/mg/

[Anonymous No.12176144]

Math, generally
>hidden sympathies edition
Talk math.

[Anonymous No.12176251]

>>12166845

[Anonymous No.12176252]

First for von Neumann.

[Anonymous No.12176264]

[Anonymous No.12176274]

>>12176252
>It's opposite
Fuck's sake.

[Anonymous No.12176292]

>>12176144
TSUKOYOMI!

[Anonymous No.12176293]

>>12176274
Wow, I had never noticed that.

[Anonymous No.12176294]

>>12176269
http://www.ms.uky.edu/~carl/ma330/html/egyptgeo.html

[Anonymous No.12176304]

>>12176252
>the statement itself, or if it states something false, its opposite
If the statement is unprovable, how could we know whether it states something true or false? I would say "unable to prove either certain statements or their negation." I think this covers it.

[Anonymous No.12176306]

>>12176264
pls more

[Anonymous No.12176360]

>Robert Lee Moore (November 14, 1882 – October 4, 1974) was an American mathematician who taught for many years at the University of Texas. He is known for his work in general topology, for the Moore method of teaching university mathematics, and for his racist treatment of African-American mathematics students.
hello based department?

[Anonymous No.12176373]

What's the difference between a norm and a metric? Does the scalar multiplication property make a big difference anywhere?

>>12176360
Sounds like a faggot if the claim isn't just slander

[Anonymous No.12176376]

>>12176360
How sad

[Anonymous No.12176383]

>>12176373
>if the claim isn't just slander
It most definitely isn't. The man hated the idea of blacks doings maths.

[Anonymous No.12176389]

>>12176304
>>12176252
Wait, he's a platonist though.

[Anonymous No.12176390]

>>12176304
There are true statements which cannot be proven.

[Anonymous No.12176395]

>>12176306
I don't have more. Read the book, Logicomix. It's great, a biography of Russell.

[Anonymous No.12176401]

>In 1973, the University of Texas honored him by giving the name Moore Hall to a new building housing the physics, mathematics, and astronomy departments. Because of Moore's racism, in one of the changes following the George Floyd protests the University changed the name to Physics, Mathematics, and Astronomy Building.[1]
>>12176373
He was a texan whose father fought on the Confederacy side in the american civil war. I think it's highly likely but that shouldn't erase or overshadow his contributions.
To answer your question, a metric applies to two points and is defined as(iirc) [math]||x-y||[/math]. In a sense a metric is like a generalization of a norm. Also each norm induces a metric, but not every metric is derived from a norm.

[Anonymous No.12176424]

>>12176373
A norm only makes sense if you have a vector space. Metrics are on metric spaces, where there is not necessarily any such thing as scalar multiplication.
Even on a vector space, it makes a massive difference. A norm on a vector space (over Q, R, or C) will always lead to an unbounded metric space, but it's very easy to get a different metric which is actually bounded. for example, if || || is a norm, the metrix defined by d(x, y) = ||x - y|| / (1 + ||x - y||) is bounded.
Note that every homogeneous, translation invariant metric is actually given by a norm, if you just set ||x|| = d(0, x).
There are plenty of metrics which are not homogeneous or are not translation invariant, like bounded ones or ones which expand/contract certain parts of your space irregularly.

[Anonymous No.12176427]

>>12176401
>a metric applies to two points and is defined as(iirc) ||x−y||.
If this is what your notation for a metric is, it's shit notation. Not every metric space comes with a group structure, x - y might make no sense.

[Anonymous No.12176431]

>>12176390
Proof?

>>12176401
>Texan
I agree it shouldn't overshadow his contributions. I am racist on a statistical level, but being racist to individuals who have shown competence, is just retarded

[Anonymous No.12176440]

>>12176431
https://www.miskatonic.org/godel.html
ctrl+F for Rudy Rucker

[Anonymous No.12176444]

>>12176427
sorry, it should've been a "can be defined as" since I was referencing a norm

[Anonymous No.12176478]

>>12176440
I'm gonna have to think on that for a while

[Anonymous No.12176492]

Does anyone have experience applying to German universities as a foreigner? There's this thing called the DoSV and it's unclear whether it applies to foreign applicants or only natives.

[Anonymous No.12176559]

thoughts on non-boolean logic with LEM

>>12176360
moore method is unorthodox, but much closer to a work environment than any class, would be interesting to see it used today

[Anonymous No.12176586]

How can I get back into math?
I'm shit at calculations. I don't have interest in calculus. My interests are in abstract algebra,logic, set theory, proofs.
I hate computer science so nothing like that to be honest.

[Anonymous No.12176621]

>>12176586
If you want to do abstract algebra you're gonna need to be good at doing calculations and having a decent knowledge of calculus will be helpful, although it's not strictly necessary if you avoid certain kinds of questions or algebraic structures.

[Anonymous No.12176628]

>>12176559
What do you mean? Just essentially talking about functions into a set of truth values with LEM in the metalogic?

[Anonymous No.12176643]

>>12176621
that wasn't what I ran into at all during education. I enjoyed it and didn't do any calculus. I was the only humanities kid int those classes lol

[Anonymous No.12176652]

>Geometry by Brannan, Esplen, & Gray
>Geometry: A Comprehensive Course by Dan Pedoe
>Introduction to Geometry by Coxeter
How do these books compare? Which one do you recommend?

[Anonymous No.12176655]

>>12176643
Even if you took a relatively easy introduction to abstract algebra you should've done good amount of computation to get anywhere. Didn't you see any examples of groups or rings coming from analysis?

[Anonymous No.12176657]

>>12176655
yeah I can do computation just not calculation

[Anonymous No.12176703]

>>12176657
What?

[Anonymous No.12176778]

>>12176360
based

[Anonymous No.12176785]

>>12176652
I didn't read any of them and I'm stupid as shit but I will give you my opinion because this is /sci/
Pedoe is grad level according to the intro and from going through like 2-3 chapters somewhat terse
The Brannan book requires LA and basics of Group Theory but has appendixes for them
I don't know anything about the Coxeter book. Also I won't comment on the cost of the books because you will likely pirate them anyways, probably

[Anonymous No.12176830]

>>12176264
This is great

[Anonymous No.12176884]

>>12176360

[Anonymous No.12176911]

>>12176830
Just in case: https://readcomiconline.to/Comic/Logicomix/TPB?id=124918#1

I just started it. Seems neat

[Anonymous No.12176951]

is it obvious that if [math]\pi[/math] is a cycle of length greater than 2 that [math]\pi^{2}[/math] is not identity?

[Anonymous No.12176959]

>>12176884
>racist treatment of African-American mathematics students.

Being shitty to the actually decent blacks is immature. He'd probably mock fatties at the gym for trying to better themselves.

[Anonymous No.12176977]

>>12176959
I'm just memeing bro.

[Anonymous No.12177048]

>>12176951
Is it obvious to you that if pi is a cycle with pi^2 = identity, then pi has length 2?

[Anonymous No.12177074]

My prof gave an axiomatization of incidence geometry and two of the axioms are logically equivalent
-_-

[Anonymous No.12177076]

If [math]M,N \trianglelefteq A,B[/math] respectively, is [math] M*N \trianglelefteq A*B[/math] where * is the free product?

[Anonymous No.12177113]

>>12177076
Not in general
If A and B are the infinite cyclic groups generated by <a> and <b> respectively and M and N are <a^2> and <b^2> respectively, then a^2b^4 for instance would be an element of M*N. If you conjugate it by a which is an element of A*B you would get a(a^2b^2)a^-1 = a^3b^2a^-1 which can't be an element of M*N so M*N isn't normal
Sorry about no TeX I can't get to my computer right now

[Anonymous No.12177162]

>>12176144
Lets say we come up with an explicit formula for prime numbers . What are the implications in math and the sciences ?

[Anonymous No.12177179]

>>12177162
a polynomial time algorithm?

[Anonymous No.12177200]

>>12177074
-_-

[Anonymous No.12177204]

>>12177179
yes

[Anonymous No.12177211]

What area of physics has the higher chances of being used to develop a nondeterministic computer?

Given how quantum computers cannot really tackle NP-Complete problems after all...

[Anonymous No.12177229]

>>12177211
>What area of physics has the higher chances of being used to develop a nondeterministic computer?
mine desu

[Anonymous No.12177235]

>>12176360
Can the Moore method be realistically used today?

[Anonymous No.12177240]

>>12177235
It's impossible. The students will just use the internet to find proofs.

[Anonymous No.12177252]

>>12176360
Kant wrote that "[Whites] contain all the impulses of nature in affects and passions, all talents, all dispositions to culture and civilization and can as readily obey as govern. They are the only ones who always advance to perfection.” He describes South Asians as "educated to the highest degree but only in the arts and not in the sciences". He goes on that Hindustanis can never reach the level of abstract concepts and that a "great hindustani man" is one who has "gone far in the art of deception and has much money". He stated that the Hindus always stay the way they are and can never advance. About black Africans, Kant wrote that "they can be educated but only as servants, that is they allow themselves to be trained". He quotes David Hume as challenging anyone to "cite a [single] example in which a Negro has shown talents" and asserts that, among the "hundreds of thousands" of blacks transported during the Atlantic slave trade, even among the freed "still not a single one was ever found who presented anything great in art or science or any other praiseworthy quality". To Kant, "the Negro can be disciplined and cultivated, but is never genuinely civilized. He falls of his own accord into savagery." Native Americans, Kant opined, "cannot be educated". He calls them unmotivated, lacking affect, passion and love, describing them as too weak for labor, unfit for any culture, and too phlegmatic for diligence. He said the Native Americans are "far below the Negro, who undoubtedly holds the lowest of all remaining levels by which we designate the different races". Kant stated that "Americans and Blacks cannot govern themselves. They thus serve only for slaves."

