serious-discussion

[dusty wagon]

I don’t train legs at all anymore it hurts my knees

[sleek wing]

_ _

[eager reef]

why is that emote so small

huh

[sleek wing]

_ _

[eager reef]

leg day skipper

[sleek wing]

atm I've just been training legs by involuntarily having to bike up massive fuck off hills

[eager reef]

what helps my knees is that fucking machine

what is it called

[dusty wagon]

Makes my lower body lighter for more pull ups

[eager reef]

sus machine

abductors? or is that the wrong one

i keep mixing the two up

[dusty wagon]

Bruh adductors and abductors are the worst exercise

It’s for women tho

[sleek wing]

I can't actually do a single pull up but I can do like 10 chin ups it's really weird

[eager reef]

the one that trains your thighs

[eager reef]

thats not true

lol

[sleek wing]

it crushes my GIGANTIC BALLS

[eager reef]

exercises are never for any gender specifically

but that one really helps with knee pain

[dusty wagon]

I only see women do them

[eager reef]

also static holds on leg extensions

[eager reef]

yeah because a lot of men dont train legs sufficiently

lol

[dusty wagon]

Checks out

[eager reef]

just as a lot of women dont train upper body enough

[sleek wing]

not my fault bench is just the most fun objectively

[eager reef]

im currently training my gf to bench a plate

[eager reef]

this is true

objectively

[river moon]

If I have pain in my right shoulder after doing like 4-6 BB reps could it be that I'm doing something wrong or is it just my right shoulder being weak? I'm left handed and my left shoulder is nowhere near as tired as my right by the time I do the last reps, just wondering if somebody else had a similar experience

[dusty wagon]

Buff upper body looks unaesthetic anyway on women imo

[eager reef]

do you have a video of your form

[river moon]

not really

[eager reef]

upper body training doesnt just make you buff suddenly

[dusty wagon]

Have you tried doing DB instead

[eager reef]

it tones it if you arent eating in a surplus

[dusty wagon]

Consistency shows though

[eager reef]

plus i disagree

i find upper body muscles attractive

shoulder pain while BB bench can be for a lot of reasons

[river moon]

yeah and I do both, no such pain in case of DB but I still feel sometimes that my arms aren't equally strong in like DB or pull ups

[eager reef]

imbalances, improper form

wide grip is always a candidate

a lot of people grip way too wide

[river moon]

yeah I tried narrower grip and that helped

[eager reef]

i bench pretty narrow as well

i sometimes do certain variations where i grip wider but generally quite narrow

[eager reef]

then you might've let your shoulders flare out while gripping wider

maybe on one side more than on the other

(this was where my shoulder issues came from)

but its hard to speculate on it without further information

[bright hill]

I need a new workout routine

[eager reef]

also the shoulders flaring out is a bit of a controversial topic

willingly doing it is a technique in powerlifting afaik for example

so its wrong of me to say that its generally wrong to let the shoulders move while benching

[sleek wing]

oh my bench is narrow as fuckkkkkk

[eager reef]

but i had some issues from doing it so idk might be it

[dusty wagon]

I’ll start a 14 week internship on Monday kinda nervous ngl

[eager reef]

good luck!

[dusty wagon]

Thanks

I should just try to not fuck up

[eager reef]

what kinda internship is it

[dusty wagon]

Physical modelling and coding simulation software

[eager reef]

sounds neat

[dusty wagon]

I’ve never worked for such a long time in full time haha

• i also have to move 250km for the internship

I already have an apartment in the new city though I’ll go there tomorrow

[dusty wagon]

Or do any practical work at all

Only my HiWi-Job for 1.5 years or so

[eager reef]

sounds exciting

[dusty wagon]

The weekends will be boring probably because I don’t know anyone there

Time to study or something then

[eager reef]

or maybe try to meet some people

[bright hill]

How many exercises per workout is like, normal?

I'm used to doing 4 but the first workout routine I just opened has 9 each day

[eager reef]

depends on what you're training

and how many sets you're doing

[eager reef]

can you show me the routine?

[bright hill]

Oh, sure

http://simplyshredded.com/mega-feature-layne-norton-training-series-full-powerhypertrophy-routine-updated-2011.html

Just scroll down a bit

He rambles for quite a while

[eager reef]

well i mean

its a full upper day

so you do have to do quite a few different exercises

since its a lot of different bodyparts

[little vine]

A lot of those are only 2 sets

So it might not take as long as it looks

[eager reef]

dunno

im not the biggest fan of the plan i think

[dusty wagon]

Nah I’m only there for 14 weeks anyway and I don’t like going out alone

[bright hill]

But every day is like that

[eager reef]

i mean its normal

wait let me check

my routine

[bright hill]

Can you show me your routine?

The fitness wiki is so goated tho

[eager reef]

i can't actually

its from a creator i support

[bright hill]

I see

[eager reef]

and it'd feel wrong towards him to share his product on the internet just like that

yk

its paid for

[bright hill]

Yeah, it's fine

[eager reef]

i can give you the outline tho (amount of exercises etc)

the routine itself takes your maxes on the compound lifts and creates an individual environment

i can also give you other routines of mine that i used at some point

[river moon]

I have about 7-8 exercises 4 sets each and that takes about an hour and a half to complete, so 2 sets per exercise shouldn't be too bad if you choose weights which aren't too heavy for you

[eager reef]

i have around the same

but for less sets on some

[little vine]

Do you guys like the gym? I get so bored working out

I'm envious of the people who can live at the gym

[river moon]

I'm a CS student so I spent most of my day at my PC doing something, so gym being the only place I regularly go out to and do something different from my usual life and that helps make it more entertaining

it's also a big mood boost after I get back home for whatever reason physical activity makes you happier

[little vine]

That's for sure . I do feel better after, but getting there I have to make myself

[eager reef]

I genuinely enjoy it

Just the act of being there and working out

Idk

Exercise selection is what does it for me

[dusty wagon]

With the same playlist and same exercise routine it can get boring fast

But switch some stuff up then it’s ok

[eager reef]

If my back day is made off some single arm kneeling pulldowns or whatever

Yeah I find that boring

However barbell rows and pull ups? Slap

[vivid halo]

lmfao I need to think about exercise selection at my apartment gym

usually at larger gyms things are laid out so that you don't really have to think about it so hard

[little vine]

It's nice that it's close though. I just moved close and can walk to mine which has been a game changer

[bright hill]

It's not fun per se, I think

But after a while exercises become automatic and I have enough head space to think or listen to an audiobook

And that's really pleasant

[karmic field]

Is step harder than tmua?

