/sci/ - Science & Math

Is the parametric equation for a line just the parametric vector form, with "t" (in pic related) just being the free variable in parametric vector form and the "direction vector" just being a vector parallel to a set of solutions to a linear system?

All problems in mathematics are the result of abstract concepts that have no basis in reality.

A polygon has only two dimensions. No such thing exists.
A line has only one physical dimension. No such thing exists.
A point is a dimensionless concept. Again no such thing exists.

Furthermore these concepts so not make any internal sense among themselves.
A line is defined as the distance between two points. Yet a one dimensional object can never interface with a dimensionless object.

Then we have negative numbers, considered as independent objects, yet none can exist in reality. We compound this error by extrapolating the existence of imaginary and then complex numbers.

Rather than examining the foundations of the mathematical system and reconsidering our conceptual approach, we come up with bandaid treatments which paper over the patent absurdities.

>>15379919
>A line is defined as the distance between two points. Yet a one dimensional object can never interface with a dimensionless object.
guess you gotta draw the line at some point

>>15379919
This “flawed” mathematical system took men to the moon.

>>15379919
Math is not supposed to be reality. It is supposed to model reality, which it does extremely well. Math is the reason behind the existence of this website.

Hey HEEEEY. New thread, new problem! This one comes from Solomon Golomb’s book, the guy from the previous thread. It's just a small exercise from there. Anyway, good luck to all those attempt! Feel free to ask for clarification or for hints. I appreciate anyone who attempts successful or not.

>>15379950
XD

>>15379790
Holy heck! I love complex numbers. They are so beautiful! Who would have thought it's just scaling and rotation.

>>15380140
whoops, sorry, replied to the wrong guy.
XD was meant for >>15379940

>>15379790
Have any of you read this? I am wondering if it's any good. Seems well recommended, but it's not mentioned on the /sci/ wiki.

>>15380140
Does it matter the order in which flags are put in a pole?

>No finite polynomial can be constructed that is equivalent to [math]\sqrt{1-x}[/math].
I was explaining this to a brainlet elsewhere using derivatives and it got me thinking.

Given a restricted set of some algebraic symbols and some rules for what constructions are permitted (eg polynomials) and an algebraic expression F that is not obviously one of the former (eg F=sqrt(1-x)), is there a general algorithm for showing whether an equivalent expression to the given F can be constructed or not?

>>15380177
Hello anon. Thanks for your question. Yes, the order on the pole does matter as far as I can tell, I'm fairly confident.

>>15380140
Each way to put the flags on the poles can be considered as an ordering of the flags together with a way to split the flags in the poles (see picture).

There are six ways to split five flags in three poles : you can group them 3-1-1 or 2-2-1, and in each case there are three ways to order them.

There are thirty ways to order the five flags : there are 5! permutations, but there are two red and two blue flags, so the number of possibilities is [math]\frac{5!}{2!2!} = 30[/math].

So the final number of possibilites is 6*30 = 180.

Suggest problem books on secondary school math such as intermediate algebra, planimetry, stereometry, pre-calculus and such

>>15380381
Hey anon! Nice job, that's absolutely correct. It's how I went about it as well. Though of course we don't need need to think about 3-1-1 and 2-2-1 as for larger numbers it might be challenging to find all such possibilities. We could just give one flag to all poles and then distribute the rest using stars and bars. For this case give every pole one flag. We're left with 2 flags and 2 bars, giving us 6.

Thank you a lot for your time and effort solving this problem. And an extra thank you for your image and explanation of your solution. I really appreciate it. I hope you have a fine day!

>>15380164
I have read the combinatorics part of this book. I thought it was quite good. I can't really comment on the other parts about graph theory and infinite combinatorics but the combinatorics part had good topics, examples and exercises. Has some funny jokes too, i genuinely laughed out loud while reading sometimes.

>>15380415
Is combinatorics all you ever read?

>>15380436
For now, basically yes. I've read some number theory as well though not much. I'm in last year of high school(I'm above 18, I spent a year abroad which didn't count for my graduation) and so the school subjects require enough time that I can't dedicate enough time per day towards properly studying a subject like analysis or algebra. So I just read and do combinatorics for now. I figured I'll learn other things in university. Also to be honest combinatorics just seems like the most fun.

I haven't done math since 7-8 years ago when I was in grade 10. Getting started with it again by reading this book.

>>15380516
>precalculus
Why bother?
https://lyryx.com/wp-content/uploads/2017/06/Guichard-Calculus-EarlyTranscendentals-2017A.pdf

>>15380711
>Why bother?
Because you need it?

>>15378362
>>15378374
Well I guess in particular I mean how can I get a rough idea of where a group of number that large fall on the number line. How does mathematica determine which number is greater? How does the computer even represent these values and do operations on them? Surely theres some analytic method? I cant tell at a glance which of the three numbers you listed there are the largest or approximate the difference between say [math] 5^{5}^{5} and 2^{2000} [/math]. I know we can determine the number of digits in n by taking the base 10 log of n, but my calculator said the value is too large and got an overflow error.

I know this is a dumb question, but it's one of those things I never learned, and I was wondering if there were explicit instructions or at least heuristics for this kind of thing.

>>15380945
You can take logs

>>15380945
>>15381010
Oh you mentioned that; you can easily see that [math]\log_2(2^2000)=2000[/math] and [math]\log_2(5^{5^5})=3125\log_2(5) > 2000[/math], for example. Then since logs are monotone 2^2000 < 5^5^5.

June Huh? Not July What? Or August Eh? It's already September, huh? What? It's April? Come again? Time flies, huh?

>>15379919
Ok then stop doing it

>>15381015
Thank you

>>15380178
In general the answer is no. Classifying clones even on finite sets is a hard problem and what you're asking is probably equivalent to the halting problem.

>>15380146
I was asleep. Missed it.

>>15381428
Damn, universal algebra looks neat. Thanks for introducing me to it.

>>15380164
>>15380415
It's great. I read it maybe three years ago summer after my junior year.
It really helped get me into combinatorics. Super fascinating subject, both from a pure math perspective, but also because it has a lot of neat applications in computer science, optimization, biology, network science, and related fields.

Actually, I'm getting ready to graduate with a masters degree, mainly focusing on combinatorics. Currently I'm taking a class on so-called "matroids" which are a combinatorial object closely connected to both graph theory and linear algebra. The guy in OP's pic >>15379790
is named June Huh. He was just awarded the fields medal last year for his work on matroid theory. Combinatorics is in many ways a really fundamental and really basic field of math, but it's actually relatively new and very active. The subfield of combinatorics basically didn't exist until the 1930s. In general prospects for academics today honestly suck ass compared to prior generations, but combinatorics is probably one of the healthiest areas of mathematics at the moment. Only thing more trendy is probably statistics and probability (because of all the machine learning and data science fags taking over the world).

Which Analysis Problem books does /sci/ like? I have found this three volume set by Kaczor and Nowak, I am quite pleased, but there are many problems and some are not very enlightening. I am familiar with Pólya's book as well, but it's more geared to complex analysis. Please give me your recommendations, and as a self-learner, I am only interested in books which contain solutions, or have a solutions manual.

>>15382611
I also found this set. Looks good from a first glance, but it's helpful to get feed back. I've actually studied analysis a long time ago, but want to work through problem books as a way to see what I need to refresh myself on.

Dumbass stupid ass retarded ass ChatGPT

>>15382715
Leszek Gasinski & Nikolaos S. Papageorgiou have two books that are nice

>>15382961
Thanks for the recommendation. This series seems to begin at a higher level of difficulty (in terms of material) than the others which begin with more elementary analysis. Also, what's with the Polish and creating problem books? This is the third in a row to have a Polish author. Also turned into a collage for Autism's sake (it's a scripted process anyways)

Is there a "standard" notation for the set [math]\{1, 2, 3, ..., n\}[/math]? Particularly for combinatorics/discrete mathematics.

I've seen[math]I_n, [n], \mathbb{N}_n[/math] being used..

>>15383433
From your options I'd prefer [n], which I also see regularly.
If you got a set theory proper context, you can also use n itself (being {0,1,...,n-1}). If you got code environments, then range(n) will do also.

