Workshop: Reflection in Philosophy and Mathematics
Idea & Motivation
Since Gödel’s incompleteness theorems one of the central topics in philosophy of mathematics is the question concerning the status of independent statements. In the general case there is no clear answer to the question about the truth of some independent statements in mathematics. However, there are independent statements, which can be recognised to be true. Those sentences are soundness statements of the underlying systems. Through reflection on these soundness assumptions we can extend our formal systems in well-motivated fashion.
Therefore, from a meta-mathematical perspective reflection is an intuitive method to extend sound formal systems. Reflection is related to questions of great philosophical significance, such as the question about the epistemic status of the reflection principles adopted to extend the sound systems. In particular, it is debated whether such principles are justified, and if yes, on what grounds. Another related question concerns the possibility of a priori knowledge of statements in the new extended system entirely via reflection. The process of reflection and the use of reflection principles provide a deep connection between mathematics and philosophy. The workshop is devoted to the discussion of “reflection” from mathematical, logico-philosophical and epistemological perspectives. In particular, we will be interested in the following questions: What is the epistemic status of reflection principles? What is the connection between proof-theoretic reflection principles, known from the application over arithmetical theories, and set-theoretic reflection principles? What role do non-mathematical concepts, such as truth or necessity, play in the process of reflection?