I am not anything resembling a mathematical expert. And I'm not sure if this is best asked here or maybe in a general science subreddit because I'm not sure it's really a question of math at all.

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I was watching this video recently, and as I watched it all I could think of was the 23 enigma. Or the golden ratio.

https://www.youtube.com/watch?v=d6iQrh2TK98

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It just seems like people drawing references that don't actually implicate the number itself as important. Just the human perception of what makes things important.

A prime (heh) example is in the starting premise.

"Pick a random number between 1-100. Oh look, so many people pick 37."

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But it begs the question why right? Why 1-100? Why not 1-10, why not 1-1000, why not 10-1000, why not 200-300? Why do we think randomness is found between 1-100?

Humans aren't computers, just because you ask for a random result doesn't mean they give you one. Might as well ask for a random sports team in the state of Texas and draw significance from how the animal mascot of that sports team has been used by humans across the world for one reason or another.

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And we see this in the data. When asked for the 1-100 both on the street and on reddit, 37 was not the most picked. It was a popular number, but Derek keeps having to tweak the data to make 37 truly prominent. Says he's removing "non-random numbers" like the top and the bottom. Dropping 7 because it's one-digit and apparently is no good as a result? Or 42 and 69.

People responded with memes--because humans don't really know how to generate a truly random result. Derek basically copying the magician in the video (Oh pick a random number . . . but it has to be less than 50, with two digits, and they both have to be dissimilar, odd numbers.) I don't think there was ever any randomness to be found.

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37 does have unique properties. But the same can be said of many numbers.

7, 77, and 73 were all more popular than 37, why isn't he talking about them?

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There's this section where he proposes that 37 is a practical number for humans because ~37% is the solution to the secretary problem. Except that 37% is not really a great result to begin with and when studied people who are in secretary problems don't usually abide by the 37% rule (https://pubsonline.informs.org/doi/abs/10.1287/mnsc.2014.1902) Nevermind that the secretary problem supposes a reality where optimal decision making is always on the table.

Also, that ~ doing a lot of work since 1/e is .367879441171. Not 37.

The theory was not that we pick numbers around 37. We pick 37. (Except, again, the data shows we don't.)

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At best this seems like a question of psychology to me, mostly why a particular set of numbers are considered "random." But that's not how it's approached. It just seems like disparate components just strung together.

I'd be interested if someone can enlighten me to what I'm missing about this. If this were just a video about "some interesting things about 37" I think I'd be on board. Primes feel kind of random? I can kind of follow. But, it's all around us because it has some deep, real-world importance that other numbers don't have? I'm not seeing that.

I just finished vector calculus and obviously, we have line integrals ∫ F • dr and surface integrals ∫∫ F • dS but why is there not a integral like ∫ F × dr or ∫∫ F × dS?

Source:

https://youtu.be/5M2RWtD4EzI?si=5eVXbdDPcJluPFZG

I've been thinking constantly and working towards finding a complete solution to my former Master's thesis topic, but I've been finding that the problem may just be insanely complicated to the point where I don't think I'll find a solution any time soon. I'm starting to think that it might just be better to leave the problem alone and move on with my life, to focus on more important matters.

Has anyone ever felt this way? How would you get yourself to distance yourself from a problem that might just not be worth your time?

As Professor Mike Stillman said about the Hagoromo chalk,  “one of the best-kept secrets of the math world.” I would like to know if there is something similar when it comes to pen? I would love also to ask you please (By you I mean every single person reading this psot) if you have any good or remarkble experience with any kind of pen and/or paper when doing math. From my personal experience, my flow can be affected by how the pen and paper are good or bad.

Finally, Do you know about any other kept secrets in the math world?
Thank you very much for reading and for sharing your thoughts.

I'd love to learn more about the mathematics of tilings but I don't know where to start. I can find several textbooks mentioned online, such as "Tilings and Patterns - Branko Grünbaum and G. C. Shephard" and "The Tiling Book: An Introduction to the Mathematical Theory of Tilings - Colin Adams".

What books/other resources would you recommend, and what approach do they take to the topics at hand?

