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    $\begingroup$ Mathematical logic in its modern form came into being basically as a response to Hilbert's illusions (is "wishful thinking" a better term?) about the deductive power of formal systems. $\endgroup$ Commented Jun 3, 2014 at 19:14
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    $\begingroup$ Regarding the title, do you really mean a false illusion, or rather do you just mean "illusion"? To my way of reading, a false illusion would not really be an illusion at all, but real. $\endgroup$ Commented Jun 3, 2014 at 21:22
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    $\begingroup$ So this is really about bad/outdated goals in research rather than actual mistakes? $\endgroup$ Commented Jun 3, 2014 at 22:11
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    $\begingroup$ Would be interesting, how the P=?NP question is judged if it were settled; to me it seems a good candidate for unfulfilled expectations. $\endgroup$ Commented Jun 4, 2014 at 13:52
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    $\begingroup$ nice question but dislike the vagueness of the initial paragraph. why not cite the example? sounds like you could be referring to an important open problem... $\endgroup$ Commented Jun 6, 2014 at 23:14