[Anonymous No.12177255]

>>12177252
Kant was an opponent of miscegenation, believing that whites would be "degraded" and the "fusing of races" is undesireable, for "not every race adopts the morals and customs of the Europeans". He stated that "instead of assimilation, which was intended by the melting together of the various races, Nature has here made a law of just the opposite". He believed that in the future all races would be extinguished, except that of the whites.

[Anonymous No.12177258]

>>12176264>>12176830
>>12176306

>>12176252
Reminder that Tarski truth is not truth and logic is just game of truth as symbols.

[Anonymous No.12177290]

If I have a 3D Plateau's problem with two parallel squares being the boundaries, the solution is a prism. If the boundaries are two parallel circumferences, the solution is a cylinder. However, I'm interested in the case where one boundary is a square and the other is a circumference. Is the analytic solution for that case known?

[Anonymous No.12177312]

>>12177258
>yes I'm a logician, how'd you know?
>truth? Basedjack what are you talking about these are symbols on a piece of paper
>haha basedjack come play this game I made up. You start with a few symbols on a paper and you can use these rules to fight down more symbols. The goal is to try to write down as many as you can. Wanna play?
>look skyjack, in this game I found a strategy to write down any bunch of symbols you want. I think I won!
> what do you mean principle of explosion? Invalid? Basedjack you're no fun

[Anonymous No.12177320]

>yes I'm a logician, how'd you know?
>truth? Basedjack what are you talking about these are symbols on a piece of paper
>haha Basedjack come play this game I made up. You start with a few symbols on a paper and you can use these rules to write down more symbols. The goal is to try to write down as many as you can. Wanna play?
>look basedjack, in this game I found a strategy to write down any bunch of symbols you want. I think I won!
> what do you mean principle of explosion? Invalid? Basedjack you're no fun

[Anonymous No.12177361]

>>12177074

-_-

[Anonymous No.12177482]

>first proposition of the paper
>"can be proved rather easily"
>i can't prove it

[Anonymous No.12177506]

>>12176360
A real hero.
Would probably be cancelled today.

[Anonymous No.12177507]

>>12177482
post proposition, let's see if /mg/ can do it

[Anonymous No.12177515]

>>12177507
page 1

[Anonymous No.12177517]

>>12177507
page 2

[Anonymous No.12177581]

>>12177517
Just give me the number to the suicide department

[Anonymous No.12177638]

>>12177162
Depends what form the formula has exactly. If it's something we can work with easily then it'd give the answer to a lot of unsolved problems. If it's something that allows us to factor large numbers then a lot of current cryptography is broken (although we have replacements).

[Anonymous No.12177643]

>>12177515
>>12177517
What have you tried?

[Anonymous No.12177653]

Take an arbitrary convex polygon and triangulate it, as seen in the image (it doesn't have to be a fan triangulation, any triangulation will do). There are lots of possible triangulations, but how do you find the triangulation with the largest standard deviation of triangle areas without trying every permutation?

[Anonymous No.12177655]

>>12177162
I found the formula.

[Anonymous No.12177698]

>>12177517
>Be french
>Gâteaux literally means "cakes" in french
>Skip through the article to check the proposition
>Suddenlly read about a cake-differentiable function

I thought I was having a stroke

[Anonymous No.12177782]

>>12177581
bro... don't give up...
>>12177643
i spent another 2 hours on this cancer and figured out everything except "frechet differentiability implies continuity of p(x)"
well at least i can say i made a decent effort today

[Anonymous No.12177783]

What is your favorite isomorphism theorem /mg/? Mine is the third one.

[Anonymous No.12177812]

Just started Lebesgue integration at uni. I want to kill myself.

[Anonymous No.12177830]

>>12177812
Lebesgue integration is great, just takes time to settle in your head.

[Anonymous No.12177834]

>>12177783
The numbering isn't totally standardized. Which one do you mean?

[Anonymous No.12177850]

>>12177783
I've been using the second one the most lately, so let's go with that.

[Anonymous No.12177887]

>>12177698
Made me smile, ha

[Anonymous No.12177892]

>>12177812
I didn't learn Lebesgue integration. Nevertheless, I also want to kill myself.

[Anonymous No.12177905]

>>12177892
You didn't learn Lebesgue integration?

[Anonymous No.12177908]

>>12177850
Again, what theorem is that?

[Anonymous No.12177910]

How does pi times radius squared actually get the area of a circle
t. 110 iq brainlet

[Anonymous No.12177915]

>>12177910
>>12176294
Are you gonna ask this every thread?

[Anonymous No.12177921]

>>12177915
Yes I want to know

[Anonymous No.12177924]

>>12177908
For me, they are:
1st [math]G/\ker(f) \cong \text{im}(f)[/math]
2nd [math]HK/H \cong K/H\cap K[/math] or [math](M+N)/M \cong N/M\cap N[/math]
3rd [math](G/K)/(H/K) \cong G/H[/math]

[Anonymous No.12177928]

>>12177910
there's no point in asking such questions here
google it, and i'm sure you will find multiple very detailed and illustrated explanations

[Anonymous No.12177931]

>>12177924
>no correspondence theorem

[Anonymous No.12177960]

>>12177931
Highly observant from you. Do you want medal or a kiss on your cheek?

[Anonymous No.12177963]

>>12177921
Have you considered reading the answer that was already spoonfed to you?

[Anonymous No.12177967]

>>12177960
¿What do you look like? I only flirt with trans girls if they're cute.

[Anonymous No.12177993]

>>12177967
Well, I'm unironically pretty horrendous inside and outside. However, this is off topic also, so don't hit on me, silly boy(s).

Some people were talking about locales earlier, this one's for you:
>Towards Higher Topology
https://arxiv.org/pdf/2009.14145.pdf
This could be of interest to someone:
>Construction of free differential algebras by extending Gröbner-Shirshov bases
https://arxiv.org/pdf/2009.13847.pdf
Homotopy stuff:
>The universal six-functor formalism
https://arxiv.org/pdf/2009.13610.pdf

[Anonymous No.12177998]

>>12177993
I'll just take the medal in that case. :/ Also please don't assume my gender.

[Anonymous No.12178009]

>>12177910
because Pi is defined as the number for which this is true

[Anonymous No.12178013]

>>12178009
That is wrong.

[Anonymous No.12178021]

>>12177993
Personally I'm not interested in the papers you posted right now but that's a good spirit to make something out of mg

[Anonymous No.12178056]

Is there some systematic way for me to double check my own proofs when studying, even if it only works about 70% of the time? I don't want to flood a form with proof verification requests nor do I want to try and monopolize my professor's time basically asking if I did it gud. All I find on the subject is 'be skeptical' but I find it hard to be skeptical when I might be so shit at the subject I don't even recognize I'm dog shit

[Anonymous No.12178068]

>>12178056
Find a friend to study with.

[Anonymous No.12178072]

>>12177998
Sorry, my friend.

>>12178021
<3

[Anonymous No.12178075]

>>12177921
Anon, look at this picture. Each ring has a length of 2pi * its specific radius, and an area approaching that of just multiplying by its width/radius for really small radius jumps (larger radius jumps don't mimic a rectrangle exactly because the inner portions have a smaller radius/length so it's not the same as varying a line segment along an axis, but radius varies negligibly at the limiting case). Each ring is a fixed radius apart so each one changes the circumference/length by a certain amount, and the length change goes to 0 with really small radii jumps. So lining up all these thin, infinitesimaly varying segments, we get a triangle whose base is 2pi*r, r being the outer radius. What is its height? Only the sum of all the radii jumps, the total width of rectangles, so r. By area of a triangle, (2pi*r*r)/2 = pi is squared... I mean the united states are... I mean pi are squared

[Anonymous No.12178089]

>>12178056
Honestly, I find that descartes' method of "anything you perceive clearly is true" works pretty well. Just go carefully over each claim you make. Test some examples afterwards/during too for sanity

[Anonymous No.12178100]

Is there a law of large numbers for the median? Is it the case that
the median over n iid. random Variables converges to their expected value in some sense?

[Anonymous No.12178154]

>>12178100
For continuous random variables, the sample median should converge to the median of the distribution, if that median has a positive density.

