Monthly Archives: February 2023

What makes pure math fun and interesting

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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.

Chess Thoughts

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I’m not particularly good at chess, although I’ve been playing it occasionally, to the point of joining chess clubs during my university years. The intense atmosphere that was sometimes present during chess games was thrilling. And I remember having fun moving oversized and glossy plastic pieces over cheap sheets of plastic that made up the chessboards.

Maneuvering the pieces requires a lot of care and attention on the player’s part. The idea is that you have to corral your pieces together so that they become a fully functional and working unit. But they could only move in certain directions and certain ways, so doing the above was like reading a roadmap at times. There were also arbitrary rules that one had to remember, such as when to castle, the en passant rule, and the stalemate rule.

The types of pieces on the chessboard were also important. You had knights that were shaped like horses. They moved in rather odd ways, but they were very powerful and could be unpredictable in their movements. Being short-ranged pieces, they could splinter the opposing army and provide critical reinforcements to your pieces.

There were also bishops that were donned with the headgear found on actual catholic bishops. As long-range pieces, they reinforced your army when you went in for a decisive attack. However, it is not easy to properly position these pieces and maximize their attacking potential.

And there were the rest of the pieces, with varying functionalities and purposes, such as the queen, the most powerful piece on the board, the rook that functions as a defensive resource, the pawns that represent the foot-soldiers, and the king that was the piece that had to be protected at all times.

The arrangements and configurations of the pieces were important, and people spent years playing games to fully utilize the resulting patterns. Sometimes, the formation of the pawns could make or break your game. At other times, knowing when to attack the opposing army could change the tide of the game, particularly when the opposing side’s king was vulnerable.

Like many large board games, this game comes in phases (the opening, middle, and endgame). A lot of memorization is needed to reach the higher heights of this game, especially nowadays, with games being deeply influenced by computer analysis. And to be a good player, being able to visualize and calculate with the pieces is also very necessary.

Sometimes, when looking at this game, I feel like I am looking at a snow globe. Within the globe is a liquid, and when shaken, small pieces of plastic float around, recreating the effect of a snowy day. The chess board is a little like that. It has sixty-four squares arranged into a grid. Given that this grid is limited in size— there are only so many squares to place your pieces on—the board reminds me of a snow globe ornament.

You could get lost playing games in that dimension, within this snow globe-like board, and with all of its idiosyncrasies. Thinking about this takes me back to the university chess clubs I went to, where I heard the clacking of chess pieces and played all those stimulating games.