[clear hawk]

Ive never been to a gym, but i used to workout at home during lockdown, that was in 10th grade, its been 2 years without any continuation....

[frail lagoon]

Yes, at least step 2 and 3

[gaunt latch]

song selection moment

[vivid halo]

yeah that too

[tawdry falcon]

👋 In your opinion, what are the prerequisites for multivariable calculus and topology, and which of those should I learn next?
Both seem very interesting topics and I think I may really enjoy them.

Currently I studied calc I and II and some real analysis, passively learnt some set theory and also took some boolean algebra classes in the past years of high school (up to de morgan's laws)

[mint canopy]

You're probably ready for both judging by what you say you've done.

Just choose one tbh, there's no obvious first one. Perhaps topology is the road less travelled of the two

[zealous garden]

yeah, the only thing I'd recommend is linear algebra but you can probably pick (the portions you need from) that right up as part of multivar

[tawdry falcon]

Ok, I'll watch a few lectures about both of them and see which one interests me more
Thank you so much, people

[little vine]

Topology is interesting but it's really difficult for me to grasp unless it's a metric space.

[mint canopy]

It gets easier

[little vine]

That's reassuring

[bright hill]

For multivar, you're missing linear algebra and for topology it depends on what kind of topology you wanna dig at

[zealous garden]

Good point

[bright hill]

Algebraic topology for example needs pointset and abstract algebra

[zealous garden]

pointset top you should be good for

[bronze wedge]

usually when you say "i want to do topology" you mean point set topology

algtop is boring until grad

[bright hill]

Algtop is grad tho

Isn't it?

[bronze wedge]

No

there are lots of undergrad algtop stuff

but they are combinatorical in nature

[bright hill]

Is hatcher grad algtop?

[bronze wedge]

yes

[bright hill]

I see

[bronze wedge]

theyre about like CW complexes etc

[bright hill]

That'd be awesome

(sry for taking too long to respond, my roommate arrived and I got distracted)

[bright hill]

I do 4 exercises 3 sets of 12 each

Except for deadlifts and squats for which I do 3x5

And it usually takes me an hour to 1.5

Maybe I rest too long

[tawdry falcon]

Oh I see
If linear algebra is super necessary for multivar, I'll try with point set topology and see how it goes, if I don't like it I'll go with linear alg

So many choices

[bright hill]

Both subjects are absolutely essential in almost every area in math

So you really can't go wrong

LA is way more fun imo tho

Have fun!

[little vine]

You're in for a treat if you study linear algebra

Those proofs are morsels

[zealous garden]

linear algebra is like ol bay seasoning

or frank's red hot

[little vine]

It's the garlic of math

[mint canopy]

breaking bad "jesse what the fuck are you talking about" scene here

[zealous garden]

"frank's red hot, I put that sh*t on everything"

the commercial

[tawdry falcon]

Random.org has spoken
It's time for linear algebra

[little vine]

Enjoy

Linear is such a nice property

[tawdry falcon]

I have no knowledge other than basic information about what each subject is about, so maybe it's not a good idea to overthink what to choose

[little vine]

I feel ya

Just got to pick

[mint canopy]

Definitely.

Just start

You'll find out if you like it or not once you've done a bit lol

[little vine]

Like I just had a topology test so I'm putting that down for a bit to catch up in algebra

[mint canopy]

I see so many people hemming and hawing about what book, what subject, what order

[little vine]

But if I get my study in every day it's somethin

[mint canopy]

and they end up spending more time just thinking about what to study than it would take to learn the basic definitions of those subjects

so just do it

and if you don't like it, just don't

[tawdry falcon]

Well, thanks for the solid advice
Cya!

[little vine]

Peas

[eager reef]

Im out rn, remind me when you see me next time in chat

[bright hill]

Sure

[tender tulip]

what are divisors on a reimann surface

[burnt dune]

a divisor is any finite linear combination on the integers

over the surface

algebraic geometry basically

haha

[tender tulip]

How is this useful

also apparently they form a group, which I assume is just the free abelian group

[burnt dune]

yes ur right

[burnt dune]

they are useful in geometry

like u look at some functioons that have poles at these point

or zeroe

i dont remeember tbh

it was as mall section for me in compplex analysis didnt go deep

a small*

[grim pivot]

Hey, quick question. is rotation a valid subject for asking math help?

[burnt dune]

yea sured

[grim pivot]

Aight, I'll dive in a help channel!

[sick burrow]

is there a way to graph f(x,y)=0 on desmos?

but like with a pre-defined f

[storm sage]

what are you trying to graph

[sick burrow]

well for instance I could put in y-x^2=0

[gray snow]

Bro when help

[sick burrow]

that's easy

that works

[storm sage]

yea

oh like you defined f on a separate line

[sick burrow]

but if I put in f(x,y)=y-x^2

[gray snow]

How the fuck I know I didn't pick too hard topic

[storm sage]

is there any way you can inline it?

[sick burrow]

then on a separate line put f(x,y)=0

no graph

[storm sage]

desmos is rather.. particular

[sick burrow]

what I'm trying to do is make a program that shows how projective transformations change a line

[storm sage]

the desmos server will probably be more knowledgable

they know all the weird tricks

[sick burrow]

is there a desmos server?

[storm sage]

https://discord.gg/u2AGqTpr

[sick burrow]

that's the correct one?

there isn't an official desmos?

[storm sage]

I don't think there's an official desmos discord server

but yea that's the correct one

[sleek wagon]

,calc 9^100

[fathom swallow]

Result:

2.6561398887587e+95

[sleek wagon]

hm

,calc 100^9

[fathom swallow]

Result:

1e+18

[sleek wagon]

,calc 100^99

[fathom swallow]

Result:

1e+198

[sleek wagon]

,calc 100^1000

[fathom swallow]

Result:

Infinity

[sleek wagon]

how do I get this to 1e+900

,calc 100^90

[fathom swallow]

Result:

1e+180

[sleek wagon]

bro

[chilly coral]

,calc 10^900

[fathom swallow]

Result:

Infinity

[chilly coral]

You can't

It's too big

[storm sage]

#bots

test it here

[onyx idol]

If you had to do the pde du/dx<vd^2/dy^2 and find the domain what would you do? I know you can use function decomposition at du/dx and vd^2/dy^2 being equal but how would that play a role into their discontinuous points?

