Frege: Abstract Objects

This paper is divided into two main sections. In the first, I attempt to show that the characterization of Frege as a redundancy theorist is not accurate. Using one of Wolfgang Carl's recent works as a foil, I argue that Frege countenances a realm of abstract objects including truth, and that Frege's Platonist commitments inform his epistemology and embolden his antipsychologistic project. In the second section, contrasting Frege's Platonism with pragmatism, I show that even though Frege's metaphysical position concerning truth (...) has been criticized as reproachable, I argue that it may be useful for people to think like Platonists while conducting their scientific and philosophical inquiries. (shrink)

Michael Dummett argued that Frege was a realist while Hans Sluga countered that he was an objective idealist in the rationalist tradition of Kant and Lotze. Sluga ties Frege's idealism to the context principle which he argues Frege never gave up. It is argued that Sluga has correctly interpreted the pre?1891 Frege while Dummett is correct concerning the later period. It is also claimed that the context principle was dropped prior to 1891 to be replaced by the doctrine of unsaturated (...) entities. (shrink)

H. Sluga (Inquiry, Vol. 18 [1975], No. 4) has criticized me for representing Frege as a realist. He holds that, for Frege, abstract objects were not real: this rests on a mistranslation and a neglect of Frege's contextual principle. The latter has two aspects: as a thesis about sense, and as one about reference. It is only under the latter aspect that there is any tension between it and realism: Frege's later silence about the principle is due, not to his (...) realism, but to his assimilating sentences to proper names. Contrary to what Sluga thinks, the conception of the Bedeutung of a name as its bearer is an indispensable ingredient of Frege's notion of Bedeutung, as also is the fact that it is in the stronger of two possible senses that Frege held that Sinn determines Bedeutung. The contextual principle is not to be understood as meaning that thoughts are not, in general, complex; Frege's idea that the sense of a sentence is compounded out of the senses of its component words is an essential component of his theory of sense. Frege's realism was not the most important ingredient in his philosophy: but the attempt to interpret him otherwise than as a realist leads only to misunderstanding and confusion. (shrink)

This paper argues that Carnap both did not view and should not have viewed Frege's project in the foundations of mathematics as misguided metaphysics. The reason for this is that Frege's project was to give an explication of number in a very Carnapian sense — something that was not lost on Carnap. Furthermore, Frege gives pragmatic justification for the basic features of his system, especially where there are ontological considerations. It will be argued that even on the question of the (...) independent existence of abstract objects, Frege and Carnap held remarkably similar views. I close with a discussion of why, despite all this, Frege would not accept the principle of tolerance. (shrink)

Joan Weiner has argued that Frege’s definitions of numbers constitute linguistic stipulations that carry no ontological commitment: they don’t present numbers as pre-existing objects. This paper offers a critical discussion of this view, showing that it is vitiated by serious exegetical errors and that it saddles Frege’s project with insuperable substantive difficulties. It is first demonstrated that Weiner misrepresents the Fregean notions of so-called Foundations-content, and of sense, reference, and truth. The discussion then focuses on the role of definitions in (...) Frege’s work, demonstrating that they cannot be understood as mere linguistic stipulations, since they have an ontological aim. The paper concludes with stressing both the epistemological and the ontological aspects of Frege’s project, and their crucial interdependence. (shrink)

Gottlob Frege is often called a "platonist". In connection with his philosophy we can talk about platonism concerning three kinds of entities: numbers, or logical objects more generally; concepts, or functions more generally; thoughts, or senses more generally. I will only be concerned about the first of these three kinds here, in particular about the natural numbers. I will also focus mostly on Frege's corresponding remarks in The Foundations of Arithmetic (1884), supplemented by a few asides on Basic Laws of (...) Arithmetic (1893/1903) and "Thoughts" (1918). My goal is to clarify in which sense the Frege of Foundations and Basic Laws is a platonist concerning the natural numbers.1.. (shrink)

Probably the best arguments for Platonism are those directed against its rival philosophies of mathematics. Frege's arguments against formalism, Gödel's arguments against constructivism and those against the so-called syntactic view of mathematics, and an argument of Hodges against Putnam are expounded, as well as some arguments of the author. A more general criticism of Quine's views follows. The paper ends with some thoughts on mathematics as a sort of Platonism of structures, as conceived by Husserl and essentially endorsed by the (...) author. (shrink)

According to Frege, the introduction of a new sort of abstract object is methodologically sound only if its identity conditions have been satisfactorily explained. Ironically, this ontological restriction has come to be known by Quine's criticism of Frege's intensional semantics, as the precept "No entity without identity." The aim of the paper is to reconstruct Frege's methodology of the introduction of abstract objects in detail, and to defend it against the more restrictive methodology underlying Quine's criticism of the recognition of (...) intensional objects. The main thesis is that Quine's criterion of non-circularity for the satisfactory individuation of abstract objects must be rejected. (shrink)

