Thread 16745109

[math]/\mathfrak{mg}/[/math]

Metaphysical principles of the infinitesimal calculus edition
Talk maths, previously >>16712226

>>16745109 (OP)
Your boy confuses 0 with null.

>>16745109 (OP)
Any zero element of an abelian category is a number.

Philosophical musings belong to >>>/his/ History and Humanities.

>>16745239
Kek. Troll or Schizo though? Idk.

>>16745239
Zero elements in abelian categories are not numbers; they are objects, morphisms, or identity elements in algebraic structures.
You probably struggle with wave/particle duality too.

>>16745307
>identity elements in algebraic structures
That can be a number when the algebraic structure's elements are considered numbers

>>16745310
In an abelian category, each Hom-set has an additive identity, called the zero morphism, and the category contains a zero object that serves as both initial and terminal. These 'zero' structures are not numbers, but rather structural elements enabling the addition of morphisms and the definition of kernels and cokernels.

>>16745109 (OP)
i am not sure to understand. without zero, how would you handle coordinates?

>>16745337
The elements of [math] \mathbb{Z} [/math] are considered numbers. It has an element "0", which is therefore considered a number. Checkmate

>>16745369
>the set of all integers is an abelian category
OMGosh, if troll then sucks at trolling.
More like a kid who wants to be cool. You have said nothing that needs category theory but you insist on using what you do not understand.
>but any ring can be viewed as a category with one object
This category is not additive in the way required for an abelian category unless you define addition of morphisms — but in this construction, morphism composition is multiplication, and there’s no built-in addition of morphisms.
>Checkmate
Son, you're playing checkers, and you lost.
Again.

>>16745369
>The elements of [math] mathbb{Z} [/math] are considered numbers. It has an element "0", which is therefore considered a number. Checkmate
The category of abelian groups (which includes [math] mathbb{Z} [/math] as an object) is an abelian category. But [math] mathbb{Z} [/math] itself is just an object in that category; not a category, and certainly not an abelian category.

>>16745409
> >the set of all integers is an abelian category
No one said that. You're hallucinating. Checkmate

>But mathbbZ itself is just an object in that category; not a category, and certainly not an abelian category.
No one said it is a category. Checkmate

What I said here >>16745310 is still true, so suck it

>>16745395
Even if we created the integers, god created us, so basically he created them transitively

Let X be the topological space with only 2 distinct points a,b, whose only closed sets are the empty set, X, and {a}. Is there a commutative ring whose Spec (with Zariski topology) is homeomorphic to X?

>>16745514
Wrong troll is wrong. Groups are not categories and >>16745310 is as false of a statement as it's always been.

>>16745522
Lemma: if A is an algebraic structure whose elements are considered numbers, then the identity element of A is considered a number.
Proof: obvious.

Note: the above lemma proves >>16745310 .

Example: the elements of the Abelian group [math] \mathbb{Z} [/math] are considered numbers. Hence, the identity element of [math] \mathbb{Z} [/math], which is 0, is considered a number.

>>16745522
>Groups are not categories
There is an isomorphism between the category of (small) groups and the category of (small) groupoids with one object. (Small) groupoids are (small) categories

>>16745533
>>16745535
See >>16745409.

>>16745533
>Lemma: if A is an algebraic structure whose elements are considered numbers, then the identity element of A is considered a number.
>Proof: obvious.
And has nothing at all to do with abelian categories.
Stick to sets, son. You clearly understand them.

>>16745555
Why does it need to have something to do with Abelian categories?

>>16745576
See >>16745239

>>16745584
I don't care about that, I was talking about >>16745310 and >>16745307

>>16745597
>i can't follow a post thread
nmp
scroll up and figure it out, faggot.

>>16745601
Figure what out? Lol you sound like a schizo.

>>16745622
>wait, groups aren't categories?!?
>(˶˃⤙˂˶)
Kek. Sure, easy mistake.
You look cute with that Cantor backpack. Are you sure you're 18?

>>16745622
>wait, groups aren't categories?!?
>(˶˃⤙˂˶)
Kek. Sure, easy mistake.
You look cute with that Cantor backpack. Are you sure you're 18?

>>16745109 (OP)
Does anyone have any literature recommendations for diff eq. where the paths are always on a smooth Riemannian manifold? I've been interested in exploring Brownian motion on smooth manifolds, but I figured it would be worthwhile to see about deterministic systems of differential equations on Riemannian manifolds before I dive into SDEs on manifolds.

>>16745109 (OP)
define "number"

Is the signal processing / information theory EE PhD guy here? I wanted to ask what’s the job market like for them, and what kind of cutting edge research is being done in the field.

how do i get better at modeling things? i try and idk im just ass at it. i think it'll take me too long to get decent at it.

>>16747579
Glad to know I'm genius level because I know some basic cohomology theories.

Do you dare me
to grace this thread
with a pretty drawing?

>>16747629
Yeah, I'm here. Can't seem to leave this god foresaken place.

Job market is good if you're a citizen of any of the big countries and don't mind doing glowie defense related work. Job market is kind of ass if you're trying to be in the tech sector or you can't get a security clearance. I can't speak about everywhere, but for a US citizen with a couple of decent publications, you should have no problem getting jobs that start in the $150k/yr range fresh out of grad school.

In terms of cutting edge research, there's a ton. A lot of people are still chasing the dragon of information theoretic performance bounds on deep learning based systems. Differential geometry and information geometry is slowly starting to percolate through the signal processing world as well (which is good if you ask me).

>>16747629
Oh, I forgot to mention in >>16748789

If you're looking to get a sense of what's "cutting edge" in the field, I recommend taking a look at IEEE Open Journal of Signal Processing (OJSP), the Journal of Advances in Information Fusion (JAIF), and the FUSION conference proceedings.

OJSP and JAIF are reputable and open access, so you don't have to pay for an IEEE membership or anything. The FUSION conference isn't technically open access, but the previous years always become available once the current year's proceedings hit IEEE Explore (so you'll be able to see the 2024 proceedings for free in September or whenever the most recent conference proceedings get published).

>>16745239
You ableists make me sick with your bigoted ableism

if no zero, what should be the initial state of a counter?

>>16748928
11. That's when you really need a counter. Everything before that you can just use your fingers.

>>16749134
>11.

eleven what? i don't understand. is this a private joke?

also, if you want to count the fruits in multiple containers to then have a table like a database. how would you report that some containers have nothing inside?

>>16749149
> eleven what? i don't understand. is this a private joke?

Not a private joke, just a retarded one. You've got 10 fingers. You can just count with those until you hit 11. That's when you need to start the counter.

>>16749149
>eleven what? i don't understand. is this a private joke?
ignore him he's some toeless freak, it's 21 for us normal people

>>16749157
Man, I'm not trying to take my shoes off in the middle of my work meetings to add another 10 to the count! That's barbaric!

>>16745109 (OP)
>zero is not a number
>A number is a mathematical object used to count, measure, and label.
It would appear he is retarded.

Let A be the ring of all smooth real-valued functions on [math] \mathbb{R}^n [/math].
Does A satisfy the ascending chain condition for radical ideals? I.e., does every ascending chain of radical ideals in A eventually stabilize?

How the fuck do you make the leap from "it's true in these cases" to "it's true in all cases"?

Like they show me some theorem and I plug in a bunch of values and the theorem holds for all the values I try, how do I know it's proven for all of them even the ones I don't try?

>>16749194
Is "half" a number?

>>16749481
There's a few ways you can deal with this. The simplest is to prove by contradiction. Assume the theorem is false, and see where this gets you. Often you'll find yourself ending up in a contradiction (meaning your assumption must be false).

I think people are being stupid about a^n + b^n ≠ c^n
Doesn't seem very hard to prove.
Why do they say it's hard?

>>16745237
Can't confuse things that are the same thing
x=x+0, null additives are exactly the same as 0 additives, by definition.

>>16745250
Why yes, you are an anus, you dont have to demonstrate the fact.

>>16749481
>what is a proof

>>16749134
No, you can get to 24 counting finger segments (its why a day is 24 hours), 28 if you count thumb segments too.

>>16749636
Proofs aren't brute force, they are logical extrapolations, if they had to be done by brute force, you couldn't even prove whole numbers since you would have to wait until you counted the last one.

>>16747579
can you post the other one

>>16749530
Interesting, thanks.

>>16749734
is this bait? seriously can't tell. i know that some people out there are actually that stupid, so is this one of these cases?

>>16749613
Never initialize, Anon. Keep being you; plus or minus an unknown non-value.

Has anyone looked at ancient symbols for math cheat codes?
Seems like a quick way to level up once you figure out the encoding. Beware fad fakes serving as forest to hide the trees.
Schizos are also mostly part of the forest.

>>16748789
>>16748798
Thanks anon.

