Complexity and Prediction Part V: The crisis of mathematical paradoxes, Gödel, Turing and the basis of computing

Before the referendum I started a series of blogs and notes exploring the themes of complexity and prediction. This was part of a project with two main aims: first, to sketch a new approach to education and training in general but particularly for those who go on to make important decisions in political institutions and, second, to suggest a new approach to political priorities in which progress with education and science becomes a central focus for the British state. The two are entangled: progress with each will hopefully encourage progress with the other.

I was working on this paper when I suddenly got sidetracked by the referendum and have just looked at it again for the first time in about two years.

The paper concerns a fascinating episode in the history of ideas that saw the most esoteric and unpractical field, mathematical logic, spawn a revolutionary technology, the modern computer. NB. a great lesson to science funders: it’s a great mistake to cut funding on theory and assume that you’ll get more bang for buck from ‘applications’.

Apart from its inherent fascination, knowing something of the history is helpful for anybody interested in the state-of-the-art in predicting complex systems which involves the intersection between different fields including: maths, computer science, economics, cognitive science, and artificial intelligence. The books on it are either technical, and therefore inaccessible to ~100% of the population, or non-chronological so it is impossible for someone like me to get a clear picture of how the story unfolded.

Further, there are few if any very deep ideas in maths or science that are so misunderstood and abused as Gödel’s results. As Alan Sokal, author of the brilliant hoax exposing post-modernist academics, said, ‘Gödel’s theorem is an inexhaustible source of intellectual abuses.’ I have tried to make clear some of these using the best book available by Franzen, which explains why almost everything you read about it is wrong. If even Stephen Hawking can cock it up, the rest of us should be particularly careful.

I sketched these notes as I tried to pull together the story from many different books. I hope they are useful particularly for some 15-25 year-olds who like chronological accounts about ideas. I tried to put the notes together in the way that I wish I had been able to read at that age. I tried hard to eliminate errors but they are inevitable given how far I am from being competent to write about such things. I wish someone who is competent would do it properly. It would take time I don’t now have to go through and finish it the way I originally intended to so I will just post it as it was 2 years ago when I got calls saying ‘about this referendum…’

The only change I think I have made since May 2015 is to shove in some notes from a great essay later that year by the man who wrote the textbook on quantum computers, Michael Nielsen, which would be useful to read as an introduction or instead, HERE.

As always on this blog there is not a single original thought and any value comes from the time I have spent condensing the work of others to save you the time. Please leave corrections in comments.

The PDF of the paper is HERE (amended since first publication to correct an error, see Comments).


‘Gödel’s achievement in modern logic is singular and monumental – indeed it is more than a monument, it is a land mark which will remain visible far in space and time.’  John von Neumann.

‘Einstein had often told me that in the late years of his life he has continually sought Gödel’s company in order to have discussions with him. Once he said to me that his own work no longer meant much, that he came to the Institute merely in order to have the privilege of walking home with Gödel.’ Oskar Morgenstern (co-author with von Neumann of the first major work on Game Theory).

‘The world is rational’, Kurt Gödel.