Frege: Definitions and Conceptual Analysis

Joan Weiner (2007) has argued that Frege’s definitions of numbers are linguistic stipulations, with no content-preserving or ontological point: they don’t capture any determinate content of numerals, as they have none, and don’t present numbers as preexisting objects. I show that this view is based on exegetical and systematic errors. First, Idemonstrate that Weiner misrepresents the Fregean notions of ‘Foundations-content’, sense, reference, and truth. I then consider the role of definitions, demonstrating that they cannot be mere linguistic stipulations, since they (...) have a content-preserving, ontological point, and a decompositional aspect; Frege’s project of logical analysis and systematisation makes no sense without definitions so understood. The pivotal ontological role of elucidations is also explained. Next, three aspects of definition are distinguished, the informal versus the formal aspect, and the aspect of definition achieved through the entire process of systematisation, which encompasses the previous two and is little discussed in the literature. It is suggested that these insights can contribute to resolving some of the puzzles concerning the tension between the epistemological aim of logicism and Frege’s presentation of definitions as arbitrary conventions. Finally, I stress the interdependence between the epistemological and ontological aspects of Frege’s project of defining number. (shrink)

Frege and Peano started in 1896 a debate where they contrasted the respective conceptions on the theory and practice of mathematical definitions. Which was (if any) the influence of the Frege-Peano debate on the conceptions by the two authors on the theme of defining in mathematics and which was the role played by this debate in the broader context of their scientific interaction?

This paper aims to show that Frege’s and Hilbert’s mutual disagreement results from different notions of Anschauung and their relation to axioms. In the first section of the paper, evidence is provided to support that Frege and Hilbert were influenced by the same developments of 19th-century geometry, in particular the work of Gauss, Plücker, and von Staudt. The second section of the paper shows that Frege and Hilbert take different approaches to deal with the problems that the developments in 19th-century (...) geometry posed for the traditional Kantian philosophy of mathematics. Frege maintains that Anschauung is a source of knowledge by which we acknowledge geometrical axioms as true. For Hilbert, in contrast, axioms provide one of several correct “pictures” of reality. Hilbert’s position is thereby deeply influenced by epistemological ideas from Hertz and Helmholtz, and, in turn, influenced the neo-Kantian Cassirer. (shrink)

What, in Frege’s view, makes definitions fruitful? In Grundlagen §70, Frege offers an answer: Unfruitful definitions are definitions that “could just as well be omitted and leave no link missing in the chain of our proofs”. The §70 passage, however, poses an interpretive puzzle as its characterization of fruitfulness appears to conflict with other conditions that Frege imposes on definitions, namely, eliminability and conservativeness. It appears that the only way to resolve this conflict is to attribute to Frege a notion (...) of fruitfulness that is trivially satisfied and, hence, poorly motivated. I argue that this worry is misplaced. This is because Frege distinguishes between two roles of definitions, namely, between definitions qua explanations of concepts, and definitions qua resources of a proof system. I use this distinction to argue that a fruitful definition, for Frege, is a definition that plays both roles, and that to play both roles, the definition has to be used in the proof of sentences containing the term so defined. Starting from §70, I develop and defend this reading of Frege’s notion of fruitful definition. (shrink)

This paper describes both an exegetical puzzle that lies at the heart of Frege’s writings—how to reconcile his logicism with his definitions and claims about his definitions—and two interpretations that try to resolve that puzzle, what I call the “explicative interpretation” and the “analysis interpretation.” This paper defends the explicative interpretation primarily by criticizing the most careful and sophisticated defenses of the analysis interpretation, those given my Michael Dummett and Patricia Blanchette. Specifically, I argue that Frege’s text either are inconsistent (...) with the analysis interpretation or do not support it. I also defend the explicative interpretation from the recent charge that it cannot make sense of Frege’s logicism. While I do not provide the explicative interpretation’s full solution to the puzzle, I show that its main competitor is seriously problematic. (shrink)