>>15383507
We're talking mathematics not computer science you worthless code monkey

>>15383507
Thanks. I've also seen and understand the set-theoretic definition of natural numbers, however it is highly impractical in any other area.

I guess I'll just stick to the [n] notation, although I'd want some approval from my combinatorist /mg/ friends over here.

If all norms on a vector space [math]V[/math] are equivalent, does this mean the metric induced by the norm is Cauchy-complete in [math]V[/math]? Is this vector space necessarily of finite dimension and thus isomorphic to [math]\mathbb{R}^{\text{dim}(V)}[/math]??

>>15382611
>>15382715
>>15383040
>Almost exclusive Polish mathematicians
based

>>15383040
I dunno, but it's also how some people I know learned probability (one by M Capinski & T Zastawniak and one by Z Brzeźniak & T Zastawniak). They are not too advanced but certainly nice. Plenty of Hungarians and Russians have problem books, so it might just be a slavic thing.

>>15380140
Interesting. How would you approach/find the answer to something like this, in the simplest cases?

Number theory is so fucking boring holy shit.

>>15383568
Personally, I always use [n]

>>15383433
[math] \{ i \}_{i=1}^n [/math]

>>15383767
[math] \mathbb{R}^{ \operatorname{dim}(V)} [/math]

how to prove that = is transitive

>>15385072
It's a consequence of substitution, an axiom of first order logic.

>>15381062
XD
we made similar jokes when someone shared news saying he won the fields medal.
>>15382552
That's amazing anon. Like I mentioned here >>15380472 I'm still in the last year of high school. Do you have any tips? I know that's a bit vague and to be honest I'm not sure what I'm asking for either. But either way I'd appreciate if you could tell more about how your undergraduate studies went, how your masters is going and what you intend to do in the future. And any advice you might have.
>The guy in OP's pic
June Huh was a good choice, glad you noticed!
>because of all the machine learning and data science fags taking over the world
Is combinatorics ever useful for machine learning?
Anyway, I appreciate if you can answer my question. Have a nice day!
>>15383433
I like [n] simply because it's what a friend of mine used and I liked it. It's also common and intuitive enough that most people would understand. When in doubt {1,2,...,n} just does the job.
>>15383507
Using range(n) is quite based. If I saw someone using it I'd be admiring honestly.
>>15383568
>approval from my combinatorist /mg/ friends over here
Well, Stanley's Enumerative combinatorics book uses it and I think that's a pretty good approval.
>>15384346
Well the simplest cases happen when you have just 3 flags. Since every pole needs at least one flag, you have no choice but to put 1 flag on each pole. And then depending on how those 3 flags are colored, your job is quite simple. If they're all different colors, there are 6 different ways, if 2 are the same color but 1 is different there are 3 ways and if they are all the same color, there is just one way. When we have 4 flags it's a bit more complicated but not much. Can you try to do what happens if we have 3 flag poles, 2 blue flags and 2 red flags and every pole must have at least one flag. Try to do this first and see if you can, after that try the problem I shared. If you get stuck ask for help! Good luck! I believe in you chap.

>>15380140
To apply a sequence of 3 types of flags of arbitrary length>0 to a pole, this is represented by the generating function:
(r+b+w)/(1-(r+b+w))
To do this to three distinct poles you get the generating function:
[(r+b+w)/(1-(r+b+w))]^3 = Sum[ (n C 2)*(r+b+w)^(n+1), n>=2]
Since 5 flags are used, the n=4 term is what we want.
The coefficient of (r^2)(b^2)(w^1) in (4 C 2)*(r+b+w)^5 is the answer.
To extract the coefficient, differentiate wrt r twice and divide by 2, b twice and divide by 2, w once and divide by 1 and set r=b=w=0.
You get (4 C 2)*5!/(2)^2 = 180.

>>15385720
You can generalize this to P poles, pole p requiring at least f(p) flags, T types of flags, type t occurring exactly n(t) times.
Let F = f(1)+...+f(P), N = n(1)+...+n(T).

The generating function is:
(x1 + x2 + ... + xT)^F / (1- (x1 + x2 + ... + xT))^P
= Sum[(m C P-1)*(x1 + x2 + ... + xT)^(m+F-(P-1)), m>=P-1]
The m = N+P-1-F term corresponds to N total flags.

The answer is (N+P-1-F C P-1)*(N C n(1),n(2),...,n(T))

>>15384494
It's because it's hard to connect it to "real world" problems, so it requires a special kind of autism.

I'm a comp sci major in college and their forcing me to take math classes and I declare that math is not about intelligence, just non-stop route memorization

>>15386222
I'm a math major who had to take a couple of compsci classes and I declare that the only thing that there is to compsci is
public static void main(String[] args)

>>15386222
then you're either taking math classes taught by brainlets, or you yourself are a massive brainlet

>>15386222
Let me guess, you're being filtered by linear algebra?

>>15386271
calc 2

>>15379790
got my final on optimization in 12h
how do i learn matlab in 12h?
gotta some wizardry on that for the final

>>15386275
You're probably in a plug n' chug program then. You'll see more intelligence required in analysis courses.

>>15386313
he is a csi faggot
these are the dumbest retard in stem
barely above social studies nigger
I swear all comp sci faggot i've seen couldnt do a basic proof
>t. phys-math

>>15386325
It really depends on the school, and branch of computer science. Some of the topics in advanced compiler design are very demanding, as are topics like computer graphics (algorithms, not design).

Suppose I just want to self study math for myself and gain as much insight in as little time as possible. Would you recommend working through Spivak’s calculus, Zorich’s Mathematical Analysis, something else? Should I bother staring at every problem for hours, or to look up the solution after an hour or so?

>>15386603
And more than this, should I even do every single problem in a section before moving on?

>>15386614
Doing every single one may be difficult, but you should be convinced you know how to solve each problem before going on. Doing problems is the only way you can tell that you've actually understood the material, rather than merely just having followed the author along line by line.

What is your motivation to learn mathematics? If you're self-studying without anything to drive you along, it's very easy to give up the moment things become challenging.

>>15386622
Motivation is to eventually learn higher level and new mathematics. I just wonder if I’m really developing much from staring at a problem for hours

>>15386635
I'm not a tutor, but I think if you're not making any progress in a few hours, it's time to lookup the answer. Analysis can be especially tricky as there's a lot of identities that you're just "assumed to know" in order to answer questions of the form "Show that...." rather than "calculate".

I really like Zorich's book, but the questions are very challenging, and it's not always obvious why they relate to the chapter material. I imagine in a school setting, the TA offers students a lot of help in such cases.

Check out the problem books above, especially:
>>15382611
>>15382715

I think they're very good if you don't have a tutor to help you. Polya's "How to Solve it" might come in handy as well. Another short book which is good is called "Counter examples in Analysis".

>>15386644
Thank you. I’ve skimmed through pdfs of the books by Kaczor and Novak, and I think I will order physical copies soon.

>>15386663
just divide

>>15386663
I can't find lambda that satisfies the equation, am I retarded? pls help

>>15386677
Those books are good. The other one is quite nice too. At first it looks really basic, but by volume 3 they're covering some of the more advanced topics in analysis, that you'll find in Zorich's Volume II.

Got a question about platonism, intuitionism, etc

Regardless of the answer to the question, is the question of whether real numbers exist equivalent to the question of whether complex numbers exist? Do they have the same status?

>>15387052
What the fuck does that even mean?
They exist because we defined them.

>>15386663
[eqn] \frac{2-3i}{1-2i} = \frac{8 + i}{5} \\
\frac{4+i}{2-i} = \frac{7 + 6i}{5} [/eqn]

>>15386706
To me, it seems there is no such lambda. Note
from >>15387064 that lambda changes from the
division of corresponding components.

What is the geometric object that results from the intersection of two hyperplanes in R4?

>>15387210
A 2-dimensional plane in R4 if the hyperplanes aren't identical or parallel.

Do you guys know of a solution set to Zorich vol 1 or 2?

>>15387216
Thanks. This is what I suspected but I wanted to check before spending the time to demonstrate this with a detailed argument.

Do i ACTUALLY need this or does my professor just want to torture me?

>>15387377
And this isn't even proofs where understanding it gives you a new point of view and increases your analytical thinking.
No, this is just made-up bullshit with the specific purpose of making other made-up bullshit more complicated.