[dusty wagon]

,w 10^1000

[fathom swallow]

Results provided by WolframAlpha

Click here to refine your query online
Upgrade to WolframAlpha Pro!

Requested by Theta#9545

[dusty wagon]

@sleek wagon

[sleek wagon]

yes?

woah

[neat lintel]

Is it possible to prove that a function is constant on an interval if the derivative is 0 everywhere on this interval, without using the mean value theorem?

Omg hi i saw 💬

[deep mango]

If you know the fundamental theorem of calculus, you could use that.

I think there are also ways to prove that the only function with derivative 0 is a constant, using properties of the product rule and stuff. I don't remember exactly how to do that

[tender tulip]

Actually, lets start with this idea @golden pendant

Do you know Euler's formula

[golden pendant]

following you

e^ix=cosx+isinx

[tender tulip]

$e^{xi} = \cos(x) + i\sin(x)$

yeah

[fathom swallow]

Mizalign

[tender tulip]

well, what does this curve look like when traced out on C

(it's fairly simple)

[golden pendant]

circle

But in order to "find" them, we isolate them by integrating f(z)(z-a)^n around a closed path in C, i am still lost on this

[tender tulip]

I'll get to that

[golden pendant]

find what

Ok sry continue

[tender tulip]

In total

$e^{x+yi}$ is another way of showing polar form

[fathom swallow]

Mizalign

[tender tulip]

where the radius is e^x

[golden pendant]

and e^yi is the unit circle

[tender tulip]

So now we have this under our belt

now the next thing to do is

What happens if we try to integrate 1/z^n around the unit circle

[golden pendant]

we write z^(-n) as re^-ion and do the integration

[tender tulip]

Yep

[golden pendant]

on o from 0 to 2pi

[tender tulip]

z = re^it, dz = ire^it dt

dz/z^n = it * re^(n-1)it

now we have an integral we can work with

$i\int_{0}^{2\pi}{e^{(n-1)ti} dt}$

[fathom swallow]

Mizalign

[tender tulip]

Well, the amazing shit here is that

[golden pendant]

Ok i think i am getting where this is going

Still i need a few more pieces

[tender tulip]

Well,

that integral

is 2πi ONLY for n = 1

and 0 otherwise.

[golden pendant]

fucking right

[tender tulip]

which is awesome

[golden pendant]

is there a reason for that, other than the mathematical calculatiion

[tender tulip]

Natural log shit that's hard to explain

[golden pendant]

is it because its an orthonormal base?

[tender tulip]

yes

[golden pendant]

or something like that

[tender tulip]

e^x is a fantastic base

for C and you can do hilbert shit with it

Chebyschevs come from it

[golden pendant]

okay then keep going

[tender tulip]

but yeah

[golden pendant]

wdym

[tender tulip]

SO

[tender tulip]

the primative/antiderivative of 1/z is ln(z), but because e^2πi = e^0, there is "multiple branches" or multiple inverses, each differing by a multiple of 2πi, and because you integrate "all the way around" back to where you started, you collect that 2πi term in the end as you "step up" a branch. This doesn't happen with 1/z^2 or higher, because it's just scaled by -nz

the complex part of ln(z) looks like a helicoid, so when you go around, you go to the next "floor", like a spiral staircase

[golden pendant]

oh waw

okay keep going

[tender tulip]

So now,

We can say every function with countably many isolated "singularities" (points that explode to infinity) can be represented around these singularities in a "ring" shape via that laurent series

[golden pendant]

okay

[tender tulip]

so we can just set z -> z - a to move the singularity to 0, and represent it by

$f(z) = \sum_{n = -\infty}^{\infty}{c_nz^n}$

[golden pendant]

Kinda like taking the limit of (z-a)^nf(z)/n! for the taylor series, i also want to know wdym here cause ik taylor series but i am not sure what taking the limit is

[fathom swallow]

Mizalign

[tender tulip]

Taylor series use a "different method" to essentially isolate the coefficients via linearity

[golden pendant]

yeah

[tender tulip]

But,

so if you essentially take the integral of f(z)/(z)^(n+1),

[golden pendant]

we isolate coeficient by the derivative, and then evaluating at 0 no?

[tender tulip]

yea

[tender tulip]

it will return 2πi * c_n

which we can just keep doing this for ALL the coefficients

when we integrate AROUND 0

but here's the fucking AWESOME part

We can substitute z for re^2πti, right.

in both the integral and series

and almost like magic

We have the complex fourier series.

which you can do some algebra via euler's formula if we know the function is Real for the reals, aka real coefficients to derive the real fourier series

Ultimately, the fourier series is like finding the laurent series for f(ln(z)) in a way

Instead of considering a "disc" or a ring around a point

we can represent it as a sliver of the complex plane, the real value being the log of the radius, the complex value, the angle

and find the function of the polar form of the aforementioned values

Which this fact was awesome for me when I first realized this

I realized this one day during Precalc

[golden pendant]

i am still lost

trying to fit my head into all of this

[tender tulip]

Okay so there's also the orthogonal basis way

which honestly the whole fact that we have BOTH sine AND cosine confused me

the answer was complex functions.

but essentially

We do the same orthogonal basis thing

I'm trying to think of the best way to go about this

Okay so

@golden pendant you still here

[golden pendant]

Here you jumped

[tender tulip]

I have an idea

[tender tulip]

I realized there's a better way

latex time

[golden pendant]

Keep going

[tender tulip]

Essentially

the reason is

The integral distributes over addition, right

[golden pendant]

Nvm i understood it

[tender tulip]

yea

[golden pendant]

yes

[tender tulip]

it's like for taylor series

[golden pendant]

everything will go to 0

yeah

[tender tulip]

You want to know something else sick

[golden pendant]

yes

[tender tulip]

We can define an inner product on L^2 C functions I think

[golden pendant]

whats L^2

[tender tulip]

functions that are square-integrable

[golden pendant]

oh okay

i know square sum series

[tender tulip]

the absolute value of it's square you can integrate

[golden pendant]

Okay keep going

[tender tulip]

$\int_{0}^{2\pi}{\frac{u}{v}dt} = u \cdot v$

if u and v are functions of t

WELL

let u = e^nxi, v = e^mxi

[fathom swallow]

Mizalign

[tender tulip]

It's 0 unless n = m

aka

inner product :)

[golden pendant]

yes

I learned another inner product, its the integral of the complex times the conjugate of the other

are those 2 related?

just the coefficients are different right?

