serious-discussion

[gaunt quest]

I would argue that lies in the realm of belief

which immediately makes it clear that it's not enough to just say "we know where the line is"

[burnt ledge]

if i had information abt the house then id be able to say

[zealous garden]

I would say knowledge correlates pieces of information into something of meaning when interpreted properly

Truth of the contained information or meaning is irrelevant

Like, if the contained meaning is false

That itself carries additional meaning and information

And there still is knowledge there

[gaunt quest]

right but the point is that information is usually incomplete, so a probabilistic view like I'm suggesting makes sense to me

but then no true knowledge of real things :(

[burnt ledge]

but even tho knowledge is usually incomplete kts very often sufficient to gain certainty

i dont have total informstion about my house but i know certainly it is one

[zealous garden]

I don't think those notions are meaningful at a philosophical or metaphysical level

Think about it like special relativity

There is no one version of "now"

[gaunt quest]

but that would be true information? so you are still deriving knowledge from truth how I'm reading that

[zealous garden]

Different observers have different notions of simultaneity

[zealous garden]

Knowing that a proposition is false carries information, how about that

You can still use it to derive something

[gaunt quest]

a high probability would correspond with that idea. being sure enough to act off something wouldn't be the same as "knowing"

[burnt ledge]

i dont believe that

[gaunt quest]

right but only because it implies something would be true

there would be no information in complete nonsense

like 2 + 3 = apples

no axioms, no nothing that's a meaningless statement

if I define axioms which make that true, then it has meaning

[zealous garden]

Something being true only is helpful because it shows something false

[gaunt quest]

or if I define axioms which make it false, that's only because there is something that is true

[zealous garden]

So it's irrelevant if I care about what's false versus what's true then

[fringe summit]

Incompleteness

[zealous garden]

(A -> B) = (!B) -> (!A)

[gaunt quest]

hi godel

[zealous garden]

So it really doesn't matter if you give me true or false

Knowledge is knowledge

[gaunt quest]

right but I'm saying then that your knowledge is supported on the existence of true statements. you're just defining it contrapositively to me

like not B -> not A is a true statement

[zealous garden]

It could be a false statement

[fringe summit]

“You cannot prove the existence of strong inaccessible cardinals in ZFC, assuming consistency”

Fit this into your weird negation scheme

[gaunt quest]

I think the negation stuff is just confusing and unnecessary

[fringe summit]

I think considering knowledge philosophically is dumb

[zealous garden]

I think a certain level of philosophical consideration of knowledge is necessary

But not this

[fringe summit]

I do not think highly of such philosophical ideas to begin with, however

[zealous garden]

I like philosophy

[fringe summit]

“Do I really exist”

[zealous garden]

Despite not having read enough of it

[gaunt quest]

I think you don't need it to be effective, but it is similar to me in principle as saying "I don't need analysis if I can do calculus"

[gaunt quest]

like we don't need a basis of knowledge to be able to say we know things, but without trying to find one we have a poor idea about what that knowledge is or how it should work in edge cases

[gaunt quest]

oh dear

[wet mortar]

I subscribe to clarence0 on youtube

[zealous garden]

What's that even supposed to mean

[fringe summit]

I believe this is wrong because no answer on “what is knowledge” will modify how things are handled

[zealous garden]

Exist is such a fuzzy word in the first place

I rate the question irrelevant and meaningless

Closed for off-topic discussion

[gaunt quest]

I think there are real analogues to gettier problems in things like ethics

if someone had no knowledge of a consequence for example, we might consider them innocent

so it matters what we consider knowledge, is it enough that they had the necessary information and is it their failure to interpret it correctly?

is interpretation a barrier for knowledge in the first place, so we can argue that they didn't have knowledge. if not, then we can argue they did have knowledge

things like that

[fringe summit]

I do not care whether they knew or didn’t, I’d just question whether it is believable that they acted in a manner consistent with what would be expected

[gaunt quest]

that's very vague to me I don't like that

[fringe summit]

Asking if they do or don’t know isn’t something you could show anyway

[zealous garden]

Yeah, knowledge isn't what you should question

[fringe summit]

“Did you know” “no”

[neat lintel]

please stop talking about philosphy

[zealous garden]

You should question reasonable expectation or suspicion

[neat lintel]

it has been hours

[gaunt quest]

reasonable is predicated on many assumptions

[fringe summit]

Yes

[gaunt quest]

so that doesn't help anything

if there's a disagreement you can't just say "my reasonable is right yours is wrong"

[zealous garden]

Every measurement you could possibly take will include many assumptions

[fringe summit]

You say help like what you’re asking for makes sense

[gaunt quest]

I'm not asking for a single consistent definition of knowledge

[zealous garden]

If there's a disagreement I can argue your measurement tool isn't properly calibrated

[gaunt quest]

but more ways of thinking about it can provide frameworks to resolve conflict where that becomes an issue

[fringe summit]

I do not think any definition is usable

[zealous garden]

Knowledge, well, you just know

[fringe summit]

It’s just justifications and arguments all the way down. There is no knowledge

[turbid pebble]

that's a really dismissive attitude toward epistemology, but that's okay

however

[gaunt quest]

we have agreed on ways of describing what measurements are and aren't acceptable though, and we can use that more explicit framework to more robustly resolve these issues

[turbid pebble]

i think that there's a piece of wisdom you can pick up

[fringe summit]

Epistemology is bad

[turbid pebble]

as much as you think it's annoying for people to talk about philosophy

you know what's even more annoying

[long matrix]

no u

[gaunt quest]

there being no knowledge is fine to me, doesn't mean we can't develop theories of it

[turbid pebble]

people who go out of their way to make it known how much they think philosophy is dumb

that's more annoying than people talking about philosophy

[gaunt quest]

each theory just has issues and we select the theories based on which issues are considerable as irrelevant in this case

[turbid pebble]

like bro if you think it's a boring subject go do something else

[long matrix]

what war have i walked into...

[turbid pebble]

try shutting up or something some time

[zealous garden]

Knowledge exists, but it's an observer-dependent phenomena

[turbid pebble]

but all it show me if you walk in saying "philosophy bleh bleh" is you don't actually care

[zealous garden]

Delusion is also observer-dependent

[fringe summit]

Rather than resolving the issue of vagueness in “reasonably expected” you just moved it to making choices on what theories you subscribe to

[turbid pebble]

so why should i care about your rebuke of it

[zealous garden]

Combine them and we can study the observer-independent knowlusion

[gaunt quest]

except now we have a way to independently discuss what the issues of the theory are, and the people involved can make an informed agreement. it's not perfect but it's better

[fringe summit]

Aggressive

[turbid pebble]

2:03 PM]The eternal Sharp: Epistemology is bad

and this?