>>15387377
Of course you need this. Do you really want to assume an ultrafilter and construct the hyperreals?

>>15387384
You're it isn't proofs where understanding it gives you a new point of view and increases your analytical thinking. It is a definition.

>>15385371
>Is combinatorics ever useful for machine learning?
Not really. Combinatorics isn't super useful for machine learning. ML is mostly statistics and linear algebra.

However matroid theory basically provides a combinatorial analog of some of the ideas in linear algebra. It turns out that matroids are closely connected to optimization, greedy algorithms, and linear programming. This ends up being useful for machine learning, because ML often involves optimization algorithms that can be better understood using matroids. That being said, I don't know much about the topic. I don't ML algorithms are doing a lot of combinatorics or matroid theory or anything like that. I haven't looked to dep into it, but from what I understand matroid theory ends up being a useful tool, not for actually executing optimization algorithms, but rather for proving that these algorithms in fact work in the general case. So it more so has to do with machine learning theory, rather than machine learning implementation.

My undergrad education was at a very small liberal arts college. I was a good student and I think the professors were great, but the math department was very small and class choices were limited. Also, this was unfortunately during COVID lockdowns. However, I definitely learned a lot of cool stuff.

Masters program was at a much larger university, and it has really been fun, and this fall I'll be starting my PhD. I'm actually moving to a philosophy department, but I'll still be doing a decent bit of quantitative work, because I'm going to be studying game theory.

Anyway, my biggest piece of advice is to get to know your professors. This is super fucking important, and something I have struggled with, since I'm not great in social groups and I have difficult understanding normie psychology. Letters of Rec. are probably even more important than raw grades for grad school and the job market, so get to know your profs.

>>15387524
>Combinatorics isn't super useful for machine learning. ML is mostly statistics and linear algebra.
But combinatorics is useful in statistics?

>>15387217
Does not exist as far as I know. I have come across an unfinished solution blog where multiple people suggested solutions, but I think your best bet will be to look it up on math stack exchange, and to ask the question yourself if it is not on there.

Is there a continuous function [math] f : \Bbb{R}_{>0} \rightarrow \Bbb{R}_{>0} [/math] such that for every positive real [math] k [/math] ,
(1) [math] \lim_{x\to\infty} \frac{f(x)}{x^k} = \infty [/math] , and
(2) [math] \lim_{x\to\infty} \frac{f(x)}{e^{kx}} = 0[/math] ?

>>15388073
Oh I just solved it, but I'll leave it up as an exercise

>>15387052
Well that the jump from real to complex number treatment is very algebraic in nature and thus fairly tame. In this sense, yes.
Of course the ontological questions, the meaning of "exists", is a bottomless pit that even goes beyond mathematics. Empirically, one can do math without addressing them.

>>15379790
Haven't had to integrate partials like this in a while, is there something obvious u-sub I'm missing?

[eqn]
\int \left( f(x) \frac{\partial^3g(x)}{\partial x^3}-\frac{\partial g(x)}{\partial x}\frac{\partial^2f(x)}{\partial x^2}\right)dx
[/eqn]

Answer key has it as:

[eqn]
\left[f(x)\frac{\partial^2g(x)}{\partial x^2} - \frac{\partial g(x)}{\partial x}\frac{\partial f(x)}{\partial x}\right] - \int \left(\frac{\partial f(x)}{\partial x}\frac{\partial^2 g(x)}{\partial x^2} - \frac{\partial^2 g(x)}{\partial x^2}\frac{\partial f(x)}{\partial x} \right)
\\
= \left[f(x)\frac{\partial^2g(x)}{\partial x^2} - \frac{\partial g(x)}{\partial x}\frac{\partial f(x)}{\partial x}\right]
[/eqn]

>>15388523
it looks like integration by parts I'm just not sure whats u and v

>>15388523
>>15388527
split the integral into two along the subtraction and use integration by parts on each half separately, and then recombine

>>15387052
This is a really interesting question, and I think the equivalence of these questions DOES depend on the answer to each of them.

The only time I think these questions wouldnt be equivalent is if you think the real numbers exist because theyre "real" in a physical sense. Ie, real numbers can be assigned to physical quantities in some countable way, one whole apple (Z+), gaining negative $10 (spending $10) (integers), eating half of a cake (rationals), wrapping a length of paper around a can (irrationals). Complex numbers are only special because of imaginary numbers, which don't cleanly correspond to physical quantities in a way that can't equivalently be expressed without them. Even quantum mechanics can be formulated solely in terms of sine and cosine. So if you think real numbers are "real" numbers, and imaginary numbers are "fake" or "constructed" numbers, then these questions are not equivalent.

Most people, myself included, don't view the existence of numbers in this way, but its an interesting thought

>>15387057
>regardless of the solution to equation A, is equation A equivalent to equation B?
>WHAT DO U EVEN MEAN THE SOLUTION TO EQUATION A IS X
boy

>>15386325
Physics math double major here, I went to a great university with a great compsci department, did not have this experience. Most of them could do rigorous proofs I hadn't seen simply because we're in different fields (proving turing completeness, halting problem, etc) while I could do ones they couldnt (analysis n shit). If none of the cs people you know do this, either you have a weak cs department or you arent looking hard enough (likely the latter).

They out-earn us anyways lol but thankfully if you can learn physics you can probably teach yourself the skills needed to pull down six figures in tech

>>15388537
Fuck youre totally right. Thanks lad my calc is rusty, need to take a math methods class or something

Is there a name for this?
vector space over [math]\mathbb{C}[/math]
vector space over [math]\mathbb{R}[/math]
vector space over [math]\mathbb{Q}[/math]
module over [math]\mathbb{Z}[/math]
??? over [math]\mathbb{N}[/math]

>>15389074
It wouldn't have an additive inverse, so you'd end up with some sort of semiring/rig underlying, but I don't think that that'd generalise very well to a module-like structure

>>15389074
vectors can be integer-valued, they needn't be made up of reals.

>>15389087
>generalise very well to a module-like structure
this is precisely what module means compared to vector space. A module needn't be defined over a proper field; a ring or group or (iirc) set is fine.

>>15389074
>module over Z
>??? over N
N forms the smallest ring containing the natural numbers. Z is also a ring. So, you decide.

are there any textbooks on structural set theory?

Just finished my first two semesters. I feel like it would be the responsible thing to do to earn credit for a few courses this summer like numerical analysis or differential equations or multi-variable calculus but I also just want to be a lazy fuck and marathon video games while maybe casually working through an introductory programming or combinatorics book. It's best that I tackle these courses earlier than later and that I be held accountable for my education but I also really just love video games...

>Before abstract algebra midterm
>"God? Its me, I know I haven't been your humblest servant, but I will renounce all earthly vices if I could just mercifully be allowed to receive a 65%, which would leave me with a 70% average and is the bare minimum to continue my studies. Your always humble servant..."

>After midterm
>"GOD?!!? As your angriest, scorned, denizen, I am DISGUSTED you found it suitable to humiliate me with an 89%. Is this your sick idea of a joke?!!? I reject you, I deny you, I will dedicate my life to your destruction!"

Uh, any of you other homos know this feel?

>>15389244

>I don't understand this concept. This is it. This is the point where I am exposed and filtered as a brainlet.

*gets the concept*
>Naturally. I am one of today's greatest minds, the reincarnation of Euler.

rip MJR

I read about mathematicians in the 20th or 19th century who had contrary views on the philosophy of mathematics. Like intuitionism, constructivism, finitism.

Are these different views still recognized in academia as an actual debate and relevant and important in some way?

>>15385720
Hello anon! Thanks for your solution, it is of course correct! Always fun to see people using generating functions. How did you know to differentiate to extract the coefficient? Sorry, I'm not very familiar with generating functions still, I've been meaning to get around to it sometime soon. Is what you did explained in generatingfunctionology?

But again, thanks a lot! Amazing work and I hope you have a just as amazing day.
>>15385860
Nice generalization!
>>15387524
>Not really. Combinatorics isn't super useful for machine learning.
Well I suppose that's fine.
>this fall I'll be starting my PhD
Best of luck to you, anon. I hope whatever you face you overcome it.
>my biggest piece of advice is to get to know your professors.
Thank you for this advice. I hope I can put it to good use. In highschool I've managed to be close with teachers to the point one of our math teachers brings problems he can't solve to me and asks for geogebra/desmos help. In university that might be more challenging.
Again, good luck on your PhD journey and thanks for your advice.

