The first variant of this rather fragile topic touched upon several existential topics, so in this post we will explore several topics that are more conventional, and are perhaps slightly easier to talk about. This of course, being, first, the idea of mathematical maturity.

Now, there are hundreds of accounts defining the meaning of mathematical maturity, and I highly recommend simply searching for them: the answers you will find there are more meaningful than whatever I can produce here, as I myself am still in the early stages of achieving mathematical maturity. In fact, I would argue that “achieving mathematical maturity” as a goal is not truly possible. While there may be a threshold for which one’s mind becomes slightly less competent (studies have shown that there likely is no correlation between an increase in age and scientific productivity/ingenuity: see Age and Scientific Performance, Stephen Cole 1979 or Age and Achievement in Mathematics: A Case-Study in the Sociology of Science, Nancy Stern 1978), an academic (should) never stop evolving and learning, meaning that the state of maturity is a constantly shifting constant. Mathematics in particular, as a field that is really just entirely unpredictable, demands constant flexibility from the doers of, and there’s just no time to grow old if one hopes to experience it to its fullest. So now, to speak of mathematical maturity on an elementary level, it is fair to say that this can really only first be considered upon entering university for the vast majority of future mathematicians. There are a considerable few who have already known of both their abilities and interest prior to this milestone: for instance, I had the pleasure of attending a summer class at the Euler Circle; this program teaches students a variety of advanced undergraduate and even some graduate topics in mathematics. Even though I produced something of decent value at the end of this summer experience, the mathematical maturity of my peers far exceeded my own, and it was clear that they were just on a different level. I was just inspired by the competence and passion those people had for mathematics. Several of the participants in the Euler Circle had also participated in and (I believe) one had won the Regeneron Science Talent Search a few years prior. This is nothing short of extraordinary, especially when you consider how rare pure mathematics is in the general scientific community. When I think of mathematical maturity, I think about these people. It’s the ability and confidence to pick up a textbook and understand its contents without much consideration for prerequisites. I believe that mathematical maturity is the ability for one to confront new mathematical knowledge with an open and qualified mind. As I said earlier, this is impossible to achieve, as built into its very definition is the notion of “new mathematical knowledge”: nothing is really learnable in the first attempt, and there’s always bound to be something so creative and unique that even the weird geniuses will struggle to understand it. I mean isn’t that how new areas were invented - weird geniuses who stumbled upon something extraordinary?

Now let’s talk about taking a break. Is it okay to take a break? As with most things, taking a break allows us to stand back and look at the bigger picture and appreciate life when it becomes too chaotic. Yet, I am at a point where it feels unproductive to not constantly be thinking about mathematics. Many have told me that taking a break is not a good idea, as it reduced my capacity to learn more, but I somewhat disagree. While a break takes away valuable time to learn mathematics, it is also an opportunity to expand my range of view. While the most dedicated (complete respect to them) are hammering away at hard problems regularly, I can sit back and expand the bigger picture, allowing myself to 1) not experience burnout too soon and 2) think about what areas need improvement and most importantly 3) think about what I want to learn and what I need to do to learn it. Burnout is a severe issue in teenagers and even in academics: how dreadful would it be to fall out of love with the area you love most? Taking a break is a good way to step back and do other things: then you can return refreshed and ready to explore new areas or do new work. As for improvement, while this is still considered “thinking about math”, I find it almost therapeutic to envision what areas you are weakest in, and determine exactly what must be done to achieve your objectives. More often than not your limitations are just your weakpoints, and you can confront them directly after your break. Finally, in concert with the second point, learning new stuff is just spectacular, especially when it’s a new area that you hadn’t explored much (combinatorics for me cough cough ). But, diving straight in is what causes burnout, so I’m stepping back to repeat the second point and wait until I have more opportuntities to really do significant work. This allows me to maintain a healthy relationship with mathematics and my learning.