/mg/ maths general

[Anonymous No.12531655]

The definition of a set edition
>By an “aggregate” (Menge) we are to understand any collection into a whole (Zusammenfassung zu einem Ganzen) [math]M[/math] of definite and separate objects [math]m[/math] of our intuition or our thought. These objects are called the “elements” of [math]M[/math].

Talk maths, formerly >>12516628

[Anonymous No.12531712]

CANTOR THE PLAGIARIST

Did you know that Cantor was a serial plagiarist? His famous indexing of the algebraic numbers is due to Dedekind, but Cantor gives him no credit in “Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen”. Moreover, one of his most famous techniques, diagonalisation, is clearly inspired by the work of du Bois-Reymond (that Cantor was knowingly aware of), who applied it to a different problem (orders of growth of real functions). Really makes you rethink Cantor, huh?

[Anonymous No.12531722]

>>12531712
damn, can we get a nose check on that?

[Anonymous No.12531742]

>>12531655
>>12531655
>definite and separate objects m of our intuition or our thought
that shit sounds almost esoteric compared to how it's lol.

[Anonymous No.12531765]

>>12531429
Please respond

[Anonymous No.12531788]

>>12531765
Which topics does the geometry section of your book cover? Things like conformal mapping?

[Anonymous No.12531904]

>>12531712
>>12531722
Cantor came from a Jewish family that converted to Lutheranism.

[Anonymous No.12531918]

>>12531904
fucking unreal

[Anonymous No.12531941]

What is the class of all classes that are similar to a given class?
>think about it; don't Google

[Anonymous No.12531956]

What's the cardinality of the set of all groups?

[Anonymous No.12531967]

>>12531956
Depends, is every subset of a group also a group?

[Anonymous No.12531981]

>>12531956
Uh oh, you just ran into a heckin problemerino with the """definition""" of sets: https://math.stackexchange.com/a/226416
It's okay though, we can instead call this collection of things a """class""" to avoid any paradoxerinos so that we can continue to rely on the ZFC axiomerinos without defining what a """set""" is.

[Anonymous No.12531989]

For natural integers n<m, we have a canonical inclusion of symmetric groups
S_n <= S_m.
Is the limit S_infinity interesting at all?

[Anonymous No.12532011]

>>12531989
Permutations of the naturals that don't fix finitely many points, if you consider that interesting.

[Anonymous No.12532034]

It is commonly said that it's very hard to assign uniform measures to subsets of integers numbers, in a translationally invariant way, because you can express every set as a countable union of singletons. This presupposes a kind of a bottom up approach to measure. I wonder if anything can be done with a top-down approach. Assign m(Z)= 1.
We want
m(2Z)=m(2Z+1) = 1/2
In general we require if A,B partition Z into two sets, then m(A)+m(B)=1.
m(X)=m(X+n) for any n.
m(X)=m(X') whenever the symmetric difference between X and X' is finite.
m(A)<=m(B), whenever A is a subset of B, where we may restrict A,B to some particular set of subsets
Is there any m which captures this notion?

[Anonymous No.12532062]

Is this an open problem?

[Anonymous No.12532064]

>>12532034
turn your sets into 0-1 sequences
then put m(A) = banach limit of cesaro averages of A
that's not very constructive but i don't think you can do anything very constructive here

[Anonymous No.12532069]

>>12532062
we solved it in the previous general, now quantamagazine is going to write an article about 4chan

[Anonymous No.12532073]

>>12532064
I thought about that, but the limit exists only for a very small set of subsets. I wonder if it's possible to extend it to a larger set.

[Anonymous No.12532074]

>>12532062
not solvable in zfc
currently developing new grundlagen

[Anonymous No.12532080]

For any quadruple of natural numbers can you find four points, arranged as a square in [math]\mathbb{C}[/math] such that the escape times of those points in the mandelbrot set is the quadruple? Can you find a counterexample?

[Anonymous No.12532088]

>>12532080
Does it have to be ordered? Consider the square abcd, where a,c=/=0 and b,d=0. The entire disk of radius 2 about the origin has an escape time>0.

[Anonymous No.12532093]

>>12532073
i didn't say "limit" i said "banach limit"
look it up, it's an extension of normal limit to all bounded sequences

[Anonymous No.12532116]

You are now remembering the time when you mentioned some "high level" math concept to your professor.

[Anonymous No.12532201]

>>12532116
not once have i done this

[Anonymous No.12532237]

>>12532201
>He never butted in a prof's office to talk about his newly discovered triple integral formula

[Anonymous No.12532245]

>>12532116
Never happened, thank the Lord.

[Anonymous No.12532247]

>>12532116
Not ashamed of doing it because I'm not a beta.

[Anonymous No.12532252]

>>12532116
"Hey I was doing some research could you explain what a Banach space is"

[Anonymous No.12532259]

Have there been any ideas in mathematics in the last 30 years?

[Anonymous No.12532264]

>>12532259
Only schizophrenic delusions

[Anonymous No.12532383]

Is caring about foundations a sure sign of mental illness? Never met a productive, creative mathematician who cared about foundations (they are intuitively clear and fundamentally irrelevant), only Schizos or absolute midwits.

[Anonymous No.12532423]

>>12532383
>Is caring about foundations a sure sign of mental illness?
not "caring" as much as "challenging" the foundations
my personal favorite type of schizo are those who claim that mathematical entities need to "exist" or "be useful" without any clear definition as to what those entail

[Anonymous No.12532451]

>>12532259
I'm sure we could list a bunch, say knot-theory-via-TQFT, or muh perfectoid spaces, or let's even say univalent types or ...Feynman path integrals to estimate the relearning distance in weight space of a deep neural networ. I'd say Hairers stochastic tools, but I don't know too much about it and I don't want all my examples to come from the same corner).
But they are all of the form that require a huge amount of background. As always, that which is deemed relevant will be cleared up and, made learnable and look not as obstroose in decades to come. Time progresses and new textbooks appear the proof old theorems via the new tools and suddenly they look fundamental. Say Noethers ideal theory for talking about numbers.
Computational complexity, Hilbert space theory or scheme theory or theory all looked extremely obstroose, once. Some ideas that didn't gain momentum from those times still look weird, because they were not developed.

That's my take for today.
Here's some plans for 2021:
https://youtu.be/UXTz-eFDmjc

[Anonymous No.12532469]

>>12532264
thats all of mathematics tho

[Anonymous No.12532503]

>>12531788
No it’s literally baby geometry like deriving standard formulas for conic sections in the complex plane and finding complex roots.

[Anonymous No.12532543]

>>12532503
>it’s literally baby geometry like deriving standard formulas for conic sections in the complex plane and finding complex roots.

[Anonymous No.12532585]

Has anyone here genuinely tried applying a quasi-scientific process to themselves to find the most optimal system for studying mathematics/reading a mathematical text? Would they care to share their experience?
If not, I'll try doing it myself over the years.

[Anonymous No.12532609]

>>12532585
>applying a quasi-scientific process
i.e. trial and error?
I'm sure everybody does that automatically.

[Anonymous No.12532614]

>>12532543
Instead of bullying me can you tell me why I’m going through all this trouble?

[Anonymous No.12532620]

>>12532609
>I'm sure everybody does that automatically.
People (myself included) tend to settle for what they view as "good enough". Genuine experimentation is rare.

[Anonymous No.12532626]

>>12532620
Yeah, might well be.
On a related note, I think you have an idealize view on science. Science, in praxis, is indeed to try until you get the result you want (if possible) and publish before the next guy does

[Anonymous No.12532634]

>>12532626
Holy fucking shit who cares.

[Anonymous No.12532639]

>>12532626
Hum, not really, no. Science is about understanding the laws and structures of the universe. We're not playing a game or competing against each other here, we're trying to understand the universe through the collective effort of a selected number of human beings who chose to dedicate their lives to it.

[Anonymous No.12532644]

>>12532639
>Science is about understanding the laws and structures of the universe
HAHAHAHAHAHAHAHAAHAHAHAHAHAH
tell me when you reach middle school

[Anonymous No.12532647]

>>12532644
Honestly what do you think science is then? If you got into saying with another mindset then you're in the wrong field, son.

[Anonymous No.12532649]

>>12532614
Complex Geometry studies complex manifolds (smooth manifolds, except instead of asking for the transition maps between charts to be smooth we ask for them to be holomorphic (holomorphic generalizes to more than one complex dimension))(Riemann surfaces are the simplest examples of these), complex analytic sets (basically the holomorphic equivalent of an algebraic set) and related concepts i.e. complex Lie groups, complex spaces, plus most likely some other stuff I don't know about.

[Anonymous No.12532655]

>>12532649
When I wrote "complex Lie group" I was actually thinking "complex orbifold."

[Anonymous No.12532661]

>>12532639
Mhm, well I said you have an idealist view of science and went on to describe science in praxis.

Yes, there is a scientific method that many people have an idea of, but whether that translates to the academic publish-and-perish culture, where lecture halls are named after the companies who bought their way into the institution, is another thing. University today is job training and Professors are but working on moonshot programs. If they get themselves into the limelight, they skip ahead the standard track by years and promoted as mascot for their unis (see the recent knot theory woman who took an afternoon to solve a problem, or look at Higgs who did one thing in the 70's and was kept at the uni for it, for 50 years without doing anything else worth speaking of)

[Anonymous No.12532671]

>>12532655
>This terminology should not be blamed on me. It was obtained by a democratic process in my course of 1976–77. An orbifold is something with many folds; unfortunately, the word "manifold" already has a different definition. I tried "foldamani", which was quickly displaced by the suggestion of "manifolded". After two months of patiently saying "no, not a manifold, a manifoldead," we held a vote, and "orbifold" won.
>Thurston (1980, section 13.2) explaining the origin of the word "orbifold"
https://en.m.wikipedia.org/wiki/Orbifold

Lol mathematicians man... Smh...