>>15389244
>>15389253
Do all mathfags have BPD?

>>15389637
Why are you so racist? This is not the first time you have been racist.

>>15389398
amongst philosophers of mathematics yes
amongst other professional mathematicians no
von Neumann talks about it in one of philosophy of science papers how all the mathematicians after Gödel's result basically said who gives a fuck if non-intuitionistic math is good enough for physics it's good enough for us otherwise large quantities of the mathematical literature especially from analysis and geometry have to go into the bin which nobody wants to do

>>15390236

>>15390287
>otherwise large quantities of the mathematical literature especially from analysis and geometry have to go into the bin which nobody wants to do
Assuming the Godel(not a real mathematician) was right. I don't find his methods valid, they only apply to a few types of formal systems, and have no practical application in actual mathematics. It seems there is some huge interest in propping the guy up, for what reason I don't know.

>>15390282
I'm sorry, I swear I'm not actually racist. It's just a meme I made when it came to my mind.While I still think it's funny, I l totally understand that it's racist. Apologies for giving the wrong impression about myself but I assure you that I am not racist, quite the opposite in fact. I'll try to avoid this in the future, I could use apolitical images instead.
>>15390339
interesting quote.

>>15380472
>I'm in last year of high school
Nice, do you bottom?

>>15390287
Damn, I was hoping it would actually be useful. I guess for me at least it feels insightful to understand math better, knowing about philosophies of math.

Why in the world is the proof to basis representation theorem so convoluted? Just subtract two representations to equal 0. It equalling 0 implies that all the differences of coefficients are 0, sums of powers of positive number is always positive, hence proved.

>>15389637
>How did you know to differentiate to extract the coefficient?
It is pretty much just from the definition of taylor series at 0.
f(x) = Sum[(x^n)*f^(n)(0)/n!].
The coefficient of x^m is f^(m)(0)/m!.

>Is what you did explained in generatingfunctionology?
I'd hope so. You just need to know what a geometric series represents and what multiplying generating functions represents.
It should be in any book using generating functions.
I don't have generatingfunctionology but I know it is in
Enumerative Combinatorics by Richard Stanley
Analytic Combinatorics by Philippe Flajolet and Robert Sedgewick

>>15390339
but manic depression is like ADHD
it's just another meme
math is here to stay

Suppose R is an integral domain and S is a cancellative semigroup. Prove or find a counterexample: the semigroup ring R[S] has no zero divisors.

>>15390287
Have there been any other changes in how people have viewed mathematics since then? Will there ever be any more changes or has it just been completed now?

>>15390287
von Neumann was a jew
jews always say things like "all of the goys like X are like Y"
they're prejudiced
you can ignore it
for example
I was never exposed to the difference between "intuitionistic" and "non-intuitionistic" and I don't know anybody who has, either
von Neumann can have his say, but what he said came out of a jewish mouth or was penned by a jewish hand

>>15390653
Cute, I'm reading a chapter of integral domains right now in gallian's text.
product of nonzero elements in an integral domain is nonzero and S is cancellative. You are kinda told everythig else.

>>15390669
you don't have a math degree
you're an idiot
you should not post here
go be an idiot somewhere else

>>15390767
Is that why you're asking a softball homework problem on /mg/? Because you have a math degree, are smart, and should post here? Somehow, I don't see that being the case. You okay anon?

>>15390831
third time somebody said this about the question
getting real tired of your bullshit, internet
it isn't a homework question
shut your lying mouth

>Universitext
>GTM
which side are you on?

>>15391282
I'm GTM all the way.
My mom got me Lang's Algebra when I was 15, and it was all I used in the psych ward after a suicide attempt. Since, I've developed a certain level of nostalgia for GTM. That, and they often go on sale a lot.

>>15389074
Free monoid is where my mind goes.

>>15390414
Well, intuitionistic logic is useful in formal verification and in topos theory.

>>15391358
you are full of it and you know it

Stumbled across this hilarious review for Stillwell's Mathematics and its History. Reads like a 4chan post, so which one of you wrote it?

>>15391483
kek

How much do you actually remember from the texts you read? Or do you mostly just remember an intuitive idea/implicit model of the main theorems/definitions/problems?

>>15391303
How did you try to do it?

>>15391502
Obviously his calculations were off. His mom just wanted to help him out for the next attempt.

>>15391483
you get this problem in general when attempting to use math texts outside degree-granting institutions
what's the point
internet points?

>>15391370
how are they full of it?

>>15391358
Huh? Wouldn't that just be using different rules rather than a different view in philosophy of math? By "it" being I meant knowing philosophies of math, not "using (intuitionist for example) logic".

>>15391593
The point, edification! Only an NPC stops trying to learn things once their prescribed education program comes to and end.

>>15391618
I'm an NPC
And I regularly send armed wombats (with shoulder mounted laser canons) out to take care of anything that needs to be taken care of in the event that I discover that I have been learning, which I take as a domestic emergency

>>15391631
Alright...

>>15379790
What's the most interesting thing for you in mathematics, /sci/? Is it something general or something specific?

>>15379804
120+ replies and none of you useless faggots answered my question. Literally the first post in the thread. I hope you all fucking die.

>>15392076
go to office hours.

>>15392084
Lol, I actually did and my professor told me I was exactly right

How do you guys stay focused during lectures?

My lectures are often very abstract and it's difficult to follow along very step. I will make a remark in my notes to go back later and justify a step or statement but I'm lazy and rarely do (because I prioritise working problems and problem sets).

>>15392238
For example, lecturer makes a remark at the conclusion of some set of statement that summarises what she has been showing. The sentence is full of technical jargon. I can break down this sentence and understand what she is saying but I'm a bit slow. I can't do this in real time. My eyes glaze over a little bit.

>>15379790
Stupid question but how is algebra use for convex optimization?

>>15392354
It's simple. Just consider the etale-cohomology of pre-sheaves assigned to derived categories induced by the homotopy of minimal solutions

I've gone through 90% of Lang's Basic Mathematics and then my study habit broke off for some reason. Any tricks to getting back into it?

One problem is that I kinda lost my motivation. I originally started learning because I got a job as a data analyst, but by now I've learned that being an analyst doesn't involve any "heavy" math at all. Once while making a model a senior colleague asked me about some statistics terminology since he's already forgotten all the school stuff, but besides that it's all just data wrangling.

Let [math]a, b \in \mathbb{Z_{>0}}[/math] be coprime integers.
What is the largest integer not in the set
[eqn] \{ax + by | x,y \in \mathbb{Z_{>0}} \}[/eqn]
?

>>15392687
look up the coin problem

>>15379790
The Metallic Ratios are a myth
Go on
GOLDEN ratio? Then silver then bronze?
Gold R(5) something something phi
Silver R(8) something something pythagorean narcissism
Bronze R(13) ok checks out looks fibonacci enough we should expect the next root to be the sum of 8 and 13 but WHAT'S THIS?
4 + Root (20)
CANNABIS?
What spiral does that even produce?
I have not learned enough to draw the smoking weed ratio but man if someone could animate the spirals that would be incredible.

>>15380937
This. I'm in the same boat. I forgot all trig.

I would pay for webassign practice problems but I would not pay university stupid level costs.

>>15392756
euler's formula is all I ever used in 4 years, everything just rolls out

>>15392783
what do you mean by rolls out?

>>15392596
Good work. If you've already got through 90% of it, do you really need to finish it? Maybe study something that is relevant to your interests or work. Have you studied calculus and probability theory?

I've personally found that a little set theory and propositional logic goes a long way with data wrangling.

>>15390503
I see. Thanks a lot anon. This is very helpful.
Although I should say Stanley's text is just.... scary! I've looked at the exercises section of the first chapter before and out of the hundreds of questions there, I was able to pick out do very few. And the text itself looked very intimidating too. I'll look at it at a later date when I'm more prepared. I'll give Sedgwick a quick look. Thanks a lot again for your reply. I hope you have a great day.
>>15392063
Problems in general. Sometimes they have such suprising solutions that it truly deserves to be called interesting in my opinion. For example squaring the square.