[Anonymous No.12178174]

>>12178100
We define the median by declaring that
[math] M(X_1, ..., X_n) \geq \epsilon \iff |\{1 \leq k \leq n: X_k \geq \epsilon\}| \geq \frac{n}{2}. [/math]
The question is then if for [math]X_n[/math] iid. with [math] \mathbb{E}(X_0)=\mu[/math] we have that [math] \lim_{n} M(X_1, ..., X_n) = \mu [/math] almost surely.
We assume WLOG that the expected value is 0.
Let us assume the opposite is the case. Then there must exist an [math] \epsilon > 0[/math] for which [math] \mathbb{P}(\{Y_n \geq \epsilon \text{ for infinitely many k }\}) > 0[/math], where [math] Y_n = |\{1 \leq k \leq n: X_k \geq \epsilon[/math].
We could get a contradiction via Borel-Cantelli if we can show that
[math] \sum_{n=0}^\infty \mathbb{P}(Y_n) < \infty [/math].
What now?

[Anonymous No.12178186]

>>12178174
>What now?
All that's left to do now is Wake me up when September Ends

[Anonymous No.12178244]

>>12178174
Why don't you try a numerical experiment with a skew distribution, so that the mean doesn't equal the median?

[Anonymous No.12178246]

just enrolled in a maths msc with 0 research experience and no idea what to write my dissertation about but it's gonna be fine, right?

[Anonymous No.12178248]

>>12178174
Hey, I think it holds given then [math] \mathbb{P}(\{X_1 \geq \epsilon\}) < \frac12[/math], i.e. that (including the negative case) the individual variable has a median of zero.
In this case we set [math] \mathbb{P}(\{X_1 \geq \epsilon = p\})[/math] and have that
[math] \mathbb{P}(\{Y_n \geq \epsilon\}) = \mathbb{P}(\{\exists m \geq \frac{n}{2}, I \subset \{1, ..., n\}, |I| = m: X_i \geq \epsilon \forall i \in I\}) = \sum_{m = \frac{n}{2}}^n {n \choose m} p^m \leq {n \choose \frac{n}{2}} n p^n \leq C 2^n n^2 p^n [/math], which is summable. (Stirling's approximation was used in the end).
[math] \mathbb{P}(

[Anonymous No.12178255]

>>12178244
yes I realized I was retarded, it can't work without further conditions.

[Anonymous No.12178298]

>>12178246
Yes.

[Anonymous No.12178329]

thoughts on reverse polish notation?

[Anonymous No.12178335]

>>12178329
Parenthesis exist, no need to replace perfectly good notation with retarded compsci bullshit.

[Anonymous No.12178337]

>>12178335
There is nothing wrong with compsci.

[Anonymous No.12178346]

>>12178337
Indeed. CSs classes are one of the only places to pick up cute traps with like zero competition from the other guys.

[Anonymous No.12178351]

>>12178337
Get out

[Anonymous No.12178354]

Is there anything between a vacuous truth and the principle of explosion?
>vacuous
>"if x is in {}" aka if a contradictory situation is true
>explosion
>given that a contradictory situation is true
But doesn't "if" act like a local "given"?

[Anonymous No.12178358]

Is there any way to become smarter for maths?
Or are the tisms gifts that naturally come from god?

[Anonymous No.12178361]

>>12178358
Whatever you do don't fall for the study meme.

[Anonymous No.12178365]

>>12178351
but computational complexity is based
https://arxiv.org/pdf/2009.13949.pdf

[Anonymous No.12178367]

>>12178358
>Is there any way to become smarter for maths?
Watch Michael Penn videos religiously.

[Anonymous No.12178372]

>>12178358
each person learns at a different pace. you learn most math by doing math, so crack open a book and get to it.

[Anonymous No.12178379]

>>12178367
Michael Penn seems kind of smart but also kind of retarded. A lot of times I notice him overcomplicating stuff, taking a lot of time for proving something that a simple quick observation would solve.

[Anonymous No.12178384]

>>12178372
But I want to be a genius.
I don't want to learn at a slow pace like a pleb. That's what the plebs are for. I want to rise above them.

[Anonymous No.12178386]

>>12178384
Then you need memes my friend
Dank memes

[Anonymous No.12178392]

>>12178354
Are they giving people computer privileges at the insane asylum now

[Anonymous No.12178395]

>>12178384
then fumble your way to the top. just don't cry when you come crashing down.

[Anonymous No.12178408]

>>12178361
Explain.

[Anonymous No.12178413]

>>12178395
Geniuses don't fumble.

[Anonymous No.12178426]

>>12178367
>Watch Michael Penn
No watch Penn & Teller

[Anonymous No.12178428]

>>12178413
you couldn't be more wrong.

[Anonymous No.12178503]

>one of my letter of recommendation just died of corona and now I only have two good ones
oh god I'm fucked

[Anonymous No.12178526]

I studied Pure Mathematics 8 years ago at university level.

Is there a good study list for Analytical Number Theory?

As I want to learn and study Riemann Hypothesis.

[Anonymous No.12178621]

>isomorphism theorems
>not just isomorphism theorem and isomorphism corollaries

[Anonymous No.12178670]

>>12176959
IT'S JUST A PRANK BRO

[Anonymous No.12178687]

>>12178379
true, but I think his point (most of the times) is to use a more abstract reasoning. For example see https://math.stackexchange.com/questions/337937/why-sum-k-1-infty-frack2k-2 the second answer gives you a elementary way to solve the problem, the first abstract it with generating functions.

[Anonymous No.12178689]

>>12178351
I love beowulf dude

[Anonymous No.12178712]

>>12178392
I'm not insane. Just think about what I wrote. An "if.. then" statement just has you imagine a situation where the proposition is true, a situation which is equivalent to a universe where the proposition is taken as an axiom.

Funny thing is, I actually have been in a mental hospital before. Just for two days though, I used my charisma to convince them I didn't need to be there. Truth is, mental wards are shite.

[Anonymous No.12178741]

>>12176360
welp time to withdraw my application to UT

[Anonymous No.12178753]

>>12178741
"-October 4, 1974)"

[Anonymous No.12178789]

>>12178753
They didn't scrub everything he did there off the face of the earth
Going to UT is tantamount to supporting him every time he made an African American's student's life worse
Fuck him and Fuck UT

[Anonymous No.12178799]

>>12178741
>>12178789
>Implying a nigger like you could enter university

[Anonymous No.12178801]

https://youtu.be/60OVlfAUPJg

Whorah math marines!!!

What is your profession....

MATH!!!!!

[Anonymous No.12178802]

Guaranteed (You)s.

[Anonymous No.12178882]

>>12177834
sorry my dumbass thought it was standardized but mine is the same with the anime poster
>>12177850
I don't think that I completely understood the second isomorphism thm tho. What are you using it on?
>>12178056
Reading my proofs again and again until they don't mean anything works for me. I think finding a friend to study with is really important for at least undergrad math, so you can check each other's proofs.

[Anonymous No.12178928]

Is this a good book for category theory?

[Anonymous No.12178936]

>>12178928
I skimmed it when it was posted. It's meant to be light reading for laymen, it's not a textbook. If you're looking for a basic category theory text you still can't go wrong with Mac Lane.

[Anonymous No.12178938]

>>12178928
lol

[Anonymous No.12178941]

>>12178928

[Anonymous No.12178955]

can i ask you smart asses a question about my homework?

[Anonymous No.12178966]

>>12178928
she's a militant feminazi but everybody defends her because she uploaded a few basic category theory videos back in 10 years ago when there were next no online resources.

[Anonymous No.12178971]

>>12178955
Sure.

[Anonymous No.12178976]

>>12178955
You just did.

[Anonymous No.12178979]

>>12178882
I constructed a freaky chain complex and was interested in its homology. Start with a module, take its injective envelope, then project onto this injection's cokernel, then quotient out all the fixed points under a certain subgroups action on the cokernel. Then take the injective envelope of this quotient of the cokernel and define your differential to be the composite of these two projections and finally the injection. The image was easy because that is just two surjections followed by an injection, but the kernel was a bit more tricky. It is actually the sum of the kernel of the first projection and the preimage of the kernel of the second projection, call those [math]K_1, K_2[/math]. Then I wanted to consider the quotient [math](K_1 + K_2)/K_1 \cong K_2/K_1\cap K_2[/math]. There were some technicalities here, like how I wanted a certain subgroup to act trivially on this, and indeed it does. The trivial action was far from obvious in the sum form but the action is trivial on [math]K_2[/math].

[Anonymous No.12178980]

>>12178966
>next to no online resources for category theory in 2010

This is what Zoomers actually believe.

[Anonymous No.12178981]

>>12178971
can i compose the function f: N -> Z with itself? im guessing not because Z "is not in the domain" with N.

[Anonymous No.12178996]

>>12178980
No, anon, we know.

[Anonymous No.12178999]

>>12178354
Given a set [math]A[/math], let [math]\phi_A(x)=x\in A[/math].
(Conversely, given [math]\phi[/math], use class notation [math]A_\phi=\{x\mid \phi(x)\}[/math].)

Writing
[math]\forall (x\in A). \psi(x)[/math]
is just notation
[math]\forall x. x\in A\implies\psi(x)[/math]
which, with the above, we may write as
[math]\forall x. \phi_A(x)\implies\psi(x)[/math]

The defining property for the empty set is the one that is false for any x:
[math] \phi_{\{\}}(x)=\bot [/math]
So it appears the vacuous truth notion in this bounded context derives from Explosion.

