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The math books that made the list are ones that always rate highly on these lists, but I think they are uniformly bad choices for people with a more passive or

The math books that made the list are ones that always rate highly on these lists, but I think they are uniformly bad choices for people with a more passive or recreational interest in math. A lot of textbook publishers have series for undergraduates which are a better choice. For example, here’s Springer’s: https://www.springer.com/series/3423 https://www.springer.com/series/3423. For someone who has a college level education in science or engineering, but not a math degree, I think they would have more fun looking through a series and picking a title that seems cool and interesting to them. I honestly just have to wonder how many programmers have unread copies of Principles of Mathematical Analysis sitting on their shelves unread.

titanomachy 3y ago
Do you think the Princeton Companion is good for an amateur/casual audience? I've been thinking of picking up a copy. I've taken like 8-10 undergrad math courses, but no advanced topics.
- rramadass 3y ago
  Definitely! I have both the Princeton Companion to Mathematics and Princeton Companion to Applied Mathematics and both are well worth the money. One of the difficulties in studying the breadth of mathematics is that the huge amount of jargon/notation of the different branches/models/tools all seem complex/unrelated and hence a single volume where you can jump back and forth between different views is very much necessary to appreciate the unity of Mathematics. It is also an invaluable aid in understanding new terminologies/techniques that one may come across while studying other subjects. The volumes are truly comprehensive w.r.t. "Modern Mathematics" and should be a part of every educated person's "personal library".
  - sfpotter 3y ago
    Definitely unnecessary for them to be a part of every educated person’s library. This is a pretty absurd viewpoint. The reality is that there are loads of educated people who contribute vast amounts to the world without even the most passing familiarity with “modern math”. I also don’t agree with the sentiment of the rest of the post, sorry. If someone is casually or recreationally interested in math, building a comprehensive view of modern math just… isn’t important. If you’re interested in spending more time with math, just do it—don’t let the perfect be the enemy of the good. There’s no right approach other than the one that makes you happy.
    - rramadass 3y ago
      Hard disagree. These books are not textbooks but Companions thus giving one history, development, concepts, ideas etc. of Mathematics. This is very similar to Encyclopedias/Dictionaries that every family generally has in its "personal library" but just more detailed w.r.t. Mathematics. By "comprehensive" i meant the books are comprehensive and not that one has to study every bit of it. Pick and choose topics/terminologies as you come across them at whatever level you want. You are adding to your education and not necessarily trying to become a "Mathematician". Every "educated person" (STEM or not) with curiosity to understand more about the World (and all-pervasive technologies in particular) has to have some idea of "Modern Maths" at some level. This is why you have so many "popular science and mathematics" books. While these companion books are not "popular maths" books they make a very good reference for when one wants more breadth and depth but not necessarily a full-blown textbook. These are also great "motivators" for students by sparking their curiosity (and perhaps inspire some to become Mathematicians)) since they catalog so much within one pair of covers.
- sfpotter 3y ago
  I haven’t looked at it. If it seems cool and fun, why not? If you’re near a university with a decent math library, I would start there. Wander around the stacks and see what you find anything that engages you. If not, don’t sweat it—do something else.
laichzeit0 3y ago
Yeah I laughed a bit when I saw Rudin’s book on that list. Understanding Analysis would be a better choice, even the end chapters of Spivak’s Calculus are a great into to Real Analysis.
- harshreality 3y ago
  I agree and I don't know why the myth persists that [baby] Rudin is the best way to learn analysis. It's certainly not so for a lot of people who could learn analysis perfectly well from friendlier (but not necessarily any less rigorous) sources. Rudin requires a lot of self-motivation. In addition to Abbott which you mentioned, "How to Think About Analysis" (Alcock), good gentle introduction for anyone who tries to dive into an analysis textbook and hits concrete. "Real Mathematical Analysis" (Pugh), good, rigorous approach that doesn't require quite as much effort to power through as Rudin does, although some of the exercises are tough. "Analysis I" and II (Tao), is good and starts from scratch building numbers from the Peano axioms, so there are no other unproven assumptions underlying everything later. "Counterexamples in Analysis" (Gelbalm & Olmstead), very helpful for understanding some pathological cases that break theorems one might intuitively believe are true. There's no reason today why someone can't grab pdfs of all of the above (even Rudin, although it's never been digitally typeset), from any friendly internet library, and use them all to build a better understanding rather than relying on one alone.

