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.     

From qualitative differences to a continuum of development: commentary on Leslie Smith's epistemological principles for developmental psychology

In his article Epistemological principles for developmental psychology Leslie Smith helps to re-open some of the key issues Piaget explored through his genetic epistemology. Smith shows the important parallels between logician Gottlieb Frege's understanding of rational thought, and the way in which Piaget developed such notions in his own theory. But while Frege's theory helps set the parameters for whether thought can be judged as rational, or if it even should be judged as rational, it also shows the logicians' disdain for exploring anything resembling development of rationality. Thus Frege might have an important, but necessarily mediated impact on the field of human development. Piaget's carefully crafted theory of epistemological development potentially serves as such a mediating device. It can be argued that Piaget crafted together arguments of logicians such as Frege, and epistemologists such as Lévy-Bruhl, to develop his extraordinary achievement of a genetic epistemology that leads to an understanding of the human condition. One of Piaget's accomplishments was to develop a continuum out of the logicians' dichotomy between non-logical and logical in which the non-rational flows into the rational.     

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.     

Experimental Studies of Children's Learning

This article compares John Dewey's theory of inquiry with Jean Piaget's analysis of the mechanisms implied in the increase of knowledge. The sources for this paper are Dewey's studies on logic and the theory of inquiry and Piaget's historical-critical and psychogenetic investigations. Three major conclusions result from the comparison: first, there are significant convergences between the two theories; second, Piaget's developmental analysis makes explicit what is programmatic in Dewey's investigations; and, finally, Piaget is incorrect in characterizing Dewey's pragmatism as a method that does not meet the criteria of intelligent activity.     

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.     

Harré on Piaget's Sociological Studies

Piaget's Sociological Studies is largely taken up with questions of sociology, epistemology, moral, political and legal theory. Roughly, these are by-passed in Harré's (2000, this journal) interpretation of Piaget's social account. Our critique is in two parts. In the first part, we explain the basis of our disagreement with Harré's interpretation of six specific issues. These are Piaget's model of social exchange, sociology, concept of egocentrism, response to Wallon, psycho-social parallelism, distinction between the concrete and the abstract. In the second part, we challenge two central theses invoked in Harré's review, namely (A) all psychological activity is a joint activity, and (B) any society is based on irreducible differences in psychological activity. We have tried to set the record straight as far as Piaget's social account is concerned. Harré's review shows a general lack of acquaintance with Piaget's account and our aim has been to compensate for this.     

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.     

Developmental stages,Piagetian stages in particular: A critical review

Developmental stages in general and Piaget's stages in particular have given rise to considerable controversy. Much of this controversy revolves around the responses that have been given to the following five central questions: (1) Do developmental stages exist? (2) If they exist, where are they? (3) What features define a developmental sequence as a sequence of developmental stages? (4) What psychological processes underlie developmental change? (5) Should we abandon the concept of developmental stages? The main goal of this paper is to present a critical review of such responses, while arguing for a strong conception of development and a “non-received” view of Piaget's theory. After an introduction section, we elaborate on each of the five questions. Finally, we present several reasons why this paper often appeals to Piaget's theory, and why his theory has been greatly misunderstood.     

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.     

The Priority Principle from Kant to Frege

In a famous passage (A68/B93), Kant writes that “the understanding can make no other use of […] concepts than that of judging by means of them.” Kant's thought is often called the thesis of the priority of judgments over concepts. We find a similar sounding priority thesis in Frege: “it is one of the most important differences between my mode of interpretation and the Boolean mode […] that I do not proceed from concepts, but from judgments.” Many interpreters have thought that Frege's priority principle is close to (or at least derivable from) Kant's. I argue that it is not. Nevertheless, there was a gradual historical development that began with Kant's priority thesis and culminated in Frege's new logic.     

Avoiding misinterpretations of Piaget and Vygotsky: Mathematical teaching without learning,learning without teaching,or helpful learning-path teaching?

This article provides an overview of some perspectives about special issues in classroom mathematical teaching and learning that have stemmed from the huge explosion of research in children's mathematical thinking stimulated by Piaget. It concentrates on issues that are particularly important for less-advanced learners and for those who might be having special difficulties in learning mathematics. A major goal of the article is to develop a framework for understanding what effective mathematics teaching and learning is, because doing so is so important for struggling students and for research about them. Piaget's research had a fundamental influence on the on-going tension between understanding and fluency in the classroom, supporting efforts toward increasing understanding. But in some countries, misinterpretations of Piaget led to practices that are counterproductive for children, especially struggling learners. Such misinterpretations are identified and a more balanced approach that also draws on Vygotsky is described—a learning-path developmentally-appropriate learning/teaching approach.     

