What is meant by “mathematical maturity”?

I have often heard people talk of "mathematical maturity", sometimes in the sense of the maturity required to understand an area of mathematics or in the approach to a problem or proof.

However, it's not very clear to me what exactly is meant by this, though I get the feeling that it's not just knowledge of some basic or advanced areas of mathematics. Is it an ability for abstraction, or a mathematical intuition, or something else entirely?

• – lhf Nov 29 '13 at 17:03
• I think Terry Tao's post "There’s more to mathematics than rigour and proofs" is relevant. – Rahul Nov 29 '13 at 17:04
• So in some sense, would you say that mathematical maturity is an ability to combine intuition with rigour? – Milind Nov 29 '13 at 17:10
• What's it like to understand advanced mathematics quora.com/Mathematics/… – GTX OC Nov 29 '13 at 17:12
• I as I understand the term, I would put it into opposition to knowledge of any specific area of mathematics. It basically means you are used to the mathematical way of thinking and can follow abstract arguments. Thats my 2 cents. – Adam Nov 29 '13 at 23:54

Example Consider the the futile attempt to at first to percieve $\Bbb R^4$. If you were to construct some wooden axis in your room next to you; then when you tried to think how to attach an additional 4th axis so that it is perpendicular to the others yet independent, I know I at least would probably run into trouble. But if you let go for a second, of your intuition. You might find you would actually get further and even have a better intuitive understanding. If you instead focused on natural extensions and what it requires. Or studied current developed notions on $\Bbb R^4$, you may get farther.