We discuss a passage from Grundgesetze der Arithmetik that raises doubts regarding Frege’s attitude towards platonism. First, we motivate a platonist interpretation of Frege’s mature philosophy of mathematics and outline his conception of the aims of definition. We then present the passage which prima facie raises doubts about a platonist interpretation of his logicism. We then survey and discuss readings of this passage by other interpreters. Finally, we present an interpretation that renders the passage compatible with a platonist interpretation of Frege and offers an explanation of Frege’s rather uncharacteristic concessive attitude in the passage.

Frege’s Grundgesetze der Arithmetik received four short reviews during his lifetime (by H. Courbe, R. Hoppe, C. Th. Michaëlis, and C. Faerber) and a longer critical discussion by G. Peano. We present an introductory essay and English translations of these five pieces.

I characterize a variety of mathematical Platonism according to which it is true that for the number of the planets to be eight just is for there to be eight planets. I then argue that the Neo-Fregean program is best developed as a version of such a view.

We discuss the typography of the notation used by Gottlob Frege in his Grundgesetze der Arithmetik. §1. Background to the Grundgesetze der Arithmetik. Grundgesetze der Arithmetik was to have been the pinnacle of Gottlob Frege’s life’s work — a rigorous demonstration of how the fundamental laws of classical pure mathematics of the natural and real numbers can be derived from principles which, in Frege’s view,were purely logical.His logical system, calledBegriffsschrift, i.e., “concept-script”, was first introduced in 1879 in his book with this title [19]. It includes the first occurrence in formal logic of quantifiers,1 with which multiple and embedded generality could be expressed — no earlier logical system was capable of this. It also offers the first formulation of a logical system that contains relations rather thanmerelymonadic predicates. In addition, Frege here presents his celebrated definition of the ancestral of a relation. Taken together, these developments made logic expressively adequate for mathematics for the first time in history [13, pp. xxxv–xxxvi]. Begriffsschrift is thus widely acknowledged as the greatest advance in logic since Aristotle — as W.V. Quine put it [43, p. vii]: Logic is an old subject, and since 1879 it has been a great one. In 1884 Frege published the book Die Grundlagen der Arithmetik [20] in which he formulates and argues for logicism— the idea that arithmetic (and analysis) is reducible to logic. The principal aim of Grundlagen is to provide philosophical arguments for logicism, but Frege also offers proof-sketches of how Peano’s axioms for arithmetic can be derived from entirely logical principles, based on an explicit definition of “cardinal number” and taking extensions of concepts as primitive. It was to be the task of hismagnumopus, Grundgesetze der Arithmetik [21,23], to show conclusively the purely logical nature of mathematics by presenting gapless proofs of the axioms of arithmetic and real analysis in his formal system, using only explicit definitions Received October 21, 2014. 2010Mathematics Subject Classification. 03A05, 01A55, 03Bxx, 00A30.

All contributions included in the present issue were originally presented at an ‘Author Meets Critics’ session organised by Richard Zach at the Pacific Meeting of the American Philosophical Association in San Diego in the Spring of 2014.

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This paper discusses Bernard Linsky and Edward Zalta’s Neo-Logicist programme. In the first section we offer a brief summary of the formal framework – third order object theory – in which a mathematical theory is to be embedded. The second section focuses on their claim that mathematics can be known by re-interpreting it within that formal framework. We argue that they fail to offer a satisfying explanation of mathematical knowledge. The third section discusses their conception of mathematical truth and highlights various shortcomings for their view. In the concluding section we argue that their approach should not to be regarded as a Neo-Logicist project.

Eli Hirsch recently suggested the metaontological doctrine of so-called quantifier variance, according to which ontological disputes—e.g. concerning the question whether arbitrary, possibly scattered, mereological fusions exist, in the sense that these are recognised as objects in our ontology—can be defused as insubstantial. His view is that the meaning of the quantifier ‘there exists’ varies in such debates: according to one opponent in this dispute, some existential statement claiming the existence of, e.g., a scattered object is true, according to the other it is not. This paper argues that Hirsch’s proposal leads into

In 1885, Georg Cantor published his review of Gottlob Frege's Grundlagen der Arithmetik. In this essay, we provide its first English translation together with an introductory note. We also provide a translation of a note by Ernst Zermelo on Cantor's review, and a new translation of Frege's brief response to Cantor. In recent years, it has become philosophical folklore that Cantor's 1885 review of Frege's Grundlagen already contained a warning to Frege. This warning is said to concern the defectiveness of Frege's notion of extension. The exact scope of such speculations varies and sometimes extends as far as crediting Cantor with an early hunch of the paradoxical nature of Frege's notion of extension. William Tait goes even further and deems Frege 'reckless' for having missed Cantor's explicit warning regarding the notion of extension. As such, Cantor's purported inkling would have predated the discovery of the Russell-Zermelo paradox by almost two decades. In our introductory essay, we discuss this alleged implicit (or even explicit) warning, separating two issues: first, whether the most natural reading of Cantor's criticism provides an indication that the notion of extension is defective; second, whether there are other ways of understanding Cantor that support such an interpretation and can serve as a precisification of Cantor's presumed warning.