[eager reef]

subtle flex

[tender tulip]

i fucked it up

[tender tulip]

At this point I had 3 years of self study

[eager reef]

what age is precalc

[tender tulip]

I was 16

[eager reef]

ah

[tender tulip]

yeah

this was what I meant

Well

[golden pendant]

Ah

[tender tulip]

the conjugate of e^ix

is, well

e^-ix

[golden pendant]

ok i am listening then

[tender tulip]

so we essentially have an orthogonal basis

e^inx forms an orthogonal basis.

[golden pendant]

yes

[tender tulip]

soooooo

back to that series form

$\sum_{n=-\infty}^{\infty}{c_n e^{inx}$

right

[fathom swallow]

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

[tender tulip]

well, that's like representing a function in the "basis" of the e^inx's

and we use the inner product (LIKE THE DOT PRODUCT)

to get the coefficients

by taking f(x) dot e^inx

and basically, doing some algebraic manipulations gets the sine and cosine forms

since after all

[golden pendant]

yeah

[tender tulip]

$\cos(x) = \frac{e^{ix} + e^{-ix}}{2}$

[fathom swallow]

Mizalign

[tender tulip]

$\sin(x) = -i\frac{e^{ix} - e^{-ix}}{2}$

[fathom swallow]

Mizalign

[tender tulip]

so essentially

the "sines" and "cosines" are just "grouping" terms with their negative

and shortening it using sine and cosine, which are purely "real for real" complex exponentials

[golden pendant]

couldnt we have used the same logic without complex? arent cos(nx) and sin(nx) also an orthonormal basis?

[tender tulip]

Yeah, I never understood why there's like, 2 "basises"

sines and cosines

turns out they're just kinda "paired exponentials" which are the TRUE orthonormal basis

[golden pendant]

okay

thanks a lotttt. Ill try to impress my prof monday xd

[tender tulip]

I remember looking at the sines and cosines and remembering euler's formula

and going

w a i t

[tender tulip]

okay so, lets say we have a multivariate polynomial F(x,y)

such that F(x,y) when viewed as univariate polys of x or y has nonzero discriminant

And we consider the projective curve of it

can we find a group law for it

IF

we “solve” the differential equation F(y(t),y’(t)) = 0, y(a) = b

and find an addition formula for y

[golden pendant]

who are you talking to?

[tender tulip]

And substitute

no one in particular, seeing if anyone knows

[golden pendant]

AH i thought you were explaining

[tender tulip]

oh god this is going to be a pain in the ass

mmmm i don’t think you can reverse this

[sterile summit]

Hello, on what website can you learn ks3 and ks4 maths for free?

Year 9, 10 and 11

[woven whale]

what's ks3/ks4

[sterile summit]

Key stage 3, 4

Which encompasses what I wrote below the original comment

Ks3 is year 7 8 and 9

[tender tulip]

so I’m going to do some absolutely whackass calculus here

[sterile summit]

But idc about 7 and 8

[woven whale]

heh?

that tells me literally nothing

[sterile summit]

I describe it below

I just told you what the abbreviation is

[woven whale]

'year 9 10 and 11 content' doesn't tell me anything either

you gotta name topics and stuff like that

[mint canopy]

This is the UK curriculum, illuminator

[sterile summit]

Yah

[mint canopy]

It's very clear for anyone in the UK what is meant.

@sterile summit The best I can find is this textbook: https://mrvahora.files.wordpress.com/2014/10/gcse-1-9-textbook.pdf

I'm not sure what the context of your question is, but perhaps you should just find a teacher

[tender tulip]

is this even possible to solvr

[frozen merlin]

,rotate

[sterile summit]

yah that's cool, but i was thinking something like this: https://uk.ixl.com/maths/year-9

problem is, it's behind a pay wall

thought someone here might know an alternative

[mint canopy]

It's the best I could find.

[sterile summit]

yah it will be good enough I think. My situation is a little weird rn, but what I was mainly trying to learn was terminology and word problems.

so I think this'll be fine in that reagrd

thanks

[tender tulip]

Let M_C be the group of meromorphic functions on C under multiplication. Is this group isomorphic to the direct product of free abelian group of points of C AND C

since every meromorphic function is uniquely determined by it’s poles and zeroes up to a scaling

[next dew]

Say you have probabilities for N events and for each combination of 2 events, you determined some events are conditionally dependent.
How do you know if is it safe to remove one of the conditionally dependent events from the data used for analysis?
Say, A and B are dependent. How would I know if I would lose some information if I removed A or B? Do you look at all the other probability tuples and see if they are interchangeable? (Like P(A|C)= P(B|C)), and if not, then neither can be removed, or what?

[next dew]

For example, D and E are conditionally dependent

P(D) = 0.57
P(D | E) = 0.57
P(E) = 0.47
P(E | D) = 0.47

and P(C | E) does not equal P(C | D)

P(C) = 0.58
P(C | E) = 0.66
P(E) = 0.47
P(E | C) = 0.53

and

P(C) = 0.58
P(C | D) = 0.62
P(D) = 0.56
P(D | C) = 0.60

Does this mean statistical analysis would be less effective without E or D

If P(C | E) = P(C | D) and P(E | C) = P(D | C), it would be safe to remove either in cases where I only want to analyze other event's relationships to C?

[open aspen]

you should probalby post this in a math channel

otherwise youll get clowned on

like this:
mathematicians cant analyze their own relationships, you expect us to analyze yours?

[severe swallow]

People who decided to pursue pure math during the later stages of college... do you sometimes ask yourself, "wtf am I even talking about?" And if so, how often

[neat frost]

Wdym “wtf am I even talking about?”

