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?
I would describe mathematical maturity as being like muscle-memory for sports. When I play squash there are certain things that are so ingrained it takes thought and effort to do them wrong: when I serve I know where my racquet is and where the ball is to the extent that I don't look at either of them, I look at where I want to put the ball. Mathematical maturity is similar: when I start reading a paper or text there are things that are automatic in my head and don't need thinking about or explaining by the author.
For example, this morning I was reading through a proof in the theory of bases for Banach spaces. First there was a choice of $\varepsilon '$ made so that $$\varepsilon '= \frac{\alpha \varepsilon}{2(1+\alpha)}$$ Mathematical maturity here is noting that this means that $\varepsilon '<\varepsilon$ for $\alpha >0$ and thinking that there is going to be a point later in the proof where $\alpha$ is critical for the argument; someone without mathematical maturity is likely to worry about where this $\varepsilon '$ comes from and why it's written like this. Second there was a requirement for a set $S$ where $0$ lies in $\bar S$ in the weak-* topology but not the norm topology. Here mathematical maturity is knowing about both topologies, how they're defined and instinctively noting that this is only true in infinite dimensional spaces and not needing to look up definitions or prove this.
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".
