In our end of year post I threatened to write more math reviews, and multiple people in the comments egged me on. So now, with Jane laid up in the final stages of pregnancy, I have seized control of the Substack for a very special lightning round of math textbooks I recently enjoyed. No, wait! Don’t close the tab! I promise that some of these will be fun for non-mathematicians as well.
As a non-mathematician, I found this post absolutely delightful to read, so thanks!
In my economics undergrad as a French, I was able to read some introductory math classics from famous French mathematicians. I suspect that many of them has never been translated to English. This leads me to speculate that there might be a treasure trove of Russian, German, Spanish, Chinese, etc., books that are "the very best introduction to undergrad topic X across all languages" and have never been translated into English because they came out before the age of computers. Does that sound plausible to you?
Great as usual. Given your liking for quality mathematical writing I suspect Timothy Gower's "Princeton Companion to Mathematics" and Evan Chen's "An Infinitely Large Napkin" would both be of interest to you, despite having little in common. The Companion surveys pure mathematics as it stood at the start of the 21st century and represents donnish mastery of the literature at its best. Gowers wrangled the great and the good into producing the entries and the breadth and depth of knowledge shows. The Klainerman entry for example is the best short explanation I've ever read as to why PDEs should be treated as a unified object of study and not a massive pile of hacks.
The Napkin by contrast is a a one volume gallop through the entire standard undergraduate sequence and a chunk of graduate algebraic geometry written by an IMO gold medalist weaned on xkcd comics, Paul Graham essays and internet writing of every dye and hue. Conveying the intuition behind the central ideas via plentiful examples and problems is the name of the game, everything peripheral is dispensed with. This method is often revelatory; using group actions to overkill AIME problems or the Frobenius element to solve an IMO number theory problem is just great pedagogy. Is it even readable if you didn't go through some IMO training? Possibly not but it works even as a cultural artefact. Mathematical culture is slowly migrating online and alongside neo-bourbaki distributed formalisation efforts indebted to the open source software movement you get crazy textbooks like this written by people who spent their youth in the Art of Problem Solving forums sourcing binders full of shortlisted IMO problems.
Raymond Smullyan is my favorite math writer, after Martin Gardner. (I mean who really understands these Galois, Grothendieck or Gödel guys? :^) His book with the vampires, knights and knaves is great, by the end of it (if I'm not mixing up books) he tricks you into Gödel's incompleteness theorem. ("What is the Name of this Book")?
For me, one part of good math books (or physics books) is that they ask you to do the work. And so part of what makes a good book that is trying to teach you something, is that it leaves the 'right' amount for you to discover for yourself while working through the problems. Which is a hard task, because the writer doesn't know where the reader is starting. "Gödel, Escher Bach" was perfect for me in this regard. And Penrose's "Road to Reality" failed me, 'cause I got stuck in chapter 3 or 4, and I needed help. There is a further problem in recommending good learning books, and that is that once you learn a topic, you've then lost the ignorance to learn it again. I've more to say, but the dog is ready to come in and I'm ready for bed. Best wishes to Mrs. Psmith, more coddling is fine. And I have a theory that boredom is good for kids, at least those who are introverts like me.
Thanks for this. I love math and love reading about it. But I never got very far along in it because I switched to dance. I found dance harder and I needed the challenge.
Funny timing. Was just visiting with a mathematician who is some kind of topologist (he’s been a professor for 54 years, still working full time — his son was my childhood best friend) but I only got through three years of college engineering math so I’ve never quite understood what he did.
You write that there are always more subsets of any set than the number of elements in the set itself. How about {}? Zero elements, one subset = the null set? Seems like cheating. B
Okay, I'll admit that I managed to finish 3/5 of these reviews, which coincidentally seems to match John's completion rate of the books themselves.
Anyway, I hope that the mathematicians are able to eventually complete their trials and return to their home planet victorious.
As a non-mathematician, I found this post absolutely delightful to read, so thanks!
In my economics undergrad as a French, I was able to read some introductory math classics from famous French mathematicians. I suspect that many of them has never been translated to English. This leads me to speculate that there might be a treasure trove of Russian, German, Spanish, Chinese, etc., books that are "the very best introduction to undergrad topic X across all languages" and have never been translated into English because they came out before the age of computers. Does that sound plausible to you?
Great as usual. Given your liking for quality mathematical writing I suspect Timothy Gower's "Princeton Companion to Mathematics" and Evan Chen's "An Infinitely Large Napkin" would both be of interest to you, despite having little in common. The Companion surveys pure mathematics as it stood at the start of the 21st century and represents donnish mastery of the literature at its best. Gowers wrangled the great and the good into producing the entries and the breadth and depth of knowledge shows. The Klainerman entry for example is the best short explanation I've ever read as to why PDEs should be treated as a unified object of study and not a massive pile of hacks.
The Napkin by contrast is a a one volume gallop through the entire standard undergraduate sequence and a chunk of graduate algebraic geometry written by an IMO gold medalist weaned on xkcd comics, Paul Graham essays and internet writing of every dye and hue. Conveying the intuition behind the central ideas via plentiful examples and problems is the name of the game, everything peripheral is dispensed with. This method is often revelatory; using group actions to overkill AIME problems or the Frobenius element to solve an IMO number theory problem is just great pedagogy. Is it even readable if you didn't go through some IMO training? Possibly not but it works even as a cultural artefact. Mathematical culture is slowly migrating online and alongside neo-bourbaki distributed formalisation efforts indebted to the open source software movement you get crazy textbooks like this written by people who spent their youth in the Art of Problem Solving forums sourcing binders full of shortlisted IMO problems.
An enjoyable read, as always! I wish I had the time to read all of these.
If you could recommend *one* high-level math to for an educated person wanting to expand his mathematical horizons, what would it be?
Really nice set of books. Thanks for the write-up.
Glad to find another Quanta reader on the internet. Good times!
Raymond Smullyan is my favorite math writer, after Martin Gardner. (I mean who really understands these Galois, Grothendieck or Gödel guys? :^) His book with the vampires, knights and knaves is great, by the end of it (if I'm not mixing up books) he tricks you into Gödel's incompleteness theorem. ("What is the Name of this Book")?
For me, one part of good math books (or physics books) is that they ask you to do the work. And so part of what makes a good book that is trying to teach you something, is that it leaves the 'right' amount for you to discover for yourself while working through the problems. Which is a hard task, because the writer doesn't know where the reader is starting. "Gödel, Escher Bach" was perfect for me in this regard. And Penrose's "Road to Reality" failed me, 'cause I got stuck in chapter 3 or 4, and I needed help. There is a further problem in recommending good learning books, and that is that once you learn a topic, you've then lost the ignorance to learn it again. I've more to say, but the dog is ready to come in and I'm ready for bed. Best wishes to Mrs. Psmith, more coddling is fine. And I have a theory that boredom is good for kids, at least those who are introverts like me.
Thanks for this. I love math and love reading about it. But I never got very far along in it because I switched to dance. I found dance harder and I needed the challenge.
Funny timing. Was just visiting with a mathematician who is some kind of topologist (he’s been a professor for 54 years, still working full time — his son was my childhood best friend) but I only got through three years of college engineering math so I’ve never quite understood what he did.
You write that there are always more subsets of any set than the number of elements in the set itself. How about {}? Zero elements, one subset = the null set? Seems like cheating. B