To me, pure math is quite peculiar. While precise and structured, it is, at the same time, fluid and expressive. It is almost as if pure math lives within the tension between being structured and being expressive.

If a theory is structured to the point of being concrete and rigid, then it becomes an entirely different pursuit, such as in chess, a game in which the possible actions that one can take are set in stone and where calculation and memorization become critical skills. For me, such theories would end up being restrictive and rigid.

On the other hand, if a theory is so fluid that there is no structure present, then anything goes, and we could make up whatever theory we wanted. That would, in a sense, be an ultimate expression of creativity. However, such theories would then be just speculations. On this point, I recall a set theory book mentioning that useful theories restrict the number of such speculations.

The fact that pure math is, in a sense, both structured and expressive allows me to explore my imagination and complexity in ways that I think are unique to this subject.

I feel inspired when I gaze at images of liminal spaces or otherworldly pictures of imaginary realms. To channel such feelings in meaningful ways, I seek challenging and stimulating problems in pure math. Through these challenges, I sometimes feel as if I am transported into these images.

I think these imageries lend themselves well to pure math since there are theories and problems that take place in settings that can be thought of as imaginary mindscapes. Such settings can also far exceed, or be far removed from, the practical limits of everyday life. For instance, there are results in pure math that involve quantities so large that, outside of pure math, they will probably only be found in works of fiction.

Something that I find mysterious is that complexity in pure math lies somewhere between a board game and real life. In a board game, everything is fixed on an eight by eight grid. Anything that needs to be expressed in that domain has to be through that board. On the other hand, there is no such restriction in math. Topics of discourse can be discussed, and things are not so set in stone. However, in math, there appear to be limits on how far you can go with complexity. In pure math, complexity is largely revealed by proofs. Beyond pure math, and more generally, in life, it is impossible to prove things. Perhaps that impossibility is because our reality is too complex to draw definitive conclusions. Pure math is simplified, when compared to our reality, to the extent that proofs are possible. However, this relative simplicity belies the enormous complexity within this subject. As a result, mathematical proofs can be powerful and insightful.

A world within itself is how I see pure mathematics after all these years. It is a world that is both structured and expressive, as well as imaginative and complex. This world is endless. Engaging with this subject can be compelling, from writing precise arguments to exploring my imagination.