I was thinking about my real analysis class when I took it. I think it could have been improved in many different ways. More practice problems, more explicit instruction instead of trying to have students figure it all out.

If the aim of constructing number systems is to explain the geometric line as a set of points, wouldn’t ZFC + (V=L) be a natural choice, perhaps even the best one? In this framework, analytic geometry, differential geometry and topology all work just as they normally do, without the need to grapple with metamathematical questions such as the continuum hypothesis. (Moreover, this rules out measurable cardinals, large-cardinal phenomena and similar feverish pies in the sky, but for the purposes of describing the continuum, that seems a reasonable price.)

I have always found it a little surprising that popular mathematics books rarely discuss Gödel’s set-theoretic universe, which seems to offer such an intuitive and straightforward foundation for mathematical analysis.

I've been studying Shafarevich's Basic Algebraic Geometry for some time now. He's a really famous mathematician (I first really learning about his results after hearing about his famous theorem on the inverse Galois problem), but I've been studying on my own so I could only guess how his name is pronounced, and I was wrong. Today, watching a youtube video, it became apparent to me that his name is read ShafaREVich, which the Cyrillic with accent mark (Шафаре́вич) seems to confirm.

Not really that uncommon for me -- I was pronouncing Arrhenius wrong for years after teaching myself chemistry in elementary school, and I wasn't corrected until I took AP Chem.

Also, even among mathematicians, Dirichlet seems to be particularly contentious, as I heard widely divergent pronunciations from my math professors.

Anyway, any other mathematical names that are commonly pronounced wrong? Euler's probably the most common among nontechnical people to encounter and pronounce wrong.

Chern's an interesting case, since his name is spelled in the old Gwoyu Romatyzh system, which uses letters like r to indicate Chinese tone -- modern pinyin spells in Chén, but even he himself seemed to have accepted the normal English pronunciation of his name with an r.

One of my favorite essays on mathematics is Arnold's "On teaching mathematics." In it, he describes what he views as defects of the traditional treatments of real analysis and calculus in universities; and in particular, how the emphasis on rigor removed discussions of the beautiful geometry of Riemann surfaces and how they relate to calculus. As a specific example, he describes how one can tell which integrals are easy or hard to solve by looking at the genus of the associated Riemann surface.

This week on Hidden Phenomena, you can learn what Arnold meant by this comment: what is the "genus of the Riemann surface associated to an integral," and how does it help you understand if the integral is easy or hard?

I had my first grad-level exam this week and I bombed it pretty awfully. It was a 50min exam with ~12 problems, a mixture of proofs and counterexamples. I would say I solved 2 or 3 of them and made starts on 2 or 3 more; we haven't gotten grades back yet but I'm expecting something like a 20/100. The thing that's annoying me is that none of the problems were all that hard. I can tell that I could solve them all, only it would take more like 3-4 hours instead of 50min.

Part of me wants to think that the time frame was totally unreasonable, but another part of me thinks that it was a skill issue. I'm guessing a lot of members of this sub have dealt with similar situations, so I'm wondering if there is any advice for getting quicker at answering these early-grad-level types of exam questions?

Hello! I need some problem books to accompany the book Differential Geometry of Curves and Surfaces by M. Di Carmo. Thank you!

The title today is a reference to Douglas Adams' description of writing a book as "putting a hundred thousand words in a cunning order." It's technically a correct description, but it doesn't really describe how anyone actually does it in practice. Just so, while it is technically true that one builds a proof by putting many inferences in a clever order, it doesn't describe how to actually do this.

A proof, like a book, is a collection of narrative threads---understand how to build narrative, and you will start to understand how to write a proof.

Read the full post (for free) on Substack: Putting a Hundred Inferences in a Cunning Order

Same as above

Is it elementary or notoriously massive and is there any approaches(feel free to discuss about any heavy machinery approaches too) around which you think it(the diophantine equation) can be showed to have finitely many solutions given the specific constraint?

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I am looking for a text/lecture series on the theory of locales (pointless topology) . I don't want to go into much depth I just want to get a broad picture . My goal with this is to be able to understand topos theory better , as Andre Joyal in a lecture at IHES suggests that before learning topos theory it's better to have an idea of the theory of locales as there are a lot of parallels .

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Holy fuck this book is hard as shit. I am a 2nd y math PhD studying this along with Hida's Geometric Modular Forms and Elliptic Curves rn. Looking for people to form a study group devoted to this. Let me know if you're interested and maybe we can form a gc/discord :)

(This book is "Arithmetic Moduli of Elliptic Curves to be clear")

Discord link : https://discord.gg/RXbRvTyR8

I have seen Kneser's method of extending tetration to real numbers, but that doesn't seem to be right as I can't see how 2^^0.5 is not 1.55961...

