Intuition and ingenuity: Alan Turing’s work and impact

Don't blame Alan Turing
Don’t blame Alan Turing

Legend has it that Apple’s rainbow-coloured logo showing the apple with a bite out of it is in homage to Alan Turing “the father of modern computing”. Turing died of cyanide poisoning on 7 June 1954, two years after being convicted of homosexuality and accepting chemical castration instead of prison. A half-eaten apple was found next to him, and one theory is that he’d laced it with cyanide, his own homage to the wicked queen in Snow White, his favourite Disney cartoon. Another theory is that he died accidentally after inhaling cyanide fumes from apparatus he had in his bedroom for electroplating spoons. A third explanation is that he really did commit suicide, but set up the apparatus so his mother would think it was an accident. The coroner didn’t test the apple for cyanide, so we’ll never know for sure.

If there are doubts about Turing’s death, his life is fairly well-known, or at least some aspects of it. His most noteworthy exploit for the general public was helping to break the code of the Enigma machines the Germans used to communicate with their submarines during the Second World War. If you’d like to get some idea of how he did it, take a look at the excerpts from the “Enigma Paper” in Alan Turing, His Work and Impact, just published by Elsevier. Cryptography is the second of four parts of this thousand-page overview presenting Turing’s most significant works from the four-volume Collected Works along with comment, analysis and anecdote from leading scholars. The other three parts are on Turing’s contributions to computability, artificial intelligence, and biology.

That simple naming of the parts already gives you some idea of the breadth of Turing’s influence, and we could also add economics. I actually got the Elsevier book thanks to Professor K. Vela Velupillai who wrote for us about Turing’s economics here.  That article described the foundations of computable economics, while here at the OECD the project on new approaches to economic challenges was being launched. “New Approaches” revisits some of the fundamental assumptions about the functioning of the economy, and the implications for policy. It also addresses how to extend the capabilities of existing tools for structural analysis and analysing trends over the long term to factor in key linkages and feedback – for example between growth, inequality, and the environment.

Vela cites Turing’s Precept, an idea that should be kept in mind by economic theorists, analysts and policy makers everywhere: “the inadequacy of reason unsupported by common sense”. There’s a corollary to that in how you present the reasoning, best summed up by the German mathematician David Hilbert at the 1900 conference of the International Congress of Mathematicians in Paris. Presenting a paper on 23 unsolved problems that would help set the research agenda for mathematics in the new century, Hilbert quoted an old French mathematician as saying: “A mathematical theory is not to be considered complete until you have made it so clear that you can explain it to the first man whom you meet on the street”. And Turing himself claimed that “No mathematical method can be useful for any problem if it involves much calculation.”

Gregory Chaitin recalls Hilbert’s remark when he presents Turing’s “Solvable and unsolvable problems”, that ends with Turing’s Precept. In this “lovely paper” Turing explains the notion of computability and proves the unsolvability of a decision problem without using any mathematical formalism. His “models” are two puzzles that were popular at the time: a picture made up of a number of movable squares set in a frame, with one square missing so you can move the squares around to form the image; and two pieces of intertwined wire you can separate without bending or breaking them.

That said, much of what’s presented is for specialists and you’d need a good grounding in mathematics to follow it. But there’s still plenty even for a non-mathematician like me, some of it surprisingly moving, for example when Bernard Richards describes how he presented his and Turing’s work on morphogenesis to Turing’s mother shortly after his death. Some of it is intriguing – why does the UK government still refuse to declassify the two 1946 papers “Report on the applications of probability to cryptography” and “Paper on statistics of repetitions”? But no matter how well you know the life and work of Turing, you’ll learn something from this book.

By the way, that Apple story at the beginning is only a legend. Rob Janoff who designed the logo explained that he was asked to come up with something simpler than the (hideous) picture of Newton sitting under an apple tree that was the company’s first logo, and the bite was just to show that it was an apple, not a cherry or a tomato. The gay-friendly rainbow was to advertise the colour graphics capabilities of Apple’s computers. On British TV show QI XL, Stephen Fry recalled asking  his friend Steve Jobs about the Turing story “It isn’t true, but God we wish it were!” was Jobs reply.

Useful links

Models used in the OECD Economics Department

Turing’s Economics: A Birth Centennial Homage

Click to go to the Alan Turing Year website

This month marks the centennial of the birth of mathematician Alan Turing, the “father” of modern computing and artificial intelligence. To celebrate the occasion, we’ll be publishing a series of articles on modelling and economics. The series starts with a contribution from Professor K. Vela Velupillai of the Algorithmic Social Sciences Research Unit at Trento University’s Economics Department, and Elected Member of the Turing Centenary Advisory Committee.