[deep mango]

sure, probably every few weeks

but usually i'm used to it

especially because i spend a lot of time thinking about tactile pictures of the things i'm working with

[tender tulip]

yo stf is a principal divisor

i know for a reimann surface a divisor is just a formal sum of points on the surface but what makes it principal

[severe swallow]

You have to feel it to know what I mean.. sometimes the things you do feel already super niche, even though the rabbithole goes down so much deeper... and I guess you sometimes also ask yourself how much the things you do matter in reality... its hard to describe

[neat frost]

Like I think I understand what you’re asking

But why the word talk

[tender tulip]

rabbit holes more like mental zip bombs

[severe swallow]

Because you tend to talk about math as well. Not only write. This was more metaphorical though. Also talking in your head counta

[neat frost]

Oh I don’t

Okay but I see your point

[severe swallow]

Tactile pictures?

[neat frost]

No I’ve convinced myself that my degree is the best possible degree for my future

And I am therefore 100% comfortable with it

[severe swallow]

This is not my point. Dont get me wrong. I think its also the best choice for me. Its not that I dont like it. These are seperate thoughts.

[deep mango]

like the point of this is that the objects are abstract and it's hard to come to grips with working with things that aren't really real or have any physical embodiment, right?

[neat frost]

Oh you’re having an existential crisis?

Those are fun

Enjoy them

[deep mango]

but if you take time to develop that kind of intuition for the work you're doing, it doesn't feel like that anymore

[severe swallow]

This isnt quite what I was getting at. It sounds like you're talking about developing an intuition for abstract objects

[deep mango]

well in my eyes being unable to cope with working with abstract objects all day is where the feeling comes from

[severe swallow]

For me its kind of like.. "why am I comfortable with this?"

[deep mango]

and when i ask that, the answer provided by my intuition to me "look, you can see it all right there, it's working as intended and it's real. that's why you're comfortable with it."

[severe swallow]

See it where? I still have a hard time getting an idea of what tactile pictures are

Maybe cuz I'm not a native english speaker

[deep mango]

Hmm

It's like objects I see in my mind which move around and do whatever, almost as though I'm planning to draw a picture of them

but not actually going through with the drawing

[severe swallow]

Well ok for a basic example. What does that do when you have a group?

[deep mango]

I view the group as acting on an object

I mean, I don't care about algebra anyway

I'm an analyst so topology is always important and often well-structured

Idk how algebraists do this aside from seeing things move around diagrams

[velvet dagger]

I feel like my intuition is more audial than visual somehow

And I think about a lot of objects in terms of what they do

[odd narwhal]

Aural btw

[neat frost]

Good morning shin!

[burnt dune]

i have no intuition

i just write down definitions

and use theorems

🙂

[untold sage]

I don't like the word intuition. It's so vague

[eager crescent]

"intuition" just means knowing what is going on

So yeah it's super vague

[deep mango]

well when i do problems i can tell pretty clearly the difference between "i'm mindlessly symbol pushing and throwing spaghetti at the wall" versus "i have a picture of what i'm trying to do, and i have a plan for how to get there"

whether or not i can properly define those states of mind is not super important to me

the only time i ever feel like "wtf am i doing with my life, what is this nonsense" is when i'm working the former way, and that's a sign for me that i need to figure out how to transition my understanding towards the latter way

and then usually i feel good about it

[untold sage]

The second one just seems like you're pretty sure you know how to solve the problem

[eager crescent]

Also it's wrong to think that "intuition" is innate or something. Intuition is something you can work on, by thinking more deeply about the concepts rather than just manipulating symbols like Ryc described

[burnt dune]

how many times did your picture of what your trying to do actually help you reach the correct order of symbolic pushing to prove something?

[untold sage]

I think intuition just means geometric understanding most of the time

[burnt dune]

not alot for me

[deep mango]

probably like 3 or 4 times a week at least

[burnt dune]

only like 3 times

[eager crescent]

Not necessarily geometric

[burnt dune]

wow u must be so smart

how do u picture stuff in like

[deep mango]

i can barely ever solve a problem if it's symbol pushing

[burnt dune]

advanced stuff

[deep mango]

it's not about being smart

it's about knowing that i 100% need that

the smart people are the ones who can do fuckin diagram chasing and shit and keep track of what's going on

[eager crescent]

Someone with intuition will generally be able to reformulate the problems/concepts using a variety of analogies

[burnt dune]

yea i meant those examples

like okay sometimes when doing some point-st toopology yea i can think of R and like balls and shit

theyre easy

[deep mango]

idk all the math i do is physicsy, it's like PDEs and differential geometry and fourier analysis and shit

[burnt dune]

but like

[eager crescent]

So trying to reformulate the definition/theorems is a good way to train your intuition

[burnt dune]

in algebra and stuff they are just like no way

exact sequences

[deep mango]

i don't do things that can't be visualized

[burnt dune]

tensor products

[deep mango]

you can visualize that stuff though

like

understanding how something manipulates symbolically

is semi-visual for me

the symbols move around in my head and i get comfortable for how and where they're supposed to move

[burnt dune]

isnt that symbol pushing

but in your head

instead of on papepr

paper*

[zealous garden]

yeah that's not really visualization

[burnt dune]

intuition is just putting what ur doing insidee an easier framework

an analogy

inuition is analogy

[untold sage]

The main point of writing stuff down is so you don't have to try and remember it

[burnt dune]

yea ig

[eager crescent]

No, that is simplifying too much

[burnt dune]

idk what inuition can be other than that tbh

[zealous garden]

no, the main point of writing stuff down is to train the information deeper into your brain by forcing you to play a performative role in the learning effort

[burnt dune]

people confuse "knowing alot of theorems" with intuition

[eager crescent]

Intuition is just having a deep insight into the material

[burnt dune]

like i know that this must be false cuz i know this would contradict some shit that i know

is that intuition xd

[untold sage]

I agree

[zealous garden]

That's logical deduction innit

[burnt dune]

yea tahts what i want to say

[untold sage]

Well it all just depends on defintions

[burnt dune]

all math is just symbolic pushing

deductive reasoning really

[untold sage]

How you define intuition

[severe swallow]

Bruh.. well that can make things a little more... intuitive

[zealous garden]

I would say intuition is first and foremost, informal

[burnt dune]

intuition is just some cheat way to get insights on what order u must do to prove something

[zealous garden]

it might formalize to something that is just true, but it's informal

[eager crescent]

Paul lévy would be a counterexample that comes to mind. He often forgot theorems which did not have his direct attention at the moment apparently, and he was renowned for his intuition in probability

[burnt dune]

and for me the more u go in algebra the harder it gets

[burnt dune]

maybe he is just a genius + probability is fairly intuitive ig

idk probability but

maybe im saying this cuz i like algebraa

and i got this feeling when i learnt about tensor prodcuts of modules + exact sequences

[eager crescent]

I mean he was considered a genius yes but he also lamented about the lack of place dedicated to working on intuition in education

[zealous garden]

I'm gonna ask VMM if probability is intuitive next time I see him

[burnt dune]

what intuition do u get about that lmaooo

other than understanding them in another field

that would be cheating

cuz that means u know about them beforehand

just not that in that particular langauge

haha

so yeah

math is bascially symbolic logic

no wonder its incomplete bs

lmlao

[deep mango]

Why not?