What makes certain definitions fruitful? And how can definitions play an explanatory role? The purpose of this paper is to examine these questions via an investigation of Frege’s treatment of definitions. Specifically, I pursue this issue via an examination of Frege’s views about the scientific unification of logic and arithmetic. In my view, what interpreters have failed to appreciate is that logicism is a project of unification, not reduction. For Frege, unification involves two separate steps: (1) an account of the (...) content expressed by arithmetical claims and (2) the justification of that content. The distinction between these steps allows us to see that there are two notions of definition at play in Frege’s logicist work, viz., one concerned with conceptual analysis, the other concerned with the construction of gap-free proof. I then use this discussion to explain how Frege employs his definitions to defend an epistemological thesis about arithmetic, and to clarify Grundlagen’s fruitfulness condition of definitions, and thereby address two interpretive puzzles from the recent literature. (shrink)

Logical analysis is in Frege primarily not an analysis of a concept but of its sense. Five Fregean philosophical principles are presented as constituting a framework for a theory of logical or conceptual analysis, which I call analytical explication. These principles, scattered and sometime latent in his writings are operative in Frege's critique of other views and in his constructive development of his own view. The proposed conception of analytical explication is partially rooted in Frege's notion of analytical definition. It (...) may also be the basis of what is required of a reduction of one domain to another, if it is to have the philosophical significance many reductions allegedly have. (shrink)

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.

Neo-logicism is, not least in the light of Frege’s logicist programme, an important topic in the current philosophy of mathematics. In this essay, I critically discuss a number of issues that I consider to be relevant for both Frege’s logicism and neo-logicism. I begin with a brief introduction into Wright’s neo-Fregean project and mention the main objections that he faces. In Sect. 2, I discuss the Julius Caesar problem and its possible Fregean and neo-Fregean solution. In Sect. 3, I raise (...) what I take to be a central objection to the position of neo-logicism. In Sect. 4, I attempt to clarify how we should understand Frege’s stipulation that the two sides of an abstraction principle qua contextual definition of a term-forming operator shall be “gleichbedeutend”. In Sect. 5, I consider the options that Frege might have had to establish the analyticity of Hume’s Principle: The number that belongs to the concept F is equal to the number that belongs to the concept G if and only if F and G are equinumerous. Section 6 is devoted to Frege’s two criteria of thought identity. In Sects. 7 and 8, I defend the position of the neo-logicist against an alleged “knock-down argument”. In Sect. 9, I comment on Frege’s description of abstraction in Grundlagen, §64 and the use of the terms “recarving” and “reconceptualization” in the relevant literature on Fregean abstraction and neo-logicism. I argue that Fregean abstraction has nothing to do with the recarving of a sentence content or its decomposition in different ways. I conclude with remarks on global logicism versus local logicisms. (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)

In Frege's Conception of Logic Patricia A. Blanchette explores the relationship between Gottlob Frege's understanding of conceptual analysis and his understanding of logic.

In his Foundations of Arithmetic, Frege aims to extend our a priori arithmetical knowledge by answering the question what a natural number is. He rejects conceptual analysis as a method to acquire a priori knowledge . Later he unsuccessfully tried to solve the problems that beset conceptual analysis . If these problems remain unsolved, which rational method can he use to extend our a priori knowledge about numbers? I will argue that his fundamental arithmetical insight that numbers belong to concepts (...) is based on the recognition that different sentences express the same thought. In Frege's philosophy of arithmetic, propositional analysis does the main work. How it can do this work will be discussed in sections 3, 4 and 5. Sections 6 and 7 explore this approach further. (shrink)