I'm an undergrad math student currently interested in algebraic topology and a bit of differential geometry. Lately, I've been curious about abstract harmonic analysis and Kolmogorov complexity and computer science related theories too, ofc category theory itself. I've noticed that many of my peers are heavily into algebraic geometry, even though the fundamental AG courses offered by uni are at in grad level, they still have taken the time to self-study this field, also continuously advocating its central role in mathematics, suggesting me to start my AG journey as soon as possible. They often mention how it intersects with various branches of math, hinting at its fundamental importance.

Although I see algebraic geometry as a potentially valuable tool for future research, I'm still on the fence about diving into it. I guess I'm looking for more concrete motivations or applications that resonate with my current interests and can be grasped at the undergraduate level.

Could anyone share examples or fundamental reasons why studying AG might be beneficial for someone like me, whose primary focus has been on topology and analysis? I'm particularly interested in how AG concepts might connect to the areas I am studying or planning to explore.

I was hoping to gather the collective wisdom of this sub in finding "unconventional" math texts. The kind I have in mind is something like Visual Complex Analysis by Tristan Needham, Finite Dimensional Vector Spaces by Halmos or Fourier Analysis by T. W. Körner.

I am trying to integrate Chat GPT in my studies but sometimes it starts to hallucinate which made me question this approach, what do you people use in your work/learning and in which ways you make sure to have better answers and not just garbage?

I just deciding rn to buy this book or not. First chapters kinda slow, but last ones look promising. Does it worth to buy it?

hi r/math,

i'm a sophomore in high school (15-16 years old for non-americans) who has a deep passion for mathematics. i genuinely live and breathe math. while i do have a few friends who also enjoy the subject, i feel like their interest doesn't quite match my own. for them, it seems to be more about being good at math in school, whereas i actively engage in self-study because i truly love the subject.

i recently applied to a summer program called promys, with decisions coming out on the 30th. i'm hoping that this could be a great opportunity to meet like-minded individuals.

in the meantime, i was wondering if anyone here knows of any resources or ways for me to connect with other math people around my age? any suggestions would be greatly appreciated.

thank you for taking the time to read my post!

This might be a silly question, and moderators will forgive me if this is off topic, but l'm interested in being a quant after I get my master's degree, and l've recently been watching a lot of Jane Street/Citadel job interviews that involve logic-based questioning and so on. I was curious to know if you guys do anything in your spare time to keep your brain elastic and active that also helps in your career in developing logic-based skills. I feel like, as most in my generation, as much as I want to be a quant, l'm slowly burning my dopamine receptors and, similarly, reducing my logic-based skills through excessive use of social media (mostly doom scrolling lol) and so on. I've gotten into coding games, suduko, online chess, reading, etc. (typical "brain games"), but I just thought it'd be best to ask some of you!

I'm looking to learn first learn about homomorphic encryption broadly and then focus on fully homorphic encryption. For context, I'm familiar with computational complexity theory and I have an undergrad in maths (although I didn't do much abstract algebra). I'd appreciate you sharing useful resources/recommendations/insights.

This is a very basic question, Is there any specfiic way or approach to read a math book? For example, if I'm studying Calculas, is there any tips or tricks that you would give on how to tackle the book?

This might seem like a dumb question, but i feel like i need to get the basics strong and check if my approach in going through a book right or wrong.

Info: 17M preparing for math and physics olympiad

Is there any website of sorts which gives us a random shape or structure and then challanges us to replicate it in graphs? Sounds like a competitive game, as everyone would have a different approach.

As a student, I usually preferred algebra or discrete math courses over analysis courses. The reason was that the homework and exam problems for algebra often were very simple and easy to write after you had the aha moment. Whereas a lot of analysis problems were more tedious, tracking epsilons and using triangle/minkowski/holder inequalities , dominated convergence theorem, etc over and over again.

However, from the perspective of research, I actually enjoyed working with analysis more and have more of an appreciation for analytic number theory which I used to think was quite dry. It's nice to be able to more consistently make progress even if some of the work is tedious.

I still enjoy elegant proofs, but looking for a nice proof for a homework problem when you know one probably exists is different from working on an open problem that may or may not have a solution at all. With analysis, you can work and work and gradually make progress whereas with algebra, sometimes it's more stop and start. The breakthroughs are awesome and you make huge progress all at once, but then the rest of the time you're banging your head without much to show for it. (these are all huge generalizations of course).

