3 ms·

Maybe I’m just not pure enough but I find the whole concept of proof by counterexample to be elegant, and I don’t see why proving that something must be true is

by taneq 3mo ago

Maybe I’m just not pure enough but I find the whole concept of proof by counterexample to be elegant, and I don’t see why proving that something must be true is superior to proving that it can’t be false.

bananaflag 3mo ago
You mean proof by contradiction, which is something different.
- taneq 3mo ago
  Hmm, I think I might have conflated the two. Thanks!
chowells 3mo ago
It's elegant if all you're concerned with is whether a conjecture is true or false. Answered, move along! But mathematics is not a collection of facts. Mathematics is the study of abstraction. And what do you learn from a single data point? What can you abstract from that? That's why just being a counterexample isn't really interesting. There has to be more than "counterexample" for there to be something to abstract. Was it generated from an analysis of the problem? Can the counterexample be generalized to explore the problem further? Is the counterexample a surprise in a way that suggests something is missing from current understanding? Being a counterexample doesn't mean that something isn't interesting to a mathematician. But it's also not the interesting part.
- ATMLOTTOBEER 3mo ago
  You’re smuggling in a frame here which isn’t obviously true: that mathematics is not just a collection of facts
  - chowells 3mo ago
    You have a weird definition of "smuggling". I say it outright. Because I follow it up with a description of the actual practice of math: it's the study of abstraction. That's not a frame. That's just what math is.
    - im3w1l 3mo ago
      I think the nature of mathematics is an interesting question without one clean answer. To present a radically different view of mathematics: It's a game of string transformations, where the goal is to produce specific strings given a set of rules. The (syntactically valid) strings would correspond to statements, a producible string a theorem, and the production the proof.
      - chowells 3mo ago
        This is a description of one portion of math... one that's very easy to get tunnel vision towards when undertaking a very formal undergraduate mathematical education. And that's especially true if it was alongside a computer science education, which is precisely the branch of math concerned with formal systems being used in calculation. I've done both of those things. I know what you get taught. But I've kept my math education going for the 25 years since then. I've talked to practicing mathematicians about what they do. I've learned a lot about the scope of math. As an aside: most people really dislike it when I say that they should be much more precise about different numerical systems. The integers are not a subset of the rationals. They are entirely different constructions, but there is an isomorphism between integers and a subset of the rationals that preserves the integers' ring structure within that subset of the rationals and a few other aesthetic concerns. You can see why no one wants to communicate like this, even if they acknowledge it's technically correct. So I know all about pushing symbols around. But I also know that pushing symbols around isn't the whole story. Pushing symbols around is only useful as a final check. Do you want to validate that 1+2=3? Pushing symbols around can help. But how do you decide that the ideas behind 1, 2, 3, +, and = are worth having precise and compact representations? Math doesn't just use formal systems to generate proofs. It's not enough for symbols to be arranged neatly according to some rules. Math is also the process of creating the sets of symbols and their rules and communicating to other people why this set of rules and symbols is interesting. What ideas get preserved when you are working with this system? What is it an abstraction over?

by taneq 3mo ago

bananaflag 3mo ago

You mean proof by contradiction, which is something different.

taneq 3mo ago
Hmm, I think I might have conflated the two. Thanks!

chowells 3mo ago

It's elegant if all you're concerned with is whether a conjecture is true or false. Answered, move along! But mathematics is not a collection of facts. Mathematics is the study of abstraction. And what do you learn from a single data point? What can you abstract from that? That's why just being a counterexample isn't really interesting. There has to be more than "counterexample" for there to be something to abstract. Was it generated from an analysis of the problem? Can the counterexample be generalized to explore the problem further? Is the counterexample a surprise in a way that suggests something is missing from current understanding? Being a counterexample doesn't mean that something isn't interesting to a mathematician. But it's also not the interesting part.

ATMLOTTOBEER 3mo ago
You’re smuggling in a frame here which isn’t obviously true: that mathematics is not just a collection of facts
- chowells 3mo ago
  You have a weird definition of "smuggling". I say it outright. Because I follow it up with a description of the actual practice of math: it's the study of abstraction. That's not a frame. That's just what math is.
  - im3w1l 3mo ago
    I think the nature of mathematics is an interesting question without one clean answer. To present a radically different view of mathematics: It's a game of string transformations, where the goal is to produce specific strings given a set of rules. The (syntactically valid) strings would correspond to statements, a producible string a theorem, and the production the proof.
    - chowells 3mo ago
      This is a description of one portion of math... one that's very easy to get tunnel vision towards when undertaking a very formal undergraduate mathematical education. And that's especially true if it was alongside a computer science education, which is precisely the branch of math concerned with formal systems being used in calculation. I've done both of those things. I know what you get taught. But I've kept my math education going for the 25 years since then. I've talked to practicing mathematicians about what they do. I've learned a lot about the scope of math. As an aside: most people really dislike it when I say that they should be much more precise about different numerical systems. The integers are not a subset of the rationals. They are entirely different constructions, but there is an isomorphism between integers and a subset of the rationals that preserves the integers' ring structure within that subset of the rationals and a few other aesthetic concerns. You can see why no one wants to communicate like this, even if they acknowledge it's technically correct. So I know all about pushing symbols around. But I also know that pushing symbols around isn't the whole story. Pushing symbols around is only useful as a final check. Do you want to validate that 1+2=3? Pushing symbols around can help. But how do you decide that the ideas behind 1, 2, 3, +, and = are worth having precise and compact representations? Math doesn't just use formal systems to generate proofs. It's not enough for symbols to be arranged neatly according to some rules. Math is also the process of creating the sets of symbols and their rules and communicating to other people why this set of rules and symbols is interesting. What ideas get preserved when you are working with this system? What is it an abstraction over?