[Anonymous No.12532688]

>>12532661
I don't have an idealistic view of anything, you're the one who have a pessimistic view of science.
>where lecture halls are named after the companies who bought their way into the institution
Never even heard of that ever happening. Also why would a company do that? Do you think the masses would buy their product because they saw the company's name on the lecture hall of their institutions? Lol.
>University today is job training
Yeah, because we live in a capitalistic world, no shit Sherlock.
>If they get themselves into the limelight, they skip ahead the standard track by years and promoted as mascot for their unis (see the recent knot theory woman who took an afternoon to solve a problem, or look at Higgs who did one thing in the 70's and was kept at the uni for it, for 50 years without doing anything else worth speaking of)
Because that's highly common, huh? You can count on the fingers of one hand the amount of people like that and they totally deserve it. Imagine if you prove the riemann hypothesis, your contribution to science and math would have been so great that being awarded with a lifetime easy job would be the least they could do.

[Anonymous No.12532701]

Post favourite olympiad styled problems

[Anonymous No.12532711]

>>12532701
I don't know what are my favorites, but the worsts are the ones who ask you to calculate sin/tan/cos(random number°), those kind of problems are so retarded...

[Anonymous No.12532722]

>>12532711
If you work in the real number system it's actually very easy to calculate.

[Anonymous No.12532733]

>>12532722
Oh, it's so very easy huh? Then surely you'll have no problem solving: https://brilliant.org/problems/interesting-trigonometry-imo-problem/

I give you 15 minutes, mr-intelligent.

[Anonymous No.12532734]

>>12532701
1. Let [math]a, b[/math] be integers not both zero, [math]|a|, |b| < 2021[/math]. Prove that [math]|a - b \sqrt{2021}| > 2021^{-2021}[/math].

2. For a prime number p, how many consecutive pairs of quadratic residues are there?
(e.g. quadratic residues mod 7 are 0,1,2,4, so there are two pairs 0,1 and 1,2)

[Anonymous No.12532748]

>>12532733

[Anonymous No.12532750]

>>12532671
sounds like polyfold would have been the goto name

[Anonymous No.12532761]

>>12531655
It’s arguable that the green text here doesn’t describe anything in reality. Your subcategorization into elements ignores the inherent change in properties involved in any “aggregate”, ie, “is a constituent part of M” which would not be present otherwise, since the object which holds the property is not necessarily both definite and separate.
Now consider play-do, green slime, and other such objects which can be broken apart into portions and squeezed back together. Does it have these “elements”?

[Anonymous No.12532787]

>>12532761
Poeple always like to cite that Cantor definition. Of course it's from decades (maybe 2) before the impredicativity discussions

[Anonymous No.12532919]

large cardinals are a shitshow
there is no reason to say that strongly inaccessible cardinals exist except "dude imagine this shit exists lmao"
this is 100% cultural and social construct no relation to any truth or fucking nothing
we could as well throw out the axiom of well-foundedness and say "dude theres a set inside a set and so on infinitely many times lmao"
fuck woodin and fuck all the set-theory california jews

[Anonymous No.12533137]

>>12532919
>we could as well throw out the axiom of well-foundedness
Well yes, and we should, it's mostly a trick to get strong induction on transfinite objects. Regularity is used to little anyway, otherwise.

We shouldn't adopt either of those existence statement and you make a good case for it.

>and say "dude theres a set inside a set and so on infinitely many times lmao"
We shouldn't do that, however, since it spoils the set theoretic models that only has the Hereditarily finite sets.

[Anonymous No.12533190]

>>12532787
Her sister's hotter

[Anonymous No.12533219]

This is going to sound stupid, but bare with me. starting from 2, what's a formula that I can use with "2" so that it steps down from 2 incrementally and then when it reaches 1 it stays at 1.

For example, 2 - 0.1 will reach 1 in ten steps, but fails this because if you do it an eleventh time it'll be 0.9. It needs to be a formula that will step down to 1.0 exactly and then stay 1.0 even if I do it again or hundred times.

[Anonymous No.12533237]

>>12533219
f(x) = 2-x if 0<=x<1 and f(x)=1 if x>1 or x<0

[Anonymous No.12533331]

>>12531655
The definition of a set is a basket

[Anonymous No.12533336]

Does anybody understand how this book got from the first part to the second (I drew in red what I don't get)? Was it just algebraic manipulation that I'm not seeing or was something plugged in?

[Anonymous No.12533337]

>>12532761
The space of interactions
Play do in a mixed shape might not carry the same patterns if you overlay them in euclidean space but there is some space where any effect can be morphed into it's causes

I agree the OP definition is weak, I'd say a set is the simplest form of connecting pieces into a whole, a basket if you will where calling the set S also calls each element s, un-tied to eachother other than being called at the same time by S

[Anonymous No.12533340]

>>12533336
"Choose n0 large enough"
He's saying pick a number

[Anonymous No.12533345]

>>12533340
I see that but I don't get where the form of the inequality near my question marks came from

[Anonymous No.12533355]

>>12533345
Its a separate track of thought
Look at the next line x/n becomes x/n0 and 2-a^2 comes from logic

[Anonymous No.12533383]

>>12533331
That's an okay metaphor to look at it. However, a basket can be filled, emptied and one basket cannot be the member of two other basket at once.

The one thing where the basket or bag analogy is seemingly helpful is when pondering about whether an empty set can or should exist (and e.g. pic related is not a fan of it), but this also only helps because indeed the basket metaphor is so physical and an empty basket still is something.

[Anonymous No.12533409]

>>12532919
you can say that to any axiom, really...
What is intriguing is how, despite infinite, empty etc sets being something we just accept because we want to, mathematics is useful in real life for engineering and so on.
Applications don't imply any real existence of mathematical concepts;models, but it's interesting to ask which are applicable and why.

That's if you care about any sense of "reality", because there's no reason to be a platonicist otherwise. The purest of the pure mathematicians is just toying with cool concepts because it's fun or challenging.

[Anonymous No.12533416]

Honestly let's just replace set theory with the tao te jing and say hilbert's program is done

[Anonymous No.12533424]

Broke: infinite sets are absurd
Joke: the empty set is absurd
Woke: singletons are absurd

[Anonymous No.12533436]

>>12532919
All mathematics is finitistic symbol pushing games at its core, even the most egregious claims about unreal infinite objects is actually some real finite fact about syntax and computation, even when mathematicians think otherwise.
Mathematics was born as a shorthand to reality and as a shorthand to reality shall it perish.

[Anonymous No.12533437]

>>12532734
1 is easy, the exponent could be -2, even close to -3/2.
2 is cool, still not hard if you know the basics of congruences and quadratic residues.

I remember thinking about the same problem but with 3 consecutive residues and I did find it harder, what you think?
There's an old IMO pdf showing how to compute a general sum of legendre symbols of a degree 3 polynomial evaluated on all residues mod p which would solve the problem, but the result for degree 2 is an overkill for your problem, maybe the same happens for degree 3 and we don't need to go so far.

[Anonymous No.12533442]

>>12533436
>All mathematics is finitistic symbol pushing games at its core
Trivially true, since mathematicians don't actually have access to the infinite. Finite descriptions are required.
>even the most egregious claims about unreal infinite objects is actually some real finite fact about syntax and computation
This is not true, by Godel. Infinitist mathematics cannot be reduced to facts about finite things, since if that were so, it would be finitistically justifiable, but we know there cannot exist such a justification.

[Anonymous No.12533448]

>>12533437
>1 is easy, the exponent could be -2, even close to -3/2.
How do you show it then?

[Anonymous No.12533471]

>>12533448
Multiply by [math]|a+b\sqrt{2021}|\ne 0[/math]. RHS will be a nonzero integer, hence greater than or equal to 1. The limitation on [math]a[/math] and [math]b[/math] makes the multiplying number lesser than [math]2\cdot 2021\sqrt{2021}[/math].

[Anonymous No.12533474]

>>12533471
>RHS
I meant LHS.

[Anonymous No.12533510]

>>12533442
You don't understand Godel. He proved that syntax is bound to the limits of syntax, and that the logical formula "for all P, P or not P" cannot be read straightforwardly, thus it means something else: it must interpreted.
Mathematical symbols must be interpreted, and the direct interpretation is not the only possible one.
Finite descriptions of infinite things are a misreadings of finite descriptions of finite things.
No mathematics is ever a description of anything real other than the symbols it pushes around, we relate it to the real world by analogy, because the symbols and their manipulation mimic what we believe the abstracted portion of the real world in question to behave like. So you either reject all of mathematics, or have to agree that infinitist mathematics isn't meaningless, but misinterpreted.

[Anonymous No.12533587]

>>12533510
>He proved that syntax is bound to the limits of syntax
I don't understand the meaning of this sentence.
>and that the logical formula "for all P, P or not P" cannot be read straightforwardly
He only showed truth cannot be equated with provability, if it exists at all. Pretty sure he was a Platonist itself, so "for all P, P or not P" makes perfect sense for him in the most straightforward interpretation (of mathematical truth).
>Finite descriptions of infinite things are a misreadings of finite descriptions of finite things
Godel showed that this isn't actually true. As I said, finite things are real in the strict sense and so if you could genuinely exhibit this correspondence you would prove (in a finitistic way) that arithmetic is consistent, contradicting Godel's result. This is exactly the point. You cannot interpret infinitist mathematics in a finitist way: there will always be the notion of non-halting, of a never ending task, which is not a finitistic notion and of which you cannot get rid.
>No mathematics is ever a description of anything real other than the symbols it pushes around
This is so wrong I'm amazed you would type this out. Physicists, engineers, computer scientists regularly use mathematics that describe real things. It doesn't *equal* to things in the real world, but it's still a description.
>So you either reject all of mathematics, or have to agree that infinitist mathematics isn't meaningless, but misinterpreted.
Some infinitist mathematics is meaningful. You're falling for the phantom of transparency if you think all of it is.