[fringe summit]

Is saying it is bad

[turbid pebble]

why should i bother with decorum around someone who went out of their way to disrespect a field

[gaunt quest]

reasonably expected is incredibly heavily rooted in current societal expectations to me, it's not acceptable enough for many issues

[turbid pebble]

you earned negative respect today guy

[zealous garden]

I'm giving out a free lesson in Geometric Psycho MetaPhysics

[long matrix]

i think the usual "ethics" i see people go on about is bad

[zealous garden]

You better be taking notes

[fringe summit]

I will not respect someone talking about eugenics positively

[turbid pebble]

that's fine

philosophy isn't all old

if you care about it you can participate

then it's current

[fringe summit]

It’s all bad :^)

[turbid pebble]

man fuck you

[fringe summit]

[turbid pebble]

you're an example of what's wrong with discussion

peace out

[fringe summit]

I literally sent it with a :^)

[zealous garden]

Arbitrary and by convention

[fringe summit]

How can I be more explicit that it is a joke

[gaunt quest]

less arbitrary matters

[fringe summit]

Aggressive

[zealous garden]

Look at this guy

[fringe summit]

How

[long matrix]

i wanna join this discussion but i woke up too late...

[turbid pebble]

if it's a joke, then you're trolling and you deserve even less respect

[gaunt quest]

like I don't wanna drag sensitive issues here but reasonable is actively used rn to argue for some pretty horrible things, and people don't have tools to see why that's not inherently true

[zealous garden]

I've never seen you before, and you insult my good server mate sharp

[turbid pebble]

because it followed from what you'd said before

[fringe summit]

Bro is incapable of having a little levity

[turbid pebble]

you are stringing people along into the idea that you are a dipshit, so that you can be like "aha, i'm not tho"

nah bro

i got tired of that strat 20 years ago

[zealous garden]

I'm a dipshit and I'm proud mister

[long matrix]

cel ur probably taking #serious-discussion a tad too seriously

[fringe summit]

It is well known that my tastes in knowledge are esoteric

[gaunt quest]

I think that's the real world current case of harm that motivates a less arbitrary approach to defining knowledge, even if it's never perfect

[gaunt quest]

I don't mind it if people wanna get off the train earlier than me on it

[zealous garden]

You misspelled geometric

[turbid pebble]

this is just as dumb as "korean music is abusive"

i gotta go to an eye appt

i hope you improve

bye

[fringe summit]

There is no non-arbitrary reference frame imo

[fringe summit]

This is a question of morality and/or ethics

Things can be 100% reasonable but evil

[zealous garden]

Thanks, I appreciate that. I hope you improve too!!

That guy's alright

[fringe summit]

It’s reasonable why I would crush the competition in business

Just not a good thing

[gaunt quest]

I think that's a bit reductive, since people are arguing over knowledge and definitions of things. when they have a bad theory of knowledge, then I can't begin to argue ethics or morality with them

[fringe summit]

Is and ought to be are wholly independent

Is-Ought problem

You will not solve this with questions of knowledge

[jagged forge]

the problem of the common good (edit: actually i think this is slightly different)

[fringe summit]

Simply the wrong question

[zealous garden]

"Reasonable" depends on context

[fringe summit]

Yep

[zealous garden]

And it depend on you deciding what you want out of your reasoning

[neat lintel]

Not really

[fringe summit]

Reasonable is a question of “ought”

[fringe summit]

Ah true, antitrusts

[neat lintel]

You'll get more done by collaborating with competition

That's what most bussinesses do

[zealous garden]

The reasonability of an action depends entirely on whats going on and what you want out of the situation

That latter part means you can't make it objective

[jagged forge]

collaborating with competition? or collaborating with other orthogonal institutions that specialize in something you need?

[fringe summit]

Was the Cold War reasonable?

[neat lintel]

what?

[fringe summit]

If you’re collaborating they aren’t the competition

[zealous garden]

I think you can get close if you start talking about Agent goals and Information

[long matrix]

sully did all this shit start here

i was awake back then

[jagged forge]

business collaborate when the other business produces something they need at a better cost. if they are producing the same thing and targeting the same market it's very unlikely they are collaborating

[neat lintel]

Have you ever heard of price fixing?

OPEC?

[gaunt quest]

I think so

[long matrix]

i still have no clue what the focus of this discussion is rn

[fringe summit]

They’re not so much the composition there but fair

[jagged forge]

isn't opec not going so well lately since the saudis are throwing a hissy fit

[gaunt quest]

I just wanted to play devil's as a joke but idm that this became a more serious discussion

[neat lintel]

No? Americans are throwing a hissy fit

[gaunt quest]

still interesting

[neat lintel]

OPEC is making money

[fringe summit]

Wow I can’t believe you would just string someone along or whatever he said

[gaunt quest]

I'm an awful person ik Q_Q

[fringe summit]

[long matrix]

ur a blue cat.

[gaunt quest]

how do you know

[long matrix]

sharp should rename yourself the yellow sharp

ik.

[neat lintel]

When OPEC decreases oil output they are doing that to raise prices.
When they do that everyone else cries because they have to pay more

[zealous garden]

Me too

[jagged forge]

in any case i don't buy that situations like opec are the norm

there are plenty of examples of business crushing competitors in the us

[zealous garden]

Blue, what kind of knowledge does this statement contain

[long matrix]

knowledge is silly

[solar hawk]

The yellow flash of the leaf

[long matrix]

know that you know not

[solar hawk]

🍃

[gaunt quest]

the only thing I can say I know is that you probably feel the same way as whatever you replied to

:]

[zealous garden]

Exactly this, absolutely based

[gaunt quest]

the missile knows where it is because it knows where it isn't

[zealous garden]

No, the missile knows where it is because it has a GPS guidance system

[gaunt quest]

I mean if you could know everywhere that you weren't, then you're left with where you are

so who's to say really

maybe that's assuming law of excluded middle or something idk the formal logic on this Q_Q

[gaunt quest]

but yeah the missile could know that, you don't know 😤

who says it's using the GPS system

[zealous garden]

Who's to say my screen is even plugged in right now, that I'm not intuiting everything you say because you're so predictable

[gaunt quest]

your eyes don't even work

you're just absorbing the information through your skin

[zealous garden]

After all I only need an estimate of what you're saying to respond

[long matrix]

everyone else on the internet is fake...