Understanding things by imagining picking them up and moving them around in the right spots

[zealous garden]

if it's still just symbols instead of some concept or structure the symbol represents you're not really visualizing it are you? that's like saying you visualized a book when you were imagining (literally) the words on the page

[untold sage]

Another quite different defintion of intuition is subconscious reasoning

[deep mango]

like if you show me an SES i imagine a function coming from the left embedding something into a space, and then the quotient on the right, and then i can pick things up from each part of it and move them around

[burnt dune]

yea

thats not visualizataion

thats just u being smart

[deep mango]

but the space and the functions and stuff are represented by the symbols

[zealous garden]

that's just symbolic manipulation though

[deep mango]

i feel like our definitions of visualization are different

[burnt dune]

its like ur saying u visualize numbers when multiplying when no ur just using ur brain as paper

[zealous garden]

I'm certain they are

[deep mango]

but i don't do that

[burnt dune]

ur just smart

[deep mango]

if you tell me two numbers

i don't need to visualize them to multiply them

they just multiply

i don't need to move them around or feel them or anything

sure, i might add up some numbers in my head

but i'm not like

[eager reef]

Give me two numbers to multiply and I’ll give you the wrong result

[deep mango]

imagining the process of adding on paper

[eager reef]

That’s my special talent

[zealous garden]

I too have my times tables as well wired shortcuts in my brain

but how do you visualize multiplication

[eager crescent]

From Paul Lévy (in french) :```Il m'est souvent arrivé de demander que l'on fasse dans l'enseignement un plus grand usage des méthodes qui favorisent l'intuition, et de m'entendre répondre qu'il s'agissait avant tout de développer le sentiment de la rigueur. Cette réponse implique un malentendu qui m'étonne toujours; c'est à mon avis une erreur d'opposer ainsi l'intuition et la rigueur. L'intuition favorise la découverte et aide à comprendre; une fois qu'elle nous a suggéré un énoncé, il s'agit de le démontrer rigoureusement. Mais l'intuition est aussi ulile pour arriver à ce résultat que les yeux le sont au promeneur qui pourtant se servira finalement de ses pieds pour arriver au but, et il est aussi absurde de la combattre qu'il le serait de recommander à ce promeneur de fermer les yeux pour arriver plus sûrement au but. L'intuition et le sentiment de la rigueur sont deux qualités qui se complètent, et s'il arrive parfois, je le reconnais bien volontiers, que le développement excessif de Tune chez certains savants nuise à l'autre, ce n'est pas une raison pour ne pas les cultiver toutes les deux, et le rôle de l'éducateur doit être de développer le sens de la rigueur chez ceux à qui il fait défaut sans pour cela prendre l'attitude de combattre le sens de l'intuition partout où il le rencontre.

Une des meilleures manières de développer l'intuition est sans doute de développer le sentiment du parallélisme entre la théorie abstraite et son image concrète, et de favoriser le langage qui suggère cette image. Cela ne doit en rien nuire à la rigueur. Quelquefois deux images différentes sont possibles.```

[zealous garden]

if you stop to think of it as a concept

[eager reef]

[burnt dune]

exactly

[untold sage]

Repeated addition on the number line lol

[burnt dune]

good question

no1 has intuition for the number 7 lmao

[deep mango]

i imagine an area usually, or some kind of dimensional analysis

[burnt dune]

u just symbol push it

its just a symbol

[burnt dune]

no way

hahaha

[zealous garden]

for example, I generally think of multiplication as a sort of nesting

[deep mango]

well that's what it always means

so why wouldn't i do that

oh or i think of it as a composition of functions

[zealous garden]

like multiplication is what happens whenever you take every "point" of object A, and make it a (displaced) copy of object B

[deep mango]

like, sometimes it's most useful to think of multiplication of numbers as stretching one number by the other

[burnt dune]

multiplication is just multiplication

its too abstract

to have intuitoin for

even if we have some vague analogy

no 1 can use this analogy

to do actual problems

u just symbol psh

and only after u get ur result

[bronze wedge]

depends on the field

[deep mango]

that's not how i work at all

[untold sage]

People seem to use "abstract" and "vague" interchangeably

[burnt dune]

then you can say oh this makes sense cuz of <insert analogy>

@deep mango yea cuz ur smart

[bronze wedge]

multiplication is not "too abstract" in any way

[zealous garden]

I don't think multiplication is too abstract to have an intuition for

[burnt dune]

well excuse my word ig

but maybe

too simple

too structureless

[untold sage]

Literally just repeated addition tho

[burnt dune]

like elementary set theory proofs

[zealous garden]

multiplication is structureless

[burnt dune]

they are just too easy

[bronze wedge]

Algebraists working in [insert name] algebras in shambles

[eager crescent]

Part of the message translates to:
One of the best ways to develop intuition is no doubt to develop a sense of parallelism between the abstract theory and its concrete image, and to favorize the language which favorizes this image. This must by no means affect rigor. Sometimes two different images are possible.

[burnt dune]

proving some set theory identity

yeea you can have some intuition using venn diagrams

or some shit

but to do acutal work u just let x be here

show its there

with the right order of symbol pushing

thats all i want to say

or ask

[untold sage]

I suppose rigour always involves symbols

[zealous garden]

yes, language is unavoidable if you want to express your ideas

but I'm concerned in trying to figure out what the ideas themselves are like

[burnt dune]

my point being is the higher u go the harder intuition will be of good use to produce a proof

[zealous garden]

in your head, how you experience thought, contemplation, meditation, and problem-solving in mathematics

[bronze wedge]

probably the least intuitive fields are the more "mathematical physics heavy" stuff

[zealous garden]

is it really just mechanical symbol pushing to you?

