Russel Quotes

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.

Throughout the Middle Ages, Jews had no part in the culture of Christian countries, and were too severely persecuted to be able to make contributions to civilization, beyond supplying capital for the building of cathedrals and such enterprises. It was only among the Mohammedans, at that period, that Jews were treated humanely, and were able to pursue philosophy and enlightened speculation. Throughout the Middle Ages, the Mohammedans were more civilized and more humane than the Christians. Christians persecuted Jews, especially at times of religious excitement; the Crusades were associated with appalling pogroms. In Mohammedan countries, on the contrary, Jews at most times were not in any way ill treated. Especially in Moorish Spain, they contributed to learning; Maimonides (1135–1204), who was born at Cordova, is regarded by some as the source of much of Spinoza’sphilosophy.Mohammedan civilization in its great days was admirable in the arts and in many technical ways, but it showed no capacity for independent speculation in theoretical matters. Its importance, which must not be under-rated, is as a transmitter. Between ancient and modern European civilization, the dark ages intervened. The Mohammedans and the Byzantines, while lacking the intellectual energy required for innovation, preserved the apparatus of civilization—education, books, and learned leisure. Both stimulated the West when it emerged from barbarism—the Mohammedans chiefly in the thirteenth century, the Byzantines chiefly in the fifteenth.

Bertand Russell