> glowie defense related work
man, is AI/ML research roles a no go? glowie jerbs don’t seem to have an exponential raise in compensation like the tech bros.

>>16748798
danke! My uni does have a sub to IEEE so accessing journals isn’t an issue.

can you guys explain these 'infinities' to a layman like me? what is ''huge'' for example (except for OP's cock intake)
they're from the vsauce video linked here: https://youtu.be/SrU9YDoXE88?t=1252

Need a good theory book for multivariable differentiation theory and good problems, ie Rudin ch 9 but at a higher level for more quality problems


Covering stuff like inverse function theorem, implicit function theorem, mean value theorem, lagrange multipliers, multivariable Taylor series theory etc

Can anyone here explain the origins of diagrammatic categories to me in a clear and concise way? I understand that ribbon categories have nice ways of displaying morphisms with arrows and boxes and all that, but how did we even get here to begin with?

>>16750135
Duistermaat and Kolk, goes in depth on those topics and it has an insane amount of quality exercises, though many are quite hard IMO.

>>16747629
another EE PhD here, the field is so sprawling and has found its way into almost every single engineering application that there's always industry jobs that are available
the current gov't admin is shitting all over most research, though, so right now there's a lot of funding uncertainty in academia and research institutions like national labs. however, defense and nuclear research seems healthy right now

>>16750173
Thanks anon, it looks good, just that the notation feels too posh /modern but I will get used to it

>>16750083
It's pretty common for starting salaries at any of the UARC's to be 160-200k USD for recent PhD graduates. That's not the top of Google or Microsoft salaries, but you're doing pretty good for yourself. You'll just have to accept wearing a sleeping mask so the glowing in the dark doesn't disrupt your sleep.

>>16748789
>>16750184
NTA but what are your guys' thoughts on the future of photonics research?

>>16750132
i have no idea, but it's quite possible that they are just a byproduct of a philsopohyically incoherent theory of sets that shouldn't be taken too seriously to begin with, that they are just the phantasms of set theorists who are way too attached to that faulty and overbearing theory.

there is a serious, albeit rather fringe notion that the power set axiom makes the notion of set in formal set theories incoherent. turns out that you can still do most of mathematics if you drop the power set axiom and just try to formulate stuff more directly. power sets are replaced by power classes, which are not sets, and the ontological distinction between sets and classes is made more explicitly. i've grown very sympathetic to these kind of ideas. i've encountered them on logic questions on mathoverflow, a guy named nik weaver seems to be the most prominent proponent of this view, which he calls "mathematical conceptionalism", you can read about his views in his many papers on the topic on arxiv.

It nullifies structure.
You make the quadratic part zero and it becomes cubic. Make one part of a cube totally flat and it becomes a plane.

[math]E\subset \mathbb{R}^d[/math] is a non-empty closed box and [math]f:E\to\mathbb{R}[/math] is continuous.
I know that [math]f[/math] gotta be uniformly continuous since [math]E[/math] is compact. How can I prove that [math]G_f = \{(x,f(x)):x\in E\}[/math] is Jordan-mensurable with measure 0?

how the hay do I get intuition for topological groups, I did some of the munkres exercises but it still feels like "mostly just a topology but with some extra automorphisms and regularity". at least most of the nice properties carry over to the quotient spaces

>>16750736
It's enough to show the outer measure is 0.
Fix some [math]\varepsilon[/math] and take the [math]\delta[/math] you get from uniform continuity to subdivide [math]E[/math] into compact boxes [math]B_1,...,B_k[/math] of diameter [math]\leq \delta[/math] and some leftover set [math]R=E\setminus (B_1\cup...\cup B_k)[/math] of volume at most [math]\varepsilon[/math].
On each [math]B_j[/math] [math]f[/math] will attain its minimum [math]m[/math] and maximum [math]M[/math], and [math]M-m \leq \varepsilon[/math] by uniform continuity.
Then [math]G_f \cap (B_j \times \mathbb R) \subseteq B_j \times [m,M][/math] which has volume at most [math]\varepsilon \text{vol}(B_j)[/math].
Then the total content on the boxes is at most [math]\varepsilon \text{vol}(E)[/math] while the graph on [math]R[/math] is contained in [math]R \times [-L,L][/math] with [math]L= \sup_{x\in E} |f(x)|[/math] (vol [math]2\varepsilon L[/math]).
Then you let [math]\varepsilon \to 0[/math].

Also:
>Jordan measure
what century is it?

>just use a bold letter for your variable
no. why are they like this? what are my alternatives?

>>16750980
What is the context?

>>16751001
vectors, matrices, and tensors

>>16751007
Ah. The alternative is to not write them in bold. Never needed such conventions to tell what is what in an expression such as [math]Ax + b[/math].

>>16751020
it gets confusing when you're dealing with multiple of them

What is a sheaf

I've had 10 different extensive videos on this playing in the background.
It appears to be some kind of link between a topological space and geometry that you can embed in it.

It's called a "sheaf" because it's infinitesimally small on one end and on the other end it splays out. Yeah?

>>16750132
https://en.wikipedia.org/wiki/Large_cardinal
https://en.wikipedia.org/wiki/List_of_large_cardinal_properties
https://en.wikipedia.org/wiki/Huge_cardinal

>>16750636
>philosophically incoherent theory of sets
care to explain what you mean by that

>>16751178
i suppose, like a bundle of wheat

>>16750132
Your introduction to "infinity" is that "measure" doesn't make any sense as you know it.

Get used to thinking more in terms of functors where something can either map to something else indefinitely or it terminates (an endofunctor).

"Larger infinities" are where mathematicians start fucking around more and more with the structure of how the idea of a quantity makes any sense. The current logical model says that you can embed a function in itself as long as it's self-similar (or something) so you can now take infinity and embeded in a function infinite times. Or something, I'm just making something up.

>>16751222
Is there any way to describe this that isn't extremely abstract or extremely dense with 10 different symbols.

I have a decent background in category theory and topology and differential geometry and what this is supposed to be is still up in the air.
It might help to explain what it's used for.

>>16751178
It is what it does

>>16751296
I didn't know it was a verb

>>16751306
A sheaf is a functor so there is a natural verb interpretation.

>>16751324
>In mathematics, a sheaf (pl.: sheaves) is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them.
Okay gotcha. A sheaf is anything that transforms anything into anything else off of the most abstract notion of math in existence.

Sheaf farmers live simple lives, but they enjoy reaping the benefit of harvesting arbitrary sets of non-degenerate abelian rings on the surface of a unit disc.

Thank you for your service.

>>16750537
I don't know much about photonics beyond the tiny bit of optical communication coverage in my digital comms course in undergrad. There are a lot of grad students at my alma mater that study the photonics side of EE, and they seem to be well funded.

>>16751178
jeez are you that guy who posted that question in another thread recently?

did you catch my answer? here's what you need to do: first understand that a presheaf is nothing a contraviariant functor to set. then restrict to the case where the source category is the category of open sets of a topological space (which is just the partially ordered set of open subsets of that space, viewed as a category). this immediately gives you the family of prototypical examples of presheaves of which sheaves are just an abstraction, namely: presheaves of maps with a certain property that is stable under restriction. if you have a topological space X and a given target space Z for every property P of maps X –› Z such as "bounded", "constant", "locally constant", "continuous", "smooth" or whatever (depending on how much structure X and Z have) the functor F_P : V(X) –› Set (where V(X) is the partially ordered set of open subsets of X, viewed as a category) given F_P(U) = {s : U –› Z; s has property P} and sending inclusions U' –› U'' to restriction of maps F_P(U') ‹– F_P(U''). the elements of F_P(U) are called "sections".

keep these things in mind. these specific instances of presheaves are then called sheaves if you can, for any collection C of open subsets of X, glue a compatible system of sections on the corresponding open subsets (meaning a system of such maps s : U –› Z for the open subsets U of that collection C which agree with each other when restricting on common subsets) together (in a unique fashion) to a section on the union of all these open subsets of the collection C.

>cont. ..

>>16751406

so imagine a bunch of maps of open subsets which are all compatible with each other. clearly you can glue them together as maps. but: does this glueing preserve a given property? that's not the case for say bounded maps. take the reals and on the collection of open subsets (-n .. n) the inclusions (-n .. n) –› R, which are all bounded. these are compatible and you can be glued together to the identity map on R itself, but that's not a bounded map! so the presheaf of bounded maps on R is not a sheaf. but that preservation of properties does work for the properties "continuous", "smooth", "uniform".

ok, so in general a sheaf on a topological space is just an abstraction of these specific sheaves that naturally arises when you try to capture that glueing condition in a categorical way.

once you get that, familiarize yourself with the construction of the étalé space of a presheaf. for this you need to first understand the very important und fundamental concept of a stalk of a presheaf. in any case, the étalé space gives you a way of constructing a topological space Z from the information of a given presheaf F on X such that the presheaf F has a natural map of sheaves into the sheaf of continuous maps X –› Z such that this map is an isomorphism of sheaves when F was a sheaf to begin with. this is the sheafification construction, which associates to any presheaf a sheaf in a natural way. and what also follows is that. you can view any abstract sheaf on a topological space X as the sheaf of continuous maps from X to the étalé space of F.

that shows you that the sheaf of continuous maps is the prototypical example of a sheaf and then sheaves are just abstractions of the idea of "continuous maps on X to a given space Z", which are useful because they provide a reification of properties (like continuity, smoothness etc) as a mathematical object. (this is also why they can be used for alternative definitions of, say, smooth manifolds.)

