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/math/ - Mathematics

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25 Dec 2021Mathchan is launched into public
Catalog Archive

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  No. 900
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I'm sorry if this belongs rather to the /phil/ section. I just want to discuss this topic and I guess, it has to do with math anyway.

Some groups like the "New Atheits" and/or the so called "Intuitionists" (and the late Wittgenstein, too) think that you can make the following assumption:
"p is true" = "There is a proof for p [within a formal system]".

If it is impossible to provide a proof for p in principle, then you can claim that p is neither true nor false. Its just "undecidable".

My question is:
After the results of Tarski and Kurt Gödel, can we really still hold this assumption?

We see that "p is true" must be something different from "there is a proof for p" as there are some true statement of which the proof doesn't exist. We cannot define "truth" within a formal system and need some relation to something without the system itself. A semantic model.
In the light of this insights, the position that identified true with provability makes a lot less sense to me and many others.

Anyway, as far as I read, the critique of the inuitionists on the rule of double negation and the law of excluded middle relais a lot on the hidden premisse that "truth" and "can be proofen" is the same.
So, there viewpoint seems much less plausible anymore.

What do you think, anon?
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  No. 961
I agree with you, OP.

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  No. 960
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What do you think, anon?
Made up Bullshit or genial insight?
Usufull or not?

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  No. 937
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Are there any other people here who researched/studied zeta regularization (products)?

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  No. 935
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>>936
I'm interested in making a game with simple 3D graphics (only consisting of simple shapes like cubes and pyramids) without using a 3D engine but I don't know the name of the specific math/geometry stuff I need to study in order to do that, can anyone orient me?
>t. don't even remember high school math and only got the captcha right by chance, but i'm good at learning things
(Please refrain from "just use an engine"-type replies)
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  No. 936

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>>935
Linear algebra as well as analysis if you plan to have physics or complicated motion.

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  No. 925
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>>931 >>933
Topics which are not in high school syllabus but might help in JEE Maths or Physics questions.

Book recommendations or advice for JEE Maths or Physics
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  No. 931 >>933
>>925
http://sheafification.com/the-fast-track/
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  No. 933
>>925
>>931
teri-ma-ka-rape.com

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  No. 351
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>>376 >>490 >>492 >>929
(please move to /adv/ if this belongs there)

I started with category theory a week ago with Lane's book. It's a bit hard and there are some examples (like lie groups) that I don't quite get. Related to this I've read Fraleigh's book on abstract algebra. Is this enough background or should I wait a bit before getting into category theory?

Thanks
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  No. 571 >>573

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>>570
Due to the way things end up being I'd say it's not an unfair analogy to draw. However primarily I'd say they're a priori just two different languages for describing maths.
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  No. 573 >>575
>>568
>Not >565 but category theory comes in handy to formulate many concepts, it's a useful language to have.

And why is it better than set theory, HOL or mereology?
(Or even some kind of formal ontology in the informatics)

As far as I understand, you could easly define a predicate "x is isomorph to y" := Iso(y,x) and work with this.
Okay, I see the advantage of a graphical picture.

What I have see in categories looks suspicies like a usual powerset.

Or I'm just to simple-minded to get it at all?

>>571
Okay.

As far as I see, the central relation in set theory is the "being part of", maybe in the HOL is more "impled".
What is the categories about?
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  No. 575
>>573
First of all I wouldn't say it's "better" than set theory or higher order logic.
Second of all you can't just easily define a predicate "is isomorphic to", well you can, but it's useless and you're missing the point. You want to define it in a way for it to have the desirable properties and if you just define a predicate, then you have to figure out how to piece it into the rest of the theory.
Third of all higher order logic is fundamentally different from set theory or category theory, as it's part of the deductive system of your theory, not part of the actual things you talk about. So indeed I'd say yes you're being too simple minded/delusional to get it right now.

