Ignoring the fact that it's technically impossible to conclusively prove anything, it is maddeningly difficult to prove a negative but it is possible.
Poll: Can you prove a negative ?
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The essence of the statement "you can't prove a negative" is, I understand it, to be just the bastardisation of the more technical "one man's modus ponens is another man's modus tollens". That is, there is always another way to read the outcome of an argument in such a way as to deny that a contradiction one has reached cannot somehow be accepted, or to take something as patently ridiculous and hence that one should deny one of the premises of the argument.
It's perfectly possible to prove a negative, but it requires acceptance of either the logical rule we call the Excluded Middle (that every statement is either True or False) or that the deductive system allows us to build new arguments from more basic ones by shifting the negations from conclusions to premises (more often called left and right rules for negation's proof-theoretic behaviour).
The Oxford Philosopher Michael Dummett once tried to show that neither of these were rationally justified, but while he was convincing in regards to the former, he couldn't show that it was unsafe for a valid scheme of deduction to use the relevant transformation rules on proof structures.
It's perfectly possible to prove a negative, but it requires acceptance of either the logical rule we call the Excluded Middle (that every statement is either True or False) or that the deductive system allows us to build new arguments from more basic ones by shifting the negations from conclusions to premises (more often called left and right rules for negation's proof-theoretic behaviour).
The Oxford Philosopher Michael Dummett once tried to show that neither of these were rationally justified, but while he was convincing in regards to the former, he couldn't show that it was unsafe for a valid scheme of deduction to use the relevant transformation rules on proof structures.
Mathematics doesn't physically exist. But as for the halting problem, who knows, perhaps with advances in quantum computing and artifical intelligence, we might be able to solve it. I can't say for certain, but anything is possible.TheApatheticDespot said:
Depends. If you say something like "Not all dogs are black." - thats quite easily proven. If you say however something along the lines of "Blue dogs do not exist" - thats quite different entirely. You would have to gather all the dogs in the world in one giant cage and sort through all of them to prove it.
I think the biggest problem we run into is determining what proof we are talking about. Mathematically and philosophically, a proof is anything that expels all all possible doubt. However, the term proof has been used more loosely in scientific terms to mean we are expelling all reasonable doubt as opposed to all doubt.
In the field of science, I'm certain there are a ton of proofs for negatives. Same is true for mathematics. However, philosophically, which is math proofs for the real world, I don't think is able to prove any negatives. In fact, from what I know from philosophy, the only proofs that I have heard of are for your own existence and things that you sense exist.
In the field of science, I'm certain there are a ton of proofs for negatives. Same is true for mathematics. However, philosophically, which is math proofs for the real world, I don't think is able to prove any negatives. In fact, from what I know from philosophy, the only proofs that I have heard of are for your own existence and things that you sense exist.
To your first point, depending on how philosophically bloodyminded I'm feeling my response would be either "so what?" or "how so?". I'm not a big fan of this sort of philosophical circumlocution, but it turns out to be very difficult to assert that conceptual entities don't exist while maintaining that things like fire exist.BlackWidower said:
As for your second point, I'm afraid that's just flatly wrong. Quantum computers, despite their sometimes dramatic superiority over digital computers in complexity theoretic terms, are actually equivalent to digital computers in computability theoretic terms. Likewise, though superficially dramatically different in concept and operation, neural networks are equivalent to digital computers in terms of computability. Alan Turing proved the halting problem undecidable in 1936 using a variation on Georg Cantor's diagonalization argument. The proof is fascinating, beautiful and actually fairly accessible to the non-expert, I encourage you to look into it if you have an interest in either computability theory or mathematical logic.