[Anonymous No.12533610]

>>12531981
Based

[Anonymous No.12533612]

>>12533587
>It doesn't *equal* to things in the real world, but it's still a description.
When he said "No mathematics is ever a description", I'm sure he meant that the mathematical notation does not describe a particular thing and one that, in reality.
So what was not up to debate is that math can be used as a tool in the formulation of physics (and thus, in a realist philosophy, describe physics)

In any case, I don't know how I feel about these set theory discussions. On the one hand it's spam in the sense that some people simply want to argue. On the other hand, the situation on /mg/ is better than around 6 week ago, where no math was posted at all

[Anonymous No.12533627]

>>12533612
>When he said "No mathematics is ever a description"
This is very interesting. Where did he say this? It seems to contradict the fact that he was a Platonist.

[Anonymous No.12533643]

>>12531989
I know people care about actions of S_inf in ergodic theory and descriptive set theory. Kechris has a expository paper on this. I don't know if group theorists care about it though.

[Anonymous No.12533647]

>>12533627
I was not talking about Gödel, I was about the poster you replied to. (I'm not him)
You quote that line yourself.

[Anonymous No.12533652]

>>12533647
Oh, I thought I was speaking to Gödel for a moment.

[Anonymous No.12533738]

>>12533437
my solution for 2 is roughly:
after some thinking, it's enough to evaluate the sum [math]\sum_{i=0}^{p-1} \left( \frac{i}{p} \right) \left( \frac{i+1}{p} \right) [/math] mod p,
that's equal to [math]\sum_{i=0}^{p-1} (i^2 + i)^{(p-1)/2} [/math], we expand:
[eqn]\sum_{i=0}^{p-1} \sum_{j=0}^{(p-1)/2} {(p-1)/2 \choose j} i^{(p-1)/2 + j}[/eqn]
and the point is we can change the order of summation, and use 1^k + 2^k + ... + (p-1)^k = 0 if p-1 doesn't divide k, otherwise it's p-1.
for 3 residues, it would be enough to compute [math]\sum_{i=0}^{p-1} \left( 1 + \left( \frac{i}{p} \right)\right) \left( 1 + \left( \frac{i+1}{p} \right)\right)\left( 1 + \left( \frac{i+2}{p} \right)\right) [/math] and i think reusing the trick from the solution above will lead to a solution... but that's going to be long and ugly

Here's one more problem i like, although it's a bit less "math-olympiad-style":
3. find all rational numbers p/q such that [math]\cos( \pi p / q)[/math] is rational

[Anonymous No.12533786]

>>12533587
I don't care what Godel thought of his result, being a Platonist he's always fooling himself on the meaning of the symbols he writes by mapping them to something that can't be instantiated, in the Platonic realm in his head. But he also showed that they can be instantiated in more honest mathematics after all, by the negative translation.
Even if he was a Platonist, he understood well enough the syntactic game to expose it to other mathematicians, who made better use of it.

[Anonymous No.12533819]

>>12533786 (cont)
>phantom of transparency
If you have read Girard, you should be able to understand where I'm getting at.
Girard, while more on point than other philosophers, jumps from a dead-end to another with his quantum-inspired derealism. All that he says about subjects (us) is absolutely right, but the object (the world) does exist indipendently from us, after all. We are limited in probing it, we don't have transparency, and to pretend otherwise makes us hallucinate "phantoms" (the Platonic realm, infinity, ...). This is the sign of a leaky abstraction, that the abstraction is not doing what it is saying it is doing. Then what is it doing?
Girard recognizes the importance of cut-elimination, that guarantees that the symbolic manipulation instructed by logical implication terminates.
No mathematics is truly infinite, because no mathematician died evaluating a proof, leaving to his successor to keep it going to check whether it terminates. All proofs are either instructions for finite manipulations or erroneous.
They are all finite objects with finite behaviour, and so are mathematical systems that describe any "infinite object", whose meaning is in the computation performed.
The monstruous objects Girard laments, such as the graph of a function {(x,f(x)} on an infinite domain, cannot exist, but what can exist is the instruction to compute such function f(x) on any potential element.
But now that you know how the function f can exist, you know how to interpret the meaning of the ramblings of the infinitist: when he says that he "actually has" an infinite set {(x,f(x)}, he's saying he has a function and can construct the pair (x,f(x)) at will, ignoring potentiality. That abstraction will leak and has been leaking for a century.
But all abstraction is leaky, non-halting computation, as you implied, is non-instantiable: in practice, we will check for finitely many, but arbitrarily high, steps whether it halts and then give up if it doesn't.

[Anonymous No.12533832]

>>12533819
Why is the character limit so small, I hate it.
The point of all of that is this:
All mathematics is, fundamentally, idealization, as is infinity. If you reject that there is *some* meaning in infinity, that has to be interpreted rather than taken at face value, then you reject that mathematics has *any* meaning, because you can always translate schizo ramblings about actual infinity into sane claims about potential behaviour of finite computation.

[Anonymous No.12533887]

>>12533832
>>12533819
>>12533786
Thanks for the thorough response.
>If you reject that there is *some* meaning in infinity
I don't. I explicitly asserted that some infinitist mathematics is meaningful. My point there was that, as you seem to agree, not all of it is. Probably less than mathematicians think. And if it is meaningful, it's often not in the naive (for lack of a better word) interpretation.
I pretty much agree with everything you said.
I think that whatever we conceptualize mathematics to be, we will want it to involve reasoning, an evolving construction. We don't want to restrict ourselves to high school algebra. For interesting reasoning, it's impossible to escape semidecidability/ideal infinite. In that sense, a totally finitist mathematics in which all objects have real, finite denotations is doomed to fail, simply because it would be too boring.

[Anonymous No.12533978]

>>12533340
>>12533355
K I get it now, had some beer and read some other versions of the proof online, pardon my retardation. Proof in my book was still just a little unclear IMO though

[Anonymous No.12533983]

>>12533887
We disagree, I mean that all infinitary mathematics is, in some way, meaningful, even if that meaning is far removed from the "intended meaning". It doesn't change the (real) applications of the mathematical idea, if anything gives them even more weight.
That's why it's laughable when something is found to map to "topological" concepts: a topology on a set is a double powerset with some properties, so it's just classifying some predicates on a set, why wouldn't you expect it in more places? Mapping it to abstract multi-dimensional geometry is complicating what it truly represents, even though the intuition you get from it is valuable.

Infinitary mathematics is not meaningless, it's misunderstood by the same people that produce it.
I think that the job of the finitist in modern mathematics is not rejection of the infinity, but reinterpretation and absorption, developing better tools that draw better distinctions in meaning, rather than conflating them as much as possible, as so-called classical mathematics does. Constructivists, too, are blind by mathematical purity, as they only allow complete, perfect constructions, but took a step forward, by rejecting the hypocrisy of the excluded middle, Girard's Linear Logic is another step forward, but he has his blindspots as well. Moreover, unlike constructivists like brouwer and bishop, linear logicians are less interested in formalizing mathematics in linear logic and uncovering newfound meaning for old constructions, as well as new constructions and mathematical modes of thought, and more interested about toy models and minute changes in the rules of the logic.

[Anonymous No.12534010]

https://voca.ro/1e56quJdBQyA

[Anonymous No.12534033]

>>12533983
>Moreover, unlike constructivists like brouwer and bishop, linear logicians are less interested in formalizing mathematics in linear logic and uncovering newfound meaning for old constructions, as well as new constructions and mathematical modes of thought, and more interested about toy models and minute changes in the rules of the logic.
Do you judge that as good or bad?

I'd also proactively call Brouwer an intuitionist. I feel a lot of the CS constructivists don't reject non-constructive math, rather they want the space to adopt ALL THE axioms when they want to

[Anonymous No.12534091]

>>12534033
>Do you judge that as good or bad?
I think bad, but approaching it without care would also be bad. Ideally one shouldn't do a mechanical rederivation of all mathematics, but he should strive to keep things as opinionated as possible, as the old constructivists did, forcing discovery, with rederivation as a bonus.
I think some work around linear logic to be as masturbatory as classical mathematics, it misses the point. Girard, too, fell for so many memes, like concurrency and quantum stuff. But again, I don't think it is time wasted into meaningless activities, it's just not as productive.

>I feel a lot of the CS constructivists don't reject non-constructive math, rather they want the space to adopt ALL THE axioms when they want to
It depends on what you mean by CS constructivists.
CS people rejected excluded middle on their own without even having a logical structure onto their computations: they figured out that GOTO, and then call/cc, makes for illegible spaghetti code, and moved towards structured programming (linear continuations, intuitionistic) and delimited continuations (semiclassical constructive), but in their rejection, they also found the justification for it, and found the true meaning of the hypocrisy: what LEM does is spawn a self adjusting computation with arbitrary control flow that is ill-behaved and hard to understand.
HoTT did more harm than good to CS constructivists in this regard, but it's spearheaded mostly by the mathematical relativist category theorists. How they fit in this scheme is complex, because they recognize that all that matters to mathematical structures is their structure, but, as Girard also notes, in their commutative diagrams "one side is more commutative than the other", i.e. the one that does the actual computation. They hardly acknowledge this, and are pretty much structural Platonists, which ironically lets them conjure even more abstract monsters with ever more trivial meaning, and that is their blindspot.

[Anonymous No.12534126]

>>12534091
>hat LEM does is spawn a self adjusting computation with arbitrary control flow that is ill-behaved and hard to understand.
?

