Blackwell Reference Online

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P hilosophy of mathematics Philosophical issues arise over the ontological status of numbers. The Greek Pythagoreans discovered relationships of ratio and proportion among natural numbers and even considered number to be the first principle that determines the structure of the world. The tendency of contemporary philosophy of mathematics to identify numbers with sets has led to the revival of Platonism in mathematics. The traditional position holds that numbers are used to answer questions of the form “How many X's are there?” and, hence, that a number is a property ascribed to an object or group of objects. This view was rejected by Frege , who argued that a number-statement ascribes a property to concepts rather than to objects. Hence a number is a second-level predicate rather than a first-level predicate. On this basis, Frege inferred that existence , like number, is a property of concepts rather than of objects. “The content of a statement of number is an assertion about a concept.” Frege, The Foundations of Arithmetic ... log in or subscribe to read full text