[gaunt quest]

the internet is dead fr fr

you guys are just funny words on my screen

who says you exist

[zealous garden]

And I can also just keep talking, ignoring what you're saying, because I'm cool like that

So really, anything could be happening

[gaunt quest]

eventually people will just assume you talk a bit strangely and adapt

[zealous garden]

I do talk strangely

[gaunt quest]

they truly believe you're talking with them, so who's to say they're wrong 😤

haha

[zealous garden]

I think my brain is shaped weird

[gaunt quest]

got a bump or a dip somewhere

[zealous garden]

I have all this extra knowledge stuck in this fold right here

[gaunt quest]

the only appropriate response

[long matrix]

you

[gaunt quest]

except most importantly you

we're not the ones claiming it's odd

[long matrix]

im irrelevant

[gaunt quest]

true same

we don't exist actually

so the only person who joined a maths discord is you

checkmate atheist 😎

I mean can you see that there are people on this server?

how do you know we aren't all bots?

a bot could be sending these messages

I'm just text on your screen

because nobody exists online obviously

[zealous garden]

Because it is what we the bots are designed to do

[stuck pulsar]

Embed fail

[solid snow]

<@&268886789983436800>

[odd narwhal]

אט\

ty

[stuck pulsar]

Nice

[turbid pebble]

I'm glad I had the extra minutes to get bored on my phone so I could know the culture of this server is sufficiently dismissive and condescending to leave

[neat lintel]

wdym

took him 28 days to figure that out

[neat lintel]

AHAHAHA

[storm sage]

Tbh it is kinda condescending sometimes, I don't like it

[long matrix]

sully

[storm sage]

Case in point

[long matrix]

sotrue

what else...

[small dust]

i feel like sotrue is friendly like 75% of the time at least

[long matrix]

uhhhhhhh

i dont know about that

[zealous garden]

I typically use sotrue unironically

[long matrix]

its a non-offensive reaction to the original poster

but can be

well can be abrasive or something to the people not in the know

[small dust]

sully is cruel but the temptation to sully always overcomes me

[storm sage]

It can be easily interpreted as condescending

Regardless of original intention

[mental stratus]

[storm sage]

Especially if someone is new to the server

[long matrix]

the sully makes no attempt at explaining to the op why their post was sullied

but is usually used as a low effort response rather than no response

at least they know their post was rejected rather than ignored

[storm sage]

Yeah I don't think it is a good emoji to have in the server

It is dismissive

[wet mortar]

imma be real

[long matrix]

yeah but those kind of comments/whatever will just be ignored.

is that any better than being sullied

[storm sage]

That's honestly better

[wet mortar]

I'm the reason sully is here

[small dust]

i wish it got removed so i wouldn’t be able to sully anymore

[wet mortar]

I brought it from outside

and used it here

then they added it

[long matrix]

uve been here from 2019 so i cant refute this claim

its a bold claim tho

[wet mortar]

this was like 3-4 years ago

[storm sage]

I don't think this is true

I think ng is the originator

[wet mortar]

yeah but that was in hopf

[storm sage]

Oh

[wet mortar]

I used it here

[storm sage]

I see

[wet mortar]

[long matrix]

So how does it feel at this stage

[storm sage]

You're the bane of my existence now

[long matrix]

Looking back at the corpses

and

and stuff strewn behind you

[wet mortar]

I don't use it myself anymore

[long matrix]

does it haunt you in your dreams. You check in here and think to yourself 'what have i done'

[wet mortar]

https://m.youtube.com/watch?v=VOK4NtCkNGg&t=23s

[fringe summit]

How else will I react to someone asking if 1/0 = 0/0 = 0^0 for years

[wet mortar]

since 2021 is crazy icl

[fringe summit]

The deep lore

Or ya know, someone claiming something blatantly false ofc

[long matrix]

we now have , the less aggressive

youre judging them, but not outright rejecting

[gaunt quest]

[wet mortar]

[fringe summit]

I think is less harsh

Than

[long matrix]

no way no way.

can still be lovable

as an emoji

?

no love.

[fringe summit]

This one has no love

[naive temple]

sharp

monkeys generally dont have love

or is that an ape

[fringe summit]

the name says monkey

[wet mortar]

they all have the potential for love

[fringe summit]

❤️

[prisma plinth]

mathematicians lack love, only the ability to play in the sandbox and become alcoholics :troll:

[tame egret]

lol

[storm sage]

That's only you

[neat lintel]

This is so true

Mathemicians tried to measure the limit of their madness and they got $$DNE$$

[fathom swallow]

Brandon H

[long matrix]

dn

[prisma plinth]

dichotomy of man

[deep mango]

No

you're nuts

[proud olive]

I honestly think it's the opposite

(sorry for ping)

[fringe summit]

I prefer pings for replies

[long matrix]

[proud olive]

[deep mango]

common sense and cortesy

[fringe summit]

me when I have no idea someone replied

[solar hawk]

wow ur old

[mint patio]

Is there a version of PMA that doesn’t have the worst fucking typesetting in the world

or would the texromancers like to take charge on that

[fringe summit]

@mint patio

[mint patio]

hi

[fringe summit]

hello

[mint patio]

well?

oh I meant

do you have a response to my question

LMAO

[fringe summit]

Ah

What’s PMA?

[mint patio]

[fringe summit]

Ah yeah no idk

I’ll check the one I have

But there’s also uhh

Folland analysis

[mint patio]

unluck

[fringe summit]

Yeah I’ll check in a bit just in case

[mint patio]

yeah but I’m a masochist

🥺🥺

thanks cutie

[fringe summit]

Folland might be more maso since prereqs

[mint patio]

isn’t he measure theory anyways, I want to review my real analysis

[fringe summit]

I think it’s real analysis but includes measure

[hallow blade]

real analysis includes measure theory

[storm sage]

Folland's analysis assumes you already have the knowledge from a first analysis course like rudin PMA

[fringe summit]

Aight fair

[twin moat]

I just looked up the earliest timeline in the world but how could it almost 12 hours ahead of my current time zone?

[echo tundra]

What's the origin of sully, I need some lore

[static rapids]

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[stuck pulsar]

Mmmm yeah, lets call it…
“X”

[ripe dirge]

Bot detected

[eager stag]

folland's analysis is measure theory, topology and functional analysis while rudin's pma is an intro analysis book

[storm sage]

No ChatGPT posting in here

Or irrelevant walls of text

[prisma plinth]

i gave up trying to get this to drill right. I’ll file it down, can’t get the fucking metal to stop wobbling

[mental stratus]

formal power series will be the end of me

[wet mortar]

formal power series are based tho

[prisma plinth]

give example

[tiny forge]

a power series wearing a tuxedo

[long matrix]

formal power series?
convergence?

[river moon]

$\left(\sum_{k=0}^\infty D^k\right) (f) = \sum_{k=0}^\infty f^k(x)$

[fathom swallow]

Transparent Elemental

[river moon]

[ripe dirge]

[short sail]

why there are two discussion channels ?