>>16751406
>>16751412
btw sorry for my retarded english, it's not my first language and i'm pretty tired atm and whenever i'm on 4chan i don't give a fuck about my writing to begin with, and so when things get a bit more involved, that quickly leads to serious fuckups. just let an ai correct my retardation whenever i fucked up the grammar and write incomprehensibly.

>>16745109 (OP)
at what point do i start reading papers? i've just been hitting the books hard for >2 years now and i kind of want to start attacking real problems as discussed in le literature

>>16751414
It puts letters in your mind using a white frame and subliminals anon

Is invasive af

>>16749613
0 and null are not the same. 0 is still a value, null means there is no value. Remember x=x+0 is true while x=x+null is segmentation fault.

>>16751417
i just asked gpt to do that myself and i think it did a decent job.

>>16751417
>>16751421
on second thoughts, it butchered some ideas / descriptions, like that "functions defined locally on open sets, satisfying compatibility" crap addition, what the fuck is that supposed to mean? so ok, one needs to take great care with these fuckers.

>>16751424
You defined gluing parameters and for it they need to be compatible, I think it was just checking your math

>>16751178
a category fibred in sets satisfying descent

>>16750184
>>16748789
Can you not go into finance after a signal processing PhD?

I asked G**gle G*mini for a roadmap to learn Linear Algebra, Projective Geometry, Differential Geometry, Chaos Theory and Fractals. Is this legit?

>>16751782
Didn't attach the pdf for some reason. Had to convert it to an image

>>16751784
>some reason
anon the shoah..
>is this legit
sure just start and worry about the rest of the roadmap later

>>16751782
>>16751784
not my area of expertise, but yeah, seems legit, excep that it doesn't mention real analysis as prerequisite, which it definitely is for differential geometry, differential equations and dynamic systems. furthermore i would also definitely throw in a bit of complex analysis because it is illuminating in many ways and also mandelbrot sets and julia sets are naturally fractals in the complex plane.

be aware that if you have no previous background in mathematics, this is a 1--3 year endeavour to study all this, depending on your intellect and how much time you devote to it.

>>16751802
>>16751833
I've gone through a bit of Book of Proof and done some Calc 3, Linear Algebra and DiffQs from an engineering textbook. Any books you guys will recommend?

>>16751854
I want to apply Maths to the visual arts, M. C. Escher style, hence the topics.

>>16751857
search: penrose triangle cohomology pdf

>>16751854
sorry, no. the undergrad books i've studied from are in german and tend to be in the style of pure mathematics, not applied mathematics. in any case, for analysis that would be amann / escher and forster, for complex analysis freitag, for linear algebra bosch. but i've studied many topics just from lecture notes and reading wikipedia. not sure if this is any help to you.

>>16751870
Cool stuff, I'll check it out later
>>16751878
I have Amann/Escher but I don't know if I'm the right person for it desu

>>16751910
just open the book and see if it fits you. but you should definitely not read only amann / escher on analysis, they are too autistic about it. grab some other book on analysis as well, preferably one of the classics in your native language, defaulting to english if you have none.

>>16751683
You can, but usually you don't unless you really really want out of typical career paths. Also, even though there's a lot of similarities in the mathematical background and topics (linear systems, Fourier analysis of Gaussian/Markov processes etc.) there's a few major differences in focus (in my limited exposure to finance/econometrics).

Signal processing engineers are generally used to the underlying structure for their systems coming from physics derived models. While these models can be complicated and non-linear, they tend to be consistent and reliable. Usually your focus is determining how to solve or approximate a particular SDE problem, where the central tendencies are given by theoretical physics. The complications mostly come with dealing with non-linearities and model mismatches rather than actually finding the underlying structure which works for your problem.

In contrast, the models used in finance have no such reliable structure. If you're lucky, you can apply something straightforward like a Black-Scholes model with white Gaussian increments. Usually you're stuck basically just eyeballing it and trying to fit some mixture of well-behaved SDE models. It's not a great feeling to get used to if you were trained on constant velocity models coming from basic physical limitations on rigid object motion (as an example).

is fuzzy logic worth learning about?

>>16752092
Possibility theory is kind of neat as a modeling discipline, and that uses fuzzy sets.

>>16752052
I meant that I read a bit of it and while I like the autism, I don't think it coincides with mine.

Is it just me?

>I am a useless adhd/add retard
>can barely function
>decide to start math degree
>realize I will not make it unless...
>start taking (low dose 10-25mg) crystal M

>>16752071
sounds grim desu. I just wanted a mostly-theoretical job without being a federal leech glowie.

Have you ever worshipped math in an esoteric greek way?

>>16747579
I'm kind of retarded, but I'm doing my best. This is about as far as I can safely argue with people. Beyond this and I know to keep my mouth shut otherwise I end up wearing colored socks taking strange dicks at parties I can't remember. Don't want that happening again. Not after last time.

>>16752582
Looking at your line, I realized there's kind of a dumb placement near where you drew it. They put stochastic calculus as being closer to the surface than measure theory.

You need to know measure theory to do pretty much any kind of stochastic calculus. Without measure, you don't have martingales or filtrations, which are really important to stochastic calculus and SDEs. The only stochastic calculus you can really do without measure theory are when your updates are white Gaussian increments and your process equation and measurement equations are both linear.

>>16752735
It's really not necessary to understand measure theory if you want to apply stochastic calculus.
The interpretation of a filtration as an "information set" can get you very far.
See, for example, the (somewhat) popular book by Bjork.

>>16745109 (OP)
Infinity has no precision at endless length
Nothingness has endless precision at no length

Both have an element of endlessly increasing quantity and compete absense of quantity. Because all number has precision and magnitude like a particle has position and momentum.

MMP defines the group through the oroboros operator. [Math]0 /circlearrowleft /infty = /{/} which is the new elemental notion of the empty set.

After all the more things change, the more things stay the same

Through this construction the distinction between geometry and algebra becomes clearly defined. As one inspects a quantity through the relationship of a numbers precision with its magnitude. And opens a door to allow degrees of irrationality

>>16752789
You can learn some amount of stochastic calculus without measure. My first stochastic processes course used Papoulis, which specifically doesn't use measure. That approach is quite flexible and is basically good enough to understand the vast majority of state estimation, stochastic control and SDEs work from WW2 until about 1962.

Unfortunately, for most anything interesting or non-linear, you really need measure to understand what's actually happening. If you're okay with handwavy linearization arguments you can somewhat ignore this, but you'll be relying on convergence arguments that can be pretty problematic and opaque when they don't work.

I am a mathlet. I say this with no pride at all. just the opposite, but I have always been bad at math.
I want to improve this.
What is "normie tier" math level?
where should I go over normie maths? knowing that I can't go too deep because I'm retarded.

>>16753071
Calculus

Don't try geometry requires visualization don't try economy requires having money

Arithmetic and calculus is for retards

>>16751784
I think the PDF upload feature is still disabled. Not sure when it'll be back.

>>16753071
is that really how to jump a car?

>>16753071
Learn single variable calculus and linear algebra.

That's all the math you will ever need to know. The rest is just schizo bullshit.

>>16753166
yes, what's weird about it?

>>16753071
It depends on what you mean by normie. If you mean, not a STEM major or college graduate, even a few months of single variable calculus would put you way ahead of that group.

If you mean for a typical STEM student/graduate, it's not that much higher. A standard calc sequence (differentiation, integration, multi-variable and ODEs), and a semester of linear algebra would put you at the same level of math education as like 90% of people with a STEM bachelor's. Even some lower-tier math BA programs in the US (especially those geared towards making primary school math teachers) don't go much beyond this.

Anyone here studied/studying Euclid's Elements?

Thoughts on proposition 4 of book 1?

>>16753350
what kind of thoughts do you want me to give on side-angle-side congruence?

>>16753373
Have you studied Elements? I think proposition 4 is stupid, but maybe I'm missing the point.

