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>>12919344

Runic

Talk mathematically
>>
The picture is trivial and left to the reaser to untangle
>>
Answer me fast: Does a straight line have a normal line? Or can I say that only curves that aren't straight lines have normals?
>>
>>12931507
Straight lines have technically normal lines but not *the* normal.
>>
>>12931534
But like, do they all pass through the same point or not?
>>
>>12931543
Draw a picture mate, you will realize quickly that what you just asked is a dumb question
>>
>>12931580
I know, I just wanted confirmation for my thoughts.
>>
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Did I buy a uhhhhhhhh illegal print or something
>$12 though
>>
>>12932039
HE BUYS BOOKS HAHAHAABAHAHAHAABHAHAH

THIS SHIT'S GOING TO YOUTUBE

*FAG BUYS TEXTBOOKS 1,000,000,000,000 VIEWS*

LMAO WHAT A FAG
>>
>>12932039
Yeah probably. The spine font looks exactly like that of some PID book I accidentally bought once.

Still a good deal though. Where did you get it from?
>>
imagine a chaturbate in which the anime tranny has a remote controlled butt plug and it vibrates every time she gets a donation. at the same time she read new papers from arxiv aloud.
>>
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>>12932180
Seriously, anon... What's wrong with you? Do you really think I'm not already doing that?
>>
https://www.youtube.com/watch?v=dmA0_F9XuII
>>
>>12932180
seek help dude
or go get topped by a trans escort
you seem repressed
>>
>>12932379
Liberal fafgggot voice
>>
>>12932383
who hurt you sweetie?
>>
>>12932039
>Valdimir
kek
>>
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I studied all day today but I still feel like I did absolutely no progress at all.
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>>12932352
https://www.youtube.com/watch?v=RmQHWlQpDVY

>>12932462
What did you study?

>>12932437
Based bootleg.
>>
>>12932495
>What did you study?
My main problem is with Differential Geometry right now. I thought I was doing good at it and understanding the concepts but I spent all day and noon just trying to do one question, I couldn't follow some of the solutions I saw despite understanding the theory. I'll try to do more exercises tomorrow and if I still have trouble I'll give up on it for now.
>>
>>12931140
what can I study in two weeks to know differential equations well enough for a mechanics class
>>
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>>12932507
Is it all the notation? I can honestly say it was the biggest problem for me for basically the whole course back in the days.
>>
>>12932507
I had the same problem in a different area but made a breakthrough today. I believe in you anon!
>>
Simple question: what does it mean when the minus sign is on the right? As in,
>The intervals (F(x−), F(x))
>>
Schemes: deep, profound, insightful objects of study, or just a bunch of algebraic, overly-abstract nonsense?

Should I care about them?
>>
>>12932636
Learn algebraic geometry if you want to act pretentious around other mathematicians and don't want to be understood by any of them.
>>
Guys, do you wanna solve navier stokes with me?
>>
>Riemann Hypothesis is still unsolved
*takes pencil*
Fine, I'll do it myself.
>>
>>12932740
Man I just want a million bucks,.
>>
>>12932805
For what?
>>
Can one think of the tensor product of vector spaces V_1,...,V_k as just being the space of ordered pairs (v_1,...,v_k) such that the identity map is multilinear? Or is there some other nuance here.
>>
>>12932623
means limit approaching x from below. so for cadlag functions f(x+)=f(x), f(x-) well defined but might not be equal to f(x).
>>
>>12932740
Don't talk about RH on /sci/ else you summon the schizo who thinks he solved it
>>
Going into my third year of undergrad next fall. I no longer have any required liberal arts diversity training courses to take (am burger) and I'm excited to make more room in my schedule for actual studying.
I've been told taking more than three math classes at a time is a bad idea, but I have four that I want to take plus two physics classes I need for my minor. What would you recommend doing? Keep in mind I'm very autistic and have no social life so time is of little concern as long as I don't burn myself out.
>>
>>12932039
You got Arnol'd lmao
>>
I'd like to get into measure theory, but haven't taken a rigorous math course since my undergrad maths degree. I'm a data science pleb now and want a deeper understanding of statistics. Could I just hop right in or should I ease my way in with proof practice using books like Rudin's analysis?
>>
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>>12932740
But it's already been solved anon, please see pic related.

>>12933031
>schizo
rude!
>>
>>12933036
It depends on the four. Since you are third year, they are probably not total baby classes, so be careful
>>
>>12933036
4 math + 2 physics seems like a lot depending on the level

>>12933056
assuming you've already gone through baby rudin, I'd just jump into stein and Shakarchi to learn measure theory basics. Then Durrett or similar for probability.
>>
>>12933059

I regret taking the time to read some of that, my eyes are bleeding now
>>
I'm trying to prove that the converse of Schur's lemma does not hold. That is, I'm trying to prove that there is a division ring such that it is not the ring of endomorphsims for some simple R-module. I know it has something to do with showing that the division ring is not finite-dimensional but I don't know how to use this accurately.
>>
>>12933036
>Keep in mind I'm very autistic and have no social life so time is of little concern as long as I don't burn myself out.
I'm mildly in that category and 4 math courses in a semester didn't work out. 3 + a french course was survivable, but not ideal. If physics has enough distance from math for you, then go for it, but it depends, just be careful
>>
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>be into math as a kid
>read entry tier number theory shit like Archimedes Revenge
>really liked it as a kid
>end up dropping out of high school
>mom gives me a graphing calculator for my 25th birthday
>sends me to seminar about data infographics jsut cause she thinks I will enjoy it
>I'm a factory wagie whose coworker yesterday unironically asked me what 19x10 was and I almost called him a fucking retard
Is there some way I can get back on track academically? I'm scared of going back to college cause I'll end up in debt. is there any point to trying to self study math to some point that you could demonstrate your knowledge? I feel my life slipping away and I fucked up my brain sleeping 5 hours a night the last year I hate disappointing my mother and father. I can't go back to college at this point. My friend recommended some klein book on calculus to review (I've only done up to calc I but I used to go to these 1 day lectures as a teenager so I've had faintest exposure to all ranges of higher math, just no real conception of it )

srry for blog posing
>>
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>>12933135
I'll ask you two questions:
(1) Given a unital ring [math]R[/math], what is its endomorphism ring isomorphic to?
(2) What can you say about the ideals of a division ring?
After you have answered those, one more:
(3) What can you conclude if you assume that [math]R[/math] is a division ring?
>>
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>>12933150
Free pdfs are everywhere, a calc review is not a bad idea, and you could move on to more proof based things after that if that tickled your fancy. I'm not a burger, but don't feel ashamed to go to community college to start getting back on track, nothing wrong with that. In addition, don't feel like you have to be a full time student, you can do 2 courses a semester/quarter and continue to work. If you REALLY want to save money, at most colleges/universities you can self study a subject and challenge for credit, but I personally find self studying a much longer process. Good luck, and have fun getting back on the math train!
>>
>>12933157
You mean R as a right module over itself?
>>
>>12933135

If D is a division ring, then End_D(D)=D as a module over itself. D is also simple over itself. I think what you really want is to produce an A-module M so End_A(M) is a division ring yet M is not simple. In this case I think you need to go to the non-commutative case. If you look at a Verma module, say even for sl_2(C), the irreps are given by the simple socles (so in particular these are not simple U(sl_2)-modules). But the endomorphism rings should be C.
>>
>>12931586
You can always establish a normal line at EVERY
point of the curve. In fact, the normal to a family
of ellipses are a family of hyperbolas.
>>
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>>12933179
Yes. (1) and (2) as right-sided questions, that is:
(1) [math]\text{End}_R(R_R) \cong \text{?}[/math]?
(2) What are the right ideals of a division ring like?
>>
Are inner products inherently topological? Since theyre just a set of interactions of basis vectors, they basically act like a form of counting intersections, with multipliers maybe.
>>
>>12933002
I suspected that, but it seemed kind of strange. Thanks anon.
>>
>>12933281
No, inner products are geometrical, not topological, as they define both a metric and a notion of angle.
>>
>>12933281
>they basically act like a form of counting intersections
they don't act like that at all
>>
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Any good roadmap for learning math? I'm talking from kindergarten tier to Oxford tier mathematics. Can't believe I've neglected learning for so long, especially such an important subject. I'm most interested in statistics and analysis but please post any advice at all.
>>
>>12933400
They literally do. Basis vectors ax and ay in vector z, vs bases bx and by in q. Inner product just valuates how much they overlap.
>>
>>12933363
You dont have to take a metric. Where does angle come from, other than orthogonality? Why cant angles be topological just like measures can come from sets?
>>
>>12933534
Inner products imply a metric, but the notion of "metric" doesn't necessarily define an inner product
>>
>>12933502
No, you are braindead and nongenius
>>
>>12933533
>Basis vectors ax and ay in vector z, vs bases bx and by in q. Inner product just valuates how much they overlap.
even if this made sense (it doesn't), "how much they overlap" is not intersections. intersections are topological, "overlaps" are not.
>>
>>12933534
Because there is no useful definition of angle in terms of open sets, and there never will be really, there are tons of homeomorphisms which do not preserve angles at all.
Suggestion: learn some topology before you become that one psued underclassman no one likes
>>
>>12933570
Cool.
>>
>>12933281
inner product is the most rigid structure there is. its literal purpose is to turn topology into geometry. no, it's not "inherently topological", it's the exact opposite.
>>
>>12933541
Hmm? Square roots aint the only weg though

>>12933584
0 creatiivity, into the garbage

>>12933590
Faggot just tell me the math. I literally could not give a shit about your social hierarchies, Im just a god of shitposting.

