6 ms·

All this would be way more interesting if it actually helped to demonstrate a novel mathematical fact. Right now it's more like notational play.

by kfse 3mo ago

sixo 3mo ago
I read this kind of essay as a certain part of the arc by which new thoughts are formed: an act of large-scale pattern matching, laying out a bunch of cases which resemble each other, searching for the essential basis of the resemblance. To post such a pattern allows the thought process to become distributed. Perhaps someone else will see the insight.
ajkjk 3mo ago
I happen to think that novel facts and theorems and proofs are way overrated. If you find a new fact it just goes into the giant pile of facts that are sitting around uselessly. The useful progress in math is comes from "refactoring" efforts to make things simpler and more intuitive. I don't mean that this is necessarily the case, but that it is where we are now: we have found ourself in a situation where we have way too many facts and not enough simple perspectives that make them useful and accessible. Just my opinion, though.
- kfse 3mo ago
  I agree that a lot of progress in math comes from refactorings and novel concepts that generalize neatly. My point is that those breakthroughs don't happen through refactoring in a vacuum, they happen because the refactoring is undertaken with a specific motivation. It often ends up having more general applications than the original motivation, but doing it without a specific motivation doesn't usually yield progress. It's the same as with software refactoring. If you refactor without a sense of what you want to get out of the refactor, how do you know whether you're refactoring the right things? (Also just my opinion)
  - ajkjk 3mo ago
    I don't think that argument is actually true? You can refactor for purely aesthetic reasons and it may well turn out to help. Maybe you don't even know, going in, what the goal is; just that the way it works now isn't good and maybe if you start looking you'll find a better approach. Happens all the time, I think. Of course you can also refactor for aesthetic reasons and not help, but that's a skill issue. My clue-finding and pattern-matching and such is all based on philosophical aesthetics: something feels amiss when these patterns exist without being obvious; therefore they should be extracted and examined from various sides to see if a connection is found.

All this would be way more interesting if it actually helped to demonstrate a novel mathematical fact. Right now it's more like notational play.

by kfse 3mo ago

All this would be way more interesting if it actually helped to demonstrate a novel mathematical fact. Right now it's more like notational play.

sixo 3mo ago

I read this kind of essay as a certain part of the arc by which new thoughts are formed: an act of large-scale pattern matching, laying out a bunch of cases which resemble each other, searching for the essential basis of the resemblance. To post such a pattern allows the thought process to become distributed. Perhaps someone else will see the insight.

ajkjk 3mo ago

I happen to think that novel facts and theorems and proofs are way overrated. If you find a new fact it just goes into the giant pile of facts that are sitting around uselessly. The useful progress in math is comes from "refactoring" efforts to make things simpler and more intuitive. I don't mean that this is necessarily the case, but that it is where we are now: we have found ourself in a situation where we have way too many facts and not enough simple perspectives that make them useful and accessible. Just my opinion, though.

kfse 3mo ago
I agree that a lot of progress in math comes from refactorings and novel concepts that generalize neatly. My point is that those breakthroughs don't happen through refactoring in a vacuum, they happen because the refactoring is undertaken with a specific motivation. It often ends up having more general applications than the original motivation, but doing it without a specific motivation doesn't usually yield progress. It's the same as with software refactoring. If you refactor without a sense of what you want to get out of the refactor, how do you know whether you're refactoring the right things? (Also just my opinion)
- ajkjk 3mo ago
  I don't think that argument is actually true? You can refactor for purely aesthetic reasons and it may well turn out to help. Maybe you don't even know, going in, what the goal is; just that the way it works now isn't good and maybe if you start looking you'll find a better approach. Happens all the time, I think. Of course you can also refactor for aesthetic reasons and not help, but that's a skill issue. My clue-finding and pattern-matching and such is all based on philosophical aesthetics: something feels amiss when these patterns exist without being obvious; therefore they should be extracted and examined from various sides to see if a connection is found.