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Much of what I know beyond the basic mathematics taught in elementary school I've learned on my own, and accepting this basic difficulty and learning to go slow

Much of what I know beyond the basic mathematics taught in elementary school I've learned on my own, and accepting this basic difficulty and learning to go slowly has been an absolute boon. I've simply accepted the fact that I might need to spend almost two weeks on a single proof or even definition, continually returning to it or ingesting it in different ways until I am ready to move on. I'm able to read a typical piece of somewhat difficult literature (fiction or non-fiction) in about two weeks, so it was certainly an adjustment, but as soon as I got used to this I found that my actual understanding of the concepts had increased tenfold. I think part of the problem is that mathematics is so precise that subsequent understanding is tightly coupled to a comprehension of prerequisites. In so-called "softer" disciplines, the concepts are less precisely delineated, and their relationships are fuzzier. If you don't quite understand some prerequisite concept, you can sort of muddle your way through even the later notions that depend on it. Not so for mathematics. If you fail to get a complete and precise idea of the basics you'll eventually face complete doom or confusion when working through the subsequent portions of a theory.

xyzzyz 3y ago
> I've simply accepted the fact that I might need to spend almost two weeks on a single proof or even definition, continually returning to it or ingesting it in different ways until I am ready to move on. This is exactly the correct attitude. There is no royal road to mathematics, it requires a lot of careful thinking to put everything together in your head. When I was studying mathematics seriously, getting through 4 pages of a textbook in an hour was a really good pace, but usually I only managed half or quarter that. > I think part of the problem is that mathematics is so precise that subsequent understanding is tightly coupled to a comprehension of prerequisites. 100% agreed with this and the rest of your comment. I often see people pick up a research paper, try to read it with very little understanding of the prerequisite concepts it is built upon, try to look these up, which only uncovers prerequisites of the prerequisites they need to understand first, then despair, give up, and blame arcane notation, probably because the alternative is just too humbling to contemplate.
- ChainOfFools 3y ago
  "One book opens another" - supposedly an adage from alchemical literature

xyzzyz 3y ago

> I've simply accepted the fact that I might need to spend almost two weeks on a single proof or even definition, continually returning to it or ingesting it in different ways until I am ready to move on. This is exactly the correct attitude. There is no royal road to mathematics, it requires a lot of careful thinking to put everything together in your head. When I was studying mathematics seriously, getting through 4 pages of a textbook in an hour was a really good pace, but usually I only managed half or quarter that. > I think part of the problem is that mathematics is so precise that subsequent understanding is tightly coupled to a comprehension of prerequisites. 100% agreed with this and the rest of your comment. I often see people pick up a research paper, try to read it with very little understanding of the prerequisite concepts it is built upon, try to look these up, which only uncovers prerequisites of the prerequisites they need to understand first, then despair, give up, and blame arcane notation, probably because the alternative is just too humbling to contemplate.

ChainOfFools 3y ago
"One book opens another" - supposedly an adage from alchemical literature