>>12933622
But if it turns topology into geometry... Doesnt that mean theres an inherent junction node? Where the interaction and transformation can occur. Read alchemy. Any concept is mathematical if well defined, needs fundamental definition of conception. Junctions, contradictions, elements, etc
>>
>>12933650
Meds. Now.
>>
>>12933655
Literal faggot cant handle a conversation without resorting to psyops. Whatever, Im done with you for tonight
>>
>>12933650
answer to your question is NO, get over yourself you undergrad. lmao.
>>
>>12933650
nonsensical analogies are not creativity, they're a sign of schizophrenia
>>
>>12933682
Its fair for you to assume Im egotistical. The truth is Im just shitposting, and I have pride in my shit, but yeah.

>>12933698
It is not nonsense, and also, the DSM is dangerous. Dont be a slave to the government. You are on 4chan, arent you, or is this place pozzed too?
>>
>>12933750
I'll poz your little schizo neghole
>>
Look /mg/, Im gonna be honest with you. I have been diagnosed with bipolar disorder, and even though I hate the dsm and psychiatry and all that, I understand why you think Im schiz. But I really dont wanna make a big deal of it. Its just what it is.
>>
Hey /mg/, does anyone have a proof as to why hyperoperations (e.g. extensions of operations using Knuth arrow notation) like tetration, etc. lose properties? I remember seeing a formal proof for it somewhere. I'm trying to develop a group theoretical understanding of it since I'm working with tetrations and extensions/algebra on the function (and its inverses) so this would be great.
>>
What is the most kino branch of mathematics and why is it complex analysis?
>>
>>12931140
/mg/ Is /sqt/
And /sqt/ is /mg/
>>
>>12934247
cringe
>>
>>12931140

> Can't decide whether to take this PhD assistantship
> When I wake up in the morning and take my meds (Adderall) it seems like a good idea
> Later in the day after my meds wear off it feels like a bad idea
>>
>>12934195
Which text do I start with for complex analysis
>>
>>12932091
I can't properly study unless I'm thrown into a room with just a book sadly
>>12932133
Amazon heh, I was just browsing randomly and I saw it! It's unlisted now though, it seems like the book has everything pasted on.
>>12932437
Heh, I didn't even realize that, good catch.
>>12933051
I think it's like the Spanish cover or something, the inside is English
>>
Day 1 of dedicating my life to foundations of mathematics. Just reviewing my copy of Enderton's mathematical introduction to logic. Thanks for reading my blog.
>>
DOCTOR
DOCTOR PLEASE HELP
I-I'M GETTING DILATIONS
I THINK MY SON'S GOING BACK IN
>>
>>
>>12934449
Stein and Shakarchi
>>
>>12933114
ty anon, I'm guessing this is the stein and shakarchi book? https://www.amazon.com/Real-Analysis-Integration-Princeton-Lectures-ebook/dp/B007BOK6PW
>>
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>>12934644
>Stein and Shakarchi
I've heard a lot about these two. Should I start with their first book then? The first one is on fourier -> real -> complex -> functional?
>>
>>12934714
You can safely jump around to whatever topics interest you, or go in order. Stop asking questions and just go read already fag.
>>
>>12932988
I mean sort of? It's not a space of ordered tuples because not every element of the tensor product can be written as an ordered tuple. After all, you can't turn (v, w) + (x, y) into a single ordered pair if v and x are linearly independent and w and y are also linearly independent: there's no way to combine them bilinearly. So the tensor product is generated by order pairs, but you do end up with elements which are linear combinations of ordered pairs but which are not themselves equal to a single ordered pair.
>>
>>12933036
I was fine with 3 or 4 math classes a semester, they were less trouble for me than the general requirements were. Obviously you might be different but 3 isn't that bad especially if there's overlap.
Maybe you want to do 2 math and 1 physics a semester or something.
(Also, keep in mind that if you have any plans for graduate school you should be doing your best to push yourself outside of your comfort zone with the amount of math you're doing daily/weekly).
>>
>>12933150
Plenty of ways to learn higher level math for free and to work towards a point where you can make some college worth it, or at least some sort of community college / online classes on the side of your job.
Highly recommend using things like Khan Academy or MIT OCW for reviewing calculus and then for learning Linear Algebra. The MIT OCW Linear Algebra lectures are great, and come with plenty of problems you can work on. Also use 3Blue1Brown's two youtube series on these topics to build intuition.
After calculus and linear algebra, resources become a little more sparse, but you can find youtube lectures and there's always plenty of free pdfs of textbooks online.
>>
>>12934449
>>12934714
I also recommend checking out Richard Borcherds' youtube channel, he is a fields medalist who has been making lots of quarantine lectures. He just finished a whole series on introductory complex analysis.
>>
>>12934921
Holy BASED, thank you
>>12934760
>You can safely jump
That's assuming I know real analysis right
>Stop asking questions and just go read already fag.
okay...
>>
>>12934929
>That's assuming I know real analysis right
It's not necessary, real analysis is isomorphic to a projection of complex analysis anyways.
>>
>>12934943
>isomorphic to a projection of complex analysis anyways.
Oh fuck.
I'm going to start now then.
Much love anon!
>>
>>12933188
Its endomorphism group is isomorphic to itself as an R-module, since it's generated by 1 element. Further, division rings don't have non-trivial two-sided ideals. So as a bimodule, the endomorphism group is simple.
>>
/mg/ I have a question. When you guys study texts, assuming that you have the book physically, do you write in the book itself or do you take out a piece of paper and jot notes down?
>>
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>>12935086
Let's make your words more precise. So we know that [math]\text{End}R(R_R) \cong R[/math]. If we then assume that [math]R[/math] is a division ring, let [math]I[/math] be any of its right ideals. If [math]I \neq 0[/math], there is some [math]0 \neq a \in I[/math], and thus [math]aR \subseteq I[/math]. However, there is some [math]b\in R[/math] such that [math]ab = 1[/math], so [math]1 \in aR \subseteq I \Rightarrow I = R[/math]. This shows the simplicity of the ring as a right module over itself, and should suffice to show that a division ring is the endomorphism ring of a simple right module. Similarly for a simple left module. Unless I'm missing something, this should invert the implication in Schur's lemma (which has like a million different wordings).

>>12935137
>do you write in the book itself
No
>do you take out a piece of paper and jot notes down.
Occasionally.
>>
>>12933590
What's a set?
>>
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>>12935219
HERE WE GO AGAIN!!!
>>
>>12935227
I want to know the answer though, people keep asking and they never seem satisfied with the answer. Why?
>>
>>12935227
Please don’t respond to retards
>>12935219
>>12935235
Kill yourself
>>
>>12935219
It's a collection of objects.
>>
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Alright you faggots how are we gonna solve navier stokes? We will split the prize but dude I need money.