In Part One of this dissertation I examine the relationship between reason and objectivity in Frege's thought, concentrating on the question of whether or not Frege should be interpreted as a platonist. By platonism I mean the view that objects such as numbers or propositions are objective, non-spatial and timeless, existing in a realm distinct from the external world of physical objects and the internal realm of the mind; and that statements about or involving such objects are true independently of (...) whether or not human beings take them to be true. I defend the traditional interpretation of Frege as a platonist against the claims of revisionist interpretations of Frege, offered by Hans Sluga, Thomas Ricketts and Erich Reck, that seek to deny or mitigate Frege's commitment to platonism. I also argue that similarities between Frege and Descartes further support my interpretation of Frege as a platonist. ;Part Two of the dissertation revolves around Frege's famous distinction between sense and reference. Frege introduces this distinction to explain how an identity statement can be both true and informative. I begin with an examination of Frege's first attempt to solve the puzzle of informative identity statements in his early work the Begriffsschrift, comparing it with his ultimate solution in "On Sense and Reference." I then examine the nature of the distinction between sense and reference and explore some of its consequences. Next I examine the relationship between the sense/reference distinction and three interrelated Fregean principles: the context principle the compositionality principle and the priority principle (judgments are prior to the concepts that compose them. Finally, I examine Saul Kripke's influential criticism of Frege's claim that for a proper name to adequately perform its function it must contain both a sense and a reference. I then critically examine responses by Tyler Burge and Michael Dummett to Kripke's interpretation and criticism of Frege. (shrink)

In the dissertation I seek to clarify an aspect of Frege's thought that has been insufficiently explained in the literature, namely, his notion of logical object. It is well known that the core of Frege's philosophical enterprise up to Grundgesetze der Arithmetik was the reduction of arithmetic to logic. Since Frege regarded numbers as objects, logic must have an ontological basis, i.e., an adequate class of objects to which numbers are reducible. These objects are, for Frege, extensions of concepts and (...) truth-values. ;In the dissertation I adduce some elements of Frege's philosophy that elucidate why he saw these as natural candidates for paradigmatic cases of logical objects. In particular, I argue that Frege could not have taken Hume's principle instead of Axiom V as a fundamental law of arithmetic. This would be inconsistent with his views on logical objects. First, I try to explain what are the general criteria for qualifying objects as logical, and why extensions of concepts have to be seen as the paradigmatic case of logical objects according to these criteria. There is, in my view, a connection between Frege's view on this topic and the famous thesis first formulated in ""Uber Begriff und Gegenstand " that "the concept horse is not a concept." ;Next, I examine Frege's introduction of truth-values as objects in his writings from 1891-92. As I shall argue, Frege's ontologization of truth-values is basically motivated and justified by the technical advantages that this step brought for the formalism of Grundgesetze der Arithmetik. ;Finally, I examine Frege's arguments in section 10 of Grundgesetze der Arithmetik, since this is the place in Frege's writings where extensions and truth-values are identified. I argue that Frege did not mean to employ a generalized version of the context principle in this section. Moreover, I argue that there is no incompatibility between Frege's procedure in this section and his realism regarding mathematical entities. (shrink)

For Frege the objective is independent of our inner world and the same for everyone. I argue that Frege's demand for objectivity is the result of his concern to make sense out of the practices of science. I point out that science, for Frege, requires agreement and purposeful disagreement among its participants. Without such interaction science reduces to useless, self-indulgent babble. It is the social nature of science that ushers in objectivity. I compare Frege's demand for objectivity with Locke's views (...) on language. ;Frege divides what there is into three realms according to a thing's nature and our access to or knowledge of it. The first realm contains the subjective, the second realm the objective and sensible, and the third realm the objective and non-sensible. It is the third realm to which the objects of mathematics and thoughts, i.e. bearers of truth, belong. We have access to the members of the third realm by reason. I point out that there is a tension in Frege's work. For we would expect Frege to appeal to contemplation of the third realm to justify his foundations of arithmetic and the nature of thoughts. I show, however, that it is the discipline of science, especially mathematics, and our deep rooted practices that influence Frege's conclusions. (shrink)

In the section “Validity and Existence in Logik, Book III,” I explain Lotze’s famous distinction between existence and validity in Book III of Logik. In the following section, “Lotze’s Platonism,” I put this famous distinction in the context of Lotze’s attempt to distinguish his own position from hypostatic Platonism and consider one way of drawing the distinction: the hypostatic Platonist accepts that there are propositions, whereas Lotze rejects this. In the section “Two Perspectives on Frege’s Platonism,” I argue that this (...) is an unsatisfactory way of reading Lotze’s Platonism and that the Ricketts-Reck reading of Frege is in fact the correct way of thinking about Lotze’s Platonism. (shrink)

In this paper, I seek to clarify an aspect of Frege's thought that has been only insufficiently explained in the literature, namely, his notion of logical objects. I adduce some elements of Frege's philosophy that elucidate why he saw extensions as natural candidates for paradigmatic cases of logical objects. Moreover, I argue (against the suggestion of some contemporary scholars, in particular, Wright and Boolos) that Frege could not have taken Hume's Principle instead of Axiom V as a fundamental law of (...) arithmetic. This would be inconsistent with his views on logical objects. Finally, I shall argue that there is a connection between Frege's view on this topic and the famous thesis first formulated in ‘Über Begriff und Gegenstand’ that ‘the concept horse is not a concept’. As far as I know, no due attention has been given to this connection in the scholarly literature so far. (shrink)