[Anonymous No.12534171]

>>12534126
Pseudocode, where @k -> E binds the current continuation in E:

callCC : ((A -> B) -> A) -> A
callCC f = @k -> k (f k)

callCC2 : ((A -> B) -> B) -> A
callCC2 f = @k -> f k

LEM : Either (A -> Empty) A
LEM = callCC (\k -> Right (\a -> k (Left a)))

callCC duplicates the current continuation k, calling f with k and, if it returns, applies its return value to it. Its type is Peirce's Law, which is invalid intuitionistically and linearly.
callCC2 doesn't duplicate the continuation, it calls f with the current continuation expecting it to not return. Its type is Double Negation Elimination, which is intuitionistically equivalent to Peirce's Law, but allowed in linear logic. This is because continuations are used linearly in constructive logic (it is equivalent to restricting the conclusions at the right of the turnstile to a single one), but once you bind it to an intuitionistic variable, you are free to duplicate it. You do not have such freedom in linear logic.

LEM is implemented computationally with callCC, what it does is the following:
It saves the current continuation, and tentatively returns a fake proof of ¬A which, when called with a proof of A, will discard the then-current continuation and call the stored continuation, returning to the point where it returned the fake proof of ¬A, and return the proof of A you gave it instead.
This breaks many desiderable properties: for one, it introduces a computational effect (the mother of all effects, in fact, as all monads can be expressed with continuations), so the order of execution matters for the end result, but the other is that even that result that you end up getting, you can't trust it to be a correct answer, as you can never be sure whether it is a false value that should eventually be proven contradictory.
Eventually, once all "turned out wrong" LEM-calls are resolved, you'll receive a right answer.

[Anonymous No.12534250]

>>12533738
Oh cool, your solution is different than mine.
I think mine is simpler, but it doesn't seem to generalize for triples of consecutive residues like yours do.

Let [math]i(k)\in\{0,1,2,…,p−1\}[/math] denote the inverse of [math]k\in\{1,2,…,p−1\}[/math] modulo [math] p[/math]. Then [math]\left(\frac{k(k+1)}{p}\right)=\left(\frac{i(k)^2 k(k+1)}{p}\right)=\left(\frac{1+i(k)}{p}\right)[/math], the sum of which from [math]k=1[/math] to [math]p−1[/math] is the same as [math]\sum\limits_{k=1}^{p−1}\left(\frac{1+k}{p}\right)=−1[/math].

[Anonymous No.12534321]

>>12533137
but then you need a more detailed version of extensionality

Anyway you can model any AFA in plain old ZFC, they're equiconsistent

[Anonymous No.12534564]

>>12534010
Based.

[Anonymous No.12534634]

h-hi

I'm measuring the density of something at different length scales to see at what length scale the measurement is the most consistent. I'm trying to think of a way to do this mathematically and not visually so I can automate it. Any suggestions on good keywords to search for?

[Anonymous No.12534646]

>>12534634
Use the derivative on the data chart

[Anonymous No.12534886]

>>12533738
>find all rational numbers p/q such that cos(πp/q) is rational
but this is just Niven's theorem

[Anonymous No.12534907]

I swear working in fields that require regular application of the transcendental functions is the fastest way to feel the ageing of your brain. The amount of work that relies on you remembering functional identities, series and integral representations, and all these other hocus pocus tricks make you really feel the difference between the adult brain and the youthful wunderkinder.

[Anonymous No.12535127]

>>12532423
Be computable

[Anonymous No.12535162]

>>12532787
>>12533190

https://www.youtube.com/watch?v=6YgMym9OqEY

Until now I never conceived of a world where e-thots would be gaming on chess boards. I guess... I had just separated the world of chess from the world of video games. But I guess in the end Chess really is just another form of degeneracy.

[Anonymous No.12535408]

>>12535162
fuck off hateful incel
we respect women in /mg/

[Anonymous No.12535428]

>>12535408

[Anonymous No.12535580]

>>12535428
thank you for posting this very based gentleman

[Anonymous No.12535896]

redpill me on complex multiplication

[Anonymous No.12535963]

>>12535896
You can't really give a brief redpill on this. It's the area of maths where it seems like it all comes together in one really extravagant whole.

[Anonymous No.12535974]

>>12535896
re^itheta representation
r1 * r2 for embiggen and theta1+theta2 for spinny

[Anonymous No.12535978]

>>12535974
oh, I thought he was talking about elliptic curves

[Anonymous No.12535982]

>>12535978
I was, ignore the engineer

[Anonymous No.12535990]

>>12535982
i personally found the best beginner friendly introduction to be in Freitag and Busam if that helps
the translation is shit but in some ways that can be a good thing since it forces you to actually make sure you understand it before moving on, and Freitag has a lot of great exposition about how theta functions work, multiplier systems and all that
then again if you're asking you may already know all that

[Anonymous No.12536012]

>>12531712
>Really makes you rethink Cantor, huh?
same thing with einstein plagiarizing poincaré

germans are mental midgets so they jsut steal ppl's work and try to pass as saints.

[Anonymous No.12536019]

>>12532259
>Have there been any ideas in mathematics in the last 30 years?
no, at best you have hott book

[Anonymous No.12536028]

Is there any thread for people studying maths or do they all come here?
I'm not a mathematician, I'm just looking for fellow students.

[Anonymous No.12536033]

>>12536028
this thread is 90% undergrads

[Anonymous No.12536037]

>>12536033
I'm learning multivariate calculus now and am looking for someone to study "with". I'm always a better student if I'm talking about it with someone else.

[Anonymous No.12536051]

Why is complex exponentiation defined in terms of the complex logarithm? Why can't you define it the other way around?
I'm reading Stroud and can do all the computations, but why is it this way in the first place?

[Anonymous No.12536069]

>>12531655
Hey math nerds.
Quickly tell me everything you know about infinity, I suddenly need to work with it.
What set of functions/symbols/laws/whatever do I need to work with infinity?
Just tell me the names and I'll figure the rest out myself.

[Anonymous No.12536073]

>>12536069
>What set of functions/symbols/laws/whatever do I need to work with infinity?
Set theory.

[Anonymous No.12536079]

>>12536012
Dedekind and du Bois-Reymond were Germans also.

[Anonymous No.12536189]

>>12536051
The logarithm can't be inverted in the complex plane because of the branch cut along the negative part of the real axis. This is why the logarithm can't be satisfactorily defined in terms of a complex exponentiation process, which suffers no such problem. You can't really say that the complex exponential is the inverse of the logarithm because the exponential is periodic but the logarithm is discontinuous, etc.
As for the rest of your question, I assume that you're asking why [math]a^z[/math] is defined as [math]e^{z\log a}[/math] where the logarithm takes its principal value, in which case the answer is that it is just a convenient artifice. If [math]a>0[/math] there is no issue, but for other values we may run into issues with the complex logarithm destroying the properties we normally take for granted with exponents.

[Anonymous No.12536269]

Let p be an odd prime, find the number of nonempty subsets of {1, 2, ..., p-2, p-1} which have a sum divisible by p.

[Anonymous No.12536285]

>>12535162
Let me settle this once and for all:
Alexandra is seemingly the smarter one and probably also has the better (more unique and conventionally attractive) looks.
But she also puts up fake smiles a lot, seems combatitive and all in all not fun.
If you ask who's more fun in bed, it's certainly not Alexandra, even if she has the better looks and vocalization (taller, doesn't need makeup)

[Anonymous No.12536288]

>>12536269
The number of nonempty subsets of [math]\{1, 2, \ldots p-2, p-1\}[/math] divided by [math]p[/math].

[Anonymous No.12536300]

>>12536288
Thought so but how do I prove it?

[Anonymous No.12536314]

>>12536028
>>12536037
Post this every day in /mg/ and you will find someone

[Anonymous No.12536318]

>>12536300
The entire set balances out how none of the singletons sum to zero.
Then you'll probably need to show that 1/p of the subsets of cardinality k (k not 1 or p) sums to zero. It's simple for k = 2. Best of luck on the induction,

[Anonymous No.12536326]

>>12536269
choose k and consider the k-element subsets
given a subset A, consider adding 1 to all elements, look how this affects the sum
something something bijection bijection and that's it

[Anonymous No.12536343]

>>12534091
>Girard, too, fell for so many memes, like concurrency and quantum stuff
Explain how the quantum stuff is a meme. What do you know that he doesn't?

[Anonymous No.12536417]

>>12535162
Boomer image, islamist take

[Anonymous No.12536427]

>>12531655
Is there anything special about least squares? Why aren't things like least cubes or least inf ever used? Is it because squares gives an unbiased estimator, but these other methods don't?

[Anonymous No.12536440]

>>12532451
Yo what the fuck is the point of knot theory? I get that many areas of mathematics appear esoteric but actually are really useful, but fucking knots?

[Anonymous No.12536486]

>>12533738
You use the fact that 1^k + ... + (p-1)^k = 0 when k is less than p-1. I wonder what's a nice way to prove it.
A way I came up with is using symmetric polynomial theorem,
x_1^k +... + x_(p-1)^k is a symmetric polynomial and thus is expressible as a sum of products of the elementary symmetric polynomials.
But x^(p-1) - 1 splits into distinct linear factors, so all symmetric polynomials of degree >=0 and less than p-1 evaluated at 1,2,...,p-1 are 0, thus the sum is also 0.
Do you know a better way?

[Anonymous No.12536563]

>>12536486
take g that's a generator mod p, then [math]1^k + ... + (p-1)^k = 1 + g^k + ... + g^{(p-2)k} [/math], and just use the formula for sum of geometric sequence
your idea is good too

>>12536427
least squares is simply the nicest to manipulate mathematically of all those, so people just went along with it
it's differentiable, you get to express your stuff in terms of euclidean norm/scalar product/matrix multiplication sometimes
even from a CS standpoint, it's good because it's just multiplying and adding so it's the easiest to compute

[Anonymous No.12536643]

>>12536440
Don't ask for usefulness

[Anonymous No.12536797]

Are there any good algorithms that solve Travelling Salesman when negative weights are allowed, the "distances" are not symmetric, and triangle inequlity doesn't hold?

i.e we can have
edge(u,v) < 0
edge(u,v) != edge(v,u)
edge(u,v) + edge(v,w) < edge(u, w)

(edge(v,v) = 0 holds)

[Anonymous No.12536804]

>>12536440
people tie knots all the time anon

[Anonymous No.12536816]

>>12536440
if you study hard enough, you'll be able to hang yourself

[Anonymous No.12536892]

Math is literally just schizophrenia. What compels someone to dedicate their life to this?

