I advise that you read the first 3 books on your list, and then reevaluate. If you do not know any more math than what is generally taught before calculus, then you have no idea how difficult math will be for you or how much you will enjoy it.
It is important to ask what you want to learn math for. The last four books on your list are categorically different from the first four (or at least three of the first four). They are not a random sample of pure math, they are specifically the subset of pure math you should learn to program AI. If that is your goal, the entire calculus sequence will not be that useful.
If your goal is to learn physics or economics, you should learn calculus, statistics, analysis.
If you want to have a true understanding of the math that is built into rationality, you want probability, statistics, logic.
If you want to learn what most math PhDs learn, then you need things like algebra, analysis, topology.
Thanks, I made an edit you might not have seen, I mentioned I do have experience with calculus (differential, integral, multi-var), discrete math (basic graph theory, basic proofs), just filling in some gaps since it’s been awhile since I’ve done ‘math’. I imagine I’ll get through the first two books quickly.
Can you recommend some algebra/analysis/topology books that would be a natural progression of the books I listed above?
In my experience, “analysis” can refer to two things: (1) A proof-based calculus course; or (2) measure theory, functional analysis, advanced partial differential equations. Spivak’s Calculus is a good example of (1). I don’t have strong opinions about good texts for (2).
Dummit & Foote’s Abstract Algebra is a good algebra book and Munkres’ Topology is a good topology book. They’re pretty advanced, though. In university one normally one tackles them in late undergrad or early grad years after taking some proof-based analysis and linear algebra courses. There are gentler introductions to algebra and topology, but I haven’t read them.
A couple more topology books to consider: “Basic Topology” by Armstrong, one of the Springer UTM series; “Topology” by Hocking and Young, available quite cheap from Dover. I think I read Armstrong as a (slightly but not extravagantly precocious) first-year undergraduate at Cambridge. Hocking and Young is less fun and probably more of a shock if you’ve been away from “real” mathematics for a while, but goes further and is, as I say, cheap.
Given how much effort it takes to study a textbook, cost shouldn’t be a significant consideration (compare a typical cost per page with the amount of time per page spent studying, if you study seriously and not just cram for exams; the impression from the total price is misleading). In any case, most texts can be found online.
There’s some absurd recency effects in textbook publishing. In well-trodden fields it’s often possible to find a last-edition textbook for single-digit pennies on the dollar, and the edition change will have close to zero impact if you’re doing self-study rather than working a highly exact problem set every week.
(Even if you are in a formal class, buying an edition back is often worth the trouble if you can find the diffs easily, for example by making friends with someone who does have the current edition. I did that for a couple semesters in college, and pocketed close to $500 before I started getting into textbooks obscure enough not to have frequent edition changes.)
My guess is that if you have an interest in computer science, you will have the most fun with logic and discrete math, and will not have much fun with the calculus.
If you are serious about getting into a math graduate program, then you have to learn the calculus stuff anyway, because it is a large part of the Math GRE.
I advise that you read the first 3 books on your list, and then reevaluate. If you do not know any more math than what is generally taught before calculus, then you have no idea how difficult math will be for you or how much you will enjoy it.
It is important to ask what you want to learn math for. The last four books on your list are categorically different from the first four (or at least three of the first four). They are not a random sample of pure math, they are specifically the subset of pure math you should learn to program AI. If that is your goal, the entire calculus sequence will not be that useful.
If your goal is to learn physics or economics, you should learn calculus, statistics, analysis.
If you want to have a true understanding of the math that is built into rationality, you want probability, statistics, logic.
If you want to learn what most math PhDs learn, then you need things like algebra, analysis, topology.
Thanks, I made an edit you might not have seen, I mentioned I do have experience with calculus (differential, integral, multi-var), discrete math (basic graph theory, basic proofs), just filling in some gaps since it’s been awhile since I’ve done ‘math’. I imagine I’ll get through the first two books quickly.
Can you recommend some algebra/analysis/topology books that would be a natural progression of the books I listed above?
In my experience, “analysis” can refer to two things: (1) A proof-based calculus course; or (2) measure theory, functional analysis, advanced partial differential equations. Spivak’s Calculus is a good example of (1). I don’t have strong opinions about good texts for (2).
Dummit & Foote’s Abstract Algebra is a good algebra book and Munkres’ Topology is a good topology book. They’re pretty advanced, though. In university one normally one tackles them in late undergrad or early grad years after taking some proof-based analysis and linear algebra courses. There are gentler introductions to algebra and topology, but I haven’t read them.
Great, I’ll look into the Topology book.
A couple more topology books to consider: “Basic Topology” by Armstrong, one of the Springer UTM series; “Topology” by Hocking and Young, available quite cheap from Dover. I think I read Armstrong as a (slightly but not extravagantly precocious) first-year undergraduate at Cambridge. Hocking and Young is less fun and probably more of a shock if you’ve been away from “real” mathematics for a while, but goes further and is, as I say, cheap.
Given how much effort it takes to study a textbook, cost shouldn’t be a significant consideration (compare a typical cost per page with the amount of time per page spent studying, if you study seriously and not just cram for exams; the impression from the total price is misleading). In any case, most texts can be found online.
And yet, sometimes, it is. (Especially for impecunious students, though that doesn’t seem to be quite cursed’s situation.)
Some people may prefer to avoid breaking the law.
There’s some absurd recency effects in textbook publishing. In well-trodden fields it’s often possible to find a last-edition textbook for single-digit pennies on the dollar, and the edition change will have close to zero impact if you’re doing self-study rather than working a highly exact problem set every week.
(Even if you are in a formal class, buying an edition back is often worth the trouble if you can find the diffs easily, for example by making friends with someone who does have the current edition. I did that for a couple semesters in college, and pocketed close to $500 before I started getting into textbooks obscure enough not to have frequent edition changes.)
I am not going to be able to recommend any books. I learned all my math directly from professors’ lectures.
What is your goal in learning math?
If you want to learn for MIRI purposes, and youve already seen some math, then relearning calculus might not be worth your time
I have a degree in computer science, looking to learn more about math to apply to a math graduate program and for fun.
My guess is that if you have an interest in computer science, you will have the most fun with logic and discrete math, and will not have much fun with the calculus.
If you are serious about getting into a math graduate program, then you have to learn the calculus stuff anyway, because it is a large part of the Math GRE.
It’s worth mentioning that this is a US peculiarity. If you apply to a program elsewhere there is a lot less emphasis on calculus.
But you should still know rthe basics of calculus (and linear algebra) - at least the equivalent of calc 1, 2 & 3,