Bourbaki1123 said:
We all are likely familiar with the notion of mathematical maturity, you sort of get an intuition for how to put mathematical ideas into context and make sense out of them as you do/read more and more math.
I've noticed that it's much easier for me to pick up a graduate level math text and grasp the material now than it was, say, a little over a year ago. Over the four years that I've been serious about studying pure math, I've noticed this trend continue and there has clearly been a great deal of accumulation.
My questions are (to those who wish to answer): (a) What is your experience with this sort of thing? (b) Does ease tend to vary logarithmically with how much time you put in studying mathematics? That is, I assume it can't continue to feel easier and easier to get through dense material without dropping off as the complexity of the material goes up, but the more I do/ read math, the easier it gets to look at another area of mathematics.
I will be graduating next year just so you know where I am coming from.
I have found like you that things do become easier as things go along, but only in the context of studying things that build on in some way of concepts that I have met before.
For example if I just picked up a book on advanced logic, I don't think that I would be able to see the forest from the trees in the same way that I would do in the case of understanding more statistics. I will probably have a better chance of grasping quicker than I would say a year before, but I don't think I would get the same kind of effect if I had not even explored (or at least thought about) basic ideas that the particular subject builds upon.
One thing that I have noticed is that within mathematics itself, there is a pattern to the heuristics of understand it that is invariant to the topic.
For example every area of mathematics involves assumptions and constraints. It doesn't matter if you are talking about a pivotal quantity like the t-distribution or chi-square distribution in statistics, or whether you are talking about a basic results of standard calculus (like taylor series to give an example), every result is based on assumptions. You can't use Taylor series for example if you don't have specific convergence properties, and you need some kind of normal approximations in many statistical tests.
Once you understand that, then you start to look at everything in assumptions: why is this result true under these assumptions? What other things does that imply? How flexible are these assumptions? (that is, can you use something that doesn't exactly meet the assumptions and still make some kind of sense of it?)
Without this insight into using this assumption view as a template, a lot of the learning is seemingly disconnected (or at least it has the tendency to be) and real understanding can potentially be lost.
With regards to the second question, I would say yes to that too.
It's really amazing about how our brain works. It's almost like it's tuning into knowledge like a person adjusting the tuning knob of a radio to get the right station. The fact that given enough time, effort, and conscious/unconscious thought, a lot of us can obtain a type of understanding that is really incredible: it's like magic if you ask me.
I don't think this is just limited to mathematics though. If you ask anyone who has been doing something for a long time who has done a lot of varied work (i.e. not been doing the same sort of problem day in day out), then you'll find that they see things that many amateurs would miss or not even conceive of. It's like a programmer knowing intuitively how to fix a bug and identify it quickly without too much effort in comparison to someone that is still learning to program.
One thing I do want to mention though is that especially with things like math, you are always learning different perspectives of how to view mathematics. There are possibly infinitely (or at least a large number of) ways to interpret mathematics in many contexts.
For example, I did a teaching unit where I taught high school for a few weeks and the supervisor was explaining how the second derivative explained concavity of the function. The teacher was explaining in a visual manner how the slope of the slope was changing for some function. Now it makes perfect sense when you look at it in that way, but I just never really thought about how to explain it in that way using that perspective.
It is a trivial example no doubt, but it does give you a hint about how many things in math have a variety of perspectives and explanations in a variety of contexts.