>>16751417
>Is invasive af
it is, i taught myself really bad category theory and quantum physics with LLMs and probably gave myself at least a minor psychosis, very interesting experience

>>16753378
Disregard the voices discuss your mother

>>16753379
that seems like it is supposed to make me dox, and since i cannot reasonably assume that my computer isn't compromised i don't need to dox myself because i'm already under surveillance
thus i don't need to discuss my mother, thank you

>>16753172
lmaoooo
>shizio bullshit
thats fucking hilarious anon can you redpill me on PDEs i wanna learn but my wm is fucked so creating intuition that ill remember is hard

>>16753377
>make two identical triangles
>put one on top of the other
>see? they're the same
Was Euclid tired that day or what?

>>16753076
>>16753172
>>16753265

okay, it seems to bee an agreement that I should go with calculus and linear algebra. so I'll go with that, it is enough work for a while.

Consider orbits, etc. The structure of velocity and acceleration at right angles exists until the limit, therefore 'pi' = 4, even if the universe is continuous.
Credit for this idea goes to Miles Mathis.

>>16745109 (OP)
anyone have any good math history book recommendations? i had a prof for analytic geometry who had this vividly detailed memory of the history of various fields and shit and i thought that was really cool

>>16753406
Nothing worth anything is ever free.

>>16753916
I don't have a book recommendation, but I can recommend this article https://www.sciencedirect.com/science/article/pii/S0001870897917138 if PDEs interest you.

>>16753920
>Nothing worth anything is ever free.
What about solar energy?
Every plant begs to differ.
"The best things in life are free."

Someone posted this image in my thread about Euclid, very cool. Does anyone know more about this?

>>>/pol/513091928

Gosh wikipedia jannies are so fucking stupid
>The proof they put out is handwave-y
Some underpaid grad student 10+ yrs ago replaces it with an induction proof
>Janny replies, Reeee can you keep the old proof and just correct the part that's wrong
"Your proof relies on well-ordering. Here's a paper that talks about why well-ordering is handwavey"
>Janny still pretends his proof is the better(TM) one
Scrolls down
>Given a divisor [eqn]d[/eqn], remainder is in interval [eqn][0, |d|)[/eqn] of length [eqn]|d|[/eqn]. Any interval of the same size can be used.
WTF is he saying
Noted his claim only works when [eqn]d>0[/eqn]. Noted it's not an if-and-only-if relation.
>Janny reverted it without explanation
I hate Wikipedia. Everyone on that site wants to be a king. That's how it turned into a poophole of arrogant people.

>>16754433
i weep for the shit that one worthless autist did to the polytope

Has anyone switched to Computer Science after getting blackpilled by constructivism?

>>16751178
It might be helpful to think of why it's called a sheaf to begin with. A nice example is your favorite 2D manifold--take S^2 or the plane. We want to take open sets--open discs in either case--and find a way to associate some algebraic object that corresponds to that disc. The primordial example is the set of rational functions that are defined everywhere inside that disc (these are basically the "regular functions" on that open set, which become the bread and butter we use at this level of generality.) Obviously, the denominator of the rational function can't be zero anywhere inside that disc. As a result, the bigger your disc, the less rational functions we are able to say are defined everywhere on that disc, and vice versa. This means that as we shrink our open disc to a smaller and smaller radius, we get more and more rational functions. Try and visualize a connection between discs, where we have the disc in the base space, and the corresponding "disc" of rational functions. The bigger we make the base disc, the less rationals correspond, and the smaller we make the disc, the more we get. This is like a sheaf of wheat. The more you move the binding in the middle to one side, the bigger the other side gets. This is the motivation for defining a sheaf in a contravariant way. If a disc is contained within another, then its corresponding image under the association should contain the image under the bigger disc. We're reversing directions. We also want things to "vary smoothly" in a way reminiscent of vector bundles in diff top in our base examples. If you read through the definition of a sheaf, you'll find that the OGs behind the construction were careful to capture that idea. The nice thing is, we also get a bunch of more non-intuitive examples with it. Indeed, the idea of "vector bundles on manifolds" is largely analogous to "coherent" sheaves on a variety. A sheaf is a system that captures manifolds and vector bundles and extends it.

Why the last circle?

>>16754002
There's a million and one proofs-without-words of the Pythagorean theorem and this isn't one of them, it only works when the triangle is isosceles.

>>16749481
Induction is the traditional way to deal with this.
For all n, P(n) implies P(n+1).
P(1) is true.
Then, by induction on n, P(n) is true for all (natural) n.

Whats the best edition of Euclid's Elements?

I think that I have an approach at solving a very niche and unimportant unsolved math problem. But I only have bachelor level math knowledge and never wrote a paper. How do I proceed with this? I for sure cannot publish anything, I can't even verify if what I have is none-sense or not, but I can't seem to find any holes in my solution. Do I send an email to some random math professor and ask them to check if it's legit?

I'm planning for a self-study real analysis. LLMs gave me:
>Apostol's calculus
>Spivak's calculus
>Baby Rudin
>Abbott's understanding analysis
Do I really need all of them? I guess the learning path is more like
>Apostol vs Spivak
>Baby Rudin vs Abbott
My background is pre-calculus. Right now I'm working through How to Prove it.

>>16755506
post it here and let us tear into it
worked for the Haruhi problem guy

>>16755624
I fear the very slim chance it turns out to be correct and then everyone would know it was posted on 4chan first. I think I'll join some math mailing list with my full name and post it there

>>16755277
>Euclid_Elements_Book_1_Proposition_2.gif
That's a very nice GIF. It makes me very jealous.
>Why the last circle?
I don't know, because I'm not familiar with that Proposition.

>>16755277
The end goal is probably to draw that last red circle at center point C with radius |AB|.

>>16753916
bump
>>16753935
thanks nonny, looks interesting.

>>16755561
Baby Rudin and Abbott are very different books. Abbott basically stops at chapter 6 in PMA, and covers the topics in the first 3 chapters of PMA in much higher detail but with much less general results.

Abbott is a great book for a second or third year analysis undergraduate course for people fresh out of a calculus track, who aren't sure if they want to do graduate school. PMA is much more of a "transition to graduate mathematics" kind of book, which really is at home as either a upper undergraduate elective or a first semester graduate course.

>>16753916
I'm quite fond of this paper: https://arxiv.org/abs/1006.4131
Also, Needham in his book "Visual Complex Analysis" often makes short remarks about the origin of certain concepts, and regularly refers to a book by Stillwell (Mathematics and Its History) for more detail.
I haven't read it yet, but it seems to be regarded positively.

What chapters should I study in Stewart’s Pre-Calculus to get a refresher before Calc?

>>16753916
Again, not a book, but this article https://www.sciencedirect.com/science/article/pii/0315086084900363 might be nice if functional analysis interests you.
Also interesting, at least to me, and related to FA are these links on Banach and the so called Scottish problem book they had laying in the cafe that Banach and his peers would frequent.
http://kielich.amu.edu.pl/Stefan_Banach/e-index.html
http://www.math.lviv.ua/szkocka/

what math to take for an econ major looking to do grad school?

>>16755888
So, Abbott and then PMA?
Or what would you recommend for a first real analysis text?

>>16756228
It really depends on what you're trying to do with it, and how comfortable you are with having a book kick your ass a bit. I really like PMA, but Abbott and Ross's Elementary Analysis are both a great backup if you find you need something a bit more handholdy. Maybe try giving PMA a go, and if you don't think you're understanding enough to get it, go to either Abbott or Ross? It's pretty normal to get frustrated with real analysis and have it kick your ass a bit before you get it.

I am studying and researching but how do I formulate proper alternate physics?

>>16756244
Thank you anon.

>>16755282
>it only works when the triangle is isosceles.
You're right, I didn't realize that but now I do. It looked cool but it wasn't. The other so-called intuitive proofs are so convoluted I don't see how they're better than Euclid's proof. It's funny when people criticize authors who aren't alive to respond.

>>16755693
no and none of the videos and other materials online about this proposition have the last circle, only this one gif on wikipedia

https://youtu.be/SMRcqlYCjW8

>To cut off from the greater of two given unequal straight lines a straight line equal to the less.

"The less line", that sounds weird to me, isn't the opposite of "the greater line", "the lesser line"?

I'm doing my masters in applied math. I'm stuck between going for a good GPA versus taking classes I actually want to take. I don't know what the landscape looks like for my PhD or if grades even really matter.

But I have avoided analysis a lot and it's starting to become a regret. I've only taken a very elementary level analysis course, didn't even use Rudin, and I'm now realizing I'd really love to know more analysis.

I want to take measure theory and functional analysis, but I'm scared of getting a poor grade in them since my exposure is so weak. I can get away with not taking any more hardcore rigorous theory courses but I'm thinking I'll regret it.

