I found chatgpt to be a game changer for people who want to read a math text without getting stuck on something because you are missing a gap in a math prerequisite.
Chatgpt essentially fills in the gap between a concept you do understand with a concept that is in the text.
You might not get the full treatment of the gap, but it’s often enough for you to move forward in a math text.
Also these annoying “proof left as an exercise” is now “ask chatgpt for the answer”.
I think not doing the proof exercises is problematic. Just understanding some concepts on how to vaguely go from A to B isn't really enough for math, need to be able to come up with how to do it. That is the hardest part in self-study, because skipping is easier than in some class or seminar setting (and if not in a group can also not discuss problems and see how other people approach them).
I think that is problematic for math because figuring out the way is the math. Need to be super disciplined not to ask chatGPT questions that go far into that territory but rather present it with a way devised and then discuss (if possible).
Of course passively reading ChatGPT transcripts isn’t going to help you master math.
But equally problematic is the said roadblocks that - prior to ChatGPT - simply led many learners to stop learning. We finally have a mechanism to adjust difficulty level to the mastery level of the learner. This is fantastic.
I think chatgpt's effectiveness for self-studying would depend on the subfield of pure math. For example, real analysis I believe is still best self-studied by just reading baby rudin and doing the examples and exercises. However, I really could not make much progress on topology until chatgpt walked me through what the open set axioms actually meant in the context of metric spaces (which most of the topological spaces one encounters are), otherwise they just seemed very arbitrary.
In my opinion, it is less dependent on the subfield than the textbook you use for that subfield. Unfortunately, math textbook recommendations are relatively subjective, with many popular choices unsuitable for self-study, or even study.
With regards to topology, your experience rings true. In short, anyone with knowledge of calculus / basic real analysis wanting to learn topology should read "Real Analysis" by Carothers.
Usually topology is taught after real analysis, extending many results that hold on the reals as the main motivation. But this process is quite abrupt without the intermediate context of metric space, leaving many people confused. It doesn't help that Baby Rudin is quite terrible at teaching these concepts for you. On the other hand, Carothers' book is a paragon of mathematical exposition. It excels at telling you why metric space, topological space, and all the definitions are made that way.
With regards to the parent, I have to say "Proof is left as exercise" is probably the number one thing that forces students to actually read the texts. The best way to learn is to ask ChatGPT after you're stuck, not before.
I have worked through some of Carothers myself and like it a lot.
Are there other math books you think highly of that are similar, i.e. good for explaining why the definitions are the way they are as well as teaching the material?
> Also these annoying “proof left as an exercise” is now “ask chatgpt for the answer”.
ChatGPT is horrendously bad at mathematical proofs (I've written about this before), so I fear that this is a rather dangerous approach: you could be learning things that are wrong without realising it.
I just opened up a Tao’s real analysis pdf and copy and pasted one of their exercises (didn’t cherry pick), would be great if you can point out what is bad about it.
Feel free to take another exercise from that Tao’s real analysis pdf if you think you have a good counter example, would love to know its limits on honours undergrad level math texts.
Establishing a proof of a statement in mathematics means giving a series of steps of the form
A(1) && A(2) && ... && A(n) && B => A(n+1),
where each of A(i) has been proved earlier, B is one of the axioms with which you are working, and => means derivation using some fixed rules. The axiom list for B is context dependent, so that a journal paper may be using an extended set of the form "everything already known by the community, given a reference", etc. A textbook will use lower level textbooks mentioned in the introduction as lists of such contextual axioms.
The IMHO biggest issue with this chatgpt proof is that even though correct in principle, it really misses the context of your exercise: it does not really know if e.g. the well orderedness principle had been introduced already, which exact definition of the natural numbers is being used (Peano, intuitive?), etc.
As a result, the "proof" it provides is primarily name dropping — albeit correct in principle, it still requires filling in the actual argument. So, might be helpful as a hint for a student, but requires the proof to actually be produced.
Chatgpt would have no problems introducing the well-ordered principle if prompted, so the self learner can dive into the details if needed.
Hints are probably what you want if you’re a self learner and stuck, so actually chatgpt is doing a good job to guide self learners through a text they’re stuck on.
Being stuck on one statement for days is not a strategic way to learn.
> Chatgpt would have no problems introducing the well-ordered principle
Yes, well, that is part of what I was trying to say. The statement of the principle is not important outside of the structure you are building when following one particular proof, or reading a book (so, following several proofs).
