Abstract
According to Wittgenstein, mathematics is embedded in, and partly constituting, a form of life. Hence, to imagine different, alternative forms of elementary mathematics, we should have to imagine different practices, different forms of life in which they could play a role. If we tried to imagine a radically different arithmetic we should think either of a strange world (in which objects unaccountably vanish or appear) or of people acting and responding in very peculiar ways. If such was their practice, a calculus expressing the norms of representation they applied could not be called false. Rather, our criticism could only be to dismiss such a practice as foolish and to dismiss their norms as too different from ours to be called ‘mathematics’.
Highlights
According to Wittgenstein, mathematics is embedded in, and partly constituting, a form of life
If we tried to imagine a radically different arithmetic we should think either of a strange world or of people acting and responding in very peculiar ways
Mathematics as Grammar What is the meaning of an arithmetical equation, such as ‘7 + 5 = 12’? Does it describe relations between abstract objects, numbers, as Platonists believe? Or is it rather, as John Stuart Mill held, a wellconfirmed empirical hypothesis to the effect that adding five objects to seven objects will always produce a total of twelve objects? Neither, according to Wittgenstein, who rejects the very assumption on which both opposing schools are agreed, namely that arithmetical equations, like all other declarative sentences, must be descriptions or statements of fact of some sort
Summary
One may object, do logical and mathematical propositions not express truths? – Certainly, we call such sentences ‘true’, but on Wittgenstein’s view they are merely conventional truths, comparable to:. One metre could have been subdivided into 120 centimetres. Is it really conceivable that we might use a different logic or arithmetic? One may be inclined to object that whereas different units of measurement are always possible, there can be no alternative to our logic and arithmetic. A logical calculus that allowed one to infer ‘p’ from ‘p v q’ would just be flawed; likewise an arithmetic containing the equation ‘7 + 5 = 13’. On Wittgenstein’s view such different logical or mathematical systems could not be rejected as false, they may well be impractical. Are drastically different rules of logic or arithmetic even conceivable? Is that a tenable position? Are drastically different rules of logic or arithmetic even conceivable?
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