In many cases, an algorithm can be designed and implemented to prove/disprove a mathematical axiom, but could it be accepted as proof or refutation by rigorous mathematicians?
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5$\begingroup$ I suppose that based on your thesis sentence, I am either not a good programmer or one of those exceptional good programmers who doesn't "know how to create a proper algorithm" to prove or disprove a mathematical axiom. There are programs called Proof Assistants that are intended to aid mathematicians in constructing formal proofs. $\endgroup$– hardmath ♦Nov 30, 2022 at 6:34
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3$\begingroup$ Even so, a mathematician does not think in terms of proving (or disproving) "a mathematical axiom". The axioms are a starting point for deductive reasoning, and the things that one can prove in this way are called theorems. $\endgroup$– hardmath ♦Nov 30, 2022 at 6:48
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2$\begingroup$ Maybe you meant to write "theorem" rather than "axiom"? One can't prove axioms. $\endgroup$– Federico PoloniNov 30, 2022 at 21:04
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The short answer is: it depends.
Often disproving a theorem can be done by providing a counterexample. If you find a legitimate counterexample, then the theorem is disproven. At that point it is of no importance how you came up with it. That includes numerical algorithms.
One thing to keep in mind in numerical computations is that you always deal with limited floating point precision. There is a case for rigorous analytical research over using these calculations with limited precision.
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1$\begingroup$ A great example for this question would be Lander and Parkin's 1966 paper "COUNTEREXAMPLE TO EULER'S CONJECTURE ON SUMS OF LIKE POWERS"-- often lauded as "the shortest math paper". Here is a Link to the paper and here is a link to a blog post showing the result. $\endgroup$– BrickmanDec 6, 2022 at 13:51