>>12935219
IT IS LITERALLY JUST A BASKET YOU EXISTENTIAL JEW
>>
>>12935257
I just don't want to look stupid like that person who kept using the term set in another thread and someone else kept proving he was an idiot.
>>
>>12935339
Set is a very general term though. A set is literally just a basket, what the hell did that nigga mean then?
>>
>>12935342
I don't know, I'm just afraid of being blown away by someone super smart if I use the term wrong.
>>
>>12935219
Oh boy, the real question is, What Isn't a set?
>>
>>12932379
Seems not repressed enough actually
>>
>>12932636
both
>>
>>12935400
So we're not concerned here about logical issues with the use of sets? I won't be called an idiot for using them?
>>
>>12935417
Nah bro, you're set
>>
>>12935400
The collection of all sets
>>
>>12935359
Imagine caring. Whjen u realize math foralism is just ego norms also useless
>>
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Wait, I'm going to be honest for a moment, is there another definition of set that I'm missing? I've always thought that set is the objects inside a basket, not the basket itself.
>>
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>>12935602
>I've always thought that set is the objects inside a basket, not the basket itself.
You can put a basket inside a bigger basket, but if you would put all baskets but one in the remaining basket, where would you put the last one?
>>
>>12935620
>where would you put the last one?
In your boipucci.
>>
>>12934910
Thanks anon
>>
>>12933059
at least post in fucking png faggot
>>
>>12932514
dont you solve the problems on class tho? As in they teach you the methods (such as sinx=x, only the first 2 terms of the taylor polynomial, etc.)
>>
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>>12935471
Does a set of all sets contain itself?
>>
>>12933187
what do you mean by your second statement?

I constructed a family of ellipses in desmos and at that point i realized i don't really understand what you mean by 'a normal to a family of curves'
>>
>>12935731
What are those
>>
>>12935821
Some accurate calipers
>>
>>12935620
That sounds like a big basket so I would put it in the garage
>>
>>12934645
Yup! It's a really nice book for self study. I think you probably only need the first two chapters to jump into Durrett.
>>
Is union really necessary in ZFC? It seems like everything it proves can be proved without it
>>
How do I show that pic related is equal to (0,0)?

sen (t) = sin (t) btw.
>>
>>12936119
1st: treat the two expressions around the comma as two different limits (obviously)
hint: multiply out e^(at) and write sin and cos in their exponential forms
>>
>>12936118
To clarify, I mean ZFC specifically and not ZF
>>
>>12934195
I was reviewing Complex Analysis the other week and was having this exact same thought.
It's pretty damn kino for sure. Even just the graphs and visualizations of functions are cool.
>>
>>12934875
oh yes of course, thanks!
>>
>>12936118
How do you prove that every ordinal has a successor without union?
>>
>>12936135
Thanks for the hints, I managed to make progress, but it's tending to infinity for me...
>>
>>12936119
The limit is indeterminate if [math]b \geqslant 0[/math]. There must be something information telling you that [math]b < 0[/math] if you're "supposed" to get a limit of [math](0, 0)[/math].
>>
>>12936288
Yeah, it states that a>0 and b<0
>>
Do springer hardcovers use a sleeve or are the covers printed on? I wanna get one but I'd rather have the original cover instead of the boring ass yellow.
>>
>>12936472
No, most of them are printed on demand with the worst type of paper and printing. There are a few exceptions like Axler's Linear Algebra Done Right which has good quality paper and printing.
>>
>>12936550
I mean, I can deal with shit printing, I'm just more curious about the cover. I prefer soft covers but if the hard cover is a sleeve then I could probably print off my own
>>
>>12936560
It's not a sleeve.
>>
>>12936584
fuck. thanks
>>
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>>12936218
Seems like it's provable

https://www.researchgate.net/publication/220592514_On_the_axiom_of_union

>>12936118
Seems like ZFC minus Union still proves that [math]\bigcup X[/math] exist iff [math]\{|\,x\,|\mid x\in X\}[/math] holds only finitely many infinite cardinals.
Cartesian products and all this stuff exists also without it.

I suppose you need it "already" if you want a set like [math]\bigcup_{n \in {\mathbb N}} {\mathcal P}^n A[/math], for some A, and where my index set is very moderate.

The paper seems to end up saying this: Of course in full ZFC, if [math]S[/math] is a set of Neumann ordinals, then [math]\bigcup S[/math] (which intuitively ends up just being the biggest ordinal in S) bounds S from above.
Conversely, if you want to say that a given set of ordinals S surely is bounded above, then you need exactly the axiom of union (and can then in particular say that the bound is [math]\bigcup S[/math]).

>>12935342
>>12935602
>I've always thought that set is the objects inside a basket, not the basket itself.
An issue with that is that the empty set is (ostensibly) a set too, so that intuition is itchy. Here the basket idea would resolve this. However, the basket would seem to also be a visualization of a multiset (e.g. a basket holding two 7's), so I'd feel inclined to not lean into the physical intuition too deep either.
What's clear is that basically all set theories let you form {X} from X, so a set is always something that can be inside a set too.
(I advocate the perspective that "set" corresponds to a certain class of properties.)
>>
>>12936685
Just came to me that I can be more concrete.
The set
[math]\{{\mathbb N}\to \{0,1\}, \ \ \ ({\mathbb N}\to \{0,1\})\to \{0,1\}, \ \ \ (({\mathbb N}\to \{0,1\})\to \{0,1\})\to \{0,1\}\}[/math]
which holds the characteristic functions on N, the characteristic functions on that function space, and so on,
has a tower of bigger and bigger cardinalities, and to form it's union you truely need the axiom
>>
...}
>>
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>>12936064
Based!
>>
>>12935137
it's critical for me if I'm self studying a book to work with a pen/paper to verify as much of what I'm reading on my own. Anytime theres's a theorem, try to prove it yourself first. Anytime the author says that something 'follows immediately' or that 'the general case is similar' you should work through it yourself.

This also quite efficient in my experience because by treating all of the theorems/lemmas/propositions/examples as exercises you 1) don't have to worry about finding practice problems, since many books don't have any 2) have an immediate answer key if you get terribly stuck and need a hint
>>
How does the universal property of tensor algebras work?
>>
What does this notation mean? Dot product?
>>
Fucking algebra everything sound so fucking trivial but it is a pain in the ass to prove it.
>>
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>>12937335
Let [math]k[/math] be the coefficient field we are working over. Then let [math]V[/math] be our vector over that field, and let [math]f\colon V\to A[/math] be any [math]k[/math]-linear map from the vector space to a [math]k[/math]-algebra. You should know that [math]T_k(V) = k \oplus V \oplus (V \otimes_k V) \oplus \cdots[/math], so so how do we extend [math]f\colon V\to A[/math] to an algebra homomorphism [math]g\colon T_k(V) \to A[/math]? If [math]\mu_A \colon A\otimes_k A \to A[/math] is the multiplication of [math]A[/math], consider the composite [math]V \otimes_k V \xrightarrow{f \otimes f} A \otimes_k A \xrightarrow{\mu_A} A[/math]. I'll leave checking the [math]k[/math]-linearity of this to you. Using this idea, let [math]f_n\colon V^{\otimes n} \to A^{\otimes n}[/math] be the nth tensor power of [math]f[/math]. Then, if [math]n \ge 2[/math], we compose that with [math]\mu_A \otimes 1_A^{\otimes (n-2)}[/math], then with [math]\mu_A \otimes 1_A^{\otimes (n-3)}[/math] etc. until we have reduced the tensor power to 1. Note that this is OK because both [math]T_k(V), A[/math] are associative! I'll leave checking that this gives you an algebra homomorphism to you as well. But that's pretty much how to do it. For higher powers, keep multiplying the first two until you have no tensor product left.

>>12937393
Yes, or more generally whatever inner product you have in your space.
>>
>>12937410
I have no idea why I put that [math]f_n[/math] thing there.
https://www.youtube.com/watch?v=uO_L-gEIf8o
>>
>>12937410
>Yes
I don't get why the modulus there...
>>
>>12937410
Ok so if I have [math](v_1 \otimes v_2 \otimes \cdots \otimes v_n)(w_1 \otimes w_2 \otimes \cdots \otimes w_m) = v_1 \otimes v_2 \otimes \cdots \otimes v_n \otimes w_1 \otimes w_2 \otimes \cdots \otimes w_m[/math], then that is sent to [math]f(v_1)\cdots f(v_n)f(w_1) \cdots f(w_m)[/math] by your thing. Thanks! Shouldn't you be sleeping btw? It's night in Europe.
>>
Anyone know anything about the logic of Birkhoff and von Neumann? They proposed a logic based on quantum mechanics, without negation (and the law of excluded middle). If this matches with how the real world works, why didn't this logic catch on? It's not even used by physicists. Why use a logic for mathematics and other things that doesn't work the way the real world works?
>>
>>12937481
Our logic is perfect. There's no need for another one. That's pure academic masturbation.
>>
>>12937484
Yeah, I'm reading Quine and he seems to think the same thing. But he says stuff like it's simple and works well, and that it's familiar. These seem not entirely satisfying to me.
>>
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>>12937443
I assume you have a real or complex vector space here. Whenever you have an inner product, it comes with a norm for your vectors. You define [math]\vert\vert v \vert\vert := \sqrt{\langle v, v\rangle }[/math]. This idea comes from the fact that it generalises the dot product, and in two dimensions that will give you precisely [math]v\cdot v = v_x^2 + v_y^2 = \vert\vert v\vert\vert^2[/math], and it works.