Frege begins Die Grundlagen der Arithmetik, the work that introduces the project which was to occupy him for most of his professional career, with the question, 'What is the number one?' It is a question to which even mathematicians, he says, have no satisfactory answer. And given this scandalous situation, he adds, there is small hope that we shall be able to say what number is. Frege intends to rectify the situation by providing definitions of the number one and the (...) concept of number. But what, exactly, is required of a definition? Surely it will not do to stipulate that the number one is Julius Caesar - that would change the subject. It seems reasonable to suppose that an acceptable definition must be a true statement containing a description that picks out the object to which the numeral '1' already refers. And, similarly, that an acceptable definition of the concept of number must contain a description that picks out precisely those objects that are numbers - those objects to which our numerals refer. Yet, while Frege writes a great deal about what criteria his definitions must satisfy, the above criteria are not among those he mentions. Nor does he attempt to convince us that his definitions of '1' and the other numerals are correct by arguing that these definitions pick out objects to which these numerals have always referred. Yet, while Frege writes a great deal about what criteria his definitions must satisfy, the above criteria are not among those he mentions. Nor does he attempt to convince us that his definitions of ‘1’ and the other numerals are correct by arguing that these definitions pick out objects to which these numerals have always referred. There is, as we shall see shortly, a great deal of evidence that Frege’s definitions are not intended to pick out objects to which our numerals already refer. But if this is so, how can these definitions teach us anything about our science of arithmetic? And what criteria must these definitions satisfy? To answer these questions, we need to understand what it is that Frege thinks we need to learn about the science of arithmetic. (shrink)

This entry explains Frege's Theorem by using the modern notation of the predicate calculus. Frege's Theorem is that the Dedekind-Peano axioms for number theory are derivable from Hume's Principle, given the axioms and rules of second-order logic. Frege's methodology for defining the natural numbers and for the derivation of the Dedekind-Peano axioms are sketched in some detail.

Three clusters of philosophically significant issues arise from Frege’s discussions of definitions. First, Frege criticizes the definitions of mathematicians of his day, especially those of Weierstrass and Hilbert. Second, central to Frege’s philosophical discussion and technical execution of logicism is the so‐called Hume’s Principle, considered in The Foundations of Arithmetic . Some varieties of neo‐Fregean logicism are based on taking this principle as a contextual definition of the operator ‘the number of …’, and criticisms of such neo‐Fregean programs sometimes appeal (...) to Frege’s objections to contextual definitions in later writings. Finally, a critical question about the definitions on which Frege’s proofs of the laws of arithmetic depend is whether the logical structures of the definientia reflect our pre‐Fregean understanding of arithmetical terms. It seems that unless they do, it is unclear how Frege’s proofs demonstrate the analyticity of the arithmetic in use before logicism. Yet, especially in late writings, Frege characterizes the definitions as arbitrary stipulations of the senses or references of expressions unrelated to pre‐definitional understanding. One or more of these topics may be studied in a survey course in the philosophy of mathematics or a course on Frege’s philosophy. The latter two topics are obviously central in a seminar in the philosophy of mathematics in general or more specialized seminars on logicism, or on mathematical definitions and concept formation. (shrink)

In an unpublished manuscript of 1914 titled ‘Logic in mathematics’, Gottlob Frege offered a rich account of the paradox of analysis. I argue that Frege there claims that the explicandum and explicans of a successful analysis express the same sense and that he furthermore appreciated that this requires that one cannot conclude that two sentences differ in sense simply because it is possible for a (minimally) competent speaker to accept one without accepting the other. I claim that this is shown (...) by Frege’s suggestive remarks about a cloudy grasp of a sense. I then argue that this fact calls into question a key assumption behind Frege’s master argument for the sense/reference distinction. (shrink)

This article treats three aspects of Frege's discussions of definitions. First, I survey Frege's main criticisms of definitions in mathematics. Second, I consider Frege's apparent change of mind on the legitimacy of contextual definitions and its significance for recent neo-Fregean logicism. In the remainder of the article I discuss a critical question about the definitions on which Frege's proofs of the laws of arithmetic depend: do the logical structures of the definientia reflect the understanding of arithmetical terms prevailing prior to (...) Frege's analyses? Unless they do, it is unclear how Frege's proofs demonstrate the analyticity of the arithmetic in use before logicism. Yet, especially in late writings, Frege characterizes definitions as arbitrary stipulations of the senses or references of expressions unrelated to pre-definitional understanding. I conclude by examining some options for conceiving of the status of Frege's logicism in light of this apparent tension, and outline a suggestion for a philosophically fruitful way of resolving this tension. (shrink)

In this entry, Frege's logic is introduced and described in some detail. It is shown how the Dedekind-Peano axioms for number theory can be derived from a consistent fragment of Frege's logic, with Hume's Principle replacing Basic Law V.

