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Sheafification – The optimal path to mathematical mastery: The fast track (2022)
- rfl890 1y agoThe optimal path to mathematical mastery is "Error establishing a database connection"?
- ethersteeds 1y agoThat's correct
- klipt 1y agoObviously it's not meant literally. The database represents your mind, and the error represents all the mathematical proofs you haven't written yet. By solving the error (proofs) you learn mathematics. /s
- Avicebron 1y agoSheaf residuals, each edge checks if its two endpoints line up. The squared differences add up to a single number R. That’s your “how inconsistent is the network right now?” gauge.
- bbor 1y agoOof. Not usually one to comment on technical issues, but that's a rough one to get for a (presumably) completely-static site! Dr. Sheaf, please consider serving your site via a dedicated service like S3. This is a solved problem <3
- figassis 1y agoHe probably doesn’t know how. This message is em from Wordpress. There are many WP providers where you just signup and start writing. They don’t tell you that if you get to HN front page, your db will die. I don’t know any static blog providers. He’d have to roll out his own or use Medium/Substack.
- easygenes 1y ago> I don’t know any static blog providers. Bear Blog comes to mind (no connection): https://bearblog.dev https://bearblog.dev
- stavros 1y agoThat doesn't look static, it's a Django app, no?
- easygenes 1y agoAh yeah, I guess I just assumed it was static because every time I've been to a Bear blog it has been so light and snappy. 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) [0] https://micro.blog https://micro.blog [1] https://getpublii.com https://getpublii.com
- bbor 1y agoFascinating, thanks for sharing your expertise. I've never published a blog, obviously. This just seems like such an absurdly simple use case... If blogs aren't static, what is???
- CobrastanJorji 1y agoComputer science is math. Software engineering is not math.
- heresie-dabord 1y agoAnd vibe-coding is barely coding, nevermind software engineering. (;
- stared 1y agoIt stings be a bit that people use Wordpress in times of really wonderful static sites generators AND vibe coding. They are both pleasure to write, make it easy to tweak style however you please, can cost nothing (just host it on GitHub pages), no maintenance, and if something hits Hacker News, no risk of an outage.
- deleted 1y ago[deleted]
- TimorousBestie 1y agoSafe, conventional, non-controversial choices here. Given the domain name, I expected more category theory. 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!)
- asplake 1y agoAbout the domain name, I don’t thought that too. But its stated intention is “to accelerate the development of prospective mathematical scientists”, hence the lean to physics and to certain authors.
- vallassy 1y agohttps://web.archive.org/web/20240222200132/https://sheafification.com/the-fast-track/ https://web.archive.org/web/20240222200132/https://sheafific...
- graycat 1y ago(1) Didn't see anything on the math of probability or stochastic processes, e.g., nothing from Dynkin, Neveu, Breiman, etc. (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.?
- jhanschoo 1y ago3 years still sounds optimistic to me ... even full-time, if we are talking reading through the material thoroughly. I suppose the modern student knows to pick and choose the gaps that they have, that are still relevant, and that they lack under guidance in an institution.
- lsc4719 1y agoThis is wrong. In Computer Science language, human learning works in greedy way. We make locally optimal decisions in life. We cannot learn something in globally optimal way. We learn something in locally optimal way. And by repeating that we can reach somewhere at some point.
- falcor84 1y agoHere's my take on this - you should try to organize your local space of affordances and rewards in such a way that locally optimal choices will over time get you closer towards your global objective.
- stared 1y agoI am not sure if I agree with the list. I mean, one red flag is the frequent mention of Landau & Lifshitz. It is considered a "standard textbook", but I feel it stuck there by inertia. There are quite a few choices of both less boring and more insightful. (Back when I was reading such stuff, 20 years ago, the Feynman Lectures provided orders magnitude more insight. And fun.)
- lambdas 1y agoShivlov for the one and only text in linear algebra is rough. IMO, it’s a little terse and fast paced. Efficient if you’re already well versed enough to be dangerous, but otherwise I think might slow down the beginner to a crawl in places. 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)
- mananaysiempre 1y agoThe Feynman Lectures are in a very different category from Landau & Lifshitz. You shouldn’t use L&L as your first university-level physics textbook (too difficult and divorced from the real world), but you probably shouldn’t use the FLP as your second, either, provided your first was good (too easy to be worth the time spent, though it’s still useful as pleasure reading that also fills in things you might have missed). 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.
- bamborde_zaiku 1y ago[dead]
- marifjeren 1y agoNot much attention given at all to explaining what order in which these should be read or what optimality means. This is just a list of books some guy is proud to have read
- tomku 1y agoLists like these are almost always more accurately titled "How I wish I'd been taught X" and are aimed squarely at a student who has never really existed, one who simply learns whatever is put in front of them and can therefore be steered by a heavy stack of "the best" books away from all the mistakes and dead ends and frustrations that real students face. Real pedagogy's tougher than that!
- seanhunter 1y agoThere is no royal road to mathematics[1], and it's incredibly arrogant to think that any person can provide a single optimal path. For me for example the next steps are Axler, Abbott and Herstein[2]. That's where I am at the moment, and it's way earlier than the books listed here. It would be far from optimal for me to try to bang my head stubbornly on this list. Mathematics demands you put in the work to build a foundation - you cant just skip steps. For some people those books I listed are very rudimentary. For others they are definitely too advanced for where they are and they'll need something else. 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.
- asplake 1y ago> "Topics in Algebra" by Herstein. this is a lovely book and beautifully written but some of the notation is a bit dated 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.
- graycat 1y agoI heard from somewhere that Herstein was an Emil Artin student at Princeton. 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.
- jebarker 1y ago2-3 seems wildly optimistic for reading that list and actually consolidating the knowledge. During my math PhD I ready a fraction of that number of books. It’d be more convincing if someone was provided as an example of “mastering mathematics” (whatever that means) through this curriculum and timeframe.
- ceh123 1y agoEven one of these topics I would say it would take most PhDs at least 2-3 years to “master”. I feel like at the end of my math PhD (5 years, 3 focused solely on my research area) I had just scratched the surface of mastery in my sub field, and that’s with 3 published papers. I guess you’re right though, defining “mastery” is the key missing point here.
- rokob 1y agoWhy waste time on physics if you are trying to learn mathematics? I always found those “motivations” to be a distraction.
- bwfan123 1y agoThe author misunderstands knowledge and mastery. 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.
- nathan_compton 1y agoWeird! 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.
- muhdeeb 1y agoWell put! 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.
- qoez 1y ago"There is no royal road to mathematics" https://terrytao.wordpress.com/career-advice/work-hard/ https://terrytao.wordpress.com/career-advice/work-hard/
- math_dandy 1y agoNot sure about these books as a self-study curriculum — their unifying theme seems to be that they require a reasonable level of mathematical maturity going in. But, they absolutely comprise an excellent “greatest hits” list of math books in the most influential subdisciplines. You’re guaranteed to learn a tonne if you study any one of these books.
- griffzhowl 1y agoIf you have a good grasp of multivariable calculus and want to learn differential geometry, I can't recommend highly enough a book that isn't on this list, Loring Tu's Introduction to Manifolds. 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
- bye_algebratraz 1y agoFor people who might just want to get by doing some baisic math on their computers in a fun and easy way, you should look at http://www.tmaker.site http://www.tmaker.site 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.