[Anonymous No.12536913]

>>12536343
I explained it. His position is too subjectivist to my taste, he rejects the objective world because we cannot experience it directly, and cites quantum indeterminism as a justification.
But quantum indeterminism is barely even a justification for itself, it cannot be used to justify philosophical positions beyond its scope. He went from one kind of mathematical mysticism to another.

I'm also just not hyped or interested at all about his coherent spaces.

[Anonymous No.12537017]

>>12536892
if the irrationals were countable, then the reals would also be countable

[Anonymous No.12537347]

>>12536892
>far outnumber

>>12537017
Sure, but just because e.g. [math]{\mathbb N}^{\mathbb N}[/math] isn't countable, doesn't mean that some subset [math]S\subset {\mathbb N}[/math] couldn't surject onto [math]{\mathbb N}^{\mathbb N}[/math].

:^)

[Anonymous No.12537370]

>>12537347
what does that have to do with it

[Anonymous No.12537381]

>>12537370
I call doubt to the claim that Cantor has shown that the reals outnumber the rationals.
Cantors diagonalization argument shows that possible existence claims about certain functions (bijections between some sets) would be inconsistent.

[Anonymous No.12537501]

>>12537347
if f: S -> N^N surjective
then define f': N -> N^N by
f'(x) = f(x) for x in S
= (1,1,1,...)
w00t a surjection
N^N is countable

[Anonymous No.12537504]

>>12537501
>= (1,1,1,...)
(for x not in S)

[Anonymous No.12537513]

>>12537501
>f'(x) = f(x) for x in S
If "x in S" is not decidable, then the set f' doesn't constitute a function, unless you adopt LEM

[Anonymous No.12537526]

>>12537513
LEMchad reporting in

[Anonymous No.12537542]

>>12537526
If you allow undecidable predicates in if-then-else type definitions of functions in set theory, then the required [math]\exists ![/math] to prove that a graph is a function is kinda weak.
Why even call it function.

[Anonymous No.12537553]

>>12537542
so I can count all the subsets of N, obviously

[Anonymous No.12537582]

>>12537553
What about the index set [math]I\subset {\mathbb N}[/math] of all ascii files [math]a_i[/math] (with [math]i\in I[/math]) so that

int main(){a_i;}

is compiling and terminating C++ code?
You want to adopt LEM so that this factually uncountable subset [math]I[/math] of [math]{\mathbb N}[/math] is deemed """countable""" in your theory?

[Anonymous No.12537610]

>>12537582
good luck finding a computer to do that for you

[Anonymous No.12537625]

>>12537610
do what

[Anonymous No.12537643]

>>12537625
find all the ascii files
unless you are doing that by hand

[Anonymous No.12537650]

>>12531712
Every mathematician and physicist since the beginning of time was a plagiarist. The one person who isn't and who invented the number has been forgotten through time and has no name, all that is left is plagiarists.

[Anonymous No.12537656]

there might be interesting stuff in foundations
but spamming about countable/computable/finite/infinite/uncountable is not one of them

[Anonymous No.12537663]

>>12531956
We don't ask questions like that around here I suggest you be careful in future.

[Anonymous No.12537666]

>>12537650
do you know the story of grug? he invented the number one, but his fellow cavemen started copying him and came up with two and three
fucking plagiarists i swear

[Anonymous No.12537677]

>>12537666
but gr0g invent zero
hole new concept
no copy grug

[Anonymous No.12537701]

If a field k satisfies a relation, and that relation is satisfied by some F_p, must k have a subfield isomorphic to F_p? For instance, suppose k satisfies a^4=a for all a in k, since this relation is also satisfied by F_2, must k have characteristic 2?

[Anonymous No.12537786]

>>12537643
"all ascii files" is just the enumeration of all words of a given alphabet.
We can just as well speak of all bit sequences. Those are also countable and here I can define [math]I\subset {\mathbb N}[/math] as the index set of those bit sequences which a C++ compiler could compile and run and then half.

This subset of the natural number can't effectively be counted. But sure, I'm aware that it's formally ZFC-countable.

If you reject the possibility to define subsets of the naturals via undecidable predicates at all, then that's another story.

[Anonymous No.12537845]

>>12537786
cool, go find a computer to do that

[Anonymous No.12538250]

>>12536486
If I recall correctly, given an integer [math]0 < k< p-1[/math], it can be proven without the use of primitive roots that there is an integer [math]a[/math] relatively prime to [math]p[/math] such that [math]a^k\not\equiv 1\pmod{p}[/math]. Denote the sum by [math]S_k[/math]. Since [math]a, 2a, \ldots, (p-1)a[/math] form a system of residues modulo [math]p[/math], we have [math]S_k\equiv a^k+(2a)^k+\cdots+((p-1)a)^k = a^k S_k \pmod{p}[/math] and the conclusion follows.

[Anonymous No.12538284]

Pic related makes sense for [math]\rho[/math] of compact support, but it is well known that if you take the charge density formally as a dirac delta you also get the correct electric field. How the fuck is this justified rigorously? From what I have gathered, the dirac delta can at most be extended to a linear functional over [math]C^\infty[/math] put in here we have a smooth function with a singularity at x. If we take some regularization of the dirac delta, is the limit well defined?

[Anonymous No.12538289]

When does this get interesting?

[Anonymous No.12538301]

Does anyone have a good recomendation for a primer in math? Ive been out of school for 8 years and would like to brush up but all of the textbooks i got from my friends are to advanced currently. I really need to get better at basic algebra so i can actually understand the calculus and geometry books i was given.

[Anonymous No.12538315]

>>12538284
Questions about rigour in physics are pointless. Physics is not subject to an axiomatisation process which makes mathematical rigour a moot point.
All that's needed to know is that the techniques are defined in a clear way, and the physical theories give an accurate description of observed reality. Questions about the rigorous definition of the Dirac delta just don't matter.

[Anonymous No.12538320]

>>12538315
nice non answer. It may be just a lucky coincidence notation works, my question was if there was some way to make sense of it formally.

[Anonymous No.12538327]

>>12538320
It's not a non answer so much as your question is a non question.
You can look up tempered distributions if you want but it's really utterly irrelevant to the pic you posted.

[Anonymous No.12538362]

When we describe the probability of a dice rolling 1 we say P({1}) = 1/6
But really we only know this given the model M of a dice so it should really say P({1} | M) = 1/6
Now what is P({1}) then?
If one has no information the most reasonable assumption is P({1}) = 0.5 it either happens or it doesnt
Q.E.D.

[Anonymous No.12538364]

>>12538284
>dirac delta can at most be extended to a linear functional over C∞
...no, the Dirac Delta obviously makes sense for any continuous function and can be defined there through the same integrals.
Because, you know. Continuous functions are integrable on compact sets.

[Anonymous No.12538376]

>>12538327
If you know why it is utterly irrelevant then that would answer my question you retard as that would mean distribution theory cannot justify this procedure. That is the type of question I am asking. It just something that anyone would find odd that the formal equation works for both singular and non singular densitites. Again this could be a coincidence or maybe not that is my fucking question. If you think it is useless well good for you.
>>12538364
What I have gathered is that all distributions of compact support can be uniquely extended to a continuous linear functional in [math]C^\infty[/math], can the dirac delta be extended in this sense to continuous functions? The problem is that in this problem the function has a singularity.

[Anonymous No.12538393]

>>12538376
The Dirac delta was defined specifically to work with singular densities. It's just a heuristic definition that is made precisely so you always get the correct field from a point particle.

[Anonymous No.12538413]

>>12538393
Defined how? As just the evaluation map in any space of functions no matter how irregular? The dirac measure seems to do this but can this be justified through the theory of distributions?

[Anonymous No.12538438]

>>12538289
When you start drawing compulsorily and masturbating to them.

[Anonymous No.12538502]

>>12538413
I think the point that the other poster(s) was/are getting at is that the Dirac delta was defined by Dirac as just something that worked and gave you the physically correct answer. He literally just wrote down the integral relations of the Dirac delta, and then started working with it, and that's all you need when you're aiming to represent a point particle.
I'm not sure if you're asking whether there's an issue with Coulomb's law itself, but when you say it's interesting that all these formal things work, that's just because it was defined to have these formal properties first, then the theory of distributions came up as a post-hoc justification for describing stuff like the Dirac delta.
As far as I'm aware, there is no longer considered any issue with rigor of delta functions.

[Anonymous No.12538505]

>>125383655
Doesn't this say something about physical countable infinities
If there are countably infinite rolls of 1 in the universes life then there are just as many rolls of 2-6
Probability just comes from uneven distribution

[Anonymous No.12538515]

>>12538505
Specifically the average of all subsets contain uneven ratios of 1 vs 2-6 but the whole set does not

[Anonymous No.12538547]

>>12538502
I understand that dirac delta came first and the theory of distributions came later. But my point is that from what I've seen from the theory of distributions, it is not clear how to make sense of coulombs law as the dirac delta can only be evaluated for tes functions and here it is evaluated against a function with a singularity. Now I understand you can prove rigorously using distributions what that Q1/r is the greens functio first and then you define the equation as a convolution with the greens function, but for field theory these integral actually comes up when you need to regularize.

[Anonymous No.12538563]

>>12536797
anyone?