Hello,
I will be starting my Applied math PhD this fall and will have to give a PhD qualifying exam. I graduated from my bachelors last year and because it was during COVID I wasn't able to get the depth of knowledge I need for a PhD. Now when I saw my PhD qualifying exam it was daunting I could do the problems but not rigorously. I need to practice problems and a lot of them. So what would you recommend for revisiting these courses? First course in ODEs, Calc - I, II (I can do all of the problems here and just need a reference book), Vector Calc ( I need a good book here as my instructor didn't teach us Stokes theorem) and Linear Algebra.

From my research I found the following :
ODEs - Arnold (for the theory), Boyce ( for problems )
Linear Algebra - Sheldon Axler, Linear algebra done right ( I love this book)

Calc - III - Openstax ( lots of problems )

I would also like some recommendations on the following courses as I never took them but my PhD expects me to know this:
PDEs : My bachelors supervisor recommended this: Partial Differential Equations - Google Books

Second ODE Course : couldn't find one

I am open to any other advice regarding the PhD and any additional topics you recommend I should know :).

Thanks

Edit - typo

This recurring thread will be for general discussion on whatever math-related topics you have been or will be working on this week. This can be anything, including:

• math-related arts and crafts,
• what you've been learning in class,
• books/papers you're reading,
• preparing for a conference,
• giving a talk.

All types and levels of mathematics are welcomed!

If you are asking for advice on choosing classes or career prospects, please go to the most recent Career & Education Questions thread.

Hello r/math,

I wanted to share "NSO and GSSOTC: A Two-Pager for the Logician," authored by Ohad Asor. The work dives into two sophisticated logical frameworks: Nullary Second-Order Logic (NSO) and Guarded Successor Second-Order Time Compatibility (GSSOTC). These frameworks aim to address classic limitations in logic, like Tarski's "Undefinability of Truth," and extend the capabilities of logic systems in handling self-referential and temporal statements.

Here's a brief outline of the key ideas:

1. NSO: This framework abstracts sentences into Boolean algebra elements, avoiding direct syntax access, thus sidestepping issues highlighted by Tarski. It enables a language to speak about itself in a consistent and decidable manner, leveraging the properties of Lindenbaum-Tarski algebra.

2. GSSOTC: This extends logic to support sequences where any two consecutive elements meet a specified condition. It is useful in software specifications and AI safety to ensure outputs are temporally compatible with inputs without future dependencies.

The document further delves into the interactions between these systems and their implications for theoretical computer science and logic.

The full text provides a detailed theoretical foundation and potential applications, especially in areas like AI safety and software development.

https://tau.net/NSO-and-GSSOTC-A-Two-Pager-for-the-Logician.pdf

Looking forward to your thoughts and discussions!

Hello, I am a Korean American who made the unfortunate choice of attending a Korean university not as an international but regular student, so all my courses are in Korean. I've had difficulty in Calculus and statistics (Linear Algebra seems easier). But the problem is that right now in Intro to Statistics, I am having difficulty doing some of the word problems. If I had to guess it's probably from the fact that I can't actually use the Stats textbook because it takes me too long to read, especially the problems.

Has anyone have similar experiences with math in a foreign language?

And are there any English textbooks to recommend? I'm assuming most textbooks in Stats, regardless of language, follow the same lesson format?

I did fine in HS with an intro to Stats course. I was wondering just where my problems are lying.

New Foundations is a set theory created by Quine in 1937. It has a very simple set of axioms that say that two sets are equal if they have the same elements, and that for any stratifiable formula ϕ the set {x|ϕ(x)} exists. Russell's paradox is avoided by the notion of stratifiable formulas, which are those in which each variable can be labelled with a number in such a way that 'x = y' only appears when x and y have the same number, and 'x∈y' only appears when the number on y is one more than the number on x. So the axioms do not give you the set {x|¬x∈x}, because '¬x∈x' is not stratifiable.

It has long been an open problem to prove whether or not NF is a consistent theory. (Gödel's incompleteness theorems do not produce an obstacle to using ZFC to prove the consistency of NF, since ZFC would not thereby prove it's own consistency. It was correctly suspected that NF had a weaker consistency strength than ZFC.)

