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2020-08-20, 02:53   #15
CRGreathouse

Aug 2006

2×2,969 Posts

Quote:
 Originally Posted by Gelly If you cannot prove with numerical analyses even the simple properties of your generators, then this is not a proof.
Quote:
 Originally Posted by jzakiya My background is in engineering, not as a mathematician, so I think and talk like one. That is why I appreciate the feedback because I know there is an epistemological difference how we think, and emphasize what's important.
I have a great appreciation for engineers (having both a grandfather and a father-in-law of that profession) and engineering. In terms of how to communicate results in a mathematical paper, results without proof come in several forms:
• Conjectures. These are statements which are believed to be true, but for which no proof can be found at present. (There are some subtleties here: by proposing a conjecture, you're implicitly suggesting that you think the problem is at least somewhat hard and worthy of study.)
• Claims. These are statements which the author claims to be true, but for which the author does not provide a proof. (Actually, sometimes a proof does occur at a later point, but that's a different matter.) Sometimes this is used for informal statements that could not have a formal proof; otherwise there is some reason (e.g., article page limits, or because proving the claim is an exercise in the text) for which the proof is omitted. The understanding in such cases is that the author would be able to provide more details if contacted.
• Propositions. Statements provided without comment on their importance or truth/falsity. (The term is also used, confusingly, in a sense like that of theorem or lemma. Sorry; I didn't make this stuff up.)
• Axioms. Statements which some mathematician (physicist, etc.) may choose to assume to be true and use as the basis for a mathematical system (like ZFC, NF, etc.)

I would characterize your statements as claims, and as such, they can't be used in any of your proofs (as they aren't themselves proven).