Epistemological principles for developmental psychology in Frege and Piaget

My main aim is to identify and discuss parallels between the epistemologies of Gottlob Frege and Jean Piaget. Although their work has attracted massive attention individually, parallels in their work have gone unnoticed. My discussion is in four parts and covers psychologism and epistemology; five epistemological criteria in Frege's rational epistemology under an AEIOU mnemonic, namely autonomy, entailment, intersubjectivity, objectivity and universality; the elaboration of these same criteria in Piaget's developmental epistemology; their implications for developmental psychology and epistemology. One main conclusion is that the same criteria fit both Frege's and Piaget's epistemology. A second conclusion is that Piaget's developmental epistemology can be regarded as an elaboration of Frege's rational epistemology in each of these five respects on both methodological and substantive grounds. Both conclusions are compatible with non-psychologism, which was accepted by both Frege and Piaget.     

Eight good questions for developmental epistemology and psychology

My reply to eight good questions arising from commentary is an elaboration of my main argument that there are parallels in the epistemologies of Frege and Piaget and that these parallels have distinctive implications for developmental psychology. The eight questions are: (i) was Piaget really an epistemologist? (ii) is Piaget's epistemic subject psychological or epistemological? (iii) is Frege's non-modal logic consistent with Piaget's account of necessity? (iv) does Piaget's constructivism entail realism? (v) what is the relation between thinking and thought? (vi) is Frege's concept of mind too narrow? (vii) how are cause and reason related in the interpretation of thought? (viii) what is the status of an act of judgment in the interpretation of thought? These questions are productive, and can be developed.     

II. The origin of Frege's realism

An explanation of Frege's change from objective idealism to platonism is offered. Frege had originally thought that numbers are transparent to reason, but the character of his Axiom of Courses of Values undermined this view, and led him to think that numbers exist independently of reason. I then use these results to suggest a view of Frege's mathematical epistemology.     

From Piaget's Epistemic Subject to Pascual-Leone's Metasubject: Epistemic Transition in the Constructivist—Rationalist Theory of Cognitive Development

The explanation of the transition from one epistemic theory to another is an important part of Piaget's genetic epistemology. It is argued that this epistemic transition leads to a retrodictable orthogenetic tendency toward optimizing equilibration. The objective of this paper is to establish a relationship between Piaget's epistemic subject and Pascual-Leone's metasubject and to demonstrate that the postulation of the latter can be considered as an epistemic transition between two constructivist—rationalist theories, which leads to the development of a theory with greater explanatory power. Epistemic transition in this paper refers to a progressive problemshift (cf. Lakatos, 1970), between the theories of Piaget and Pascual-Leone. Piaget builds a “general model” by neglecting individual differences, that is, studies the epistemic subject, whereas Pascual-Leone by incorporating a framework for individual difference variables, studies the metasubject—the psychological organization of the epistemic subject. Empirical evidence is presented to demonstrate that Pascual-Leone's theory of constructive operators is a model of the psychological organism (the metasubject), which is at work inside Piaget's epistemic subject. Finally, it is concluded that the greater explanatory power of Pascual-Leone's theory can be interpreted as an epistemic transition between Piaget's epistemic subject and Pascual-Leone's metasubject.     

Frege: two theses,two senses

One particular topic in the literature on Frege's conception of sense relates to two apparently contradictory theses held by Frege: the isomorphism of thought and language on one hand and the expressibility of a thought by different sentences on the other. I will divide the paper into five sections. In (1) I introduce the problem of the tension in Frege's thought. In (2) I discuss the main attempts to resolve the conflict between Frege's two contradictory claims, showing what is wrong with some of them. In (3), I analyse where, in Frege'ps writings and discussions on sense identity, one can find grounds for two different conceptions of sense. In (4) I show how the two contradictory theses held by Frege are connected with different concerns, compelling Frege to a constant oscillation in terminology. In (5) I summarize two further reasons that prevented Frege from making the distinction between two conceptions of sense clear: (i) the antipsychologism problem and (ii) the overlap of traditions in German literature contemporary to Frege about the concept of value. I conclude with a hint for a reconstruction of the Fregean notion of ‘thought’ which resolves the contradiction between his two theses.     

Thought's Social Nature

Abstract: Wittgenstein, throughout his career, was deeply Fregean. Frege thought of thought as essentially social, in this sense: whatever I can think is what others could think, deny, debate, investigate. Such, for him, was one central part of judgement's objectivity. Another was that truths are discovered, not invented: what is true is so, whether recognised as such or not. (Later) Wittgenstein developed Frege's idea of thought as social compatibly with that second part. In this he exploits some further Fregean ideas: of a certain generality intrinsic to a thought; of lack of that generality in that which a thought represents as instancing some such generality. (I refer to this below as the ‘conceptual‐nonconceptual’ distinction.) Seeing Wittgenstein as thus building on Frege helps clarify (inter alia) his worries, in the Blue Book, and the Investigations, about meaning, intending, and understanding, and the point of the rule following discussion.     