[Anonymous No.12179017]

>>12178981
you can if the image of f lies in N

[Anonymous No.12179023]

>>12179017
I wouldn't say that, but you could compose the codomain restriction of that function with itself.

[Anonymous No.12179026]

>>12179017
right, but that wouldnt be the case with natural numbers and whole numbers?

[Anonymous No.12179057]

>>12179026
if it happens that f(n) is a natural number for all n, then f(f(n)) definitely make sense for all n. but technically it's not a composition of f with itself, but rather what >>12179023 wrote.

[Anonymous No.12179070]

this is on the based list, right?

[Anonymous No.12179083]

>>12179070
Yessir.

[Anonymous No.12179086]

>>12177910
4 times the integral of the circle curve in the first quadrant

[Anonymous No.12179096]

>>12178621
Really it's all just a consequence of the universal property of quotients. Checkmate pedos

[Anonymous No.12179110]

>>12178882
Second isomorphism theorem just says if you take the quotient of two objects by one of them, it's ok, you get the other object, but dont forget to remove the part that belongs to both

[Anonymous No.12179120]

>>12179096
Quotients don't even exist in all categories. Those theorems might be generalizable but they aren't totally abstract nonsense.

[Anonymous No.12179260]

Post kino lemmas.
https://en.wikipedia.org/wiki/Dickson%27s_lemma

[Anonymous No.12179345]

>>12179260
https://en.wikipedia.org/wiki/Zorn%27s_lemma

[Anonymous No.12179366]

>>12177653
how about sequentially, where you take the largest possible triangle first, and then the smallest, etc?

[Anonymous No.12179396]

Where's the fuck Nikolaj edition?

[Anonymous No.12179533]

>>12179260
https://en.wikipedia.org/wiki/Riesz%27s_lemma

[Anonymous No.12179572]

>>12178056
in this vein, how hard is Coq to learn/work in?
half the time i hear something verified in it it's by a different person then the one that presented the proof and months afterward. Is it that tedious or are mathematicians just codelets?

[Anonymous No.12179588]

>>12179260
https://en.wikipedia.org/wiki/Urysohn%27s_lemma

[Anonymous No.12179593]

>>12179588

[Anonymous No.12179607]

>>12179572
You know what, fuck it.
I'll check it out, if it helps me practice mathematics since I suck dick at the subject, I won't bitch about having to learn what a object is

[Anonymous No.12179664]

>>12178526
>Analytical

[Anonymous No.12179672]

>>12178966
No, normal people just respect feminism.

[Anonymous No.12179680]

>>12179260
Kolmogorov made-UB lemma

[Anonymous No.12179688]

https://youtu.be/Cm-O_ZWEIGY

[Anonymous No.12179693]

>>12179260
Borel's Lemma
https://en.wikipedia.org/wiki/Borel%27s_lemma
>>12179533
This is a great one.

[Anonymous No.12179731]

>>12179607
Update
OH GOD ITS RAPING ME
ITS RAPING ME AND IT HURTS

[Anonymous No.12179757]

>>12179260
The Estimation Lemma for contour integrals (ML inequality), simple but powerful.
https://en.wikipedia.org/wiki/Estimation_lemma
Fatou's Lemma.
https://en.wikipedia.org/wiki/Fatou%27s_lemma
The Fundamental Lemma of Calculus of Variations (if you don't use this shit all the time I don't know you)
https://en.wikipedia.org/wiki/Fundamental_lemma_of_calculus_of_variations
Jordan's Lemma:
https://en.wikipedia.org/wiki/Jordan%27s_lemma
The Lebesgue Number Lemma
https://en.wikipedia.org/wiki/Lebesgue%27s_number_lemma
The Poincare Lemma on Differential Forms
https://en.wikipedia.org/wiki/Closed_and_exact_differential_forms#Poincar%C3%A9_lemma
The Riemann-Lebesgue lemma, fourier transforms of L^1 functions vanish at infinity.
https://en.wikipedia.org/wiki/Riemann%E2%80%93Lebesgue_lemma
Schwartz's Lemma
https://en.wikipedia.org/wiki/Schwarz_lemma

[Anonymous No.12179764]

Don't forget the Five Lemma, the Nine Lemma, the Snake Lemma, Yoneda's Lemma, Schur's Lemma, Burnside's Lemma...

[Anonymous No.12179767]

>>12179764
What about Lemma's Lemma?

[Anonymous No.12179824]

>>12179260
Rasiowa–Sikorski lemma, basically the forcing version of the Baire Category theorem. Also Konigs Lemma, for any infinite cardinal [math]\kappa[/math], that [math]\kappa^{\text{cf}(\kappa)}>\kappa[/math]. This is used everywhere in cardinal arithmetic.

[Anonymous No.12179834]

>>12179572
Okay so one of the big issues I'm having using COQ as a proof verification system is simply that it seems like no one is really using it to verify proofs, and I'm sensing that they tend to use it more for assisting with making proofs instead.
The only source that has been any kind of help so far has been 'Coq in a Hurry' by Yves Bertot(last updated in 2006 jesus fucking christ...), all other documentation effectively starts assuming you know Coq already and a quick check on youtube for any tutorials only brings up people presenting what Coq is. I'll continue to pluck away at it because I feel using it to verify your proofs IS there but its beneath a pile of long forgotten presentations and half written mostly forgotten manuscripts

[Anonymous No.12179838]

>>12179834
to be fair doesn't it go back to the 60s?
not surprising most of the material is old and a pain to find.

[Anonymous No.12179846]

>>12179757
>https://en.wikipedia.org/wiki/Jordan%27s_lemma
This is a nice one.

[Anonymous No.12179851]

I took group theory as a sophomore, I am well versed in proofs by now (direct, contradiction, counterexample, inductions and contrapositive).

How fucked am I? What's your wisdom?

[Anonymous No.12179855]

I'm a physics major who's trying to get into discrete dynamical systems including proofs. I already took real analysis, what can I expect?

[Anonymous No.12179869]

>>12179838
It looks like they started development in 1985 and released in 1989, so yeah the language is old. While we have old languages like C(1972) that gets alot of learning material published on it because the industry complex demands it, Coq users base is relatively small given its a subniche group of Mathematicians and Computer Scientests who use it but I'm still a bit surprised not one of them has sat down and started to work on an updated beginers guide, especially since the Coq its self is still getting stable releases

[Anonymous No.12179874]

>>12179869
https://coq.inria.fr/documentation
lists some books and tutorials within the last 5 years
where have you been looking?

[Anonymous No.12179876]

Currently taking two versions of Real analysis 1 at the same time (scheduling mishap). How do I get fast at writing math proofs, or will i get fast from all this practice?

[Anonymous No.12179885]

>>12179876
pretty much you'll get all the practice you'll need from analysis. Just make sure you spend a lot of time on the problems.

[Anonymous No.12179888]

>>12179876
The latter.

[Anonymous No.12179889]

>>12179874
>where have you been looking?
up my ass apparently, thanks for the heads up, having a look at a few of em now

[Anonymous No.12179890]

>>12179572
>Is it that tedious
Absolutely. We should be farming out proof verification to Pajeets. I know someone who worked for 8+ months to formalize a famous proof in Coq. Most of the time he spent trying to find the right data structures to represent modules of the proof. It is a big waste of most mathematicians time.

[Anonymous No.12179894]

>>12179885
>Just make sure you spend a lot of time on the problems
that seems to be happening whether I like it or not
>>12179888
sick trips

[Anonymous No.12179896]

>>12179876
Practice.

>>12179260
Nakayama, horseshoe.

[Anonymous No.12179897]

>>12179876
Just learn the triangle inequality and develop intuition for what in means to have upper and lower bounds on an interval so that you can use that to your advantage when taking limits.

You'll most likely start by building the axioms that make lR a field, then you'll learn about absolute values and the cauchy-shwarz inequaliy, pay close attention in the function analysis bit since statements about properties of functions will be useful when testing for continuity by epsilon/delta definition. Introduction to differentiation shouldn't be that difficult since most basic proofs will require induction.
Be wary of the concept of infimum and supremum because you won't want to mess up the supremum of a set with its maximum (same goes for the supremum and minimum).Analysis is pretty accesible in comparison to formal introductions to algebras so if you've taken a proof oriented linear algebra course you'll be fine.

[Anonymous No.12179898]

>>12179824
I always forget what cofinality means - is there a good way to remember it?

[Anonymous No.12179899]

>>12179851
>How fucked am I?
What is your hurdle?
>>12179260
Squeeze lemma

[Anonymous No.12179907]

>>12179899
What d you mean by my hurdle?

[Anonymous No.12179910]

>>12179851
>I took group theory as a sophomore
been there done that.
It isn't too hard, most proofs follow directly from the group axioms
It gets more screwy as you go on but there's only ever a few proofs that go beyond a handful of manipulations

[Anonymous No.12179913]

>>12179898
Think of it like minimum step size. Cofinal means unbounded. The cofinality of a cardinal is the least cardinality of a cofinal subset. Regular cardinals have cofinality equal to themselves, think sucessor cardinals. Singular cardinals have cofinality less than themselves. Think small limit cardinals like [math]\aleph_{\omega}[/math]. This has countable cofinality since to get to [math]\aleph_{\omega}[/math] you can just take a step on each cardinal less than [math]\aleph_{\omega}[/math].