[Anonymous No.12538703]

>>12536797
If the system has an outer ring and the metric gives that outer ring some form of lowest value then yes
Its just if the metric is informational enough to track paths
More answers maybe possible like counting vertices

[Anonymous No.12538756]

>>12538284
I'm sure you can find a sequence of functions even on a bounded subset that converges to the direct delta

[Anonymous No.12538769]

>>12538301
What do you mean by basic algebra? Do you have difficulty finding x such that x+1=2? If not, you’re ready to jump into spivak or apostol calculus.

[Anonymous No.12538778]

>>12538769
Nah not that bad. Id basically just like a quick review on exponents and some more practice with polynomials and some formulas.

[Anonymous No.12539147]

>>12538289
Starting DG is a filter for most beginners/intermediate students. Except if you already know why you study it. DG is analysis and PDEs """""without"""""" coordinates, meaning that you essentially learn how to write equations down in different coordinate systems. Furthermore you look for geometrically canonical stuff like the Levi-Civita connection. For that example, try writing down the LCC in of the flat [math]\mathbb{R}^n[/math] in fucky coords and you'll automatically start to recognize Christoffel symbols.
tl;dr it's analysis+geometry and examples are essential

[Anonymous No.12539167]

Sick and tired of these threads. Posts here are useless, it's either basic idiotic doubts by stupid people or some off topic bullshit with an anime girl pic attached for some reason or some trendy shit people read on Quanta Magazine. Where the fuck are the actual productive discussions about mathematics? And I'm talking real mathematics, research level, not some undergrad shit. That's why I'll stop posting here, it's a complete waste of my time.

[Anonymous No.12539182]

Putnam threads are back: >>12539169

[Anonymous No.12539196]

>>12539167
Quite a few posters have been either bitched at for posting higher level maths or just haven't got any response to an attempt at starting discussion and they all left.
Every now and then I lurk hoping for an interesting problem to be posted in my field but it seems there are no discussions at all now.

[Anonymous No.12539233]

What are some of the best books for someone starting Real Analysis this year? I've heard some great things about it and want to get a head start before my classes start up.

[Anonymous No.12539296]

/hnymg/

>>12539167
>That's why I'll stop posting here, it's a complete waste of my time.
See you tomorrow, honey.

>>12539196
What is your field? Mine is essentially (finite) group cohomology. Any overlap?

[Anonymous No.12539304]

>>12539296
analytic number theory
not that I expect anything to ever come up on that subject kek, but I got my hopes up since I saw some problems being posted from an intro to modular forms book a while ago

[Anonymous No.12539512]

>>12539296
>would you like to compute H*(SL(2,9); F3) with me
no i would prefer sucking on your boyclitty and then fucking your asshole with all my strength

[Anonymous No.12539557]

>>12539304
At least you have pointed your existence out now, so maybe ANT friend spots you and you can have a discussion on such matter. Mayhaps.

>>12539512
Please do not lewd me.

[Anonymous No.12539647]

>>12538362
>the model M of a dice
i love how atheists can stop cramming the word model in their narrative because they heard it in popsci youtube

[Anonymous No.12539798]

>>12539304
>analytic number theory
What's your favorite introduction book to the subject?

[Anonymous No.12539800]

>>12538250
You're right.
>hat there is an integer a relatively prime to p such that ak≢1
This can be done by looking at the polynomial x^k-1 and noting it can have at most k roots.

[Anonymous No.12539892]

Why is [math]N! \ \sum_{k=0}^{n} \frac{1}{k!} = \lfloor e \ n! \rfloor[/math] ?

[Anonymous No.12539895]

>>12539892
Think about it.

[Anonymous No.12539900]

>>12539892
I mean it's evidently in the ballpark since the sum on the right hand side is the exp(1).
Are you asking for details on why it's that exact integer and not plusminus one?

[Anonymous No.12539915]

>>12539895
>>12539900
Yes, specifically I'm stuck at proving that [math]N! \ \sum_{k=0}^{N} \frac{1}{k!} > N! \ e - 1[/math]

[Anonymous No.12539961]

>>12539798
It's a pretty big subject so it depends on what part you want an introduction to.
Overall, Apostol's books are great. I like his book on modular forms a lot, but it's one of those books where you really should do the exercises.
Titchmarsh has one of the best introductions to the theory of the Riemann zeta function
Serre's Arithmetic is a huge hit as well obviously
For getting into modular forms more generally I liked Diamond and Shurman, and there are a few things from Don Zagier which are really insightful.
Also a bit of a tangent, but Gauss' Disquisitiones Arithmeticae is actually a surprisingly good read. It makes you feel like a total brainlet though since he did it all at 21.
Berndt also has a fun little booklet Number Theory in the Spirit of Ramanujan - not really a great introduction to the theory, but a really fun read.

[Anonymous No.12539979]

>>12539915
Just bound the remaining series by a geometric one. EZ

[Anonymous No.12539984]

>>12539915
You want to show
1/N + 1/(N)(N+1) + ... < 1 for N large enough.
But this is obvious. For N>2 this is smaller than
1/2 + 1/4 + 1/8 + ... = 1

[Anonymous No.12539990]

>>12539915
ok so I got
[math] N! \ \sum_{k=0}^{N} \frac{1}{k!} > N! \ e - 1 \\
\iff \sum_{k=0}^{N} \frac{1}{k!} > e - \frac{1}{N!} \\
\iff \sum_{k=0}^{N} \frac{1}{k!} + \frac{1}{N!} > e \\
\iff \frac{1}{N!} > \sum_{k=N+1}^{\infty} \frac{1}{k!}
[/math]

How do I prove that last inequality?

[Anonymous No.12539994]

How do you actually specialize/choose a field/choose a masters thesis topic.
I am starting a masters at a european university and I have specialized in the sense that I am mostly attending lectures about Analysis of PDEs. But in this area I am really just learning the things everyone else learns. Then how does one get an idea for a thesis topic? Do you have to start reading research papers and exploring various books?

[Anonymous No.12540005]

>>12539994
it certainly helps if you are browsing arxiv or reading books, but as a beginner to research, it's hard to pick a problem for yourself
for most people, their advisor gives them a topic (or a list of topics to choose from)
masters theses are often just expository writing without any new results

[Anonymous No.12540022]

>>12539990
aaah I see now, just multiply both sides by N and proceed as >>12539979 >>12539984
Thanks!

[Anonymous No.12540095]

>>12539990
also, holy shit the last part really shows how fast N! grows.

[Anonymous No.12540175]

>>12532062
What is a group implied to be? If its the set with the operation (rotation matrix group for example) i can think of a counterexample so i guess ir means something else

[Anonymous No.12540188]

>>12539800
Yes but this proof is supported by field theory.
There is a proof through just elementary number theory. I'm just not 100% sure of how much about primitive roots it uses, but I don't think it's everything because that would be too easy with them, and I remember it being a bit complicated.

[Anonymous No.12540279]

>>12540175
Better write it up, you might be considered for a fields medal with a breakthrough of that magnitude.

[Anonymous No.12540369]

>>12539167
>coming here for serious math instead of for my occasional OC

[Anonymous No.12540392]

>>12540369
kek, I think top panel has actually been posted in /mg/ before

[Anonymous No.12540434]

>>12540369
Good post. Captures how I feel about this very well.

[nkun No.12540452]

>>12540369
The problem with the H* one isn't that it's not interesting math, but that it's posted as self-indulgent gimmick without others having much from it. Math is too broad to have one general be rewarding for most people, while taking an effort for /mg/ might also be wasted time, if you talk into a wall.
We haven't solved the math engagement problem, so it's just animeposter banter

[Anonymous No.12540569]

>>12540369
Quite good OC.

[Anonymous No.12540712]

>>12535428
Based ALPHA male energy

[Anonymous No.12540715]

>>12535162
>Until now I never conceived of a world where e-thots would be gaming on chess boards
Anon, women are social creature so of course they would turn everything they touch into match up of e-thottery.

[Anonymous No.12540909]

>>12539304
>since I saw some problems being posted from an intro to modular forms book a while ago

That was probably me. I did a course in analytic number theory a while ago but I'm far from attempting research on the subject, even though I'd probably like to. That would require a good advisor and a lot from me (in particular because I don't want it to be my primary interest), neither is going to happen in the next few years.
But maybe I'll post some related non-research problems here.

[Anonymous No.12540923]

Why do we teach calculus and linear algebra before mathematical analysis? Isn't that kind of putting the cart before the horse?

[Anonymous No.12541049]

>>12540923
If you are american, then because you are a bunch of dumb fat faggots who think analysis is just calculus with more rigour. If you are nor well because you need mathematical maturity in order to understand the process of abstraction. You will find out that analysis (except for maybe measure theory) doesn't really makes calculus a lot easier because the point is to generalize these technics and apply them to more general scenarios. Its the same reason in any field you don't start with advanced stuff retard

[Anonymous No.12541061]

>>12541049
You're kind of mean anon. Do you want to get a drink or something? It's on me.

[Anonymous No.12541093]

>>12541049
One course of calculus should be enough for math majors, and there is no need to skip proofs - they should do a proof course right in the first term.The only purpose is to make students be quite familiar and good with calculations, one course is more than enough. Then specific calculations can (and should) be introduced as complementary exercises in all other subjects, being less in volume and more oriented towards "create a method to calculate this" rather than repeating the same recipe hundreds of times.

[Anonymous No.12541124]

Mathematically speaking, what does bitcoin do that money can't?

[Anonymous No.12541296]

>>12534250
>but it doesn't seem to generalize for triples of consecutive residues like yours do.
Actually the resulting sum isn't so trivial to deal with in this case.


Here's the general problem for degree 3:

Let [math]p>2[/math] be a prime, [math]f\in \mathbb{Z}[X][/math] with leading coefficient not multiple of [math]p[/math].
Compute [math]S_f := \sum\limits^{p-1}_{k=0} \left( \dfrac{f(k)}{p} \right) [/math].

