Matthias Schirn (LMU) gives a talk at the MCMP Colloquium (26 Jan, 2012) titled "Logical abstractions and logical objects in Frege: a critical approach". Abstract: In this talk, I shall critically discuss some key issues related to Frege’s notion of logical object, his paradigms of second-order abstraction principles (Hume’s Principle and Axiom V; see my handout), his logicism and, if time allows, the position which has come to be known as neo-logicism. Although the notion of logical object plays a key role in Frege’s foundational project, it has hardly been analyzed in depth so far. I shall begin by explaining the connection between logical abstraction and logical objects. A schema for a Fregean abstraction principle can be stated as follows: Q(a) = Q(b) « Req(a, b). Here “Q” is a singular term-forming operator,a and b are free variables of the appropriate type, ranging over the members of a given domain, and “Req” is the sign for an equivalence relation holding between the values of a and b. I call an abstraction principle logical if the equivalence relation, denoted on its right-hand side, can be defined in second-order or higher-order logic. I shall argue that Frege’s principal motive for introducing extensions of concepts into his logical theory is not to be able to make indirect statements about concepts, but rather to define all numbers as logical objects of a fundamental and an irreducible kind in order to ensure that we have the right cognitive access to them qua logical objects via Axiom V. Nonetheless, reducibility to extensions cannot be the ultimate criterion for Frege of what is to be regarded as a logical object. In the second part, I shall briefly examine Frege’s problem of ref erential indeterminacy of numerical singular terms in The Foundations of Arithmetic (1884) and of course-of-values terms in his opus magnum The Basic Laws of Arithmetic (vol. I, 1893/vol II, 1903). This problem arises from what is usually called “the Julius Caesar problem”. In The Foundations, Frege attempted to introduce cardinal numbers as logical objects by means of Hume’s Principle (after having explored an unsuccessful inductive definition). The attempt miscarried, because in its role as a contextual definition Hume’s Principle fails to fix uniquely the reference of the cardinality operator “the number which belongs to the concept j”: it does not place us in a position to decide whether, say, the number of planets is identical with Julius Caesar. I argue that the Caesar problem which is supposed to stem originally from Frege’s tentative inductive definition of the natural numbers in The Foundations, §55 is only spurious; that the genuine Caesar problem deriving from Hume’s Principle is a purely semantic one and that the prospects of removing it by explicitly dcfining cardinal numbers as extensions of concepts (as equivalence classes of equinumerosity) or otherwise are presumably poor. Moreover, I intend to show that in The Foundations Frege could hardly have construed Hume’s Principle as a primitive truth of logic and used it as an axiom governing the cardinality operator as a primitive sign. When Frege comes to introduce his prototype of a logical object, namely courses-of-values of first-level functions — which include extensions of first-level concepts and of first-level relations as special cases — via Axiom V in Frege 1893, he encounters a variant of his old indeterminacy problem from The Foundations, now clad in formal garb. I shall confine myself to making some critical comments on Frege’s attempt to overcome the referential indeterminacy of course-of-values terms. In the third and final part, I shall try to shed new light on several aspects of Frege’s logicist programme from the point of view of both his theory of the cardinals and his theory of the reals. One issue that I focus on is Frege’s original plan, only mentioned in Frege 1884 but never carried out by him, to introduce the real numbers (tentatively) by abstrac...