New PDF release: Wittgenstein, Finitism, and the Foundations of Mathematics

By Mathieu Marion

ISBN-10: 019823516X

ISBN-13: 9780198235163

This pioneering booklet demonstrates the an important value of Wittgenstein's philosophy of arithmetic to his philosophy as a complete. Marion strains the advance of Wittgenstein's considering within the context of the mathematical and philosophical paintings of the days, to make coherent feel of rules that experience too frequently been misunderstood simply because they've been provided in a disjointed and incomplete approach. particularly, he illuminates the paintings of the ignored 'transitional period' among the Tractatus and the Investigations.

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This pioneering ebook demonstrates the the most important significance of Wittgenstein's philosophy of arithmetic to his philosophy as a complete. Marion lines the advance of Wittgenstein's pondering within the context of the mathematical and philosophical paintings of the days, to make coherent experience of rules that experience too frequently been misunderstood simply because they've been offered in a disjointed and incomplete manner.

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Additional info for Wittgenstein, Finitism, and the Foundations of Mathematics (Oxford Philosophical Monographs)

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For example, Σ is used in the definition of an integral as an infinite sum of all values of a function. 34 Wittgenstein, Finitism, and Mathematics By the same token, there is no reason to believe that Wittgenstein had any qualms with quantification over an infinite domain. Are there such domains in the Tractatus Logico-Philosophicus? In other words, are there infinite totalities in that work? 2211). 535). In virtue of the name–object relations in the Tractatus Logico-Philosophicus, an infinity of names implies an infinity of objects.

Since Ω0’x is identified with x, one can say that 0 stands for the formal property constituted by the number of times (in this case, zero) the operation is applied to produce such an expression, and it provides the meaning of the numeral 0 in arithmetic. Similarly, ΩS0’x is identified with Ω’x, and one can say that S0 stands for the formal property constituted by the number of times the operation is applied to produce such an expression (in this case a single application to the base) and it provides the meaning of the numeral S0 in arithmetic.

So, in the above example, for N = 2, we have: (λx. 3x2 + 1)2 = 13 A typical operator is that of identity or I, which is defined thus:22 I ≡ λx. x It is worth noticing that there is no equivalent for this operator in systems of set theory such as Zermelo-Fraenkel’s, where each set has an identity function but there is no universal identity function that can be applied to everything. Differences of this sort are due to the 1987). One finds a direct echo of Molk in Weyl’s book when he speaks of the ‘completely vague concept of function which has become canonical in analysis since Dirichlet’ (Weyl 1987: 23).

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Wittgenstein, Finitism, and the Foundations of Mathematics (Oxford Philosophical Monographs) by Mathieu Marion


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