I address the philosophical debate over whether the mathematical theory of probability arose on the basis of empirical observations or of purely theoretical speculations. The debate tends to pose a strict dichotomy between empirical problem-solving and pure theorizing. I alternatively suggest that, in the case of mathematical probability, an empirical problem-context acted as an enabling condition for the possibility of mathematical innovation, but that the activity of the early mathematical probabilists gradually became the study of a theoretical system of ideas. (...) This case has some implications for a more general philosophical view on the context of mathematical discovery. ---------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------- ----------------------------------------. (shrink)

: C.S. Peirce defines mathematics in two ways: first as "the science which draws necessary conclusions," and second as "the study of what is true of hypothetical states of things" (CP 4.227–244). Given the dual definition, Peirce notes, a question arises: Should we exclude the work of poietic hypothesis-making from the domain of pure mathematical reasoning? (CP 4.238). This paper examines Peirce's answer to the question. Some commentators hold that for Peirce the framing of mathematical hypotheses requires poietic genius but (...) is not scientific work. I propose, to the contrary, that although Peirce occasionally seems to exclude the poietic creation of hypotheses altogether from pure mathematical reasoning, Peirce's position is rather that the creation of mathematical hypotheses is poietic, but it is not merely poietic, and accordingly, that hypothesis-framing is part of mathematical reasoning that involves an element of poiesis but is not merely poietic either. Scientific considerations also inhere in the process of hypothesis-making, without excluding the poietic element. In the end, I propose that hypothesis-making in mathematics stands between artistic and scientific poietic creativity with respect to imaginative freedom from logical and actual constraints upon reasoning. (shrink)

I articulate Charles S. Peirce’s philosophy of mathematical education as related to his conception of mathematics, the nature of its method of inquiry, and especially, the reasoning abilities required for mathematical inquiry. The main thesis is that Peirce’s philosophy of mathematical education primarily aims at fostering the development of the students’ semeiotic abilities of imagination, concentration, and generalization required for conducting mathematical inquiry by way of experimentation upon diagrams. This involves an emphasis on the relation between theory and practice and (...) between mathematics and other fields including the arts and sciences. For achieving its goals, the article is divided in three sections. First, I expound Peirce’s philosophical account of mathematical reasoning. Second, I illustrate this account by way of a geometrical example, placing special emphasis on its relation to mathematical education. Finally, I put forth some Peircean philosophical principles for mathematical education. (shrink)

What understanding of mathematical objectivity is promoted by Peirce’s pragmatism? Can Peirce’s theory help us to further comprehend the role of intersubjectivity in mathematics? This paper aims to answer such questions, with special reference to recent debates on mathematical practice, where Peirce is often quoted, although without a detailed scrutiny of his theses. In particular, the paper investigates the role of intersubjectivity in the constitution of mathematical objects according to Peirce. Generally speaking, this represents one of the key issues for (...) the philosophy of mathematical practice, whereas – with regard to Peirce – these two aspects (intersubjectivity and objectivity) have been studied for a long time, but mostly as unrelated topics. To reconstruct the connection between Peirce’s reflection on intersubjectivity and the objectivity of mathematical theories, the paper is divided into three parts: (1) the first part introduces Peirce’s view of mathematics; (2) the second analyzes his peculiar “pragmatist realism,” which overcomes common dichotomies in the philosophy of mathematics; (3) the third illustrates the role that Peirce attributes to intersubjectivity in science, and investigates to what extent the intersubjective dimension is also essential in mathematics. (shrink)

: This essay shows that Peirce's (more or less) final classification of the sciences arises from the systematic application of his Categories of Firstness, Secondness and Thirdness to the classification of the sciences themselves and that he does not do so until his 1903's "An Outline Classification of the Sciences." The essay proceeds by: First, making some preliminary comments regarding Peirce's notion of an architectonic, or classification of the sciences; Second, briefly explaining Peirce's Categories of Firstness, Secondness and Thirdness; Third, (...) examining how Peirce classifies the sciences in 1902 and 1903—and specifically how the 1903 classification utilizes the Categories; Fourth and finally, showing that he is only led to classify the sciences in this fashion as result of his philosophical inquiries during those intervening months—especially as a result of his Harvard Lectures on Pragmatism. (shrink)

The classification of the sciences is one of the most discussed and analysed aspects of Peirce’s corpus of work. I propose that Peirce’s attempt at systematising the sciences is characterised by a distinctive historicity, which I construe in two complementary senses. First, I investigate Peirce’s classification as part of a broader nineteenth-century move toward classifying the sciences, a move that was at the same time motivated by social and epistemological goals. I claim that this re-contextualisation adds an entirely new layer (...) to the otherwise distinctively internalist readings of Peirce’s classification. I then look at how Peirce’s scheme, especially in the form it displayed in the early 1890s, relates to his own historical writings, particularly his history of science. Looking at Peirce as a historical actor in his own right through the lens of his classification, I claim, is indispensable to understand the contemporary relevance of his contributions to the history and historiography of the sciences. (shrink)

Peirce's determination to understand matter, the cosmos, and "the grand design" of the universe remain relevant for contemporary students of science, technology, and symbolic logic.