How to self study pure math – a step-by-step guide [video]
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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”.
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.
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?
Chatgpt helps remove these roadblocks so you get shown the way and then figure out how it was done after.
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.
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.
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.
https://math.unm.edu/~crisp/courses/math401/tao.pdf
https://chat.openai.com/share/6026ae38-e7f6-439f-b91f-110046...
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.
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.
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.
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.
It sucks at 3-years old level novel logic, let alone math proofs.
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...
See also similar concept maps from the other books: https://minireference.com/static/conceptmaps/math_and_physic... and https://minireference.com/static/conceptmaps/linear_algebra_...
It has solutions for many exercises.
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.
Sidenote: I recommend linear algebra done wrong over Axler’s text. But that was just my personal preference. Both would be good introductions.
Summary available here: https://www.askyoutube.ai/share/6533668f27f67cda427ab90f
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.
https://www.amazon.co.uk/Mathematics-Its-History-Undergradua... by Stilwell
Which is a history of mathematics but with serious examples to work on, and may appeal to self-study maths fans.
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.
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.
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.
Being disconnected to reality is a feature in pure math.