Lichtenberg-Kolleg - The Göttingen Institute for Advanced Study

Mathematics Beyond Theorems (22 to 23 March 2012)

Since Euclid it seems widely accepted (in Western culture) that mathematical knowledge is recounted in theorems, and that proving (new) theorems is the essence and the motor of doing and furthering mathematics. Today’s highly explicit and formalized communication of mathematical research appears to corroborate this representation of mathematics.

For historians – and for philosophers – of mathematics this suggests lots of questions :

1. What is a theorem? For instance: When are two theorems equal – esp. if the two statements are found at different times or places of the historical process?

2. What about mathematical practice which cannot adequately be coached in terms of theorems?

L. Wittgenstein at the end of the 1920s proposed a verificationist answer to the first question.
Over the last decades, several historians of mathematics, starting with H. Sinaceur, C. Gilain, and C. Goldstein have explicitly addressed the first question in detailed case studies. Extra-European mathematics – the earliest being cuneiform texts describing procedures to solve certain types of problems – provide obvious examples (whose detailed analysis may not be obvious) of mathematical knowledge without theorems. This aspect has been emphasized by Karine Chemla, Jim Ritter, and several younger historians.

The history of mathematics also presents non-algorithmic phenomena which are nevertheless not oriented towards provable statements. A fairly recent and not yet well-known contribution in this direction, which will be presented at the workshop, is due to A. Herreman who uses semiotic methods to introduce the notion of “inaugurational texts” in the history of mathematics; their central concern is to propagate a thesis which by its very nature cannot possibly be proven. His prototype is Church’s thesis for the theory of computable functions. Texts that Herreman has analyzed along these lines include Descartes’s Géométrie and Fourier’s Théorie de la chaleur. Euclid’s Elements do not belong to this class of texts.

Attempts to exploit for the history of mathematics ideas about tacit knowledge – a concept originally proposed by M. Polanyi to describe scientific practice, among other things – also naturally focus on mathematical activities beyond theorems.

Last but not least, a variety of approaches to the history of mathematics using methods from social history have been pursued for several decades now, and continue to give a quasi sociological dimension to the very notion of theorem. Examples of such studies will be presented and discussed at the workshop.

The Workshop was initiated by Prof. Dr. Norbert Schappacher (Fellow 2011/12) and Prof. Dr. Felix Mühlhölzer (Philosophisches Seminar, Göttingen).