This paper investigates the relation of the Calculus of Individuals presented by Henry S. Leonard and Nelson Goodman in their joint paper, and an earlier version of it, the so-called Calculus of Singular Terms, introduced by Leonard in his Ph.D. dissertation thesis Singular Terms. The latter calculus is shown to be a proper subsystem of the former. Further, Leonard’s projected extension of his system is described, and the definition of an non-extensional part-relation in his system is proposed. The final section discusses to what extent Goodman might have contributed to the formulation of the Calculus of Individuals. 1 The Calculus of Individuals In 1936, Henry S. Leonard and Nelson Goodman presented a joint paper at the meeting of the Association for Symbolic Logic which was held at the meeting of the Eastern Division of the American Philosophical Association in Cambridge, Massachusetts. Eleven years later, they published an elaborated version of this paper under the title “The Calculus of Individuals and its Uses” [12]. The calculus they introduce in this paper is today usually taken as a basis for the study and use of formal part-whole relations (often called “mereology”) in analytic metaphysics, sometimes mediated by Goodman’s presentation of the calculus in his The Structure of Appearance [7]. Goodman used the Calculus of Individuals in his Ph.D. dissertation thesis A Study of Qualities of 1940 [6], which eventually became The Structure of Appearance. As in his joint paper with Leonard, he used the calculus as an addition to set theory to solve a problem known as the difficulty of imperfect community in Rudolf Carnap’s Aufbau [1]. Only in Structure Goodman abandoned set theory and presented a nominalistic construction that used only the Calculus of Individuals.1 The focus of this paper, however, will be an investigation of the system that Leonard presents in his Ph.D. dissertation thesis Singular Terms [10] of 1930, which is the first See [2], 121–139, and [3], §3.2, for a discussion.

