The Happy Enthusiast (original) (raw)
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).
Social processes, program verification and all that
Mathematical Structures in Computer Science, 2009
In a controversial paper at the end of 1970's, R.A. De Millo, R.J. Lipton and A.J. Perlis argued against formal verifications of programs, mostly motivating their position by an analogy with proofs in mathematics, and in particular with the impracticality of a strictly formalist approach to this discipline. The recent, impressive achievements in the field of interactive theorem proving provide an interesting ground for a critical revisiting of those theses. We believe that the social nature of proof and program development is uncontroversial and ineluctable but formal verification is not antithetical to it. Formal verification should strive not only to cope, but to ease and enhance the collaborative, organic nature of this process, eventually helping to master the growing complexity of scientific knowledge. †
In Monsters, monstrosities, and the monstrous in culture and society, Diego Compagna & Stefanie Steinhart, edd. (Wilmington, DE: Vernon Press), 2019
Monsters lurk within mathematical as well as literary haunts. I propose to trace some pathways between these two monstrous habitats. I start from Jeffrey Jerome Cohen’s influential account of monster culture and explore how well mathematical monsters fit each of his seven theses (Cohen 1996). The mathematical monsters I discuss are drawn primarily from three distinct but overlapping domains. I will describe these in much greater detail as they arise below, but here is a brief preview. Firstly, late nineteenth-century mathematicians made numerous unsettling discoveries that threatened their understanding of their own discipline and challenged their intuitions. The great French mathematician Henri Poincaré characterised these anomalies as ‘monsters’, a name that stuck. Secondly, the twentieth-century philosopher Imre Lakatos composed a seminal work on the nature of mathematical proof, in which monsters play a conspicuous role (Lakatos 1976). He reconstructs the emergence during the nineteenth century of a proof of the Euler Conjecture, which ascribes a certain property to polyhedra. Lakatos coined such terms as ‘monster-barring’ and ‘monster-adjusting’ to describe strategies for dealing with entities whose properties seem to falsify the conjecture. Thirdly, and most recently, mathematicians dubbed the largest of the so-called sporadic groups ‘the Monster’, because of its vast size and uncanny properties, and because its existence was suspected long before it could be confirmed.