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This isn't a meaningful question. There isn't a distinction between what is mathematics with a proof and what is mathematics without a proof. The examples that

by jordigh 11y ago

This isn't a meaningful question. There isn't a distinction between what is mathematics with a proof and what is mathematics without a proof. The examples that others have given about Ramsey numbers are, in fact, a proof. The proof could consist of a gigantic computation or it could be some other deep insight. But in the end, saying the Ramsey number R(5,5) is, say, 47, is not very meaningful without a convincing argument that establishes why it's that number. And those convincing arguments are a proof.

Houshalter 11y ago
That's not true. There is a whole class of problems that are trivial to prove an answer is correct, but very difficult to find the answer. Like factoring the RSA numbers, or finding a large prime number, etc.
- jordigh 11y ago
  You're thinking of computer things like P vs NP. Any mathematician who develops some sort of algorithm for factoring large numbers would announce this algorithm and a proof of its correctness as the interesting thing to announce, not merely that some large number was factored, without disclosing how. Or perhaps they would keep it secret, but it would be a huge cultural faux pas to have a method to factor large numbers without also explaining (and thus proving) how it works.
  - Houshalter 11y ago
    It was just an example of a problem where the hard part is finding an actual answer, not just proving that the answer is correct. Proving things seems really boring and uninteresting to me, and there is no guarantee the task is even possible.
    - jordigh 11y ago
      There is no guarantee it's possible to factor large numbers quickly either. "Proving" isn't distinct from "mathematics". That's what I'm trying to get at. In a sense, a proof is equivalent to a computation (and it's possible to make this precisely true in some contexts). The public at large has gotten some impression that proofs are something only some kind of mathematics needs, probably after being traumatised by high school geometry classes. Proofs are all that mathematics is.
      - Houshalter 11y ago
        You can model many problems as proofs, but not all, and it's not usually a useful observation. Most interesting math problems have nothing to do with proving things. The problem is people have already solved all the interesting problems. Proofs are just what's left. One interesting math problem someone told me just the other day was to come up with a series of forward and backward steps that would keep you from falling off a cliff, if a malicious person only chose each 2nd, 3rd or 4th step. Or to find the resistance between two points on an infinite grid of resistors. Or to find a series of bridges to get between several island without ever crossing a bridge twice. Or coloring a map with only 4 colors. Or in machine learning, tons of unsolved problems involving approximating intractable inference problems. That's a pretty wide variety of problems just off the top of my head, none of which are proving things. Sure some of the problems might not have a solution, but at least I'm not asking people to solve the halting problem, which is what half of those "unsolved math proofs" require. Even if you do solve them, the result is uninteresting. You could, at least in theory, quickly explain fermats last theorem to almost anyone. Or the four color theorem, or most of the problems I mentioned. But only a few mathematicians understand the proofs, and sometimes no one understands them (the ones proved computers.)
        jordigh 11y ago
        Finding that resistance in the grid of resistors involves performing a computation. That computation is a proof. You have it in your mind that proofs are something that you don't like and completely distinct from computations. You don't have a clear distinction in your mind between what you like and you don't. It seems to me that some mathematics is just unfamiliar to you, and when that happens, you call it "proofs".

by jordigh 11y ago

Houshalter 11y ago

That's not true. There is a whole class of problems that are trivial to prove an answer is correct, but very difficult to find the answer. Like factoring the RSA numbers, or finding a large prime number, etc.

jordigh 11y ago
You're thinking of computer things like P vs NP. Any mathematician who develops some sort of algorithm for factoring large numbers would announce this algorithm and a proof of its correctness as the interesting thing to announce, not merely that some large number was factored, without disclosing how. Or perhaps they would keep it secret, but it would be a huge cultural faux pas to have a method to factor large numbers without also explaining (and thus proving) how it works.
- Houshalter 11y ago
  It was just an example of a problem where the hard part is finding an actual answer, not just proving that the answer is correct. Proving things seems really boring and uninteresting to me, and there is no guarantee the task is even possible.
  - jordigh 11y ago
    There is no guarantee it's possible to factor large numbers quickly either. "Proving" isn't distinct from "mathematics". That's what I'm trying to get at. In a sense, a proof is equivalent to a computation (and it's possible to make this precisely true in some contexts). The public at large has gotten some impression that proofs are something only some kind of mathematics needs, probably after being traumatised by high school geometry classes. Proofs are all that mathematics is.
    - Houshalter 11y ago
      You can model many problems as proofs, but not all, and it's not usually a useful observation. Most interesting math problems have nothing to do with proving things. The problem is people have already solved all the interesting problems. Proofs are just what's left. One interesting math problem someone told me just the other day was to come up with a series of forward and backward steps that would keep you from falling off a cliff, if a malicious person only chose each 2nd, 3rd or 4th step. Or to find the resistance between two points on an infinite grid of resistors. Or to find a series of bridges to get between several island without ever crossing a bridge twice. Or coloring a map with only 4 colors. Or in machine learning, tons of unsolved problems involving approximating intractable inference problems. That's a pretty wide variety of problems just off the top of my head, none of which are proving things. Sure some of the problems might not have a solution, but at least I'm not asking people to solve the halting problem, which is what half of those "unsolved math proofs" require. Even if you do solve them, the result is uninteresting. You could, at least in theory, quickly explain fermats last theorem to almost anyone. Or the four color theorem, or most of the problems I mentioned. But only a few mathematicians understand the proofs, and sometimes no one understands them (the ones proved computers.)
      - jordigh 11y ago
        Finding that resistance in the grid of resistors involves performing a computation. That computation is a proof. You have it in your mind that proofs are something that you don't like and completely distinct from computations. You don't have a clear distinction in your mind between what you like and you don't. It seems to me that some mathematics is just unfamiliar to you, and when that happens, you call it "proofs".