You could do just as well with Zorn's lemma or axiom of choice as you would with the well-ordering; what if your course introduces one of these, but not the well-ordering principle, and then asks to solve this particular exercise? In that case, the gist of the exercise would actually be to re-derive, say, the (axiom of choice)=>(well-ordering) implication for the natural numbers. A point that would be thoroughly lost on chatgpt without the course context.
When ChatGPT’s output looks correct, it usually just means that it actually met the problem already, and “learned” the answer verbatim, and now it just applies some transformations to its output to fit your context better, giving the illusion of something more than a search engine would have done.
It sucks at 3-years old level novel logic, let alone math proofs.
This is about computability, not analysis, but I think the point still applies: ChatGPT is quick to give you an answer that sounds plausible but is actually complete nonsense.
I like the selection of topics and the resources recommended, but I'd love to see some recommendations of "front the ground up" resources, like start with ELI5 stuff like variables, math operations, expressions, equations, functions, notation conventions, etc. In my experience as a tutor, many people have gaps in their "basic math" knowledge, so it makes learning pure math topics more difficult to access (like if you don't know what the symbols are, you can't see the beauty of the thing).
I've been working on addressing this problem through my books (links in profile), because I think it's important to consider the general audience, which means including all the "high school math prerequisites in each book.
For anyone who is wondering how to fill in these basics skills, the khan academy videos + interactive exercises are probably the best way to go: https://www.khanacademy.org/ If you prefer a book on the topic, check mine: https://nobsmath.com Here is the concept map from the book that lists the various topics normally covered in high school math that I think everyone should know: https://minireference.com/static/conceptmaps/math_concepts.p...
Oh wow, that's a great book indeed. It covers exactly the topics I was referring to (high school math), but also touches on some more advanced stuff (e.g. permutations).
I find the lecture series on the mathmajor youtube channel extremely well made, with very clear explanations, and giving a good feel for what it means to study pure maths.
I really like the MathMajor channel too but for the moment it's very Abstract Algebra centric. For those of you who speak French I can also recommend https://m.youtube.com/@MathsAdultes with videos made by a teacher whose students are future Maths teachers and cover a broad variety of topics like Series, Number theory, Measure theory (from Lebesgue to the dominated converge theroem),...
The latest videos are about Topology.
Did a lot of that, before, during, and/or
after formal courses, and the results were
good.
In one case, in a subject I'd carefully
studied on my own, I was required to take
a course given by the department Chairman.
The course was mostly filtering, not
education, and as I did really well
totally torqued the Chairman, angry that
he was not able to filter me.
Eventually I got my degree, and he got
fired -- his attacks on me were not the
only reason he got fired.
Here is something of a standard
sequence:
(1) first year algebra
(2) plane geometry
(3) second year algebra
(4) trigonometry
(5) solid geometry
(6) first, second year calculus
(7) abstract algebra
(8) linear algebra
(9) analysis
(10) multi-variate advanced calculus
(11) differential equations
(12) measure theory, Banach space, Hilbert
space, Fourier theory
(13) Probability, stochastic processes
(14) statistics
Keys: Get the best books, 1-3 books per
subject. This point is crucial since
actually most of the books are not very
good and the best books are MUCH better
than the rest.
Then read the definitions, theorems, and
proofs, and work some or all of the
exercises. Doing this, obtain intuitive
explanations and the reasons the material
is important.
While it is good for each proof to see why
it works, commonly people are flexible on
just how hard they study the proofs.
Maybe as a check on the quality of your
progress, sit in on a few sessions of a
few courses at a good math department.
This sequence (1) -- (14) is a good
start for more specialized topics and
research and/or applications.
One good research career direction is to
have such a sequence and then leave math
for research in some other field where can
apply the math. E.g., I had a fellow
student who did that and ended up with a
good career applying his math in biology
and ecology in an Ivy League university.
For a research career, need to get a sense
of what constitutes good research: The
material in (1) -- (14) is the finely
filtered and polished result of 200 years
or so of math and too good for most
research now.
In nearly any applied field, take one of
their problems, look a little more closely
than usual, and will see where want some
new tools and results.
On a surface level this is easy (I should say straightforward) to do, however the time investment is significant. There's so many other things I might want or even have to do with my time that working deliberately through 3 books for each of 14 topics just isn't feasible.
I like the video’s focus on resources. I’ve worked through several of those texts myself and can vouch for them. Though I was hoping a step by step guide might touch on things like making outlines, reviewing notes at a later date, etc.
Sidenote: I recommend linear algebra done wrong over Axler’s text. But that was just my personal preference. Both would be good introductions.