>>12937462
Yes, exactly. That's how you turn a morphism of vector spaces into a morphism of algebras over the same ground field. I probably should be sleeping, but I slept like 7-15 and 20-00 yesterday, so my cycle is a bit off.
>>
>>12937406
Just wait until you get to Topology and learn how involved it is to prove the "obvious" fact that [math]\mathbb{R}^n[/math] and [math]\mathbb{R}^m[/math] aren't homeomorphic (for [math] n \neq m[/math])
>>
>>12937527
Prove ur claim fggt
>>
>>12937527
I think it's easy to do with higher homotopy groups. I might be remembering wrong though, been a while.
>>
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>>12937549
You don't even need homotopy. Homology is sufficient. Suppose there is a homeomorphism [math]f\colon \mathbb{R}^m \to \mathbb{R}^n, 1 \le m < n[/math]. Remove the origin and [math]f(0)[/math] respectively to get homeomorphic spaces, homotopy equivalent to [math]S^{m-1}, S^{n-1}[/math], so those two will now be homotopy equivalent. This implies [math]0 = H_{n-1}(\mathbb{R}^m \setminus \{0\}) \cong H_{n-1}(\mathbb{R}^n \setminus \{ f(0)\}) \cong \mathbb{Z}[/math].
>>
>>12937592
Do you use discord?
>>
So its basically just... At the origin its a few axes which implies a dimension and Q=(you cant mix dimensions)

But why Q?
>>
>>12937608
I do. Dapta#4167 not anitr
>>
>>12937625
If I add you will you post lewds?
>>
>>12937634
Depends who you are. Im a guy though
>>
>>12937640
I'm a guy too...
>>
>>12937645
Well then I most likely wont post lewds unless youre magic. Come add me anyway mate
>>
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>>12937608
Nope, sorry. Besides, I would not post lewds.

>>12937640
>>12937645
The love story off our time.
>>
>>12937621
>>12937651
Help nerd
>>
>>12937651
>off
I'm retarded.
>>
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>>12937653
Can you write it in homo sapiens? I can't read australopithecus.
>>
>>12937663
Like the nonhomeomorphism is about dimensions about the origin, right? But what are dimensions and how does thet effect homeomorphics
>>
>>12937592
Well there you go, either way, I knew it wasn't very 'involved' to do.
>>
Number theory is just trivia about primes, change my mind
>>
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>>12937667
Well, the origin of [math]\mathbb{R}^m[/math] can be replaced with any point [math]x[/math], and then you remove [math]f(x) \in \mathbb{R}^n[/math]. You can then restrict your homeomorphism to a map between these two punctured Euclidean spaces. Things to check:
>continuity of the restriction
>the restriction is bijective
>the inverse is continuous
I'll leave those to you. After that, you can deform both spaces to spheres of one dimension less. This is where the important part happens, the removed points don't really matter. [math]S^d[/math] has non-trivial homology only in degrees [math]0, d[/math], but two spaces being homeomorphic or homotopy equivalent means they have the same homology groups. Since the dimensions differ, this will not happen to the punctured spaces (because it doesn't happen to the spheres), so they can't be homeomorphic, which contradicts the assumption that the original spaces were homeomorphic. The dimensions in this case just mean how many coordinates you have.

>>12937670
Yup. I think the homological version is cleaner, as the homotopy groups of spheres are crazy. However, you could still do essentially what I did, but instead take [math]\pi_{m-1}[/math] of both spaces. One would be trivial and the other would either consist of two path components ([math]m=1[/math]) or be the integers.
>>
>>12937495
Did you actually watch WEP? I thought of picking it up because I enjoyed Flip Flappers and it seems to have some similarities.
>>
>>12937731
Yes that's the one I was thinking of. Obviously I think of it because homotopy groups were covered later than homology in my course so thinking of that was fresher in my mind. I'm no algebraic topologist, so... haven't thought much about it since.
>>
>>12937801
What do you think about these days anon?
>>
>>12937805
Number theory, and logic. I know there's arithmetic topology and such, but I don't know much about anything like that. So I haven't really used the material from algebraic topology since I took the course.
>>
>>12931507
>Answer me fast: Does a straight line have a normal line?
Points on curves have normal lines. Not lines. A straight line is a curve so yeh every point has a normal and its trivial
>>
>>12937816
Is there some overlap between those topics? Or are they just disparate interest of yours?
>>
>>12937825
There is overlap some places, but I'm no specialist. I don't do research or anything. I just read stuff and work on problems. I like philosophy which is where my interest in logic comes from but I didn't mention that since it's not that math related. Most mathematical problems I do are from number theory textbooks. I'm not advanced enough to formulate my own questions in the field.
>>
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My brain is fried and I can't understand simple things right now but I don't get this at all. I've reread the stuff about inner products over and over again.
>>
>>12937838
Fair enough. Do you ever plan to do research some day or is it just a hobby for you?
>>
>>12937852
Assuming we're working over the reals,
[eqn]\|\mathbf{x} - \mathbf{y}\|^2 = (\mathbf{x} - \mathbf{y})^T (\mathbf{x} - \mathbf{y}).[/eqn] Take it from there.
>>
>>12937860
Hobby, I couldn't get into a good enough PhD program to be worthwhile. The only job I'd get at the end of it is teaching idiots at a community college and that doesn't appeal to me.
>>
>>12933036
I agree 3 math classes is the threshold, even for autists.
>>
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>>12937791
Not yet. I've heard positive things about it, but also felt a bit too sad to watch anything sad lately. Flip Flappers is really nice, though!

>>12937801
I get what you mean. It's easy to learn using the "bulimia method". Then you binge on the course material, puke it on the exam paper and then you are empty again. Funnily, it is always the very basics and the stuff in the end, but never the middle part that leaves any traces in one's mind, it seems.

>>12937816
Not that I would know anything more than the very basics of model theory, but can you give some dumbed down overview of what logic stuff you are doing? Is it some set theory stuff, or is it something like dependence logic?

>>12937852
Do what the other anon told you, then split the thing into inner products you know the values of and the one you are after.
>>
>>12937872
I shouldn't say idiots, that's bad of me. I just mean I don't want to teach 'college algebra' to people who don't really want to take it for a living.
>>
>>12937879
Just the basics. The only formal course I took was a philosophy course, and I've studied set theory books (like Jech). The book I'm trying to get into right now is Simpson's
Subsystems of Second Order Arithmetic. For some reason the reverse mathematics thing is appealing to me. Don't expect great insights or conversation from me. I just muddle through this stuff because I enjoy learning about it.
>>
>>12937865
Still lost.
>>
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>>12937892
I see. Have you made much progress yet? What drove you into logic, if I may ask? Are you there in pursuit of Gödel's theorems and then trying to see what comes after them? I had my own little logic is cool phase back in the days before I got filtered by the first (and luckily only compulsory) logic course. It was sparked by reading about Gödel in G.H. von Wright's book about how logic developed from the Greeks to the writing days, roughly 1960's.

>>12937928
[math](x -y)^T(x-y) = x^Tx -x^Ty - y^Tx + y^Ty[/math]. Then remember that [math]\vert\vert a \vert\vert = a^Ta[/math]. You can do it. Proof: even I can do it.
>>
>>12937934
Oops [math]\vert\vert a\vert\vert^2 = a^Ta[/math]!!!!!! Sorry.
>>
>>12937934
I'll assume you're not asking me questions to prove I'm an idiot, I can save you time and admit that I'm an idiot and that I know nothing right now if that's the case.
Otherwise, the course I took did prove Godel's theorems. I liked mathematics, and philosophy, so I took the course. The course left me with a lot of quesitons so I kept reading material that's all there is to it.
As for progress, I'm just reading the seciton on the Weak Konig Lemma (specifically it is equivalent to Godel's Completeness Theorem). Sadly the book doesn't have problems, but I typically write out the theorems and try to expand them to as much detail as possible. Then in a few days I try to prove them myself to see if I still can. That's the best I can do with books that have no exercises.
>>
>>12937995
I think another fun thing to do is relax or remove assumptions for a theorem and see if it’s still possible to get the conclusion.
>>
>>12938002
Yeah that's a good technique. Any way of interacting with the things you read is good!
>>
>>12937489
Lattice theory was canceled anon.
>>
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>>12937974
Huh, I'm glad I don't have to attend any lectures or tutorials online... Luckily my English is less broken than your instructor's.