As to what category theory is "about" is I'd say morphisms, the same way set theory is abour membership, since morphisms are what ought to be defined to speak of a category and draw the beloved arrows.
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  No. 928

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So did you end up doing category theory?
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  No. 929
>>351
you lack basically all motivation. read bott tu and brown's topology first. and then vaisman's book on cohomology as an intro to cats

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  No. 863
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>>864 >>866 >>918
Hello everyone. I have seen a lot of threads(book threads, IMO advice...) that seem to exist solely so people give advice and give comments on your study material.
So I had an idea for a thread where we: state our goals, name the books(video series and other material) we plan to cover in a certain order, give an approximate deadline and other mathchan users can give suggestions, meaningful comments, advice and pointers to others!
I'll start:
My goal is to develop enough knowledge before I go to university where I hope to take advanced classes and spend more time on advanced topics while simultaneously getting myself ready for the national math competition(not my main focus, just for fun)
《Elementary math 1 i 2 notes》
-notes from my local university where they want to "bridge the gap between high-school and university math" (introductory logic, set theory, relations, functions, number theory, Euclidian geometry, vector spaces and analytic geometry)

《A Transition to Advanced Mathematics by Smith, Eggen, and St. Andre /// 101 problems in algebra》
-covers similair material as the already meantioned notes but in more detail

--------------I am here----------------

《Analysis I notes》
-notes from the local university, this time in calculus/analysis (what European universities call analysis is almost always just rigorous calculus or very elementary analysis since we cover calculus in HS)

《Linear Algebra Shilov》

《Šime Ungar, Analiza 3》
-same deal as before, rigorous calculus/ begginer analysis book for R^n

《101 problems in various topics, Andreescu books, generatingfunctionology, Problem solving strategies by Engels》

《Elements of Set Theory by Enderton》

《Introduction to Logic: and to the Methodology of Deductive Sciences by Alfred Tarski》

《An Introduction to the Theory of Numbers by Niven, Zuckerman, and Montgomery》
-mainly for competitive math but also as an introductory text

《The USSR olympiad problem book》

《Algebra by Artin》

《The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities by Steele》
-half for competitive maths, half since I heard analysis problems demand a lot of inequality knowledge

《Amann and Escher Analysis series》
-all three books

I hope to cover everything stated above in a 1.5-2 years. I covered the things above the "I'm here linePost too long. Click here to view the full text.
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  No. 864 >>901
>>863
Amann Escher do both analysis and calculus, so putting it at the end of your list isn't really ideal imo. Shilov has no prerequisites either. I'd recommend reading both earlier/now instead. The first chapter of Amann Escher is also a nice introduction to proof-based math in general.
For algebra, I read Gaitsgory's Math122 and Math123 notes (both are on AMS) combined with Lang's Algebra (didn't complete it though). Gaitsgory's notes on Math122 can also be used as a somewhat abstract course in linear algebra. I think he actually based his Math55a course on them, of which you can also find notes online. I also read the first few chapters of Gorodentsev but can't say much about it as I never finished it.
For Topology, I really like Brown's book "Topology and Groupoids". It has few prerequisites (if any) and is very interesting as it emphasizes applications in differential topology, algebraic topology and algebraic geometry (for instance, it introduces the Zariski topology early on in an exercise) whereas e.g. Munkres is far too focused on point-set topology, which you'll find in analysis books like Amann Escher anyway.
If you want a book that treats the differential geometry stuff in amann escher 3 in more detail, I recommend Sternberg's lectures on differential geometry. Kobayashi-Nomizu is the go-to book if you want to specialize in it (note that it'd most likely be a very difficult first read).
If you're interested in geometry I also highly recommend learning physics.
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  No. 866 >>901
>>863
i intend, wholely and fully, to have a good time
do tell if any of those problems are fun, i would be fine to check them out
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  No. 901 >>920
>>864
As you said, I am probably going to cover Amann and Escher quite a bit sooner. I think people scared me into thinking that analysis is more demanding than it probably is.
I also thought that set theory and logic are more basic in some sense than they are. But it seems to me like you only need basic logic (basic propositions, quantificators and so on) and just the very basics of set theory (sets, operations on sets, partitions and such) to cover math to a decently advanced level.

Thank you for the other recommendations as well.

