Substrate

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.