# How can one make a persuasive argument using numerics? [closed]

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?

• 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. Commented May 2 at 12:07
• @Trunk Physicists, especially mathematical physicists, can definitely find themselves trying to prove such conjectures. 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 ? Commented May 2 at 16:11
• @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". Commented May 3 at 2:25

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.

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

I'm not sure why a physics major is trying to prove already established iff theorems in math. Maybe it's a computational physics course and your professor is leading the class in exploring the limits of computational "proofs".

My approach - and this is not math rigorous - would mirror what is done in software testing with equivalence tests. So you would divide the input data into sets which spanned continuous data domains. If discontinuities are known to exist between the continuous data domains you have to create separate input data sets for these values. It is also normal to include data values just beyond the discontinuity ("edge" or "boundary") values both less and greater than the discontinuity values.

Suppose the theorem is writable as: F1(x) <=> F2(x).

Write code to randomly generate a suitably large number of input values within all of the continuous data set boundary values. Add the generated input data to those on, or adjacent to, the discontinuity input data values.

Write more code implementing the LHS and RHS function of the theorem so as to generate an ordered list of outputs for a list of input data.

Write some additional code to highlight any input data values that produce different LFS and RHS outputs for a given input.

Then run the entire input data set on the code.

Repeat this process many times since each run produces new randomly generated inputs in the continuous data regions.

To your question as to how this process can ever be persuasive:

It's never as persuasive as a clean math proof. But it will give great heart to people who may be designing equipment or processes that seek to apply the theorem when calculating safe operating parameters for the process or equipment provided the test data fully covers all envisaged operating parameter values.

For the mathematicians, doing numerical "proofs" can give them motivation to seek a purely math proof for the same reason that the second team to climb a mountain already know that, however difficult, it is at least very likely possible.

• Why do you say it's "already established" in math? I don't think OP has given any indication of that. Commented May 2 at 12:56
• He did say "One direction is provable. The other direction is difficult to genuinely prove". As it's not computation theory, I interpret provability to indicate that the other approach allows the theorem to be proven and that someone has proven it. While this may not be the case, i.e. if no one has yet proven it, OP would hardly say that one way was provable and another is hard to prove when both approaches in fact made it hard to prove. Commented May 2 at 15:57
• OP is talking about an IFF. I interpret their post as indicating that, say, "if x, then y" is proven ("One direction is provable"), but "if y, then x" is not ("The other direction is difficult to genuinely prove"). I do not understand at all what you are attempting to argue logically in your comment or what you mean by "approaches". Commented May 2 at 16:06
• @Bryan Krause That would explain the use of the word "direction" for sure. If we accept your interpretation then it's likely much easier that proving the iff's validity in one direction by disproving the case when the supporting condition (y) is false than proving that the supporting condition being true always implies that some big consequence (x) occurs. Like getting the monthly payment amount on a 20 year mortgage when the APR is known being an easy formula but finding the APR from a known monthly payment needs numerical analysis. Commented May 2 at 16:33