Stupid question: I read that the number of clouds necessary to fill the plane is equal to 2 + n where the cardinality of the continuum is aleph-n. Does that mean an uncountably infinite number of clouds is necessary in models where the continuum is aleph-omega-1? If so, I think that's a pretty good argument for ontological maximalism.

>>15392956
>ontological maximalism.
Are you studying math in college?

>>15393273
No, I can't afford college.

>>15390653
This isn't a homework problem.
You're just calling it a homework problem because you don't know the solution.

>>15392354
say
https://en.wikipedia.org/wiki/QR_decomposition

>>15389398
>Are these different views still recognized in academia
Mathematicians don't hold genuinely strong views anyway. People are used to the principles they learn and then use them to do math and write papers.
Lots of formal frameworks are studies - e.g. in topos theory, as has been pointed out - but this is of course all fringe compared to, say algebraic geometry and functional analysis, which is full with choice. Topics close to computer science and topics where uncountability plays less of a role (say combinatorics), is naturally done in a more or less constructive way, and would be in line with what e.g. Brouwer did.

Why do computer scientists care so much about publishing conference papers? All discussions about comparing the academic output of different computer science institutions boil down to conference publications (stuff like popl, lics, etc). Do they not publish in journals like how it's done in mathematics?

>>15380140
Curious why it's always counting. Not a good or bad thing, but it does make me wonder. Are you autistic? I feel like counting problems are sort of an autistic thing to attach oneself to (see: rainman's toothpicks). Again, no criticisms for autism. We owe a lot to y'all.

>>15387377
This is one of the most basic things in analysis, epsilon/delta proofs. And this is the more useful form for computation, but you'll learn to write the equivalent proofs more generally using neighbourhoods which I think are more intuitive.

>>15392596
https://en.wikipedia.org/wiki/Recreational_mathematics#Further_reading
or >>15380164

Filthy undergrad here, which field should I go into if I want to see a little bit of everything in Math? The internet said Operator Algebras.

>>15394582
You'd probably want to explore your math department's strong areas so that you have a course-guided experience. Maybe you'll see some usage of operators.

>>15392753
Try plotting cos(10x)+sin(x) in polar coordinates.

>>15379790
Good Evening/Sci/entists!

I started reading a lot about Combinators because I like it when Computers get a lot of symbols. Please tell me nice books about Combinators. In one book a guy uses a lot of birds that are Combinators to build arithmetic and logic! Are there other books that describe changing other math into Combinators? This was the most interesting thing I have seen in a math book.

Thank you /sci/entists for reading my post.

I've been reading von Neumann's/Morgenstern's 'Theory of Games' book and have been struggling to understand the Reduction of Compound Lotteries axiom algebraically.

Expanding terms, it makes sense why [math] \alpha (\beta u + (1 - \beta)v) + (1 - \alpha) v = \gamma u + (1 - \gamma) v ; \gamma = \alpha \beta [\math] but for some reason trying it the other way, i.e. [math] \alpha u + (1 - \alpha) (\beta u + (1 - \beta) v))[\math] doesn't work. What am I doing wrong?

>>15395135 LaTex fix:
[math] \alpha (\beta u + (1 - \beta)v) + (1 - \alpha) v = \gamma u + (1 - \gamma) v ; \gamma = \alpha \beta [/math]

[math] \alpha u + (1 - \alpha) (\beta u + (1 - \beta) v))[/math]

sorry i don't often post to /sci/

>>15394736
>>15394555
"I only wish that when I was a student beginning to learn combinatorics there was a textbook available as attractive as Bona's. Students today are fortunate to be able to sample the treasures available herein." - Richard Stanley, author of Enumerative Combinatorics, on A walk Through Combinatorics

You can also explore Sipser's or Hopcroft's books on Theory of Computation to see what graphs spawned. Alternatively http://landoflisp.com/

>>15394398
>Curious why it's always counting
It's for a couple reasons I suppose. First of all, checking the answers is very easy. If I asked other types of problems I wouldn't have the confidence to say if an answer was correct or not. Secondly I really just like counting problems. Not sure why but I find them fun!
>Are you autistic?
I genuinely don't know. I could be but I wouldn't want to label myself with a mental disorder without a medical professional diagnosing me. I do have ADHD and heard that it has similarities to autism so maybe you're onto something here.

But yeah, I share counting problems mainly because I myself like them the most and also because of practicality. I don't think I have autism but do have ADHD which may look similar. Thanks for your questions, I hope I answered them to your satisfaction.
Do you like counting problems? If not, what sort of math do you like?
Have a nice day!
>>15394555
I second this, it's a great book and very fun.
>>15380404
I think the book "Mathematical Circles" could be useful to you. Give it a look.

>>15395354
Thank you for telling me about Combinatorics. I like counting things and I think that numbers getting counted and going up more is the most important problem in Computer Science. I will probably get this book (I already have Land of LISP and Realm of Racket though).

I am looking for information on Combinators, which confusingly come from Combinatory Logic rather than Combinatorics.

Most of my books are pretty old. I am hoping for a newer book that talks about things like Iota Combinator and Super Combinator and shows how to make math other than arithmetic or logic with it. I am also looking for something stronger than an introductory text. I would be happy to get multiple books to achieve these goals.

I have the following books about it.

To Mock a Mockingbird
Elements of Combinatory Logic
An architecture for Combinator Graph Reduction
Introduction to Combinators and Lambda Calculus

I also have approximately 100 pages of notes I have taken on these books which I might add margin maids and publish at some point.

I found a nice way to use books where I copy the book by reading it and then typing it into Latex. After I finish with Combinators/Maid Phone I am going to go back to the Set Theory book that vampire maid from touhou told me and try the same tactic.

I would make links, but zoomers broke z-library by posting about it excessively on Tiktok.

Thank you /sci/entists for reading my post.

>Independently, approximate groups occurred in the theory of mathematical quasi-crystals. Originally developed by Meyer [Mey72] with a view towards application in number theory and harmonic analysis, they rose to prominence after the discovery of materials with quasi-crystalline structure in the 1980s (see [BG13] for a bibliography with hundreds of references). In the language of the present book, mathematical quasi-crystals are (translates of) uniform approximate lattices in abelian locally compact groups.

Have you anons heard of approximate groups?

Why the fuck is it not standard to have an index of symbols? I hate this book. Half the time I am going back trying to find in which page is the notation defined.

>>15396314
Have you considered trying to remember things when you read them?

>>15396314
I blacklist any book that doesn't have an index for symbols. Probably written for Indians who memorise everything from start to finish.

since you guys keep talking about it: where do i start learning about mathematical foundations? my uni doesn't offer any classes covering logic or set theory or whatever else falls under that term.

>>15396431
Do you mean foundations as in foundations of mathematics itself (model theory, proof theory, axiomatic set theory), or foundations for learning mathematics? I don't know about the former, it's mostly autistic stuff, with some use in CS I think. But for the latter you can start with How To Prove it, by Velleman. The foundational subjects to learn after that are:
>Number theory & Algebra
>Analysis
After that you could really study whatever you like.

>>15392879
I mean any trig identity or similar formula I ever used I was able to derive from euler's formula with no issues

>>15392783
I think Euler's Formula is the epitome of the fact that learning things in a very abstract pure math environment, makes the original thing so much easier to understand. Complex numbers are such an extremely effective way to describe trigonometry. Look at the proofs in a Euclidian geometry book; holy shit simple identities require so many steps and thinking, but with Euler's formula, it's a piece of cake.

>>15396566
I very much agree -- I am very lucky to not have taken a course in multivar calc, so that all the nonsense (div / grad / curl) was explained nicely with the generalized stokes' theorem when I learned it, as another example.

>>15396582
And I am even luckier for not having an undergraduate degree in math or anything remotely mathematical. Never studied calculus. Started math proper from Algebra & Analysis in my graduate degree. It's beautiful.

>>15396314
It does have one, they just fuse it with the index. Also get used to jumping around if you use Amann. They'll often give one line proofs like,

This is clear by applying II.3.14] to example b in 2.23.3 which follows from 3.13 and 3.3d

I don't use very many symbols in my Fractional Distance paper and I did have a index of symbols as Section 1.2 in my 66 theses paper. Even in the latter, the index wasn't needed since there really weren't that many symbols.

