There 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.
I 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.
Math genealogy project says Herstein was a student of Max Zorn (the guy the lemma is named after), but Zorn was a student of Artin! Zorn doctorized under Artin in Hamburg in 1930, before Artin moved to Princeton.
Of course there is no single best road to mathematics, mainly because the way humans are able to think isn't uniform.
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.
I 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.)
The 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.
Sure, if one wants to move deeper then Feynman Lectures are a nice entry point - at least for some people, everyone has different background and taste.
For L&L - well, not sure which ones I read (or: try to read), but likely electromagnetism and classical mechanics.
Shivlov 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)
Linear Algebra and Geometry by Igor Shafarevich, coupled with Linear Algebra by Friedberg, Insel, and Spence; the latter has great problems to work with, whereas the former is for a lucid exposition.
I dislike Feynman, frankly. So many of his weird hand-wavy metaphorical descriptions of shit obscure more than clarify the really important elements of the physics.
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.
In a way Feynman was also the 3b1b of his generation. He was very good at coming up with and communicating with visual metaphors. To the laypeople at the receiving end, these give a strong illusion of understanding.
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.
Not 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
Lists 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!
This 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.
Here'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.
3 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.
2-3 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.
Even 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.
If 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
Not 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.
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.
Safe, 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!)
About 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.
(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.?
It 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.
He 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.
Ah 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)
Fascinating, 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???
Obviously 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.
Sheaf 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.
https://web.archive.org/web/20240222200132/https://sheafific...
There 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.
> "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.
I 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.
Math genealogy project says Herstein was a student of Max Zorn (the guy the lemma is named after), but Zorn was a student of Artin! Zorn doctorized under Artin in Hamburg in 1930, before Artin moved to Princeton.
https://www.genealogy.math.ndsu.nodak.edu/id.php?id=6526
Have an apoplectism
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
Of course there is no single best road to mathematics, mainly because the way humans are able to think isn't uniform.
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.
I 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.)
The 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.
Sure, if one wants to move deeper then Feynman Lectures are a nice entry point - at least for some people, everyone has different background and taste.
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-...
Shivlov 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)
What would you recommend as a supplement to Shilov's LA book?
Linear Algebra and Geometry by Igor Shafarevich, coupled with Linear Algebra by Friedberg, Insel, and Spence; the latter has great problems to work with, whereas the former is for a lucid exposition.
I dislike Feynman, frankly. So many of his weird hand-wavy metaphorical descriptions of shit obscure more than clarify the really important elements of the physics.
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.
In a way Feynman was also the 3b1b of his generation. He was very good at coming up with and communicating with visual metaphors. To the laypeople at the receiving end, these give a strong illusion of understanding.
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.
Not 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
Lists 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!
"There is no royal road to mathematics" https://terrytao.wordpress.com/career-advice/work-hard/
This 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.
Here'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.
How can you ensure this if you don’t know where the global optimum is?
My impression is that most people don't have a global objective.
3 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.
2-3 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.
Even 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.
If 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
Not 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.
The 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.
Weird!
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.
Safe, 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!)
About 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.
Why waste time on physics if you are trying to learn mathematics? I always found those “motivations” to be a distraction.
(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.?
The optimal path to mathematical mastery is "Error establishing a database connection"?
It 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.
Computer science is math. Software engineering is not math.
And vibe-coding is barely coding, nevermind software engineering. (;
Oof. 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
He 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.
> I don’t know any static blog providers.
Bear Blog comes to mind (no connection): https://bearblog.dev
That doesn't look static, it's a Django app, no?
Ah 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 [1] https://getpublii.com
Fascinating, 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???
That's correct
Obviously 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
Sheaf 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.
[dead]