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John D. Barrow quote ✍️

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Turing attended Wittgenstein's lectures on the philosophy of mathematics in Cambridge in 1939 and disagreed strongly with a line of argument that Wittgenstein was pursuing which wanted to allow contradictions to exist in mathematical systems. Wittgenstein argues that he can see why people don't like contradictions outside of mathematics but cannot see what harm they do inside mathematics. Turing is exasperated and points out that such contradictions inside mathematics will lead to disasters outside mathematics: bridges will fall down. Only if there are no applications will the consequences of contradictions be innocuous. Turing eventually gave up attending these lectures. His despair is understandable. The inclusion of just one contradiction (like 0 = 1) in an axiomatic system allows any statement about the objects in the system to be proved true (and also proved false). When Bertrand Russel pointed this out in a lecture he was once challenged by a heckler demanding that he show how the questioner could be proved to be the Pope if 2 + 2 = 5. Russel replied immediately that 'if twice 2 is 5, then 4 is 5, subtract 3; then 1 = 2. But you and the Pope are 2; therefore you and the Pope are 1'! A contradictory statement is the ultimate Trojan horse.
John D. Barrow The Book of Nothing: Vacuums, Voids, and… logic math mathematics philosophy russel

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MATH 344 clues
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PHILOSOPHY 31 clues
MATHEMATICS 19 clues
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Unscramble JOHN4-letter words that start with J4-letter words that end in N4-letter words containing OH4-letter words containing HN4-letter words with J and O
Poet’s word, with inverted comma, for the end of the day
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E
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E
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N
Hint 1 meaning
The old plural of Eye.
Hint 2 anagram
ENE
Hint 3 another clue
AND STILL ___
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