851-0125-63L  Images of Mathematics

SemesterAutumn Semester 2016
LecturersM. Hampe, A. Schubbach
Periodicitynon-recurring course
Language of instructionGerman
CommentParticularly suitable for students of D-MATH

AbstractThe lecture series "Images of Mathematics" deals with the formalization of the objects and the logical language of mathematics from Hilbert to Gödel and considers its consequences in view of our conception of mathematical practice and knowledge, the limits of calculability and computability in mathematics, and the relation between the logical proof procedures and the involved intuitive aspects.
ObjectiveThe lecture series will present philosophical problems of theoretical mathematics in the 20th century and will discuss the consequences of formalization and axiomatization. It aims at a critical reflection on the modern images of mathematics.
ContentHow we understand Mathematics is probably strongly influenced by the Mathematics lessons we participated in during our school days. The common image of mathematics is therefore often characterized by the impression of a very stable form of knowledge with clear-cut problems and suitable recipes for finding the solution. It is a very static image which is very much in conflict with the rapid series of innovations that the discipline has experienced especially since the 19th century: Mathematics as a field of research has been highly innovative and even revolutionary as few other scientific disciplines in the last 200 hundred years.

These mathematical innovations did not only contribute to a progress amassing more and more knowledge. They very often changed how mathematicians conceived of their discipline. Even a contribution to a specific research question that appears at first sight to be minor can sometimes establish new connections to other fields, found a whole research field of its own or introduce new methods thereby changing the whole image of mathematics in the same way that a small addition to a picture can alter radically what we take it to represent.

The lecture series "Images of Mathematics" deals with a few moments in the history of the scientific discipline since the middle of the 19th century when the image of mathematics changed. In particular, it focuses on the consequences of the fact that in the 19th century mathematics started to not only reflect on their own conceptual and methodological foundations in a general manner (which had been done since the dawn of mathematics and was especially a philosophical task), but to formalize them in a strict, mathematical way: the objects of mathematics, its logical language and its proof procedures. Through Cantor's set theory, the mathematical treatment of logic since Boole and especially through Frege and the formalization of its axioms in a wide ranging discussion involving Zermelo, Fraenkel and others, this self-reflexive stance came to the fore.

Yet, the deeper mathematics dug into its foundations, the more radical the problems became. Finally, the optimistic Hilbert program of laying the foundation of mathematics within mathematics and of proving its own consistency as well as its completeness contributed to clarifying of the foundation of mathematics primarily insofar as it was doomed to failure. Gödel proved his famous incompleteness theorems and thereby dismissed at the same time the formalist attempt to reduce mathematical truth to logical provability. His work resulted in detailed insights in the precariousness of the foundation of mathematics and further numerous of productive consequences within mathematics.

Moreover, Gödel's theorems open many far-reaching and intriguing questions in view of our image of mathematics, questions concerning the conception of mathematical practice and knowledge, the limits of calculability of mathematics and the possible role of computability and machines in mathematics, the relation between the logical proof procedures and the involved intuitive aspects. In short, the image of mathematics is not as static as we sometimes expect it to be, it was radically redrawn by the mathematicians of the 20th century and has since then again been open to diverging interpretations.
LiteratureFor further reading (optional): Mark van Atten and Juliette Kennedy, Gödel's Logic, in: Handbook of the History of Logic, Vol 5: Logic from Russell to Church, ed. by Dov M. Gabbay and John Woods, Amsterdam 2009, 449-509; Jack Copeland et al. (eds.), Computability. Turing, Gödel, Church, and beyond, Cambridge 2013; Ian Hacking, Why is there philosophy of mathematics at all? Cambridge 2014; Pirmin Stekeler-Weithofer, Formen der Anschauung. Eine Philosophie der Mathematik, Berlin 2008; Christian Tapp, An den Grenzen des Endlichen. Das Hilbertprogramm im Kontext von Formalismus und Finitismus, Heidelberg 2013.