Type: Seminar

Distribution: World

Auth: afish@p617.pc (assumed)

Joint Colloquium: Henrik Kragh Sørensen -- Abel’s exception, history of infinity and cognitive accounts of mathematics: One chance fumbled and another one missed Speaker: Prof. Henrik Kragh Sørensen Time: Fri, 24/01/2014 - 2:00pm - 3:00pm Room: RC-4082, Red Centre Building, UNSW ----------------------------------------------- Title: Abel’s exception, history of infinity and cognitive accounts of mathematics: One chance fumbled and another one missed Abstract: In 1748, Euler initiated a new approach to the "analysis of the infinite" studying functions through their representations by analytic means. These representations included infinite series and products, thereby shifting the focus of analysis from the study of curves to the study of "functions" defined through these means. Euler’s approach proved very successful for the eighteenth century, but beginning in the 1820s doubts and objections began to emerge in the form of Cauchy’s ban of arguments by "the generality of algebra", and of divergent series in particular. This was emphasized by Abel’s observation, that certain behavior familiar for finite numbers of operations ceased to obtain when infinite series were considered. Thus, the intuition so skillfully mastered by Euler was seriously questioned. This development can be said to have come to a head when Weierstrass in 1872 presented his example of an every-where continuous, no-where differentiable function which thus defied basic intuitions about the connections between curves and functions. Importantly, Weierstrass’ example was defined through an infinite series. Over the past two decades, cognitivist accounts have provided us with new insights and perspectives on embodied foundations of basic arithmetic leading on to higher mathematics. In order to argue for the so-called Basic Metaphor of Infinity, proponents of the cognitivist account of conceptual metaphors have invoked arguments drawn from the history of mathematics. Based on the my previous work to historically contextualize Niels Henrik Abel’s (1802–1829) “exception” to a theorem by Augustin-Louis Cauchy (1789–1857), this talk argues that their use of history of mathematics is inadequate and that they actually miss out on a better argument. Special emphasis is given to the reflections about the permissibility of drawing inferences from the finite domain into infinite operations such as series and products. In so doing, I provide both an overview of the standard narrative of rigorization of analysis as well as a historical framework for critically discussing recent cognitive-historical analyses of Abel’s exception to an important theorem by Cauchy and of Weierstrass’ monster function. This will lead to a brief discussion about the relative merits of cognitive-historical analyses in the sense of Núñez et al. as compared to more traditional history of mathematics. ----------------------------------------------- Joint Colloquium web site: http://www.maths.usyd.edu.au/u/SemConf/JointColloquium/index.html