>>12937995
>I'll assume you're not asking me questions to prove I'm an idiot, I can save you time and admit that I'm an idiot and that I know nothing right now if that's the case.
Fear not, for my intention is not to torment you. I don't think you are an idiot, though.
>Otherwise, the course I took did prove Godel's theorems. I liked mathematics, and philosophy, so I took the course. The course left me with a lot of quesitons so I kept reading material that's all there is to it.
I see. I understand this kind of action very well. Have you found answers to your questions? They tend to be like the heads of Hydra. Answer one and get two new.
>As for progress, I'm just reading the seciton on the Weak Konig Lemma (specifically it is equivalent to Godel's Completeness Theorem). Sadly the book doesn't have problems, but I typically write out the theorems and try to expand them to as much detail as possible. Then in a few days I try to prove them myself to see if I still can. That's the best I can do with books that have no exercises.
Funny how there is that connection. Is it somehow related to considering natural deductions as trees or something similar? I think your style is quite good (quite because I do the same, so it must have flaws).
>I think another fun thing to do is relax or remove assumptions for a theorem and see if it’s still possible to get the conclusion.
This too. Relax the assumptions and see if the conclusion has to be weakened.
>>
>>12938067
I haven't found answers, no. It always seems like I get this feeling that something is missing or eluding me. The best solution I have is to just keep learning and hope I either just get used to something or have some revelation (unlikely).
I'm still figuring out the WKL -> Completeness theorem thing. Unfortunately it's a statement of 6 things being equivalent and you know how that goes. Plus several lemmas first. It's a mess to detangle, I'll be reading this section for months probably. I'm pretty slow.
>>
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>>12938090
>I haven't found answers, no. It always seems like I get this feeling that something is missing or eluding me. The best solution I have is to just keep learning and hope I either just get used to something or have some revelation (unlikely).
I'd like you to promise me you won't stop until you find the answers or death claims you. That way you will or will not find the answers, but at least you wouldn't give up on the way to higher understanding. Make a pact with me, anon.
>I'm still figuring out the WKL -> Completeness theorem thing. Unfortunately it's a statement of 6 things being equivalent and you know how that goes. Plus several lemmas first. It's a mess to detangle, I'll be reading this section for months probably. I'm pretty slow.
Ah, I see. It doesn't even have to be as hardcore as 6, even 3 is enough (for example, Dugundji's proof of AC -> Zorn -> Zermelo -> AC in which it's far from obvious how Zorn's lemma itself would imply the axiom of choice) to make things confusing. Do you have anyone to talk to about it and get answers to questions about unclear things? Maybe SE. At least you aren't competing against anyone, so being slow is OK. Please remember that instead of throwing books to the walls and crying under the blanket when stuck...
>>
>>12938168
I dunno why you'd want to, but I'll make the pact. I don't plan to stop reading things that's for sure. Unless I get dementia or something. I don't really talk to anyone about stuff, that's why I'm bad at communicating it to others.
I thought of making videos explaining stuff since teaching is the best way to learn something but I'd just make a fool of myself.
>>
>>12933187
>>12935739
What I was talking about is called orthogonal
trajectory. Take a member of the family of
ellipses. Establish a tangent line at a point
on the ellipse. Then there is a normal that runs
perpendicular to the tangent line crossing into
the ellipse. Note the slope of the tangent at
that point.

The other side of the ellipse has a point where
the slope of the tangent line is the negative of
what you noted. Place the tangent line there
and you will have a normal exiting from the
ellipse. Connecting the two normals inside
the ellipse makes a hyperbola shape, a member
of the family of hyperbolas.
>>
>>12938228
The hell is this 50 columns bullshit
>>
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>>12938220
Because I just want you to keep doing the things you enjoy. It's so easy to give up. Are you sure you'd make a fool of yourself? Obviously, you could apply >>12938002 to your videos. Now you are assuming you would make videos for teaching things. What if, instead of that, you would make videos about your thoughts on the stuff? Something like choosing the main topic of a video, going through the theory and then telling the viewer your interpretations and observations? Instead of teaching, a sort of dialogue invitation. I could watch those, unless they go too deep immediately.

But anyway, it's been morning for ages. Time to (try to) get some work done. Take care, friend(s)!
>>
>>12937481
You mean
https://en.wikipedia.org/wiki/Quantum_logic
?
This has a negation.
It's studied. It's also studied without LEM.
>>
>>12937670
Proving homology is homotopy invariant is pretty involved
>>
>>12938228
>>12938237
Step by step instructions to draw one orthogonal
trajectory of the ellipse.
>>
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>kids doing review work since the ap exams are looming near
>Using the chalkboards for my own work since they are just going about their own shit quietly
>Proving the snake lemma
>"Teacher, what are you doing?"
>"Some math, did you get stuck on a problem?"
>"No, it's just what you've drawn on the board doesn't look like math at all"
I don't know what feel this is
>>
>>12938860
Tell them that what they learn in school isn't mathematics and say they should demand their teacher teach them from Bourbaki.
>>
>>12938860
Some math doesn't look like math just like some girls look like men. It's the new normal.
>>
>>12938860
Probably making fun of you, oh look at this guy he thinks he's doing mathematics but he's just on another diagram chase.
>>
>>12937527
This is obvious, you can easily come up with an intuitive argument essentially equivalent to a homology computation which proves that they are nonhomeomorphic (similar to the idea in anime avatar's proof: put an S^(n-1) around the origin in both R^m and R^n, m>n, and one can be slid off to infinity, the other is locked around the origin. Of course the precise proof of this is precisely the homology argument anon posted, but my point is that you can explain the homology argument in words like I did to a 9 year old).
>>
Why does the adjoint of a linear operator T need not be the same as the adjoint of T restricted to a T-invariant subspace? I can construct such examples easily, but still don't understand this fact.
>>
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In linear / convex programming, do Brits call it a "linear program" or a "linear programme"? Google gives conflicting results.
>>
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applied maths precedes pure maths and pure maths is a subset of applied maths, fight me
>>
How do I prove pic related?
>>
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do you understand it, guys?
>>
>>12939230
what have you tried?
>>
>>12939240
No, I don't know Polish, sorry.
>>
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>>12939242
here it is in Reverse Polish, if that helps
>>
>>12939241
NVM Figured it out, was lamer than I tought
>>
>>12939241
I tried nothing yet, do you have any hints?
>>
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>>12939230
Write out what it means to be an element in the union of complements. Write out what it means to be an element in the complement of the intersection. Compare.

>>12939240
Yes. You don't?

>>12938860
The proper way to chase diagrams is by using your index fingers (or whatever you have protruding out of your arms, pardon my ableism). You place one of them on the diagram on the object from which you start, and whenever you lift something somewhere, you put the other pointing organ on that object. Then you just keep moving them around your diagram until you have proved what was to be proven.
>>
>>12939572
Use your slender fingers to chase my dick
>>
>>12939624
I don't think that's possible in subspaces of dimension 0.
>>
>>12939572
>or whatever you have protruding out of your arms, pardon my ableism
Did you just assume I have arms?
>>
I am trying to trend a group of capacitors over time to catch individuals that might be trending toward failure.
Each stack is outdoors and contains at least 100 cap cans connected together. I have each measured yearly, and from the factory.
So I made a spreadsheet that compares the most recent value to factory, and set it to conditionally format it if the % difference is off by 3%. It is effective at identification of failed cans, but not for spotting issues before hand.
Is there an effective way to trend groups of data points with respect to time?
A 100 point xy plot seems like a bad idea
>>
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>I would like to remind you that there will be an Oberwolfach seminar on:
>"Cellular E_k-algebras"
>from 23 - 29 May 2021. (Postponed from its original dates, 31 May - 6 Jun 2020).
>The lectures will be given by the organizers, and the seminar will be either hybrid or fully online.
>This seminar is an introduction to the homotopy theory of E_k-algebras in spaces and chain complexes, aimed towards applications to >the homology of moduli spaces.
>We will cover foundational topics including cellular E_k-algebras, derived E_k-indecomposables (i.e. topological Quillen homology), the Hurewicz theorem in this context, the relationship between derived E_k-indecomposables and iterated bar constructions, the description of the homology of free E_k-algebras, and several spectral sequences associated to cellular E_k-algebras. These techniques will then be applied to examples given by various moduli spaces, where the phenomenon of homological stability can be easily understood from the perspective of E_kcells. The highlight of this seminar will be the recently discovered ?secondary homological stability? for mapping class groups of surfaces. In addition to the homotopy-theoretic techniques for working with cellular E_k-algebras this requires detailed input specific to mapping class groups, which we shall survey.
>The application deadline is 11 April 2021 and for more information on how to apply, see https://www.mfo.de/document/2121a/Poster-2121a.pdf.

