Why is math intuition hard for some areas and less for others?
Enrico Bombieri is one of the world leaders in many areas of mathematics, including number theory, algebraic geometry, and analysis. He has won many awards, and is a Fields Medalist.
Today I want to talk about the notion of intuition in mathematics. I am curious what it is, how to get it, and how to use it.
One story, perhaps an urban legend, is that a senior expert in real analysis was once sent a paper that “proved” a surprising theorem. The expert looked at the proof, and was immediately skeptical. The “theorem” seemed to be too surprising—his intuition based on his great experience, was that the theorem could not be true. Yet even after hours of studying the proof he could not find any mistakes. But his intuition continued to bother him. He finally looked even more carefully, and found the problem. The author of the proof had used a lemma from a famous topology book. He had used the lemma exactly as it was stated in the famous textbook. But there was a typo in the book. Somehow the words “closed” and “open” had been exchanged in the statement of the lemma. This made the lemma false, caused a gap in the proof of the surprising theorem, and left the poor author with a buggy paper.
Proving theorems is not mechanical; proving theorems does require formal manipulation. Yet proving theorems also requires the use of intuition, the ability to see what is reasonable or not, and the ability to put all these together. Blindly using a lemma from even the most famous textbook can be dangerous, as the story shows.
I once lost several months of hard work trying to use a published theorem to solve an open problem. I almost had a proof, but eventually like the story I found a bug in the published result—the result was the sole result of a friend’s Ph.D. thesis. Oh well. This is not an urban legend, I was there, but I will discuss it another time.
Mathematical Intuition—What Is It?
Published
Iron in the Sand 
