow]

yeah that works up to a point

[leaden skiff]

ctrl f some key words

[rancid meadow]

but sometimes it's like... "what do I type in"

or wahtever

[signal kestrel]

Right now math just makes for a very depressing and upsetting day every day I go to class

I dont want to have to go to tutoring every single day, would make me feel like a dumbass

Sure if u work harder now you might get a way easier time in life after college, but I dont even think I can mentally do it

[leaden skiff]

I don't really understand how someone can enjoy programming but not like math

like they aren't the same

but to me they like, use the same parts of my brain

[rancid meadow]

they use the same parts of your brain but they are different to most people insofar as math feels sort of pointless

but programming always has a goal and a visible outcome

i mean maybe besides leetcoding

but even then it feels a bit more tangible

[surreal sapphire]

also many math majors hate programming

[leaden skiff]

what

I cannot understand that

[surreal sapphire]

which is weird to me since i like to use sage a lot

[sick burrow]

Programming feels more chill than math imo

[surreal sapphire]

but it is definitely the case

[leaden skiff]

I cannot fathom how you can like one but not the other

maybe stuff like Web dev

sure

[surreal sapphire]

i said math majors tbf, not "people who like math"

i dont know how big the intersection is

[leaden skiff]

OK sure

I guess math majors study very abstract math

I doubt category theory comes up in programming a lot

[sick burrow]

Because with math you're basically always doing something hard and if you're not then you should be doing harder math

[rancid meadow]

yeah

[surreal sapphire]

also this is anecdotal but i have been teaching C to math majors for a few years

[rancid meadow]

I agree

[leaden skiff]

programming can be very tedious yeah

it'd not often like hard to understand what you're doing really not like math

[sick burrow]

But with programming there's a lot of chill downtime where you're just "running through the motions" of things you already know how to do

[leaden skiff]

that's why they hate programming

[sick burrow]

And it also feels like programming is more horizontal than math if that makes sense?

Like the things you need to know have less prerequisites

Or maybe that's wrong

[leaden skiff]

yeah

very often you don't need to understand how stuff works

only how to use it

[sick burrow]

I've always enjoyed programming and been good at it, but I have no real desire to do it all that much

I like cs much more than I like programming

[leaden skiff]

I mean, I like game dev

and also making simulations

idk if pure programming appeals to me

haven't really tried it

[surreal sapphire]

because they are taught by me?

[leaden skiff]

no, you're teaching them C

[sick burrow]

My issue with programming is the working with actual computers part

Because if you want to program anything significant then a lot of the work is just familiarizing yourself with the relevant language and packages and reading documentation

[signal kestrel]

Math is just depressing for me

[sleek wing]

same

[leaden skiff]

just learn python 4head

[surreal sapphire]

i would teach them sage, but the programming class has no math prereqs

also C is a fine language to learn especially for a short class

[sick burrow]

Python

Python still has packages and shit though

[neat lintel]

Lige ma balls

[sick burrow]

Idk I'd just rather spend my time putting pieces together than learning what those pieces are in the first place

[ancient flame]

in what course does someone learn about the generalized stoke's theorem

[dawn bridge]

as long as they make it clear what the syllabus is u cant blame them

ur expexted to learn shit urself at university

tuition is a scam

[dawn bridge]

90% of software engineering is webdev

[leaden skiff]

yeah but saying you hate all programming feels like a stretch

there's lots of programming

[dawn bridge]

it depends on what kind of programming theyve been doing i suppose

if u throw project euler at math people i think they would generally enjoy it

[tiny marten]

Python is amazing I love it

Sage is a lot of fun

Symbolic computation platforms are always fun imo

[dire mulch]

Same, it's a very nice interpreted programming language.

I will always chose it over the rest.

[modest rune]

analysis

(one also learns the True statement of stoke's theorem in difftop)

de Rham's theorem bravely asks the question, what if Stoke's theorem was a lemma for a better theorem about topology

[ancient flame]

what type of analysis

[rose dock]

Analysis on manifolds

[modest rune]

Real analysis

you don't need manifolds for stokes

[bronze pelican]

multivariable analysis

[ancient flame]

damn

[ancient flame]

graduate level?

[odd narwhal]

I mean you do for actual stokes' theorem

And not """""stokes' theorem"""""

[deep mango]

[tiny marten]

black box that sucker and gimme the scare quotes one 😎

why know more math when little math do good

anyone can prove stuff with topology but proving stuff with calc 3, that's a challenge

i reject mathematical modernity

[cinder flume]

proving stuff with calc 3

[cobalt cliff]

why is the real projective plane called projective

[deep mango]

projective = quotient by scalar multiplication

well maybe there's a more etymological reason than that

[cobalt cliff]

I thought it might have something to do with projective geometry

[deep mango]

yes, definitely

i think it's not terribly inaccurate to say that projective geometry is basically euclidean geometry quotient scaling

but there's probably more to it than that

[modest rune]

not necessarily

we did it in my undergrad class

(i certainly have not taken grad level analysis and never plan to)