Gottlob Frege famously rejects the methodology for consistency and independence proofs offered by David Hilbert in the latter's Foundations of Geometry. The present essay defends against recent criticism the view that this rejection turns on Frege's understanding of logical entailment, on which the entailment relation is sensitive to the contents of non-logical terminology. The goals are (a) to clarify further Frege's understanding of logic and of the role of conceptual analysis in logical investigation, and (b) to point out the extent (...) to which his understanding of logic differs importantly from that of the model-theoretic tradition that grows out of Hilbert's work. (shrink)

In this short monograph, John Horty explores the difficulties presented for Gottlob Frege's semantic theory, as well as its modern descendents, by the treatment of defined expressions. The book begins by focusing on the psychological constraints governing Frege's notion of sense, or meaning, and argues that, given these constraints, even the treatment of simple stipulative definitions led Frege to important difficulties. Horty is able to suggest ways out of these difficulties that are both philosophically and logically plausible and Fregean in (...) spirit. This discussion is then connected to a number of more familiar topics, such as indexicality and the discussion of concepts in recent theories of mind and language. In the latter part of the book, after introducing a simple semantic model of senses as procedures, Horty considers the problems that definitions present for Frege's idea that the sense of an expression should mirror its grammatical structure. The requirement can be satisfied, he argues, only if defined expressions--and incomplete expressions as well--are assigned senses of their own, rather than treated contextually. He then explores one way in which these senses might be reified within the procedural model, drawing on ideas from work in the semantics of computer programming languages. With its combination of technical semantics and history of philosophy, Horty's book tackles some of the hardest questions in the philosophy of language. It should interest philosophers, logicians, and linguists. (shrink)

The book begins by focusing on the psychological constraints governing Frege's notion of sense, or meaning, and argues that, given these constraints, even the ...

For both Gottlob Frege and Bertrand Russell, providing a philosophical account of the concept of number was a central goal, pursued along similar logicist lines. In the present paper, I want to focus on a particular aspect of their accounts: their definitions, or re-constructions, of the natural numbers as equivalence classes of equinumerous classes. In other words, I want to examine what is often called the ‘Frege-Russell conception of the natural numbers’ or, more briefly, the Frege-Russell numbers. My main concern (...) will be to determine the precise sense in which this conception was, or could be, meant to constitute an analysis. I will be mostly concerned with Frege’s views on the matter; but Russell will come up along the way, for illustration and comparison, as will some recent neo-Fregean suggestions. (shrink)

Frege wanted to define the number 1 and the concept of number. What is required of a satisfactory definition? A truly arbitrary definition will not do: to stipulate that the number one is Julius Caesar is to change the subject. One might expect Frege to define the number 1 by giving a description that picks out the object that the numeral '1' already names; to define the concept of number by giving a description that picks out precisely those objects that (...) are numbers. Yet Frege appears to do no such thing. Indeed, when he defends his definitions, he does not argue that they pick out objects that we have been talking about all along-the issue never comes up. The aim of this paper is to explain why. I argue that, on Frege's view, our numerals do not, antecedent to his work, name particular objects. This raises an obvious question: If (like 'Odysseus') the numerals do not name particular objects, how can Frege write (as he does) as if sentences in which numerals appear state truths? One central concern of this paper is exegetical-to answer these questions. But my aim is not solely exegetical. For these questions direct us to something that, I believe, creates only an apparent problem for Frege but an actual problem for many contemporary philosophers: the assumption that singular terms appearing in statements about the world must actually have referents. Another aim of this paper is to suggest that the problem-as well as a solution that can be found in Frege's writings-should be of import to contemporary philosophers. (shrink)

