Going to try to write as much as I can and still be coherent. I've been going to bed late because of poor self-discipline, and then sleeping in for the same reason (one of the bad things about not having a fixed daily schedule). Today I forced myself to get up early to go on a nice long walk, but I'm in something of a sleep deficit now, so this will be shorter than usual so I can get to bed early.
Mathematics demands an uncontrolled hunger for abstraction, philosophy a very controlled one.
Nassim Nicholas Taleb, from The Bed of Procrustes
Mathematics is something that I struggled with as a child, despite being relatively adept in many ways with the subject. While I certainly didn't enjoy learning math and I have a propensity to make errors in mental math (solution: write it down and use calculators), I find many of the concepts to be tremendously easy, at least in terms of visualizing and comprehending them.
As a game designer, I'm a fan of math for the simple reason that it leads into good clean designs.
I think that some of this is because it's abstract. When you're making a game, you're really searching for the platonic ideal of something, and it's not always even something that really exists.
The result of this is that you create broad overarching systems so that each individual event can be represented within those systems. Of course, you don't necessarily need to do this with great resolution (I write tabletop roleplaying games, so for me I leave almost all of the specifics to the people who play my games), but you do need to have it be coherent in the final picture.
In reality, this coherence is absent. There are broad overarching abstracts (for instance, the concepts of honesty and entropy which illustrate both philosophical and physical abstract concepts), but there is no individual "ideal" because there is no individual who fits the rule.
Even those who fit the rule may actually be nothing more than the creation of a new and individual rule; there is no path to guarantee anything because the universe has never been the same as it is in this moment.
Don't mistake this for there being no paths; there are paths, and they generally lead in the direction they are supposed to. However, a great hero can be felled by a tragic flaw, and the wicked may be saved by some virtue that is hidden in the depths of their hearts waiting for the right call.
In philosophy, one can't pass judgment on the basis of abstraction. Montaigne is great about this, because he will find the "general path" (i.e. where something usually leads) and then present both examples and counter-examples in his essays.
I think that there's a commonality here with the concept of squaring the circle.
The problem with this is that it's a matter of precision. You want to try and get a square whose length is equal to the square root of pi, but pi is not something which can be neatly calculated as such (it has infinite length, unless our understanding of it is incorrect).
Squaring the circle is one of the great classical problems of geometry. It has also taken on mystical connotations over the years, as a perceived impossibility, and was one of the common goals of late medieval and Renaissance alchemy.
I think it's a great illustration for a key point:
Sometimes you have to accept the limits of knowledge.
I want to clarify, because I don't think that's necessarily true.
One of the distinctions between an alchemist and modern scientists (and the rational scientism that many espouse) is that the alchemist sought out cosmic mystery ("as above, so below"), and was aware that much of what they knew was unknowable.
There's something of value to this, because when I say one should accept the limits of knowledge I don't mean that one should stop dreaming of greater knowledge.
At the same time, it's simply not always possible to achieve the results one desires dearly. No alchemist successfully completed their magnum opus (and likely none ever will, unless we see people start using particle colliders for alchemy in some weird future), and if they did they were wrong about what they created from a chemical standpoint.
The best example we can get here is that in mathematics, you can hit a "good enough" for the problem of squaring the circle, especially with modern computer-based calculations. It won't be perfect, but it'll be within tolerance for all but the most demanding applications (and then the better course is error correction as needed, or merely increasing the precision until it's satisfactory).
In philosophy, however, you can't get around the abstraction. Squaring the circle is an important concept for a philosopher because it represents the pursuit of the unknowable. The medieval alchemists were trying to find God (at least, the later ones) as much as they were trying to find gold; their texts were esoteric to protect them from a society which failed to appreciate the independent contemplation of the divine and to force them out of comfort in their understandings. They saw in terms of value, not particles.
Philosophy always must in the end pursue the individual. It cannot be abstracted, because within the individual lies meaning.
Know when to follow the rule, know when to see the exception.
Don't be afraid to try either way to square a circle.
Remember that every outcome, good or bad, is unique. Be thankful for the good, overcome the bad.
