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comment Is the max of two differentiable functions piecewise-differentiable?
Perfect answer. Combined with the zero-accumulation point example, it basically answers all versions of the question.
comment Is there a more general concept than position and space?
"In my understanding, in general, the position of an object should be absolute." -- why ?? What does it even mean? There are no coordinates in "real life".
comment Why do mathematicians sometimes assume famous conjectures in their research?
These days a lot of mathematicians do research so they can publish their results, so they can get money and other benefits. The mathematical community considers it acceptable to publish interesting results relying on an assumption of a famous conjecture. Thus, this increases the set of potential topics one can do research on and then write about: you can prove a new theorem, or you can prove a new theorem while assuming a famous conjecture. Given the number of people who do research these days, many available niches end up being filled.
comment Probability of picking specific balls
I may have a few more.But I hope you get the idea.
comment Probability of picking specific balls
I have a few mistakes, actually. Fixed
comment Probability of picking specific balls
This is my understanding of the question. There are 10 events. At each event you pick one red and one blue, and discard both. An event is "good" if both balls have stars. If we try 10 times, we can expect between 0 and 7 "good" events. What is the probability that there will be exactly five "good" events out of 10.
comment What is the best way to develop Mathematical intuition?
Lastly, I don't see that much difference between the statement about a circle and the statement about A and B. I don't think that you really need to prove that a circle divides the plane into two parts. Maybe you do need it if you are building all geometry from scratch using a system of axioms - not Euclidean axioms, these will not be sufficient to prove this statement. Euclid did not define things like a point on a line lies between two other points. On the other hand, tautologies may still need a proof. (Well, it depends on which logic/model/proof system we are working with.)
comment What is the best way to develop Mathematical intuition?
"One important property of a mathematical proof is that it does not use inductive reasoning (not to be confused, of course, with induction). I don't think this fact is obvious by any means." - well, I don't think that this fact is relevant. People who don't understand proofs, don't understand what is "inductive reasoning" either. My point is that you should not try to explain the theory behind proofs, unless you are teaching advanced mathematical logic. Just expose people to simple proofs, and they will pick it up.
comment Proving that a right (or left) inverse of a square matrix is unique using only basic matrix operations
"...and thus there's only one B sufficing the equation." How did you conclude that? Why do you need to show that A is row equivalent to I? Focus on the question: uniqueness.