How do i learn advanced calculus without wanting to shoot myself in the dick?

>>15396706
Holy shit, you are right. This'll make it a lot better. Thanks.
>This is clear by applying II.3.14] to example b in 2.23.3 which follows from 3.13 and 3.3d
That's fine.

>>15396806
Oh it's slick as hell, I can't even imagine how difficult it was to write that book.

>>15396431
honestly there's really no reason to just learn foundational set theory anymore. better to get good at algebraic topology and/or geometry, get into categories and then into topos theory and foundations of compute science, its much more fruitful and it's highly active (unlike set theory).

Mathfags, pls help me. If I have the function [math]f(x,y)[/math] equal the multiplication of 2 numbers [math]x[/math] and [math]y[/math], then is [math]f(c, a+b)=c(a+b)[/math] true? Pls, I was never explicitly told how parentheses worked.

>>15396951
Is it ALWAYS true? My problem is with the parentheses notation.

>>15396951
It's true by definition. You said y is a number, so (a+b) has to evaluate to a number.

>>15396986
But is that the definition? Is the notation
[math]c(a+b)[/math] the same as [math]f(c, a+b)[/math]?

>>15397001
Where [math]f(x,y)[/math] equals the product of 2 objects [math]x[/math] and [math]y[/math]?

you're making so much more convoluted than it is
if f(x,y) = xy, then set y = a+b on both sides and you get the desired

>>15397009
on the other hand f(x,y+z)=f(x,y)+f(x,z) is a consequence of the definition if that's what you're asking

>>15397009
You don't see what I mean, this is question about notation. If we treat substitution as an operation on our expressions then you are saying it's true because:
[math]f(c, a+b) = f(c,y)\right|_{y=a+b} = cy\right|_{y=a+b} = c(a+b)[/math]
Then you are also saying that in our notation:
[math]cy\right|_{y=a+b} = c(a+b)[/math]
right?

>>15397032
I dunno how to get latex to display the 'evaluate at' symbol. The string "\right|_{a=b}" is suppose to means 'substituting b for a, for the expression on the right'. IE: f(x,y)\right|_{y=z} = f(x,z), where z is any string the function can evaluate (so the string shows up in the places where z is in the previous equation).

>>15396951
let [math]d=a+b[/math]
then
[math]f(c,a+b)=f(c,d)=c(d)=c(a+b)[/math]
so, yes, it is trivially true

>>15396582
>>15396566
>>15392783
All i remember from college is how mindless calc and diff eq was (memorize + plug/chug). What should I study so I can re-learn and appreciate the subject matter?

>>15396601
What book did you use for the Algebra and Analysis course?

Is this the best book to learn calculus for an oldfag brainlet?

>>15397709
iirc most people go with James Stewart's Calculus, but Id say it depends on what you need calculus for.

>>15397554
First math book I studied was Abbott's Analysis, though I think reading a book on proofs before it would have been better. I followed it up with Amann & Escher's Analysis. Still reading it.

I haven't studied Algebra (except Linear) deeply since I am not in pure math, mostly encountered it through study of numbers and polynomials. I encountered it first in Barbeau's Polynomials, then in Niven's Number Theory, and then in Parsolov's Polynomials. I also studied Linear Algebra (matrix focused) from Rao & Bhimasankaram. Aluffi is on my reading list. Amann & Escher has a bit of Algebra as well.

>>15397032
[math]f(c, a+b) = f(c,y) \rvert_{y=a+b} = cy \rvert_{y=a+b} = c(a+b) [/math]

[math]
cy \rvert_{y=a+b} = c(a+b)
cy \rvert_{y=a+b} = c(a+b)
[/math]

>>15397795
So, it's true?

not even once

>>15396928
>honestly there's really no reason to just learn foundational set theory anymore. better to get good at algebraic topology and/or geometry, get into categories and then into topos theory and foundations of compute science, its much more fruitful and it's highly active (unlike set theory).

>>15397554
For calc / diff q I recommend Pugh's analysis book.

Is there any reason to study chapter 1 from Amann & Escher. The whole thing seems completely pointless.

>>15397923
I never realized what happened to him.
Guy seemed to answer every other AG, AT or CT question on MSE/MO just a couple of years ago.

>>15384494
I don't agree with you. Number theory is actually one of the most interesting branches of math, in my opinion.

>>15398556
It's foundational material for a 3 volume book. What is pointless about it?

>>15379804
You said 'just' a lot and your question is worded strangely, maybe read more about it.

>>15382746
It's a machine and it's doing it's job, it's job is not to give you the exact answers to every mathematical equation. Study the machine more, or stop meme-ing on me.

>>15383519
Computer science has the ability to facilitate plenty of mathematical endeavors in any field of study, and is binary mathematics at it's core.

>>15399081
How is knowing what groups and rings are and the fact that there is only one unique operation that satisfies the familiar properties of natural numbers addition going to be useful in Analysis?

>be me
>10yo
>family moves to a town 1250 miles away
>completely black out on school for the next 4 years until we move back home
>have been completely retarded when it comes to middle school math since then
>fast foward another 8 years
>3rd year economics undergrad
>can understand all the concepts of behind calculus, linear algebra and optimization
>entirely incapable of actually solving problems because I fuck up basic operations with fractions, don't know what the fuck to do when roots are involved, consistently fuck up when multiplying polynomials, unironically too dumb to solve linear systems using elementary row operations
how would you go about rectifying this?

>>15398841
A few years ago he started just talking about his incel life, sex therapy and did twitch streams with the astrology crowd. There was an episode where some tabloids reported on his twitter rant about being mad about his parents giving him $100k so that he didn't have to struggle with anything in life.
But to be fair he's still posting on StackExchange. It's not like he forgot AG, he just got stuck with his PhD and said fuck it, then trying some random research jobs and talking with irl people. It's great to see someone who's good at math talk normally with people and no bring up mathy bs with them. Quite humble and inspiring in this sense.

>>15397923
>not even once
Well but why tho.
It's easy to make fun of him - I sorta just did. But in the end I think he's right. There's not many autists who jump over their autistic shadow and actively remove time from math and start dedicating time and brainpower towards all the other things. In the end, there no pathway for anybody to get anywhere with math - it's just entertainment like video games. There's like 20 problem that you could get famous from (and by famous I mean beyond the math community in that particular subject.) So fame can't be the thing you do it for. We just do math because we enjoy it and the autists who can't get away from it don't live a particular happy life, I dare to speculate.

>>15399143
This stuff is the "common knowledge" of mathematics. Rings, and especially groups are important in analysis as well. Groups are pretty much everywhere in mathematics.

>>15399143
Because Amann's analysis focuses on abstract mathematics, and treats the reals as a special case. Groups, and Rings are just part of the machinery for doing so. Basically you are front loading complexity to avoid repeating things. A more traditional analysis course will start with operations on reals, and move to more general objects later on, but having to repeat things already proved.

I only recommend Amann for turboautists. And I say that with the acknowledgement that all analysis is for autists. If you don't like the abstract approach, Zorich's book is good.

>>15399149
Gelfand's Algebra. It should fix all of your problems.

>>15399174
Do you think a greater proportion of analysts or of algebraists are on the spectrum?

>>15399187
If I'm forced to answer I'd say analysts, but eventually they become the same person as they move to algebraic topology.

>>15399195
Is algebraic topology the apex of mathematics?

>>15399220
I think it's surjective, but something like Algebraic Geometry has to be up there. It pulls together all of the other really hard topics, so the preliminaries to begin study are quite overwhelming.

>>15399250
*subjective. ffs.

>>15399220
Homologies are pretty cool.

>be sperg always reviewing my degree plan
>notice that the applied math program has been updated for the 2023-2024 year starting this fall
>new courses have been added
>Immediately email academic advising and request to have my degree contract updated to the new one so that my financial aid pays for it
More FREE math classes, and the ones I already had under my belt that are no longer a hard requirement, I still get to keep!!! I just keep winning!!