>>16757683
I took topology before analysis
Topology was great

>>16756228
>So, Abbott and then PMA?
This is what I followed, and I highly recommend it. Also, Folland's Analysis is a great followup to PMA, and a much better alternative to Papa Rudin. I abandoned non-constructive math before I could reach Grandpa Rudin, but it seems a good followup to Folland.

>>16751415
bmpu

Man I really fucked up not taking this maths shit seriously in my life, just memorized my way to the second year of this engineering degree Im pursuing and Im barely scraping by now.

How do I actually get out of this situation I placed myself in?

Let [math]X[/math] be a vector space with a norm. Prove that if [math](B_n)_n[/math] is a sequence of balls in [math]X[/math] s.t. [math]B_{n}\supset B_{n+1}[/math] for each [math]n[/math], then the ball's centers form a Cauchy sequence.

>>16758272
Bruh, I did my bsc in math drunk and hung over. did good here and there. managed to get a research gig somehow. i was a mixed bag. graduated and drank for 10 years straight now I'm back in my msc getting straight As. you can ALWAYS turn things around, you just start by doing as good as possible in the classes you're in currently and review ONLY WHERE NEEDED. do not attempt to go backwards by wasting time reading an old textbook.

and about memorization.... a lot of math is just memorization bud. not sure what to tell you. i get As on exams because i simply rote memorized all the material. i have never had trouble in research or solving problems.

Man I love math. Just finished Conway's "A course in functional analysis", functional analysis is just fucking great.

>>16757683
You can also do these courses on your own, take the prescribed material and do it yourself. You only "need" to follow them if you want """proof""" of your mastery of the subject for universities i.e. are interested in research positions related to the topics. You can even do them after you finished your masters. To note, an introductory course in measure theory can be followed just with set theory and a little bit of topology. Functional analysis will be quite tough with little analysis and a graduate course impossible.

>>16758396
Let [math]x_n[/math] be the center of [math]B_n[/math] and [math]r_n[/math] be the radius of [math]B_n[/math].
The sequence of radii of the balls converges since it's montonously decreasing and bounded below by 0 and
[eqn]\|x_n - x_m \| \leq |r_n - r_m |[/eqn]

>>16758490
>[math]\|x_n - x_m \| \leq |r_n - r_m |[/math]
I don't get it

>>16758493
nta, but draw a picture of one ball inside another.

>>16758224
Nice. Thank you.

the only way to agi is RL activation functions bro

>>16758830
we know samefag lmao

>>16758831
awww shieet discovery by the alleged 100 FSIQ anti memer

>>16758832
"mother fuck" "hazaaaa"
-erlich bochman

>>16758396
Nice topology question
I don't remember any of it

>>16745109 (OP)
been reading this and just finished chapter 5
this shit has been brutal, but it's been enlightening

>>16758427
Aight thanks, ill just focus on solving more problems

Recommend me good functional analysis textbooks

If the powerset P(X) of a set X has a larger cardinality, and you can easily find an injection into the interval [0,1] from the powerset of the naturals (order the subset e.g. {1, 3, 10, 15,...} and map it to 0.131015...), why would you need a diagonalization argument to prove that the cardinality of the reals is larger than that of the naturals?

>>16759017
If you have lots of mathematical maturity then the book by Brezis or Granpa Rudin are considered good.

>>16759028
>(order the subset e.g. {1, 3, 10, 15,...} and map it to 0.131015...)
If you do it like this then it's not an injection as for example both {1, 2} and {12} will get mapped to 0.12.
Also showing that |X| < |P(X)| usually involves a diagonal argument.

Michael Crowe's A History of Vector Analysis

>>16759632
whoops, i meant to reply to >>16751415

I've became a midwit, I will never live up to my own expectations. I will pursue teaching as a career and become less than mediocre in math while pretending otherwise.

>>16760061
I feel you brother. Even as I am working towards finishing my dissertation, I spend more days than anyone should feeling like a total retard who doesn't know his ass from a hole in the ground.

Y'all not gonna talk about how division by zero is possible

https://en.wikipedia.org/wiki/Cayley%E2%80%93Dickson_construction
>each step up on the Cayley Dickson constructions loses some fundamental property of number theory
>complex numbers have no order
>quaternions stop commuting
>octonions neither commute nor associate, no one really knows what the fuck they are
>sedonions and onwards are no longer alternative (whatever that means, anyone?) and also support zero divisors

Infinity and zero are not opposite, and a number divided by zero is not infinity, but they go hand in hand.
A slope with infinite curvature is an orthogonal straight line.
A rotating sphere revolves around an infinitely thin pole with zero radius.
A black hole reaches a critical mass that spacetime can't support and then just has infinite density with an event horizon with a 0% chance of escape.

What the fuck is going on when you can divide by zero and have infinity show up in this finite dimensional set of numbers.

>>16754685
Okay that definitely got me somewhere, thanks mate.

There's a good chance that I'm overthinking it and that if it just sounds abstract the it probably is. It involves mapping functions from something to something else and it sits between a bunch of very nondescript notions like space, topology and functors.

>>16760209
Normies don't get infinity
And I blame it on college instructors who treated their students like little babies
Universities should teach GenEd students proof writing in the first year
Limits are not a difficult concept to grasp
https://www.youtube.com/watch?v=1SguKALJji8

I'm now back at the point where I'm trying to figure out what the fuck a differential equation even is. I've been going through this SDEs on Riemannian Manifolds book (because a relevant research application problem I'm looking at basically comes down to a non-linear SDE system on a spherical manifold). Every time I feel like I understand differential equations, I find out there's another layer of hell below the surface.

>>16760879
How is infinity supposed to be specific to the concept of a limit.
It's not. A sphere spinning around an axis doesn't gradually get slower for all eternity.

Also, does the concept of a "limit" not inherently multiple dimensions because it requires a ratio?

Silence plebeian. This is beyond your thinking paygrade.

>>16757126
This page says "lesser", weird.
https://elements.ratherthanpaper.com/1.3

>>16760209
Are you asking why 'division by 0' can happen in sedonions? Or do you mean like in the real numbers in all the examples listed. The answer is going to be vastly different.

>>16760945
>Also, does the concept of a "limit" not inherently multiple dimensions because it requires a ratio?
>This is beyond your thinking paygrad
holy mother of undergrads

Just noticed that different materials/sources say different things about the propositions and their proofs in Euclid's Elements. I'm going through and comparing all the videos and materials I can find on proposition I.4.

https://archive.org/details/euclid_heath_2nd_ed/1_euclid_heath_2nd_ed/page/n260/mode/1up?view=theater

http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI4.html

https://farside.ph.utexas.edu/Books/Euclid/Elements.pdf

https://elements.ratherthanpaper.com/1.4

https://youtu.be/GP6K-2nvZ-Q

https://youtu.be/ejfAWgxydUo

I have done propositions I.1-I.6. I was impressed with 1.1-1.3 and thought they were fun, I was unimpressed with 1.4-I.6 and they were so boring. Hopefully I'll find something interesting and fun about them when I compare all the different sources. Anyway the first three propositions are apparently of a sort called "problems" while the subsequent three propositions are of the other of the two types of propositions in the book, namely "theorems". Problems are so much more fun than theorems. The problems are propositions 1-3, 9-12, 22, 23, 31, 42 and 44-46. The theorems are the rest of the propositions.

https://libgen.li/ads.php?md5=1ed3fb67cac34480609d924f9dc37c7e

>>16761628
These Greek words sound very kawaii desu :3

How come nobody in this thread is interested in Euclid's Elements?

Should mathematics be founded on physical reality rather than abstracts?

>>16762453
No. What you're describing is an entirely separate field that already exists, called physics.

>>16762453
Probably not. Firstly, physics already exists. Secondly, our understanding of physical reality often requires the application of "abstractions" to simplify a complicated but real phenomenon. If you restricted your understanding of mathematics to only things you could directly touch/count, you'd throw out the vast majority of engineering and technical work of the last 100 years, including all of the communication networks you use daily (which rely on quite abstract non-physical representations of information to function).

>>16762453
>physical reality

>In several dialogues, most notably the Republic, Socrates inverts the common man's intuition about what is knowable and what is real. While most people take the objects of their senses to be real if anything is, Socrates is contemptuous of people who think that something has to be graspable in the hands to be real. In the Theaetetus, he says such people are "eu a-mousoi", an expression that means literally, "happily without the muses" (Theaetetus 156a). In other words, such people live without the divine inspiration that gives him, and people like him, access to higher insights about reality. Socrates's idea that reality is unavailable to those who use their senses is what puts him at odds with the common man, and with common sense. Socrates says that he who sees with his eyes is blind.

