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when there's no solution or when the proof doesn't make sense. That's uh my guess.
this is a contradiction a^-a thus a mathematical proof will never not be a proof.
False premise, true conclusion?
when the statement is an axiom
axioms can be proved
If you can prove an axiom, it isn't an axiom....
this is tantamount to asking when is 2 not equal to 2
either way, something cant be both true and false at the same time, and this is what the question asks
when you are proving something is false
still a proof...
When it is a valid proof but not yet accepted by the mathematical community. Or accepted as valid but wrongly.
am I reading this question wrong? anyone? explain why, what I'm saying, is not the only logical answer?
Logically, p and not p is a bad thing (obviously) I am sure that is not the intention behind the question.....(maybe)
hmm, I disagree. I think if someone is just learning about proofs... a question involving a contradiction seems like an obvious one to ask a student.
A proof not yet 100% accepted by the mathematical community..
Bertrand Russell's "Principia Mathematica" is a relevant book. It deals with the construction of what a mathematical proof is. Mathematical proofs are constructed using objects, maps, and equivalences. Maps are relationships between objects. Objects are defined to have properties. Equivalences are to determine what we can say are the "same". A contradiction usually doesn't pass under mathematics, but it works in fuzzy logic and paraconsistent mathematics. (For a famous example, Godel's incompleteness theorems on the extendability of arithmetic axioms.) So, in short, what passes as a mathematical proof is technically what mathematicians have agreed on to make sense. The equivalences might have Platonic existence outside of this agreement, in which case the mathematicians have made a mistake; but in the end, instrumentally it is the intuition that decides what can be called the same. But this doesn't mean that you can just run around making any proof you want. There is still a lot of technical subtlety, and as you learn mathematics you will create ad hoc modifiers to what constitutes a good proof. You will have different levels of rigor for different kinds of proofs, because in the end, we're looking for math to be intellectually useful. But, in short, if you're given that \(A=B\), a strong heuristic is that any map on \(A\to C(A)\) should also map as \(B\to C(B)\). And by "strong heuristic", I mean it is right in virtually every scenario except for the most contrived, philosophically demanding ones.
If it is not accepted by the Mathematical community, it is not a proof, so something cant be a proof unless its accepted by the mathematical community. So we still have a contradiction, even by your definition...
An addendum, the contradiction paradox is usually a good way of determining falsehood, but as has been demonstrating by the "quantum eraser" experiment http://grad.physics.sunysb.edu/~amarch/ you need to be careful about what constitutes a contradiction. @zzr0ck3r You aren't reading what I wrote clearly enough.
Hum... a proof has no independant (ie logical) existance?
@estudier We call that Platonic existence. It might, but this is up to a lot of debate.
Bertrand Russell, Plato, and many modern mathematicians believe it does, but in a form we don't know of yet. Wittgenstein, Hume, Kant, and the peerless mathematician Godel (along with plenty, but not the majority, of Ivy mathematicians) believe that it doesn't.
I don't agree that this is a case of Platonistic argument (which I don't subscribe to).
Go on? Platonic existence is defined as something that can be objectively verified in the most extreme of cases no matter what (generally, I'm a bit rusty). This seems to me to be a clear cut case of what constitutes a proof. I won't press my own views, I'm just giving names to read. What they have to say is much better than what I do.
A proof is a logical binding together of possibly Platonic mathematical objects but not such an object itself.
Well, I don't think I can pretend to know enough about quantum theory to understand what this all means, but I have a new answer. A proof is not a proof when know one is there to witness the proof.
If it is a very small proof...
One hand clapping.....
This is true, but a proof is also constrained by axioms, which are maps.
@nincompoop Why are you deleting your posts?
He's a ninja.
Of course, a disclaimer is that I'm not a philosopher, and this is veering very close to philosophy, so I might've just been misinterpreting all the books I've read. :P
This is only philosophy. :)
is an incomplete proof a proof?
so i like answering your question with an oder question : when a thesis is not a thesis ? so just till not is proven ,because till not get an accepted proof so till then is just a conjecture this is OK ?