[eager crescent]

It's not cheating at all. If you know of something from some other field, it can help with intuition

[burnt dune]

yes

most of the time

only after i have proof

or only after i solved a problem

only then i can think about what it actually means

but i cant reverse it

i cant think about something visually or like have intutiion

then the magical proof pops out

that just doesnt work

at all

[zealous garden]

"magical proof pops out" is rarely a part of the story

[burnt dune]

yea thats all what i want to say

math is symbol pushing

untill its not

[zealous garden]

unless you already know where you're going

[burnt dune]

yea thats my point

[untold sage]

Getting mvt from rolle for example is very visual

[burnt dune]

proving Hom is left exact functor

i wont forget this one

when i did that for the first itme

i was like ????????? XD

symbol pushing literally

literally symbol pushing i swear

[zealous garden]

I think I get what you're saying now

[burnt dune]

yea

shamroock helped me

[eager crescent]

Doing things for the first time is rarely intuitive

[burnt dune]

proof that an epimorphism is something in set category

it was literally symbol pushing

but once i get the result

and prove it

only then i can attach to it the analogy that fits my idea

of what it means

[dapper badge]

I think part of it is that these were probably the first examples you considered when exposed to things like epimorphisms

[burnt dune]

if f has left inverse its an epimorphism

[eager crescent]

The first time you walked, it probably wasn't very intuitive to baby you and yet here you (probably) are walking without a second thought.

[dapper badge]

These are very abstract notions and of course you won't have intuition for them the first time you see them

[burnt dune]

"of course you won't have intuition for them the first time you see them"

[bronze wedge]

they tend to be interested in very specific spaces (usually spaces of generalized functions) and as such (without physical intuition) it is hard to understand why they chose those spaces in the first place

[burnt dune]

such sad truth

ur right

but doesnt it feel sad

[dapper badge]

Not really

[burnt dune]

cuz thats what i literally said "math is symbol pushing untill it isnt"

[dapper badge]

That's not what I'm saying

[burnt dune]

wdym

[eager crescent]

It's not symbol pushing

[dapper badge]

I'm saying it takes a lot of work to generalize a concept to the point where we can define something like an epimorphism

[eager crescent]

Symbols are just a way we use to communicate

[zealous garden]

yes

[dapper badge]

And the road to that generalization isn't paved with symbol pushing, but with intuition, examples, connections, and a lot of hard work

[eager crescent]

You can just "choose" a simpler language and then translate it into mathematical symbols if it makes it easier

[burnt dune]

so your saying

[eager crescent]

Not many mathematicians think solely in terms of symbols

[burnt dune]

that to do this

people go from intution to symbol pushing

[eager crescent]

They think in terms of concepts most of the time

[burnt dune]

?

like thats how they like defined a topology for example?

give an example

[untold sage]

What does symbol pushing even mean

[burnt dune]

plus im talking for my own pov here

[zealous garden]

yeah this term has been used a lot without a proper definition

[untold sage]

That's how a lot of debates tend to go

[burnt dune]

symbol pushing

[untold sage]

And it's all just pointless

[burnt dune]

like set theory proofs

u can do all of them

without picturing shit

just let x be here

then x is in here and here

etc

just symbol pushing

symbolic logic

ig

[zealous garden]

I visualize set theory

[untold sage]

I'd say that's just being rigorous

Also it requires a lot of reasoning

And thinking

[burnt dune]

yea good for you but my problem is ur visualizations most probably wont help you prove something elementary , u will first start with symbol pushing

[zealous garden]

yeah that's not just symbol pushing

[untold sage]

Even though you can't visualise it

[burnt dune]

okay lets give examples

for my own pov

it started with topology

i could actually visualize this shit

but then

as the problems get more harder

visualizing this shit wouldnt help me and the solution would just be "use uryshons lemma on this weird absurd bla bla"

it has to do with the fact ( we all know this ) is that alot of beautiful theorems that make absolute sense have the most weird shit proofs in human kind

do u get me

also

[zealous garden]

there are very weird proofs sure

[burnt dune]

alot of things are just not visualiziablee XD

things in algebra

for sure

[untold sage]

Ig if you wanna come up with some of those proofs yourself, just try stuff

[zealous garden]

I refuse such a proposition

[bronze wedge]

topology is when visualization ended for me lol

[burnt dune]

every measurable function has a sequence of simple functions converging to it

yo this shit makes sense for you right?

but the proof is just some weird fuckery magic sequence of ismpled functions

that came out of nowhere

[bronze wedge]

that seems pretty easy to visualize

[burnt dune]

yes

exactly

ur visaulization wont do u jack shit to prove it

[zealous garden]

it might show you where to start

[burnt dune]

yea for the easier problems thats true i agree

connectdness problems for me in point-set ig

they were like easy to envision and shit

but as i get more deep that just doesnt work for me

[eager crescent]

I disagree. While it is possible to do just that and prove elementary theorems by just "pushing" symbols logically without much intuition, you can also just understand what is going on at each step.

[untold sage]

Sometimes you just do stuff even when you don't know where it will take you

[burnt dune]

yes but only AFTER your done with the step

[bronze wedge]

while you may think it is "black magic" or whatever, once you start to understand the structure/etc of these types of things it kinda comes naturally (eg. in probability often to prove that the set of omega s.t. a process is continuous is in a sigma algebra, you use a countable union over the rationals (which doesnt make sense when you look at it for the first time))

[burnt dune]

@untold sage amazing point

i agree too

and infact

its most of thee time not only sometimes for me

[untold sage]

Like people just decided why not make this connection between elliptic curves and modular forms and boom you have flt

[burnt dune]

yes but thats exactly what walter said and that takes years t

thts likee after being ap rofessional

this god feeling

i know it

i had it with basic group theory where i knew if something was right or wrong

[eager crescent]

No, not after. Just pushing symbols can limit you in your reasoning, since there can be multiple logical developments from a statement

[burnt dune]

but that was like after tons of cramming lamo

lmao*

like someone whos a profeessor in analysis would probably have this feeling

[eager crescent]

I'd argue that understanding what is going on is more important that just mindlessly pushing symbols

[burnt dune]

but im talking about average intelligence students like me

who are learning this shit for the first time

[eager crescent]

Ok

[burnt dune]

u would be right

i am tlaking in the context of solving problems

[eager crescent]

Then doing it over and over again (cramming) can be one way to come to understanding

