When I do math it’s basically like my internal monologue narrating a 3blue1brown video.

Immanuel Kant said a sentence which I find very pertinent for this video : "Concepts without intuition are empty, intuition without concepts is blind ". I truly think that there should be a moderate state of combining rigour and intuition when dealing with mathematical objects.

Decartes: argues that imagining is not important
Also Decartes: invents a way to imagine functions

0:29  Nothing came to my mind 😂
Edit: Thank you guys for 66 likes. I never got this much in my life.

I think it is very hard to ask someone who understands the proof completely how they thought about it the first time they saw it.  The experience of understanding something the first time is psychologically very different than something that has become routine.

I tend to split mathematicians into two broad camps. On one side, holists - people who see mathematics as a unified, geometric and often physical structure(e.g. Felix Klein, Hermann Weyl, Vladimir Arnold, etc.). They lean heavily on intuition, symmetry, and global structure, using geometry and physics (though calling them "intuitionists" would be misleading given the formal meaning of that term).
On the other side, axiomatists - those who prioritize formal systems, logic, and rigorous derivation. The Bourbaki group is the canonical example. Here, clarity comes from structure and proof, not physical intuition in the first place.
Personally, coming from a soviet-style education, I was drawn to the Arnold-type approach. I tend to think of things geometrically or physically. 2:24  I instinctively visualize as a layered geometric object (say, nested hyperspheres with varying densities, where limit of sum is an integral)
But here’s the paradox, in the Soviet system, training is extremely rigorous and disciplined. You’re often discouraged from mixing ideas too early.
In the American system, teaching is more relaxed and holistic from the start. And yet, outcomes seem inverted, the rigid system often produces thinkers who later become holistic, and the flexible system often produces mathematicians who end up valuing disciplinary rigor more.
Maybe I’m overfitting a pattern. But it’s a pattern I keep noticing.

This is so cool. Thinking about how you think about mathematics. Now I'm thinking about how I think about how I think about mathematics. And onward.

Some months ago we had a quiet period at work. Four of us sat at a table, a mix of students and tutors, with experience ranging from first year maths through to postgraduate level. Each of us 4 found it "easiest" to think about Taylor series in a completely different way. Finding ways for us to "bridge" these four approaches was a fascinating exercise.
Thankyou for your time and effort creating these presentations!

Whenever I'm learning anything, I always make schemas then break them then remake etc. my goal is to vascilate between rigour and intuition to sharpen my intuition by practicing critical analysis

When I do math in my head, I compare it to cartography : the semantics and mathematical constructions are the terrain, my intuition is the sense of direction/navigation and the rigour becomes a clear map, that I (or somebody else) has drawn (that match the mathematical constructions with the points on the map(syntax definition, plays a role of  coordinate system) with the proof being a corresponding trajectory both on a semantic-terrain path and syntax-map path).

I am an engineer and MBA. My take on this. Intuitive insights come only after mastery of a discipline. But how we arrived there have to be logically explained. Otherwise, nobody will take us seriously.

I kinda hate the split of "intuition" and "rigor" like they are opposite and we must "combine" them. In my mind they are not only not mutually exclusive, but they support each other. After learning some math history I found that there were very few mathematicians who were able to do math without precision. They had very strong intuition which somehow aligned with the precise notions that came later, so they were able to do math with their intuition. But most people who tried to do math simply made mistakes, or at least didn't find anything interesting, and were forgotten. We only remember the ideas of the few who were able to have intuition so strong they didn't need the rigor, but most people end up writing nonsense when they do that.
Rigor makes things more accessible, in the sense that when your intuition suddenly breaks, you have something to hold on to which makes sure you are not making mistakes. When doing lots of math and seeing lots of examples and proofs you end up building an intuition which is grounded in the rigor. Without rigor, developing the intuition would require you to have a mind that simply "gets" the ideas, and for most people it simply doesn't work (they may think it works when they watch 3b1b videos, but the second they try to write a proof on their own they end up getting confused and writing nonsense).

The video has a point about what would happen if we teach things precisely first. It says that because "most functions are ugly and useless" we'd have to start by teaching about the ugly functions before the useful ones. This is obviously false, as the fact that you don't use the pretty functions in proofs doesn't mean you can't see lots of examples. I think an example of a book who does this well is Spivak's Calculus (BTW many of the things I'm writing here are inspired by the preface to that book). He starts by describing functions using an intuitive "thing that maps elements from a set to another" description. Then when showing examples he starts with "obvious" functions which are easy to visualize, then slowly going to more "weird' examples. He then discusses ambiguities with the intuitive definition and gives the set-theoretic definition with ordered pairs. By that point the reader already has a good idea of what a function is, is able to recognize "nice" functions, but is aware that not all functions are like that. Then Spivak asks how exactly we define which functions are "reasonable", and uses that question as the motive for the rest of the book. The book turns to be basically an adventure, looking at lots of examples of functions and potential definitions of what a "reasonable" function is (things like continuity and differentiability are motivated that way).
When going through the book, the definitions, proofs and examples allow the reader to develop a very strong intuition for the different concepts, and the intuition is always based on rigorous things he saw. At a certain point when looking at a property of a function you don't think "seems reasonable" when thinking of a general function, but you just know "okay if I assume differentiability then it's probably true" (you then have the tools to write down a proof, but sometimes the intuition is so strong and precise that you don't even feel the need to do that, and you still are probably correct).

I once tried to teach C++ syntax using formal languages and production rules thinking that the most "logical" is the easiest way to understand... And well, not a single student asked me about it nor mention the meta-syntax I gave them that day 😔
At least it worked as a kind of experiment to see if an engineer student would like that kind of approach.

Excellent content. I've tried asking people who study math many times about how they think.  Never got much out of them

Styles of thinking form a space and you do math by shifting the problem around in this space, deforming it to reveal deeper structure. Sticking to one style makes you blind to certain things, so it's useful to try and learn how others think.

Intuition gives an area to search for answer while rigor gives the exact direction.

I'm chemist, so I definitely start with the intuitive approach. Thanks to combinatorial explosions.

Intuition is THE mechanism of understanding. If you can't understand it mechanically, you'll be stuck plugging numbers into black boxes without knowledge of what's really going on inside. Mathematics are for quantifying, relating, and describing our understanding of something; not for learning how to understand it. Mathematicians might be a special case, since the math they're often concerned with isn't applied to anything. It's just finding mathematical relationships that might maybe apply to something in theory at some point. Engineers and scientists are typically the ones applying mathematics to real observed principles.

Not sure which I am. In my school days we learned that one apple plus one apple equals two apples in grade one. By grade eight we were doing complicated things like amortizing mortgages still using only arithmetic. Only started algebra in grade nine, learning that one X plus one X equals two X's. Felt to me that we were back to grade one again only adding X's together instead of apples. Some of my fellow students had a terrible time with this though. Then in grade ten we started Euclidean geometry (continued in grade twelve). This combined logical reasoning with picturing shapes. I excelled in this kind of problem solving especially in grade twelve.

The best approach is the synthesis of the two methods. For in the end, the difference between a diagram and a well-formed-formula is syntax: the glyphs we use to communicate concepts. As long as a diagram and a well-formed formula are semantically equivalent, then there is no formal reason to prefer one mode of syntax over the other. Instead, such appraisals are left  to a person's aesthetic disposition or their reckoning of utility. Reality bears witness to this in our books of mathematics having many diagrams, formal propositions, and use of natural language, which together form a tapestry of, ideally, coherent thought.