>>12940223
Yes... Sorry I'm the worst...
>>
>>12940420
You are the absolute worst.
>>
Bros... i did maths...
>>
>>12940630
Cool... i did your mom...
>>
>>12940642
y-you too?
>>
>>12938228
So i kind of got what you meant.
https://www.desmos.com/calculator/nsuacsmpi3
So did you mean the two intersecting normals create the assymptotes of a hyperbole?
>>
is math in uni actually just about the concepts rather than numbers? i had the equivalent of B because of retarded stuff. i had no problem with problems or understanding the subject but my retarded brain would always fail at shit like 17+17, (x+1)^2, forgetting a component of the chain rule, flipping the bayes theorem the wrong way, etc which you are supposed to do without giving it a thought
>>
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>>12940497
Honey, I'm fabulous.

>>12940884
It is about concepts that can be applied to numbers. Then you go deeper and deeper until you have seemingly lost all connection to the number side, but you can then start peeling off the layers of generalisation and abstraction to get back to 1+1.
>>
>>12940983
How the fuck are categories supposed to generalize numbers?
>>
>>12941227
Just see the construction of the natural numbers in Mathematics Made Difficult. This is basic stuff anon.
>>
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>>12941227
Let's begin with natural numbers. We will include 0, because discrimination is a no-no. This makes [math](\mathbb{N}, +)[/math] an example of a monoid, a structure based on the ideas that we can take any elements in our set, operate them and the result will still be in our set, that we may perform the operations in any order, and that there is an element operating with which does absolutely nothing. The first two are satisfied by the natural numbers under addition, and the third if we include 0. That will be our number connection. Then we turn our attention to categories. First, assuming there is precisely one object in our category, one can easily check that the morphisms satisfy all three of those properties, so as long as there is a set of those and precisely one object, we have a categorified monoid. Next, suppose we have an arbitrary set of objects, possibly empty, for each of which the endomorphisms are sets. This generalises the monoid thing by becoming a "monoidoid" if you will. Finally, drop all size limitations to generalise even further. This way any category is a generalised version of [math](\mathbb{N}, +)[/math]. Buried under all the abstract nonsense, the numbers lie.
>>
>>12941326
>We will include 0, because discrimination is a no-no
It's like you don't even like the free semigroup on one generator. :(
>>
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>>12941424
There is nothing wrong with having a deficit like missing the 0. Let's all get along even if we are different. Even monoids and proper semigroups. We are all valuable.
>>
>>12941493
I just wish that algebra and topology hadn't split so hard in the 1950s. H-spaces are topological unital magmas and operads are abstract clones. Everything has (at least) two names now.
>>
>>12941493
>We are all valuable.
This is true. You may be a man in a dress but even you have some value. Keep it up bro.
>>
>>12940777
>>12938228
Yes, these normals frame the hyperbola you will
draw as it enters and exits the ellipse. Also, since
hyperbolas come in pairs there is one on the
other side of the ellipse. There you have it.

As a challenge, make a series of ellipses and
the hyperbolas that passes through them at
certain points on the ellipse. You'll see that the
family of ellipses is orthogonal to the family of
hyperbolas.

Next question: What is orthogonal to a family
of circles?
>>
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>>12941518
Even the brightest of us are humans, humans are simple, and therefore a team of humans will be semisimple. When these simple humans form a ring and start working on something, that something will necessarily split. Sadly, there is no glue to actually unify them again. I guess one could try to formulate some sort of "universal topology", and then somehow connect it to universal algebra via some metatheorems or something, but that would be so awfully abstract and probably mostly useless or at least avoided by many. Alternatively, the higher category theorists could find the correct formalism for their stuff, and then there would be the possibility of having some higher enriched categories unifying them, but that would also be super abstract and debatable whether it would be worth the time to actually get used to their formalism etc. I don't see a unified future.

>>12941531
Love you too, sister.
>>
Why do we give a fuck about Pfaffians again?
>>
>>12941662
It's an invariant of some thing - why wouldn't it be relevant?

iirc I encountered it as [math]\vec E^2 - \vec B^2 [/math] in my Master thesis, the symmetric form being [math]F={\mathrm d}A[/math].
>>
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>Theorem: The Law of Excluded Middle is equivalent to the statement that subsets of finite sets are finite

Ummmm constructivists, explain yourselves.
>>
>>12941547
Updated version:
https://www.desmos.com/calculator/ypfikfva5x
i have heard of this property, altough im still struggling to make it work.

Im thinking of fiber bundles of radial lines. But if i treat circles as ellipses, it has to relate to hyperboles.
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>>12941326
>Set
What's a set? Don't you know I can just say I define a champocle! That doesn't mean it exists, too bad you're not a brilliant contrarian genius like me. You will never understand you're just doing fake mathematics.
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>>12941890
Hi Wildburger.
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>>12942035
I'm making fun of the guy who always says stuff like that. He always uses some made up word as an example for why things don't exist.
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If I want to go back to get a PhD in math at age 29, how difficult would that be? I have a bachelor's degree in math/stats and a masters degree (non thesis) in operations research
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>>12941877
Subsets are richer constructively, because predicates are.
If [math]A[/math] is a set and [math]P[/math] and unary predicate, then [math]\{x\in A\mid P(x)\}\subset A[/math].

E.g. if [math]A=\{3,8\}[/math] then the four decidable predicates "always always true", "is even", "is odd", "is always false" give rise to the found subsets [math]A=\{3,8\}[/math], [math]A=\{8\}[/math], [math]A=\{3\}[/math], A={}, [math]\{3,8\}[/math]. Generally, on a set of cardinality n, there are 2^n characteristic functions and thus at least 2^n subsets.

But since predicates don't collapse to mere characteristic functions if you adopt excluded middle ("let's make everything decidable"), the constructive power class, i.e. the class of all subsets [math]\{x\in A\mid P(x)\}[/math] of A, is generally far greater than the function space of characteristic functions.
You might think of subsets of sets as corresponding to described but not necessarily terminating processes, whereas "function" means the nice terminating case.

Constructively, to prove a disjunction requires you to be able to prove one of the disjuncts. If you got an undecidable predicate, let's say the continuum hypothesis [math]{\mathrm{RH}}[/math] is one in our context, then classically [math] {\mathrm{RH}} \lor \neg {\mathrm{RH}} [/math] is true (per axiom LEM), while without LEM it's just as undecidable as [math] {\mathrm{RH}}[/math] itself.

As a consequence, if you got a singleton, say [math]A=\{7\}[/math], then [math]\{7\mid {\mathrm{RH}} \lor \neg {\mathrm{RH}}\}[/math] is classically just [math]\{7\}[/math], but constructively we can neither judge it to be [math]\{7\}[/math] nor [math]\{\}[/math]. Subsets are richer constructively, because predicates are.

To be finite means to be in bijection with an n in N. If we don't know about any of the elements of a subset, we can't put it into bijection with any set. So the subset of A may fail to be finite, even if A is.
LEM trivializes predicates and thus subsets.
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>>12942081
My bad, that was almost like what Wildburger would say. I think he used unicorns as an example in some video. Sadly he is correct. They don't exist.
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>>12942129
Why don't they exist?
>>
which area of mathematics would you find it most likely that a non-mathematician would make a huge breakthrough, mostly because the relevant problem concerns broader creative thinking? graph theory?
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>>12942159
Most likely such a breakthrough would come from a physicist or computer scientist (moreso computer scientist I think). So whatever areas they work closest to I guess.
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>>12942133
Proof of non-existence: If they did, I would have one as a pet. I don't have a unicorn, so they don't exist.
Reason for non-existence: The world is a horrible place and life is unfair. If I could tweak the parameters a bit, there'd be mythical beasts and people would be less broken inside. I can't so we'll just have to try and survive in a world without horned horses and fire breathing lizards.
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>>12942183
Oh you mean unicorns, not sets. So we all agree sets are OK?
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>>12942108
some erratas:
shouldn't have copypasted the
A=
for the four subsets, obviously

CH would be the appropriate name for the hypothesis, not RH

Might have also written, more properly
{x in {7} | RH or not RH)}
for the subset of {7} that's none of the classical/binary cases
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>>12941877
It's worse than that, there are many non-equivalent ways to be finite constructively. Your theorem actually depends on the definition of finiteness.

