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Difficulties are raised for views that explain consensus in mathematics using only sociological pressure. Mathematical proof is sociologically very peculiar, when compared to other socially constrained practices. A preliminary analysis of the factors that have been at work historically in the “benign fixation of mathematical practice” are then exhumed: dispositions, implicit applications, an implicit logic, all play a role.

Popper’s world-3 doctrine is invoked to argue that characterizing mathematical developments as social processes is not incompatible with insisting on the objectivity and partial autonomy of mathematical knowledge. The argument is illustrated and supported by a historical case-study of the interplay between social and conceptual change in and after the French Revolution.

Mathematical writing, chiefly of proofs, is compared with the telling of stories. Contrasts are also noted. The positive analogy is used to support the view of mathematics as being about relations rather than objects obviating a need for ontological commitment to mathematical objects. The negative analogy is used to deny some philosophers’ identification of mathematics with fiction.

The Very Idea Of “Mathematical Practice” Implies, Beyond The Transparent Social Turn In Philosophy, Pedagogy, And Didactics Of Mathematics, A Theory Of Mind. Theories Of Mind May Be The Informal Folk Theories Of Our Everyday Lives Or The More Formal Theories Of Professional Students Of Mind. My Conjecture Is That Folk Theories Of Mind Are At Present Still More Influential In The Work Of Students Of Mathematics And The Mathematics Classroom Than Are Professional Theories. The Problem Is That Whichever Theory Prevails In Any Given Setting Or Study, Or For Any Given Researcher, It Is More Likely Than Not To Locate The Mind In The Brain And In The Person. So One Question I Want To Pose Is: Which Theory Or Theories Of Mind Are Built Into Our Theories Of And Approaches To Mathematical Practice? If Turning Our Attention To Mathematical Practice As Opposed To Focusing On Questions Of Foundations Is A Turn To The Social, Perhaps We Should Be Alert To The Possibility Of A Social Turn In Our Theories Of Mind.

In this paper, as part of an argument for the of revolutions in mathematics, I argue that there in incommensurability in Mathematics. After Devising A Framework Sensitive To Meaning Change And To Changes In The Extension Of Mathematical Predicates, I Consider Two Case Studies That Illustrate Different Ways In Which Incommensurability Emerge In Mathematical Practice. The Most Detailed Case Involves Nonstandard Analysis, And The Existence Of Different Notions Of The Continuum. But I Also Examine How Incommensurability Found Its Way Into Set Theory. I Conclude By Examining Some Consequences That Incommensurability Has To Mathematical Practice.

Well-chosen languages contribute to problem-solving success in the history of mathematics. Innovations in notation may not constitute progress on their own for philosophers who reject formalism. Yet successful languages are important concomitants of progress insofar as they enable mathematicians to state claims more broadly and recognize obscure relationships between different branches of mathematics. Platonists recognize the importance of Poincare’s ’happy innovations of language’ as a means whereby ever more mathematical reality is revealed. However, the distinction between what mathematics is about and the formal means used to study mathematics can rarely be made precise outside of isolated historical contexts. Understanding mathematical progress as increased scope is thus an alternative to Platonism’s “mathematical progress as genuine discovery” and formalism’s “mathematical progress as clever invention.”

This paper considers the birth of algebraic proof by looking at the works of Cardano, Viète, 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.

Informal and formal logic are complementary methods of argument analysis. Informal logic provides a pragmatic treatment of features of argumentation which cannot be reduced to logical form. This paper shows how paying attention to aspects of mathematical argumentation captured by informal, but not formal, logic can offer a more nuanced understanding of mathematical proof and discovery.

A study of the epistemologies of practising research mathematicians provides data with respect to the imaginative narratives (Bruner, 1986) used by these mathematicians when reflecting on their research practices. Unlike the paradigmatic narratives of formal mathematics, imaginative narratives involve the members of the mathematical community in active engagement, collaboratively, together with an acknowledgement of the holistic nature of knowing, thinking and feeling. The theoretical distinction is drawn between a contingent repertoire used within the imaginative mode when researching and the objectivist repertoire used to situate mathematics publicly as impersonal, separate and independent of the human or social. Focusing on the mathematicians’ narratives, attention is drawn to how the mathematicians use transcendental or operational functions to support a simple objectivist stance and this is compared with the complexity of the contingent view couched in metaphoric and analogous reasoning. The chapter ends with a discussion of the implications of this analysis for mathematics education.

In this paper we explore how the naturalistic perspective in and the situative perspective in while on one level are at odds, might be reconciled by paying attention to actual mathematical practice and activity. We begin by examining how each approaches mathematical knowledge, and then how mathematical practice manifest itself in these distinct research areas and gives rise to apparently contrary perspectives. Finally we argue for a deeper agreement and a reconciliation in the perspectives based on the different projects of justification and explanation in mathematics.