[long matrix]

you're asking the wrong question

[short sail]

why ?

[long matrix]

omit the two.

[short sail]

i don't understand

[storm sage]

so there can be multiple conversations going on

[short sail]

okk thanks it's a good idea

[calm vessel]

When you integrate |x|

you obtain x^2 sgn(x)/2

why exactly does sgn(x) come about?

[rocky shuttle]

because it's x^2/2 if x > 0 and -x^2/2 if x < 0

[mental stratus]

aight did some rings, formal power series, polynomials and some topology today

maybe tomorrow is for linear algebra

[neat lintel]

sgn = sign = + or -

the sign of 1 is + the sign of -1 is -

[calm vessel]

I think I'm struggling with trying to understand the meaning of the indefinite integral

because if you plug in -2 you get 4 sgn(-2) /2 = -2

but the area under the |x| graph is positive?

[prisma plinth]

what specifically lol

[prisma plinth]

your area “flips” signs if the direction of integration is backwards

so integrating from 0 into the negatives is “negative area”

[storm sage]

I'm assuming you're integrating from 0 to some other point. If so, plugging in a negative value would mean going from 0 to a negative value, meaning your integral is backwards, so it flips sign.

[calm vessel]

Yeah it feels like the meaning of plugging in -2 into the indefinite is integral is actually evaluating the definite integral from 0 to -2 which is backwards

but to consider all indefinite integrals as definite ones in disguise also feels weird to me. Because then how did the constant of integration get a chance to sneak back in 0-o?

[calm vessel]

I don't have endpoints in mind

[gaunt quest]

maybe interpret it thinking of |x| as the derivative then, you have a positive slope on both sides, so you expect thst the integral should be generally increasing. x^2 decreases as you increase x -> 0, then increases for x > 0

so you need to introduce the sign change

[calm vessel]

OH

that's very nice

[storm sage]

The endpoint is determined by what value of C you choose

Since it's really x^2 sgn(x)/2 + C

If you choose C to be zero, then the function is zero when you plug in zero, so you can interpret that as doing a definite integral starting at zero and then ending up at whatever value you plug in for x

[calm vessel]

Yeah when I plug in smth like say b, it would be computing F(b) = F(b)-F(0) since F(0) is 0.

So if there is a root and it's continuous you can shift around the endpoint of "direct" evaluation using the C

[storm sage]

Yeah exactly

[calm vessel]

hmm that also means if there are multiple roots integrating with any of them as the starting point should be the same

[storm sage]

Yup

That's correct!

(And if there aren't any roots then you can't really write it as a definite integral)

[calm vessel]

why not?

[gaunt quest]

would it be accurate to say that the definite integral hides the constant of integration in the subtraction, while the indefinite integral doesn't since it's just a function that's evaluated (so needs the constant to cover all antiderivatives)?

[storm sage]

(In this case though there's one obvious root so writing it as a definite integral is somewhat natural)

[storm sage]

Because definite integration from c to x always has a root (x=c)

[calm vessel]

oh you mean the indefinite evaluation cant be viewed as a definite one

so if I chose the integral of x to be x^2 + 1. Then evaluating that (x^2 + 1) at some point is meaningless

[gaunt quest]

generally I don't yeah

which is why I have never thought too hard about this

[calm vessel]

I needed to construct a function that's exactly k times differentiable

So integrating |x| works

[gaunt quest]

ah I see

that's an interesting problem

[prisma plinth]

integrate weierstrass’s function 5 times :troll:

[topaz swift]

Anyone here seen this?

[solar hawk]

Nope

[tame egret]

nope

[neat lintel]

I can see it

[topaz swift]

what forgetting 1 letter does

[neat lintel]

🫡 Calculus exam tomorrow

I do not feel fully prepared

[storm sage]

🫂

[neat lintel]

Good luck💀

TY Hoping i get the easy questions. I need to get a 100%

[cinder zephyr]

noooooooooooooooooooo

you've jinxed yourself

now you're definitely going to get the hard ones

[neat lintel]

Let them come. I will be prepared

[uncut socket]

@heady girder You are a helper because you have the helper role

[zealous garden]

Ping

[uncut socket]

You can remove it by going id:customize and unselecting, "I want to help people with math"

[fringe summit]

[long matrix]

im helpful because im not paid

[neat lintel]

.

hi

[strong hedge]

anyone know how to solve

log8^27 x log49^16 x log3^343 without using calc

[mental stratus]

Ugh, these linear algebra problems will be the end of me

[languid marten]

Use log properties

[strong hedge]

the ans is 6 tho

[neat lintel]

the answer is "pick a basis" if the vector space is finite-dimensional and trivial if it's not

hope that helps 👍

[mental stratus]

??

today has been a bad math day

although i blame the book's framing of the problems, im moving on to other books for the exercises now

and i fu****g hate number theory exercises

[wet mortar]

halmos

[mental stratus]

halmos is so old tho

[wet mortar]

and goated

[mental stratus]

🐐

[neat lintel]

only half trolling

"finite-dimensional" in an exercise or theorem or something is basically begging for you to check if it's true in any vector space. if it's not then the key is usually to pick a basis, or do something with a dimension count

there isn't much to do in infinite dimensional vector spaces in basic linear algebra - these only really get interesting when you study their topology (say, when you put a norm on them)

[wet mortar]

the most useful general thing is probably projections

[rare sonnet]

are all vector spaces of the same (finite) dimension isomorphic

[wet mortar]

yes

as are those of the same (infinite) dimension

[prisma plinth]

calc problems and linear alg problems are monotonous tedium

most abstract alg problems make you use your gnoggin and I find that more enjoyable

[echo tundra]

yes you can easily form a bijective map between the basis vectors from (say) U to V

once the basis is fixed, everything else gets fixed

[neat lintel]

sounds like you're just doing the wrong calculus and linear algebra problems

it's not so hard to come up with a ton of abstract algebra exercises which are "monotonous tedium"

[prisma plinth]

for classes

[neat lintel]

what exactly makes them monotonous tedium

[prisma plinth]

D & F hasn’t really had many monotonous problems that take way too long

[prisma plinth]

The best example was this one limit

$\lim_{x \rightarrow 0}{\frac{1-x \cot(x)}{x^2}}$

[fathom swallow]

Mizalign

[prisma plinth]

problems that just babushka doll into using the same rules and shit 50 times, where it’s easy to carry small errors into the final answer

[neat lintel]

oh no, computation, the horror

[prisma plinth]

doesn’t make it any less tedious and annoying

[wet mortar]

ok here's an algebra problem

classify all groups of order at most 16

up to isomorphism

[neat lintel]

big cope

sometimes you have to compute

i don't really see what's tedious about this, you just need to take a few derivatives

this is on the super light side of computations tbh

[prisma plinth]

touche

[neat lintel]

you cannot avoid doing small things like this. might as well get good at them

[prisma plinth]

flashback to the 4 page exponential matrix bs

[sleek wing]

C_2, C_2^2, C_2^3, C_2^4, C_4, C_4^2, C_2xC_4, C_2^2xC_4, C_8, C_2xC_8, C_16, D_8, Q_8, D_16, SD_16, Q_16 uhhhhhhhhhh S_6 uhhhhhhhhhhhhhhhhh

UHHHHHHHHHHHHHHH

[wet mortar]

[flint basin]

C_3, C_5, ...

