Math Self-Study Progress Report 3 (and Gemini's Betrayal, Sort Of)

I just finished Chapter 3 of Stephen Abbott's Understanding Analysis.  It's only around 25 pages, but I feel that some of the concepts are going to be important in later chapters that I took time to digest them.  Another reason that I went slow was because I tend to struggle with set theory.  I still remember when I first got the taste of it back in Form 7.  It was unlike any other material I've studied before.  I was told that that was real math, and what I had been studying before was merely computation (not that computation is easy).  Anyway, to me, definitions and theorems in set theory always read like legal documents where every word matters.

Oh! I finally got the physical book from Amazon.  It makes my study so much better.  I still prefer physical books, especially for studying, because I usually jump back and forth for reference.

Chapter 3 started by introducing the Cantor set, one of the most non-intuitive mathematical objects I've encountered.  It then went through open and closed sets, compact sets, open covers, perfect and connected sets, and finally concluded with Baire's Theorem.

There are certain things in math that I still remember from secondary school, like how to derive the quadratic formula, or how to expand $\sin{(\alpha+\beta)}$, or integration by parts etc.  I think Cantor set will likely be added to that list.  It's uncountable, yet nowhere-dense, and free of any isolated points.  How can this happen?  In my opinion, Cantor sits right next to Newton, Euler and Gauss as the greatest mathematicians of all times.

I've been using Gemini Pro throughout, with occasional use of DeepSeek when I needed a "second opinion."  This time, I didn't prompt Gemini with things like "Be my TA", but just asked him to check my answers and clarify concepts.  It certainly toned down the unnecessary praise and I felt much more comfortable.  Again, Gemini has become indispensable in my self-study because it corrected a lot of mistakes I made in exercises that I would otherwise miss.  And of course, there are exercises that I would not be able to complete without Gemini's hints.

Speaking about hints, there was one exercise where I was asked to come up with a way to construct a perfect set of irrational numbers.  It gave me two hints, each of which pointed to a slightly different direction.  Here's the screenshot.  Click to enlarge.

So, I thought about #1 for around a day and asked for help again.  Here's Gemini's response.

So, I explored the closure of $F$, thinking that I would get what I desired.  But then when I tried to prove it, I got stuck.  So, I asked Gemini again.  This time, the response felt like betrayal.  I was quite disappointed when I read it.

Did Gemini intentionally lead me through a dead end path or what?  I don't know.  Anyway, I didn't feel too bad because I don't think it's wasted time.  My brain burned through some cycles, useful cycles, to think through the problem.

I'm quite surprised that I've gotten this far.  Next would be Chapter 4 about continuity.

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There's been a lot of controversy and discussion around math and AI ever since OpenAI announced a solution to the Naiver-Stokes equation.  Terrance Tao, along with 25 Fields medalists, signed an open letter titled "A Severe Misalignment of AI in Mathematics."  I actually wanted to sign, but I don't want to create my own ORCID.  Just too many accounts already.  I agree with the general sentiment of the open letter.  I also have a number of things I want to say, as an enthusiast.

First, I believe the most important things in math are not problems and proofs, but mathematical ideas.  Like what I've just studied regarding Cantor set, it raises questions about the size of sets, and thus the ideas of cardinality, measure, and density were born.   They all describe a set's size but are independent concepts.  Calculus is a mathematical idea, so are complex numbers, and irrational numbers.  The proof that $\sqrt{2}$ is irrational is important, but not as important as the idea of irrational numbers themselves, that there are things beyond fractions waiting to be explored.

Second, the Millennium Problems should not be treated as Math Olympiad.  These problems raised questions that invite further exploration.  They're not meant to be homework sets, which I feel is what OpenAI has been treating them.  In Chinese, "數學不是刷題".

Lastly, I'm optimistic though that AI will not replace mathematicians anytime soon because of the exact same reason.  New mathematical ideas need time to develop.  When Cantor first published his proof that the real numbers are uncountable, it was met with enormous opposition, including from his teacher.  It took the mathematical community 25-30 years to accept Cantor's ideas, which have become the cornerstone of what we're all studying today.  Or take calculus as another example.  Newton and Leibniz started in the 17th century and it took around 300 years to lay a solid foundation, which became what I'm self studying today - real analysis.  AI may be very good at solving problems but I have yet to see any new mathematical ideas developed.  

One last analogy or food for thoughts.  Let's give AI all the knowledge up to the 1500s, and ask it to find a solution to the equation $x^2 + 1= 0$.  I bet AI would come back to say there's no solution.  It takes a human to invent complex numbers.