
An Infinitely Large Napkin - throwawaymath
http://web.evanchen.cc/napkin.html
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throwawaymath
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...](https://usamo.files.wordpress.com/2019/02/napkin-v15-20190220.pdf)

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axiom92
Am I missing something here? The link you've shared is exactly what the post
is about.

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todd8
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.

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derefr
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?

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todd8
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.

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kangnkodos
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.

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joycian
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?

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jerf
"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.

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pingucrimson
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).

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vortico
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.

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LeonB
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?

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berbec
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.

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improv32
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.

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berbec
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?

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improv32
Here's a cool link I just found going over this exact scenario:
[https://www.mathpages.com/home/kmath530/kmath530.htm](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.

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

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Cerium
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.

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joe-collins
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.

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uptownfunk
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.

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Isamu
Thank you, this is awesome, very readable.

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