[ancient flame]

oh damn interesting

I didn't know it went that deep

[tiny marten]

it does it's just that what ryc means is a projective thing

it's sorta in the background if your idea of projective is "forms a loop when the base space doesnt otherwise"

like i am doing a bunch of stuff on this space and its projective but it just looks like a simplex

but in the background is that quotient

compactifying or turning something into a loop or whatever is like the coarse overview

the coordinates in projective stuff reflects this

[cobalt cliff]

I should ask what is projective geometry, I was thinking of it as projecting through some surface or curve

@tiny marten

[tiny marten]

oh okay, most people learn a sort of basic model of a projective dealy

for the reals what you do is identify positive and negative infinity

so that things "wrap around"

[cobalt cliff]

ok

[tiny marten]

and then the projective real plane is a sphere

alternatively, the projective complex plane

what you gotta do for this is some topological stuff in the background that does the identifying

and you get this new value to track, whether a point is or is not at infinity

which requires a specific coordinate system

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

it's really not that crazy in simple examples

it's kind of a bookkeeping thing in practical terms

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

the riemann sphere is the "projective complex line"

you kinda think of it as one complex dimension sometimes

on the riemann sphere a line is really just a specific circle, since everything comes back around

[bronze pelican]

Riemann sphere is a 🏀

[open aspen]

We didnt bother doing it in grad real analysis. we are doing it in grad complex analysis though

[modest rune]

what

thats wild

[open aspen]

generally speaking we didnt really care about riemann integration in real

only insofar as it related to lebesgue integration or riemann-stieltjes

[tiny marten]

yeah in my complex analysis class we did all riemann stuff

[open aspen]

we used theorems and stuff about R-Integration, such as for L^p spaces, but didn't really prove anything about R-integration itself

[tiny marten]

and then in real was measures and some functional analysis

due to specific prof (i liked it)

[open aspen]

they had you doing complex then real?

[tiny marten]

i did that

[open aspen]

thats wild

[tiny marten]

i am an undergrad so i just take what i want

i already took the undergrad versions too

rn i am just in a topics course for that level content

[cobalt cliff]

how do we relate the riemann sphere with homogeneous coordinates

I am looking at the wiki

[tiny marten]

well luckily when you actually do complex analysis you dont really have to because it is given to you in the form of a theory of mobius transformations on the sphere

however with the riemann sphere in the two variables we have points (x,y) in C but then you modify it just like you had R^2

i think, i have never personally used projective coordinates on complex projective stuff

yeah so it seems that it just applies, at least that is what the article says. should be the same procedure

https://en.wikipedia.org/wiki/Riemann_sphere#As_the_complex_projective_line

you can do everything you need to do using automorphisms of the sphere directly, which is what i know i dont do AG or anything

[cobalt cliff]

so in C^2 the set of all one dim subspace is the Riemann sphere

[tiny marten]

set of equivalence classes of relation by there exist so and so constant

the points related by that constant existing so like (x,y)=(az,aw) are identified into individual points

that part is less super important unless you are doing algebra on it or something

or trying to mess with a metric i guess

this is technical background that is a bit tricky

when you do this relation it changes what a line is relative to the new space's curvature

[cobalt cliff]

so we can get projective n-plane by treating the n-plane as a hyperplane in (n+1) space and doing stereographic projection from the point perp to the hyperplane (n) sphere

[tiny marten]

oh okay yeah something like that

so we have these nice models of the projective spheres because 3 dimensions is fine and now we can look down at a special model of 2 dimensions

it's a good way to look at it, living in a space 1 dimension higher

projective geometry is a bit easier to get a grasp on than hyperbolic, which has a bunch of models

but it has similarish principles including "points at infinity"

[cobalt cliff]

I see the picture now, associate each point (x_1,.....,x_n) in n-space to (1,x_1,.....,x_n) i.e. translate it upward by 1 , look at the n-sphere centered at (1,0,......,0), then look at a one dim subspaces of (n+1)-space the each 1-dim subspace associate a point on the plane to a point an the sphere with (0,0,........,0) on the sphere corresponding the the point at infinity.

but now how do we get the topology of a projective plane

[deep mango]

it's a quotient topology

[tiny marten]

yeah that sounds right, the topology is a specific quotient as ryc says

which is tricky if you havent seen it yourself

well, not that tricky but it's simpler to see it

[deep mango]

(the quotient from R^(n+1) under the equivalence relation which identifies all elements in one-dim subspaces)

[cobalt cliff]

I mean RP^2 should be the 2 sphere connected sum with a mobius band

[deep mango]

sure

[cobalt cliff]

I was thinking how can we see that from what info we have so far

[deep mango]

it should be tied together by observing that both of these are equivalent to taking the 2-sphere and identifying antipodal points

1. from R^3 quotient by lines, you get this by first quotienting rays down to the sphere and then identifying opposite rays

2. for sphere connect sum mobius band, you get this by cutting S^2 into two domes and a ring, identifying the two domes (on antipodal points), and then identifying the ring on antipodal points. then you get a mobius band and a sphere with a hole cut out, which you proceed to reglue.

