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?

(MG)

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1. The equation can be rewritten as min(a,b) = (a/b), so b*(a/b)^s >= b*a/b = a. If b is smaller, (a/b)^s >= 1, so b*(a/b)^s >= b*1 = b. The case where they are equal is also easy.

With (2) I am confused as to why they didn’t just write that mu({}) is mu-measurable. The other cases where mu(A) = 0 seem like a slog to work out.

whoops, something got eaten there. let’s try again.

1. The right hand side of the equation can be rewritten as b*(a/b)^s, which makes it much easier to understand. If a is smaller, (a/b)^s >= (a/b), so b*(a/b)^s >= b*a/b = a. If b is smaller, (a/b)^s > 1 so b*(a/b)^s >= b*1 = b. The case where they are equal is also easy.

With (2) I am confused as to why they didn’t just write that mu({}) is mu-measurable. The other cases where mu(A) = 0 seem like a slog to work out.

Small correction, (a/b)^s > 1 should be (a/b)^s >= 1 because s can be 0.

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Mathematical maturity is hard to acquire and hard to describe if a person acquired mathematical maturity at a young age which would probably be most mathematicians. You basically have to try to separate logical thinking from rational thinking and hit a wall enough times to figure out how to separate the two from each other and how make use the two perspectives to find your answer. It is NOT fun, it takes a long while, and while attempting this psychologically you will be unstable because that is not how brains work on a day to day basis. Honestly it is a wonder that I was stubborn enough to acquire the blasted thing BUT it is a requirement for “getting” most mathematics. When a person acquires mathematical maturity entirely depends on how resistant a person is to working with, trusting, and creating perceptions that have reasons why they don’t match up with what is “real”. Something I resisted enough that my first solution using mathematical maturity was that trying to rationalize logic until I pass out for 16 hours is not rational because i tried that passed out for another 8 hours and am about to pass out again if i keep trying. I have my doubts that maintaing the maturity was worth the trouble but because I have it I’ll use it.

The way that solution solved the problem was that if you pass out then you solve nothing and my attempt to rationalize logic and passing out were at that point directly related.