Unfortunately this sentiment is mirrored among most programmers who claim to be interested in mathematics. Mathematics is fascinating and useful and doing it makes you smarter and better at problem solving. But a lot of programmers think they want to do mathematics, and they either don’t know what “doing mathematics” means, or they don’t really mean they want to do mathematics. ​ Honestly, it sounds ridiculously obvious to say it directly like this, but the fact remains that people feel like they can understand the content of mathematics without being able to write or read proofs. ​ So read on, and welcome to the community. ​ I honestly do believe that the struggle and confusion builds mathematical character, just as the arduous bug-hunt builds programming character. ​ I’m talking, of course, about the four basics: direct implication, proof by contradiction, contrapositive, and induction. These are the loops, if statements, pointers, and structs of rigorous argument, and there is simply no way to understand the mathematics without a native fluency in this language. ​ And so it stands for mathematics: without others doing mathematics with you, its very hard to identify your issues and see how to fix them. ​ And finally, find others who are interested in seriously learning some mathematics, and work on exercises (perhaps a weekly set) with them.

(See PDF in reMarkable for annotations and notes).

You might suppose a great mathematician such as Kolmogorov would be writing about some very complicated piece of mathematics, but his subject was the humble equals sign: what made it a good piece of notation, and what its deficiencies were. ​ even great mathematicians – perhaps, especially, great mathematicians – thought it worth their time to engage in such deepening. ​ I’m still developing the heuristic, and my articulation will therefore be somewhat stumbling. ​ Every piece should become a comfortable part of your mental furniture, ideally something you start to really feel. That means understanding every idea in multiple ways, ​ People inexperienced at mathematics sometimes memorize proofs as linear lists of statements. A more useful way is to think of proofs is as interconnected networks of simple observations. ​ For someone who has done a lot of linear algebra these are very natural observations to make, and questions to ask. But I’m not sure they would be so natural for everyone. The ability to ask the “right” questions – insight-generating questions – is a limiting part of this whole process, and requires some experience. ​ Conventional words or other signs have to be sought for laboriously only in a secondary stage, when the mentioned associative play is sufficiently established and can be reproduced at will. ​ So, my informal pop-psychology explanation is that when I’m doing mathematics really well, in the deeply internalized state I described earlier, I’m mostly using such higher-level chunks, and that’s why it no longer seems symbolic or verbal or even visual. I’m not entirely conscious of what’s going on – it’s more a sense of just playing around a lot with the various objects, trying things out, trying to find unexpected connections. But, presumably, what’s underlying the process is these chunked patterns.