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I am currently a physics undergraduate, but I will be pursuing a physics PhD next fall.

One part of the current research I am doing is to build up evidence for a mathematical "if and only if" theorem. One direction is provable. The other direction is difficult to genuinely prove, and I was advised to build up numerical evidence for the statement instead.

This made me wonder: how can numerical data for this situation ever be persuasive?

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    Persuasive to whom?
    – J W
    Commented May 1 at 5:33
  • @J W To the scientific public, presumably. And especially to OP's research peers. My question would be more related to the nature of this "theorem" and what a physics major is doing proving a math iff theorem . . . I think elaboration on this would help find a proper answer for you.
    – user104446
    Commented May 2 at 12:07
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    @Trunk Physicists, especially mathematical physicists, can definitely find themselves trying to prove such conjectures.
    – Anyon
    Commented May 2 at 14:07
  • @Anyon A lot of people "can find themselves trying to prove" all sorts of things ! The question is do physicists - experimental or theoretical - have to find answers to such questions as proving an iff theorem in order to advanced some model or theory of a phenomenon that they are working on ?
    – user104446
    Commented May 2 at 16:11
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    @Trunk I don't know how to address whether they "have to" do that as the only possible path to making progress, since it would presumably depend a lot on the circumstances. But it's hardly unheard of for theoretical physicists to define some mathematical object, study its properties, and in so doing, prove statements that may be of the "iff" variety. Physics journals routinely publish theorems, but I have no sense of what proportion would be "iff theorems".
    – Anyon
    Commented May 3 at 2:25

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A conjecture without a proof is always a conjecture, but you can establish evidence for the conjecture without a proof. Take for example elementary number theory and something like the Goldbach conjecture. We do not have a proof, but we know that the smallest counterexample would have to exceed 4*10^18.

This makes a theorem of the following type interesting:

  • If the Goldbach conjecture is true then this interesting fact is true.

Historically, there were conjectures (such as about Mersenne primes) that were shown to be false with an explicit counter-example, though believed to be true previously. The counter-example I have in mind was just within the reach of calculation by hand.

What your advisor is asking you to do is to make certain that there are no simple counter-example.

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