Digital infinity

Digital infinity is a technical term in theoretical linguistics. Alternative formulations are "discrete infinity" and "the infinite use of finite means". The idea is that all human languages follow a simple logical principle, according to which a limited set of digits—irreducible atomic sound elements—are combined to produce an infinite range of potentially meaningful expressions.

'Language is, at its core, a system that is both digital and infinite. To my knowledge, there is no other biological system with these properties....'

‘It remains for us to examine the spiritual element of speech ... this marvelous invention of composing from twenty-five or thirty sounds an infinite variety of words, which, although not having any resemblance in themselves to that which passes through our minds, nevertheless do not fail to reveal to others all of the secrets of the mind, and to make intelligible to others who cannot penetrate into the mind all that we conceive and all of the diverse movements of our souls.’

Noam Chomsky cites Galileo as perhaps the first to recognise the significance of digital infinity. This principle, notes Chomsky, is "the core property of human language, and one of its most distinctive properties: the use of finite means to express an unlimited array of thoughts". In his Dialogo, Galileo describes with wonder the discovery of a means to communicate one's "most secret thoughts to any other person ... with no greater difficulty than the various collocations of twenty-four little characters upon a paper." This is the greatest of all human inventions, Galileo continues, comparable to the creations of a Michelangelo...

'Digital infinity' corresponds to Noam Chomsky's 'universal grammar' mechanism, conceived as a computational module inserted somehow into Homo sapiens' otherwise 'messy' (non-digital) brain. This conception of human cognition—central to the so-called 'cognitive revolution' of the 1950s and 1960s—is generally attributed to Alan Turing, who was the first scientist to argue that a man-made machine might truly be said to 'think'. The idea of a thinking machine had previously been considered absurd, having been famously dismissed by René Descartes as theoretically impossible. Neither animals nor machines can think, insisted Descartes, since they lack a God-given soul. Turing was well aware of this traditional theological objection, and explicitly countered it.

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