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An Infinitely Large Napkin (evanchen.cc)
148 points by throwawaymath 7 months ago | hide | past | web | favorite | 32 comments

Adding this here as it may be of related interest for those who enjoyed the massive math cheat sheet on the front page recently. Evan Chen, a math student at MIT, wrote up what would be considered field notes for higher mathematics. The full PDF is here[1], complete with a dependency graph showing what you need to know before reading any particular section.


1. https://usamo.files.wordpress.com/2019/02/napkin-v15-2019022...

Am I missing something here? The link you've shared is exactly what the post is about.

I'm a computer scientist, not a mathematician (but I've taken around 25 college math courses spread over many years). Nevertheless, I do have a couple of suggestions and observations that I will address to the author that I hope is seeing this.

First, it would be a very sophisticated high school student to tackle topology and some of the other areas of abstract mathematics. I really like the topics you've picked for your book, but they do seem to require quite a bit of mathematical sophistication (e.g. Topology).

Secondly, I feel that there are a few important fields that you might consider adding to your napkin: Combinatorics, Statistics, Differential Equations, and Logic.

The usefulness and the importance of understanding statistics is pretty obvious in today's data dominated world. Statistics seems to fall outside of Mathematics at some (most?) universities, but I keep my statistics books right next to my math books.

Combinatorics is full of interesting results some esoteric (the friendship theorem) and some practical (stars and bars). The proof techniques of combinatorics are also worth studying for their own sakes (like the probabilistic method).

I've always felt a love hate relationship with Differential Equations. Theoretically, they are disappointing ("oh hey, let's try this, surprise its the solution!") but practically they are needed everywhere.

One of the best math experiences that I had in high school was a logic course that I took one summer with two other students. What fun and it always served me well in course 18.

The point of the "napkin" isn't to be a generic Maths textbook; it's to trace up the prerequisite chain from category theory until it connects with high-school-level maths. Do you need statistics or differential equations to understand category theory?

I like the book but the contents seem to go well beyond the prerequisites of category theory—-it’s almost 900 pages.

I got the impression that the author was not simply attempting to connect high school math to category theory but was providing a broader survey of higher math. I interpreted the author’s remarks about the path to category theory as the inspiration for embarking on the project that has turned out to be a wide survey of higher math that might benefit young mathematicians.

> suggestions and observations that I will address to the author that I hope is seeing this

The author's site has a contact page (http://web.evanchen.cc/contact.html), you could send them your feedback directly.

This is supposed to be aimed at high school students.

I majored in math in college, and yet there were things which I could not decipher in the first few pages of chapter 1. For example, on page 43, what is "nonzero residues modulo p"? I guess you start with something, divide by p, and get a remainder, or residue. But what is that something? Going back to page 41, I see the hint that Z is the set of integers. I vaguely kind of remember that this was a thing that you learned once and just used forever. I had long forgotten that Z is the set of integers. I don't really see where this is clearly stated in this book.

If I was writing this for a high school student to skim through, I would make a big deal that Z is the set of integers, and Z is going to be used many times going forward, and it will always mean the same thing, the set of integers.

Someone who just learned all this stuff would be able to skim through it.

But the way it's written now, it's going to take a lot of intense work for a high school student, or someone who majored in math many years ago, to work through the whole thing.


Edit: I see now. The prerequisites are in Appendix E. Technically, it's in there. But it's still problematic. High school books can just be read from beginning to end. For college level texts, it's OK to tuck things in Appendix E, and require the reader to go back and forth. So, no, I still would not say this is really aimed at the high school level.

From the current Preface:

"I initially wrote this book with talented high-school students in mind, particularly those with math-olympiad type backgrounds. Some remnants of that cultural bias can still be felt throughout the book, particularly in assorted challenge problems which are taken from mathematical competitions. However, in general I think this would be a good reference for anyone with some amount of mathematical maturity and curiosity. Examples include but certainly not limited to: math undergraduate majors, physics/CS majors, math PhD students who want to hear a little bit about fields other than their own, high school students who like math but not math contests, and unusually intelligent kittens fluent in English."

I would clarify and say that it's aimed at high school students with a serious thirst for competition math and math in general, Evan hangs around a lot (or at least he used to) on the Art of Problem Solving forums, and I think it's these students who spend their day pouring over math discussions, and math problems, that the book is aimed at.

These students would already know that Z is the set of integers, and if they didn't I don't think it would be a deterrent to pouring over the book.

The fact that Z denotes the set of integers is a sort of mathwide standard. It comes up a lot (number theory, group theory) Did you major in some specific area of maths? Or is it just the span of years between?

"The fact that Z denotes the set of integers is a sort of mathwide standard."

I'm pretty sure I saw that in my high school in the early 90s, but it was a one-off event where we discussed ℕ, ℝ, ℤ, and ℚ, but we never used them for anything. I'm sitting here trying to remember our high-school set theory (which is getting cognitive interference from my college training on the topic), but my memory is claiming I either never had to write {x | x ∃ ℤ} in high school, or if I ever did, we blipped over it really quickly.

High school math generally implicitly takes place in "casual ℝ". I call it casual because the only time it even gets close to really hammering on the characteristics of real numbers is in the limit discussion. I certainly never heard "Dedekind cut" in high school.

We used Z a lot when dealing with modular arithmetic and complex roots of unity, mostly just to quantify our variables. I can't recall ever using N or Q in high school, though.

