Brendan Larvor

Why did Kuhn’s Structure of scientific revolutions cause a fuss?

Abstract After the publication of The structure of scientific revolutions, Kuhn attempted to fend off accusations of extremism by explaining that his allegedly “relativist” theory is little more than the mundane analytical apparatus common to most historians. The appearance of radicalism is due to the novelty of applying this machinery to the history of science. This defence fails, but it provides an important clue. The claim of this paper is that Kuhn inadvertently allowed features of his procedure and experience as an historian to pass over into his general account of science. Kuhn’s familiar claims, that science is directed in part by extra-scientific influences; that the history of science is divided by revolutionary breaks into periods that cannot be easily compared; that there is no ahistorical standard of rationality by which past episodes may be judged; and that science cannot be shown to be heading towards the Truth—these now appear as methodological commitments rather than historico–philosophical theses.

How to think about informal proofs

It is argued in this study that (i) progress in the philosophy of mathematical practice requires a general positive account of informal proof; (ii) the best candidate is to think of informal proofs as arguments that depend on their matter as well as their logical form; (iii) articulating the dependency of informal inferences on their content requires a redefinition of logic as the general study of inferential actions; (iv) it is a decisive advantage of this conception of logic that it accommodates the many mathematical proofs that include actions on objects other than propositions; (v) this conception of logic permits the articulation of project-sized tasks for the philosophy of mathematical practice, thereby supplying a partial characterisation of normal research in the field.

What Philosophy of Mathematical Practice Can Teach Argumentation Theory About Diagrams and Pictures

There has been a rising tide of interest among argumentation theorists in visual reasoning. In the hands of the leaders of this development the effort has been to assimilate visual reasoning to verbal argumentation. At the same time, there is a more mature but still advancing literature on the use of diagrams in mathematical reasoning. There have been efforts to bring the two together. In this paper, I wish to use the philosophy of mathematical practice to identify a severe limitation in the attempt to assimilate visual reasoning to verbal reasoning, and by extension to criticise the approach to reasoning that treats all reasoning as if it were verbal reasoning.

Mathematical Cultures : The London Meetings 2012-2014

Brendan Larvor, Ed., Mathematical Cultures: The London Meetings 2012-2014 (Switzerland: Springer 2016), ISBN: 978-3-319-28580-1, DOI: 10.1007/978-3-319-28582-5

From Euclidean Geometry to Knots and Nets

This paper assumes the success of arguments against the view that informal mathematical proofs secure rational conviction in virtue of their relations with corresponding formal derivations. This assumption entails a need for an alternative account of the logic of informal mathematical proofs. Following examination of case studies by Manders, De Toffoli and Giardino, Leitgeb, Feferman and others, this paper proposes a framework for analysing those informal proofs that appeal to the perception or modification of diagrams or to the inspection or imaginative manipulation of mental models of mathematical phenomena. Proofs relying on diagrams can be rigorous if (a) it is easy to draw a diagram that shares or otherwise indicates the structure of the mathematical object, (b) the information thus displayed is not metrical and (c) it is possible to put the inferences into systematic mathematical relation with other mathematical inferential practices. Proofs that appeal to mental models can be rigorous if the mental models can be externalised as diagrammatic practice that satisfies these three conditions.

What Are Mathematical Cultures

In this paper, I will argue for two claims. First, there is no commonly agreed, unproblematic conception of culture for students of mathematical practices to use. Rather, there are many imperfect candidates. One reason for this diversity is there is a tension between the material and ideal aspects of culture that different conceptions manage in different ways. Second, normativity is unavoidable, even in those studies that attempt to use resolutely descriptive, value-neutral conceptions of culture. This is because our interest as researchers into mathematical practices is in the study of successful mathematical practices (or, in the case of mathematical education, practices that ought to be successful).

Authoritarian Versus Authoritative Teaching: Polya and Lakatos

Lakatos argued that a proof, when presented in the usual “Euclidian” style, may leave the choice of theorem, definitions and proof-idea mysterious. To remove these mysteries, he recommended a “heuristic” style of presentation. This distinction was already present in the work of Polya. Moreover, Polya was directly concerned with teaching and consequently paid attention to the emotional and existential experience of the student. However, Polya lacked Lakatos’s account of proof analysis and was not a fallibilist. Therefore, the question of whether Lakatos advanced pedagogy from where Polya left it reduces to two questions: (1) does proof analysis have a place in the classroom? and (2) does fallibilism have a place in the classroom? In this paper, I argue that the answers are (1) Yes and (2) No.

Between History and Logic

“The original publication is available at www.springerlink.com”. Copyright Springer DOI: 10.1007/978-1-4020-6127-1_11

Proof in C17 Algebra

This paper considers the birth of algebraic proof by looking at the works of Cardano, Viete, Harriot and Pell. The transition from geometric to algebraic proof was mediated by appeals to the Eudoxan theory of proportions in book V of Euclid. The crucial notational innovation was the development of brackets. By the middle of the seventeenth century, geometric proof was unsustainable as the sole standard of rigour because mathematicians had developed such a number and range of techniques that could not be justified in geometric terms.

Re-reading Soviet philosophy: Bakhurst on Ilyenkov

“The original publication is available at www.springerlink.com”. Copyright Springer. DOI: 10.1007/BF00819094 [Full text of this article is not available in the UHRA]