Sheafification – The optimal path to mathematical mastery: The fast track (2022)
sheafification.com
sheafification.com
Even more so is the idea that you can actually cover the material listed in that page in 3 years. If you were to blast through it in that time you would only be skimming the very surface of the topics. There's simply no way you could possibly do all of those subjects justice in that time.
[1] As Euclid is supposed to have said about geometry to the Pharoah Ptolemy when Ptolemy said he wanted to learn geometry but because of all the concerns of his kingdom he didn't have time to read the Elements.
[2] "Linear Algebra done Right" by Sheldon Axler
"Understanding Analysis" by Stephen Abbott
"Topics in Algebra" by Herstein. this is a lovely book and beautifully written but some of the notation is a bit dated. I have two more recent algebra books but they are a bit advanced for me until I work through Herstein. They are Aluffi "Algebra Chapter 0" which is a good modern algebra book which introduces category theory at the start and Hien I forget the title but it's a springer one that he claims is good for an introduction but it's definitely not. It assumes you know a lot. It's very good though.
Now you make me feel old! I had the second edition just as it came out in 1984. A long time ago but I remember it as one of my favourites.
Herstein went to the extra trouble to make his linear algebra also work for finite fields.
In the back he has group representation theory is a small nutshell.
Also in the back he does linear programming, but his treatment is obscure and for no good reason. Since then nearly every treatment, beginner or advanced is not obscure at all.
https://agorism.dev/book/math/ag/royal-road-algebraic-geomet...
Also iirc, various books titled Royal Road to Geometry, which might make more sense given that IMO geom was the first to get crushed by chatbots
Ymmv, but denying existence of royal road to math seems like a narcissism limited to the most elite layity
Not sure I fully understand your point about narcissism and laity. If Euclid and I are laity then I think I'm happy to be called that though. For me the "royal road" refers to the idea that you can take shortcuts and understand things without putting in the work. That just does not ring at all true to my experience with maths anyway.
There are people with more or less Aphantasia, so people who can't or have a hard time forming mental images. Then there are others who can rotate 3 bezier curves in their head and plot the intersections.
For students of the latter category relying on mental images is a great way to teach them, for the former it is catastrophical.
Anybody who has thought anything mathematical should be aware of the fact that different people prefer different ways of learning.
(Back when I was reading such stuff, 20 years ago, the Feynman Lectures provided orders magnitude more insight. And fun.)
Same for Hartshorne’s Algebraic Geometry. Neither of these are bad textbooks at all, they both have a place on my bookshelf, but certainly better options have appeared through the years (for AG, I’d be remiss to mention Ravi Vakil’s fantastic The Rising Sea, due for a physical publishing October, and Ulrich Görtz & Torsten Wedhorn two part series)
I have to admit I like the FLP less than the typical reader—it’s immensely fun in the moment, but I’ve always found the material too disjointed to build a coherent picture (ah, and now we have the tools to understand this random fun thing that I’ve never mentioned before and never going to mention after). As far as classic introductory books, the Berkeley course covers less, but the things it does cover fit together much better.
As for L&L, it’s just very uneven in quality. General relativity is great; electromagnetism is kind of bad. (And the two are in the same book!) Theoretical mechanics is adequate but way too much of a slog despite how short it is. Elasticity is surprisingly good. QM is OK but there’s a dozen approaches to QM depending on your background and it’s a toss-up whether this one will work for you. QED is good at the things it covers but like half of those things are relegated to the status of obscure specialist topics these days, while the point of view is something you should be aware of eventually but definitely not at your first go through the subject. And so on.
For L&L - well, not sure which ones I read (or: try to read), but likely electromagnetism and classical mechanics.
For quantum mechanics, I used to suggest these: https://p.migdal.pl/blog/2016/08/quantum-mechanics-for-high-...
To name an example: Feynman is the source of the popular idea that in special relativity we can think of a particle as having a constant 4-vector with length c and that movement changes the direction of the four vector into the spatial directions, thus "slowing" the speed through time.
This is a very strange way of thinking about this stuff because the entire point of special relativity is that there is no objective state of affairs about velocity at all. It's meaningless to talk about the velocity of a single particle because velocity is a relative quantity. Also, I'm just generally suspicious of all this "hyperbolic rotation" stuff. I mean its true as far as the mathematical structure is concerned, but most of the time metaphors which try to get us to think of a minkowski space as being a lot like a normal 4d euclidean space confuse us or at least hide the real interesting structure, which is that in a minkowski space much of the 4d structure implied by a set of events is redundant. That is, spacetime is less than space and time together, not more.