>>16762453
Physicists exist. Applied math exists. Quant traders exist.
Mathematicians are happy with theory :) IMO

im 27, my knowledge doesn't go beyond basic high school stuff like functions, limits, simple integrals, and i still struggle sometimes with some algebraic properties because i don't really remember them.

for as long as i can remember, from around when i was 7 or 8, i always felt an obligation to like and be good at math, but i never did, i get the fascination, but i just don't experience it much myself, i have other interests.

now this feeling of obligation became stronger in the last years, it has become my main thought, i feel extreme guilt and dread, i am not good at math, i think i have an average iq, nothing exceptional, i am also quite lazy, i feel pressured to enroll to my local university (free in my country), but i tried studying for 1-2 hours a day the past week and it seems impossible to me, everything is so dry and difficult, and its only basic pre-required material.

why am i like this? am i just traumatized by something? i think its very clear i will never succeed in math, as i said i don't even -want- to, i just feel this incredible guilt for not being into it, does it make sense for someone?

its clear by the way i type i am not very bright, i struggle even with basic leetcode puzzles, why the hell do i have this thing for math?

>>16762725
I dare say its becasue while your grandfather was sodomizing you he made you repeat the times table.

Let X,Y be same-dimensional compact topological manifolds both with nonempty boundary. Assume X is compact and Y is connected.
Let [math] f:X\rightarrow Y[/math] be a continuous map such that f maps [math] \partial X [/math] homeomorphically onto [math] \partial Y [/math].
Then is f necessarily surjective onto Y?

>>16763084
Sorry, typo in the first sentence. X is compact, Y need not be.

>>16762725
The good thing about math is you don't have to be enrolled in school to learn it per se. In fact, I would even argue against it until you get a bit more foundation down. If you have an interest in mathematics, you might find a certain joy in solving problems at the more basic level. Revisit those old "basic" high school problems. If you know limits and simple integrals, you're off to a great start. Maybe try harder problems. Do you know L'hospitals rule? The method of substitution? Try some of these out online as a fun challenge. Look up calc curriculums and see how much you can do, and for what you can't--slowly try to increase your knowledge. If you feel a pressure to know math, it might hinder you. You will always be battling the feeling of not being good enough. You seem to have said this. Perhaps then, you can try battling this thought, and just see math as something enjoyable. No one is forcing you to do it, but you also have some capability to dispatch it--namely by getting good. You don't have to enroll into a college to do this, but maybe it'll help. Idk anon, but the answer to your problem might just be that you don't know where to start and are dealing with navigating the ocean of doubt in front of you.

>t. started my PhD in math at 25 after a power gap of doing nothing with life and slowly re-studying again

>Suppose [math]U[/math] is a neighborhood of 0. Since scalar multiplication is continuous, [math]\exists\delta>0[/math] and a neighborhood [math]V[/math] of 0 s.t. [math]\alpha V\subset U[/math] whenever [math]|\alpha| <\delta[/math]
Why?

>>16763215
Pick [math] \epsilon>0 [/math] small enough that [math] B_\epsilon(0) [/math], the [math] \epsilon[/math]-ball centered at 0, is contained in U. Then take [math] \delta = 1 [/math] and [math] V = B_\epsilon(0) [/math].

>>16763215
It follows quite directly from the definition of continuity of scalar multiplication, and the fact that opens of the reals contain open intervals. Maybe write down what kind of domain scalar multiplication has as a function?

>>16763224
Not every TVS is a metric space.

>>16763226
>opens of the reals
I'm talking about a topological vector space

>>16763231
I tried making the idea clear to you but alas. Almost as if open intervals trivially generalize to balls in [math] \mathbb{C} [/math] or any metric space for that matter, which are the standard choices for a field in a TVS.

>>16763231
>>16763238
You seem to have misunderstood what I said btw, we have [math] *: \mathbb{K} \times X \rightarrow X [/math], which one of these spaces has a metric? The general topological space or the field?

>>16763227
I see, I hadn't thought of that. I'll have to think about it some more.

>>16763275
Note that [math] *(0, 0) = 0 \in X [/math].
Let [math] A \in \mathcal{N}(0) [/math], i.e., a neighborhood of 0 in [math] X [/math]. Since the scalar multiplication map [math] *: \mathbb{K} \times X \rightarrow X [/math] is continuous, the preimage [math] *^{-1}(A) [/math] is open in [math] \mathbb{K} \times X [/math].

By the definition of the product topology, there exist open sets [math] W' \subseteq \mathbb{K} [/math], with [math] 0 \in W' [/math], and [math] V \subseteq X [/math], with [math] 0_X \in V [/math], such that:
[math] W' \times V \subseteq *^{-1}(A) [/math].

Since [math] W' [/math] is an open subset of the metric space [math] \mathbb{K} [/math] (either [math] \mathbb{R} [/math] or [math] \mathbb{C} [/math]), there exists a radius [math] \delta > 0 [/math] such that the open ball [math] B_\delta(0) \subseteq W' [/math], where:
[math] B_\delta(0) = \{ \alpha \in \mathbb{K} \mid |\alpha| < \delta \} [/math].

Therefore,
[math] B_\delta(0) \times V \subseteq *^{-1}(A) \iff *(B_\delta(0) \times V) \subseteq A [/math].
This implies:
[math] \forall \alpha \in \mathbb{K}, ; |\alpha| < \delta, ; \forall y \in V, \quad \alpha y \in A [/math].

>>16763347
Oh right, of course. I understand now, thanks anon

>>16763365
As homework practice to understand TVS properties you should try to prove the following by using what we just proved:

Suppose that [math] U [/math] is an open neighbourhood of [math] 0 \in X [/math]. Prove that there exists an open neighbourhood [math] V [/math] of [math] 0 [/math] such that [math] V = -V [/math] and [math]V + V \subseteq U [/math].

>>16763227
trump vaginal syndrome

>>16763243
Both? I mean, you can induce a metric on [math]X[/math] by making [math]||x-y|| = d(x,y)[/math]

>>16763379
Please refer to this image and read the definition of a TVS again.

>>16763372
I think I got it, here is a proof:

By continuity of addition, we can find open nbhds W,W' of 0 in X such that [math] W + W' \subseteq U [/math]. Let [math] W'' := W \cap W' [/math]. Now [math] \forall x,y \in W'' [/math] we have [math] x+y \in U [/math].

Since negation is continuous (and its own inverse) we know [math] - W'' [/math] is open. Let [math] V := W'' \cap (-W'') [/math]. Then V is an open nbhd of 0 in X, and clearly [math] V = -V [/math].
And [math]\forall x,y \in V [/math] we have [math] x+y \in W'' + W'' \subseteq U [/math].

This thread sucks. Just kids posting their homework.

>>16745109 (OP)
0, 1, and 2 are not numbers.

>>16764034
Feel free to post something good yourself

>>16763215
Topological vector spaces have, by hypothesis, a continuous map [math]m: K \times X \rightarrow X[/math], K a field, X your topological vector space.

Therefore, as [math]m(0,0)=0[/math], it follows by a definition of continuity that for every neighborhood [math]U[/math] of [math]0[/math], there exists a neighborhood [math]W=(-\delta,\delta) \times V[/math] of [math](0,0)[/math] so that [math]m(W) \subset U[/math]. But then for $|\alpha|<\delta$, [math]\alpha V=m(\alpha,V) \subset f(W) \subset U[/math]. This completes the proof.

First time using LaTeX on here, hope it comes out right.

The important point is recognizing continuity here is an axiom of the multiplication map and breaking down what that means.

I want to learn more about topological spaces, I always get distracted and get clingy to this book that is probably not a good idea.

Has nobody in this thread studied Euclid's Elements? Seems like everyone is just talking about their school work, and nobody studies Euclid's Elements in school these days.

>>16764297
We have more general, effective ways to study various things that take forever and take contrived approaches in Euclid. Like for instance the concept of parabolas and quadratics equations as part of Descartes analytic geometry, that is much more readily applicable for the physical sciences.

Like it's still interesting but just not very useful.

>>16764311
You didn't answer my question.

>Has nobody in this thread studied Euclid's Elements?

Have you studied it or not?

>>16764318
No I haven't.

>>16764334
Then what exactly are you basing all the opinions in >>16764311 on? What other people have told you about it?

>>16764340
I haven't *studied* Euclid's elements in detail, but I'm aware of how it works and its methods and some of its main results. It constructs everything from circle and line, not just geometry but we in modern day would call algebra and number theory. For instance Book II is meant to be describing algebra. But modern algebra and hell *numbers* totally trivialize this.

All the geometry are based on really precise connection of circle and line that while cool, is essentially a shadow of the Greek philosophy at the time-no other reason. But descriptive algebra methods are just more useful and there is no real modern use for restricting to this circle and line construction. Parabolas, cubic equations, or sin(x) are all not constructible via circle and line and all lean eventually into the infinitely more general and applicable study of calculus

The only real argument I can see for its use in teaching is that it teaches proofs, but I'm almost certain there is a more modern, less 'philosophically shoehorned' approach for this. For instance, perhaps combinatorics.