objects are those, that are not possibly concrete; and ordinary objects are those that are possibly concrete. The notion of an ordinary object allows Zalta in other projects to propose a theory of merely possible and also of fictional objects.9 This, however, will be of no concern here. Abstract objects enter OT via a comprehension schema for abstract objects (OC): (OC) ∃x(A!x ∧ ∀F (xF ≡ φ)), where ‘x’ is not free in φ This axiom schema asserts that for any formula φ (minding the restriction on free variables), there exists an abstract object that encodes all and only those properties F that satisfy φ; or, expressed in a more sloppy way, for any collection of properties, there is an abstract object encoding them. OC guarantees that any (abstract) object that is described by an expression of the form ‘ix(A!x ∧ ∀F (xF ≡ φ))’ exists (where there is no free ‘x’ in φ). So, there is, for example, an abstract object that encodes the property of being Zalta (or being identical to Zalta):10 ix(A!x ∧ ∀F (xF ≡ ∀y(Fy ≡ y = Zalta))) OT was originally developed as a formal theory of fictional, abstract, and intensional objects inspired by the work of Meinong’s student Ernst Mally: see (Zalta, 1983). All of the following examples are, of course, dependent on the English names and predicates entering the formal language in some way. How this is done for mathematical terms is described below. Moreover, identity is a defined notion in OT. So, strictly speaking, one would have to specify that the identity relation referred to in our examples is identity between concrete, rather than abstract, objects. What is the Purpose of Neo-Logicism? 41 An abstract object that encodes being either Linsky or Zalta: ix(A!x ∧ ∀F (xF ≡ ∀y(Fy ≡ (y = Linsky ∨ y = Zalta)))) An abstract object that encodes all the properties Zalta has: ix(A!x ∧ ∀F (xF ≡ F (Zalta))) Note that Zalta himself is not identical to any of these objects (since he is concrete and not abstract). He exemplifies, rather than encodes the respective properties. Sherlock Holmes, on the other hand, is an abstract object, viz. the abstract object that encodes all the properties that (the fictional character) Sherlock Holmes has according to the stories by Arthur Conan Doyle. (The devise for fomalising this will be introduced below in the discussion of mathematical theories.) There is also an abstract object that encodes being a square circle: ix(A!x ∧ ∀F (xF ≡ ∀y(Fy ≡ (y is a circle ∧ y is square)))) Moreover, there is an abstract object that encodes being a set that contains all and only those sets that do not contain themselves. In order to avoid inconsistency, the second-order comprehension schema for predicates:11 ∃X∀x(Xx ≡ φ(x)), where X is not free in φ For simplicity’s sake we only give the comprehension schema for monadic second-order variables. The restrictions apply in the same way for the general formulation for polyadic variables. We here use the common formulation of second-order logic introduced in (Church, 1956); the current bible of second-order logic is (Shapiro, 1991). Linsky and Zalta use an equivalent formulation that employs λ-conversion, which requires an analogous restriction. 42 Philip A. Ebert & Marcus Rossberg (and likewise the thirdand higher-order comprehension schemata) has to be restricted. It has to be demanded of the standardly unrestricted second-order comprehension schema that φ does not contain any descriptions or “encoding subformulae”. So, the fully explicit formulation of φ must not contain subformulae of the form pxY q, i.e. subformulae containing the encoding mode of predication.12 Identity between abstracta, ‘=A’, is a defined relation. Two abstract objects are identical if, and only if, they necessarily encode the same properties: x =A y =df A!x ∧A!y ∧2∀F (xF ≡ yF ) With this criterion for identity at hand, we can see that the abstract object introduced above which encodes being Zalta is distinct from the object encoding all of Zalta’s properties: the latter encodes using a Mac while the former does not. So much for the formal background. Linsky and Zalta now suggest that mathematical theories can be identified as those abstract objects, that encode all the mathematical propositions that are true according to them.13 This needs some unpacking. First, encoding was introduced as a mode of predication, i.e. a second-level relation that holds between an object and a property. In order for mathematical theories to be able to encode propositions, they are handled as zero-place properties. Any proposition p thus gives rise to a property being such that p; using the notation of λ-conversion, this can be expressed as: ‘[λy p]’. One might complain that object comprehension, OC, is suspect on the grounds that with the introduction of OC the well established second-order comprehension schema, considered logical by many, needs to be restricted to avoid inconsistency. We will not follow this criticism here. See (Linsky and Zalta, 1995), pp. 538–539, and (Linsky and Zalta, 2006), pp. 89–90. What is the Purpose of Neo-Logicism? 43 Being true according to a mathematical theory t can then be characterised using the resources of OT: it is simply defined as t encoding that particular truth:

A mathematical theory T is categorical if, and only if, any two models of T are isomorphic. If T is categorical, it can be shown to be semantically complete: for every sentence φ in the language of T, either φ follows semantically from T or ¬φ does. For this reason some authors maintain that categoricity theorems are philosophically significant: they support the realist thesis that mathematical statements have determinate truth-values. Second-order arithmetic (PA) is a case in hand: it can be shown to be categorical and semantically complete. The status of second-order logic is a controversial issue, however. Worries about the purported set-theoretic nature and ontological commitments of second-order logic have been influential in the debate. Recently, a number of authors – most notably Vann McGee and Shaughan Lavine – have argued that one can get some of the advantages of secondorder axiomatisations – categoricity, in particular – while walking free of the standard objections against second-order logic. In so arguing they appeal to the notion of an open-ended schema. In “open-ended arithmetic”, induction holds not only for formulas in the current language, but for formulas in any extension of that language. Both McGee and Lavine stress that this is not merely a variant of second-order logic. In this paper, we discuss the defensibility of this claim on McGee’s part. McGee’s account involves a metatheoretic rule that provides information about extensions of the language are legitimate. We argue that in the presence of this rule open-ended schemas are as ontologically suspect as standard second-order quantification.

Both firstand second-order logic (FOL and SOL, respectively) as we use them today were arguably created by Frege in his Begriffsschrift – if we ignore the notational differences. SOL also suggests itself as a natural, and because of its much greater strength, desirable extension of FOL. But at least since W. V. Quine’s famous claim that SOL is “set theory in sheep’s clothing” it is widely held that SOL is not proper logic – whatever this is taken to be by different authors – but some kind of mathematics. Even contemporary advocates of SOL like Stewart Shapiro point out its mathematical character, albeit without regarding this as problematic. Recent criticisms focus both on the ontological commitment of SOL, which is believed to be to the set-theoretic hierarchy, and on the allegedly problematic epistemic status of the second-order consequence relation.