I started my academic life as a biologist, and when I switched to mathematics, nothing annoyed me in my new chosen discipline more than the expressions “we will start from scratch”, which still fills me with foreboding, and “mathematical maturity”, as in: “The reader does not need to know , , and , but she is expected to have mathematical maturity”. After more than 25 years of doing mathematics, I can report that I still often (very often; too often) find myself lacking in this desirable quality, and that it does stand for something tangible.
So what is it? It seems to me that it is the ability, or the knack, for intuitive mathematical thinking, the equivalent of leaning your body into a curve as you ride a bicycle, and the knowledge, without measuring, how much coffee to put in a сafetière. By intuitive mathematical thinking I mean a set of heuristics one brings to bear on a problem unconsciously.
Let me give two examples that certainly can be done by conscientious slog, but which are immediate if you have this mythical mathematical maturity, or at least if it visits you when you are confronted with a problem. The key here, like in a cryptic crossword, is first to see what the solution is, and then work out why.
1. A move from Falconer’s book. Kenneth Falconer is an authority on fractal geometry; his book with that title is a classic, and is in particular used to teach the subject in a master’s course in the Open University. The book requires quite a bit of mathematical maturity to make sense, even though it develops everything from scratch. Reading one of Falconer’s proofs, you realise that he is using the following result:
If are positive numbers, and ,
Why is this true?
2. A remark from Evans and Gariepy. In their wonderful book Measure Theory and Fine Properties of Functions, L. C. Evans and R. F. Gariepy introduce (outer) measures on a set : a mapping is a measure on (an outer measure really) if and whenever . It follows from this definition (check) that if , . Then they make another definition: A set is -measurable if for any set ,
Immediately after this definition, they make a remark, which they do not bother to prove: if then is -measurable. Can you see why?