The Wikipedia page is worth reading: <a href="https://en.wikipedia.org/wiki/What_the_Tortoise_Said_to_Achilles" rel="nofollow">https://en.wikipedia.org/wiki/What_the_Tortoise_Said_to_Achi...</a><p>Particularly this quote:<p><i>The Wittgensteinian philosopher Peter Winch discussed the paradox in The Idea of a Social Science and its Relation to Philosophy (1958), where he argued that the paradox showed that "the actual process of drawing an inference, which is after all at the heart of logic, is something which cannot be represented as a logical formula ... Learning to infer is not just a matter of being taught about explicit logical relations between propositions; it is learning to do something" (p. 57). Winch goes on to suggest that the moral of the dialogue is a particular case of a general lesson, to the effect that the proper application of rules governing a form of human activity cannot itself be summed up with a set of further rules, and so that "a form of human activity can never be summed up in a set of explicit precepts" (p. 53).</i>
Is the point here that the very structure of syllogism itself can be denied? That however inexorable a "If A and B, then C" argument is, someone else could always argue that it's not quite valid yet? It kind of reminds me of the point made (turgidly, but still) by Yudkowsky in The Simple Truth[1] - sometimes you just have to throw up your hands and declare the counter-arguments specious.<p>[1] <a href="http://yudkowsky.net/rational/the-simple-truth" rel="nofollow">http://yudkowsky.net/rational/the-simple-truth</a>
Reading this story in Hofstadner's GEB destroyed my ability to accept mathematical proofs as "proven". I just don't find them convincing; but more like using authorised forms o argument within an artificially stylised tradition (like English Literature). And I wonder if alien mathematics will reveal our mathematics as embarassingly parochial - and not the universal common ground usually assumed.<p>So, instead of proof, I have to fall back on intuition and working code, with their severe limitations.<p>However... studying mathematical proof has at times informed and grown my intuition, by revealing new ways to see a problem and new (bizarre and unintuitive) ways to decompose it.<p>I might have been better off never having seen this story.