Is there a proof for no proof ?
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2$\begingroup$ a proof for no proof? ... This question is nonsensical, maybe it's just that you weren't clear in your description. Please describe further. $\endgroup$– Michael HoffmanCommented Nov 5, 2009 at 16:38
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1$\begingroup$ I'll consider reopening this question if it becomes a real question. As is, people can only speculate about what it means and then answer based on their speculations. $\endgroup$– Anton GeraschenkoCommented Nov 5, 2009 at 17:55
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$\begingroup$ As long as there are logical rules, there are proofs, because each logical rule is a proof. $\endgroup$– Zirui WangCommented Jun 11, 2010 at 15:09
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2 Answers
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If I understand your question you are asking can something be proved as unprovable? If so I'd suggest that Gödel's incompleteness theorems for a starting place.
"Gödel's incompleteness theorems state that any effectively generated formal theory in which all arithmetic truths can be proved is inconsistent; hence, any such consistent formal theory that can prove some arithmetic truths can't prove all arithmetic truths."
http://en.wikipedia.org/wiki/Gödel's_incompleteness_theorems
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Presuming you mean "can we know certain things absolutely without proof", the answer is "no".
There was a philosophical movement called "logical atomism" in the early 20th century (the most famous practitioner being Bertrand Russell), which claimed the world consisted of ultimate "logical atoms" and that these things could be known absolutely without proof.
Eventually (especially after Gödel) the philosophers came back round to deciding there was no such thing as "certain knowledge".
For an accessible introduction I would highly recommend the graphic novel Logcomix: An Epic Search for Truth which follows the life of Bertrand Russell.
answered Nov 5, 2009 at 16:59