It's one of those fields that was seriously studied at the time, but we've since found far more efficient subjects essentially encapsulating it. This happens a reasonable amount: it also happened with the study of elliptic equations and maybe general topological spaces (as much as I love it).

>>16764354
I'm not interested in your second hand opinions on Elements. I asked if somebody had studied it. You haven't. The reason it's not being taught isn't that everything we have now is superior. Elements was taught in schools for over 2k years until the 1950s. It was removed because it teaches logic and they don't want that in public school, which is all about creating unthinking slaves.

>>16764297
>>16764357
I did do Elements in high school in a "talent" program. We did not prove every little lemma or theorem but it was used to teach the ideas of axioms and proofs. The final test consisted of proving 3 geometric theorems and explaining if they hold in non-euclidean (hyperbolic and spherical) geometry. I don't see why you're so fervently obsessed with the book but you be you. There are more condense ways to learn logic and proofs and even geometry.

>>16764357
lmaoooo, sure buddy

>>16764134
I get it now. Thanks!

>>16764361
>obsessed
Retard, what suggests I'm obsessed? I asked if someone had studied it, because I'm studying it now, and nobody has responded to my posts regarding it.

>>16764366
stfu public school teacher

https://www.latimes.com/archives/la-xpm-2006-jan-30-oe-crease30-story.html

>>16764381
Uni student in grad classes, fervent studier of mathematics up till then. I'd say I know a fair bit of math and hence aware of how little relevance Euclid's elements has in it today and why I am not going to buy into 'but the classics' talk'. Out of the major core fields still studied today, it really only has relation to galois theory. I mean it obviously will have relation to systems of geometry (projective, non-euclidean, finite), how important the study of specific constructions are in that though, I don't know.

Proof is cool and if Euclid inspires you to learn it, by all means go for it. I think there are other ways of teaching it though and my worry is it would be forcing the teaching of something that will be completely irrelevant for students in the future. Hell, at least do a modernized version: Hartshorne has a book in geometry, I have one from Perdoe that utilizes vector algebra too as it is an efficient way of doing geometry.

>>16764381
Please calm down, you have been trying to derail discussion for days just to talk about Elements. I would regard that as obsessed yes. I answered your question and explained that yes it is still taught in public schools. This is not some conspiracy, logic and proofs are seen as "difficult" to an average student (even in undergrad) and thus avoided. The math curriculum has a lot of problems, but it is not a big conspiracy to keep you a slave.

Just like that other anon is trying to tell you there are other ways to learn the things you desire. Elements is only "essential" from a constructionist perspective which is something you have probably not yet encountered.

>>16764395
>constructionist
I meant constructivist, my apologies.

>>16764395
I think the major shift was in the attitudes of what a school should be. Schools (including college now) are seen as more compulsory and required for jobs. Math in particular is designed essentially to prep for that in college. Basically schools have less a reputation of 'learning what you need to be considered an educated gentleman to the elite few' and more like 'learn what you need about basic subjects, shit you may need for college, gauge your ability as school performance' for a job.

But I'm just taking a guess and pulling things out of my ass.

On one level I'm a bit sympathetic with the person replying: must suck studying a subject you take personal interest with no one, not even the school system, to share interest in. Basically me with general topology.

>>16764389
Idgaf about you or students. You don't want to read or discuss Elements, then stfu. We have the Prussian education system, which is divided in three parts, the top 0.5% of the population are taught how to think, 5-10% are taught to be highly skilled slaves with no critical thinking skills, the rest are prepared to be manual labor. You are in the middle tier. Elements is for the top tier.

>>16764395
>it is still taught in public schools
no it isn't, stfu

https://www.zhibit.org/diemythographer/die-mythographer-die/occasional-letter-number-one-2006

https://archive.4plebs.org/pol/thread/512644823/

https://archive.4plebs.org/pol/thread/513091928

>>16764408
>Elements is for the top tier.
Elements is not for the top tier, it is basic logic, introductory proof and "intermediate" constructivist geometry. Do not fool yourself. You dismissing that other anon as "middle tier" for not having is incredibly dense. With a little bit of study he could probably understand it faster than you without any real math background.
>no it isn't, stfu
I just told you from literal self hand experience that yes it is. It is simply not a required course, but many schools still allow you to take it as an elective course. Especially, outside of the US. I am even supportive of you studying logic and proofs on your own because they are valuable skills but this fixation on Elements is just foolish.

>>16764408
>complains about being slave to system
>makes a nationalistic defense of your education system

Do you hear yourself? Whatever arbitrary means you need to use to justify your perceived intellectual superiorty...

>>16764417
You know nothing. The seven liberal arts (which include logic and geometry) are today reserved for elite private schools, yeshivas and Freemasonry.

https://archive.4plebs.org/pol/thread/503426619/#503426619

>>16764423
>>makes a nationalistic defense of your education system
wtf are you talking about retard

>>16764408
Also its so tempting to mention more but its kind of rich calling me middle tier.

I will say this: there is a group of people that worship the classics, based on the elitist appearance that its what smart people do. Its posturing and purely matching to an appearance.

But smart people dont need to do that: they can understand the much more varied kinds of intellectual pursuits outside the classics. In a given field, have a broader awareness of its history and various nuances at different points in time.

>>16764430
Will admit what I said here was stupid and I misunderstood, thought you were referring to like your unique education system and it having this categorical division.

>>16764297
I did a decent amount of geometry in school, and I've never directly studied Euclid's elements. We did cover Euclidean and projective geometry in the undergrad elective I took on "geometric methods for science and engineering." Unfortunately, I've never really encountered a reason to study specifically Euclid's elements from the classic source. If I want to study Euclidean geometry, I'll usually just look through a more modern book (O'Niell's elements of differential geometry has a pretty great Euclidean geometry section, as an example).

>>16764432
>rich calling me middle tier.
Get over yourself faggot, all I'm saying is you're not a Rothschild, you didn't go to Eton College, and you're not a Freemason, nobody on this website is the top 0.5% of society. I don't study the classics for an "elitist appearance" but because I want to acquire the tools of thought which the elite have suppressed for everyone but themselves.

>>16764454
>I've never directly studied Euclid's elements

Then maybe it's time to start now.

https://elements.ratherthanpaper.com/1.1

https://www.youtube.com/playlist?list=PL2V76rajvC1I2TrbPMRLcTqhdcbha4sDE

desmos.com/geometry

>>16751419
>You get 0 bitches
>You get null bitches
So, that is the difference?

>>16764357
Elements is not rigorous logic. Euclid assumes things without stating, and also, tries to circularly define primitive notions when they should just be left undefined. This is why Hilbert's geometry has way more axioms, things like "betweenness".

>>16764479
Who the shit cares about Eton College? And man you are seriously delulu.

>>16764636
>>16764648
>t. public school brainwashed faggots

so /pol/ discovered the Elements huh

>>16764953
Well I feel you've just outted yourself as a frustrated insecure highschool grad because who tf cares about Eton College when it comes to the grand scheme of math.

Might have set one up for a better college and education sure but it makes no one an authority on anything in academics.

>>16765005
lol the brainwashing goes deep, fuck off public school teacher

>>16765005
Also, I wasn't talking about preparing you for mathematics study, I said we have the Prussian education system where the top 0.5% of the population are taught how to think while the rest aren't, and I said this group are taught the seven liberal arts, including Elements.

>>16765008
What even kind of insult is that? Come on you can do better.

>>16765031
I'm not here to play your silly games. You can keep studying your braindead shit, being brainwashed and having no critical thinking skills, I don't care.

>>16765037
Nothing to me reeks of uncritical thinking more then shilling for a book because its seen as more intellectual as it is what you think the other elites use. Isn't that sheep behavior? I'm being the critical one: weighing my experience to the book, comparing it to others I know, thinking for myself what's best.

There are like over 50 graduate textbooks on a wide variety of topics in math I've studied from, of my own volition... and the highest math you can think of is Elements. Please tell me more about braindead shit.

>>16765077
You are viewing this from a math perspective while he is clearly not. He is not interested in math at all maybe in logic but importantly he is fixated on being a "free thinker" and believes that Elements will teach him that because the """elite""" also supposedly learn from it (see his /pol/ links). There is no point in discussing this with him. Once he has finished Elements (a book taught to literal slaves) he will either release that it is just a neat geometry book or he will convince himself he has become a master of some sort.

>>16765081
You're right but its easy to get roped into arguments on here. I've seen this situation before with sci.math google groups. I don't really see much value for me to be on here and yet the psychological impulse remains.

Like there was a long lengthy argument I had about Bohmian mechanics, although that was interesting, it was a huge time sink for me.

