Turing and the Universal Machine by Jon Agar

Author:Jon Agar
Language: eng
Format: epub
ISBN: 9781785782381
Publisher: Icon Books Ltd

• CHAPTER 15 •

CRISIS LOOMS

Non-Euclidean geometries were the first suggestion that mathematics was not merely the steady accumulation of true theorems, and their discovery directed attention towards foundations. Similar anxieties surrounded analysis – the mathematics that underpinned calculus, and therefore also of immense practical significance. As the nineteenth century progressed, the temperature of debate rose. A rearguard action was fought. Some philosophers – such as the Englishmen Bertrand Russell and Alfred North Whitehead, and the German Gottlob Frege – proposed that secure foundations could be found in the surely indubitable rules of reliable thought, and began a mammoth effort to reduce mathematics, including geometry and analysis, to logic. Yet again, paradoxes and problems arose: even a concept as apparently simple as a ‘set’ turned out to be far more complex than initially hoped. A fin-de-siècle crisis loomed, yet some remained optimistic. In 1900, the Göttingen mathematician David Hilbert addressed the International Congress of Mathematicians in Paris. The turn of the century provoked him to seek hope for the future by looking to the past:

History teaches the continuity of the development of science. We know that every age has its own problems, which the following age either solves or casts aside as profitless and replaces by new ones. If we would obtain an idea of the probable development of mathematical knowledge in the immediate future, we must let the unsettled questions pass before our minds and look over the problems which the science of today sets and whose solution we expect from the future. To such a review of problems, the present day, lying at the meeting of the centuries, seems to me well adapted. For the close of a great epoch not only invites us to look back into the past but also directs our thoughts to the unknown future.

Hilbert then put to his audience of the world’s most distinguished mathematicians 23 unsolved problems. They were not chosen at random but pinpointed the weakest spots in the crumbling foundations. If solved, mathematics would again be secure, and Hilbert was certain they would be:

[The] ‘conviction of the solvability of every mathematical problem is a powerful incentive to the worker. We hear within us the perpetual call: There is the problem. Seek its solution. You can find it by pure reason, for in mathematics there is no ignorabimus [we will not know].

One particular concern of Hilbert’s was to be of great significance to Alan Turing and the history of computing. Hilbert described his own philosophy of mathematics as ‘formalist’, accepting that, as with non-Euclidean geometries, what mattered was that you chose the rules of the game, and as long as the game of mathematics had certain properties, then the mathematics was sound. In particular, Hilbert was asking questions of the axioms of mathematics, ones he stated even more precisely in 1928: given a set of axioms, could we be certain that the system of theorems derived possessed three properties: completeness, consistency and decidability? Mathematics would be complete if, starting from the axioms, either a proof or a disproof could be found for every meaningful mathematical statement.

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