What about the 'nuts and bolts' of integration, differentiation, etc. Why is linear algebra covered but no calculus? Even if calculus is covered in real analysis, I think calc needs its own course, and is the most important one by far as a building block before you dive into the other stuff. Otherwise, you will be stuck and not knowing any of the notation. No mention of partial differential equations either, which is also very important. You need to learn how to master change of variables and substitution--those come up constantly in advanced concepts.
Linear Algebra courses in pure math degrees are typically used as a gentle introduction to theoretical math, different from the applied linear algebra courses that engineers and the likes would take.
Just based on my experience with UCLA's pure math undergraduate program (which I assume is similar to other top tier math departments), the listed subjects in the video cover the "core/required" courses of a solid pure math degree. PDE's and differential equations in general are optional electives (edit: usually taken by the applied variants of math degrees).
It should also be noted that calculus is a pre-req for even declaring the major, and hope the video should list that as a pre-req for self-study as pure math.
Fwiw, I believe the romanticization of advanced mathematics needs to end. I'm 15-20 years out of university, and I had thought I might delve into some specific area of math after graduating. I studied EECS and only took courses in abstract algebra and real analysis. But pursuing further math never materialized for me. Moreover, all of my friends who majored in math transitioned away from it within five years of college. Most became developers, a few became product managers, but I haven't seen anyone maintain or even express an interest in advanced math outside of academia, especially in their 20s and 30s.
I have a PhD in pure math. Nothing I learnt specifically in that PhD is needed for my job, and in fact, I do a job where most people who do the same thing as me don't have a PhD. So its value even for signalling is limited.
However, I'm glad I did the PhD.
The 4 years I spent on it were time in my life well spent. I enjoyed the work and it didn't bother me to be earning less than I otherwise would have.
The main skill I gained from the PhD was being able to read technical papers - typically involving stats or financial math. I have needed to do this on and off for my work, and I've found that lots of people who have good high school or college math find this much much harder than I do.
I study math for fun. I find the ability to do this precious and life affirming. When I read an interesting article in Quanta, I can look at the papers cited, if it's a field I have some background in, and make more sense of them than the average reader. In ten years of doing this, I have once published a short paper which added a small improvement to a recreational problem. So studying math isn't really about external achievement - I am the equivalent of someone who plays the guitar at home but will never have a gig or record a song. But I feel very lucky to exist in a time period when I had access to this educational opportunity, and when so much interesting math is available to read and study essentially for free.
You see that here on HN with all the articles on Category Theory. CT appeals to people who majored in Comp Sci but long for some advanced and abstract mathematics to get away from the daily dose of Node. It is simplistic enough that you can read the first few chapters of a CT book and feel like you're really getting some deep math topics.
I'm a huge fan of Haskell and the language uses many concepts from category theory, such as monads, functors, and applicatives. However, these are all abstractions which are simple enough to not need any basis in mathematics. I tried to read resources on category theory to understand its motivation, but unlike real math where theorems actually seem to contain new insight, I never found such a thing in category theory.
The entire body of CT has just a small handful of what anyone would consider theorems or lemmas. It’s mostly just stacked definitions.
This is a set. This is a mapping. This is a set of sets. This is a mapping from elements to elements. This is a mapping from elements to sets. This is a mapping from sets to sets. This is the inverse mapping…. Chapter after chapter.
But then there’s “insight” that all of mathematics can be cast as CT, because… all math is just things that map to things! Whoa, far out, dude. Mind is blown.
I think what's happening is a lot of people who studied very little or no math are realizing a few 100 level courses in things like linear algebra, probability, stats, calculus etc would be very helpful. They might not realize that the cost/benefit beyond that is minimal. It's similar for many subject areas. A couple 100 level courses in accounting is super helpful, same for finance, law, mechanical engineering, electrical engineering, computer science, biochemistry and on and on.
Nothing you said supported your first sentence. If people enjoy pure math why shouldn't they study it? Now people should not feel inadequate if they decide not to study it.
What math helps with is signaling. Employers care about your problem solving skills (as opposed to actual mathematical knowledge) and there’s virtually no better signal than IMO/Putnam or their easier variants. On the other hand a mediocre math phd is not all that useful at all outside of publishing mediocre papers in mediocre journals, since the student probably just has a lot of knowledge but not many problem solving skill.s
When something is advertised as a guide to "study pure math," why is everyone's kneejerk reaction to recommend that the guide include resources that aren't "pure math."
Chatgpt essentially fills in the gap between a concept you do understand with a concept that is in the text.
You might not get the full treatment of the gap, but it’s often enough for you to move forward in a math text.
Also these annoying “proof left as an exercise” is now “ask chatgpt for the answer”.