“How did the serpent of inconsistency enter Frege’s paradise?”

The paper explores the alleged connection between indefinite extensibility and the classic paradoxes of Russell, Burali-Forti, and Cantor. It is argued that while indefinite extensibility is not per se a source of paradox, there is a degenerate subspecies—reflexive indefinite extensibility—which is. The result is a threefold distinction in the roles played by indefinite extensibility in generating paradoxes for the notions of ordinal number, cardinal number, and set respectively. Ordinal number, intuitively understood, is a reflexively indefinitely extensible concept. Cardinal number is not. And Set becomes so only in the setting of impredicative higher-order logic—so that Frege’s Basic Law V is guilty at worst of partnership in crime, rather than the sole offender.

The Proof of Hume’s Principle

Beginning in Grundgesetze §53, Frege presents proofs of a set of theorems known to encompass the Peano-Dedekind axioms for arithmetic. The initial part of Frege’s deductive development of arithmetic, to theorems (32) and (49), contains fully formal proofs that had merely been sketched out in Grundlagen. Theorems (32) and (49) are significant because they are the right-to-left and left-to-right directions respectively of what we call today “Hume’s Principle” (HP). The core observation that we explore is that in Grundgesetze, Frege does not prove Hume’s Principle, not at least if we take HP to be the principle he introduces, and then rejects, as a definition of number in Grundlagen. In order better to understand why Frege never considers HP as a biconditional principle in Grundgesetze, we explicate the theorems Frege actually proves in that work, clarify their conceptual and logical status within the overall derivation of arithmetic, and ask how the definitional content that Frege intuited in Hume’s Principle is reconstructed by the theorems that Frege does prove.

Double Value-Ranges

From the time of Begriffsschrift onwards, Frege treated functions of two or more places on a par with those of one place. This included the treatment of relations (Beziehungen) as a special case of polyadic functions in the way that concepts (Begriffe) were a special case of monadic functions. By the time of Grundgesetze (and unlike in Begriffsschrift), Frege dealt with relations largely through their extensions, which were what he called “double value-ranges” (Doppelwerthverläufe). This is in some ways a misnomer, since double value-ranges are simply a special case of single or ordinary value-ranges, namely value-ranges of functions derived from the value-ranges of monadic functions with additional saturated places. Frege’s treatment of the extensions of relations (which he came to call simply “Relationen”) thus embodies a move analogous to the treatment of polyadic functions as functions of functions, a device invented in 1920 by Moses Schönfinkel and since (unfairly) known in combinatory logic as “currying”. This paper considers the details of Frege’s Grundgesetze treatment of relations via their extensions, exhibits its grammar, and indicates its formal elegance by comparing it with other possible treatments.

Definitions in Begriffsschrift and Grundgesetze

Frege’s definitions in Part III of Begriffsschrift introduce novel forms of variable-binding and quantification. Frege’s commentary, however, shows that he did not fully grasp the logical significance of his notation, treating the new variables as themselves somehow defined. In Grundgesetze, such issues are avoided by exploiting the value-range notation as a substitute for functional abstraction, relying on the inconsistent Basic Law V. In presenting Frege’s Theorem without appeal to Basic Law V, Richard Heck reinstates a generalized form of the notation of Begriffsschrift by using a higher-order logic employing generalized variable-binding to form names of higher-level functions. Heck and Robert May suggest that by the time of Grundgesetze Frege had achieved the necessary understanding of variable-binding to appreciate Heck’s notation. However, even the later Frege had only a piecemeal characterization of what we now see as falling together as variable-binding. For him, different forms of variable-binding do different logical work, marked by distinct ranges of variables. Eliminating the value-range notation in his definitions after the manner of Heck would require rethinking the role of variable-binding operators. This would not be philosophically cost-free for Frege, though there is some slight evidence that Frege may have begun to move in this direction toward the end of his career.

Second-Order Abstraction Before and After Russell’s Paradox

In this article, I discuss certain aspects of Frege’s paradigms of second-order abstraction principles, Hume’s Principle and Basic Law V, with special emphasis on the latter. I begin by arguing that, contrary to a widespread view, Frege did not express any dissatisfaction with Basic Law V before 1902. In particular, he did not raise any doubt about its assumed logical nature. I then show why Frege nonetheless fails to justify Basic Law V as a primitive logical truth along the lines of the semantic justification that he provides for the other axioms of his system. In subsequent sections, I argue (a) that Frege could not have chosen Hume’s Principle as a logical axiom, neither before 1902 nor after 1902; (b) that even if in the wake of Russell’s Paradox Frege had accepted Hume’s Principle as a logical axiom, such an axiom could not have replaced Basic Law V which was designed to introduce logical objects of a fundamental and irreducible kind and to afford us the right cognitive access to them; (c) that Frege most likely held that the two sides of Basic Law V express different thoughts; (d) that for Basic Law V or for any other Fregean abstraction principle that is laid down as an axiom of a theory, the case in which both real epistemic value and self-evidence are given their due is ruled out. I make a proposal as to how Frege might have escaped this epistemic dilemma.

