At the end of the nineteenth century, with the advent of non-Euclidean geometry and other ideas, a strand of pessimism had crept into mathematics. “Ignoramus et ignorabimus,” the pessimists seemed to say (a Latin maxim that means “we do not know and will not know”).
The great mathematician David Hilbert made an impassioned speech in 1900 in which he entreated fellow mathematicians to reject intuition and look for rigorous proof as the yardstick for mathematical truth. He stated that mathematics should be purely and strictly logical, bereft of contradictions, and for this the foundations of mathematics should be made totally certain.
“In mathematics,” he declared, “there is no ignorabimus“.
It was in this exhilarating period of great hope in the certainty of the laws of mathematics that Russell started writing his The Principles of Mathematics. This led him to the discovery of the famous Russell’s paradox: “Does the set of all sets which do not contain themselves contain itself?” An explanation of this paradox is given in Logicomix by the means of an analogy:
“Imagine a town with a strict law on shaving. By it, every adult is required to shave daily. But it is not obligatory to shave yourself. For those who don’t want to, there is a barber. In fact the law decrees: ‘Those who don’t shave themselves are shaved by the barber.’
“It sounds innocuous…However, if taken literally, it leads straight to paradox! For, you see, the question arises: ‘Who will shave the barber?’ He obviously cannot shave himself, for being the barber, it would mean that he is shaved by the man who shaves only those who don’t shave themselves! But he can’t ‘go to the barber’, for again, that will mean he’ll shave himself, which the barber isn’t for!”
The publication of the paradox at the time was greeted with joy by mathematicians such as Henri Poincaré, who believed in intuition, and with dismay by others like Hilbert, who swore by contradiction-free…