[tiny marten]

huge brain ryc

[cobalt cliff]

in R^2 I am seeing it like this , we could see infinity as a circle and when you pass through it your orientation changes

[deep mango]

yes

[tiny marten]

beautiful reasoning, 11/10

[cobalt cliff]

so we have a mubius band at/beyond infinity

at least as far as I am concerned

[neat lintel]

Anyone wanna help me with this assignment?
https://i.imgur.com/u4DSnXc.png
I will be providing you with a set of questions to answer.

[twilit harbor]

out of curiosity what is the smallest value you guys can find with this equation

only using digits 1-9?

into the negatives

[cold needle]

banned

[mint patio]

what happened metal

:o

[cold needle]

pron

[jade crane]

SEND

[cold needle]

no it was

not like

not like that

[jade crane]

[jovial ember]

So you’re keeping it for yourself?

Wtf bro

Sharing is caring

Wtf

Mods always keep all the free nitro to themselves

[ancient flame]

yeah send me some

[jovial ember]

All the pron to themselves

WTF

[cold needle]

bru i cant

[jovial ember]

Why

[charred mortar]

Oi my free nitro

[open aspen]

you're already breaking the rules (arguably) by allowing this conversation to take place in #serious-discussion insetad of #chill, whats one more rule broken 😈

[long matrix]

sin + sin = sin + epsilon

[wise vigil]

Epsilon is a small value right? I only know it in the context of approximations

[long matrix]

Usually.

[open aspen]

its used frequently in analysis, for example to show that a function is continuous

or a limit exists, etc

[long matrix]

Well you could always start a proof with 'let epsilon be large...' but then you wouldn't be taken seriously

[open aspen]

usually looks like let $\varepsilon > 0$ ... since epsilon was arbitrarily small [conclusion]

[fathom swallow]

Migillope

[wise vigil]

I see ok. One would just place it as a variable to signify the infinitesimally small value.

Is that right?

[open aspen]

"let epsilon be very large ... then 1/epsilon ..."

[open aspen]

sorta. infinitesimals are a little tricky but yeah that's the idea

[leaden skiff]

what's the symbol for a large number

[wise vigil]

I first heard of this when I went to a zoom lecture on Diophantine approximation. Most of it went over my head but we were doing continued fractions so I think they somehow connected to that

[wise vigil]

inf lol

[leaden skiff]

if epsilon is for small nums

[long matrix]

1/epsilon

[wise vigil]

oh

[open aspen]

dont get people started on "is infinity a number"

[long matrix]

we observe this function as it approaches 1/epsilon... jk

[open aspen]

its a recipe for disaster

[wise vigil]

lol

[long matrix]

but infinity is a number

u just add it to your set, ez

R u {inf}

[wise vigil]

My teacher went off on this kid the other day when he was trying to prove that it's a number

inf + 1

[leaden skiff]

it is sometimes

[ancient flame]

beuh

wartime

[leaden skiff]

projective spaces

[shy snow]

i fail math

[open aspen]

R u {apple} makes apple a number, welcome to the numberable fruits club

[leaden skiff]

{apple} is a number, its 1

fun fact

[open aspen]

I asked my professors this at the start of my first semester, and they all gave different answers

[leaden skiff]

{apple}! = {banana}

[wise vigil]

I mean how can you even treat it as a single number? In what case can it be used as a number and not an idea?

[leaden skiff]

it depends on what you mean by infinity, and also like what field you're studying

[open aspen]

whats your level of maths?

[flat harbor]

excuse moi my level of maths is "exquisite"

[leaden skiff]

in projective geometry all points at infinity are considered the same point. so like, parallel lines will intersect at infinity

[wise vigil]

I am in calc 1 rn but I jump everywhere. Don't hold back lol

[flat harbor]

single point compactification and its consequences to humanity

[rose dock]

I love compactifications

[open aspen]

the one point compactification of R^2 with the typical topology usually makes infinity considered a "number"

[charred mortar]

Isn’t the notion of a “number” not particularly well-defined anyways

[open aspen]

but here is a stack exchange question about it https://math.stackexchange.com/questions/36289/is-infinity-a-number

the accepted answer is very good

[rose dock]

I'm surprised that the question was not deleted upon post

[leaden skiff]

I would say a number is an element of a ring

[open aspen]

and mentions this, among other things

[flat harbor]

are real number even real...? there is no infinite... so how can there be.. infinite decimal...? pi is not real.. or sqrt2.. or 1/3...even......

[long matrix]

yes, the first line is all thats needed to be said of that answer

[leaden skiff]

are there any number like objects that aren't rings, or elements of rings that aren't really numbers

[open aspen]

also like... cardinal numbers are a thing. its a complicated question mainly because it isn't well posed

people usually mean "is infinity a real number" when they ask that, in which case the answer is no

[wise vigil]

Yikes. I have so much latin reading to do. Gl on the debate

[leaden skiff]

it's a fake number

[charred mortar]

Scam

[open aspen]

real by not being complex

[flat harbor]

behold, the humble none one many counting system

[leaden skiff]

idk what that means

[flat harbor]

there is none, there is one, or there is many

[leaden skiff]

is that a ring?