titanomachy 3y ago

Do you think the Princeton Companion is good for an amateur/casual audience? I've been thinking of picking up a copy. I've taken like 8-10 undergrad math courses, but no advanced topics.

rramadass 3y ago
Definitely! I have both the Princeton Companion to Mathematics and Princeton Companion to Applied Mathematics and both are well worth the money. One of the difficulties in studying the breadth of mathematics is that the huge amount of jargon/notation of the different branches/models/tools all seem complex/unrelated and hence a single volume where you can jump back and forth between different views is very much necessary to appreciate the unity of Mathematics. It is also an invaluable aid in understanding new terminologies/techniques that one may come across while studying other subjects. The volumes are truly comprehensive w.r.t. "Modern Mathematics" and should be a part of every educated person's "personal library".
- sfpotter 3y ago
  Definitely unnecessary for them to be a part of every educated person’s library. This is a pretty absurd viewpoint. The reality is that there are loads of educated people who contribute vast amounts to the world without even the most passing familiarity with “modern math”. I also don’t agree with the sentiment of the rest of the post, sorry. If someone is casually or recreationally interested in math, building a comprehensive view of modern math just… isn’t important. If you’re interested in spending more time with math, just do it—don’t let the perfect be the enemy of the good. There’s no right approach other than the one that makes you happy.
  - rramadass 3y ago
    Hard disagree. These books are not textbooks but Companions thus giving one history, development, concepts, ideas etc. of Mathematics. This is very similar to Encyclopedias/Dictionaries that every family generally has in its "personal library" but just more detailed w.r.t. Mathematics. By "comprehensive" i meant the books are comprehensive and not that one has to study every bit of it. Pick and choose topics/terminologies as you come across them at whatever level you want. You are adding to your education and not necessarily trying to become a "Mathematician". Every "educated person" (STEM or not) with curiosity to understand more about the World (and all-pervasive technologies in particular) has to have some idea of "Modern Maths" at some level. This is why you have so many "popular science and mathematics" books. While these companion books are not "popular maths" books they make a very good reference for when one wants more breadth and depth but not necessarily a full-blown textbook. These are also great "motivators" for students by sparking their curiosity (and perhaps inspire some to become Mathematicians)) since they catalog so much within one pair of covers.
sfpotter 3y ago
I haven’t looked at it. If it seems cool and fun, why not? If you’re near a university with a decent math library, I would start there. Wander around the stacks and see what you find anything that engages you. If not, don’t sweat it—do something else.

laichzeit0 3y ago

Yeah I laughed a bit when I saw Rudin’s book on that list. Understanding Analysis would be a better choice, even the end chapters of Spivak’s Calculus are a great into to Real Analysis.

harshreality 3y ago
I agree and I don't know why the myth persists that [baby] Rudin is the best way to learn analysis. It's certainly not so for a lot of people who could learn analysis perfectly well from friendlier (but not necessarily any less rigorous) sources. Rudin requires a lot of self-motivation. In addition to Abbott which you mentioned, "How to Think About Analysis" (Alcock), good gentle introduction for anyone who tries to dive into an analysis textbook and hits concrete. "Real Mathematical Analysis" (Pugh), good, rigorous approach that doesn't require quite as much effort to power through as Rudin does, although some of the exercises are tough. "Analysis I" and II (Tao), is good and starts from scratch building numbers from the Peano axioms, so there are no other unproven assumptions underlying everything later. "Counterexamples in Analysis" (Gelbalm & Olmstead), very helpful for understanding some pathological cases that break theorems one might intuitively believe are true. There's no reason today why someone can't grab pdfs of all of the above (even Rudin, although it's never been digitally typeset), from any friendly internet library, and use them all to build a better understanding rather than relying on one alone.