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.

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

  1. Its ok quoting Gobel, Turin and Einstein.
    Also ok if you are a university boffin writing yet another 3,000 word thesis of mind boggling crap and jargon that only another boffin understands.
    Where does all this equate with a Politician who patronises 18-25 year olds saying they all like things in chronological order? Where are we going? Prediction or imagination?
    I’m just joe blogs at the other end of all this money and time wasting everything to me is just COMMON SENSE, in which there is no degree – like the ordinary people at the Grenfel Tower disaster – I am merely a 58 year old pawn. Mother of 4 grown children, at the mercy of highly paid, time wasting idiots!

  2. Predicting human behaviour without reference to axiomatic presuppositions about that same human behaviour? Surely, this was what Gödel’s pulverisation of Hilbert’s attempt at a totalitarian foundationalism for mathematics ruled out wasn’t it? Surely, educational, political, psychological, economic and sociological enquiries are unsuitable objects of THE scientific method because they lack the required ontological stability – being (as the Greeks would have said) merely matters of OPINION (doxa). Opinion is cataclysmically unstable and hence no basis for the predication required of KNOWLEDGE (science/episteme). Respect to you, but I’m amazed you have spent so much time on such a pointless quest.

  3. The Outher Limits of Reason by Yanovsky is a fabulous book in this domain. All those of a ‘big data is the solution’ bent would do well to read it especially macro economic tinkerers. Even fairly simple puzzles with a few parameters can be unsolvable regardless of resources devoted to the solution. Very accessible book (must have been if I read it!)

  4. Great and interesting. It is fun to see your mind at work, and this intellectual history is without a doubt one of the hidden mysteries. Thanks for exploring.

    I’d recommend Charles Peltzer’s tremendous “The Annotated Turing,” which takes the 1936 paper sentence by sentence and explains it all. It gives a primer on the diagonal proof, Cantor’s work, &c. The book is in itself a major intellectual achievement. It is much more understandable even than the Franzen book on Godel that you often (nicely) cite.

  5. One correction so far: “An algebraic number is a number that can be produced by a finite series of algebraic operations.” is false. E.g. the roots of a quintic may not satisfy this, but they do fit the standard definition (any root of a polynomial with integer coefficients).

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