In Our Time: Bertrand Russell

Bertrand Russell was, among many other things, one of the influential British philosophers of the 20th Century. The experts who discussed his life & ideas on In Our Time were A. C. Grayling (New College of the Humanities, London and St Anne’s College, Oxford), Mike Beaney (University of York) and Hilary Greaves (Somerville College, Oxford). The programme concentrated on his mathematical work and on his philosophical ideas. They started the programme with a brief description of Russell’s early life – he was born in 1872 as a part of an aristocratic family & didn’t go to school until his teens, just before he went to study at Cambridge University. The bit that stuck in my mind from this part of the programme is that he wasn’t sent to school because he was thought of as the sensitive sort – his brother went to school, but that was because he was more into sports & more robust.

He started out studying mathematics, and then moved onto philosophy, and worked in both fields over his life. In mathematics he was particularly concerned with building the whole of maths up from logic alone – so in arithmetic instead of accepting as an axiom that 1+1=2 you first have to prove that. This was partly because of a philosophical point of view that why should you accept those axioms on trust, and partly because if your system of deciding if a proof is valid or not depends at any point upon intuition then it’s possible for different mathematicians to disagree about the validity of proofs.

So to derive something like 1+1=2 from logical principles alone he first had to define the numbers based on logical principles and the operation of addition. He used the then new idea of sets – called “classes” at the time. I think the idea for how to use sets to define numbers worked as follows: the idea of identity (something is identical to something else) is a logical principle. The idea of non-identity is also such a principle. If you have a set that contains all things that are not identical to themselves, then you have a set with nothing in it – this is the null set, or zero. This set is a singular object. If you have a set that contains the null set and nothing else, then it is a member of the set of all sets that only contain one thing. Which you can use as the definition of the natural number 1. Now you have two objects (the null set, and the definition of 1) and can use those in the same way to define the natural number 2. I am a little confused here why this isn’t using the number to define itself – but I suspect the confusion arises from me (and the experts on the programme) using words to discuss something that’s better done symbolically. They didn’t cover how Russell used sets to define the operation of addition, but I suspect that’s even more complicated.

But using sets to define the basic logical underpinnings of arithmetic introduces a paradox – called Russell’s Paradox, because he described this flaw. If you have a set that contains all sets that do not contain themselves, then does that set contain itself or not? The word picture they used to make the paradox more clear was to say imagine there’s a barber in a village who shaves all the men who do not shave themselves, and only the men who do not shave themselves. If he does not shave himself, then he is a man who does not shave himself and so must be shaved by the barber. But the barber is himself, so if he shaves himself then he is ineligible to be shaved by the barber. But the barber is himself, so now he is not shaved by himself and so must be shaved by the barber etc etc. So Russell took his theory back to pieces and tried to rebuild it without this flaw (and ultimately failed, I think they said). He tried to categorise the sorts of objects that can exist into a hierarchy – there are objects that aren’t sets, then there are level one sets that contain objects that aren’t sets, level 2 sets can contain objects that are level 1 sets etc etc.

And I was reminded that I should re-read “Gödel, Escher, Bach” at some point 🙂

On the philosophy side of things Russell was the founder of something called Analytic Philosophy, which is apparently the dominant philosophy in the English speaking world these days. He was reacting against Idealism which was the dominant philosophy when he was studying at Cambridge. I think the key thing was that the Idealists thought of the mind as the dominant thing, the world exists as it is perceived – essentially a sceptical philosophy where you don’t know if anything is real except that which you have perceived yourself. Russell was more of a Realist (technical term, I think) who was on the side where if you can express a thought about something then that thing must in some sense exist (even if what you are saying is “fairies don’t exist” then the very fact you can conceive of fairies means they do somehow exist even if not actually in the actual real world). Analytic philosophy isn’t as far from Russell’s mathematics as one might imagine at first glance – a large part of his system is breaking down language into logical components and using this to express ideas with clarity. I have a feeling I’ve completely mangled this explanation, and looking at wikipedia hasn’t helped. I do remember the example they gave of the sort of thing he was talking about, which is that the sentence “The present King of France is bald” is actually made up of three logical sentences. In words this would be “There is a thing that is the King of France”, “All things which fit the definition of this thing are this thing” and “The thing that is this thing, is bald.” So when you look at the original sentence it’s hard to tell if it’s true or false – and Russell wanted this to break down to a binary system, either a statement is true or it’s false. The original sentence is actually quite complex – with no King of France, is he bald or not? But if you look at the three logical sentences that make it up, then you can assign it to the “false” category because the very first logical part of it is clearly false (there is no thing that is the King of France).

They ended the programme by saying that Russell did lots of other things as well as mathematics & philosophy – for instance he was heavily in politics, wrote several popular books. But clearly there just wasn’t enough time in the programme to do more than scratch at the surface of his life. And even then it felt like one of the more complex episodes of In Our Time that we’ve listened to.