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.     

Grundlagen,Section 64: Frege's Discussion of Definitions by Abstraction in Historical Context

I offer in this paper a contextual analysis of Frege's Grundlagen, section 64. It is surprising that with so much ink spilled on that section, the sources of Frege's discussion of definitions by abstraction have remained elusive. I hope to have filled this gap by providing textual evidence coming from, among other sources, Grassmann, Schlömilch, and the tradition of textbooks in geometry for secondary schools (including a textbook Frege had used when teaching in a Privatschule in Jena in 1882–1884). In addition, I put Frege's considerations in the context of a widespread debate in Germany on ‘directions’ as a central notion in the theory of parallels.     

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.     

Remarks on Independence Proofs and Indirect Reference

In the last two decades, there has been increasing interest in a re-evaluation of Frege's stance towards consistency- and independence proofs. Papers by several authors deal with Frege's views on these topics. In this note, I want to discuss one particular problem, which seems to be a main reason for Frege's reluctant attitude towards his own proposed method of proving the independence of axioms, namely his view that thoughts, that is, intensional entities are the objects of metatheoretical investigations. This stands in contrast to more straightforward interpretations, which claim that Frege's hesitancy is mainly due to worries concerning the logical constants or what counts as a logical inference.     

La pensée post-formelle en gérontagogie ou l’après-Piaget

Jean Piaget, one of the most influential theorists in developmental psychology, assumed that formal thought, characterized by the development of an individual's logical capacities, was the last stage of adult thinking. According to Piaget's theory, adults undergo an unavoidable decline in their intellectual functions after formal thought. However, here we review how the brain evolved, describing its main structures, and examining each cerebral hemisphere's specific functions. Evidence is also provided for the production of new neurons and new connections between them, forcing a revision of old theories about the decline of intellectual functions in the elderly. We then consider Jones’ theories X and Y, and the different definitions of intelligence (fluid vs. crystallized, and qualitative vs. quantitative), and how these perspectives have influenced the way we see intelligence. Evidence supporting the addition of another stage, named post-formal thought, is examined in the context of gerontagogy. Dialectical thought characterizes this stage, and developing wisdom is its main goal. We examine the two basic principles of dialectical thought, namely the principle of contradiction and the principle of relativity of everything. It is suggested that the learning of wisdom should be the focus of future university programs to educate the elderly.     

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.     

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.     

Two studies of Piaget's theory of moral judgment

The relationship between conservation skills and subjective responsibility (SR) in moral judgment was examined in two studies. The Kohlberg ontogenetic priority hypothesis stating that conversation is a necessary but not sufficient condition of SR was pitted against the common structural basis hypothesis advanced by Damon. Piaget's notion of heteronomy as a combination of preoperational thought and unilateral respect was studied by using the behavioural independence tasks by Subbotsky as indicators of unilateral respect. A teaching experiment using Galperin's and Obukhova's method to induce conservation indicated significant transfer from conservation to SR. SR was found to be more susceptible to regression over a 3–month period than conservation. Conservers were less likely to regress on SR than nonconservers. Unilateral respect was related to objective responsibility but not to conservation. Taken together the data fail to support Kohlberg's ontogenetic priority hypothesis and only partially support the Piaget and Damon model of a common underlying structure. It is concluded that SR judgments largely reflect individual judgmental strategies and the effect of conservation on SR is intepreted in terms of increasing cross-situational consistency in the use of these strategies.     

Primitive religious thought and the evolution of religion

In an article in this journal about the ideas of Levy-Brühl, Don Wiebe reviewed ideas from anthropology to argue that there is indeed a primitive mentality, and that mythopoeic religious thought is an instance of this mentality. Wiebe ends by declaring religious thought generally to be mythopoeic. The anthropological literature does support well Wiebe's thesis that primitive thought is primitive, in particular those studies which use Piaget's theory of cognitive development as a guide for interpreting primitive and other thought styles. But Piaget's theory can be extended to interpret other stages of culture and thought. This aids in recognizing that religious thought develops in stages also, from primitive folktales to archaic complex myths, and then to classical rationalized theologies and most recently to symbolic theologies.