Strange Loops

James Lighthill

  • Gödel, Escher, Bach: An Eternal Golden Braid by Douglas Hofstadter
    Harvester, 777 pp, £10.50, August 1980, ISBN 0 85527 757 2

Mathematicians have always prided them selves on being poised half-way between the arts and the sciences. On the one hand, mathematical theorems share with artistic works the features of beauty and unexpectedness – qualities which general readers may have been enabled by Hardy’s Mathematician’s Apology to perceive in this simple but profound theorem: a prime number that leaves the remainder I when divided by 4 can be expressed as the sum of two squares in one and only one way. Mathematical proofs, from those of Euclid onwards, as Hardy again illustrated through one of Euclid’s best, may brilliantly combine the artistic merit of economy of means with these features of beauty and unexpectedness. Yet, above all, it is the imaginativeness that great mathematicians showed as they went beyond the numbers to create far more complex logical structures – where profound insights were nevertheless used to discern and demonstrate powerful simplicities of interconnection – that links them to the great artists with their gifts for imaginative invention.

On the other hand, many of the greatest mathematicians such as Archimedes, Newton and Gauss have combined with these artistic gifts a very different type of genius: one enabling them to use mathematics to solve practical problems in the physical world, and to carry further the massive growth in knowledge of the physical sciences. These successes were achieved, above all, by mathematicians who were able to acquire keen insight into the experimental method. Collaborating with mathematically-minded experimental physicists and observational astronomers, they progressively uncovered an extraordinary fact: that all those indefinitely repeatable regularities in observation and experiment which we call physical laws take a mathematical form. Every physical law, as it is probed more and more deeply, needs the numerical and logical structures of mathematics to express it with precision. Of course, physicists themselves created their own new imaginative inventions: ‘physical ideas’ of great subtlety which must be grasped in depth by anyone who seeks’to apply mathematics effectively in that field. Nevertheless, after experiment and imaginative thought have generated these powerful physical ideas, deep mathematical investigation has added its contribution to establishing that the laws governing them are of an inherently mathematical form.

The dual claims of mathematics, as the framework for natural laws and as an important field of imaginative expression, increasingly attracted a special sort of prestige. Here, it was felt, poised between the arts and the sciences, lay one of the most characteristic and powerful faculties of the human mind. Naturally, mathematicians themselves were among the most forward in arguing that to study the mind’s mathematical modes of operation could be a matter of very special significance.

By the First World War, such analysis seemed already to suggest that mathematics and pure deductive logic were essentially the same faculty. Not only did mathematics necessarily have a structural form consisting of theorems and proofs, and leading back ultimately, by a chain of deductive logical argument, to a very small number of axioms: but, also, logic itself could be expressed in a mathematical formalism (the ‘propositional calculus’ of Peano, with its own axioms); and, conversely, number, and all the other protagonists of the mathematical scene, could be defined, as Frege showed, in terms of purely logical concepts.

It was natural, then, for the logically deductive mode of thought to inherit the special prestige already attracted by mathematics. To Bertrand Russell, for example, the establishment of logic’s invincibility as a deductive weapon seemed an aim of supreme importance. In the atmosphere of excessive hopes generated by Russell’s early successes in modifying the rules of logic (through the Theory of Types) to avoid a whole class of difficulties typified by the classical Epimenides paradox, Kurt Gödel’s demolition of Russell’s aim in a conclusive argument known as Gödel’s Theorem could be seen as a spectacular event.

Gödel’s Theorem is equally impressive as a piece of mathematics. The imaginative powers he used to build up its proof illustrate perfectly the fact that a mathematician’s skills extend far beyond formal logic. This is no mystical contention, contrasting the ‘reductionism’ of seeing a proof as a single long chain of deductively logical steps with the ‘holism’ of identifying a higher entity in the chain as a whole. It is concerned, rather, with recognising how a great step forward in mathematics involves the mind’s ability to create structures of a completely new kind through an imaginative activity closely akin to that of the great artist.

The primary purpose of Gödel, Escher, Bach is to explore such kinships of the imagination, while also exploiting Douglas Hofstadter’s own artistic imagination to the maximum possible extent, so as to make all the essential ideas underlying Gödel’s achievement accessible to readers without specialised mathematical knowledge. Previous popular introductions to the beauty of mathematics have been like Hardy’s, which used a splendidly clear and interesting style of writing but required no other artistic aids because Hardy attempted to explain only very simple proofs. In mathematics, Hardy’s book scaled, as it were, the Chiltern Hills of haute vulgarisation, but Hofstadter attempts the Himalayas.

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