En este artículo se argumenta que Frege noes el metafísico platónico sobre matemáticasq u e s e c o n s i d e r a n o r ma l me n t e . S emuestra que el proyecto fregeano tienedos etapas distintas: la identificación delo que es verdadero en nuestras nocionesordinarias, y luego la provisión de unaexplicación sistemática que comparte losaspectos identificados. Ninguna de estasetapas involucra mucha metafísica. Elartículo critica en detalle la interpretaciónque hace Dummett de l (...) os parágrafos§§55-61 del Grundlagen. Estas seccionesestán bajo el encabezado ‘Todo númeroes un objeto auto-subsistente’ y Dummettlas describe como las que contienen lospeores argumentos planteados por Frege.Se ar guye que, esenci al ment e, t odoslos puntos interpretativos de Dummettson erróneos. Finalmente, muestro quel os pl ant eami ent os de Frege sobre l aindependencia de las matemáticas conrespecto a los humanos y sus actividadestampoco lo comprometen con ningunaposición metafísica particular.This paper argues that Frege is not themetaphysical platonist about mathematicsthat he is standardly taken to be. It isshown that Frege’s project has two distinctstages: the identification of what is trueof our ordinary notions, and then theprovision of a systematic account thatshares the identified features. Neither ofthese stages involves much metaphysics.The paper criticizes in detail Dummett’sinterpretation of §§55-61 of Grundlagen.These sections fall under the heading‘ E v e r y n u mb e r i s a s e l f - s u b s i s t e n tobject’ and are described by Dummetta s c o n t a i n i n g t h e wo r s t a r g ume n t sput forward by Frege. It is argued thatessentially all of Dummett’s interpretivepoints are mistaken. Finally, I show thatFrege’s claims about the independenceof mathematics from humans and theiractivities does not commit him to anyparticularly metaphysical position either. (shrink)

A feature of Frege's philosophy of arithmetic that has elicited a great deal of attention in the recent secondary literature is his contention that numbers are ‘self‐subsistent’ objects. The considerable interest in this thesis among the contemporary philosophy of mathematics community stands in marked contrast to Kreisel's folk‐lore observation that the central problem in the philosophy of mathematics is not the existence of mathematical objects, but the objectivity of mathematics. Although Frege was undoubtedly concerned with both questions, a goal of (...) the present paper is to argue that his success in securing the objectivity of arithmetic depends on a less contentious commitment to numbers as objects than either he or his critics have supposed. As such, this paper is an articulation and defense of both Frege's analysis of arithmetic and Kreisel's observation. (shrink)

In this paper, the author derives the Dedekind-Peano axioms for number theory from a consistent and general metaphysical theory of abstract objects. The derivation makes no appeal to primitive mathematical notions, implicit definitions, or a principle of infinity. The theorems proved constitute an important subset of the numbered propositions found in Frege's *Grundgesetze*. The proofs of the theorems reconstruct Frege's derivations, with the exception of the claim that every number has a successor, which is derived from a modal axiom that (...) (philosophical) logicians implicitly accept. In the final section of the paper, there is a brief philosophical discussion of how the present theory relates to the work of other philosophers attempting to reconstruct Frege's conception of numbers and logical objects. (shrink)

Frege is celebrated as an arch-Platonist and arch-realist. He is renowned for claiming that truths of arithmetic are eternally true and independent of us, our judgments and our thoughts; that there is a third realm containing nonphysical objects that are not ideas. Until recently, there were few attempts to explicate these renowned claims, for most philosophers thought the clarity of Frege's prose rendered explication unnecessary. But the last ten years have seen the publication of several revisionist interpretations of Frege's writings (...) — interpretations on which these claims receive a very different reading. In Frege on Knowing the Third Realm, Tyler Burge attempts to undermine this trend. Burge argues that Frege is the very Platonist most have thought him — that revisionist interpretations of Frege's Platonism, mine among them, run afoul of the words on Frege's pages. This paper is a response to Burge's criticisms. I argue that my interpretation is more faithful than Burge's to Frege's texts. (shrink)

Un trait du langage qui menace de saper la sûreté de la pensée est sa tendance à former des noms propres auxquels aucun objet ne correspond. [...] Un exemple particulièrement remarquable de cela est la formation d’un nom propre selon le schéma «l’extension du concept a», par exemple «l’extension du concept étoile». À cause de l’article défini, cette expression semble désigner un objet; mais il n’y a aucun objet pour lequel cette expression pour-rait être une désignation appropriée. De là les (...) paradoxes de la théorie des ensembles, qui ont porté un coup mortel à la théorie des ensembles elle-même. J’étais moi-même sous cette illusion quand afin de fournir une fondation logique aux nombres, j’ai tenté de construire les nombres comme des ensembles. (shrink)