Joint Colloquium: Henrik Kragh Sørensen -- Abel’s exception, history of infinity and cognitive accounts of mathematics: One chance fumbled and another one missed

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 

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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.  

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Joint Colloquium web site: 

http://www.maths.usyd.edu.au/u/SemConf/JointColloquium/index.html