[burnt dune]

u can understand but not solve problems

[eager crescent]

But not the only way

[burnt dune]

do we agree?

or are u all grotheendeick

[eager crescent]

Just taking the time to think a lot about basic definitions and properties can help

[burnt dune]

idk man

it helps in a large scale

but when doing examples or problems

[zealous garden]

yeah symbol pushing is usually how I'd refer to symbolic manipulations with little motivation or forethought, or that are carried out mechanically/mindlessly

[burnt dune]

idk

maybe im stupid

[eager crescent]

There is no universal answer of course

[burnt dune]

as @untold sage said, just doing shit and not knwoing wheere ur ssupposed to go

[zealous garden]

yes

[untold sage]

You may define symbol pushing as something that a computer can do (without AI)

[burnt dune]

thats right too

[zealous garden]

Agree

[burnt dune]

is it

a coincidence

that

computers can do proofs better than we all can

yet we have intuition XD

[zealous garden]

no

we have intuition

intuition can fill the gaps

leading to bad proofs

computers are dumb

they have no intuition

[eager crescent]

Visualizing is not the ony way that intuition manifests btw. Just knowing that a certain type of problem can be solved via Urysohn's lemma, to take your example, qualifies as having intuition for this certain type of problems.

[zealous garden]

so they must reach it through rigorous bricks of logic

[burnt dune]

so do we

how do u prove something with intuition

[zealous garden]

by having it fill a gap without you noticing

[burnt dune]

yes this is just experience and developing the bag of tricks

after solving many exercises

100% true

[burnt dune]

and i ask again : does this really happen XD

like does this really happen that often

it rarely happens with me

tbh

especially with exercise problems

and probleems on qualifyign exams

[untold sage]

That's just subconscious reasoning tho

[burnt dune]

not big general problems or theory

maybe ur refering to research math

[zealous garden]

haven't the foundations undergone revolutions when we decided that intuition filling a gap was no longer acceptable?

[burnt dune]

yees

algebraic geometry did go through one

the italians were like this

and didnt provee jack shit

[zealous garden]

lmfao sounds right

[burnt dune]

ik

[eager crescent]

It does if you have no gaps in your foundational knowledge.

[burnt dune]

wiki the italian school on algebraic geometry

[eager crescent]

Not with all problems ofc

[burnt dune]

thats just extreme cuz no 1 has no gaps

learning new math is filling up ur gaps in older math

are we talking about the same avg intelligent student here?

[zealous garden]

I have no idea what "average intelligence" means

[eager crescent]

I suspect that you just don't realize that you're often filling in the gaps, but if it really never happens to you, that may be a sign that you need to focus on your foundations

[burnt dune]

it happens

but on the easier problems

once the problems get more hard and abstract

i just yolo it --> grab my textbook look for theorems of similar ideas

and try to put things together

commutative algebra

is 99% like this

literally

[eager crescent]

Imo you need a good grasp of all the main theorems in the domain you're studying

[zealous garden]

expand this statement

[burnt dune]

maybe ur right

im a below avg student ig

[zealous garden]

Don't put yourself down

[burnt dune]

do u know functional anlaysis

[eager crescent]

If you have to grab your textbook it means you haven't assimilated them perfectly (and it's fine, working with my notes in front of me is something I do sometimes as well to speed up assimilating them).

[zealous garden]

not yet

[burnt dune]

okay

do u know measure theory

[burnt dune]

yea working with notes infornt is so much better to leanr the stuff tbh

[zealous garden]

No, but I'm familiar with some of the ideas

[burnt dune]

hmm okay

here is an example

do u know commie algebra

[zealous garden]

no

[burnt dune]

okay u know real analysis right?

[zealous garden]

yes

[burnt dune]

at baby rudins lvl

okay so like

say u have this problem

and u see this vaguee sense of mean value theorem

like u see the main expression

f(a)-f(b)/(a-b)

so i look for the mean value theorem (incase i forgot it for example )

and try to piece shit out

brute forcing the logic

untill i have something that is rigourous enough to be called a proof or a solution

thats what i mean by piece shit together

i dont have any intuition whatsoever

i dont understand jack shit

[candid cloud]

Done

[burnt dune]

i may after i did the problem

[candid cloud]

Can someone please help me

[burnt dune]

or i may even beforee doing the problem but that wont help me

yo u know

[eager crescent]

If you have a sense that it looks like mean value theorem, then you have some intuition on the problem

[burnt dune]

grothendeick one said " no 1 must prove shit unless this shit is obvious to him or her "

[candid cloud]

I’ve answered some just let me know if it’s right plz

[zealous garden]

#❓how-to-get-help

[candid cloud]

5 is a rational number

[eager crescent]

There're varying degrees of intuition

[burnt dune]

this wouldnt be a good quote unless the stateement that most peoplee prove shit that is not obvious is true

do you get me

[eager crescent]

And as I said earlier, it's something to work on. People rarely have lots of intuition when learning a topic. If you did a few more similar problems, it would probably become second nature to find the solution

[burnt dune]

yes i will definitely try to work more on examples

examples build stuff

computations are far more important than theory

[eager crescent]

Yeah, though everyone works different

You mentioned Grothendieck earlier, and he's an extreme example but alledgedly he was often unable to think of even simple examples in mathematics

[burnt dune]

damn he stupid

lmao

how did he pass exams

[eager crescent]

[zealous garden]

lmfao

[burnt dune]

most exams are like

produce a counterexample to this

or true or false

how did he do them

or did he just write im gorthendeick lmao and pass

[eager crescent]

There's a that famous moment (legend?) of Grothendieck being asked to give an example of a prime number and he said something obviously not prime

I dunno if this is just a legend though, cuz iirc some say the number was 57 and other people said it's one number or another

[zealous garden]

bootstrap paradox

[burnt dune]

yea i know that one

thats so me

im grothendeick

[eager crescent]

Then why you say you need examples

[burnt dune]

im just joking

i want to pass exams

and do problems

grothendeick wont do shit to help me

[zealous garden]

what do you mean by "do problems"

[burnt dune]

do exercises

problems that u see in an exam

not research problems

[zealous garden]

just homework and exams?

[burnt dune]

yes

research is still far from me

and i wont be able to do research if i cant pass shit

[zealous garden]

it's never too soon to start reaching

even if you know you can't reach it yet