[sleek wing]

yeah I did the easy ones first...

[long matrix]

so if we apply collatz, I believe we are done.

[flint basin]

Professor can we have a different homework problem

[sleek wing]

I do think they're all the 2 groups + S_8 tho
no wait the non-split extenstions my despised

[wet mortar]

no

[long matrix]

classify all groups of order at least order 16

would that be better

[sleek wing]

are you insane

p^8 isn't even known

[long matrix]

whats p^8

yeah i dont know it.

[sleek wing]

groups of order p^8

[long matrix]

what for any prime?

[sleek wing]

we know them for some primes but not in general

we know p^7 for all primes

[long matrix]

lolol

trolololool

[flint basin]

Finite group theory moment

[sleek wing]

I personally know p^3 and like a third of p^4

[long matrix]

here's an algebra problem: prove the set of endomorphisms of an abelian group naturally is a ring

[sleek wing]

ok that's an actual exercise though

abelian groups are modules so just take the endomorphism ring :trollface:

[long matrix]

prove the endomorphism ring is a ring

[wet mortar]

no

[sleek wing]

why would we call it that if it wasn't one...

[long matrix]

u could be lying

[sleek wing]

I would never

[long matrix]

well prove the endomorphism field isn't a field...

[sleek wing]

the map sending everything to 0 isn't invertible

[long matrix]

but i called it endomorphism field

[sleek wing]

hmm, perhaps cope some more?

[storm sage]

normal curve meme

• mathematics is all about computing stuff
• mathematics is about understanding beautiful abstract connections and generalizations between different structures
• mathematics is all about computing stuff

[sleek wing]

only algebra heads go through the middle path

[river moon]

*get stuck at

[long matrix]

i dont understand the normal curve meme unless its a picture. sorry.

[prisma plinth]

i’m eventually going to get into alg top which is going to be all computation at a certain point horror

[sleek wing]

does he know....

lets just say... this space?? Ain't Q-good

[eager reef]

Alg top computations are based

[storm sage]

mizalign you should write down a presentation of the hawaiian earrings group

[eager reef]

Real

[storm sage]

I actually have no idea what one would look like

[sleek wing]

yeah sure lemme start <a_1, a_2, a_3, a_4, a_5, a_

[storm sage]

Has anyone come up with one

[sleek wing]

I doubt it, you only really care about it's homology right

I will google

oh goody, the countably infinite free group is a proper subgroup

fantastic start

[storm sage]

Moreover, this group has become quite important in infinite group theory since in many ways it is the non-abelian Specker group.

What on earth

[pure sun]

Apparently H_1(hawaiian earring) is the following:
direct sum of:
i) functions N \to Z
ii) product_p (p-adic completion of (uncountable direct sum of Z_p's))
iii) uncountable direct sum of Q's

[storm sage]

This is insanity

When the abelianization of your group is scary...

[pure sun]

https://web.math.rochester.edu/people/faculty/doug/otherpapers/eda-kawamura2.pdf

3.1

[sleek wing]

yeah a similar description is on the wikipedia page

[raven plaza]

That sounds like a pretty reasonable description

[pure sun]

So i can come up with loops which give rise to classes in the first piece (functions N to Z)

But not the other ones

Like what is a divisible class going to look like

The projection to H1(any individual loop) must be 0

So it goes around each loop a net 0 times

[storm sage]

Yeah I guess an easy one would be going around loop 1 CW loop 2 CW loop 1 CCW loop 2 CCW

[pure sun]

That is zero in homology

[storm sage]

Oh I can't read what you said

[pure sun]

If you do that “to infinity” like, go around all loops CW then all loops CCW, then maybe that would work? Idk i have zero intuition for pure singular homology

[storm sage]

I thought you were looking for something that was 0 in the homology

My bad

[pure sun]

And this isnt a cw complex

No i want something divisible

In the Q part of homology

[prisma plinth]

I know nothing about homology lol

In the topological context

[mental stratus]

today has been awful, i woke up wayyy earlier than yesterday and i feel like i got half the math done

[wet mortar]

today I did no maths at all

I woke up and cooked

[neat lintel]

this is good

math doesn't need to be done every day

[wet mortar]

yes

[long matrix]

collatz is waiting

[neat lintel]

THE TENSOR PRODUCT IS ESSENTIAL TO THE EXISTENCE OF BLACK HOLES

[mental stratus]

no i need to do math everyday

[neat lintel]

the grass is outside, waiting to be touched

[flint basin]

who needs to touch grass

there is math to be done

[fringe summit]

Exactly

Grass is for the weak

[long matrix]

@brittle socket

[mental stratus]

i wanna be touched

[long matrix]

by blo

[wet mortar]

yeah people don't hug enough nowadays

it's not good for long term mental wellbeing

[burnt ledge]

guys what the hell is homology

[mental stratus]

apparently size doesn't matter

neither does position

[wet mortar]

topology?

[mental stratus]

Because for balls in R^n translations and dilations are homeomorphisms

[oblique beacon]

I have a cactus at home

[wet mortar]

number of holes of a given dimension measured by abelian group

hope this helps

[storm sage]

It measures how far a sequence is from being exact

[burnt ledge]

How do you measure a hole with an abelian group ?

[storm sage]

In topology, the lack of exactness means the presence of a hole

[burnt ledge]

and whats a hole "of a given dimension"

and whats an exact sequence

[storm sage]

A circle has a hole of dimension one lower than a sphere

[storm sage]

Do you know what a group is

[neat lintel]

The study of holes

[long matrix]

1, 2, 3, 4, ... whats the homology

[burnt ledge]

Yeag

[storm sage]

Imma just redirect you to read the start of chapter 2 of hatcher's algebraic topology then

It's very good and free online

[long matrix]

evil eric

[storm sage]

(By start, I mean the part of chapter 2 which comes before section 2.1)

[burnt ledge]

thx for edit, i was gonna say that hatcher is probably just the name of some guy. okay ill have a look

[storm sage]

Wait this was a serious suggestion

[mental stratus]

is spanier good

[vivid halo]

if you're talking about singular homology, this is measuring how you can map simplices into your space. The intuition is that if you have some sort of "hole" in your space, then you should be able to detect this by the presence of non-contractable simplices surrounding this hole

[storm sage]

Start at "The Idea of Homology"

You can ignore the mentions about homotopy groups and then feel free to ask here what any unfamiliar terms mean

[storm sage]

I've heard it's very good

I've never read it

[mental stratus]

is algebraic topology a hard prerequisite for something like bott tu diff geo or a lee Smooth manifold?