In particular, the case [math]f(X) = X(X+1)(X+2)[/math] is interesting for counting triples of consecutive quadratic residues.

Any ideas?

[Anonymous No.12541350]

>>12540369
Keyed and unlocked.

[Anonymous No.12541450]

>>12540392
Probably.
It was one of the first few results that showed up when I googled "integral ramanujan".

[Anonymous No.12541688]

I'm on the verge of suicide right now, this semester was my last chance to make something useful and actually learn the essential subjects like abstract algebra and analysis. I had an entire week to study those subjects, but all I did in that week was play videogames, read manga or just browse this cursed site randomly. I'm already too old and my grades are not good and I can't even study properly, the first thing I do when I wake up is turn on my phone and start reading random shit on 4chan and this goes on for 10 hours straight, until it's time to sleep. I'm not asking for advice, nor am I trying to make people feel sorry for me. I dig my own grave, I'm just leaving my thoughts here, perhaps my last ones. I can't see a light at the end of the tunnel for me anymore, only darkness and I was the one who made that choice, damn this is frustrating. I wish time travel was possible, but even then I guess I would end up making the same pathetic mistakes, that's how fucked up I am. Now it's too late for me, classes are going to return tomorrow and I didn't study a single thing, this will be again another semester of failures and terrible grades. It's the end for me, I had a good run I guess though. I hope everyone here can succeed in academia from the bottom of my heart, failing the way I did is very pathetic and the worst feeling of all.

[Anonymous No.12541712]

>>12541688
Don't do it anon. Were you intending to become an academic? You can always change your focus and career. Nobody is too old. In the future you will look back upon this episode, how scary it seemed at the time, and will laugh.
Academia is a dying institution anyway. Very few people actually succeed. There are plenty of other options of what to do.

[Anonymous No.12541741]

>>12540909
Yeah there is a really high barrier to entry which is why I said I never expected research level discussion on 4chan but broadening the topics we broach in /mg/ is always good

[Anonymous No.12541842]

>>12541688
Do you really think that any possible life outside of academia for you is unworhty? Really?

[Anonymous No.12541849]

>>12541688
Let me give some advice for you anyway, ditch the smartphone and get a flip

[Anonymous No.12541856]

>>12532734
Is it:
1 if p = 2
1 + (p-3)/4 if p = 3 mod 4
2 + (p-5)/4 if p = 1 mod 4 ?

[Anonymous No.12541937]

>>12541712
>>12541842
It's pathetic enough for me to have to abandon my dreams of becoming an academic one day, but I could do it, I really could. The problem is not that, the real problem is that I already spent too much time and after all that time I keep gathering failures. So it's not only about finding a job, it's about even graduating. I'm so pathetic I can't even study, out of all the subjects I've completed so far, I didn't study seriously for a single one of them, I just got lucky. This semester was my last oppotunity to change, but now I see that I'll be like this for the rest of my life, there's no saving me.
>>12541849
I know that, but it's like a drug, the first thing I do when I wake up is get it and browse random sites. Sometimes I manage to control myself, but eventually I turn it on and from there it all goes downhill.

[Anonymous No.12541939]

>>12541856
Only if you count 0 as quadratic residue and (p-1, 0) as a pair of consecutives when p-1 is a quadratic residue.

[Anonymous No.12541950]

>>12541937
>I know that
If you know it then why haven't you done it yet? I used to have the same problem, although perhaps not to the same extent. Spend 60 bucks on a flip -- alcatel works for me -- and the compulsion will absolutely pass.

[Anonymous No.12541955]

>>12541937
I'll be real with you, if you really can't make yourself change, then yes you should give up on academia and maybe even graduating. But that doesn't mean you can find a good life without those things, there are people who are happy with way, way less than what you're trying to achieve.

[Anonymous No.12541975]

>>12541955
Sure, but I can't change it now. My family spent a lot of money on me, they have huge expectations, even now they still do because I don't tell them how much I failed. They think I'm a really intelligent person who will definitely succeed and I helped build that image, you see? There's no path left now. Besides, even if I could give up on math, that would be very sad and depressing for me. This semester is my last chance and so far I'm sure to fail, there's no path left for me anymore, all roads were closed a long time ago, it's either success or suicide now, I'm really on my limit here.

[Anonymous No.12541988]

>>12541939
Yes, that’s what I had in mind.
The idea is that we can parameterize the set [math]S = \{(x,y) \in \mathbb F^2_p, x^2-y^2 = 1\}[/math] by the map [eqn]\begin{array}{rcl} \mathbb F_p^{\times} & \to & S \\ u & \mapsto & \left(\frac{u + u^{-1}}{2}, \frac{u-u^{-1}}{2}\right)\end{array}.[/eqn]
Now, the set of pairs of quadratic residues is simply the set [math]\{(x^2, y^2), (x,y) \in S\}[/math]. A pair [math](x^2, y^2)[/math] is generically the image of 4 pairs unless either x or y is zero, in which case it is the image of two pairs, which yields the above count.

[Anonymous No.12541998]

>>12532639
>Science is about understanding the laws and structures of the universe
Lol

[Anonymous No.12542010]

>>12541937
do you like math, or the idea of being a mathematician?

[Anonymous No.12542021]

>>12541998
?

[Anonymous No.12542032]

>>12542010
I love math.

[Anonymous No.12542053]

>>12542032
Based and lovepilled.

[Anonymous No.12542060]

>>12542032
Do you love that it exists, or do you love working with it?

[Anonymous No.12542092]

>>12542060
Both? I really like the feeling of solving a really hard problem.

[Anonymous No.12542120]

>>12531655
Does anyone have any good examples of cases where a conjecture that initially seemed true intuitively was later found/proven to be false? I was trying to give examples to a friend to emphasize the importance of rigor.

[Anonymous No.12542161]

>>12542120
Rigor is only important because we defined it as such.

[Anonymous No.12542162]

>>12542120
The most classical example is eighteenth century mythology about how "Functions are mostly differentiable and stuff."

[Anonymous No.12542181]

>>12542120
Also Fourier series stuff.

[Anonymous No.12542208]

>>12542120
Well the consistency of naive set theory is the goto example.

But the interplay between truely geometric theories and their axiomatiziation is such that, in the worst case, the axiomatization should bend to the geometric intuition. So it's hard to be "intuitively false" about geometry. You might just fuck up with your intuition about the real number line and definitions build on it (e.g. topology).

The number theoretical examples with large counterexamples exist often, but those are rather such that you can't speak about intuition but merely have a false expectation based on nobody having found such counterexamples.

But to make your case, rigour can also be used to settle disputes - given people at least agree on the formal definitions

[Anonymous No.12542216]

>>12542161
Elaborate?
>>12542181
>>12542162
Thanks. I also found these 3 examples on the Wikipedia page for Collatz Conjecture, which is what started the discussion about rigor in the first place.

[Anonymous No.12542217]

>>12542120
Anything that has to do with exchanging limits
>A limit of continuous functions is not continuous
>The integral of the limit of a converging sequence of functions is not necessarily the limit of the integrals
>etc.

[Anonymous No.12542232]

>>12542208
>consistency of naive set theory
Do you mean for example Russel's paradox?
>rigoour can also be used to settle disputes - given people at least agree to formal definitions
Indeed, although in my experience with people arguing, most people do not argue logically or see any purpose in even attempting it. I usually just step back whenever I see people arguing politics or something for that reason. It is a useless endeavor if they won’t agree to some specific terms of discussion. But typically, people argue because they want to be "right", not because they have a genuine desire to discover the truth.

I like number theoretical examples only because it is easier to show/explain to someone uninitiated. Set theory would require a bit more explanation to someone who has no experience with maths

[Anonymous No.12542234]

>>12542120
your friend can always argue that those results were all intuitively false and that everyone else including you was a brainlet.

[Anonymous No.12542287]

>>12541988
Yeah, I know this solution. I wrote a little bit differently in terms of equivalence classes, where the relation is z ~ w iff zw = 1 (mod p). Then you send a class to inverse(2) (x+y) etc.

This idea works for the problem of calculating [math]\sum\limits^{p-1}_{k=0} \left( \frac{f(k)}{p}\right)[/math] where f is a degree 2 polynomial over integers. I wish there was a similar solution for the case of degree 3, but it looks harder. Even if it is something like [math]f(X) = X^3 - a[/math], doesn't seem obvious to me.

[Anonymous No.12542296]

>>12541688
Don’t forget to livestream your suicide so at least you will be useful by entertaining us as you die.

[Anonymous No.12542304]

>>12541124
Fuck banks

[Anonymous No.12542311]

>>12542120

anything in probability btfos intuitionists

[Anonymous No.12542315]

>>12541975
Your situation is not the greatest, but even if you fail at math it's not over. The other roads won't close. Some people bounce back with way less than what you'll have, even if your whole family cut ties with you.
It is going to feel horrible if it happens, but getting to the bottom means new opportunities to begin new journeys from zero - where you pretty much have nothing to lose.

If I were you, I'd tell everything to my family as soon as you finish your tests, regardless of results (expected or not). The chance of them forgiving you and even helping you will only decrease if you don't miraculously recover.

[Anonymous No.12542320]

>>12541975

Maybe your parents should have monitored your video-game / computer time, it's not your fault you got addicted.

[Anonymous No.12542392]

>>12541975
Try and relax dude, treat this as a wake up call for you to sort your life out. Sometimes you need circumstances to slap you round a bit to get yourself on track for becoming who you want to be.
I came very near absolutely bombing in both undergrad and my PhD and I turned out ok. This is an opportunity for you to sort yourself out and improve yourself.