[Anonymous No.12179944]

>>12179910
I really struggled to keep up when the prof was proving that any permutation is a product of disjointed cycles

[Anonymous No.12179955]

>>12178979
that is so cool anon, I don't have the preliminaries to understand what you did exactly but after a few google search conceptually it looks so cool.
>>12179110
thank you I noted this on my book, writing theorems in this type of sentences really helps my understanding
>>12179851
are you saying you are fucked because you took group theory so early? why would that be a problem? To tell you the truth I wish I took group theory as a fucking kindergartener

[Anonymous No.12179981]

>>12179913
I'm not 100% I'm following.
Sounds like the cofinality is the smallest prime factor of sorts, and regular cardinals are like primes.

?

[Anonymous No.12179989]

Are there any websites where you can learn math with illustrations and text? I'll prefer it over video.

[Anonymous No.12179993]

>>12179944
do you mean understanding it formally, or the intuition to it. because it felt obvious to me.
Informally speaking here's my intuition:
a permutation is essentially a bijection, so everything gets uniquely mapped to every element once. call it [math]f[/math]. then note [math]f(f(f(f\ldots)))[/math] must also be a bijection since it's a composition of bijections. As for the intuition as for why a certain number of repeated [math]f[/math] will get you back to simply [math]f[/math] (for permutations of finite elements), just note there's only so many permutations (for permutations of finite elements). By pigeonhole, you must eventually get back to [math]f[/math]. Finally since you can follow some element [math]g[/math] as you repeatedly apply [math]f[/math] you can begin to see why this larger cycling behavior is made up of smaller disjoint cycles. if [math]f(f(f(\ldots g)))[/math] eventually gets back to [math]g[/math] then every [math]t[/math] in the cycle also has [math]f^{-1}(t)[/math] in the same cycle as [math]g,t[/math]. If we have some other element [math]k[/math] such that [math]k[/math] is not reached by [math]f(f(f(\ldots g)))[/math] then [math]f(k)[/math] can't be in that cycle either since we established for every arbitrary [math]t[/math] it's inverse is also in the cycle. Since it's injective, the inverse is unique and we can't have [math]f(k)=f(f^{-1}(t))=t, k\ne f^{-1}(t)[/math]
I ended up writing more than I expected but I feel that about sums up the thinking

[Anonymous No.12180014]

>>12179993
Oh yeah, the intuition is all there since its obvious that if a permutation isn't a cycle and we know that the elements aren't fixed then there must be at least two cycles that don't mess with each other (which is what disjointed cycles are) such that the composition of them describes the permutation completely.

The formal proof uses construction arguments that I have a hard time wrapping my head around while following the whole strong induction premise.

[Anonymous No.12180025]

>>12180014
yeah, induction can be screwy to follow even when its your own proof, much less someone elses.

[Anonymous No.12180026]

>>12179907
What would be fucking you? In what way could you be fucked?

[Anonymous No.12180052]

>>12180025
Well, I hope I can get a bit more practice throughout the semester. Since we're all in lockdown here exams seemed kind of pointless so they left us 20 propositions to prove that I have to turn in come november.

>>12180026
I took some other not so easy classes because I wanted to have a full schedule (differential equations, non linear dynamics and analysis in Rn) so if this class demands lots of my spare time I might have to choose which subject would I be okay with if I flunked.

[Anonymous No.12180082]

>>12180052
>they left us 20 propositions to prove that I have to turn in come november
Sounds like an interesting class style. Honestly think I'd prefer that to an exam.
>I took some other not so easy classes because I wanted to have a full schedule (differential equations, non linear dynamics and analysis in Rn)
Sounds like a mix of pure and applied. which of those is more of a pain in the ass is mostly a matter of personal perspective, though professor, format, and text will often do more to make a class easy or difficult.
Honestly if you can reach intuitive understanding of a subject, everything else is easy. Ive had classes Ive slept through and aced and classes I've followed closely and barely nicked. Honestly, i know more about the subjects I slept through because those were the ones I really -got- and knew on a deeper level than a set of rules and manipulations. until you reach that point you're just crunching numbers and shifting symbols mindlessly.
Obviously some things like probability are easier to build an intuition than some other subjects like PDEs, but I believe you can reach that point with any subject if you try or are lucky enough to have a natual affinity with the subject.

[Anonymous No.12180138]

>>12180052
>I took some other not so easy classes because I wanted to have a full schedule (differential equations, non linear dynamics and analysis in Rn) so if this class demands lots of my spare time I might have to choose which subject would I be okay with if I flunked.
Meh, I found basic algebra pretty easy, I'd be wayyy more worried about the other ones, depends on the prof though. Like >>12180082
said

[Anonymous No.12180150]

god damn fuck coq, just trying to see if it will verify a simple set proof and as far as I can tell, it simply can't.

[Anonymous No.12180176]

>>12180082
>Interesting class style
Yeah I'd much rather have that than 2 hours to squeeze out a proof about something I've already done but forgot.

>Sounds like a mix of pure and applied-->
Im a physics bachelor but I'm planning on a double major since I enjoy rigorous math sometimes more than hand wavy jargon that barely models real life.
I can relate to passing a subject when I barely attended class and flunking another that I actually had to put effort in (real analysis vs proyective geometry). I haven't had taken anything remotely close to group theory so Idk what to expect, that's my fear.

[Anonymous No.12180195]

>>12180176
>hand wavy
hate this so fucking much
in an intermediate stats and the professor refuses to explain where the formulas are coming from and why they can be trusted. He insists that that's a "mathematicians thinking" and that we should be thinking "like staticians". Makes everything seem disconnected and impossible to follow
leson learned: never take a professor with a 2 on rate my professor
>real analysis vs projective geometry
projective geometry is some interesting stuff. Would love to take a class on it. Was it taught with linear algebra as a foundation or directly? directly seems more intuitive, but w/ linear algebra feels like formal proofs would go over easier since eveything is described in symbols. I'd like to hear the critiques you have with the way you learned it.

[Anonymous No.12180203]

I'm getting in a rabbit hole with irrational numbers, give me books that talk more deeply about them

[Anonymous No.12180209]

>>12178799
Dumb fuck

>>12178789
What are your evidence on the implication of your post, namely that UT has remained at this position or it retains this position in any other way than historical facts-keeping?

[Anonymous No.12180211]

>>12180203
That's pretty irrational brah, maybe you should stick with the rationales for now just to be safe

[Anonymous No.12180241]

>>12180195
if basic statistics isn't trivial to you, don't bother with mathematics

[Anonymous No.12180242]

>>12176252
>>12176264
Good comic, it gives good motivation to further study into the subject of logic if you are mostly unfamiliar with it

[Anonymous No.12180253]

>>12180241
basic stats was, this is intermediate stats. when he's throwing around formulas that optimally guess what inputs variables were given to the underlying distribuition based on the resultant sample without explaining how that optimization problem gets solved to result in the formulas he shows you're essentially left to memorize the formulas and process with no intuition as for -why- they work. It's still not a hard class, but I can tell I'm learning next to nothing.

[Anonymous No.12180268]

>>12180195
I totally get you, my intro to real analysis prof didn't stop to try and give intuition to the e-d definition of a limit and would just ramble about its usefulness instead of breaking it down to us, he had a 3 on my profs but first years here aren't allowed to pick their teachers, only 9 of us got a passing grade out of the 60 in the class. Hand wavy-ness just fucks things up.

No foundations whatsoever, I just stuck around because I enjoyed the notation. I don't see how there's a connection to linear algebra since proyective space lacks any sort of measure. I wish I had taken a class on euclidean geometry before getting myself caught in that mess, I barely remember anything other than the definition of a flag and the duality theorems for points, lines and their generalization. If I had to choose between getting kicked in the balls and taking another class I'd rather have my balls turned to mush.

[Anonymous No.12180270]

>>12179981
I don't know if that is a super accurate statement. Think about [math]\omega_1[/math]. If you take countably many hops up ordinals less than [math]\omega_1[/math], you will never get to [math]\omega_1[/math]. You can only get to [math]\omega_1[/math] by taking [math]\omega_1[/math]-many hops, hence the cofinality of [math]\omega_1[/math] is [math]\omega_1[/math]. This is not the case for [math]\aleph_{\omega}[/math], since you can hop on [math]\aleph_1[/math], then [math]\aleph_2[/math], then [math]\aleph_3[/math] and so on for [math]\omega[/math] many steps. This sequence if cofinal in [math]\aleph_{\omega}[/math], hence the cofinality of [math]\aleph_{\omega}[/math] is [math]\omega[/math]. Cofinality is just the least number of steps you need to take to reach a cardinal.