That's ok if you are not going to compute or design or build anything with it. But they are very inadequate when it is time to shut up and compute.
Feynman (and I am sure) Grant Sanderson could/can operate at a virtuoso level at both the visual imagery and the compute layers. But their popularity with the masses is because of the visual imagery they could conjure up.
On the other hand for those who can already compute for themselves, the metaphors can be a big help for building intuition as long they think in the same style.
I dunno about “two or three years,” though. It would take me about a year to get through all three volumes of _The Quantum Theory of Fields_ alone (~1500 pages of extremely dense physics!)
I guess you’re right though, defining “mastery” is the key missing point here.
I'd tried a few well-regarded diff geo books (Isham's Modern Differential Geometry, Morita's Geometry of Forms, Schutz's Geometrical Methods) but always got to a stage where there was some lack of clarity, something seemed assumed that wasn't explicitly mentioned, or just inscrutable notation that didn't seem to have been explained previously. In contrast, Tu's book is smooth and pellucid in its clarity. Very enjoyable
(2) That it's important to study carefully that computing is about "switches" is something like knowing the details of the amazing work in the chemistry of rubber tires is crucial for truck driving or we should all early in our education achieve good mastery of each of the proteins in our DNA.
(3) After working through all those books, cancelling nearly everything else in life for some years, just what is the result, the payoff, the reason? Academic research or something in the mainstream economy, technology, etc.?
It is the one and only two dimensional spreadsheet that works in arithmetic with no need for algebra or its notation. Kind of like a hand calculator running around a blank sheet of paper.
Mastery comes from problem solving and practice - not from reading books. So, I would advise students to limit what you read, but spend a lot of time in problem solving. Get the basics in place. Start with euclid's elements and master that first.
I think Euclid is fine as a historical document and interesting in a broad sense, but its kind of silly to start with a document from 300 BCE when you could start with, for example, "How to Solve It" (Polya). And even that text could use a rewrite to make it much more readable.
I have completed substantial education in both mathematics and physics and I would say one of the weaknesses of the standard system of courses in physics is that it more or less recapitulates the development of physics in a historical shape, which substantially obscures mathematical structures which are shared between disciplines developed at different times. For example, unless you were lucky or very curious you might never appreciate that the bras and covectors relate to kets and vectors in fundamentally the same way. You might only have had a vague sense that physics involves making a lot of sandwiches.
Don't get me wrong, I'd be delighted if my kid's school broke out The Elements, but I just don't think its an obvious pedagogical strategy to start math instruction (self or otherwise) there.
You make a very good point. The historical trajectory of a field's development is an extremely haphazard presentation of the ideas. It belongs in a separate curriculum, and I'm grateful that the history of science exists as field for that purpose. It's nice to study the history after one has understood the material; that way we can see what ideas prospered and faltered, and what might be ready for reinterrogation.
Why should we study antiquated and easily falsifiable models before we get to our modern and less-easily falsifiable models?
Drawing on common intro chemistry: the plum-pudding model of the atom is cute in historical context, but a real distraction from what our best understanding of what atoms are, for which we have much better evidence than the helium-nucleus scattering experiment that first suggested a dense, charged nucleus in gold atoms. We really only need the old plum pudding as a counter example, yet fail to explain the experiment in enough detail to justify including it. What probably began as a fastidious attempt to provide full context to a landmark experiment has at this point completely degenerated into historical trivia about the structure of British desserts, and yet remains prominent in educational material.
Math is a bit better off in this respect.
I meant that the foundations of what it means to do math can be found in as elementary a text as the Elements. Most importantly it shows how an abstract world (what we feel as space around us) can be modeled as Geometry with a set of axioms, along with a way to demonstrate statements without doubt. Just understanding this very deeply (in your bones) with a lot of problems would make you a better mathematician than if you read all the books listed above, at least in my view. Other areas of mathematics model different kinds of abstract worlds, but the activity is largely similar.
The problem I see is the focus on gathering knowledge without properly assimilating it (ie reading a lot of books). By assimilating I mean, being able to explain how something is constructed from the very basic first principles of how the conceptual structure was built.
/s
Dr. Sheaf, please consider serving your site via a dedicated service like S3. This is a solved problem <3
Bear Blog comes to mind (no connection): https://bearblog.dev
Just did a little looking. Micro.blog [0] is a true hosted static site offering; just hosted Hugo. There's also Publii [1], an open source cross-platform desktop static CMS that has one-click push to a variety of cloud CDNs (e.g. Netlify or Github Pages)