Mathematics and logic are indispensable in science, yet how they are deployed and why they are so effective, especially in the natural sciences, is poorly understood. In this paper, I focus on the how by analysing Jean Piaget’s application of mathematics to the empirical content of psychological experiment; however, I do not lose sight of the application’s wider implications on the why. In a case study, I set out how Piaget drew on the stock of mathematical structures to model psychological (...) content, namely, the operations of thought involved in reasoning. In particular, I show how operations of thought form structured wholes that initially resisted modelling by either lattices or groups but could be modelled adequately by modifications of these mathematical structures. Piaget coined the term ‘grouping’ for the modified structure, and I conclude that it represents a non-canonical application of mathematics to the empirical content of experimental psychology. I also touch on the role external factors played in Piaget’s development of the grouping. According to the genetic epistemology conceived by Piaget, the origin of intelligence lies in the biological organism and develops in stages over time, and, via the grouping, Piaget established a genetic relationship between two stages of reasoning. I show how this relationship explains why mathematics and logic fit the psychological content of reasoning whilst simultaneously making their successful deployment in the natural sciences more mysterious. Finally, I turn to the explanation Piaget envisaged for the unreasonable effectiveness of mathematics in the natural sciences and consider some consequences for naturalism and Pythagoreanism. (shrink)

Tarski’s conceptual analysis of the notion of logical consequence is one of the pinnacles of the process of defining the metamathematical foundations of mathematics in the tradition of his predecessors Euclid, Frege, Russell and Hilbert, and his contemporaries Carnap, Gödel, Gentzen and Turing. However, he also notes that in defining the concept of consequence “efforts were made to adhere to the common usage of the language of every day life.” This paper addresses the issue of what relationship Tarski’s analysis, and (...) Béziau’s further generalization of it in universal logic, have to reasoning in the everyday lives of ordinary people from the cognitive processes of children through to those of specialists in the empirical and deductive sciences. It surveys a selection of relevant research in a range of disciplines providing theoretical and empirical studies of human reasoning, discusses the value of adopting a universal logic perspective, answers the questions posed in the call for this special issue, and suggests some specific research challenges. (shrink)

Many experiments with infants suggest that they possess quantitative abilities, and many experimentalists believe that these abilities set the stage for later mathematics: natural numbers and arithmetic. However, the connection between these early and later skills is far from obvious. We evaluate two possible routes to mathematics and argue that neither is sufficient: (1) We first sketch what we think is the most likely model for infant abilities in this domain, and we examine proposals for extrapolating the natural number concept (...) from these beginnings. Proposals for arriving at natural number by (empirical) induction presuppose the mathematical concepts they seek to explain. Moreover, standard experimental tests for children's understanding of number terms do not necessarily tap these concepts. (2) True concepts of number do appear, however, when children are able to understand generalizations over all numbers; for example, the principle of additive commutativity (a+b=b+a). Theories of how children learn such principles usually rely on a process of mapping from physical object groupings. But both experimental results and theoretical considerations imply that direct mapping is insufficient for acquiring these principles. We suggest instead that children may arrive at natural numbers and arithmetic in a more top-down way, by constructing mathematical schemas. (shrink)

How do people make deductions? The orthodox view in psychology is that they use formal rules of inference like those of a “natural deduction” system.Deductionargues that their logical competence depends, not on formal rules, but on mental models. They construct models of the situation described by the premises, using their linguistic knowledge and their general knowledge. They try to formulate a conclusion based on these models that maintains semantic information, that expresses it parsimoniously, and that makes explicit something not directly (...) stated by any premise. They then test the validity of the conclusion by searching for alternative models that might refute the conclusion. The theory also resolves long-standing puzzles about reasoning, including how nonmonotonic reasoning occurs in daily life. The book reports experiments on all the main domains of deduction, including inferences based on prepositional connectives such as “if” and “or,” inferences based on relations such as “in the same place as,” inferences based on quantifiers such as “none,” “any,” and “only,” and metalogical inferences based on assertions about the true and the false. Where the two theories make opposite predictions, the results confirm the model theory and run counter to the formal rule theories. Without exception, all of the experiments corroborate the two main predictions of the model theory: inferences requiring only one model are easier than those requiring multiple models, and erroneous conclusions are usually the result of constructing only one of the possible models of the premises. (shrink)