>>16765077
You don't know what critical thinking is. You think it's something you can freestyle, because you were brainwashed in public school. Critical thinking is the Trivium. We have the Prussian education system which is specifically designed to teach the Trivium to the top 0.5% of the population while suppressing it for the bottom 99.5%. The Prussian education system is in three tiers. The middle tier is for 5-10% of the population. They are not taught the Trivium, ie critical thinking, ie the tools of thought, ie how to think. This is the tier all engineers and doctors are in. It doesn't matter how much mathematics you studied, you are still in the middle tier. Elements was removed from the bottom and middle tiers. Logic and geometry are part of the seven liberal arts, calculus isn't.

>>16765093
Then leave. Go be a corporate cocksucker somewhere else. You study for money and status, I study for truth, we are not the same.

>>16765097
mate that is the most uncritical regurgitated shit I've heard.

>>16765105
again you don't know what critical thinking means

Attention to all /pol/ tourists: the axiomatic-deductive method is the standard approach in all mathematics books of interest to /mg/. You're not impressing anyone here with your Elements shilling.

>>16765110
Nobody here has studied Elements, fuck off. Also college is just a credential factory, people don't go there to actually learn anything, you're not special because you have a degree or are in college.

>>16765106
Then have you asked why the 'seven liberal arts' are what is required for critical thinking? Why only those methods lead to critical thinking, like why geometry? Why any of combinatorics, number theory, real analysis, topology doesn't count? Have you asked yourself that? Could you give a well reasoned argument of why that is how you need to learn how to think?

Again you're just a child, I understand your (misattributed) frustration.

>>16765117
well im guessing you are anyway based off your your focus on highschool and totally lack of comments on university

>>16765117
Because engineers and doctors aren't the elite, they're just highly skilled unthinking slaves. The first three of the seven liberal arts are critical thinking, how to think, the tools of thought; the trivium; grammar, logic and rhetoric. Elements is a book about both logic and geometry. You didn't learn how to think in college, you learned how to push buttons in a more elaborate fashion than the manual laborers at the same factory.

>>16765129
You didn't give a well reasoned critical argument, you just restated your premise. Don't they teach you this in logic?

Tell me oh critical thinker, *why* is geometry required for this? Why is this the *only* way to learn critical thinking?

>>16765132
When you ask for an argument for a proposition you ask for premises which support the proposition in question. You don't even know what a premise is. I posted links, read them >>16764408, inside the second link is another link, https://archive.4plebs.org/pol/thread/512205412/#512207515, research that.

Read
>>16764410
>>16764428

>>16765145
>re not taught the Trivium, ie critical thinking, ie the tools of thought, ie how to think.
lolwut are you saying.

No, you justify your premises and then make a logical argument from the premises. You're assuming 'seven liberal arts is the sole way to learn critical thinking', I asked why geometry is included in this, and you said 'because engineers and doctors aren't elite' which is technically a nonsequitur but go into what the seven liberal arts, how you dont think in college...

ie doing nothing to explain why geometry in the seven liberal arts is required for critical thinking.

>>16765151
Again, I said the Trivium is the critical thinking part of the seven liberal arts, that's my last post to you. I gave you links. Read them. I have better things to do than talking to you.

>>16765155
You sound like a broken recorder.

>>16765163
Not my fault you're a brainlet. Geometry is part of the quadrivium. The seven liberal arts are the trivium and the quadrivium. The tools of thought are the trivium, not geometry. Elements by Euclid is both geometry and logic. Logic is part of the trivium. This is my absolute last post to you.

>>16765167
Absolute absolute last you retarded midwit poser?

Anyone have good textbook recommendation for dynamical systems from a formal/pure math perspective.

/g/ here

They call them "branches" of mathematics, but are the branches on a graph or a tree.

A graph implies that math is a bunch of disparate nodes that are getting connected.
A tree implies that there's a sense of order and that there's one thing connecting all of it.

>>16758061
The topological definition of continuity is more elegant. Perhaps they should try using it before the analytic one.

>>16765243
You don't seem to know what either a tree or a graph is.
A tree is just a graph with no loops.

>>16765114
Can you compute an integral? Are you able to understand multivariable calculus and linear algebra? Did you finish highschool?

>>16766282
I meant "generic graph". I don't know what the name is or if it's just "graph".

Is there a sense of order? or is there more than one disparate node. That people know of.

>>16766463
Why are you trying to impart some precise, technical meaning onto a loose and vague description, when you don't even know the definitions?

>>16766349
You study for credentials, for corporate cocksucking, I study for truth, we are not the same.

>>16766697
You don't have to go to uni to study calculus, linear algebra, discrete math, or differential equations. You can go on MIT OCW or Khan Academy right now. You can also find syllabuses for respective courses, download pdfs of textbooks, do problem sets on your own, and etc. Are you able to understand highschool precalc at least? Are you also familiar with the putnam competition?

>>16766757
You're the one who was talking about high school retard.

>>16764482
What is the particular reason you believe Euclid's elements, out of all the other geometry sources and topics, to be of particular importance? What, in particular, is gained from studying Euclid's Elements specifically, in comparison to any other axiomatic proof-based geometry book or topic?

It's a very strange hangup to have over an important/influential, but fairly archaic book on a very well understood topic. Why not have the same autism over Euler's elements of algebra, or Bayes An Essay Towards Solving a Problem in the Doctrine of Chances? All of these works are important for a math historian, but probably not the best way for a student who actually wants to understand geometry to spend their time.

>>16766816
Elements was taught for over 2k years, then all of a sudden it became outdated in the 1950s. You've been brainwashed.

>>16766799
Yes. Did you finish HS? Did you pass your precalc classes?

>>16766877
>muh khan academy
>muh high school
stfu

>>16766883
Are you a dropout? You can go back and finish. It's not too late. Again, do you understand single variable calculus?

>>16766890
>muh school
Stfu brainwashed faggot. Go to your school, get your credentials and go suck corporate cock when you're done. I study for truth, you study for corporate cock, we are not the same.

>>16766932
Thoughts on set theory and harmonic analysis?

>>16766981
Thoughts on Euclid's Elements is what I was asking for. You haven't read it so stfu.

>>16766993
I wasn't that anon you asked but the impact of the axiomatic method on mathematical modeling is pretty cool. Are you able to understand set theory or harmonic analysis? Did you even finish HS?

>>16764354
what replaces elliptic equations and topology?

>>16766862
Yes, I'm not sure if you are aware of this, but the field of mathematics has developed a lot in the last 100 years. It's not a "brain washing" issue. It was also pretty common for basic algebra to be a topic which required collegiate education prior to the modern standardization of math curricula. If you're at the point where you're still needing to cover methods for factoring when you've reached college now, you're in remedial classes.

>>16767080
Elements wasn't abolished because math evolved beyond it, it was abolished because there's logic in it, to make the masses unable to think, to make society more efficient, in response to pressure from the Soviet Union. It's the same reason they started the Prussian education system in the 1800s, Prussia lost against Napoleon, Prussian soldiers were learning the Trivium, they were learning how to think, so they needed to put them in indoctrination camps known as Volksschule and Realschule, public school, to create more efficient soldiers and workers, and hence a more efficient society. You spent 13,000 hours in public school. That brainwashing isn't going to come off overnight.

Where did the /pol/ schizo come from? lmao
Does anyone have recommendations for complex analysis?

>>16767062
Apologies, I meant elliptic functions I think. Or something like that.

By general topology, I mean the study of topological spaces through purely point set means and various topological spaces that act as counterexamples. Dimension theory was one of the key first subjects of topology, continuum theory too but these are practically not studied. Almost always study some nicer variation of a topological space whether it be manifold, simplical/CW complex, etc.

Most the questions of point set topologies are highly set theoretic in nature, with most suspecting many are independent of ZFC. Normal moore space conjecture is one.

Lattice theory is another example.

I think what happens is you come up with this new class of objects that may be useful but doesn't well provide much information to classify shit on their own.

>>16766682
The meaning of life?

You know what, nevermind. This is beyond your thinking paygrade. I'm going to stop replying before you "ackshually" and tell me that a tree in math doesn't count because it isn't made out of electricity.

>>16765243
I mean it wouldn't graph theoretically be a branch if it were interconnected graph but I get what you are asking but it'd be like a graph, all branches of math combine together in various fashions.

new >>16767261

>>16767102
That's quite a strange take on things. Elements was never "abolished." The entire field of synthetic geometry (i.e., geometry not with respect to any specific defined coordinate system or algebraic structure) just became "out of style" once equivalent concepts were developed in analytic and algebraic geometry.

People still learn the process of rigorous proofs. That just generally happens in the context of a non-geometric course like real analysis, or algebra (either of the abstract or linear varieties). The main reason Euclid's elements went out of style in schools is because everything it offers can be learned from analytic and algebraic approaches to geometry, and synthetic geometry like Elements is significantly less useful for applied geometry problems.