Husserl and mathematical practice: Eidetic variation, anticipations and Wesenschau.

(Logic & Philosophy of Mathematics, English)

n this talk, we want to sketch crucial parts of the philosophy of mathematics of Edmund Husserl and analyze in how far it fits into the quasi-empirical viewpoint on mathematics. We will argue that his approach draws a picture very close to mathematical practice and that this picture allows us to import concepts from philosophy of science into philosophy of mathematics in a very useful manner.

In 1931 Kurt Gödel shows that every consistent, effectively axiomatized theory containing ''some arithmetic'', is incomplete. This means that there is a formula in the language of the theory, which cannot be proven and whose negation cannot be proven either. We often say that this proves the impossibility of the fulfilment of Leibniz' dream of a lingua characteristica and a calculus ratiocinator. Gödel himself drew in his posthumously published notes ''The modern developments of the foundations of mathematics in the light of philosophy'' (1961?) a rather different conclusion. While we cannot decide every meaningful sentence purely algorithmically Gödel thought that we can extend our axiom systems by reflection on the basic concepts whenever we need to decide such statements. Gödel thought for instance that we will find extensions of ZFC settling the continuums hypothesis (CH). To do this Gödel refers to some ideas by Husserl contained in his ''Formal and Transcendental Logic'' (1929).

Husserl believed that what reaches our sense organs is always underdetermined and, in a sense, informed by our history. This does not only hold for concrete objects but for abstract objects as well. Hence those objects -- according to Husserl -- might be underdetermined, we might form anticipations, which can be confirmed or falsified. We will argue that this allows us to draw parallels from the development of theories in physics and mathematics. Both in foundational endeavors, especially modern set theory but also while defining new notions.

Chair:

Time: 17:00-17:30, 18 September 2019 (Wednesday)

Location: SR 1.007

Deniz Sarikaya

(University of Hamburg, Germany)

I am currently doing my PhD studies in Philosophy at the University of Hamburg (UHH). I studied philosophy (MA 16 & BA 12) and mathematics (MSc 19 & BSc 15) 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 Education and think about Mathematics from a linguistic perspective.

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