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"Given any proposition in arithmetic, the mind can decide its truth value." Really? I don't see what reason we have to believe that this is true.
by voyou 14y ago
"Given any proposition in arithmetic, the mind can decide its truth value."
Really? I don't see what reason we have to believe that this is true.
- leephillips 14y agoI guess I didn't really mean that. What I should have said was that in any system of axioms (or within any algorithmic structure) there will exist undecidable propositions (unreachable by the algorithm). The mind can determine the truth value of some of these, so the mind can not be encompassed or modeled by an algorithm.
- duaneb 14y ago> The mind can determine the truth value of some of these, so the mind can not be encompassed or modeled by an algorithm. Do you have an example of this? I have an extremely difficult time comprehending it. Also, I was under the impression that there weren't necessarily propositions that are not algorithmically possible to reach, but rather, there are always propositions a given algorithm cannot solve.
- leephillips 14y ago> not algorithmically possible to reach I just meant undecidable propositions. > Do you have an example Maybe check out the conclusion on p.76 of this book[1] by Penrose (and the argument supporting it that begins on p.72.): "Human mathematicians are not using a knowably sound algorithm in order to ascertain mathematical truth". His example is our knowledge that a certain computation will not halt, and the impossibility of constructing an algorithm that can prove that fact. More flippantly, you know that you can examine the soundness of your own reasoning, but no algorithm can reach conclusions about its own correctness. [1]http://www.amazon.com/dp/0198539789 http://www.amazon.com/dp/0198539789