In this paper I develop a philosophical account of actual mathematical infinity that does not demand ontologically or epistemologically problematic assumptions. The account is based on a simple metaphor in which we think of indefinitely continuing processes as defining objects. It is shown that such a metaphor is valid in terms of mathematical practice, as well as in line with empirical data on arithmetical cognition.

This paper represents such an amateur approach; hence any comments backed up by professional erudition will be highly appreciated. Let me start from an attempt to sketch a relationship between professionals’ and amateurs’ contributions. The latter may be compared with the letters to the Editor of a journal, written by perceptive readers, while professionals contribute to the very content of the journal in question. Owing to such letters, the Editor and his professional staff can become more aware of the responses (...) of educated public to the journal’s output. (shrink)

Logical psychologism is the position that logic is a special branch of psychology, that logical laws are descriptíons of experience to be arrived at through observation, and are a posteriori.The accepted arguments against logical psychologism are effective only when directed against this extreme version. However, the clauses in the above characterization are independent and ambiguous, and may be considered separately. This separation permits a reconsideration of less extreme attempts to tie logic to psychology, such as those defended by Mill and (...) Boole. It also provides the basis for a reexamination of the relationship between logic and psychology, and raises the possibility of a deeper investigation into the nature of logic itself. (shrink)

In this paper, we offer a Piagetian perspective on the construction of the logico-mathematical schemas which embody our knowledge of logic and mathematics. Logico-mathematical entities are tied to the subject's activities, yet are so constructed by reflective abstraction that they result from sensorimotor experience only via the construction of intermediate schemas of increasing abstraction. The axiom set does not exhaust the cognitive structure (schema network) which the mathematician thus acquires. We thus view truth not as something to be defined within (...) the closed world of a formal system but rather in terms of the schema network within which the formal system is embedded. We differ from Piaget in that we see mathematical knowledge as based on social processes of mutual verification which provide an external drive to any necessary dynamic of reflective abstraction within the individual. From this perspective, we argue that axiom schemas tied to a preferred interpretation may provide a necessary intermediate stage of reflective abstraction en route to acquisition of the ability to use formal systems in abstracto. (shrink)

Many contemporary logicians acknowledge a plurality of logical theories and accept that theory choice is in part motivated by logical evidence. However, just as there is no agreement on logical theories, there is also no consensus on what constitutes logical evidence. In this paper, I outline Jean Piaget’s psychological theory of reasoning and show how he used it to diagnose and solve one of the paradoxes of material implication. I assess Piaget’s use of psychology as a source of evidence for (...) logical theory in light of reservations raised by psychologism, and I highlight some ramifications for exceptionalism and anti-exceptionalism about logic by considering his use of psychology as logical evidence in the framework of genetic epistemology, Piaget’s research programme. I conclude that Piaget’s psychological theory of reasoning not only plausibly serves as a source of evidence for logical theory but also makes a strong case for anti-exceptionalism about logic. (shrink)

In the paper we discuss the problem of limitations of freedom in mathematics and search for criteria which would differentiate the new concepts stemming from the historical ones from the new concepts that have opened unexpected ways of thinking and reasoning. We also investigate the emergence of category theory and its origins. In particular we explore the origins of the term functor and present the strong evidence that Eilenberg and Carnap could have learned the term from Kotarbiński and Tarski.

