An Infinitely Large Napkin [pdf] (2019)
venhance.github.io
venhance.github.io
Step 1. I don't know how X works.
Step 2. I collect several sources about X and try to understand it.
Step 3. I put in a lot of effort to understand X by reading all these sources repeatedly. I try to do exercises, do calculations, etc. I'm desperately seeking the moment it "clicks".
Step 4. I finally kinda sorta "get" X.
Step 5. I feel, "why didn't anybody simply explain X in this way?" / "why was everybody so overly formal?" / "why was everybody so overly informal?"
Step 6. I'm motivated to write a short note about X that makes it (allegedly) easier to understand X.
Step 7. I write it, and I realize it's actually hard to weave together a narrative that doesn't over- or under-assume prerequisites, that captures nuance, that has good examples, etc.
Step 8. "There are 15 competing standards."
Step 9. Find the next topic X and go back to Step 1.
The Infinitely Large Napkin is a really cool consolidation of a ton of undergrad/early grad pure math topics. It's so incredibly expansive in its scope and, if it were in book form, I'd have been ecstatic to have it as a 16 year old.
But paging through it, I find that they're very much in the style of quasi-formal lecture notes. A lot of topics are mentioned by their formal definition, and it's followed by a very anemic (if any) discussion, sometimes preceded by a very informal (sometimes humorous) introduction. Often such definitions are immediately followed up by a relatively technical exercise that presumes a fully synthesized understanding of material preceding. This can make it very difficult to learn from as a primary/sole source. It does make it fun to flip through, though, when you already have familiarity with the topics.
In my view, this isn't the kind of book you work through. It's not "math distilled". Instead, it can serve as a great diving-off point for a new subject, or an inspiration to know where to look further on a given subject, or even a useful document to find a topic that piques your interest. Other books like this are those of yore that were encyclopedic in nature, such as:
- VNR Concise Encyclopedia of Mathematics (1975–1989) edited by Gellert et al. The math here doesn't get terribly advanced (complex and numerical analysis), but it's a good, expansive treatment to dive into.
- Mathematics From the Birth of Numbers (1997) by Jan Gullberg: This is another grand tour of math, albeit "only" to differential equations. It's refreshingly written by somebody who was a surgeon/anesthesiologist and amateur mathematician.
- The Princeton Companion to Mathematics (2008) edited by Timothy Gowers. This is a massive book that covers just about everything, up to and including some of the latest problems in mathematics. It's 1000 dense pages. (There's also the Princeton Companion to Applied Mathematics edited by Nicholas Higham.)
- The CRC Encyclopedia of Mathematics (1999) by Eric Weisstein. This is an anti-digitization of the Wolfram MathWorld into book form. Expansive, and also famous for some of the drama around its copyright. :)
https://byorgey.wordpress.com/2009/01/12/abstraction-intuiti...
I think it's not that there's a difference per se between an example and an analogy, but rather that one has to recognize that even the best-chosen analogy or example won't save other people from having to go through the struggle to come to grips with a difficult concept. That is, from my point of view, the problem with "it's so simple, a monad is a burrito!" isn't especially the "a monad is a burrito" part (although I'd argue that that's an overly fmap-flavored analogy, it's not really the point); it's the "it's so simple" part. From this point of view, which is mine as a teacher, teaching isn't a process of finding the path that lets students avoid every difficulty, but one of helping them to avoid the avoidable ones, manage the unavoidable ones, and build their own mental maps so that they can eventually understand and navigate the terrain themselves.
Aha, so already they're studying induction.
Last time was Feb 11 2022 :D
ps. That diagram is just fantastic.
So to those who do enjoy it, have fun!
> light
1,044 pages.
I can only recommend this as a good starting point for anyone without the time for a full scale education, or in preparation for such, especially if you have some experience with math olympiads (as mentioned also in the introduction).
I only wished I had spent some more time with stuff like category theory before starting my studies, and had had the guts to take more advanced courses directly in my first semester.
With just a bit of prep from these notes I'm pretty convinced that it's possible to directly take e.g. Algebraic Topology, Differential Geometry, Category Theory, or Algebra during the first semester (don't know about number or measure theory, or anything requiring lots of functional analysis, have not engaged enough with those topics to know how good these notes would be as prep).
Why is it important/useful to take advanced classes so early on?
https://news.ycombinator.com/item?id=30302291 (2022, 18 comments)
https://www.amazon.co.uk/All-Mathematics-You-Missed-Graduate...