[Anonymous No.12180276]

>>12180209
>Dumb fuck

[Anonymous No.12180297]

>>12180268
>I don't see how there's a connection to linear algebra since proyective space lacks any sort of measure
modern formal treatments of it are done entirely in the language of linear algebra.
Consider a plane at [math]z=1[/math]. Then consider lines and planes through the origin, they will strike this [math]z=1[/math] plane in a single point or a line respectively. a line through the origin parallel to [math]z=1[/math] will never and these are the points and line at infinity. Intersection and such can be checked with by treating these lines and planes as row and column vectors. The correspondence between lines and points in projective geometry is then simply the realization that you can often swap row vectors and column vectors in these operations and get similar results.
I'll try to find the video i watched on it

[Anonymous No.12180306]

>>12180268
fuck, found it just after hitting submit. Here's the projective linear algebra video i remember seeing
https://www.youtube.com/watch?v=puMYfJTFdgQ
This guy's funky, he's a finitist and insists reals don't exist. He has a passion for geometry though and he's funny to watch as he quietly dodges using real numbers in a fair few of his lectures.

[Anonymous No.12180309]

>>12180209
just say you're racist. Its fine here, I mean, you're literally one click away from a board full of nothing but white supremacists here

[Anonymous No.12180316]

>>12180309
lmao, look up /pol/ meetup pics. seems like most aren't white supremacists, rather they're non-white ethnomasochists.
or i suppose you could call them white supremacists just not in the usual sense of a white person who is a supremacist and instead in the sense of them believing white people are supreme even though they themselves aren't white.

[Anonymous No.12180372]

>>12177320
Did onions onions and based get duckrolled last night or something

[Anonymous No.12180376]

>>12180372
Ok I guess s o y is just getting switched with a bunch of shit

[Anonymous No.12180378]

what are these? dont need the formulas just need the greek letters so i can look them up

[Anonymous No.12180387]

>>12180376
It's been like that for almost two years.

[Anonymous No.12180390]

>>12180378
[math]\Gamma[/math] = Gamma
[math]\zeta =[/math] zeta

[Anonymous No.12180430]

>>12180390
ty

[Anonymous No.12180442]

Why is there such a thing like the correlation between gender dysphoria and affinity to category theory?

[Anonymous No.12180509]

>>12180442
High IQs.

[Anonymous No.12180528]

>>12180442
gender dysphoria and liking category theory are both mental illnesses

[Anonymous No.12180591]

>>12180528
Cope.
Category theory is the final frontier of math.

[El Arcón No.12180595]

>>12179731
>OH GOD ITS RAPING ME
>ITS RAPING ME AND IT HURTS

[Anonymous No.12180616]

>>12180442
an eschewing of reality is required to justify their choices

[Anonymous No.12180617]

>>12180591
>Mt. Infinity
>isn't infinitely tall
I want the money I didn't pay back

[Anonymous No.12180635]

>>12180591
I'm right here

[Anonymous No.12180668]

>>12180306
Didn't expect to enjoy it as much as I did kek, thanks for the vid. I didn't get to see any relationship to linear algebra on my classes, maybe I would've enjoyed it more if I had.

Finitists are weird, they're the same kind of people that'll argue that open intervals have max and min

[Anonymous No.12180678]

>>12180668
fuck off

[Anonymous No.12180682]

>>12180668
np, he's a fun watch. It's funny stumbling across him, deciding he's an eloquent lecturer, deciding to watch more, and slowing to realize he's very bizarre as you see him using more and more esoteric techniques to dodge the reals. he refuses to use R, trig functions, angles, pi, derivatives, and probably a dozen other things, it's hilarious how much math he needs to reinvent to do even basic things.
>>12180678
keep seething finitist

[Anonymous No.12181002]

>>12176360
Teichmüller from the IUT was also a serious Nazi and told Landau not to teach Aryans.

[Anonymous No.12181122]

Hi /mg/. I am trying to learn some basics of linear algebra. I have this little system of equations modelling traffic flow in four intersections.
[eqn]
\left\{
\begin{array}{c}
\begin{align}
&x_1 = 330 - p \\
&x_2 = 170 - p \\
&x_3 = 210 - p \\
&x_4 = p
\end{align}
\end{array}
\right.
[/eqn]
In this case x_4 is a free variable and its value can be chosen arbitrarily, . I'd like to know if there's any way to calculate a value for x_4 that results in the most balanced flow of traffic throughout the intersections?
So far I've experimented that setting x_4 to 100 seems to reduce the sum of differences [math]|x_1-x_2|+|x_2-x_3|+|x_3-x_4|[/math] to 210. Are there any better ways to optimize this kind of problem?

Here's a Youtube video of a similar problem that I've used as a study resource:
https://www.youtube.com/watch?v=8Kg21jBCm-k

[Anonymous No.12181126]

>>12180668
>Finitists are weird, they're the same kind of people that'll argue that open intervals have max and min
I mean, obviously they do. There's only so much resolution you can get out of the rationals. Take the largest integer [math]M[/math], and let [math]h = 1/M[/math] (the fundamental mesh size of the universe). In the interval [math]\{x \mid x < a\}[/math] you're eventually going to find an element [math]b[/math] such that [math]b + h = a[/math], so [math]b[/math] is the maximum.

[Anonymous No.12181292]

Is there any way to improve how undergraduate maths education is structured? I’ve heard basically all my professors say they never went to lecture as students and just learned from the book. If this is true, why do we continue doing lectures in this way?

For example, a typical undergrad analysis class follows Rudin. If we have all the proofs already, in basically a perfect form, what is the point of the professor reading them aloud and copying them onto the board at a furious pace, just so the students can scribble them into their notebooks to study later? Books like Rudin, D&F, Munkres, etc.. are already very readable to begin with.

[Ouch No.12181302]

>>12181292
Rudin’s proofs aren’t even remotely close to perfect nor is his book thoroughly “readable”.

[Anonymous No.12181308]

>>12181126
isn't that ultrafinitism?
the lad in the video at least seems to think you can get arbitrary resolution out of the rationals (ergo no largest integer exists). he has said stuff like "[math]10^10^10^10^10^10^10^10^10^10+1[/math] is meaningless, you can't do things like factorize it, etc. it only has the appearance of a number" to roughly paraphrase

[Anonymous No.12181311]

>>12181308
fucked up the tex. should be this maybe? :
[math]10^{10^{10^{10^{10^{10^{10^{10^{10^{10}}}}}}}}}+1[/math]

[Anonymous No.12181321]

>>12181292
rudin is the least readable of the four analysis texts I have. have you opebed it?

[Anonymous No.12181324]

>>12181308
I'm pretty sure Wildberger is an ultrafinitist. From what I've watched of his videos he's kind of vague about what he thinks of arbitrarily large but finite quantities (like, he hasn't said that there exists a largest integer), but certainly he finds them troublesome and doesn't think we can do arithmetic with them similarly as we can with more reasonably sized quantities.

[Anonymous No.12181330]

>>12181324
interesting. this man's autism has no bounds it seems.
still not as wacky as that one ultrafinitist who when asked if 100 exists during an interview sat there quietly for a second to think about it and eventually said yeah (presumably after counting up to it in his head).

[Anonymous No.12181337]

ultrafinitism, not liking law of excluded middle, all other kinds of this shit - it's mostly wankery and attention whoring

[Anonymous No.12181341]

>>12181330
That would be Yessenin-Volpin. To quote Harvey M. Friedman "Philosophical Problems in Logic":

I have seen some ultrafinitists go so far as to challenge the existence of [math]2^{100}[/math] as a natural number, in the sense of there being a series of “points” of that length. There is the obvious “draw the line” objection, asking where in [math]2^1, 2^2, 2^3, \dots, 2^{100}[/math] do we stop having “Platonistic reality”? Here this … is totally innocent, in that it can be easily be replaced by 100 items (names) separated by commas. I raised just this objection with the (extreme) ultrafinitist Yessenin-Volpin during a lecture of his. He asked me to be more specific. I then proceeded to start with [math]2^1[/math] and asked him whether this is “real” or something to that effect. He virtually immediately said yes. Then I asked about [math]2^2[/math], and he again said yes, but with a perceptible delay. Then [math]2^3[/math], and yes, but with more delay. This continued for a couple of more times, till it was obvious how he was handling this objection. Sure, he was prepared to always answer yes, but he was going to take [math]2^{100}[/math] times as long to answer yes to [math]2^{100}[/math] then he would to answering [math]2^1[/math]. There is no way that I could get very far with this.

[Anonymous No.12181347]

>>12181341
lmao what the fuck

[Anonymous No.12181352]

>>12181337
when it's done by actual mathematicians it's an interesting mix of lol-cow bullshit and legitimately impressive math as they build back to familiar territory. It's also interesting from a reverse mathematics perspective, being able to define derivatives without limits (or abitrarily small numbers for that matter) is a rather bizarre thing. You would think that would be something of a requirement. Every now again one of these weirdos stumbles across something like that and it's honestly pretty neat.
There's also no reason why they can't. all axioms are arbitrary, some sets of axioms are just more interesting than other sets. if they want to drop the Archimedian principle and its equivalents and see how far they can get it's not like they're betraying mathematics by doing so.

[Anonymous No.12181376]

>>12176492
I don't know much about DoSV specifically but generally speaking applying to university as a foreigner doesn't have any additional requirements (besides perhaps language skills) as long as you're an EU citizen. Do you want to study math?

[Anonymous No.12181378]

>>12179955
If you really feel like that is cool, then I recommend you walk the path of the algebraist. Some abstract non-sense and a bit of topology and then you get to play with group (co)homology and attend nice seminars with me.