>>15399187
i'm nitpicking here "symptoms" of high functioning autism are the same as those observed in gifted children and most "autists" in academy are not really so
this exarcebated by old descriptions of autism describing the condition along then lines of "an extreme form of the male brain, obsessed with systems and patterns"

>>15399280
>i'm nitpicking here, but a lot of "symptoms"
fix'd

>>15399174
>Because Amann's analysis focuses on abstract mathematics, and treats the reals as a special case. Groups, and Rings are just part of the machinery for doing so.
No they don't. They even literally say in the book, none of it matters; the only field useful in Analysis is Real and Complex. It's all just fisted in for no reason aside to feed the authors' ego. Rudin's chapter 1 & 2 covers everything that is required for analysis. The whole point of Analysis is restricting yourself to specific types of structures. The whole point of Algebra is generalising as much as possible. They are incompatible.

>>15399154
Yes, they are also in kindergarten math, yet they don't teach them. What's your point?

>>15399339
Do the integrals form a group? BTW Amann Escher goes a lot further than Rudin.

How do you mathfags cope with knowing that your theorems could be BS due to flaws in the axioms?

Guys, I have a confession.
I can't write an infinite sign without turning my paper sideways and writing an 8

>>15399187
try looking at which has a higher percentage of male students?

>>15399390
Because men have higher rates of autism in the general population? Is it probable that the decision to study mathematics is independent of being autistic? It seems possible that any woman entering mathematics must be much more likely to be autistic on average, compared to men.

>>15399339
>the only field useful in Analysis is Real and Complex
An abstraction. Teaching real and complex analysis together requires a lot of generalizations, since as you are no doubt aware, these are often taught as separate subjects. The really extensive use of algebra will be in volume 2 and 3

>>15399362
A formally derived statement is not more or less than a formally derived statement. Gödel doesn't affect this.

>>15399362
By laughing at pseuds who don’t understand Gödel like you. Lmao retard

>>15399403
that was just the first estimate i thought of that maybe there could be data on
>Because men have higher rates of autism in the general population?
the idea being men have a higher average level of autistic traits (like the autism quotient test), so if one area has more males it has a (perhaps only slightly) higher level of autistic traits

If an area being harder (like if students that study it really have higher test scores) also makes it have more males, then control for how hard it is/how high test scores of students studying that are

similar to this study about autism quotient measured for stem professionals, humanities professionals, men, and women etc.
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4619566/

>>15399421
>If an area being harder (like if students that study it really have higher test scores) also makes it have more males, then control for how hard it is/how high test scores of students studying that are
This doesn't do away with the possible selection bias of any woman entering mathematics (needing to) be so autistic that the results skew. We'd need data like in your study on women and men in specific mathematical subfields as well as outside any math-y environment at all, in addition to AQ and general control variables.

>>15399383
turn your hand sideways, then

>>15399507
I don't think that would work, I draw according to where my vision is positioned that would just be painful and awkward.

>>15399179
Thanks. What would you recommend for trigonometry?

The correct answer is in the top right
what am I doing wrong, I was told that my last equality was wrong. In what Way

>>15399418
>mathlet never heard of Gödel's second incompleteness theorem.

>>15399596

Question for the good math fags here:
If you've absorbed all advanced math topics and are on par with the math elite, isn't it in the end just creativity that makes you come up with a solution to a complex problem?

>>15399618
There's no "math elite", i.e. there's nobody who's an expert in 5+ different of the 30++ math subfields. Nobody who publishes successfully in algebraic geometry also publishes successfully in both stochastic differential equations and complexity theory.
Creativity is important, but also knowing people, and choosing your topic, as well as managing other people and, if you want to make anything from it, some politics.

>>15399545
The same guy again, using the trig book in that series. There's a geometry one too.

i guess this is more a physics or chemistry question,
but if you have two moles of hydrogen, and one mole of oxygen, and that takes up some volume,
and then you burn them in a closed system, making water.
is the volume taken up by that 1 mole of water vapor less than the volume taken up by the three moles of gas? or does the heat given off by the reaction increase the pressure and therefore the volume to more than the gaseous reactants?

>>15399871
try the questions/homework general >>15379528

>>15399903
ah fuck cheers yeah i checked the archive but there wasn't one listed with any of the keywords i searched

So there is nothing wrong with studying analysis instead of reading Spivak's calculus right? will I really gain anything from Spivak

>>15399934
If you could go straight into analysis you would not be asking that question here. Just go through Stewart and get the exposure and practice you need. Get a cheap copy of baby rudin and look through it during your basic Calc I and II courses with Stewart.

>>15399939
I've finished calc 1-3, but I just don't feel like I gained ang real insight from it.

>>15399943
After you take an intro to proof writing course you are ready to go into analysis. That's literally it. Yes it will be extremely painful. Try a number theory course or abstract algebra beforehand if you want to make your ass hurt less. It'll feel really easy in hindsight though.

>>15399934
Take a look at Zorich's book. You'll get Calculus, Analysis and some applications to physics.

Is there a minimal cranium size required to excel at Analysis? Zorich looks like a pretty big-brained guy.

Question for the logic/foundational autists since I don't know what keyword to google: A common theme in everyday math is to define a function by showing that there exists a unique thing satisfying some property for every argument and then just considering the function that maps an argument to that unique thing. I don't doubt that this is allowed but what is going on behind the scenes here/what allows us to do this?

Example: Fixing any algebraic structure A, one can show that for every subset X of A there exists a unique smallest subalgebra containing A (the subalgebra generated by X). After that has been established author's then tend to define a map taking subsets of A to that unique subalgebra generated by the subset (and write something like <X>)

>>15399958
Yes, a 25 inch cranial circumference should be fine as the minimum. Any less and you are wasting your time.

>>15399974
Now I need to find a tape ruler, to see if I measure up.

>>15399339
You sound like an American. Amann & Escher is written for European mathematicians. Go read Rudin instead.

>>15399980
Alright, no tape ruler, but I improvised by using a test-lead wire and regular ruler: 23.5" rough, I guess I shouldn't have ever bothered to go into mathematics.... if only someone had told me sooner.

Apparently the average is 22.5"

>>15399988
Most unfortunate.

>>15399966
>what is going on behind the scenes here/what allows us to do this?
Axiom of choice: we're taking the choice function of a family of singleton sets, which you can quickly confirm to be unique (and hence well-defined) using extensionality.
Though if you don't subscribe to the classical + ZFC view, then you might do it slightly differently.

how to earn money out of doing foundations?

>>15399966
>>15400163
If you know the associated thing is unique, then you don't need AC but Replacement is enough. Specifically, replacement implies the axiom of unique choice. I think the only higher order theory where unique choice is not available is secure and order arithmetic with a very restricted function comprehension. And, of course, if in some categories.

>>15400924
secure and = second

>>15400723
live on neetbux and do independent research

My goal is read all of the Princeton Lectures in Mathematics. But it doesn't seem like Zorich prepares you for it since it barely talks about complex numbers. What to read after Zorich?

>>15401174
Why is that your goal?
but you could read Cartan's Complex Variables book once you have a grounding in real analysis

>>15401174
They're introduced in Vol I, but used more in depth in Vol 2. I honestly doubt you'll need more to prepare you, at least based on what I can see on wikipedia.

https://en.wikipedia.org/wiki/Princeton_Lectures_in_Analysis
>The series emphasizes the unity among the branches of analysis and the applicability of analysis to other areas of mathematics.
Kek, >>15399339 won't be pleased, seems super problematically egocentric.

logic and semantics are deeply intertwined.
i'd go as far as to say they're the same field
logos also means speech aswell but that just seems like a coincidence. maybe its not.

>>15401174
>>15401292
They're great; if you have ever taken a first course in analysis (or comparable) you can just start. They're not too advanced but have really nice exposition.

>>15401292
>Kek, >>15399339 (You) # won't be pleased, seems super problematically egocentric.
What the hell are you talking about?

>>15399986
Europeans use Rudin lol. Amann & Escher is not even popular in Germany according to Germanon, and rightfully so, it's awful. Awful exposition and even worse exercises.

>>15401613
>Europeans use Rudin lol.
I don't know anyone in my country (in yurop) who had to read Rudin for an analysis course (and that's a good thing).

>Without complex numbers, real polynomials are not closed
Is there an x,y such that the following applies:
>Without x numbers, complex y are not closed

>>15402545
Without quaternions, equations of the form [math]a^{2} =b^{2} =c^{2}= abc=d \; , \;d \neq 0,1[/math] have no solutions.

