= 27d567bc2d48d9f40e9ccc20ea370b15)
cursed comments on Open Thread for February 11 - 17 - Less Wrong
You are viewing a comment permalink. View the original post to see all comments and the full post content.
= 783df68a0f980790206b9ea87794c5b6)
Loading…= b715d02e85f7b3b4ae65ca3d3e450868)
Subscribe to RSS Feed
Powered by Reddit
= f037147d6e6c911a85753b9abdedda8d)
You are viewing a comment permalink. View the original post to see all comments and the full post content.
Comments (325)
I'm interested in learning pure math, starting from precalculus. Can anyone give advise on what textbooks I should use? Here's my current list (a lot of these textbooks were taken from the MIRI and LW's best textbook list):
I'm well versed in simple calculus, going back to precalc to fill gaps I may have in my knowledge. I feel like I'm missing some major gaps in knowledge jumping from the undergrad to graduate level. Do any math PhDs have any advice?
Thanks!
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?
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,
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).
For what it's worth, I'm doing roughly the same thing, though starting with linear algebra. At first I started with multivariable calc, but when I found it too confusing, people advised me to skip to linear algebra first and then return to MVC, and so far I've found that that's absolutely the right way to go. I'm not sure why they're usually taught the other way around; LA definitely seems more like a prereq of MVC.
I tried to read Spivak's Calc once and didn't really like it much; I'm not sure why everyone loves it. Maybe it gets better as you go along, idk.
I've been doing LA via Gilbert Strang's lectures on the MIT Open CourseWare, and so far I'm finding them thoroughly fascinating and charming. I've also been reading his book and just started Hoffman & Kunze's Linear Algebra, which supposedly has a bit more theory (which I really can't go without).
Just some notes from a fellow traveler. ;-)
"Not liking" is not very specific. It's good all else equal to "like" a book, but all else is often not equal, so alternatives should be compared from other points of view as well. It's very good for training in rigorous proofs at introductory undergraduate level, if you do the exercises. It's not necessarily enjoyable.
It's a much more advanced book, more suitable for a deeper review somewhere at the intermediate or advanced undergraduate level. I think Axler's "Linear Algebra Done Right" is better as a second linear algebra book (though it's less comprehensive), after a more serious real analysis course (i.e. not just Spivak) and an intro complex analysis course.
Oh yeah, I'm not saying Spivak's Calculus doesn't provide good training in proofs. I really didn't even get far enough to tell whether it did or not, in which case, feel free to disregard my comment as uninformed. But to be more specific about my "not liking", I just found the part I did read to be more opaque than engaging or intriguing, as I've found other texts (like Strang's Linear Algebra, for instance).
Edit: Also, I'm specifically responding to statements that I thought referring to liking the book in the enjoyment sense (expressed on this thread and elsewhere as well). If that's not the kind of liking they meant, then my comment is irrelevant.
Damn, really?? But I hate it when math books (and classes) effectively say "assume this is true" rather than delve into the reason behind things, and those reasons aren't explained until 2 classes later. Why is it not more pedagogically sound to fully learn something rather than slice it into shallow, incomprehensible layers?
I think people generally agree that analysis, topology, and abstract algebra together provide a pretty solid foundation for graduate study. (Lots of interesting stuff that's accessible to undergraduates doesn't easily fall under any of these headings, e.g. combinatorics, but having a foundation in these headings will equip you to learn those things quickly.)
For analysis the standard recommendation is baby Rudin, which I find dry, but it has good exercises and it's a good filter: it'll be hard to do well in, say, math grad school if you can't get through Rudin.
For point-set topology the standard recommendation is Munkres, which I generally like. The problem I have with Munkres is that it doesn't really explain why the axioms of a topological space are what they are and not something else; if you want to know the answer to this question you should read Vickers. Go through Munkres after going through Rudin.
I don't have a ready recommendation for abstract algebra because I mostly didn't learn it from textbooks. I'm not all that satisfied with any particular abstract algebra textbooks I've found. An option which might be a little too hard but which is at least fairly comprehensive is Ash, which is also freely legally available online.
For the sake of exposure to a wide variety of topics and culture I also strongly, strongly recommend that you read the Princeton Companion. This is an amazing book; the only bad thing I have to say about it is that it didn't exist when I was a high school senior. I have other reading recommendations along these lines (less for being hardcore, more for pleasure and being exposed to interesting things) at my blog.
I feel that it's only good as a test or for review, and otherwise a bad recommendation, made worse by its popularity (which makes its flaws harder to take seriously), and the widespread "I'm smart enough to understand it, so it works for me" satisficing attitude. Pugh's "Real Mathematical Analysis" is a better alternative for actually learning the material.
I would preface any textbook on topology with the first chapter of Ishan's "Differential geometry". It builds the reason for studying topology and why the axioms have the shape they have in a wonderful crescendo, and at the end even dabs a bit into nets (non point-set topology). It's very clear and builds a lot of intuition.
Also, as a side dish in a topology lunch, the peculiar "Counterexamples in topology".
Keep a file with notes about books. Start with Spivak's "Calculus" (do most of the exercises at least in outline) and Polya's "How to Solve It", to get a feeling of how to understand a topic using proofs, a skill necessary to properly study texts that don't have exceptionally well-designed problem sets. (Courant&Robbins's "What Is Mathematics?" can warm you up if Spivak feels too dry.)
Given a good text such as Munkres's "Topology", search for anything that could be considered a prerequisite or an easier alternative first. For example, starting from Spivak's "Calculus", Munkres's "Topology" could be preceded by Strang's "Linear Algebra and Its Applications", Hubbard&Hubbard's "Vector Calculus", Pugh's "Real Mathematical Analysis", Needham's "Visual Complex Analysis", Mendelson's "Introduction to Topology" and Axler's "Linear Algebra Done Right". But then there are other great books that would help to appreciate Munkres's "Topology", such as Flegg's "From Geometry to Topology", Stillwell's "Geometry of Surfaces", Reid&Szendrői's "Geometry and Topology", Vickers's "Topology via Logic" and Armstrong's "Basic Topology", whose reading would benefit from other prerequisites (in algebra, geometry and category theory) not strictly needed for "Topology". This is a downside of a narrow focus on a few harder books: it leaves the subject dry. (See also this comment.)
Maybe the most important thing to learn is how to prove things. Spivak's Calculus might be a good place to start learning proofs; I like that book a lot.
I'm doing precalculus now, and I've found ALEKS to be interesting and useful. For you in particular it might be useful because it tries to assess where you're up to and fill in the gaps.
I also like the Art of Problem Solving books. They're really thorough, and if you want to be very sure you have no gaps then they're definitely worth a look. Their Intermediate Algebra book, by the way, covers a lot of material normally reserved for Precalculus. The website has some assessments you can take to see what you're ready for or what's too low-level for you.