[Anonymous No.12181467]

>>12180270
Okay, thanks, I get a feel for it now. Hopefully I'll not forget it too soon, though.
But still, it feels like there's some vague tie to ideas. Especially in how the cofinality of [math]\aleph_\delta[/math] jumps between big and small when you're on a successor vs. a limit in the the ordinal. Although I don't really know what ordinals are between such alepha.

>>12180306
27:15 - 31:15
is fun to watch

>>12181337
There's nothing wrong with working explicitly without the excluded middle, I don't know why this is so misunderstood.

Let [math]{\mathbb N}^{\mathbb N} [/math] be the functions on the naturals.
Let [math]P \subset {\mathbb N}^{\mathbb N} [/math] be the functions that can be coded up (in C++, say).
Let [math]C \subset P [/math] be the functions that, on the inputs 0 say, are total / terminating / computable (excludes e.g. infinite while-loops).
Let [math]e \colon {\mathbb N}\to P [/math] be an enumeration of all codes.
Let [math]I_C := \{i\in {\mathbb N}\mid e(i)\ {\text {is computable} } \} [/math] be the index set of terminating functions.

The property of termination is not decidable, i.e. provably there’s no algorithm that you could give any C++ code so that it tells you whether it would terminate
So [math]I_C [/math] cannot be decidable either.
As a result, there is no computable surjection
[math]{\mathbb N}\to I_C [/math].
As a result, there is no computable surjection
[math]{\mathbb N}\to C [/math].

If you want to logically reason about this scenario, then you must tune your logic to respect this situation.

LEM plus a few set theory axioms imply that any infinite subsets of the nats is in bijection with it, but this is misleading. If ZF proves some set is "countable", then this word just means a formal existence statement
[math]\exists f. \dots[/math]
It's similar to a choice principle.To claim that every infinite subset of the naturals is "countable" is merely to adopt this as an axiom. Not that there's anything wrong with it.

[Anonymous No.12181470]

(In fact you only need to drop [math] {\sf LEM} [/math] and the Powerset axiom and then it's already consistent for there to be surjections from some subsets [math] S\subset {\mathbb N} [/math] into all of [math] {\mathbb N}^{\mathbb N} [/math]. With [math]S[/math] undecidable, preventing the Cantorian would-be counterexample to be a total function. But this is another story: It’s studying another theory that overrules the mathematics with claims about non-realizable claims, just LEM does or just like ZF does.)

[Anonymous No.12181483]

>>12181378
Hey babe don't groom him to be your boy slave. Analysis > algebra faggotry.

[Anonymous No.12181487]

>>12180306
>Here's the projective linear algebra video i remember seeing
>https://www.youtube.com/watch?v=puMYfJTFdgQ
the fact that all gemoerty and analysis can be done without the reals is a huge blow to classical real analysis.
It's a Copernican revolution: you remove a huge slab of vocabulary from the explanation of geometry, just like scientists have removed all theistic vocabulary to explain measurements.

When you don't need something to explain a thing, then the thing you don't need is mental masturbation called metaphysics.

[Anonymous No.12181491]

>>12181487
>all gemoerty and analysis
has someone done integrals with a finitist perspective in a way that isnt just defining them using antiderivatives?

[Anonymous No.12181500]

>>12181483
Ehh... I wouldn't consider it grooming anyone to be my slave if I give a recommendation like that. Analcysts truly are barbarians. Your post showed that once again.

[Anonymous No.12181534]

>>12181470
What is he so apologetic about anyway??

[Anonymous No.12181574]

>>12181534
I think his SE said this long before his sex therapy, drug excesses and the Silicon Valley situation. These hot takes are definately younger

[Anonymous No.12181578]

[Anonymous No.12181582]

feels

[Anonymous No.12181586]

>>12181574
BASED incel apologist. Not many men are brave enough to stand up for incels.

[Anonymous No.12181597]

>>12181500
uhm... excuse me but... i'm an analyst... could you pound my boypussy with your big algebraist cock?

[Anonymous No.12181617]

>>12181597
Nope.

[Anonymous No.12181675]

>>12181378
I would very much like to be an algebraist thank you. Calculus never felt like real mathematics to me and I'm also very fond of set theory. I hope I can manage to understand all that abstract non-sense and attend comfy algebra seminars with you one day^^
>>12181483
heyy don't fight now, you can both groom me as much as you want

[Anonymous No.12181681]

Why is there a rivalry between analysis and algebra? They don't even compete to solve the same kinds of problems, unlike for example Bayesians and Frequentists in statistics.

[Anonymous No.12181708]

>>12181675
If you follow "her" advice just make sure you don't become a similar alcoholic attention whore and blog about your depression in these generals.

>>12181681
Algebra is jewish bs and hardly math. If you don't even actually compute anything then why call it algebra instead of just some diagram masturbation? Arrows go brrr Q.E.D. my ass.

[Anonymous No.12181736]

>>12181681
there's no rivalry, just /mg/ memes

[Anonymous No.12181737]

>>12181675
What have you done so far, or have you even started yet? If you know the basics of set theory, you can already get started with groups.

>>12181681
I don't think it is that serious. At least, it's not serious for me.

>>12181708
>similar alcoholic attention whore and blog about your depression in these generals
Heh, I'm sorry about those. At least I've been [math]T_{1 \frac{1}{2}}[/math] for a month now.

[Anonymous No.12181748]

>>12181708
>he doesn't know that Rudin was jewish

[Anonymous No.12181749]

>>12181582
This is why I never understood the current PHD system. Over the course of four years you should gain the capacity and capability with dealing with mathematics that overshadows 99% of the Human population
but you're literally cock blocked by a single problem which maybe so difficult that it might be effectively unsolvable. Or worse, you get scooped meaning all your work goes into the trash. For a piece of paper from somewhere saying 'yeah he's okay at math I guess'

[Anonymous No.12181772]

>>12181737
No one cares.

>>12181748
And? There are more jews in algebra than analysis but of course there are some in analysis too.

[Anonymous No.12181783]

>>12181675
You're very fond of set theory but you don't like analysis?? What is it you like about set theory?

[Anonymous No.12181784]

>>12181737
well, I'm a second year undergrad technically. I have studied set theory this summer and now taking abstract algebra. I planned my curriculum so that at the end of the fifth semester I had taken groups, rings, galois etc as electives, but not sure what I will be doing after that.

[Anonymous No.12181793]

>>12181749
it's your final trial by fire
Having to make a thesis is hardly the worst part of the PhD system
I've heard horror stories and almost all of them involve shitty advisors

[Anonymous No.12181801]

>>12181784
>technically
???

[Anonymous No.12181804]

>>12181783
I didn't said that I don't like set theory. I was just thinking about taking integrals and so on. I don't know much about the relations between set theory and analysis, I should do some reading on that later. I will also check Rudin's offspring of analysis lecture.
>>12181801
I'm taking so much more than a regular 2nd year for my uni, that's why I wrote technically.

[Anonymous No.12181812]

>>12181804
>I'm taking so much more than a regular 2nd year for my uni, that's why I wrote technically.
so standard setup for a mathlad then
I think every math major I know is set up to graduate in a total of 3 years by a mixture of AP and packing on more classes than reccommended. even the math major I know that wants to get into finance afterward that I expected to switch majors after the 1st year is doing this shit.

[Anonymous No.12181823]

>>12181784
I see. Have you taken topology? Algebraic topology is a good way to get into abstract non-sense, as then you would be playing with all sorts of functors from the categories of topological spaces or pairs of spaces etc. to the category of sets, groups, abelian groups, modules etc. You would learn the basics of homological algebra at the same time, exact sequences, derived functors and such. You could be interested in that.

>>12181804
>I don't know much about the relations between set theory and analysis
For example sigma algebras would be a good way to learn a bit about those connections. https://en.wikipedia.org/wiki/%CE%A3-algebra

[Anonymous No.12181893]

>>12181749
Yes kinda, but I have no better proposal than to organize a PhD around a topic or problem or set of problems.

In a way, everybody is responsible on their own, to live a good life. To make a nice life out of their life.

[Anonymous No.12181921]

>>12181812
yeah I guess you are right.

>>12181823
>Have you taken topology?
I will be seeing topology for like a month this semester within a some sort of advanced calculus class, besides that I am not well-versed on topology but it looks fun for sure.
I bookmarked the sigma algebra page.
I'm gonna go study now. See you later lads.

[Anonymous No.12181926]

>>12181122
Hmm, you want to balance all 4 numbers? Maybe you should be trying to minimize the variance then? Look up the formula for variance (not standard deviation, variance will do the same and is easier) and don't simplify anything, you're just going to want to apply "Langrange multipliers" to this formula with your linear constraints. This is very tame since you're just differentiating a big quadratic and some linear functions. As long as you don't try to expand anything before differentiating it should be the easiest technique.
The solution to the Lagrange multipliers problem will give you your minimal x_i.

[Anonymous No.12181937]

>>12181582
Filtered.

[Anonymous No.12181942]

>>12181681
It's not a real rivalry but also the algebraists usually got filtered in some analysis class and us analysts usually got filtered in some algebra class. So there's a mutual sort of envy.