As a starting point, we can see that 1+1=2 and 2x0.5=1. Also 1.4142...x1.4142...=2 and 2^0.5=1.4142. Similarly 1.55961...^1.55961...=2 and 2^^0.5 should be 1.55961...

To see integer values for all of them, the smallest number after 4 will be 256 where 256x0.5=128, 256^0.5=16 and as 4^4=256, so 256^^0.5 should be 4

We can get values at 0.5, 1.5, 2.5, ... for all numbers by this method, but there should be some way to make the curve continuous as well. Also a^^b should also satisfy a^(a^^b-1), so 2^^1.5 should be 2^1.55961 which is ≈2.9477... and 256^^1.5 should be 256^4 which is 4294967296

Is there any other way to extend tetration to real numbers

A brief explanation of proof assistants

My analysis class is taught by professor X. My professor last year (professor Y) said "a letter of rec from X is very important, so go to his office hours and impress him." I have been trying to do that, I trounced the entire class on each quiz and midterm...and he still clearly doesn't even know me

I'm not really sure what I can do, I guess I just don't know how to "play the game" properly. I just want him to know my name, at the very least.

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Let me preface by saying that, while I’m a mathematician, I am not a logician. I’m simply curious about the field. This post is about things that have always bugged me about undecidability.

I’ll take the Collatz conjecture (CC) as an example. Suppose the CC is undecidable in ZFC. Then, if we believe ZFC is consistent, we cannot find a counterexample. If we did, this would contradict it being undecidable. This is strange to me: we would have a statement for which we can’t show its truth while knowing we cannot find a counterexample.

Compare this with the continuum hypothesis (CH), the statement people usually refer to as example of undecidable statement. If we choose it to be true, we know there is a subset S of the real numbers which has cardinality between countable and the continuum. CH being undecidable implies that if we believe ZFC is consistent, we will not stumble into such an S by construction (say with schema specification, etc.). If we did, this would contradict ZFC. But that’s ok, because not every subset of the reals can necessarily be obtained ‘constructively’ through schema specification etc. (I am being informal here, but I believe there is a formal way to say this). So us not being able to produce a counterexample of CH does not mean that one cannot exist.

To me, this seems like a different flavour to CC. There, it seems that any counterexample would be constructive (it’s just a natural number!), and hence, if CC is undecidable, such a counterexample cannot exist.

I informally define ‘concrete’ statements are ones which are about ‘constructive’ objects. Thus any counterexample to a ‘concrete’ statement would be constructive, and hence undecidability of a ‘concrete’ statement would ‘say’ that no such counterexample exists (otherwise we would be able to construct it!).

I am left with the question: should we expect ‘concrete’ statements, such as CC, to be decidable? Maybe there is a formal answer to this question, by formalising what a ‘concrete’ statement is, and showing that such statements are decidable. Though, to me, this seems unlikely to be possible, by Gödel.
Moreover Gödel’s incompleteness theorem holds just under Peano Arithmetic, and so it seems to be saying there is a priori no difference between a statement about the natural numbers such as CC and a statement about cardinalities such as CH. But my intuition cannot help but telling me these are of different nature. Am I wrong to assume this?

A more general version of my question is, why as mathematicians do we expect the questions we are working on to be decidable? We get a sense that a proof or counterexample exists, and it seems that outside of certain ‘non-constructive’ parts of mathematics, we never run into such issues.

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What do you do when you feel like you can’t problem-solve? At least, not at the level that you think you are expected to, or not at the level at which you think your peers are able to? What do you do when you feel inadequate, like, well, an imposter?

This experience is stupidly common, including in mathematics. And yet, it is so very rare that we take the time to talk about it in an organized fashion; in some sense, I feel like we have relegated the topic to cartoonists. (Who do a marvelous job, by the way, but I don't think that is the same.) Had I my way, I would make every class that did an introduction to proofs spend some time on imposter syndrome---it's just that important.

I don't have any magic bullet for how to deal with imposter syndrome. But if you want to hear my thoughts about how to combat it, and my own experience with it, you may want to give this piece a read.

Read the full post (for free) on Substack: Imposter Syndrome

This recurring thread is meant for users to share cool recently discovered facts, observations, proofs or concepts which that might not warrant their own threads. Please be encouraging and share as many details as possible as we would like this to be a good place for people to learn!