In praise of intuition

There are two ways to study mathematics. One is based on emphasizing rigor, and I’ll abusively refer to this as the mathematician’s approach. The other one, possibly less well known or at least less well received, is more based on intuition. Abusively, I’ll refer to this at the physicist’s perspective. With this blog post, I aim to explain why you should try to incorporate more intuition into your mathematics.

At first, it might seem impossible for intuition to be a useful skill in mathematics. Indeed, if there is one idea that is central to mathematics, it is the idea of rigor. One perspective on math is that it is a game of manipulating initial truths (postulates) through the rules of logic in order to yield new truths. The key goal of this game is that every manipulation that we do be perfectly rigorous, which is why we willfully bind ourselves to the very constraining rules of logic and to very precise definitions, constructions, etc.

Intuition is the exact opposite of rigor. Rigor carefully and exhaustively travels from point A to point B and makes sure that no stone is left unturned. Intuition instead makes far reaching leaps which are based on nothing but hot air and bravado. It would then seem extremely counter-productive to try to use intuition in mathematics which is essentially rigor embodied.

However, thankfully, mathematics is not only about rigor. Indeed, while mathematics is focused on only making statements that are true, it’s also focused on making statements that are interesting. Indeed, consider the following two statements:

• $\displaystyle{ x^2 = 4 \text{ is solved by }x=\pm 2 }$
• $\text{Any equation of the form: }a x^2 + bx + c = 0 \text{ has two solutions in } \mathbb{C} \text{ (you know the formula so I won't write it)}$

Both are true. However, one of them is trivial (not completely so: checking that these two solutions are correct is easy; proving that this exhausts the solutions is more difficult), while the second one isn’t. In fact, the second statement contains infinitely many trivial statements of the first kind. It thus constitutes a much more interesting true statement than the first one !

We then need to consider what makes a true statement “more interesting” or “less interesting”. One key factor is the generality of the statement. If one statement is a particular case of another one, then the second statement is clearly “more interesting”.

This is where intuition comes into play. Intuition enables us to identify patterns. With intuition, we can observe that statements a1, a2, a3 are true, and then conjecture that a general statement A, for which a1, a2, a3 are particular cases, might also be true. However, we don’t stop there, since mathematics is a game of rigor. We return to full mathematical rigor and try to prove that our conjecture is indeed true.

When I was a kid, I loved labyrinths: I had a few books full of them which I loved to go back to. One lesson I learned on them actually nicely illustrates my point: I quickly noticed that it is often useful to actually try to go through the labyrinth the opposite way, starting from the end and returning to the beginning. This is precisely the role that intuition can play in mathematics: it enables us to start from conjectures and to backtrack from them to statements that we know to be true.

In practice, I find it even better to mix the two approaches together. I’ll often start from an intuited conjecture and trying to prove it. In trying to prove it, I improve my understanding of the problem, and I identify problems with the initial conjecture. This means I can sharpen my intuitions in order to build better conjectures which I then try to prove, etc.