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> That sentence is essentially a rewording of an old-fashioned delta-epsilon proof. So, how does that indicate a flaw in contemporary mathematics at all? The f

> That sentence is essentially a rewording of an old-fashioned delta-epsilon proof. So, how does that indicate a flaw in contemporary mathematics at all? The first flaw is that it is not a proof but only a reduction. The second flaw is that contemporary math does not make the slightest attempt at making its claims easier to verify. What is wrong with an invitation/challenge in the form of a program asking to show the claim wrong?

gizmo686 14y ago
Contemporary math does not attempt to make its claims verifiable in the sense you describe, because it proves them. There might be a flaw in the proof itself (and there is active research in creating programs to check proofs). Sometime proofs in math our of the form you describe. For example, the (most well known) proof that sqrt(2) is irrational is a challenge to find the simplist fraction of to intergers that equals sqrt(2). The proof then goes on to show that any possible response you can provide to the challege is false. Simmilarly, for any mathametical claim, if you can provide a single counter example then you have dissproven the claim. Not being able to provide a counter example does not prove the claim.
- eriksank 14y ago
  Well, I have to insist that math does not prove a statement true. Math only proves that the statement does not contain original untruth. > Not being able to provide a counter example does not prove the claim. It is never possible to prove a claim anyway. It is only possible to prove that it does not contain original untruth.
  - lutusp 14y ago
    > Well, I have to insist that math does not prove a statement true. Be careful to define your terms. Most people, including mathematicians, will insist that Euclid's prime theorem proves that there are an infinity of primes -- that the assertion is "true". http://en.wikipedia.org/wiki/Euclids_theorem http://en.wikipedia.org/wiki/Euclids_theorem But it will always be possible to redefine "true" arbitrarily to reject any such claims, which partly explains philosophy's low standing among intellectual disciplines. > It is never possible to prove a claim anyway. You've shifted ground in your wording, and contradicted the idea of a mathematical proof. There really are mathematical proofs. In general science, your claim is correct -- a theory cannot be proven true, only false. But in mathematics, proofs exist.
    - eriksank 14y ago
      Math and science use exactly the same method of reducing their claims to their basic claims and backing those with a failed search for counterexamples. Consequently, all untruth is invariably always the result of the untruth hidden in the basic claims. The untruth in the basic claims obviously exists but it is sheer impossible to pinpoint precisely where exactly it is.
      - lutusp 14y ago
        > Math and science use exactly the same method of reducing their claims to their basic claims and backing those with a failed search for counterexamples. This is false. You're overlooking a very important difference between math and science. Science is empirical, which means any scientific theory can be overthrown by new evidence arising in nature. This is not true for mathematics -- once a conjecture becomes a theorem, it's irrevocable. Your viewpoint is a philosophical one that fails to account for the key difference between math and science.
        eriksank 14y ago
        Probably worse than that. I simply refuse to make any such key difference between math and science. In my opinion, both math and science are empirical.
        gizmo686 14y ago
        In what way is math empirical? If one wants to prove that 1+1=2, one does not take one apple, add another apple, and observe that one now has two apples. One defines: x + 0 = x x + successor(y) = successor(x) + y 1=successor(0) 2=successor(1) One then takes the true true statement 1+1=1+1 | Identity And finds: 1+successor(0)=1+1 | Definition of 1 successor(1) + 0 = 1+1 | Definition of + 2 + 0 = 1+1 | Definition of 2 2=1+1 | Definition of + I made no observation to prove 1+1=2. What I do observe is that if we map the counting numbers to the count of how many apples I have, and we map the addition operation to combing to groups of apples, then the mathematical result is consistent with reality. However, if we empirically discover that the mathematical result is not constitent with reality, than that does not mean that the math is wrong, it simply means that the reality does not behave in a way that corresponds to the math.

gizmo686 14y ago

Contemporary math does not attempt to make its claims verifiable in the sense you describe, because it proves them. There might be a flaw in the proof itself (and there is active research in creating programs to check proofs). Sometime proofs in math our of the form you describe. For example, the (most well known) proof that sqrt(2) is irrational is a challenge to find the simplist fraction of to intergers that equals sqrt(2). The proof then goes on to show that any possible response you can provide to the challege is false. Simmilarly, for any mathametical claim, if you can provide a single counter example then you have dissproven the claim. Not being able to provide a counter example does not prove the claim.

eriksank 14y ago
Well, I have to insist that math does not prove a statement true. Math only proves that the statement does not contain original untruth. > Not being able to provide a counter example does not prove the claim. It is never possible to prove a claim anyway. It is only possible to prove that it does not contain original untruth.
- lutusp 14y ago
  > Well, I have to insist that math does not prove a statement true. Be careful to define your terms. Most people, including mathematicians, will insist that Euclid's prime theorem proves that there are an infinity of primes -- that the assertion is "true". http://en.wikipedia.org/wiki/Euclids_theorem http://en.wikipedia.org/wiki/Euclids_theorem But it will always be possible to redefine "true" arbitrarily to reject any such claims, which partly explains philosophy's low standing among intellectual disciplines. > It is never possible to prove a claim anyway. You've shifted ground in your wording, and contradicted the idea of a mathematical proof. There really are mathematical proofs. In general science, your claim is correct -- a theory cannot be proven true, only false. But in mathematics, proofs exist.
  - eriksank 14y ago
    Math and science use exactly the same method of reducing their claims to their basic claims and backing those with a failed search for counterexamples. Consequently, all untruth is invariably always the result of the untruth hidden in the basic claims. The untruth in the basic claims obviously exists but it is sheer impossible to pinpoint precisely where exactly it is.
    - lutusp 14y ago
      > Math and science use exactly the same method of reducing their claims to their basic claims and backing those with a failed search for counterexamples. This is false. You're overlooking a very important difference between math and science. Science is empirical, which means any scientific theory can be overthrown by new evidence arising in nature. This is not true for mathematics -- once a conjecture becomes a theorem, it's irrevocable. Your viewpoint is a philosophical one that fails to account for the key difference between math and science.
      - eriksank 14y ago
        Probably worse than that. I simply refuse to make any such key difference between math and science. In my opinion, both math and science are empirical.
        gizmo686 14y ago
        In what way is math empirical? If one wants to prove that 1+1=2, one does not take one apple, add another apple, and observe that one now has two apples. One defines: x + 0 = x x + successor(y) = successor(x) + y 1=successor(0) 2=successor(1) One then takes the true true statement 1+1=1+1 | Identity And finds: 1+successor(0)=1+1 | Definition of 1 successor(1) + 0 = 1+1 | Definition of + 2 + 0 = 1+1 | Definition of 2 2=1+1 | Definition of + I made no observation to prove 1+1=2. What I do observe is that if we map the counting numbers to the count of how many apples I have, and we map the addition operation to combing to groups of apples, then the mathematical result is consistent with reality. However, if we empirically discover that the mathematical result is not constitent with reality, than that does not mean that the math is wrong, it simply means that the reality does not behave in a way that corresponds to the math.