>>15402556
Is that the only thing that Quaternions and the entire Cayley-Dickson family can do that complex numbers can't? Create solutions to unit circle problems with arbitrarily many variables? Is that the only point?

I'm about to get my undergrad and I'm struggling to keep a 3.0, am I retarded? Applied math and stats

>>15402637
Protip: Outside of academia, no one cares about your GPA.

Am I a brainlet if I consider stochastic calculus very hard?

>>15402637
>I'm struggling to keep a 3.0, am I retarded?
Of course not.
>Applied math and stats
Never mind. yes you are.

I was curious about a theory i had.
lets say I had a breeding experiment where each pair produce a male and female offspring I then move the male offspring to mate with another female either through rotation or randomised.
I want to calculate the average relatedness of the population for a given population size of S.
I tried calculating it through a recursive algorithm but it was more time consuming than I anticipated, so I wondered if there were a better mathematical solution to estimate it?

>>15402609
https://en.wikipedia.org/wiki/Frobenius_theorem_(real_division_algebras)

Finite-dimensional associative division algebras over the real numbers with 3 independent directions?

>>15402886
Markov

>>15399943
Are you American? Americans get treated like retards through calculus 1-3 and their lectures are filled with solving problems.

Imagine learning calculus at university without being at least exposed to analysis through your lectures and assigned readings.

>>15399943
To answer your question seriously, Spivak's Calculus is essentially an introductory analysis book but the presentation of topics does not mirror how other introductory texts do.

I personally find Zorich's exposition much better but there aren't solutions, unlike Spivak which is more popular.

Ross's Elementary Analysis is nice and concise and has hints for selected exercise.

Spivak has a great problem bank though.

Is every continuous bijective map from [math] \Bbb{R}^n [/math] to itself necessarily a homeomorphism?

>>15403613
More generally, is every continuous injective map [math] \Bbb{R}^n \rightarrow \Bbb{R}^n [/math] necessarily a homeomorphism onto its image?

>>15403613
I think this is the sort of question we should start to first pitch to an AI, double check the keywords on StackExchange, and only then ask in a forum

>>15403613
>>15403620
In fact I also used to answer this question.
>>15402609
>>15403061
I knew some classification of those algebras exists, I just needed some vague pitches to find the proper answer.

The age of human interaction and excellence is over, bros. It's just eternal capitalist flag line from here on. Lock yourself into a room with your sex doll and eat the bugs.

>>15403613
>>15403618
Yes
https://en.wikipedia.org/wiki/Invariance_of_domain

>>15403620
That function it gives is not even injective. Stupid bot still has a long way to go

>>15403636
Perfect thanks

>>15403620
The AI was wrong and useless, thanks lmao

>>15403646
I don't think it's useless - you can use it jointly with e.g. a google search, fishing for keywords.

Can there be a topological field which is connected but not path-connected?

>>15399934
Spivak is a waste of time. I have no idea who that book is for. Just jump and to Abbott then Rudin, and don't listen to anyone who recommends Zorich or Escher, they have awful exercises. However, if you are interested, they may be worth reading the material since they treat integrals, vector fields and all that physics stuff in depth, but at that point you are better off reading a book specially on those topics. You may also read Escher if you are interest in construction autism, but otherwise stay away.

is there something like a (math) textbook search engine? the libgen search is shit

I have concluded that I hate the modulo operation.
It makes perfect sense, but I hate that it looks like there should be a local maxima, and there just fucking isn't.

>>15403664
Spivak, Escher and Zorich are all awful? Strange, considering that each of these books has developed a consensus following. Sounds like a you problem, and many, many skill issues.

>>15402854
Ha, I literally just wanted a job after college. Stats is interesting sometimes but awfully hand-wavy.

>>15404131
Consensus following by people who have never read the book

>>15403439
I appreciate the suggestion

>>15401174
Are these worth reading from a European perspective? Consider the paucity of quality American mathematics education, even being for Princeton I have to wonder if there are better European alternatives. I mean no disrespect to the authors, they seem to be non-burger, but if their target audience were poorly prepared amerifats who had to be drilled easy problems for 3 years of basic calculus (LOL), then it's probably not worth reading. If a say a student at ETH Zurich had all 3 volumes of Amann/Escher, would they profit from the Princeton Lectures?

>>15404145
Zorich is used not only in Russia, but Germany and all over Asia as well. Millions of people can testify to having used it. The first version of Zorich came out in 1980, and still it is actively used. Yeah, I guess no one has ever read it. Please, stop being obtuse. Personal preference is one thing, but you're writing nonsense.

>>15399261
We should start calling injective maps subjective to fuck with normies even more.

>>15404165
Noooo, it's ONE-to-ONE and ONTO! Stop using odd words.

To follow up on the other anon's question, is there a bijective continuous map from R^q to R^z for z !=q ?

>>15404172
No, this can be proved with e.g. homology.
Also, why the letters z and q ?

>>15404161
Rudin is the book that is recommended in every single english speaking country, not just America, and it has been since it was published, for a good reason. Both Zorich and Escher are barely recommended in the English world, don't lie to yourself. The alternatives usually are Burkill or Bartle. Literally open any chapter of Rudin, and any of the other, and compare the exercises. Only one book (guess which) has insightful exercises, the other has basically glorified riddles. Go ahead and see for yourself. The only other book that comes close to Rudin's exercise is Abbott but that is barely an Analysis book.

It makes sense though. All Europeans and Asians do is waste time solving millions of riddles in their millions of mandatory reading and boasting about how hard their courses are, like you already have one here >>15404157, while Americans and I guess Japanese too, already pick a super specific specialisation by the end of their third year and begin publishing on it. Just look at either of those books, several chapters on things which 99% of readers won't use. The brevity of Rudin is another reason why it is far superior.

Also you are getting mad about something I did not even say, so learn to read.

>>15404229
Oh I've ran into you before. You're the guy that thinks you should start publishing after one year of undergrad education, right? I remember you complaining that Amann/Escher is bad because it's an "old fashioned" book for people in 4 year long undergraduate programs.

Personally I think Zorich is the best option for most (serious) people, since it supplies motivation and more engaging exposition than a dry text like Rudin. Amann/Escher is very nice to review, but it would be taxing for a student unprepared to encounter algebra. I have no problem with Amann's exercises. The exercises in Zorich aren't bad either, it's just that they assume more of the student than the exposition implies. You can remedy this with a separate problem book.

New thread please

>>15404165
Just bite the bullet and take a page from category theory. Switch from injective and surjective to monic and epic, respectively.
yes yes I know epic functions are not actually quite generalisations of surjectivity shut up
>>15404172
No. See also Netto's theorem.

>>15404612
I gotchu

New thread!
>>15404877

>>15404229
You sound like someone who had to sit through three semesters of Khan Academy exercises dressed up as Calculus 1-3.

Being so opinionated about which books people should use for their first 1-2 courses in analysis is dumb. Different students and readers will click more or less with different authors and volumes.

Any interested student should be exposed to books used in different education systems because they have different pedagogical goals.

>It makes sense though. All Europeans and Asians do is waste time solving millions of riddles in their millions of mandatory reading and boasting about how hard their courses are, like you already have one here >>15404157, while Americans and I guess Japanese too, already pick a super specific specialisation by the end of their third year and begin publishing on it.

What? Americans are the ones that spend three semesters in chum calculus before learning what calculus really is. Europeans don't differentiate between analysis and calculus at the undergraduate level; their calculus classes are analysis, and their first year analysis sequence would probably kill the average American math major.

>Rudin is the book that is recommended in every single english speaking country

Rudin is common but this isn't the case. If you insist on "every", your statement is false.

At Australian universities with real math departments, lecturers (usually Europeans who "wasted time solving millions of riddles" i.e. learning analysis and linear algebra properly in their first two semesters of university) usually have bespoke lecture notes that act as a textbook. When a primary reading is assigned, it is usually the lecturer's notes, or a relatively unknown analysis book that they have chosen for a specific reason. However, university is for adults, so any class will have a number of secondary readings: here you might find Spivak, Rudin, Tao, Zorich, and some lesser known texts, and you're expected to use them as you need them.