[flat harbor]

but to be srs identifying numbers with rings lets in a few non numberish cases

[leaden skiff]

like, 0,1,2

[flat harbor]

integral domains are way better

[open aspen]

you can arbitarily create a ring with things that are not numbers. ring of fruits

[leaden skiff]

well, yeah but like they behave as numbers

[open aspen]

I = {tomato, apple, orange}. total ordering defined by the order i put it in, let addition be cyclic and define multiplication explicitly

[flat harbor]

if u can embed an operation on the fruits

[open aspen]

ring isomorphic to Z/3Z

[flat harbor]

then the fruits represent a concept that can be considered as number

[ancient flame]

does R^n make sense for non-integer values of n?

[leaden skiff]

yeah, so any ring is isomorphic to a "number" system

[open aspen]

yep, this is precisely what I did

[leaden skiff]

that's what I'm saying

all rings are numbers up to isomorphism

[flat harbor]

then the fruits are number

as would a filthy math major would say, FRUITS ARE NOT NUMBER

STOP ABSTRACTING

[leaden skiff]

afaik, not at all

[ancient flame]

rip

[open aspen]

you mean all rings are isomorphic to a ring which is defined on "numbers" as common sense would define?

[leaden skiff]

also it doesn't even make sense for integral values

what is R^-1

[long matrix]

There are definitions which extend.

[leaden skiff]

there are???

[long matrix]

ofc there are

[ancient flame]

yeah I wanted to say non-natural number but that sounds weird

[long matrix]

why would there be not.

[ancient flame]

there are always extensions lol

R^0.5

[leaden skiff]

what's $\sqrt[]{\bR}$

[fathom swallow]

woog 2: electric woogaloo

[ancient flame]

lmao

[flat harbor]

a space upon an operation can be defined wrt a vector in R that returns the trivial set

[open aspen]

well thats easy to answer

[long matrix]

idk, look it up

[open aspen]

thats just R

[ancient flame]

why

[leaden skiff]

all rings behave in the way we would expect numbers to

that's what I'm saying

[flat harbor]

no they dont

numbers commute

most rings dont

[leaden skiff]

hmmm

ehhh I guess

but no

[open aspen]

we frequently use things like this to represent a set. For example, 2Z+1 are even numbers

[leaden skiff]

quaternions and octonions are numbers

they don't commute

[flat harbor]

debatable

[ancient flame]

2Z+1 is the set of evens?

that's interesting

[open aspen]

uhhhhh

in many ways

[ancient flame]

why sully

[leaden skiff]

wait sorry let me sully the source

[ancient flame]

bruh lol

[open aspen]

in many ways, even numbers are actually odd numbers

you get the idea though lol

[ancient flame]

what

the fuck

[leaden skiff]

even numbers and odd numbers are the same up to isomorphism or something idk

[ancient flame]

odd numbers cant form a group

no identity

[open aspen]

yeah cuz 1 is even

[ancient flame]

under addition

[open aspen]

better haha

[ancient flame]

mig be trollin

[leaden skiff]

I'm confused now

[ancient flame]

oh

[leaden skiff]

my brain hurts

[ancient flame]

me too

[leaden skiff]

anyway what is R^n where n is non natural

[open aspen]

Sorry, too sarcastic for text. I meant:

we frequently use things like this to represent a set. For example, 2Z+1 are even odd numbers

It was a typo

[ancient flame]

unnatural dimensions

lmao

[leaden skiff]

what if you had a polynomial of sets

[ancient flame]

but yeah I didn't know that was possible to do

[leaden skiff]

$\bZ^2 - 2 \bZ + 3$

[fathom swallow]

woog 2: electric woogaloo

[ancient flame]

oh actually yes I did

[open aspen]

here is a discussion i found on the internet

https://www.physicsforums.com/threads/non-integer-exponents-for-cartesian-products.570847/

[ancient flame]

cosets right

maybe not

[open aspen]

cosets are often written in this way yeah

[ancient flame]

alright im not a complete idiot then

[open aspen]

Like G/H consists of gH

[ancient flame]

yeah

[leaden skiff]

hey guys can you think of any set which when squared gives you R

[open aspen]

sqrt(R)

[ancient flame]

no

oh

[leaden skiff]

well it has to be an uncountable set

[open aspen]

that is uncountable

[flat harbor]

the union of {a} and {ai}

[leaden skiff]

what are the elements of sqrt R

[ancient flame]

isn't there something that an uncountable set must cross with a countable set to form an uncountable set or smth

[open aspen]

square roots of the elements in R

[ancient flame]

