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In this paper, I shall discuss several topics related to Frege's paradigms of second-order abstraction principles and his logicism. The discussion includes a critical examination of some controversial views put forward mainly by In the introductory section, I try to shed light on the connection between logical abstraction and logical objects. The second section contains a critical appraisal of Frege's notion of evidence and its interpretation by Jeshion, the introduction of the course-of-values operator and Frege's attitude towards Axiom V, in the expression of which this operator occurs as the key primitive term. Axiom V says that the course-of-values of the function f is identical with the course-of-values of the function g if and only if f and g are coextensional. In the third section, I intend to show that in Die Grundlagen der Arithmetik (1884) Frege hardly could have construed Hume's Principle (HP) as a primitive truth of logic and used it as an axiom governing the cardinality operator as a primitive sign. HP expresses that the number of F s is identical with the number of Gs if and only if F and G are equinumerous. In the fourth section, I argue that Wright falls short of making a convincing case for the alleged analyticity of HP. In the final section, I canvass Heck's arguments for his contention that Frege knew he could deduce the simplest laws of arithmetic from HP without invoking Axiom V. I argue that they do not carry conviction. I conclude this section by rejecting an interpretation concerning HP suggested by MacFarlane.In his opus magnum Grundgesetze der Arithmetik (GGA), Frege regarded Axiom V as the linchpin of his logicist project and at the same time its potential Achilles' heel. The first statement is true even though this axiom plays no formal role when he comes to prove the simplest laws of arithmetic. The second is likewise true even though at the end of the Preface to GGA Frege declares emphatically that no one will be able to refute his project. To be sure, Axiom V is formally needed only for framing the explicit definition of the cardinality operator in purely logical terms and for the derivation of Hume's Principle (HP) from that definition. Assuming that Axiom V is, in effect, a primitive truth of logic and that the 172 MATTHIAS SCHIRN rules of inference are truth-perserving, the derivation established for Frege once and for all the logical nature of HP, and thus enabled him to take HP as the pivot of the formal derivations of the fundamental theorems of cardinal arithmetic.In what follows, I want to discuss several issues concerning HP and Axiom V which I consider to be of vital importance for assessing Frege's logicism. I begin with introductory remarks about the relation between logical abstraction and logical objects.

Hilbert developed his famous ®nitist point of view in several essays in the 1920s. In this paper, we discuss various extensions of it, with particular emphasis on those suggested by Hilbert and Bernays in Grundlagen der Mathematik (vol. I 1934, vol. II 1939. The paper is in three sections. The ®rst deals with Hilbert's introduction of a restricted o-rule in his 1931 paper`Die Grundlegung der elementaren Zahlenlehre'. The main question we discuss here is whether the ®nitist (meta-)mathematician would be entitled to accept this rule as a ®nitary rule of inference. In the second section, we assess the strength of ®nitist metamathematics in Hilbert and Bernays 1934. The third and ®nal section is devoted to the second volume of Grundlage n der Mathematik. For preparatory reasons, we ®rst discuss Gentzen's proposal of expanding the range of what can be admitted as ®nitary in his esssay`Die Widerspruchsfreiheit der reinen Zahlentheorie' (1936). As to Hilbert and Bernays 1939, we end on a`critical' note: however considerable the impact of this work may have been on subsequent developments in metamathematics, there can be no doubt that in it the ideals of Hilbert's original ®nitism have fallen victim to sheer proof-theoretic pragmatism.In the 1920s, David Hilbert developed his ®nitist metamathematics qua contentual theory of formalized proofs to defend all of classical mathematics. On the one hand, Russell's and Zermelo's discovery of the set-theoretic paradoxes shook the logical foundations of number theory and analysis. On the other hand, Brouwer and Weyl had mounted a serious attack on classical analysis, especially on the (alleged) validity of the law of excluded middle. Hilbert felt that he had to meet this twofold challenge by establishing metamathematicall y and in a purely ®nitist fashion the consistency of classical mathematics. The central idea underlying metamathematical consistency proofs was to show the consistency of formalized mathematical theories T by means of`weaker' and more reliable methods than those that could be formalized in T.It was in the light of GoÈ del's Incompleteness Theorems that ®nitist metamathematic s turned out to be too weak to lay the foundations for a signi®cant portion of classical mathematics. Hilbert and his collaborators , notably Bernays, responded by extending the original ®nitist point of view. The envisaged extension was guided by two central, though possibly con¯icting ideas: ®rst, to make sure that it preserved the essentials of ®nitist metamathematics ; second, to conduct indeed, within the extended proof-theoretic bounds, a consistency proof for a large part of mathematics, in particular for second-order arithmetic (Z2).We may characterize the nature of Hilbert's ®nitist metamathematics of the 1920s (henceforth referred to as`M') in a few words as follows:(1) M is a non-formalized and non-axiomatized`theory' with a non-mathematica l vocabulary.*With the exception of Hilbert (1928a) and (1928b) the translations of Hilbert's writings are our own. For the most part, we have ...