[Anonymous No.12181948]

>>12181921
As the analyst I second algebraic topology. Also differential topology. Both really fun and beautiful.

[Anonymous No.12181951]

[Anonymous No.12181979]

Suppose A(1)=1, A(2)=8, and A(n)= A(n-1) +2A(n) (n greater than or equal to 2),

Use strong induction to prove that A(n)=3*2^n-1 +2(-1)^n (n greater than or equal to 1)

Can help solve me this problem?

[Anonymous No.12182008]

>>12181675
It's just banter, because we eat our corn on the cob differently.

[Anonymous No.12182017]

>>12181979
Did you mean to write A(n+1) for the iteration step?

The initial case is clear,
3*2^[1-1]+2*(-1)^[1]=3*1-2*(-1)=3-2=1=A(1)

[Anonymous No.12182031]

>>12181951
i hate all forms of dedications in books on theses

[Anonymous No.12182038]

>>12182017
No sir this was the original problem, i just didn't want to post it without translating because the professor has very cryptic handwriting.

[Anonymous No.12182044]

>>12182031
why?
I always thank my mom, just because she likes it

>>12182038
Yeah, so you should thank me for spotting that what you wrote is inconsistent. The last term s +2A(n-2), whereas you wrote +2A(n).

[Anonymous No.12182050]

There are phd students who take 2 pages to thank absolutely everybody. It's ridiculous.
Some retard even thanked the people at the university gym.

[Anonymous No.12182054]

>>12182050
I fail to see the problem with that

[Anonymous No.12182056]

>>12181893
>I have no better proposal than to organize a PhD around a topic or problem or set of problems.
I'd be okay with this if its effectively the leaders of the group vouching that you have contributed enough to warrant a PHD, maybe not solving a problem contribution but making new insights into potential new formulations or some other such shit

[Anonymous No.12182073]

>>12182044
I apologize for my misunderstanding, have a nice day sir and thank you.

[Anonymous No.12182075]

>>12182050
ye what's the point of that shit
just dedicate your thesis to your favorite anime character

[Anonymous No.12182086]

>>12181302
>>12181321
Didn’t realize I was saying something controversial. First half of Rudin is basically perfect imo.

[Anonymous No.12182094]

>>12182075
That sounds unbelievably based

[Anonymous No.12182098]

>>12182056
You want a system where midwids get PhD because they sticked with someone they like for 3-4 years and the tower of corruption carries on?
To be fair I don't want to make a case to the opposite, since in a way this situation is as it is today already and inevitable.
But still, in a way, there must be a mechanism for people to fail in some way, or not stand the test.

It's not like Qiaochu would fail his PhD anyway, he is (was) just a turboautist in not yet completely established subfield and had an existential crisis, before he got in the vicinity of Berkeley rationalist lesswrong cultist + Berkeley hippy love and piece junkies. A deadly mix.

The whole categories-for-everything history in academia is already bad enough. I like the subject, but every generation in that field hates on the next and there's stupid fights, established logicans bullying with topos theorists, established topos theorists bullying with their juniors (see Camilla), now Baez et al., who complain their theis seniors already hated them, split into parts, the homotopy theorists (Lurie etc.) shit on the CS people, the CS people bully the geometers, it's a shitshow and possibly not a place to do a PhD

>>12182075
This

[Anonymous No.12182107]

>>12182086
They got filtered, ignore them.

[Anonymous No.12182128]

How would you describe these subsets?
I'm not sure my way is the simplest, most insightful or most elegant.

I'm considering that the subfunctor is literally defined on actual subsets by the way, because it's defined, the F(c)'s can just be injected into the Hom-sets through the monomorphisms. One can factor the monomorphisms as a composite of inclusions and isomorphisms (restricting to the image), then include these in the new fixed functor, etc.

[Anonymous No.12182138]

>>12181921
Point set topology has its ups and downs, but you should still know even the basics of that, regardless of what you are going to specialise in because topological arguments for stuff just pop up all over the place. Anyway, have a good studying day!

>>12182050
You should write a personal letter to everyone you wish to thank, and then include those as appendices of your thesis. That's what I'm going to do. Appendix A for mom, appendix B for dad etc.

[Anonymous No.12182148]

>>12182128
Are you aware of the Yoneda lemma?

[Anonymous No.12182150]

>>12182128
I don't quite follow your sentences, but I guess the thing is called sieve and you can check it out on Wikipedia?

[Anonymous No.12182156]

>>12182148
I've seen the statement and proof, but pretty much didn't apply yet. It's not covered before this exercise in this book.

[Anonymous No.12182158]

>>12182148
No that's not necessarily going to help here. I remembered that the representable functor is on the right slot. Forget.

[Anonymous No.12182170]

>>12182150

again, I think this will help
https://en.wikipedia.org/wiki/Sieve_(category_theory)
https://en.wikipedia.org/wiki/Subfunctor
https://ncatlab.org/nlab/show/sieve
https://ncatlab.org/nlab/show/subfunctor

[Anonymous No.12182173]

>>12182150
Oh, it's precisely a sieve on the object c, in the category in question. Looks interesting.

I can just say that the it's a collection of subsets [math]F(a)\subseteq \text{Hom}_{{\cal C}}(a, c)[/math] such that their union is closed under precomposition by morphisms of [math]{\cal C}[/math].

[Anonymous No.12182174]

>>12182156
Yeah and it was the other way around anyway, so it wouldn't solve your problem. Anyway, start with the case where the natural transformation is just the actual inclusion. Then we know what the subfunctor would do to [math]f\colon a\to b[/math], right? It will take that to the precomposition [math]f^*(\varphi) = \varphi\circ f\colon a\to b[/math] for any [math]\varphi\colon b\to c[/math], but this precomposition is restricted to the subset of [math]\mathbf{C}(b, c)[/math] determined by the subfunctor, and it should take every morphism in that subset into the corresponding subset of [math]\mathbf{C}(a, c)[/math]. This should also happen for every such morphism [math]f[/math] in the original category. And this leads to... >>12182173

[Anonymous No.12182372]

>>12182098
My only real counter is that in the current system, your advisor can set you up to fail. I've seen it before. Some of the best graduate students effectively scrub because their advisors are such dicksweeds that they really never had a chance while their inferior counter parts walk the stage because their advisors bulldozed the path of success for them. This implies that the true test of a Math PHD program isn't 'are you good at math' but rather 'how much dick can you suck, also flip this coin, if it lands on tails, no matter how much dick you suck, you're not getting a PHD'

[Anonymous No.12182399]

>>12181617
based
>>12181675
>heyy don't fight now, you can both groom me as much as you want
this
>>12181783
Not them, but man analysis filters me. Mind you, we are doing lebesgue integration later this semester and that sounds really interesting, I'm excited to see the construction. Things like functions of bounded variation and Riemann-Stieltjes integration are pretty boring imo
>>12181942
I think you're on to something...
>>12182075
>just dedicate your thesis to your favorite anime character
They did get me through most of my struggles more than anyone else, other than /gmmg/-anon of course
>>12182086
Disagree, I prefer a bit more discourse in textbooks, and I really would've liked more tangible examples that aren't the real numbers.

[Anonymous No.12182411]

https://web.math.rochester.edu/news-events/events/search/?type=15&submit=Search
Rochester Electronic Topology Seminar, a talk tomorrow if someone is interested.

>>12182399
>analysis filters me
Same. I had 5/5 from everything during my master's but then there was the compulsory advanced measure theory thing which gave me 2/5. Eww!
>other than /gmmg/-anon of course
Awww

[Anonymous No.12182429]

>>12182411
Oh and also a Turkish one https://researchseminars.org/seminar/BilTop

[Anonymous No.12182452]

>>12182429
Is it bad that I don't speak Turkish?

[Anonymous No.12182467]

>>12182452
Neither do I, but I would assume those are in (varyingly good and bad) English. At least the last time I heard Turks speak, that was actually quite decent.

[Anonymous No.12182536]

New >>12182535

[Anonymous No.12182578]

>>12182399
>Disagree, I prefer a bit more discourse in textbooks, and I really would've liked more tangible examples that aren't the real numbers.
That's exactly why most of Rudin works so well. You've seen these concepts before, so there's no need to waste time building some sort of vague intuition, be it through discourse or examples. The point is to put these basic topics on a rigorous foundation, and the best way to do that is focus on the proofs and force the reader to do a bit of work on their own.

The reason parts of the second half are generally less successful is, I think, because unlike the derivatives of real functions or convergence of sequences, the reader is unlikely to have encountered something like measure theory before. So it is at this point that more clarity/background/discourse/examples is actually useful, and that's why Stein and Shakarchi comes so highly recommended.

[Anonymous No.12182607]

>>12182578
The one chapter I am actually complaining about is the second one. It generalizes the concept of a metric space, but rarely gives examples other than the usual metric on [math]\mathbb{R} [/math]. Like, whats is a metric on a different set? What would the closure of those open balls looks like? Etc

[Anonymous No.12183703]

How the fuck do you choose a topic for your masters thesis? For the Bachelors thesis our prof gave a list of topics and I picked some almost arbitrarily. I know I can learn the material. But it seems like you need to be far deeper into a topic for a msters thesis, no?