[wet mortar]

no

[vivid halo]

for Lee no

for Bott and Tu no

although Bott and Tu gets pretty hard pretty quickly, it's reasonable to have some exposure to algebraic topology for this

but it's also meant to cover a lot of content from algebraic topology, so it covers a lot of the same stuff

[mental stratus]

apparently bott tu are two diff people and the book i meant to ask about is intro to manifolds by loring tu

[wet mortar]

I guessed

[mental stratus]

😴 can't fall asleep

[tame egret]

turn on eye care on all devices then watch something boring

[mental stratus]

ugh Z is not well ordered

[tame egret]

wdym

[long matrix]

no u

[echo tundra]

this needs some context

[wet mortar]

topology or something

[mental stratus]

balls are homeomorphic

[wet mortar]

to what

[mental stratus]

Topological properties are the things preserved by homeomorphisms

balls

[wet mortar]

yes

[long matrix]

g is isomorphic and so is h

[wet mortar]

real

[burnt ledge]

Okay so i see this little graph, and then he makes it into a fundamental group (okay) then he makes it abelian (okay) and so on

but what i don't het is

whats this graph got to do with anything

i get that its an example

but like. Hows that a topological space

[long matrix]

fundamental group

[burnt ledge]

or rsther how does a top space become that

[long matrix]

loops

elements r loops

[neat lintel]

yeah

[long matrix]

base pointed loops

[burnt ledge]

the graph in question btw is a directed graph with 2 vertices x,y and 4 edges a,b,c,d all directed x->y

[long matrix]

mm.

[burnt ledge]

How does a top space become a directed graph. Where did directed graph come into this

[long matrix]

can u drop img

[burnt ledge]

Not here Lul

[long matrix]

u r weak

is @storm sage able to grant hax

math cheats

[long matrix]

ok nvm i think i have this

so uh there are a bunch of graphs

some of which are cayley graphs

im guessing this is... not?

yay i have a hatcher on phone

page? @burnt ledge

i take it its this

well ngl, reading the chapter before is probably helpful

So X1 is meant to be a topological space, a cell complex

uh it consists of 4 intervals glued at their ends, now x and y

[flat harbor]

id imagine its a representation of the different paths u can draw from x to y

[long matrix]

mhm

[flat harbor]

whether its abelian or not wuld depend on the actual topo space and the two points

[storm sage]

It's just an example of a topological space

If you want, think of it as a subset of R^2 (or R^3)

And you want to know how many holes it has

It's obvious how many holes it has for this example, but the point is that you can come up with a way to count how many holes in general

[mental stratus]

1,2,3,1,2,3 drink

[long matrix]

What does this abelianisation even mean tho

something i dont get

i mean yeah we can abelianize the group but what precisely is going on geometrically?

Our fundamental group was of loops going through the same basepoint. Now is it a group of loops, no basepoint involved???

[dapper badge]

Well the homotopy class of a loop naturally gives you a 1-cycle, hence a homology class

So this is sort of a natural map from the fundamental group to the first homology

The fact that the kernel of this is precisely the commutator subgroup of the fundamental group is maybe not entirely obvious, but I think there's at least geometric intuition for why the commutator lies in the kernel

Namely that taking the commutator of two loops involves going about a loop and later going about it in the reverse orientation, hence sort of cancelling the two out in homology where everything now commutes

[storm sage]

When you do homotopy groups you look at loops which you concatenate sequentially, one after the other. When you do homology, you instead zoom out and look at the big picture, irrespective of what loop comes after which one because it doesn't matter for the purpose of deteting holes.

[long matrix]

hmmm

[flint basin]

This is also alleviated in higher homotopy groups 🙂

Since they are all abelian

[burnt ledge]

i thought of thinking abt it as a subset of R² like as if it were a drawing but then it dodnt make sense why the edges would have a direction

[burnt ledge]

This could be helpful but i dont want this to get out of control, this is a slippery slope to committing ti reading the whole book and i cant be bothered to do that just for a silly little offhand question

[burnt ledge]

idk what a cell complex is, i guess thats my fundamental problem here

[long matrix]

u dont need to

see what eric said

[wet mortar]

I think the recommendation was a meme

[burnt ledge]

then why does the graph have directed edge.

i guess it doesnt ma- no nvm

[flint basin]

I think that's just for the discussion in the paragraph

It's not related to the topology on the space

[burnt ledge]

i wss gonna say it doesnt matter bc he gets rid of the direction right after but he doesnt do that he just gets rid of the order in which theyre traversed

[wet mortar]

https://m.youtube.com/playlist?list=PLcaesJ30fdQ_qyizYsFvlm9LkJvj2CxxU

watch this yt playlist tbh

[burnt ledge]

a whole hour long video 😞

[wet mortar]

speed it up

[burnt ledge]

i can kind of just mkve on from this question by saying here hes not doing homology topology hes just doing homology on this graph, to explain "the idea of homology", But im basically just mkre knterested at the moment now in the why does a directed graph behave the same way as a topology

[flint basin]

You can give graphs a CW structure is one answer

[burnt ledge]

Now what on earth is a CW structure

[flint basin]

Which just means that you can treat the edges as lines (with the usual topology of the real line), and vertices as points, and essentially "glue" them together to get a space the resembles your graph

[burnt ledge]

i can feel it coming btw i can see the moment where it all makes sense in close sight

[fringe summit]

Hatcher chapter 0 😭

[flint basin]

One can also look at homotopy and homology of graphs in their own right, which is a trendy research topic nowadays

[burnt ledge]

this still doesnt make it sense why its directed thoooiioooooo

[flint basin]

It's not

It's just for discussion

[burnt ledge]

yeah it is, al the edges go from x to y

md it would have been a different group if it...

