The Limits of Abstraction

ISBN: 9780199246182 出版年：2002 页码：214 Fine, Kit Clarendon Press

Kit Fine's book is a study of abstraction in a quite precise sense which derives from Frege. In his Grundlagen} Frege contemplates defining the concept of number by means of what has come to be called Hume's principle the principle that the number of Fs is the same as the number of Gs just in case there is a one-to-one correspondence between the Fs and the Gs. Frege's discussion is largely conducted in terms of another, similar but in some respects simpler, proposal to define the concept of direction by laying down the principle (the Direction Equivalence) that the direction of a line a is the same as that of line b if and only if lines a and b are parallel. Such principles have come to be known as abstraction principles. More generally, abstraction principles are ones of the shape: VaVP (§(a) = §(P) a = P), where = is an equivalence relation on entities of the type over which a and P vary and, if the principle is acceptable, § is a function from entities of that type to objects. Since 'the direction of is intended to stand for a function from objects of a certain sort (viz. straight lines) to other objects (their directions), the Direction Equivalence is a first-order, or in Fine's terminology objectual, abstraction; since 'the number of is intended to stand for a function from concepts (or properties) to objects, Hume's principle is a higherorder, or as Fine says conceptual, abstraction. As is well-known, Frege himself abandoned the idea of defining number implicitly or contextually by means of Hume's principle adopting instead an explicit definition of the number of Fs as the extension of the concept concept equinumerous to the concept F because he thought that an adequate definition should settle the question whether Julius Caesar, for example, is or is not the number of some concept, but that the proposed implicit definition cannot do so. But interest in abstraction principles has revived in the last couple of decades, largely as a