Since 2010, Randall Holmes has suspected he has a proof that NF is indeed consistent. But the claimed proof is so convoluted that nobody has been sure whether or not it is actually valid. He has been struggling to write it in a form that people will accept. In his own words 'These proofs are difficult to read, insanely involved, and involve the sort of elaborate bookkeeping which makes it easy to introduce errors every time a new draft is prepared.'.

But now, Sky Wilshaw has formalised the most tricky part of this proof in the Lean proof assistant! This makes it much more likely that the proof is correct overall. It also makes it easier to approach understanding the proof, since one can treat the formalised portion separately and begin by understanding the simpler parts around it.

Holmes and Wilshaw have a work-in-progress paper here describing the current version of the proof. I don't know if they plan to extend the formalisation to cover the entire proof.

I have recently come across arithmetic on conic sections where four points on a circle sum up to 0, which is a natural step down from cubics. My question is, are there any decent resources/books where I could learn about the geometric side of things, not the number theory one? As in, synthetic geometric properties that can be deduced from this and such for conics in general as well as the rectangular hyperbola or parabola specifically.

Given a variance σ^(2) for a sample of size N, I would like to remove a single data point, resulting in a subsample of size N-1. I want to do this without rerunning the whole algorithm to compute the variance.

There is Welford's method which allows you to add a data point without recalculating the whole thing (aka a running variance algorithm). This is proven to be numerically stable.

However, there is not much online about using Welford's method to remove data points. I have found this which discusses how one would remove a data point similarly to how you would add one. But, there is no proof that the algorithm mentioned in that StackOverflow article is numerically stable. When I run tests in Python and Numpy, I get a different answer than expected. This makes me think that this method is not numerically stable.

The problem is, I have very little knowledge of numerical stability. Looking into it, it seems that due to the subtraction in the method, this could result in catastrophic cancellation, or something like that.

So I have the following questions:

1. Is the method described numerically stable?
2. Does the method result in something like catastrophic cancellation? And
3. If it is not numerically stable, how can I go about finding an algorithm that is in fact numerically stable? Is this necessarily possible?

Edit: It looks like there were a few bugs in my code, and the code seems to have the same output as the standard numpy implementation! At least for me, that's good enough!

Hello, I'm a Korean autodidact learning theorem proving and C programming. I've contributed to Lean's standard library, Std, and mathematics library, Mathlib, since last year.

In 2022, I chose Lean over other proof assistants because it seemed to have the most working mathematicians whose research interests don't fall into foundations of mathematics. I read "Theorem Proving in Lean 4" thoroughly that year.

I'm interested in other proof assistants and foundations of mathematics. However, formalizing classical mechanics is a higher priority for me. I haven't been able to start formalizing it, though.

While porting two Mathlib files about strings from Lean 3 to Lean 4 in 2023, I somehow became a coauthor of a file in Std containing lemmas about strings. Last December, I started proving String.splitOn_of_valid, the first of the eleven remaining theorems on the to-do list in the file.

After spending around 160 hours, I found a bug in the definition of String.splitOn on the 3rd of this month. Maurício Collares already knew this incorrect behavior of the function last year but didn't think this was a bug. Mario Carneiro fixed it.

digama0 said:

The code change here is very small, replacing a i with i - j, but it makes termination more complex so that's where the rest of the line count goes.

This bug wasn't difficult to fix but was hard to find. The incorrect definition of splitOnAux has been around since Mar 23, 2019.

I finally proved the first theorem on the 16th of this month. This work took me roughly ten additional hours.

I plan to focus on reading "C Programming: A Modern Approach" for the rest of this year. My utmost priority is developing an educational video game for learning mathematics and logic, which will be based on proof assistants like Lean and the MM0 project.

If I have more time later, I also want to explore other proof assistants that support constructive logic well because some tactics in Lean's core library use classical reasoning by default, even if a user has not opened the Classical namespace or used lemmas within it.

Are there monthly of quarterly magazines covering physics or math for university students. Something like Russian Kvant, featuring differantial equations or analysis, electromagnetism etc. Most preferbly with questions on the subject included. I can read in English, French, German and Turkish.