Frege on Creation

In §§138–47 of the Basic Laws of Arithmetic, Frege attacks the notion that mathematical objects can be ‘created’, criticising Stolz, Hankel, and Dedekind directly, and Cantor and Hilbert indirectly. This paper tries to assess exactly what Frege’s criticism criticises, concentrating particularly on Frege’s opposition to Dedekind and Hilbert. Frege’s ostensible target is arbitrariness, and the need for consistency proofs and the method of achieving them. However, the analysis here argues that the real target is the hidden existential assumptions which are called on, as well as the attempt to avoid what Frege would consider proper definitions. This is then compared to Frege’s description of his own procedure. In the light of this, the paper concludes that Frege’s criticism is unfair, that he attacks these other mathematicians for not doing what he himself is unable to do. At the end, attention is also drawn to another attack on the idea that we create mathematics, that of Gödel. Gödel’s core concerns and core arguments differ from Frege’s, many of them being rooted in the various incompleteness phenomena discovered long after Frege’s work. Nevertheless, there are parallels, which it would be instructive to pursue.

Axioms in Frege

Frege’s conception of axioms is an old-fashioned one. According to it, each axiom is a determinate non-linguistic proposition, one with a fixed subject-matter, and with respect to which the notion of a ‘model’ or an ‘interpretation’ makes no sense. As contrasted with the fruitful modern conception of mathematical axioms as collectively providing implicit definitions of structure-types, a conception on which the range of models of a set of axioms is of the essence of those axioms’ significance, Frege’s view is a dinosaur. This essay investigates some of the philosophically-important aspects of that dinosaur, in order to shed light on Frege’s understanding of the foundational role of axioms, and on some of the ways in which our current conception of such axiomatic virtues as independence and categoricity have (and in some cases have not) been informed by a move away from Frege’s understanding of the foundational role of axioms.

Translating ‘Bedeutung’ in Frege’s Writings: A Case Study and Cautionary Tale in the History and Philosophy of Translation

In this chapter I chart the history of the different English translations of ‘Bedeutung’ in Frege’s writings, setting them in context, explaining their rationale, and exploring some of the philosophical issues raised.

Frege on the Real Numbers

This paper is concerned with Gottlob Frege’s theory of the real numbers as sketched in the second volume of his masterpiece Grundgesetze der Arithmetik. It is perhaps unsurprising that Frege’s theory of the real numbers is intimately intertwined with and largely motivated by his metaphysics. The account raises interesting, and surprisingly underexplored, questions about Frege’s metaphysics: Can this metaphysics even accommodate mass quantities like water, gold, light intensity, or charge? Frege’s main complaint with his contemporaries Cantor and Dedekind is that their theories of the real numbers do not build the applicability of the real numbers directly into the construction. In taking Cantor and Dedekind’s Arithmetic theories to be insufficient, clearly Frege takes it to be a desideratum on a theory of the real numbers that their applicability be essential to their construction. We begin with a detailed review of Frege’s theory, one that mirrors Frege’s exposition in structure. This is followed by a critique, outlining Frege’s linguistic motivation for ontologically distinguishing the cardinal numbers from the real numbers. We briefly consider how Frege’s metaphysics might need to be developed, or amended, to accommodate some of the problems. Finally, we offer a detailed examination of Frege’s Application Constraint – that the reals ought to have their applicability built directly into their characterization. It bears on deeper questions concerning the relationship between sophisticated mathematical theories and their applications.

Frege’s Relation to Dedekind: Basic Laws and Beyond

Among all of Frege’s contemporaries, Richard Dedekind is arguably the thinker closest to him in terms of their general backgrounds and core projects. This essay provides a reexamination of Frege’s critical reactions to Dedekind, in Grundgesetze and some related texts. The reexamination includes documenting their interaction in some detail and putting it into a broader context, both philosophically and systematically. It also involves separating Frege’s less compelling criticisms of Dedekind from those that have deeper, more lasting significance. The essay ends with a suggestion for how to reconcile Fregean and Dedekindian forms of logicism, based on distinguishing two distinct but complementary kinds of abstraction principles.