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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?

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    $\begingroup$ See also math.stackexchange.com/questions/438284/mathematical-maturity. $\endgroup$ – lhf Nov 29 '13 at 17:03
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    $\begingroup$ I think Terry Tao's post "There’s more to mathematics than rigour and proofs" is relevant. $\endgroup$ – Rahul Nov 29 '13 at 17:04
  • $\begingroup$ So in some sense, would you say that mathematical maturity is an ability to combine intuition with rigour? $\endgroup$ – Milind Nov 29 '13 at 17:10
  • $\begingroup$ What's it like to understand advanced mathematics quora.com/Mathematics/… $\endgroup$ – GTX OC Nov 29 '13 at 17:12
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    $\begingroup$ 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. $\endgroup$ – Adam Nov 29 '13 at 23:54
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Mathematical maturity is a departure from the way we typically would percieve lower level mathematics. In multivariable calculus or real analysis for example you would typically study Euclidean space which typically is not argued nor is there any trouble making familiar connections with this space to things we feel we understand.

Higher levels of mathematics tends to pride it self on the ability to study things abstract and often the less that it is applicable to application the better.

Typically I have found in graduate text that you have reached a level of mathematical maturity when you require rigorous axiom alone to begin work in the field. You will proceed quickly through the material this way as apposed to taking the time to draw tedious connections to something familiar which frankly may not exist until the introduction of this new concept.

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.

Of course mathematical maturity does not mean to get rid of intuition or to ignore our own internal logic. Thus it is something we all strive for yet who knows which if any of us are actually "mathematically mature".

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