[flint basin]

Because it's easier to write out what a loop in this space is when you label the edges with a direction

[burnt ledge]

No...

it wouldnt have been different...

because groups have inverses

so you can just pick whatever direction you like and it doesnt matter

so the directedness is just there so u can immediately forget about it, so its fine and i dont need to worry abt it

[flint basin]

Instead of writing 'consider the loop that takes the left middle edge going up, then the right edge going down', you can just say "the loop d^{-1}b"

[flint basin]

Exactly

[burnt ledge]

Okay this is cool

and graphs are like top spaces because you can just consider them as subsets of R² or whatever, or you can do a step in between and make them a CW complex (points joined by lines, and lines joined by areas, and so on, each optional), which are also like top spaces bc theyre just in between

[flint basin]

Technically not all graphs embed into R^2, only planar graphs

[burnt ledge]

yeag thats why i said or whatever

[flint basin]

But that idea works

[burnt ledge]

incase u cant in R² then buy some more space

[flint basin]

"buy some more space"

[fringe summit]

Download more dimensions free

[burnt ledge]

embed your K5 today

[mental stratus]

[burnt ledge]

Oh dear i got another silly question immediately ...

[storm sage]

The direction is for ease of reasoning about the space. It's not inherent to the space.

Like it's not a property of the space

It's just so that when Hatcher says "a", you know he means the path that goes from there to there in the picture

[storm sage]

Yeah it isn't important to know what a CW complex/cell complex is for this

[flint basin]

yeah what i said 🙃

[storm sage]

You can safely ignore those words as long as you understand the picture being drawn

[burnt ledge]

no i answered it, it really was that silly

[burnt ledge]

yea i was just wondering why one would use a directed graph to reason about a space if the graph were not relaged to the space. But ive been answered about why actually it is related to the space, so im happy now

so the homomorphism (partial symbol) sends each chain of edges to the sum of the vertices its entering

[storm sage]

it's called the boundary homomorphism

[burnt ledge]

and the next boundary homomorphism sends each 2-cell to the sum of the edges its entering too(?)

a cycle is not the same if its ths other way round right?

like a-b isnt the same as b-a?

[long matrix]

wouldnt think so

also 2a neq a

i presume

[burnt ledge]

yeag

he does say is the free abelian group

so when he attached A to a-b, would it be different if A were attached to b-a

does that even make sense, is that even meaningful

so bc his A is clockwise, a 2-cell at b-a would be ccw

and again everything only has a direction so that you can do groups with it, and you dont really care about it, so i should just shut up again

thank me for clearing that up

[long matrix]

well i take it the hole detection mechanism is related to the generators, but idk

[blazing pawn]

the specific choice of orientation does not matter, consistency of the orientation does

and being oriented vs unoriented matters

[burnt ledge]

yeag, so if i flipped his b around, then d1 would send a,c,d to y-x but b to x-y, so d1(a+b)=0, so a+b would be a cycle and so on, but everythinf else would be the same, makes total sense

you can just assign any of the lines any orientation you want but you only get one chance and then theres no backsies

Okay sure

[burnt ledge]

but then how this. Waaaaah i dont believe you any more

[long matrix]

automorphisms

[burnt ledge]

what

[burnt ledge]

How can being orientied or not oriented matter ! what would it be if the graph werent oriented then?

[long matrix]

choice of orientation corresponds to automorphisms, but that's only if we care about the orientation

if it werent oriented uh...

well that corresponds to a quotient right

but idk what the hell happens if u do that...

a = -a

thonk

no nvm me.

[burnt ledge]

it seems to me the whole time hes only only using orientation to more easily refer to the path that a loop or chain is taking

so then it wouldnt matter wheyher or not the groah to begin with were or werent oriented, because we just pick one choice of orientation, for our own convenience, and just work from there

[long matrix]

Well we should differentiate between going one way, and then going back the other way, I think is the point?

If we didn't, then going one way then back is the same as going one way twice

Is how I interpret this convo at least

[burnt ledge]

and then sums that make a cycle get sent to zero in the boundary homomorphism, so their inverses do too, so it doesnt matter which assignment of orientation we chose

[blazing pawn]

to coherently define "unoriented homology" you need to work with Z2 coefficients

[burnt ledge]

but then, eg, 2a-2b would just be zero

[blazing pawn]

yes

[burnt ledge]

Okay well then ill defer the idea of unoriented homology until later

[blazing pawn]

perhaps wise though its not any particularly harder to work with than Z

in fact its easier

its just weaker and less informative in some sense

[burnt ledge]

and the number of holes is the number of basis elements in the kernel

[burnt ledge]

right i see, and i can also see why

i think... or at least it makes sense to me why it might be weaker in some sense

[hushed sequoia]

my math question's unansered for 30 minutes

if anyone can help

[long matrix]

well rather than detecting how many times uve gone round something (and which way), u now only care if its an even or odd number of times

[neat lintel]

what channel?

[hushed sequoia]

help 10

[burnt ledge]

okay and so when we add the 2-cell now we have the Ker d1 (= all the cycles) / Im d2 (= all the cycles that are in A); so we treat all the parts of cycles that are just A as zero, because they can be homotopied away, so we quotient it away too, and then we are left w a group that has only two basis elements, so we have only two holes now

Bevause A filled the hole!

[fringe summit]

CW complex homology

Their eyes have been opened

[burnt ledge]

yes :D

[fringe summit]

quick teach them more before they realize

[burnt ledge]

its actually really cool that theres this neat method to count homology

like it's only obvious that graph has three holes because its drawn on a plane

but if for instance it was two poles of a sphere connected by 4 meridians

it would be much less ovvious that that has 3 holes

despite obviously being the same graph

[burnt ledge]

Hahaha, well, hatw to spoil the fun but this all started because i explicitly asked whats the deal with homology. So idve realised right from the start that homology is what is being sneaked into my brain

[fringe summit]

[fringe summit]

now, that's a fun notion of "obviously the same"

so this homology stuff shouldn't depend on how it's drawn, which is quite important

[long matrix]

ur a bad drawer

[prisma plinth]

still has no idea what you’re talking about

[burnt ledge]

wait o have a new question only tangentially related if at all

wait okay immediately solved

[long matrix]

[fringe summit]

based

[long matrix]

no fun for us

[prisma plinth]

WHAT THE FUCK IS A [HOMOLOGY] *america noises*

[burnt ledge]

all of these abelian groups are free, i was gonna ask, "i can understand how an abelian group 'has its coefficients in Z' but how could it have coefficients Z2", but then i realised that actually, its just free abelian groups have coefficients in Z, and for example Cn×Cn(abelian group) has coefficients only in Zn, so that answered my question

Just basically makingh sure my question was even right immediately anawered it

[fringe summit]

well, it has coefficients in Z since 5 * 1 in C_2 still exists

[burnt ledge]

Yea i guess

but helped it become immeidaly intuitive to me

[fringe summit]

coefficients in X is a statement of like, we allow the cycles to have coefficients in terms of X or smth?

hang on lemme grab it

[burnt ledge]

Wait holy and now i see how Z2 makes it unoriented because in Z2, +1=-1, so a-b is the same as a+b and so on, so theres no orientation anymore

