18 Jun Russell found out that when a set refers to itself then this paradox emerges. (Warburton)
Bertrand Russell’s main interests as a teenager were sex, religion and mathematics — all at a theoretical level. In his very long life (he died in 1970, aged 97) he ended up being controversial about the first, attacking the second, and making important contributions to the third.
Russell’s views on sex got him into trouble. In 1929 he published Marriage and Morals. In that book he questioned Christian views about the importance of being faithful to your partner. He didn’t think you had to be. This raised a few eyebrows at the time. Not that that bothered Russell much. He’d already spent six months in Brixton prison for speaking out against the First World War in 1916. In later life he helped found the Campaign for Nuclear Disarmament (CND), which is an international movement opposed to all weapons of mass destruction.
On religion he was just as outspoken and just as provocative.
For Russell there was no chance of God stepping in to save humanity: our only chance lay in using our powers of reason. People were drawn to religion, he believed, because they were afraid of dying.
By birth he was an English aristocrat. He came from a very distinguished family: his official title was the 3rd Earl Russell. You could probably tell that he was an aristocrat just by looking at him. He had a distinguished haughty sort of look, an impish grin and twinkly eyes.
Russell had an unusual and not particularly happy childhood. Both his parents died when he was very young, and his grandmother, who looked after him, was strict and a bit distant. Taught at home by private tutors, he threw himself into his studies and became a brilliant mathematician, going on to lecture at Cambridge University. But what really fascinated him was what made mathematics true. Why is 2 + 2 = 4 true? We know it is true. But why is it true? This led him quite quickly to philosophy.
As a philosopher, his real love was logic: a subject on the border between philosophy and mathematics. Logicians study structure of reasoning, usually using symbols to express their ideas. He became fascinated by the branch of mathematics and logic called set theory. Set theory seemed to promise a way explaining the structure of all our reasoning, but Russell came up with a big problem for that idea: it led to contradiction. The way he showed this was in a famous paradox that was named after him.
Here’s an example of Russell’s Paradox. Imagine a village in which there is a barber whose job it is to shave all (and only) the people who don’t shave themselves. If I lived there, I’d probably shave myself I don’t think I’d be organized enough to get to the barber every day and I can shave myself perfectly well. And it would probably work out too expensive for me. But if I decided I didn’t want to, then the barber would be the one to shave me. But where does that leave the barber? He’s allowed to shave only people who don’t shave themselves. By this rule, he can’t ever shave himself because he can only shave people who don’t shave themselves. This is going to get difficult for him. Usually if someone can’t shave himself in this village it is the barber who does it for him. But the rule won’t allow the barber to do that, because that would turn him into someone who shaved himself but the barber only shaves the ones who don’t shave themselves.
This is a situation that seems to lead to a direct contradiction saying something is both true and false. This is a situation that seems to lead to a direct contradiction saying something is both true and false. That’s what a paradox is. It’s very puzzling. What Russell discovered was that when a set refers to itself this sort of paradox emerges. Take another famous example of the same sort of thing: ‘This sentence is false.’ This is a paradox too. If the words ‘This sentence is false’ mean what they seem to mean (and are true) then the sentence is false — which then means that what it states is true! This seems to suggest that the sentence is both true and false. But a sentence can’t be true and false at the same time. That’s a basic part of logic. So there’s the paradox.
These are interesting puzzles in themselves. There’s no easy solution to them, and that seems strange. But they were far more important than that for Russell. What they did was reveal that some of the basic assumptions that logicians all over the world had been making about set theory were wrong. They needed to begin again.
A Little History of Philosophy