Bertie Russell and the Incompleteness Theorem (Part II)

Right, where was I? Bertie has just discovered his crazy uncle and a new fear - of going insane.

Ok, so the next big event in our young hero's life is that his grandmother (finally) hires a Maths tutor for him. Mathematics opened a skylight in Bertie's life. For the first time, Bertie came across the idea of proof. In a world that was full of terrors and things that go bang in the night, he realised that if something is true, you should be able to demonstrate that it is true. And thereafter you need not agonise over it any more - you've shown it to be true. Equally, if something is untrue, you should be able to disprove it and move on.

Of course the other way in which Mathematics changed Bertie's life was that it soon became clear that he was very good at it...and he was only to get better!

At that point we will leave Bertie for a few years. He grew up in more or less the ordinary way, eventually went off to university at Trinity College, Cambridge where he steadily demonstrated himself to be more and more brilliant.


But as Bertie found out more about Mathematics, something began to worry him. Everything was so logical, up to a certain point. Theorems built on theorems. Propositions were proved or disproved. Yet something was still missing. The whole edifice of Mathematics was built on axioms. In one way or another, something foundational had to be assumed.

Given that in some ways Bertie's happiness and sanity depended on the certainties of Mathematics, he did not take kindly to this, and quickly set about trying to remedy this foundational problem in the field. Between a marriage (begun and ended), several affairs and multiple academic papers; conferences and international unrest; Bertie attacked the problem with vigour. Together with his great friend Arthur Whitehead he wrote a massive tome on the fundamental principles of Mathematics. He became extremely famous. But he still couldn't settle the foundations of Mathematics to his own satisfaction.

And then something really remarkable happened. A young Mathematician called Kurt Gödel proved what became known as the Incompleteness Theorems. One (and only one) of the shattering implications of these theorems is this: there will always exist some things that are true, but not provable. And there will always exist some things that are untrue, but not disprovable. Another way of saying this is that is impossible to have an entirely internally consistent set of axioms for Mathematics.

Wait...just think about that for a second...


Can you imagine how devastating this must have been? Bertie's life goal was crushed in one (or two) little theorems...

We will leave Bertie at this point. As a brief postscript, I am happy to report that Bertie survived this setback and went on to make many more contributions to Mathematics, Philosophy and even politics (particularly as a pacifist during both World Wars).

I leave you with the thought of the Incompleteness Theorems.

May they keep you awake at night.

They truly are remarkable.

As was Bertrand Russell.

Inaugural EOTSDS: Bertie Russell and the Incompleteness Theorem (Part I)

I have inducted a new tradition...the last-day-of-term shaggy dog story (otherwise known as the EOTSDS)! After all, no-one wants to be at school, but we all have to sit in class and try to be reasonably productive. So I've decided that the thing to do is tell a very long, moderately entertaining and somewhat educational shaggy dog story.

This is the first EOTSDS of 2011. I took it (loosely) from a remarkable book called Logicomix http://www.logicomix.com/en/, which you should all go out and beg, borrow, steal or even (gasp) buy immediately. You'll have to forgive inaccuracies in my version and kindly take into consideration the conditions underwhich this story was first told. You'll also have to fill in a lot of shaggy dog details which just don't work in text...

Are you all sitting comfortably?

Once upon a time, a very long time ago, there lived a small boy called Bertrand Russell, but everyone called him Bertie.

Now Bertie got off to a very unfortunate start in life, due to both his parents dying. So he had to go live with his grandparents, who were very important people who lived in a stately home. Think gloomy, old, important, and really not very nice to live in, especially as a very small boy.

Poor Bertie was given a large, dark drafty bedroom all on his own in this ominous dwelling, and on his first night he heard this terrible groaning noise from somewhere in the attics. Being very young, and since the night was very dark and there was no one to call, he lay in bed and trembled with terror. Was it a ghost? Was it a demon?

This nightly groaning continued to terrify Bertie for many years. All in all Bertie was growing into a very frightened little chap. His grandmother's approach to child-rearing didn't help. She was extrememly strict and extremely religious. She didn't believe that children should ask questions, and she told Bertie lots of stories about the awful things (such as burning in hell) that would happen to him if he was naughty. So of course Bertie was convinced that the groaning was a devil sent to punish him for his misdeeds...


Well, the years went by, as they do; and Bertie grew up, as one does. And soon enough, his grandmother started to hire a series of tutors for him (in those days the children of the very rich didn't go to school, they had private tutors). He fell in love with his beautiful German teacher of course, but much more important was the science teacher. He didn't exactly learn brilliant scientific method, but even the simple observations that they made together began to make Bertie think. And one of the first things he thought was about ghosts and devils. In fact not to put too fine a point on it, Bertie began to have serious doubts about the "ghost/devil theory" of the nightly groaning.

So, one night Bertie snuck off and followed the sound of groaning. He made a momentous discovery, which was to cure his fear of ghosts forever. However that fear was replaced with a far more serious one... What he discovered that the groaning came from a mad uncle whom his grandmother had been keeping hidden in the attics. True story. In grandmother's defence, attitudes to madness or mental illness in those days were such that hiding a crazy relation may have seemed like a very good idea. Mental illness was seen as hereditary, and no-one would want to marry into a family in which there was a known "madman". And the official mental asylums were so horrible that you would not want to send any family member to one of them, no matter how serious his condition (see the opening scene of the movie Amadeus...)

So Bertie's fear of ghosts was replaced by the very real fear of going crazy; a fear that would stay with him throughout his life. But at first it wasn't just a vague fear. It crippled him.


Do not fear, good readers. All is not lost. Stay tuned for the next exciting episode (my fingers are getting tired of all this typing).

Proof and Imagination

A few years ago I saw a movie called Proof. It's about an elderly Maths professor and his daughter. He spends his whole life trying to prove a single theorem, and eventually gets Alzheimers or something like that. His daughter sacrifices her academic career to care for him... Anyway, I won't give the plot away, but the idea of the film centres around the great value of proof.

Now my idea of a mathematical proof tends to be a hand-wavy argument followed by the conclusion: "so it seems clear that..." But through the years I've changed my mind. The great beauty of a true proof (as recognised by Socrates, Aristotle, Bertrand Russell, and various others at different points in the history of thought) is that a proof is irrefutable. Every step follows inexorably on from the previous steps in logical and unarguable progression.

Stripped down to its bare bones, a proof is a statement of pure logic. It takes you from what you know (or what you assume, axiomatically or otherwise) to you guess but do not yet know. In this sense proof is another (more or less accessible, depending on your point of view) form of lucid imagination.

It's imagination that has a shape.