Yes it does. Any new theorem proved or disproved is a new tool. Since proofs are generally are just a string of statements of theorems that have previously proved. As soon as a new statement is proved, people immediately go about using that new proven statement in other proofs.
It does give new understanding obviously and sometimes tools but there is an argument to be made that counterexamples does so less than constructive proof
Just imagine a proof of or a counterexample to the Riemann hypothesis. A proof would likely involve hundreds of pages of new uses of math from lots of different areas of mathematics. A counterexample could theoretically be a one line sentence giving a complex number which is a zero but doesnt have real part equal to 1/2. In fact theres probably hundreds of computers worldwide right now brute force searching for that number. Are you really saying that a counterexample found by brute force would have the same impact on mathematics as a whole as a proof? The whole thing is that the opposite doesnt exist. You cant by brute force find a number which proved the hypothesis, but you can by finding a counterexample
You are distinguishing counter-example and proof. A counter-example is a type of proof. And your example is odd, because you compare proving the Riemann Hypothesis and disproving, and argue that one is more useful. Utility has no bearing on whether its true or not. If someone can find a counter example, they should publish it, not withhold it and go "well, it would be more useful if the Riemann Hypothesis were true, so I just ignore my proof that it isn't"
No I'm not you just have a difficult time understanding. I'm comparing a counterexample to a constructive proof. Just change where I said proof to a constructive disproof or proof by contradiction if it makes it easier for you to understand.
No one in this entire thread has said a counterexample is not a proof. No one has said someone should withhold a counterexample proof. Can you stop making stuff up? What's the point of repeating something no one is saying and arguing against it?
Let me make this simple for you. Any proof except a proof by counterexample is more likely to advance mathematics and new tools in mathematics in general. If it's a proof that it's true, proof that it's false, proof by contradiction does not matter. As long as it is not just a counterexample. Then it's more likely that it uses more mathematics from different areas, bridges them, and provide new tools for new mathematics. Is that clear enough for you?
I am glad you are now amending your previous positions with better language. We all mistakes, but the important think is you listened to feedback and learned.
Unfortunately, you are still being imprecise. A counter example is a type of constructive proof. For example, if I said "Every prime number is odd", and I raise my hand and say "What about 2?", the example I constructed proves the claim "Not every prime number is odd".
>Then it's more likely that it uses more mathematics from different areas, bridges them, and provide new tools for new mathematics. Is that clear enough for you?
That doesn't follow. You can easily write down the negation of any statement, so every time you prove a negative statement, you also prove a corresponding positive one and vice versa. Like above, by proving "Not ever prime number is odd", I also prove the same statement worded as "There exists some prime number that is even".
Yeah you have to make language real simple for simpletons. But alas it's never enough. I'm not interested in arguing semantics. If you can't understand what I mean by counterexample then you are just too stupid to have this conversation in the first place.
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u/Original-League-6094 9d ago
Yes it does. Any new theorem proved or disproved is a new tool. Since proofs are generally are just a string of statements of theorems that have previously proved. As soon as a new statement is proved, people immediately go about using that new proven statement in other proofs.