Citations of:
Die Grundlagen der Arithmetik, 823
In Matthias Schirn (ed.), The Philosophy of Mathematics Today. Clarendon Press (1998)
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NeoFregean logicists claim that Hume's Principle (HP) may be taken as an implicit definition of cardinal number, true simply by fiat. A longstanding problem for neoFregean logicism is that HP is not deductively conservative over pure axiomatic secondorder logic. This seems to preclude HP from being true by fiat. In this paper, we study Richard Kimberly Heck's Twosorted Frege Arithmetic (2FA), a variation on HP which has been thought to be deductively conservative over secondorder logic. We show that it isn't. (...) 

According to the species of neologicism advanced by Hale and Wright, mathematical knowledge is essentially logical knowledge. Their view is found to be best understood as a set of related though independent theses: (1) neofregeanisma general conception of the relation between language and reality; (2) the method of abstractiona particular method for introducing concepts into language; (3) the scope of logicsecondorder logic is logic. The criticisms of Boolos, Dummett, Field and Quine (amongst others) of these theses are explicated and assessed. (...) 

We now know of a number of ways of developing real analysis on a basis of abstraction principles and secondorder logic. One, outlined by Shapiro in his contribution to this volume, mimics Dedekind in identifying the reals with cuts in the series of rationals under their natural order. The result is an essentially structuralist conception of the reals. An earlier approach, developed by Hale in his "Reals byion" program differs by placing additional emphasis upon what I here term Frege's Constraint, (...) 