Mathematical intuition is an amazing tool, but it has its limits.

Terence Tao suggests 3 stages of education, which in my opinion could be seen as 3 stages of understanding of any concept or branch of mathematics: 1) The “pre-rigorous” stage, where you are given informal explanations, trying to build a basis of intuition. A lot of approximations and inexactitudes are needed, some part of the intuition is “wrong”. 2) The “rigorous” stage, where you are given formal explanations, definitions, proofs …etc. While in this stage, you will have to go through some notions without understanding what they “mean”. 3) The “post-rigorous” stage, where you abstract the rigorous foundations and build a second order intuition that covers the complete picture of the subject, and can easily be mapped back to a rigorous definition when needed. You can navigate and find directions in the realm of intuition while being able to put down the formality anchor at every step of the way. This is good intuition.

In that sense, good intuition is something you build starting from formal understanding, and it cannot be handed to you. It’s rare that even the best pedagogues, with a strong understanding and intuition about a concept, can provide you with a shortcut to a good intuition without you ever going through the rigorous stage. What they can give you though, is a strategy to understand the concepts and go faster from one stage to the next.

Sometimes, you manipulate a formal definition so much that you can build intuition from it about other higher level concepts, without having to have good intuition about it. Intuition always starts somewhere, some people have very good intuition without being able to prove everything going back from the axioms.

How intuitive a method is shouldn’t be a criteria as to whether we should use it or not. If a method is not intuitive but works, we are good to use it. As Richard Mcelreath puts it in Statistical Rethinking: “Some people still debate statistical approaches on the basis of philosophical principles and intuitive appeal. Philosophy does matter, because it influences development and application. But it is a poor way to judge whether or not an approach is useful. Results are what matter. For example, the three criteria used to derive information entropy, back in Chapter 7, are not also the justification for using information en- tropy. The justification is rather that it has worked so well on so many problems where other methods have failed.”

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