ANALYSIS,ABSTRACTION PRINCIPLES,AND SLINGSHOT ARGUMENTS

Frege's views regarding analysis and synomymy have long been the subject of critical discussion. Some commentators, led by Dummett, have argued that Frege was committed to the view that each thought admits of a unique ultimate analysis. However, this interpretation is in apparent conflict with Frege's criterion of synonymy, according to which two sentence express the same thought if one cannot understand them without regarding them as having the same truth–value. In a recent article in this journal, Drai attempts to reconcile Frege's criterion of synonymy with unique ultimate analysis by holding that, for Frege, if two sentences satisfy the criterion without being intensionally isomorphic, at most one of them is a privileged representation of the thought expressed. I argue that this proposal fails, because it conflicts not only with Frege's views of abstraction principles but also with slingshot arguments (including one presented by Drai herself) that accurately reflect Frege's commitment to the view that sentences alike in truth–value have the same Bedeutung. While Drai helpfully connects Frege's views of abstraction principles with such slingshot arguments, this connection cannot become fully clear until we recognise that Frege rejects unique ultimate analysis.     

I. Frege as a Realist

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.     

Gottlob Frege und Rudolf Eucken—Gesprächspartner in der Herausbildungsphase der modernen Logik

Frege and Eucken were colleagues in the faculty of philosophy at Jena University for more than 40 years. At times they had close scientific contacts. Eucken promoted Frege's career at the university. A comparison of Eucken's writings between 1878 and 1880 with Frege's writings shows Eucken to have had an important philosophical influence on Frege's philosophical development between 1879 and 1885. In particular the classification of the Begriffsschrift in the tradition of Leibniz is influenced by Eucken. Eucken also influenced Frege's choice of philosophical and logical terms. Finally, there are analogous positions concerning relations between concepts and their expressions in natural language, Frege was probably also influenced by Eucken's use of the term ‘tone’. Eucken used Frege's arguments in his own fight against psychologism and empiricism.     

Frege on Indirect Proof

Frege's account of indirect proof has been thought to be problematic. This thought seems to rest on the supposition that some notion of logical consequence – which Frege did not have – is indispensable for a satisfactory account of indirect proof. It is not so. Frege's account is no less workable than the account predominant today. Indeed, Frege's account may be best understood as a restatement of the latter, although from a higher order point of view. I argue that this ascent is motivated by Frege's conception of logic.     

Conceptual Analysis and Analytical Definitions in Frege

Logical (or conceptual) 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.     

The Role of Structural Reasoning in the Genesis of Graph Theory

The concept of quantity (Größe) plays a key role in Frege's theory of real numbers. Typically enough, he refers to this theory as ‘theory of quantity’ (‘Größenlehre’) in the second volume of his opus magnum Grundgesetze der Arithmetik (Frege 1903). In this essay, I deal, in a critical way, with Frege's treatment of the concept of quantity and his approach to analysis from the beginning of his academic career until Frege 1903. I begin with a few introductory remarks. In Section 2, I first analyze Frege's use of the term ‘source of knowledge’ (‘Erkenntnisquelle’) with particular emphasis on the logical source of knowledge. The analysis includes a brief comparison between Frege and Kant's conceptions of logic and the logical source of knowledge. In a second step, I examine Frege's theory of quantity in Rechnungsmethoden, die sich auf eine Erweiterung des Größenbegriffes gründen (Frege 1874). Section 3 contains a couple of critical observations on Frege's comments on Hankel's theory of real numbers in Die Grundlagen der Arithmetik (Frege 1884). In Section 4, I consider Frege's discussion of the concept of quantity in Frege 1903. Section 5 is devoted to Cantor's theory of irrational numbers and the critique deployed by Frege. In Section 6, I return to Frege's own constructive treatment of analysis in Frege 1903 and succinctly describe what I take to be the quintessence of his account.     

Frege's Begriffsschrift is Indeed First-Order Complete

It is widely taken that the first-order part of Frege's Begriffsschrift is complete. However, there does not seem to have been a formal verification of this received claim. The general concern is that Frege's system is one axiom short in the first-order predicate calculus comparing to, by now, standard first-order theory. Yet Frege has one extra inference rule in his system. Then the question is whether Frege's first-order calculus is still deductively sufficient as far as the first-order completeness is concerned. In this short note we confirm that the missing axiom is derivable from his stated axioms and inference rules, and hence the logic system in the Begriffsschrift is indeed first-order complete.     