I forget what it is exactly

[flat harbor]

which itself isnt closed under addition

and needs to be embedded in C in the first place

[open aspen]

he didnt say it was a group/ring

just a set

[leaden skiff]

yeah

[flat harbor]

which is less of sqrt R and more of sqrt {|z|}

[ancient flame]

I can't believe im talking about square roots of sets at 1 am

goodnight y'all

[flat harbor]

like thats not even sqrt R thats the image of R under sqrt

[leaden skiff]

wait, OK just find some bijective map from R to R^2 that's like, symmetric, and then turn the axis into your sqrt

[flat harbor]

thats called a space filling curve

and its not injective

so u dont have an inverse function

[open aspen]

i dont understand the problem with sqrt(R)

[leaden skiff]

wait what no

[open aspen]

did we want something that has a group structure or something

[leaden skiff]

R and R^2 have the same cardinality

[open aspen]

and a bijective correspondance?

[leaden skiff]

I wanna know how to construct it

[open aspen]

wdym construct it?

[leaden skiff]

so there must be a bijective mapping between the 2

[leaden skiff]

well like, I want to find some set that when squared gives R

either by listing the elements

like in some formula

[flat harbor]

what does squared mean in this sense

[leaden skiff]

or just finding some way to find it

[flat harbor]

is it the multiplication operation in R

[leaden skiff]

cartesian product

[open aspen]

${\sqrt{x} : x\in \mathbb{R}}$

[fathom swallow]

Migillope

[leaden skiff]

wait I mean the cartesian product

[charred mortar]

So R as the cartesian product of a set with itself?

[open aspen]

.-.

[leaden skiff]

yes

[open aspen]

This was not clear LOL

[leaden skiff]

well I did say set not field lol

[charred mortar]

Is that possible?

[leaden skiff]

but yeah alright

I guess it wasn't that clear

[charred mortar]

I’m not sufficiently well-versed in set theory

[open aspen]

that is a set

not a field

[leaden skiff]

oh yeah

anyway

[open aspen]

but you mean does there exist a set s/t taking the cartesian product yields... what?

identically R or something set isomorphic to R?

[charred mortar]

I guess you have the rationals as a sort of cartesian product of the integers with itself

[flat harbor]

so ure trying to find a subset of R that when mapped to R^2 by t->(t,t), its image of a certain map from R^2 to R is surjective

[leaden skiff]

yields R

[open aspen]

yeilds R exactly?

[leaden skiff]

can you find some map from R to R^2 which has the 2 axes equal to each other. and then set one of the axes to be the sqrt

[leaden skiff]

wdym isomorphic to R?

[charred mortar]

Like does a bijection suffice, or do you want something that also preserves the other properties of R

[open aspen]

then no, because cartesian product gives you elements which are 2-tuples and elements of R are real numbers

[open aspen]

i.e. has a bijective correspondence

[leaden skiff]

ah OK well then I mean isomorphic

[open aspen]

Well there is a bijection from RxR to R

[charred mortar]

For a bijection alone yeah

[open aspen]

so R itself works

[leaden skiff]

so wait

[charred mortar]

Feel like it’s not the case for like a field isomorphism

[open aspen]

yes, this is why I said set isomorphic

it does not preserve group or ring structure

[leaden skiff]

$\sqrt[]{\bR^n} = \bR^m$

[fathom swallow]

woog 2: electric woogaloo

[open aspen]

what does sqrt(R^n) mean

[leaden skiff]

or we'll, isomorphic not equals

[open aspen]

set isomorphic?

[leaden skiff]

I guess

[open aspen]

isomorphic means many, many different things depending on what noun you put in front of it

[charred mortar]

Yeah

[leaden skiff]

hmm, so like, there's no way to extend cartesian products without also making S^2 and S the same

[open aspen]

so its important to specify

"the same"? as in set isomorphic? 😅

[leaden skiff]

is there any meaningful way we can define the cartesian square root of a set

that's all I wanna know

[charred mortar]

Pretty sure that AxA had the same cardinality as A for any infinite set?

[leaden skiff]

any version of isomorphic is fine

I'm not biased

[charred mortar]

So for set isos specifically I don’t think it’s very well-defined

[open aspen]

If I saw $\sqrt{\mathbb{R}}$ I would, after some confusion about why the author chose horrible notation, think $\sqrt{\mathbb{R}} = {\sqrt{x} : x\in\mathbb{R}}$

[fathom swallow]

Migillope

[open aspen]

which, is just R thats cap it very much is not

[leaden skiff]

isn't this just R^+ Union iR^+

[charred mortar]

Maybe you could define an equivalence class

[leaden torrent]

$\mathbb{Q}[\sqrt{\bR}]$

[open aspen]

?

[fathom swallow]

Namington

[leaden skiff]

isn't it?

[open aspen]

that's the most cursed thing ive ever seen namington

[leaden skiff]

case for all positive numbers you'll get R+, but for all complex numbers you get iR+. also I mean unions not +

[open aspen]

I dont understand what you said

postive reals plus i times positive reals?

