Mathematics and the Form of Life

‘It is fair to say that Wittgenstein brought about a revolution in philosophical perspective, making platonism – as traditionally presented – impossible. When a person can carry out a mathematical procedure correctly – for example, he can extend the series 2, 4, 6, 8…the platonist says he has learned to follow a rule or has apprehended a universal. We manifest our apprehension of a universal, our grasp of a rule, by at most a finite amount of behaviour, yet the grasp of a universal, or rule guarantees a potentially infinite amount of behaviour. Wittgenstein’s point is, roughly, that it is not because we have grasped a universal or rule that we behave in a certain way, but because we act in certain ways that we say that we have grasped a rule or universal. No rule or universal will guarantee that someone will not extend the series 2, 4, 6, 8, 17, 28, 1002, =, Δ…saying that at each step he is doing exactly what he did before. There is no such thing, Wittgenstein argued, as absolute sameness. A procedure is a case of doing the same thing again if it is a practice within what Wittgenstein called a ‘form of life’ – a community that shares perceptions of salience, routes of interest, feelings of naturalness – in which it is perceived as the same. If the platonist tries to step out of the form of life in order to tell those within how things really are, then he must come to grief. For outside the form of life there is nothing: no rules, no universals, no sameness, no reality.’ (Johnathan Lear, ‘Ethics, Mathematics and Relativism’, in ‘Essays on Moral Realism’, ed. Geoffrey Sayre-McCord, p. 87)

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