And yes, it's a good thing, because this refinement has computational implications.
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>>12942196
nLab lists a few
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>>12942187
Yeah, I thought you meant unicorns... But, does it actually even matter to you whether they exist or not? Life is so full acting as if something was true regardless of its actual truth value. I can't even say "I" exist for sure, but I try to believe I'm a real human bean and act as if that was the case. In the same way I'm OK with sets not necessarily even existing, but still acting like they did. I don't know, I should probably go to sleep rather than write nonsense like this.

>>12942203
That first version of Dedekind finiteness is actually what direct finiteness is based on: if [math]M = M\oplus N[/math], then [math]N=0[/math].
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>>12942242
it's nLab, everything is pathologically explained in terms of projective object or what do I know..
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>>12942242
No it doesn't matter, in fact they don't exist. He's actually right. ZFC doesn't 'define' sets. We just agree that the thing set in a natural language is what the ZFC is about. We can't define them/prove they exist because ZFC can't prove its own consistency.
However, I find it rather pointless to go around arguing with anyone that mentions function, set, number on the internet. It doesn't prove he's smart for not agreeing to what everyone else does, it just proves he is being a jerk.
Most people just want to get on with their lives and do mathematics and there's no problem with that.
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Anime tranny, I love you.
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>>12942274
I wonder if there is some [math](\infty, 1)[/math]-topos in the language of which nLab is not horrible to read. Probably not.
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>>12942326
Speaking of which, there's a nice short paper on one of the simplest non-trivial topoi and how to use it's classifying object
(the arrows into its "powerclass of the singleton [math]\Omega:={\mathcal P}\{0\}=\{\{\}, \{0\}, \dots\}[/math]")

https://arxiv.org/pdf/math/0306394.pdf

Spoiler: Despite the non-binary nature of it's logic, the truth objects can be represented by a penis
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>>12941547
>>12941878
Okay, the family of ellipses in the desmos must
have different sizes, not translations. Keep the
ellipses centered on the origin and just alter the
sizes. The hyperbolas, depending on the point
on the ellipse they enter and exit, follow nicely.

You mentioned radial lines to my question, and
that is correct. I will attach two pics of the
situation of ellipse and circle. This is the ellipse
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>>12942108
>>12942196

Yeah ok, this makes sense the more I think about it.
Person gives me an arbitrary subset of a finite set, claims its finite.
>So show me its enumeration.
Uhhhhhh... well we can just choose arbitr— *neck snap*
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>>12941547
>>12941878
...and this is the circle
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>>12942321
>No it doesn't matter, in fact they don't exist. He's actually right. ZFC doesn't 'define' sets. We just agree that the thing set in a natural language is what the ZFC is about. We can't define them/prove they exist because ZFC can't prove its own consistency.
I guess the same is true for all other axiomatisations. No matter how many meta layers you add, there will still be a notion that is left undefined, and some of those things will then be your proto[math]^n[/math]-sets. For n=1, you take classes (for example in the sense of NGB), but what are the proto-classes or proto[math]^2[/math]-sets then? Define those as a special case of something else, but what are those? You just have to pretend your thing is based on something firm enough.
>However, I find it rather pointless to go around arguing with anyone that mentions function, set, number on the internet. It doesn't prove he's smart for not agreeing to what everyone else does, it just proves he is being a jerk.
Probably also he has a list of witty counter-arguments. He asks what a set is, you give some of the usual definitions and he picks the counter for that. Like the Chinese room works. It's a way to kill one's own and waste other people's time. Some have it too much, and some too little.
>Most people just want to get on with their lives and do mathematics and there's no problem with that.
Yup. The game can be bad, but it works well enough when people are playing with the same rules.

>>12942323
Please don't. I'm toxic.

>>12942371
Nice, I'll give that benis paper a read when I wake up! I guess it shows that all logic can be "weird", as even the most mentally phallic thinkers can be non-classical hehe... Vielen Dank. Why are we still awake, by the way? The hour is so awfully late.
>>
What if there are no logical elements other than the contradiction? What if there are no causes, only effects?
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>>12942444
It starts to become a little crazy really. You can't completely define the natural numbers with PA either. You have to use full 2nd order PA, but in order to prove this you have to use a meta-system that already has PA. Anyway, philosophers have discussed this stuff to death for 100 years now like I said, he's not proving anything profound by confusing people that have a little less knowledge than he does on 4chan.
>list of witty counter-arguments
Since you mentioned that unicorn thing, I bet if I listened to wildberger I'd actually hear all his arguments. I just don't have the time to watch those videos.
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>>12942108
You have weird tastes. Are you autistic? What kind?
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>>12941878
>>12941547
Here is one more example, just because it looks
nice. To see how orthogonal trajectories
are done by calculus check this video:
https://www.youtube.com/watch?v=dlfH9iXMcig

Here's a Wiki on the same thing for Cartesian
coordinates, not too far from what you found
on desmos:
https://en.wikipedia.org/wiki/Orthogonal_trajectory#In_cartesian_coordinates

Anyway you describe the family of curves, this
process turns it to a family of a different type
of curves. Any questions?
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>>12942465
Then you do like the FSB/KGB does and invent the causes. Everyone is a criminal, but the guilt precedes the crimes sometimes.

>>12942494
And to prove that your metasystem works, you need a metasystem for that etc. I appreciate the work logicians are doing, but at the same time it just feels so doomed when focusing on these things. No matter how hard they try, the bedrock will still have dirt over it. Surely there are some reasons for them to (be funded to) do it, but those evade the mind of a simple algebraist. For me it's more like a religion without any promise of salvation in the end, but not the boring going to church style. More like a fun being one with nature and maybe talking to your ancestors kind of thingy. You just believe and do instead of thinking about it too much.
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>>12942610
Well, everything sort of depends on language, our experience in the world. We all agree based on experience/language/reality (whatever your philosophy is) we can have certain notions, 'if', or the notion of something repeating for example. Without that we have nothing, really. So this guy can just deny that language has any shared meaning (that's why he makes up words for examples) and then there's no recovery from it, he wins.
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>>12942628
I guess, yeah. The problem with him "defining" something is that he doesn't actually define anything (which is true for set theory, sure, but) nor does he connect his new concepts to anything in any reasonable way. Instead of giving stupid names like champocle to things he wants/pretends to want to make fun of, he just invents random words without connecting them to anything. In fact, champocle makes me think of some kind of bottle rack for champagne bottles that would be optimal for preserving the sparkles... But this all can just be nonsense, I've never been good at finding flaws in any arguments and would probably be outdebated by a 10 year old.
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>>12942726
Either way, we can get a lot done because MOST of us agree on what things mean, and get the same results, thankfully!
It seems logicians always have to turn to some other sort of philosophy of the world to describe things, take this quote from Godel (sorry it's long):
... there exists today the beginning of a science which claims to possess a systematic method for such a clarification in meaning, and that is the phenomenology founded by Husserl. Here clarification of meaning consists in focussing more sharply on the concepts concerned by directing our attention in a certain way, namely, onto our own acts in the use of these concepts, onto our powers in carrying out our acts, etc. But one must keep in mind that this phenomenology is not a science in the same sense as other sciences. Rather it is (or in any case should be) a procedure or technique that should produce in us a new state of consciousness in which we describe in detail the basic concepts we use in our thought, or grasp other basic concepts hitherto unknown to us.
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>>12942610
>More like a fun being one with nature and maybe talking to your ancestors kind of thingy.
Post something shamanic to listen to, you witch
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>>12942108
>Constructively, to prove a disjunction requires you to be able to prove one of the disjuncts
So is it allowed to provide a procedure that produces a proof of one of the disjuncts, and upon completion of this procedure it will tell us which disjunct we proved, but be unable to say ahead of time which disjunct it actually proves? If that makes sense.

