TITLE: What is meant by "mathematical maturity"?
QUESTION [10 upvotes]: 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?

REPLY [3 votes]: 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.