[prisma plinth]

I don’t know any of the defs of topological homologies or anything

[fringe summit]

ye

[long matrix]

me neither, just intuit it

sotrue

[fringe summit]

congrats you are enlightened

[burnt ledge]

thats okay i barely even know the definitions of a topolgy at all

[long matrix]

[prisma plinth]

n-simplicies

what

[long matrix]

oh that makes so much sense exit

[burnt ledge]

okay, bevause i remember reading somewhere that abelian groups are like Z-modules, and Z is just a ring, and so instead of having them be abelian groups you can have it be an R-module for a different R, and because we have free ab grps we will have free R-modules

[fringe summit]

oh ok so simplices are like triangles

[prisma plinth]

wtf you mean “like”

[fringe summit]

anyway ignore meme wiki screenshots don't worry about that

[burnt ledge]

Yeag i won't

Z is the coolest ring i know of anyway

[fringe summit]

as in, [0] is a 0-dim triangle, a point

[1] is a line 0->1

[2] is a triangle 0->1->2

etc

[prisma plinth]

in R?

R^n

[long matrix]

oh was this not a joke lol

[jagged forge]

initial object 😎

[prisma plinth]

was not a joke

[fringe summit]

well you can draw em in R^n which gets you the topological simplices

[long matrix]

abelian groups are z modules arent they sotrue

[prisma plinth]

For char 0 integral domains right

[fringe summit]

for any ring

[long matrix]

maybe some conditions missing monke

[fringe summit]

\Delta^n the n simplex is the subset of R^(n+1) where (x_0, .... x_n) in \Delta^n iff they sum to 1

[prisma plinth]

hmm.

[fringe summit]

[n] the combinatorial simplex is basically {0, 1, ..., n} the linearly ordered set, draw em as 0 -> 1 -> 2... -> n and consider composing the arrows

the circle is orienting it since a < b < c

[prisma plinth]

….

oh

[fringe summit]

consider that these look exactly like what you get R^(n+1) from this

(all the x_i are >= 0 btw)

[prisma plinth]

so it’s oriented in the direction the order goes

But in the 3-simplex it goes a -> d -> c which isn’t right

[fringe summit]

the orientations of the boundary maps are a lil odd

[prisma plinth]

C O N F U S E D

[fringe summit]

so rewrite it as 0 < 1 < 2

a < b so it's oriented a -> b

b -> c

[prisma plinth]

Excuse my density but is a linear order a total order

[fringe summit]

yes

[fringe summit]

but a -> c yet the circle goes the other way

[long matrix]

pick one

[long matrix]

you either do want to be touched or you dont

[brittle socket]

N o

[neat lintel]

Is this the person everyone is touching?

[fringe summit]

so 0 -> 2 is in the opposite orientation as a boundary

[prisma plinth]

wtf you mean “as a boundary”

[fringe summit]

excluding vertex j we give it the orientation (-1)^j

boundary map

[burnt ledge]

Ah i c

[fringe summit]

this is related to homology btw

[prisma plinth]

so confused

So do the boundary maps almost permute the verticies

like in one specific direction

confused

[fringe summit]

the boundary gives you all the faces

[burnt ledge]

i just thought it was simple and wait holy crap id been distracted the whole time by the fact he only put two vertices that i forgot that what about if there were more

[fringe summit]

but they alternate directions as you exclude each vertex

[prisma plinth]

i just am confused about the combinatorical def vs the euclidean def

and which one is important for homology, or both

[fringe summit]

you like having orientations in both

[burnt ledge]

I GOT ACTUVE

YESSSS

I CAN POST

THANK YOU HOMOLOGY FOR BEING ADEQUATELY CONFUSING

[prisma plinth]

??????????

violent screaming

[fringe summit]

why wouldn't you?

how else do you get inverses

[prisma plinth]

so the topological map is from the euclidean n simplex

brain hurt

[prisma plinth]

is this a formal sum?

[fringe summit]

yep

[prisma plinth]

AAAAAAAA

[burnt ledge]

NOOOO

maybe it is actually worth to put Z2 and just do away with all these boudaries

[fringe summit]

this is based on topological spaces that can be interpreted as built out of simplices

[burnt ledge]

orientations*

[prisma plinth]

This is hurting my brain so much

[fringe summit]

here's the not-so-combinatorial one

[prisma plinth]

wtf do we care about the formal sum of the simplicies

All of this is not making any sense to me

[fringe summit]

thats how you make the homology into a group?

how else do you add simplices I mean

[prisma plinth]

so why are we bringing up the combinatorical one

[burnt ledge]

@prisma plinth

[prisma plinth]

Only more confused now

[fringe summit]

this is the topological one

[burnt ledge]

oh sorry

[prisma plinth]

so the topological one is the euclidean space of strictly positivr vectors who sum to 1

and it’s boundary is a union of lower simplicies

but what even is the boundary map, and why do we care about the formal sum

[fringe summit]

the faces, where you set one term to 0

[fringe summit]

thats how you get homology

[prisma plinth]

???

[fringe summit]

the boundary turns whole simplices into boundaries

[prisma plinth]

how the fuck would you even compute anything then

[prisma plinth]

like maps points in the simplex to their boundary?

But how

[fringe summit]

no

[prisma plinth]

WHAT

[fringe summit]

it maps the simplex itself to the boundary??

and you divide out by things which are the boundary of something

[prisma plinth]

???!!? Then what the fuck space is the simplex in

where is the map from

[fringe summit]

so you find holes (where a full triangle cannot fit)

[prisma plinth]

Like what is the domain and image of the map

[fringe summit]

the space considered is functions \Delta^n -> S?

[prisma plinth]

???????

[fringe summit]

it sends those functions (the n simplices in S)

to the boundaries

(sum of n-1 simplices in S)

[prisma plinth]

AAAAAAAAAA

…. So it’s a map on the free abelian group over Hom(\Delta^n,S)

but not a homomorphism?

[fringe summit]

it is

I mean it's defined on each basis element so like

obviously it extends linearly

[burnt ledge]

wait a secont i dont like this. So a singular 2-simplex is a cts map from Triangle -> X, but it doesnt have ro be injective? So if i want X=R³, the image of my singular simplex can be (cone without bottom)? that shouldnt be allowed to be a triangle

[prisma plinth]

The whole notion of “map from a simplex” fries my brain because why is this useful

[prisma plinth]

yeah I was also going to ask about it

it need not be surjective so

like I’m starting to get the simplicial structure but I don’t get the usefulness

[burnt ledge]

Also what channel would this belong in

#point-set-topology ?

[flint basin]

here is fine, but this is (alg) top, yeah