Leibniz on Infinite Number, Infinite Wholes, and the Whole World

The Leibniz Review 11:103-116 (2001)
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Abstract

Reductio arguments are notoriously inconclusive, a fact which no doubt contributes to their great fecundity. For once a contradiction has been proved, it is open to interpretation which premise should be given up. Indeed, it is often a matter of great creativity to identify what can be consistently given up. A case in point is a traditional paradox of the infinite provided by Galileo Galilei in his Two New Sciences, which has since come to be known as Galileo’s Paradox. It concerns the set of all numbers, N: since to every number there is a corresponding square, there are as many squares as numbers. But since there are non-squares between the squares, “all numbers, comprising the squares and the non-squares, are greater than the squares alone”, i.e. there must be fewer numbers in the set of all squares S than in N. Thus N is both equal to and greater than S. This is a contradiction, so one of the premises must be given up: the question is, which one?

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Richard T. W. Arthur
McMaster University

Citations of this work

Leibniz on Infinite Numbers, Infinite Wholes, and Composite Substances.Adam Harmer - 2014 - British Journal for the History of Philosophy 22 (2):236-259.
Leibniz's mathematical argument against a soul of the world.Gregory Brown - 2005 - British Journal for the History of Philosophy 13 (3):449 – 488.
A Note on Leibniz's Argument Against Infinite Wholes.Mark van Atten - 2011 - British Journal for the History of Philosophy 19 (1):121-129.

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