The “Five Turing Classics” – On Computable Numbers, Systems of Logic, Computing Machinery and Intelligence, The Chemical Basis of Morphogenesis, and  Solvable and Unsolvable Problems– should be read together to understand why there can be something called Turing’s Economics. Herbert Simon, one of the founding fathers of computational cognitive science, was deeply indebted to Turing in the way he tried to fashion what I have called “computable economics”, acknowledging that “If we hurry, we can catch up to Turing on the path he pointed out to us so many years ago.”

Simon was on that path, for almost the whole of his research life. It has been my mission, first to learn to take this “path”, and then to teach others the excitement and fertility for economic research of taking it too.

A comparison of Turing’s classic formulation of Solvable and Unsolvable Problems in his last published paper in 1954 and Simon’s variation on that theme, as Human Problem Solving, would show that the human problem solver in the world of Simon needs to be defined – as Simon did – in the same way Turing’s approach was built on the foundations he had established in 1936-37. At a deeper epistemological level, I have come to characterize the distinction between orthodox economic theory and Turing’s Economics in terms of the last sentence of Turing’s paper (italics added): “These, and some other results of mathematical logic may be regarded as going some way towards a demonstration, within mathematics itself, of the inadequacy of ‘reason’ unsupported by common sense.”

We – at ASSRU – characterize every kind of orthodox economic theory, including orthodox behavioural economics, advocating the adequacy of “reason” unsupported by common sense; contrariwise, in Turing’s economics we take seriously what we now refer to as Turing’s Precept: ‘the inadequacy of reason unsupported by common sense’.

At another frontier of research in many of what are fashionably referred to as “the sciences of complexity”, some references to Turing’s The Chemical Basis of Morphogenesis is becoming routine, even in varieties of computational economics exercises, especially when concepts such as “emergence” are invoked. It is now increasingly realized that the notion of “emergence” originates in the works of the British Emergentists, from John Stuart Mill to C. Lloyd Morgan, in the half-century straddling the last quarter of the 19th and the first quarter of the 20th century.

A premature obituary of British Emergentism was proclaimed on the basis of a rare, rash, claim by Dirac (italics added): “The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble. It therefore becomes desirable that approximate practical methods of applying quantum mechanics should be developed, which can lead to an explanation of the main features of complex atomic systems without too much computation.”

Contrast this with Turing’s wonderfully laconic, yet eminently sensible precept in his 1954 paper (italics added): “No mathematical method can be useful for any problem if it involves much calculation.”

Turing’s remarkably original work on The Chemical Basis of Morphogenesis was neither inspired by, nor influenced any later allegiance to the British Emergentist’s tradition – such as the neurological and neurophilosophical work of Nobel Laureate, Roger Sperry.  On the other hand, the structure of the experimental framework Turing chose to construct was uncannily similar to the one devised by Fermi, Pasta and Ulam in 1955, although with different purposes in mind.

Turing’s aim was to devise a mechanism by which a spatially homogeneous distribution of chemicals – i.e., formless or patternless structure – could give rise to form or patterns via what has come to be called a Turing Bifurcation, the basic bifurcation that lies at the heart of almost all mathematical models for patterning in biology and chemistry, a reaction-diffusion mechanism formalised as a (linear) dynamical system and subject to what I refer to as the linear mouse theory of self-organisation, for reasons you can discover here.

Those interested in the nonlinear, endogenous, theory of the business cycle know that the Turing Bifurctions are at least as relevant as the Hopf Bifurcation in modeling the “emergence” and persistence of unstable dynamics in aggregative economic dynamics.

Turing’s Economics straddles the micro-macro divide in a way that makes the notion of microfoundations of macroeconomics thoroughly irrelevant; more importantly, it is also a way of circumventing the excessive claims of reductionists in economics, and their obverse! This paradox would have, I conjecture, provided much amusement to the mischievous child that Turing was, all his life.

Useful links

Pr Velupillai kindly provided this extended version of his article, including notes and comments

Computable Economics (Elgar, 2012) edited by Veupillai, Zambelli and Kinsella brings together the seminal papers of computable economics from the last sixty years and encompass the works of some of the most influential researchers in this area, including Turing

Applications of complexity science for public policy from the OECD Global Science Forum

Algorithmic Social Sciences Research Unit (ASSRU) at the Univesity of Trento

Alan Turing