[leaden skiff]

I mean Union

[flat harbor]

the thing is if you try to use bijections to define R^n/m

[open aspen]

yeah

[flat harbor]

youll always end up with a set bijective to R

[open aspen]

that is accurate, then

[flat harbor]

so if youre not interested in any structure within R

i.e. continuity

you wont get a meaningful definition of rational products

[leaden skiff]

OK what about R^n as a vector space

[open aspen]

over what

[leaden skiff]

over R

[open aspen]

then no

[leaden skiff]

what about over something else?

[open aspen]

in this case R^2 is not isomorphic as a vector space to R

yeah over Q

[charred mortar]

I guess in that case you could define it for even powers

[leaden skiff]

isomorphic to what

[open aspen]

R

[leaden skiff]

wait yeah that's the point isn't it

[open aspen]

(as a vector space over R)

[charred mortar]

Since (R^n)^2 is isomorphic to R^2n right

[leaden skiff]

if R^n was isomorphic to R then the \sqrt would be pointless

[open aspen]

what type of isomorphic

and what do you mean pointless

[charred mortar]

I’m assuming vector space isomorphism

[leaden skiff]

yeah

[open aspen]

Pointless?

[charred mortar]

So for R^2k you could define it’s square root as R^k

But I dunno about odd powers

[open aspen]

I'd say taking R over Q and R^2 over Q would have plenty of applications involving using square roots

like idk how youre gonna define unit vector without it

[leaden skiff]

no i mean like

sqrt(R^2) = R

[open aspen]

oh I mean that's nonsense anyway

[leaden skiff]

where R^2 and R are both vector spaces over R

[open aspen]

(in any typical setting)

[leaden skiff]

is there anyway to make the \sqrt not nonsense

[charred mortar]

Yeah you’d need to have a well-defined square root operation over a vector space in some sense

[leaden skiff]

and then use the sqrt we define, on R

[charred mortar]

I.e. sqrt(V): A -> B, where A and B are some collection of vector spaces and V is any vector space in A

[open aspen]

are you familiar with operator theory

[leaden skiff]

not at all

what is it?

[open aspen]

so sqrt() is a well defined operator on R

i.e., take an element of R, you can take the square root, no problem

generalizing this, you can define operators which take vector spaces as arguments instead of real numbers, for example

you'd want it to satisfy some series of conditions, most likely, but ultimately so long as it is well defined, you can make whatever definitions youd like

for example: I define an operator "eggplant" of a vector space which sends a vector space over a field k to R over k

this is a worthless definition since it doesnt really do anything, but you get the idea

[charred mortar]

So yeah for the square root thing, you could perhaps say that you want it to satisfy sqrt(V^2) = V, where the square is the cartesian product

And the equals means vector space isomorphism

[open aspen]

you can very easily just do this with projection

is it what you want? probably not because it doesnt really have anything to do with sqrt in a traditional sense

@leaden skiff is this a satisfying answer?

[leaden skiff]

yeah I guess

jusr reading the Wikipedia article for operator theory

this isnt as "nice" as I would have liked it but it isn't that surprising

[open aspen]

life rarely is

[flat harbor]

even in vector spaces as well

since isos are bijections with extra spice

the fact that trying to define R^n/m as a subset of R thru bijects will always give u R persists

if ure lucky and the bijection between R^n and R^m is an iso, since projection/cartesian producting R between R^n is a homom the subset will have the same structure

but its still just plain old R

a more interesting example would be space filling curves, which are continuous at the price of only being surjective but not injective

[uneven hound]

Any youtube channel like 3b1b but teaches physics?

Lmao not math channel physics channel

[compact tartan]

viascience?

[flat harbor]

🤮

[uneven hound]

Thank

[brave hollow]

ay this is the class I am taking rn ( not the series the subject )

[neat lintel]

Anyone wanna help me with this assignment?
https://i.imgur.com/u4DSnXc.png
I will be providing you with a set of questions to answer.

[sick burrow]

[sick burrow]

so I'm reading this paper right

and it just goes

but I don't see E(x) or S_t(x) defined anywhere earlier

[neat lintel]

simply intuit the definition

[sick burrow]

I want to email the people who told me to read this paper

and be like "yo wtf explain thyself"

and like they did say to ask any questions that come up in the reading

but I also feel like emailing them would be an admission of defeat

[gentle kestrel]

trivial

[wise vigil]

Oh nah. Maybe this is a sign to major in cs 😂

[gentle kestrel]

School going children see this

[wise vigil]

Fr

[alpine kindle]

you can't admit defeat if you have no idea what the problem is

[sleek wing]

E(x) is almost definitely the expected value of x

looks very stats-y

[sick burrow]

See you'd think that

[sleek wing]

https://cdn.discordapp.com/attachments/359052581022203914/940656085893799936/unknown.png

[sick burrow]

But then you release that x is an element of a vector space over a finite field

[sleek wing]

I stand by this claim

post the whole statement

[sick burrow]

How does that make sense in this context

[sleek wing]

it doesn't but that's exactly why I'm standing by it

[sick burrow]

There's a decent amount of context here

Also E is a subset of the vector space

Why sully

That's literally what the paper says earlier

[sleek wing]

cause now it makes even less sense

[sick burrow]

[sleek wing]

unless it's some kinda

coset type beat

[maiden bear]

yeah exactly

what exactly is the paper about tho

[sick burrow]

It's like

Machine learning sorta?