We see no grounds for insisting that, because the concept natural number is abstract, its foundations must be innate. It is possible to specify domain general learning processes that feed into more abstract concepts of numerical infinity. By neglecting the messiness of children's slow acquisition of arithmetical concepts, Rips et al. present an idealized, unnecessarily insular, view of number development.

Recent research shows that research programmes (quantitative, qualitative and mixed) in education are not displaced (as suggested by Kuhn) but rather lead to integration. The objective of this study is to present a rationale for mixed methods (integrative) research programs based on contemporary philosophy of science (Lakatos, Giere, Cartwright, Holton, Laudan). This historical reconstruction of episodes from physical science (spanning a period of almost 300 years, 17 th to 20 th century) does not agree with the positivist image of science. (...) Quantitative data (empirical evidence) by itself, does not facilitate progress (despite widespread belief to the contrary), neither in the physical sciences nor in the social sciences (education) A historical reconstruction shows that both Piaget and Pascual-Leone's research programs in cognitive psychology, follow the Galilean idealisation quite closely, similar to the research programs of Newton, Mendeleev, Einstein, Thomson, Rutherford, Millikan and Perl in the physical sciences. This relationship does not imply that researchers in education have to emulate research in the physical sciences. A major argument in favor of mixed methods (integrative) research programs is that it provides a rationale for hypotheses, theories, guiding assumptions and presuppositions to compete and provide alternatives. Similar to the physical sciences, this proliferation of hypotheses leads to controversies and rivalries, and thus facilitates the decision making process of the scientific community. It is concluded that mixed methods research programs (not paradigms) in education can facilitate the construction of robust strategies, provided we let the problem situation (as studied by practicing researchers) decide the methodology. (shrink)

Genetic epistemology analyzes the growth of knowledge both in the individual person (genetic psychology) and in the socio-historical realm (the history of science). But what the relationship is between the history of science and genetic psychology remains unclear. The biogenetic law that ontogeny recapitulates phylogeny is inadequate as a characterization of the relation. A critical examination of Piaget's Introduction à l'Épistémologie Généntique indicates these are several examples of what I call stage laws common to both areas. Furthermore, there is at (...) least one example of a paradoxical inverse relation between the two — geometry. Both similarities and differences between the two domains require an explanation, a developmental explanation. Although such an explanation seems to be psychological in nature, it is not merely empirical but also normative (since psychology is both factual and normative according to Piaget). Hence genetic epistemology need not be reduced to psychology (narrowly conceived), but rather should be seen as being both empirical and normative and thus similar to certain types of contemporary philosophy of science. (shrink)

The relation between logic and philosophy of science, often taken for granted, is in fact problematic. Although current fashionable criticisms of the usefulness of logic are usually mistaken, there are indeed difficulties which should be taken seriously — having to do, amongst other things, with different scientific mentalities in the two disciplines (section 1). Nevertheless, logic is, or should be, a vital part of the theory of science. To make this clear, the bulk of this paper is devoted to the (...) key notion of a scientific theory in a logical perspective. First, various formal explications of this notion are reviewed (section 2), then their further logical theory is discussed (section 3). In the absence of grand inspiring programs like those of Klein in mathematics or Hilbert in metamathematics, this preparatory ground-work is the best one can do here. The paper ends on a philosophical note, discussing applicability and merits of the formal approach to the study of science (section 4). (shrink)

This paper examines an example of learning with artifacts using the commonplace materials of string and knots. Emphases include research into learning processes as well as construction of objects to assist learning. The inquiry concerns the development of mathematical thinking, topology in particular. The research methodology combines participant observation and clinical interview within a constructionist framework. The study was set in a self-styled, self-constructed environment that consisted of knots and a social substrate encouraging lively exchanges of ideas about them. Comparisons (...) of certain knots helped to elicit conceptions of the fundamental topological relationships of neighborhood, continuity, and boundaries. The paper includes comments on the suitability of specific artifacts for specific kinds of thinking and learning, and emphasizes the importance for software design of considering different learning styles. (shrink)