So a program that given a number n either finds a prime factor or tells us the number itself is prime. This proves (in finite time) either n is prime or it isn't prime, but we of course don't know which one it proves, but it can at least tell us which case it is.
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>>12942760
>Either way, we can get a lot done because MOST of us agree on what things mean, and get the same results, thankfully!
Yep. I'm OK with people fighting over AC or LEM or stuff like that, as they change things quite drastically, but neither of them is really like reaaaaally that fundamental that throwing it away would make the whole thing collapse. Since (hopefully most of) other maths is independent of which axioms (if at all) one uses, the notion seems stable enough to be accepted without too much bickering. But this is coming from someone who has been able to avoid pretty much all sorts of logical arguments so far. Maybe there are some surprising pitfalls somewhere for me to fall into, but so far so good. There was, however, some group theoretical paper using the Lefschetz principle, which was a bit surprising.
>It seems logicians always have to turn to some other sort of philosophy of the world to describe things, take this quote from Godel (sorry it's long):
That's not too surprising, though, is it? The basic idea is/was to study what kinds of reasoning are valid and stuff like that. It is not a tool for describing, but for checking if arguments are valid. I had some thought but it flew away, sorry. I guess it's bed time. Good night, nice logic anon!

>>12942766
Here you go. A spring chant to the Lady of Summer meant to guarantee a good harvest. https://www.youtube.com/watch?v=OD_TIrX7zu0
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>>12942888
That was an interesting song. Good night algebra babe.
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>>12942888
>LEM
Well, getting rid of this implies other weird things other than what was mentioned here earlier. For example you can prove R has measure 0.
For sure constructivism is useful, and if we prove something using its methods that is good. However, it will never become mainstream because it overcomplicates things for little practical benefit.
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>>12942977
>For example you can prove R has measure 0.
How does this work???
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>>12942084
What is your goal? If it’s to be the next Gauss then give up now. Otherwise, it probably won’t be a problem.
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>>12942995
Well in the development of analysis in Chapter 6 of volume 1 of 'Constructivism in mathematics, an introduction' they show that an inteval in R is non-compact. I'll give an image of a remark on this.
My point is, constructivism makes things overly complex. I think the exercise is useful, though, and I'm glad it exists. But it won't catch on with mathematicians in general (I know it is very useful in some areas of regular mathematics, yes).
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Yo I thought homology was supposed to be easier what is this?!
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>>12943045
oh wait, I think I get it now. The constructive reals have measure zero when measured via Lebesgue measure. That makes sense, because the vast majority of reals are a bunch of uncomputable/undescribable nonsense.
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>>12943192
Well you're free to stick to the constructive approach if that's what you like best.
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>>12943237
I'm starting to get desensitized to these "constructivists believe such wacky things!" examples because when you unpack it all invariably it reveals a mishmash of classical and constructivist concepts.
Of course a measure meant for the reals probably isn't going to work for the very sparse subset of reals that the constructivists care about.

I like classical mathematics though, not interested in abandoning it. Topology in particular is something I'm quite enjoying at the moment, I don't even want to begin to think about what it would look like constructively. As for analysis, just out of curiousity I'm having fun exploring the basics of what constructive analysis is.
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>>12943292
Someone was talking about the LEM and how it led to a weird conclusion. I simply mentioned another fact that didn't match with regular mathematics. Then I was asked why that was so posted an excerpt. You gonna start talking about unicorns and champocles now and how that's equivalent to believing in the real numbers?
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>>12943309
Nope lol.
I fucking played myself by not realizing that "R has measure zero" shouldn't have been taken at face value, because clearly R referred to the constructive reals, and not [math]\mathbb{R}[/math] proper.
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>>12943334
Well constructivism/lack of LEM was the context, so obviously constructive R was the topic. If you think that's a useless fact that's fine. The only real point I was trying to make was that constructivism seems to just overcomplicate things for little benefit (since I don't have a problem accepting classical stuff). I guess I'll keep quiet next time someone mentions anything to do with it. It always causes conflict.
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>>12943345
Nah its cool, it was a useful fact to learn about.
Measure theory sounds difficult to do constructively.
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>>12942911
Yes. I think the didgeridoo fits surprisingly well in it. It obviously shouldn't be there, but it just suits it. It also gives that nice nature magick vibe to me. Leave food on some stone in a grove as a sacrifice and obtain something nice in return.

>>12942977
Yeah, sure. I meant that stuff is still going to stay together f you remove LEM. It will just make things extremely painful.

>>12943058
Easier isn't the same thing as easy.
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>>12943386
You sure don't sleep long.
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>the very very very long exact homology sequence
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>>12943389
Not even a second. Maybe next time.

>>12943391
How long must it be to be very very very long instead of just long?
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>>12943396
The law of very long sequences
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>>12943423
Fear not the practitioners of magick, anon. All algebra is about writing powerful symbols and drawing mighty figures.

>>12942371
The classifier was extremely approachable in this case. Nice one.
>>
I have three books I want to learn: hatcher's topology, eisenbud's commutative algebra, and rotmans homological algebra. What order should I tackle these in
>>
>past 2am
>trying to use category theory to settle my schemes
>listening to this: https://www.youtube.com/watch?v=pQq5stWTZUM
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>>12943511
I'd say you start with Hatcher because you don't really need to know too much about rings to get it done. That will introduce homological algebra to you in the setting of algebraic topology. After that, you can basically read the other two somewhat simultaneously. Rotman uses the properties of various kinds of rings in his book, while homological methods are to be expected to be used in Eisenbud's book. Not all Rotman's rings are commutative, though, so it depends on what you are actually after. Is your future to be filled with commutative or with any sort of rings? Choosing Eisenbud hints a bit towards geometry, which then points towards commutativity quite heavily. That should help you form a hierarchy of relevance between the two books

>>12943537
Be the wolf who chases diagrams.
https://www.youtube.com/watch?v=-NbvqRTNRqc.
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>>12943578
Why is Eisenbud's book so popular by the way? I'm not the person that asked what order. I just know a lot of people in my commutative algebra class had the book even though it wasn't assigned. Is it just because it has exactly everything needed for Hartshorne?
It just seems too long winded for me, seems a little unfocused.
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>>12943578
I want to do some geometry type stuff in the complex numbers, I think. Not too sure yet
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>>12943615
So you'd want to work toward reading a book like Griffiths and Harris Principles of Algebraic Geometry? You might want to look for a book on Riemannian Geometry first as well, the intro in there is a bit sparse.
Griffiths and Harris has annoying typos, they have a bit of an errata at math overflow so if you do end up reading the book check that out.
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>>12943599
I don't know. Maybe because it is steering its readers towards algebraic geometry, and that seems to be a popular field. It's just probably what you mentioned and makes people ready for being manhandled by Hartshorne.

>>12943615
Then I recommend Hatcher so that you will get to see a bit more of topology, and its algebraic side. You should learn what homotopy, homology and cohomology are, the first one to make all the integral results make sense in complex analysis (if you remember the idea of putting a loop around a hole and then saying that it doesn't depend on which loop you use as long as you can deform one into the other), the second one to give you what you need for understanding cohomology, and cohomology for understanding how to do algebraic stuff on manifolds (de Rham, Alexander-Spanier, Čech etc.). To be honest, I have no idea which theory you would use on a complex manifold. Then definitely priority on Eisenbud. If you are really to use those three books. But out of those three, I'd say H -> E -> R (if needed).
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>>12943578
You asked what is my favorite animal on january (december?). I said wolf and mentioned the lone wolf archetype. Other anons jumped in to discuss the latter, one linked the idea to winter and darkness, to which another responded saying "wolves have always been dark", with the pic of the album cover of the song you just posted.
We came full circle.
>>
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>>12943730
I remember that. It feels like it was ages ago. I posted the album cover in that thread. It fit the discussion quite well back then, and I really like the art style. Also, on the album, if it contained Path of the White Wolves, it would be the perfect Nattfog compilation.
>>
>>12943615
What you need is ANALYSIS. Holomorphic dynamics, geometric function theory, etc. None of that GAY algebra.
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>>12943908
FYI, geometric function theory was T. KACZYNSKI's specialization. So you know it is BASED.
>>
New. >>12943915
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>>12943912
If geometric function theory is so BASED, why did TED blow himself up to RETIREMENT?
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>>12942977
This is not true. Not accepting LEM leads to less theorems, not different ones. You probably refer to an anti-classical logical system such as used by Brouwer or the Russian school.

You can perfectly do standard math without LEM, you just have to be careful about the way you state it. That's the beauty of Bishop's compatibilist approach.
>>
>>12943037
I'd just like to teach at a comfy small university while solving math problems in my free time
>>
>>12942549
No, thank you for your teaching :)



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