Plato's Problem: An Introduction to Mathematical Platonism

Plato's Problem: An Introduction to Mathematical Platonism

Marco Panza, Andrea Sereni

Language: English

Pages: 323


Format: PDF / Kindle (mobi) / ePub

What is mathematics about? And if it is about some sort of mathematical reality, how can we have access to it? This is the problem raised by Plato, which still today is the subject of lively philosophical disputes. This book traces the history of the problem, from its origins to its contemporary treatment. It discusses the answers given by Aristotle, Proclus and Kant, through Frege's and Russell's versions of logicism, Hilbert's formalism, Gödel's platonism, up to the the current debate on Benacerraf's dilemma and the indispensability argument. Through the considerations of themes in the philosophy of language, ontology, and the philosophy of science, the book aims at offering an historically-informed introduction to the philosophy of mathematics, approached through the lenses of its most fundamental problem.

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obscured by clouds’: this being obviously a statement that is not about today’s weather, nor, despite appearances, about the Sun or the clouds. This view has been endorsed, for example, by more orthodox neo-positivists such as Rudolf Carnap. We will come back to it in § 2.4.1. Empiricism in the Philosophy of Mathematics An alternative option for our nominalist is to claim that, despite appearances, mathematical statements, even when taken literally, do speak Introduction 11 only of concrete

as referred to these objects should not be taken, then, in its usual meaning, according to which no type can be concrete, but rather in a different meaning, so as to concede that a type could be concrete when it bears an appropriate relation with its tokens.42 Not only sensible intuition is at stake here, also pure intuition is. This is even clearer when one realizes that the proof given above, and the corresponding theorem, concern an infinity of numbers. Nonetheless, their infinitary character

what Hellman seems to acknowledge explicitly. The main difficulty for Hellman’s proposal lies elsewhere, however, namely in his appeal to modality. Logical modality (that is, the notions of ‘logically possible’ and ‘logically necessary’) can be explained in set-theoretic terms: if ‘◊’ is interpreted as a logical possibility operator and ‘ϕ’ is a non-modal statement, then ‘◊ϕ’ is true if and only if ‘ϕ’ is satisfied by some set. Clearly, we are not dealing here with possible sets (what would

the numbers 2, 3, 5 exist. This requirement is equivalent to the requirement that the statement ‘Snow White is Nessy’s mother’ is true only if Snow White and Nessy exist; the existence of 2, 3, and 5 on the one side, and of Snow White and Nessy on the other side, are necessary conditions, respectively, for the truth of ‘2 + 3 = 5’ and ‘Snow White is Nessy’s mother’. When appropriately generalized, this claim is fairly natural, and is involved in all those arguments – including the

that necessarily encode the same monadic properties. The last definition introduces the relation of identity between monadic properties, stipulating that a monadic property F is identical to a monadic property G if F and G are necessarily encoded by the same objects. From the second of these definitions, the comprehension schema, and the axiom stating that only abstract objects encode monadic properties, it follows that for any appropriate formula ‘A’ there is one and only one abstract object

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