Posted by: Alexandre Borovik | August 30, 2018

Trigger reflex

When I see a statement like that, I just cannot stop myself from pulling the trigger:

Why does Wittgenstein want surveyability? He seems to think that to be capable of the specific use of a theorem which a new proof makes possible we must be able  to reproduce its proof. This is just false, indeed perversely so — without understanding anything about Wiles’ proof of Fermat’s Last Theorem you can use it to rule out the truth of \(a^17 + b^17 = c^17 \) where \(a\), \(b\) and \(c\) are any three integers, even hundreds of digits long — for example I know that \(123456789^17 + 12233445566778899^17\) can’t be equal to \(12345678901234567890^17\) without needing to calculate any of the three powers. [Edwin Coleman, The surveyability of long proofs, Foundations of Science, 14, Issue 1–2pp 27–43.]
Indeed, I believe most mathematicians will make an instant observation that \(17\) is a odd natural number, and therefore the last digit of \(9^17\) is \(9\), and therefore the last digit of  \(123456789^17 + 12233445566778899^17\) is \(8\) and does not equal to the last digit of  \(12345678901234567890^17\) , which is, of course, \(0\). One does not need Fermat’s Last Theorem for that (and, for the sake of historical integrity of the narrative, the case \(n = 17\) had been settled by Kummer in 1847).
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