Thanks OP for making this thread! I checked the archives last summer to see and there was a thread on maths from some years ago but it didn't go anywhere really. Hopefully this one blossoms!
Regarding maths and its practice: I would say that maths really is about how to think and how to structure your thought. This was how the concept "mathematics" was understood by the Greeks, a manner of learning and reasoning. While this might not be evident with the early stages of numbers, calculus and algebra, as soon as things become proof-based, it becomes all encompassing.
When reading maths, it's important to interrogate the material. Ask yourself: Why is done this way? Can it be done another way? Is this really the case? And prove/show it to yourself. Furthermore, be experimental and keep a good picture. Making things tangible but poking at it and visualizing what is going on can only help you.
I had an interesting thought while out having a smoke earlier, I was thinking about "quasi-random" algorithms, (algorithms that actually reduce randomness to play toward humans' perception of what random is), like shuffle features on music apps, and I started to wonder about how I could iteratively build an ordered set where at each step, the set is "as random" as it could be.
I would say that what you are getting to harkens to Ramsey theory which studies the question of given some arbitrary structure: past what extent does some sort of order appear.
"Randomness" is not so much the absence of order, or more accurately "regularity" or "pattern", but rather the absence of
perceived order. Your ordered set, which one may suppose is a sequence of, say, numbers, is as "random as it could be" if it doesn't commit to any perceivable order. There might be one, you just can't ever see it. In this sense, you are asking the converse of the central question of Ramsey theory: given some arbitrary structure, to what extent does some sort of order
not appear.
I will say that in my experience processes which generate
randomness tend to have their randomness rooted in something else - say the digits of pi or radio static or some chaotic processes. You see this in random number generators for stuff like games and ciphers.
In this sense, the problem of generating a void of order is offloaded to something where no order is perceived. Of course, in the digits of pi, there
is a pattern, we just don't know it.
Outright undecidability / uncomputability rarely seems to be an actual thing that actually crops up in day-to-day computing (with at least one
notable exception).
Gödel's theorems are largely asymptotic notions about the extents of logic and the standard models of reasoning in maths. Their reputation is well deserved, with them killing the program that tried to formalize all of maths on logic headed by Hilbert (which subsequently made him never touch logic again), but honestly I find that those who hype the theorems up rarely understand them. For the working
logician mathematician (logicians don't work), it's a matter of limits of possibilities, not the usual operation. Sure the
totality is impossible, and that's great to know, but what of the
partial. That sort of thing is more of interest.
Math is the subject of study for real men because you cannot innovate, design, or build complex things without an understanding of math.
Calculus went wrong when you made it for niggers. That's when it went wrong. It's like how many people, it's like, "Ugh it's epsilon/delta.." Ah, fuck you man. White people don't mind it, ok? The white people are like, "Yeah it's continuity. So what's your point?" The niggers are all like, "Nooo! We don't like limits!" Here's the difference in a white person and a nigger: Do you like real analysis? Ok, you're a nigger. Fuck you. Get the fuck out.
Also, being very geometrically inclined in both research and mentality: straightedge-and-compass geometry or classical Euclidean geometry has always fascinated me and the proofs tend to almost be works of art themselves. So when I get around to it, I'll see about posting their elaborations in this thread. The subject itself is a very good practice, if only mostly for leisure, as apart from architecture and art, I can scant think of its modern uses.
