show/hide this revision's text 2 added 1085 characters in body

  • Nonexistence theorems can not be demonstrated with numerical evidence. For example, the impossibility of classical geometric construction problems (trisecting the angle, doubling the cube) could only be shown with a proof that the efforts in the positive direction were futile. Or consider the equation $x^n + y^n = z^n$ with $n > 2$. [EDIT: Strictly speaking my first sentence is not true. For example, the primality of a number is a kind of nonexistence theorem -- this number has no nontrivial factorization -- and one could prove the primality of a specific number by just trying out all the finitely many numerical possibilities, whether by naive trial division or a more efficient rigorous primality test.Probabilistic primality tests, such as the Solovay--Strassen or Miller--Rabin tests, allow one to present a short amount of compelling numerical evidence, without a proof, that a number is quite likely to be prime. What I should have written is that nonexistence theorems are usually not (or at least some of them are not) demonstrable by numerical evidence, and the geometric impossibility theorems which I mentioned illustrate that. I don't see how one can give real evidence short for those theorems other than by a proof. Lack of success in making the constructions is not convincing: the Greeks couldn't construct a regular 17-gon by their rules, but Gauss showed much later that it can be done.]

    		•
    show/hide this revision's text 1 [made Community Wiki]

    1. Nonexistence theorems can not be demonstrated with numerical evidence. For example, the impossibility of classical geometric construction problems (trisecting the angle, doubling the cube) could only be shown with a proof that the efforts in the positive direction were futile. Or consider the equation $x^n + y^n = z^n$ with $n > 2$.

    2. You can't apply a theorem to all commutative rings unless you have a proof of the result which works that broadly. Otherwise math just becomes conjectures upon conjectures, or you have awkward hypotheses: "For a ring whose nonzero quotients all have maximal ideals, etc." Emmy Noether revolutionized abstract algebra by replacing her predecessor's tedious computational arguments in polynomial rings with short conceptual proofs valid in any Noetherian ring, which not only gave a better understanding of what was done before but revealed a much broader terrain where earlier work could be used. Or consider the true scope of harmonic analysis: it can be carried out not just in Euclidean space or Lie groups, but in any locally compact group. Why? Because, to get things started, Weil's proof of the existence of Haar measure works that broadly. How are you going to collect 99% numerical evidence that all locally compact groups have a Haar measure? (In number theory and representation theory one integrates over the adeles, which are in no sense like Lie groups, so the "topological group" concept, rather than just "Lie group", is really crucial.)

    3. Proofs tell you why something works, and knowing that explanatory mechanism can give you the tools to generalize the result to new settings. For example, consider the classification of finitely generated torsion-free abelian groups, finitely generated torsion-free modules over any PID, and finitely generated torsion-free modules over a Dedekind domain. The last classification is very useful, but I think its statement is too involved to believe it is valid as generally as it is without having a proof.

    4. Proofs can show in advance how certain unsolved problems are related to each other. For instance, there are tons of known consequences of the generalized Riemann hypothesis because the proofs show how GRH leads to those other results. (Along the same lines, Ribet showed how modularity of elliptic curves would imply FLT, which at the time were both open questions, and that work inspired Wiles.)