- #1
kenewbie
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Amateur question ahead, be warned.
I really dislike proofs that something cannot be done. My first gripe is that they limit the areas we are "allowed to think about", so to speak. But more importantly, I have this feeling that any proof of impossibility is unavoidably flawed because it cannot account for any openings which may be found in systems other than we are using today.
As a theoretical example, an ancient greek could could easily prove that there are no solutions to x^2 = -1. However today we can easily give solution to that in the complex numbers. Although we are working in a different system, we have meaningful answers to what was proved impossible.
The emphasis on meaningful answer is important. Although we did not technically break the greeks proof, we sort of rewrote the rules, and was able to use what the greek said was impossible to do meaningful calculations, which can help find solutions to problems.
The fallacy of our hypothetical greek was of course that he did not specify that he was talking about real solutions. But that is sort if my point. They had no concept of imaginary numbers yet, so every number was real.
Which is the exact position that anyone today is in, when they prove that something is impossible. Although they can trivially avoid the exact problem the greek had by specifying a domain, who knows in what directions our system of mathematics will expand in the future.
Is it not presumptuous to assume that we can disprove anything for anything other than the system which we currently have? Since this system is constantly evolving, any proof of impossibility would be valid only for a finite time, until the system changes, and as such it is not a proof at all.
"Positive" proofs, on the other hand, does not have this limitation. If you prove that something is possible, you helping expanding the current system. Assuming there is no fundamental axiomatic failure in math or in your positive proof itself, it cannot be undone by future expansions of the system.
So, is there really any meaning to a proof that squaring the circle is impossible? It may be impossible with any system we currently have, but who knows what tomorrow might bring.
Are there good examples of proofs of impossibility that was commonly accepted and then found to be wrong?
Did I stumble off into crackpot-land here?
k
I really dislike proofs that something cannot be done. My first gripe is that they limit the areas we are "allowed to think about", so to speak. But more importantly, I have this feeling that any proof of impossibility is unavoidably flawed because it cannot account for any openings which may be found in systems other than we are using today.
As a theoretical example, an ancient greek could could easily prove that there are no solutions to x^2 = -1. However today we can easily give solution to that in the complex numbers. Although we are working in a different system, we have meaningful answers to what was proved impossible.
The emphasis on meaningful answer is important. Although we did not technically break the greeks proof, we sort of rewrote the rules, and was able to use what the greek said was impossible to do meaningful calculations, which can help find solutions to problems.
The fallacy of our hypothetical greek was of course that he did not specify that he was talking about real solutions. But that is sort if my point. They had no concept of imaginary numbers yet, so every number was real.
Which is the exact position that anyone today is in, when they prove that something is impossible. Although they can trivially avoid the exact problem the greek had by specifying a domain, who knows in what directions our system of mathematics will expand in the future.
Is it not presumptuous to assume that we can disprove anything for anything other than the system which we currently have? Since this system is constantly evolving, any proof of impossibility would be valid only for a finite time, until the system changes, and as such it is not a proof at all.
"Positive" proofs, on the other hand, does not have this limitation. If you prove that something is possible, you helping expanding the current system. Assuming there is no fundamental axiomatic failure in math or in your positive proof itself, it cannot be undone by future expansions of the system.
So, is there really any meaning to a proof that squaring the circle is impossible? It may be impossible with any system we currently have, but who knows what tomorrow might bring.
Are there good examples of proofs of impossibility that was commonly accepted and then found to be wrong?
Did I stumble off into crackpot-land here?
k