Beltrami's model and the independence of the parallel postulate

In their correspondence in 1902 and 1903, after discussing the Russell paradox, Russell and Frege discussed the paradox of propositions considered informally in Appendix B of Russell's Principles of Mathematics. It seems that the proposition, p, stating the logical product of the class w, namely, the class of all propositions stating the logical product of a class they are not in, is in w if and only if it is not. Frege believed that this paradox was avoided within his philosophy due to his distinction between sense (Sinn) and reference (Bedeutung). However, I show that while the paradox as Russell formulates it is ill-formed with Frege's extant logical system, if Frege's system is expanded to contain the commitments of his philosophy of language, an analogue of this paradox is formulable. This and other concerns in Fregean intensional logic are discussed, and it is discovered that Frege's logical system, even without its naive class theory embodied in its infamous Basic Law V, leads to inconsistencies when the theory of sense and reference is axiomatized therin. therein.     

Updating the Baldwin effect : The biological levels behind Piaget's new theory

In 1964, Conrad Waddington (1905–1975) presented a paper in Geneva that led to an internal reassessment of the biological underpinnings of Jean Piaget's (1896–1980) theory. This in turn resulted in an overhaul of the theoretical framework upon which his stage theory of child development had been based, including his appeals to James Mark Baldwin's (1861–1934) “circular reaction.” In addition to leading to the emergence of what has elsewhere been called “Piaget's new theory,” this renovation also resulted in the update of the famous “Baldwin Effect.” Because aspects of the subsequent framework are of contemporary significance, this essay will review some of the work leading up to those updates. In reaching behind the translations to trace the sources of the arguments to which Piaget appealed, the resulting examination fills some of the gaps found in the secondary literature without quibbling over the “correct” English interpretation of translated French terms. We also go beyond how Piaget's writings have been understood in English and extract some useful additional ideas from his sources, including how to conceive of the social context in which development takes place. We see as a result how Waddington and his colleagues, including Paul Weiss (1898–1989), provided a constructive “existence proof” for the formal hierarchy of levels that Piaget had come to by other means.     

Cantor on Frege's Foundations of Arithmetic: Cantor's 1885 Review of Frege's Die Grundlagen der Arithmetik

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.     

Jean piaget and george kelly: Toward a stronger constructivism

Among constructivist metatheoretical approaches, a strong version is defined as that which reveals reality to be actively and subjectively constructed rather than passively incorporated as objective environmental or innate “facts” by the subject. Given this definition, however, ambiguities arise concerning the potential and limits of construct integration over the course of development. Piaget's stronger constructivist model is offered as a means of clarifying and broadening the strong constructivist position on knowledge evolution. Piaget's genetic epistemology model places dramatic emphasis on the organizational capacity of the subject, specifying personal development as a strongly continuous and subsuming process. Discussion of commonalities between Piaget's position and Kelly's personal construct theory concludes this article.     

Frege on Truth,Assertoric Force and the Essence of Logic

In a posthumous text written in 1915, Frege makes some puzzling remarks about the essence of logic, arguing that the essence of logic is indicated, properly speaking, not by the word ‘true’, but by the assertoric force. William Taschek has recently shown that these remarks, which have received only little attention, are very important for understanding Frege's conception of logic. On Taschek's reconstruction, Frege characterizes logic in terms of assertoric force in order to stress the normative role that the logical laws play vis-à-vis judgement, assertion and inference. My aim in this paper is to develop and defend an alternative reconstruction according to which Frege stresses that logic is not only concerned with ‘how thoughts follow from other thoughts’, but also with the ‘step from thought to truth-value’. Frege considers logic as a branch of the theory of justification. To justify a conclusion by means of a logical inference, the ‘step from thought to truth-value’ must be taken, that is, the premises must be asserted as true. It is for this reason that, in the final analysis, the assertoric force indicates the essence of logic, for Frege.     

HOW DID FREGE FALL INTO THE CONTRADICTION?

Quine made it conventional to portray the contradiction that destroyed Frege's logicism as some kind of act of God, a thunderbolt that descended from a clear blue sky. This portrayal suited the moral Quine was antecedently inclined to draw, that intuition is bankrupt, and that reliance on it must therefore be replaced by a pragmatic methodology. But the portrayal is grossly misleading, and Quine's moral simply false. In the person of others – Cantor, Dedekind, and Zermelo – intuition was working pretty well. It was in Frege that it suffered a local and temporary blindness. The question to ask, then, is not how Frege was overtaken by the contradiction, but how it is that he didn't see it coming. The paper offers one kind of answer to that question. Starting from the very close similarity between Frege's proof of infinity and the reasoning that leads to the contradiction, it asks: given his understanding of the first, why did Frege did not notice the second? The reason is traced, first, to a faulty generalization Frege made from the case of directions and parallel lines; and, through that, to Frege's having retained, and attempted incoherently to combine with his own, aspects of a pre‐Fregean understanding of the generality of logical principles.