Theory Choice and Formalization: On the Paper-thin Difference between Formal and Informal Notions in Mathematical Practice

(Philosophy of Mathematics, English)

his talk investigates the selection of axioms for (newly formed) mathematical disciplines and argue that the vocabulary developed within the ''theory choice'' debate in Philosophy of Science offers fruitful concepts for a finer philosophical analysis of the section-process.

There are competing axiomatizations and it is not clear which one does capture the intuitive notion the best and which one yields to a fruitful new field. We want to investigate how we choose a fitting axiomatization. Theory choice in physics is not solely rational, so the status of axiomatizations gets a little relativized, we could have decided to investigate slightly different mathematical theories.

We want specially to focus on what it should mean that an axiomatization fits to the data. We argue that the informal notions predating axiomatized fields can deliver such data and that an axiomatization needs to fit to informally proved cornerstone results of the new field, including those results which deliver fruitful techniques. We focus on inner mathematical thoughts (and less on philosophical reflection on the concepts, which also play a role) and test our ideas in a case-study, namely the shift to transfinite, especially in infinite combinatorics and set theory.

There is an open debate on criteria for new axioms in set theory, some of the so called extrinsic values of axioms fit to the picture the talk is drawing. Joel D. Hamkins refers directly to experiences which were made with (partly due to contingent reasons) established theories. Those theories are often not considered to capture all aspects of the notion of sets (if there is only one such notion).

All in all, this talk can be read in two ways: As a general plea for the incorporation of the mathematical practice in philosophy and second for pluralistic positions in philosophy of mathematics, importing some of the underdeterminations of physical theories to the selection of axiomatizations of (informal) fields of mathematical research.

Chair: Sara Ayhan

Time: 14:00-14:30, 12 September 2018 (Wednesday)

Location: SR 1.005

Deniz Sarikaya

(University of Hamburg, Germany)

I am currently doing my master's in mathematics and preparing my PhD studies in Philosophy at the University of Hamburg (UHH).

I studied philosophy (MA with distinction 2016, BA 2012) and mathematics (BA 2015) at the UHH focusing on philosophy of science / mathematics, logic and discrete mathematics.
My main areas of interests are Philosophy of Science: Science and Society (Wertedebatte, Wissenschaft und Demokratie), Structuralism and Mathematics from all perspectives: I am working in Philosophy of Mathematics (esp. Philosophy of Mathematical Practice), Mathematics Didactics, think about Mathematics from a linguistic perspective.

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