Michael Beaney ha sugerido recientemente que la distinción fregeana entre sentido y referencia fue propuesta para resolver la famosa paradoja del análisis. Casi diez años antes, Michael Dummett ya insistia en que Frege fue uno de los prirneros en buscar una soluci6n satisfactoria de esta paradoja. En esre articulo, discuto algunas sugerencias de Beaney y Dummett de cómo resolver la paradoja al estilo fregeano y tarnbién sus propias contribuciones no fregeanas al debate en torno de la corrección y de la (...) informatividad del análisis. Mi intención en esta discusión es mostrar que ninguna de ellas funciona. Concluyo con una propuesta de inspiración witttgensteiniana --el enfoque de los conceptos abiertos- para apoyar la idea de que la apelación a contertidos abiertos es la rnejor rnanera de disolver la paradoja.Michael Beaney has recently suggested that the Fregean distinction between sense and reference was proposed to solve the famous paradox of analysis. Not as recently, Michael Dummett has insisted that Frege was one of the first to look for a satisfactory solution to this paradox. In this paper, I discuss Beanry’s and Dummett’s various suggestions for solving the paradox in a Fregean style as well as their own non Fregean contributions to the debate around analysis’s correction and informativeness. My intention in this discussion is to show that none of them works. I conclude with a proposal inspired by Wittgenstein’s later work -the open concepts’ proposal- designed to support the idea that the appeal to open contents is the best way to dissolve the paradox. (shrink)

Husserl and Frege on the analysis of the concept of number and primitive logical concepts.

Michael Dummett has long argued that Frege is committed to recognizing a distinction between two sorts of analysis of propositional contents: 'analysis', which reveals the entities that one must grasp in order to apprehend a given propositional content; and 'decomposition', which is used in recognizing the validity of certain inferences. Whereas any propositional content admits of a unique ultimate 'analysis' into simple constituents, it also admits of distinct 'decompositions', no one of which is ultimately privileged over the others. I argue (...) that although Russell accepts this distinction between analysis and decomposition, Frege does not. In particular, I consider claims which Dummett makes regarding how 'analysis' and 'decomposition' are related to two different models Frege at least suggests in discussing the composition of thoughts, the part/whole model and the function/argument model; and I argue that in each case, while Russell accepts views which Dummett attributes to Frege, Frege does not. (shrink)

According to the standard view of definition, all defined terms are mere stipulations, based on a small set of primitive terms. After a brief review of the Hilbert-Frege debate, this paper goes on to challenge the standard view in a number of ways. Examples from graph theory, for example, suggest that some key definitions stem from the way graphs are presented diagramatically and do not fit the standard view. Lakatos's account is also discussed, since he provides further examples that suggest (...) many definitions are much more than mere convenient abbreviations. (shrink)

Frege seems to hold two incompatible theses:(i) that sentences differing in structure can yet express the same sense; and (ii) that the senses of the meaningful parts of a complex term are determinate parts of the sense of the term. Dummett offered a solution, distinguishing analysis from decomposition. The present paper offers an embellishment of Dummett?s distinction by providing a way of depicting the internal structures of complex senses?determinate structures that yield distinct decompositions. Decomposition is then shown to be adequate (...) as a foundation for the informativity and analyticity of logic. (shrink)

This paper discusses Frege's account of definition by induction in Grundgesetze and the two key theorems Frege proves using it.

This paper has three goals: (i) to show that the foundational program begun in the Begriffsschroft, and carried forward in the Grundlagen, represented Frege's attempt to establish the autonomy of arithmetic from geometry and kinematics; the cogency and coherence of 'intuitive' reasoning were not in question. (ii) To place Frege's logicism in the context of the nineteenth century tradition in mathematical analysis, and, in particular, to show how the modern concept of a function made it possible for Frege to pursue (...) the goal of autonomy within the framework of the system of second-order logic of the Begriffsschrift. (iii) To address certain criticisms of Frege by Parsons and Boolos, and thereby to clarify what was and was not achieved by the development, in Part III of the Begriffsschrift, of a fragment of the theory of relations. (shrink)