[sleek wing]

I've already opened the image deleting it does NOTHING you fool

[sick burrow]

HA I've reposted the image

[maiden bear]

wait what the fuck LOL

[sleek wing]

this is the biggest fucking cluster fuck I've ever read

[sick burrow]

There's still more context that's missing

From that screenshot

[sleek wing]

but you're right, it is a subset of a finite field vector space thing

[maiden bear]

they even use the cardinality

and then proceed to use E(x)

[sick burrow]

They litetally just

[sleek wing]

ok and then you take some function mapping C_k(t) which maps some vector to all vectors in E^k+1 where the norm of adjacent elements works out nicely ok

[sick burrow]

Pull that notation out of their asses

[sleek wing]

this is pretty hard to parse

they could've just said "C_4(T) for some T = (t, t, ..., t)"

here

but they decide to write out the entire set? why?

[toxic schooner]

😵‍💫

[alpine kindle]

hold on is that a vector space over a vector space

[sleek wing]

yeah dw about it

[maiden bear]

[sick burrow]

Is what a vector space over a vector space?

[sleek wing]

it's more of a vector space constructed from E

I think the scalars are the same (F_q)

[alpine kindle]

oh lol

[sick burrow]

Ok so basically

[sleek wing]

yeah yeah ok ok so the set

is just C_4(T) for any vector T where all components are equal

i.e. T = t(1, 1, ..., 1)

[sick burrow]

The point of this proof is you're trying to find a chain z_1 ... z_5 in your vector space where successive elements are the apart

But z_1, z_4 and z_2, z_5 aren't t apart

You don't care about the rest of the distances

[sleek wing]

yeah cause of the norm condition

[sick burrow]

And the proof appears to be some kind of counting argument

[alpine kindle]

they still haven't explained what E(x) is

[sleek wing]

yeah I'm trying to really understand this set so I can try and intuit what E(x) is

my guess is it's

[sick burrow]

That awful sigma expression seems to be a way to count the elements of the set

[toxic schooner]

Send the paper gamma

[sleek wing]

wait it's not the fucking induced norm on E of x is it

nah

no that doesn't make sense

[sick burrow]

Yeah it doesn't

[sleek wing]

I'm kinda stumpted tbh

email them

[sick burrow]

Yeah I'm just going to email them

One of the people who's having me read this is an author on the paper so he presumably knows what's going on

[sleek wing]

one would hope

[toxic schooner]

[alpine kindle]

[sick burrow]

[alpine kindle]

this is why you put your functions in a legend

before the paper

[sleek wing]

this is why you just fucking define things

[alpine kindle]

yeah

[sleek wing]

also stfu

[alpine kindle]

[sick burrow]

the result is trivial

[sleek wing]

wait a fucking second

[alpine kindle]

actually what the fuck is this paper I'm scared

[sleek wing]

this is a complete mess of a paper

is this actually published? @sick burrow

[sick burrow]

There is more context i didn't send

Which probably helps to an extent

[sick burrow]

It's on arxiv

[dense belfry]

That's not a high barrier to entry

[toxic schooner]

[sleek wing]

I shouldn't be too mean to them tbh

[sick burrow]

It might be published

[sleek wing]

if I was an expert in this field I might know what the notation means

but I'm not

[median zinc]

What is wrong with this? A circle intersecting with itself is not an important case

[sleek wing]

it's more like, wtf does a circle have to do with this

it is a clue

[median zinc]

It looks like random graphs

[sick burrow]

That is explained earlier in the paper

[sleek wing]

a clue as to the nature of E(x)

[sick burrow]

This whole thing has to do with circles in F_q^2

[dense belfry]

Nice

[median zinc]

I think circles come from all the norms coming in too

[dense belfry]

Wew lads have you ever actually done anything with the hyperreals?

Or read about their history

Or read about the debate around them

[sleek wing]

are you taking my off hand joke comment seriously

[dense belfry]

Yes

[sleek wing]

https://cdn.discordapp.com/attachments/359052581022203914/940656085893799936/unknown.png

[toxic schooner]

What is hyperreal

[toxic schooner]

This should be an emoji

[sleek wing]

it's the real numbers with infinitesimals and infinities

[dense belfry]

You're spreading rhetoric dude

Smh

[sleek wing]

you're the only person on the entire server who took me seriously

[toxic schooner]

I took unseriously a couple of times

[dense belfry]

That's because I'm one of like 3 people here who's actually done anything with the hyperreals

[sleek wing]

I meant in context of this specific hyperreal meme statement

[toxic schooner]

I see

[sleek wing]