>>866
101 problems in algebra I thought was okay. The difficulty of the problems differs quite a bit since I solved some in minutes and others left me confused even after checking the solutions.
Some of the explanations I didn't like at all. They don't explain the thought process and sometimes just begin the solution like this:

Since (critical part of the solution that greatly simplifies the problem) holds. We know that x is equivalent to y. Therefore z and therefore the statement is proved.
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  No. 918
>>863
A gentle intro will not be very beneficial and will waste time. The reason analysis is hard is because the ideas are presented in a way that is isolated and general. The same will go for algebra and topology. The books on those subjects should be read, but not be the center of your study. They are there for reference when needed.
I studied analysis and ignored everything else and it failed me. I then changed my approach, got a book on differential geometry (lee) and began to read that, referencing books on analysis, algebra, geometry and topology when it was necessary. Math is a slow process and the time you put in will significantly outweigh the knowledge you gain.
Also a little side note, get into complex analysis asap. Both (real and complex) deal with the same ideas, just complex analysis is more tangible due to its geometric nature. I recommend Ahlfors book.
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  No. 920
>>901
Logic and model theory are a bit weird in the sense that they're mostly disconnected from the rest of the math world despite seeming fundamental. Of course I am neither a logician or set theorist so my view may be biased but it is interesting how both camps often know very little about the other side. And then there's stuff like algebraic geometry in its formulation by grothendieck that seemingly required inventing/appealing grothendieck universes just to define some things rigorously. Turns out it wasn't necessary (see the stacks project, there's extensive discussion around size issues).

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  No. 688
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>>689 >>694 >>756 >>848 >>870
Please name all the books in picrel. Also review the curriculum shown. Are the books shown in the picrel enough to cover the mentioned topics. If so how long would it take.
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  No. 887 >>890
>>881
no dude the key is to just start as obvious as it may sound. If you want to do something, do it now.
I used to think I'd have no time to read during uni but that's all nonsense. Once I started doing my assignments immediately I had enough time to read through 4 books in one semester and that conversely helped me become really fucking good at math to the point of me going to far less lectures
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  No. 890 >>891
>>887
Did you do the exercises? Or only read those books. If yes then what percent of exercises like 50% or 80% or each and every one(skipping the trivial easy ones of course).
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  No. 891 >>892
>>890
Do all of them. When you get to my point in life where I've read one undergrad curriculum's worth of books, I can't remember what I need to any more. Burn it into your memory, because you can only ascend as strong as your foundation.
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  No. 892 >>919

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>>891
Can you specify which were those 4 books
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  No. 919
>>892
Federer. Geometric Measure Theory
Kashiwara. Sheaves on Manifolds
Harthshorne. Algebraic Geometry
Hotta. D-modules, Perverse Sheaves, and Representation Theory

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  No. 107
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>>169
Talk maths.
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  No. 563 >>657
>>132
Area as in its Hausdorff dimension dimensional Hausdorff measure? Or Lebesgue measure of its convex hull?
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  No. 566
>>112
same
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  No. 657 >>658

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>>132
>>563
Wouldn't it be an improper integral over the complex plane?
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  No. 658
>>657
https://mathoverflow.net/questions/453862/is-the-area-of-the-mandelbrot-set-known
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  No. 917
Rediscovered something cute

https://files.catbox.moe/l6ft4m.mp4

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  No. 91
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>>185 >>701 >>889
You can solve this, right?
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  No. 185
>>91
Yes. What's the time limit?
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  No. 700 >>703
Where is this an Enterance Exam to? I do not think I could do it. My undergraduate univeristy is all multiple choice and so I am nearly finished it having barely written a proof. It's awful.
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  No. 701
>>91
sneed
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  No. 703
>>700
I somewhat doubt it is an actual entrance exam. It reminds me of 1st year graduate course of mathematical methods in theoretical physics given that it has a bunch of maths without too much coherence.

Also, I think the ban on electronic calculators is quite futile here.
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  No. 889
>>91
the ideas here are easy but i don't know what C-convergence means
can someone link a book on the topic?

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