Also, you don't need to mention Dedekind cuts at all when dealing with R - it can be defined by the fact that it's the smallest extension of Q that's closed under limit-taking (and I think most high school math students do understand that).

There's also the shifting of high school curriculum over time. I don't have kids so I haven't kept up with it, but from what I've picked up in the news maths education has changed radically in the 30 years since I was in school. Definitions and techniques in this article may be considered common knowledge now, since testing is more standardized at a national level.

Many, many years. Also, it's a thing in college. I don't think they talked about this much in high school when I was there. If they did, they started each book with a review, "Z is the set of integers".

I agree with you that for anyone who recently took math in college, it's obvious that Z is the set of integers. But for someone just learning it? Not so much.

I would say that even though it may be commonly used it is a bad name for Integers because it is not mnemonic. "I" would be a much better name for the set of Integers I would think.

But whatever you call it should not be too much effort to state in the beginning of a presentation or chapter or book, these are the symbols we will be using:

I = Set of Integers, ....

Sure it might be redundant but it is also easy for the reader to step over such explanations if they are familiar with them, but come back if they find some symbol they are not quite sure of. Question is are we trying to make the book easy for readers to understand, or short for the writer to write.

It is a mnemonic if you're German, which is where the symbol comes from, it's the Z of Zählen!

Good point. Of course. But if I'm writing or reading in English I still would prefer "I".

This is supposed to be aimed at high school students.

That's the stated motivation, but like the 40 hours mentioned around the same place, I have always assumed it was intended to be a bit tongue-in-cheek. The material covered here would span much of an undergraduate syllabus, and it would surely take several years even for an interested and hard-working mathematics student at a top university to understand and apply all of this effectively. Indeed, there are references to concepts that you wouldn't necessarily expect to have studied in detail or perhaps even encountered at all below postgraduate level.

This is awesome, thanks! My learning style is to read summaries of things and then go deeper where I feel like. I'm not a fan of reading 20 pages of proofs that don't teach me anything new, only to reach half a page that takes me 3 hours to get through. A book which is basically a massive cheat sheet is perfect for reading as if I'm skimming a much larger work.

If there was an infinitely large napkin, floating in space for example, far away from everything else, would its own gravity cause it to crumple toward any folds/creases/imperfections, with sufficient force to create a cascading implosion of sufficient mass/density to create a black hole?

An infinitely large napkin, if made of any sort of normal matter, would have infinite mass. It's gravitational pull would propigate, at c IIRC, out from the instant of its creation across all of space/time, sucking all objects (including itself) towards it's center of mass at the speed of light (OK, just under. Literally c-0.000...1m/s). It not only would create a black hole, it would signal the end of the entire universe. The destruction of any point in space would depend simply on the distance from the napkin. For every 299,792,458 meters away from napkin center, that object would have one second of existance left. When the gravitational pull hits anything, the acceleration would be so strong, again 0-c in 0 seconds, that some crazy subatomic fusion would occur.

Any physicists out there able to flesh this out? I find the thought experiment fascinating and am sure I'm missing/misrepresenting something.

How can an infinitely large napkin have a center of mass? It seems to me that all everything would get sucked on a trajectory normal to the napkin's surface. Any horizontal force would get cancelled by an opposite force, at an opposite point on the napkin. Also, force of gravity drops off with the inverse square of distance, so the force would have a finite value, depending on the density of the napkin. Similar idea to a common homework problem for introducing electrostatics: calculate the electric field generated at some point by an infinitely long wire with some charge density.

True if the napkin is perfectly flat. Any imperfections would negate the cancelation of opposing force, no?

The drop off would only apply if the napkin has finite mass or infinite size, correct?

Here's a cool link I just found going over this exact scenario: https://www.mathpages.com/home/kmath530/kmath530.htm

Interesting to note, the napkin creates a uniform gravitational field above and below it. Meaning that the force applied to an object is the same regardless of how far away it is from the napkin! That force is 2pi * G * m * rho where rho is the mass density of the napkin.

How is that possible since the force on a test particle on the surface of the napkin is 0?

It is the same result as infinite planar light sources or infinite planar electrostatic charges. Move the test particle a small delta above or below the napkin, and the result will become apparent.

Indeed, if it wasn't flat it would not perfectly cancel.

The drop off applies because the force of gravity is proportional to 1 / distance^2, so in this case the napkin would still have infinite mass, but it would not have infinite density, so if you took a surface integral of the gravitational force provided by each point in the napkin, over the whole napkin, it would converge on a finite value, as each point contributes less and less force as you get father away.

I've got a few years of community college calculus under my belt, but the cost of university and the unstructured nature of MOOCs had both deterred me from advancing along those lines. Wikipedia's math pages typically pre-suppose a higher level of understanding than I possess, or are written in exquisitely correct but practically impenetrable fashion.

This looks like it may be the bridge I've been seeking. Thank you.

Was not at all what I was expecting from the title. Love the work, (of course I majored in math at Uni).

I totally support and encourage any efforts to make higher math more approachable and understandable. I remember the multiple hazings I went through with Rudin (both little, big, and functional analysis). The comic in the beginning is hilarious.

Thank you, this is awesome, very readable.

from the book: "With that in mind, I hope the length of the entire PDF is not intimidating." PDF is 900 pages...

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