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Philosophical method and Galileo's paradox of infinity

In Bart Van Kerkhove (ed.), New Perspectives on Mathematical Practices: Essays in Philosophy and History of Mathematics : Brussels, Belgium, 26-28 March 2007. World Scientfic (2008)

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  1. Size and Function.Bruno Whittle - 2018 - Erkenntnis 83 (4):853-873.
    Are there different sizes of infinity? That is, are there infinite sets of different sizes? This is one of the most natural questions that one can ask about the infinite. But it is of course generally taken to be settled by mathematical results, such as Cantor’s theorem, to the effect that there are infinite sets without bijections between them. These results settle the question, given an almost universally accepted principle relating size to the existence of functions. The principle is: for (...)
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  • Set Size and the Part–Whole Principle.Matthew W. Parker - 2013 - Review of Symbolic Logic (4):1-24.
    Recent work has defended “Euclidean” theories of set size, in which Cantor’s Principle (two sets have equally many elements if and only if there is a one-to-one correspondence between them) is abandoned in favor of the Part-Whole Principle (if A is a proper subset of B then A is smaller than B). It has also been suggested that Gödel’s argument for the unique correctness of Cantor’s Principle is inadequate. Here we see from simple examples, not that Euclidean theories of set (...)
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  • Hierarchical Propositions.Bruno Whittle - 2017 - Journal of Philosophical Logic 46 (2):215-231.
    The notion of a proposition is central to philosophy. But it is subject to paradoxes. A natural response is a hierarchical account and, ever since Russell proposed his theory of types in 1908, this has been the strategy of choice. But in this paper I raise a problem for such accounts. While this does not seem to have been recognized before, it would seem to render existing such accounts inadequate. The main purpose of the paper, however, is to provide a (...)
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  • An Infinite Lottery Paradox.John D. Norton & Matthew W. Parker - forthcoming - Axiomathes:1-6.
    In a fair, infinite lottery, it is possible to conclude that drawing a number divisible by four is strictly less likely than drawing an even number; and, with apparently equal cogency, that drawing a number divisible by four is equally as likely as drawing an even number.
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  • A Minimality Constraint on Grounding.Jonas Werner - 2020 - Erkenntnis 85 (5):1153-1168.
    It is widely acknowledged that some truths or facts don’t have a minimal full ground [see e.g. Fine ]. Every full ground of them contains a smaller full ground. In this paper I’ll propose a minimality constraint on immediate grounding and I’ll show that it doesn’t fall prey to the arguments that tell against an unqualified minimality constraint. Furthermore, the assumption that all cases of grounding can be understood in terms of immediate grounding will be defended. This assumption guarantees that (...)
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