Chapter 1 concerns the role non-linguistic contextual factors play in the expression of thought. It is argued that contextual factors play a role in determining what is expressed by predicates. Several strategies for avoiding this conclusion are discussed and rejected. One strategy maintains that contextual factors determine, not what is expressed, but only what is otherwise communicated. Another contends that whatever can be expressed context dependently can also be expressed context independently. The chapter concludes by suggesting that context dependence indicates (...) that attempts to analyze thought and thinking in terms of linguistic facts alone may not succeed. ;Chapter 2 considers an argument for the view that understanding a sentence that expresses a necessary truth suffices to determine its truth value. The argument maintains, first, that understanding a sentence involves meta-linguistic knowledge of its referential and truth conditional properties, and, second, that such knowledge suffices to determine the truth value of a sentence that expresses a necessary truth. It is argued that while a strong reading of the argument's first premise is false a weak, though possibly true, reading renders the argument invalid. Various defences for the argument's second premise are then considered and rejected, and two arguments against it are presented. ;Chapter 3 is an examination of two strands in Gottlob Frege views on the analysis of Sense. Frege held that the analysis of Sense plays a crucial role in the development of both the natural sciences as well as the sciences of arithmetic and logic, and he held that the analysis of Sense yields improved understanding of Sense. Frege thus viewed foundational scientific insight as continuous with conceptual clarification. Several aspects of these strands are discussed. The chapter concludes by speculating about what these views on analysis suggest about Frege's conception of understanding. (shrink)

This book claims to provide a revisionist interpretation of Frege's philosophy. According to the author, Frege uses words "that to us have odd meanings derived from a foreign and dated philosophical framework". His philosophy is said to be deeply flawed, beset by fatal errors.

Gegen die vielfach vertretene Auffassung, Frege habe die Hilbertsche Axiomatik nicht verstanden, wird nachzuweisen versucht, daß Frege die neue Methode nicht nur verstanden, sondem auch begrifflich präzise analysiert hat. Er definiert eine formale Theorie im Hilbertschen Sinn als eine Klasse von logisch beweisbaren Wenn-dann-Sätzen, die freie Variable enthalten und deren Wenn-Satz eine Konjunktion der Axiome im Hilbertschen Sinn ist. Er untersucht ferner das Verhältnis zwischen einer Hilbertschen Theorie und ihren Modellen (Anwendungen) und wendet seine allgemeinen Ergebnisse in erhellender Weise auf (...) Hilberts Grundlagen der Geometrie an. (shrink)

AB THE DIFFICULTIES RAISED BY "TRACTATUS" 3.263 AND ITS USE OF THE TERM "ERLAUTERUNG" ARE EXAMINED. LIGHT IS THROWN ON THE MATTER BY THE SYSTEMATIC USE OF THIS TERM BY FREGE IN HIS DISCUSSION OF UNDEFINABLES. RUSSELL'S VIEWS ON UNDEFINABLES ARE ALSO TOUCHED UPON. IT IS SUGGESTED THAT THE "TRACTATUS" CONCEPTION OF AN 'ELUCIDATION' CONFUSEDLY COMBINED THE INCOMPATIBLE ROLES OF EMPIRICAL STATEMENT AND GRAMMATICAL SENTENCE (AN OSTENSIVE DEFINITION).

For frege, To define a symbol is to show how to do without it. Frege originated the distinction between metalanguage and object language. But he formulates his definitions within the begriffsschrift itself, Not seeing that, According to his own account of them, They go better in the metalanguage.

Church has recently proposed a solution of the paradox of analysis as propounded by Langford in which Church makes use of Frege's distinction between the sense (Sinn) of a name and its denotation (Bedeutung). The main purpose of the present note. is to show that a, version of the paradox may be presented which is not directly solved by Church in his review but which, in turn, may be solved by using; another distinction of Frege-that between the ordinary (gewihnlich) and (...) the oblique (ungerade) use of a name. The second part of the note will be concerned with the relative merits of this second distinction of Frege and Quine's related distinction between purely designative occurrences of a name and occurrences which are not purely designative; they will be compared merely with respect to their effectiveness in removing or solving the paradox of analysis. (shrink)