I'm curious now

what did you use them for

[median zinc]

I took at look at the paper, it's not random graphs but information theory so I don't know what is going on

[dense belfry]

Lol tbh I'm just raging because I think it's funny

[sleek wing]

yeah but I'm actually curious

[dense belfry]

I've proven a bunch of theorems about them and used them in a bunch of alternative constructions of standard stuff

That's kind of what I meant

[sleek wing]

fair enough

[sick burrow]

it's neither

[dense belfry]

I realized that my wording was overstating my case lol

[sleek wing]

I can see the connection to circles now btw

[median zinc]

You can ask the author on his thoughts
https://meetings.ams.org/math/jmm2022/meetingapp.cgi/Paper/7694

[sick burrow]

oh you found the paper

[sleek wing]

it's a sequence of vectors with v_i+1 within a circle of radius t of v_i

[sick burrow]

yes...

see that part the paper explained earlier

[median zinc]

2D vectors

[sick burrow]

the issue is it still doesn't help answer the question of what the fuck E(x) and S_t(x) are

[sleek wing]

but it's a clue

[sick burrow]

it's a clue I already had

I'm just going to email them

tbh

[alpine kindle]

oh you're a proper mathematician wow 😳

[sleek wing]

yes please do and tell me wtf it means

[alpine kindle]

ping me too

when you find out

[sick burrow]

anyways

actually idek

[long matrix]

you're telling me only proper mathematicians can prove theorems? D:

what does that make me then 😢

[sleek wing]

yeah

it makes you ONE OF US

ONE OF US

[long matrix]

imaginary mathematician?

complex mathematician?

D:

no no

improper mathematician

So I must be the whole set

I am all the mathematicians.

[alpine kindle]

and i am all the jedi

[proud ginkgo]

Check out help 2

[median zinc]

I found it
For E(x) find "characteristic function"
https://arxiv.org/pdf/0801.0728.pdf
S_t also defined in
https://arxiv.org/pdf/1406.0107.pdf

[sick burrow]

ohh

[sleek wing]

like, an indicator function?

[median zinc]

Probabl

[sick burrow]

like that makes sense

[sleek wing]

that makes sense but CMON

[sick burrow]

I figured it was probably something like that

[median zinc]

I know nothing about finite fields

If it's an indicator why not just use $1_{S}$ too

[fathom swallow]

pepper

[sleek wing]

or I_s

[sick burrow]

well thank you for finding that definition

[sick burrow]

where/how did you find these papers anyways

[toxic schooner]

am i ONE OF US?

[sleek wing]

I... think so

[toxic schooner]

noice

[sleek wing]

well depends

[toxic schooner]

[sleek wing]

how many theorem you prooved in you're lifes

[toxic schooner]

0...

[long matrix]

uncountable

[toxic schooner]

depends on what u mean by theorem

[sleek wing]

yeah can't count it cause it's 0

can't count nuffin can ya

[toxic schooner]

so true

[long matrix]

i cant count it. whym i supposed to remember

uncountable.

[toxic schooner]

I have proven stuff

not theorems per say

[long matrix]

If I prove 2n is even for any integer n

How many thms is this 👀

1?

Z?

[toxic schooner]

If that stuff counts, i really have proven uncountably many theorems

[sleek wing]

1 tehroem

also it cannot be done

[long matrix]

I think we count it as Z 👀

[sleek wing]

is 0 odd? is 0 even?

[long matrix]

Why

[toxic schooner]

👀

[long matrix]

0 is even

[sleek wing]

who knows

[long matrix]

.......

[sleek wing]

0 is even what

[long matrix]

pls.

[toxic schooner]

0 isnt a natural number

[dense belfry]

Lol I was following along from a book, I haven't done anything original with the hyperreals

I'm a grad student

[toxic schooner]

[sleek wing]

I have proven precisely one theorem that I think is completely original

[dense belfry]

It's not hard to prove an original theorem

[sleek wing]

I can't find it anywhere

[dense belfry]

It's hard to prove a nontrivial original theorem

[sleek wing]

it was kinda non-trivial

[toxic schooner]

i havent done anything like that

[sleek wing]

it allowed you to determine when conjugacy classes split when mapping to a subgroup by looking at the restrictions of irreducible characters to that subgroup

iirc

it was the induced representation whoops

[toxic schooner]

[sleek wing]

I still need to run the proof by my supervisor tbh

[vivid halo]

cute notation for induced representation

[sleek wing]

ty ty

restriction is the downwards arrow

[vivid halo]

[median zinc]

He cites himself, and it's a citation chain

[sleek wing]

the frobenius reciprocity theorem looks so swag with this notation

just move the arrow lol?!?!

[vivid halo]

yup they're adjoints

this notation makes it so clear it's nice

[median zinc]

I think for non-star mathematicians, they are under pressure to keep publishing. They cannot hide in one corner for 13 years and publish a 400-page volume on Transactions

after 13 years

[sleek wing]

yeah and then you see what that arrow actually fucking means