[Anonymous No.12542406]

>>12542392
>and my PhD
share the story

[Anonymous No.12542426]

>>12542406
That in itself is nothing too unusual. Advisor was working on something, posted a preprint and I disproved a lemma he thought was solid and gave a counterexample, absolutely tanked the relationship for half a year and it never fully went back to the way it was. At one point we were close to splitting since he was very precious about being told he was wrong and he tried to block me from working on this subject like 3 years into the PhD which would have destroyed most of my thesis not to mention the scummy feeling of going along with knowingly wrong work.
Eventually it led to much better work being done with correct results and a better theory behind why this lemma was wrong but he still made snide comments about it right up until I finished.

[Anonymous No.12542436]

>>12542426
Also at the time my gf got $15k stolen from her and the retards at Big Condom totally dropped the ball so we had a pregnancy scare just after $15k vanished into thin air, so yeah tensions were somewhat high
It all worked out ok in the end, even if I developed festering rage issues along the way lol

[Anonymous No.12542460]

Hello I don't understand what's happening in the first part of the second line in this picture. I don't get what happened to make both of the first coefficients to be 3 in both pairs. After both coefficients are 3 I can see how they're added.

https://jeremykun.com/2014/01/17/how-to-conquer-tensorphobia/

[Anonymous No.12542464]

i failed my engineering thesis, guess i'm not worthy of being on /mg/

[Anonymous No.12542467]

>>12542464
at my degree mill uni no one really fails a master's thesis

what happened?

[Anonymous No.12542493]

>>12542426
Would you do anything differently if you'd go back right to when you found that counter-example? What and how?

[Anonymous No.12542504]

>>12542467
>what happened?
nothing in particular, i just didn't manage it

[Anonymous No.12542511]

>>12542493
No, it was one of the few times in my life where I knew that it wasn't me who fucked up. It's very rare that the fuck up isn't mine so I know it when it happens, and I think I approached it as carefully and properly as I could, but there was just no accounting for ego and embarrassment.

[Anonymous No.12542604]

>>12533336
The book skipped a step. The second part is equivalent to: alpha^2 + (2alpha + 1)/n_0 < 2 and that's what the text in between the first and second part was talking about.

[Anonymous No.12542707]

I am absolutely retarded. Was about to post a vehemently confused question about "left dictionary orders" until I realized they chain up meaninglessly or have Z- bijections at that slot

[Anonymous No.12542715]

Also you guys have contagious vitamin d definiency, can you go outside more for the collective mg vitamin auras? Thank you

Related song:
https://m.youtube.com/watch?v=AzlMeTxVdH8

[Anonymous No.12542742]

>>12542715
I’ve got plenty of vitamin “D” right here. *unzips*

[Anonymous No.12542766]

If you form any bijective function between Z+ and any other set S, does that mean S has an order type Z+ because you can create an ordering for S where S(1)<S(2)<S<(3)<...

[Anonymous No.12542770]

>>12542742
That's gay anon I have thick manly quadriceps and triceps

[Anonymous No.12542784]

>>12542766
And if this is true doesn't that make 4a of pic related false?

[Anonymous No.12542800]

I have an abstract algebra test (master) in 17 days, wish me luck.

[Anonymous No.12542815]

>>12542766
No, it means S can be given an order with type Z+, it can also be given an order isomorphic to any countable order

[Anonymous No.12542841]

>>12542770
Are you trying to turn me on, fag?

[Anonymous No.12542843]

>>12542010
How would you fix the latter?

[Anonymous No.12542862]

I'm trying to solve this geometric growth equation for the common ratio 'r'. But I'm not sure how to get it. I've tried Wolfram and other online solvers, and they couldn't get it.
I'm sure I'll need to use logs to get the exponents out, but I can't find a way to get r by itself.

It's from the math used in idle games & clicker games. Not homework or anything. Can anyone help me out?
[eqn] cost = b * \frac{r^k - r^{k+n}}{1-r}[/eqn]


[eqn] \frac{cost}{b} = \frac{r^k (1-r^n)}{1-r}[/eqn]

[Anonymous No.12542933]

What happens if you give the usual order relation to the projective real line

>>12542815
Goood luck anon

[Anonymous No.12542975]

>>12542862
Can't be done in finite terms.

[Anonymous No.12543040]

>>12542975
Can you please explain? How do you know?

[Anonymous No.12543077]

>>12543040
first clue is that for high enough k almost all polynomials can't be solved in radicals
the right hand side is the partial sum of a geometric series and can't give you a solution for the ratio

[Anonymous No.12543101]

>>12542715
>go outside
Nice maymay. Eat raw meat and in particular fat. Like Mett, i.e. raw ground pork.

[Anonymous No.12543106]

>>12531712
Honestly, each elementary course on Calculus, Linear Algebra and Statistics is plagiarism.

[Anonymous No.12543338]

>>12542460
i'm not tensorphobic i'm not afraid of them i just think they are disgusting and this is a natural human reaction

[Anonymous No.12543355]

>>12543077
I see. I had to study what you said for a bit, and it makes sense now. Thank you!

[Anonymous No.12543492]

Problem of the Day >>12543339

[Anonymous No.12543643]

>>12542766
No, it only means that S *can* be ordered in such a way that it has order type Z+, not that it will be well-ordered whatever the order you put on it.
Q is countable (hence in bijection with Z+) but it is not well-ordered with the usual order relation on the reals.

[Anonymous No.12543675]

>>12542460
[math]6 \otimes 1 = (2 \cdot 3) \otimes 1 = 2( 3\otimes 1) = 3 \otimes 2\\ 3\pi \otimes \pi = \pi (3 \otimes \pi) = 3\otimes \pi^2[/math]

[Anonymous No.12543720]

>>12542460
To understand tensor manipulation, here's a nice theorem.
[math]\sum a_i v_i \otimes w_i = \sum a'_i v'_i \otimes w'_i[/math]
if and only if for every bilinear map [math]f[/math] from [math] V \times W[/math] you have
[math]\sum a_i f(v_i, w_i) = \sum a'_i f(v'_i,w'_i)[/math]

[Anonymous No.12543744]

>>12542311
This. Even simple probability brainteasers fuck up smart people if they attempt to approach the problem without rigor. Happened to me countless times.

[Anonymous No.12544077]

>>12537666
>>12537677
Zero is a much later discovery than caveman times. Actually, so is one. The abstract number concept certainly wasn't discovered until humans became conscious.

[Anonymous No.12544279]

>>12542426
>Advisor was working on something, posted a preprint and I disproved a lemma he thought was solid and gave a counterexample, absolutely tanked the relationship for half a year and it never fully went back to the way it was.
he is among the top ten people int eh field or just average mathematician?

[Anonymous No.12544521]

Any good books on matroids?
(originally posted in >>12541945, sorry if these parts of combinatorics aren't welcome here, I'm not very good in actual hard math)

[Anonymous No.12544545]

almost every field has been deeply affected by race, gender, and social issues.

is mathematics the last safe place?

[Anonymous No.12544618]

>>12544545
Sadly, no. mathematics has also been used by prejudiced white men to discriminate against people of color, LGBT people and women. However things are changing for the better.

[Anonymous No.12544660]

>>12544545
No, there are plenty of paedoes in it, see >>12544618

[Anonymous No.12544666]

>>12544618
This.
Math was originally created by brown people but then used by whites to oppress people in their kyriarchy
Now it’s getting better though

[Anonymous No.12544681]

>>12544545
I think mathematics will be safe for a while because I don’t think it is seen as particularly prestigious or profitable in the eyes of people.

It won’t be a problem so long as they won’t try to make the curriculum weaker in order to cater to stupid people. Look at computer science departments in the US where they have bastardized their undergrad programs and turned it into software engineering.

[Anonymous No.12544690]

Do we really not know what the homotopy groups of spheres are at this point in time? Why haven't we simply used computers to calculate them?

[Anonymous No.12544693]

>>12544666
Based. The truth is Greeks didn’t invent or discover most of the ideas attributed to them. It would be more accurate to say they introduced the ideas to the west. Basically, Aristotle/Socrates/Plato were the ancient equivalents to Neil Degrasse Tyson and Bill Nye the science guy.

http://ckraju.net/books/Sandhan-review-Cultural-Foundations-of-Mathematics.pdf

[Anonymous No.12544700]

>>12544690
how would you use computers to calculate them?

[Anonymous No.12544712]

>>12544700
Just train a neural network bro

[Anonymous No.12544796]

>>12544712
Look at Ramsey numbers or fuggin Latin squares. Computers are totally useless for many problems.
Neural nets are just glorified heuristics for automatic simplified brute-force.

[Anonymous No.12544807]

>>12544700
Use approximations of the maps and the homotopies involved. Get a bound of how well you need to approximate so that all homotopies between maps can be checked. Then just brute force it.

[Anonymous No.12544910]

>>12544807
why don't you do that for us

[Anonymous No.12544991]

>>12544910
Busy watching anime atm. Maybe later.

[Anonymous No.12545243]

Did this thread actually find out the definition of a set?

[Anonymous No.12545260]

>>12545243
Yes
>By an “aggregate” (Menge) we are to understand any collection into a whole (Zusammenfassung zu einem Ganzen) [math]M[/math] of definite and separate objects [math]m[/math] of our intuition or our thought. These objects are called the “elements” of [math]M[/math].

New thread >>12545256

[Anonymous No.12545272]

>>12545260
That definition doesn't account for the paradoxes of set theory, or the arbitrary decision to exclude multisets.

[Anonymous No.12545277]

>>12545272
Multisets are excluded because they are a more elaborate concept, sets are more fundamental
As for paradoxes, just don't consider sets too big ("inconsistent aggregates" as Cantor called them)

[Anonymous No.12545286]

>>12545277
Inconsistent aggregates aren't in the definition provided.

[Anonymous No.12545304]

>>12545286
They are not sets.

[Anonymous No.12545487]

>>12544279
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