# The Philosophy and Hermeneutics of Mathematics team

Dr hab. Zbigniew Król, prof. IFiS PAN -Head

The team has been in existence since 1 January 2010.

The Philosophy and Hermeneutics of Mathematics team conducts interdisciplinary research in the philosophy of mathematics (ontology of mathematics, epistemology of mathematics, issues of the development of mathematical knowledge, foundations of mathematics), combining a historical and hermeneutical approach with a formal one. The hermeneutics of mathematics reconstructs and analyses hidden assumptions, background knowledge, and tacit knowledge, showing their role in the creation of even the most formalised theories of contemporary mathematics. Such hidden determinants of the creation of mathematics include – broadly defined – Platonic methods, e.g. related to the use of classical logic, non-predicative concepts, etc.

The research is concerned, among other things, with the mechanisms of development and creation of mathematical knowledge. Analyses of past historical forms of mathematics, mainly ancient Greek mathematics, modern mathematics and the beginnings of modern mathematics, allow us to determine the historical specificity of formalised modern mathematics. The result is, on the one hand, a series of new and relevant information concerning classical philosophical problems related to mathematics and, on the other hand, the necessity and possibility of analysing certain open and hitherto unanalysed mathematical problems in the foundations of mathematics, mathematical truth theory, set theory and category theory.

The team is pursuing the long-term research topic Intuitive Foundations of Mathematics. Intuition versus truth in mathematics.

The research objectives are:

- Reconstruction of the intuitive basis of the creation of mathematics in different historical epochs as well as contemporary mathematics, together with the demonstration of the role of the intuitive basis of mathematics in science and classical philosophical problems (ontology, epistemology, philosophy and methodology of science, philosophical hermeneutics and phenomenology).
- Research activities concerning the development and history of mathematics are only part of the research themes pursued within the Team. Another strand of research is the development of new systems for the foundations of mathematics, including formalised ones, and studying their properties.
- We formulate and develop – in my opinion, extremely promising and mathematically new – concepts of intuitive models in mathematics.

The work also included an NSA-funded, approximately five-year research project entitled Transformations of the Intuitive Foundations of Mathematics and the Historical Changeability of Mathematical Knowledge. The Origin of Infinitary Concepts in Euclidean Geometry and the Process of the Constitution of Mathematical Platonism as a Cause of the Emergence of Modern Science. The research results are presented in Zbigniew Król’s monograph Platonism and the Development of Mathematics: Infinity and Geometry, Publ. IFiS PAN, Warsaw 2015.

It is possible to generalise the results obtained and apply them to the study of the problem of historical variability of mathematical knowledge and to give completely new, substantively justified schemes of the development of mathematics, especially geometry. The research carried out in the Team shows that geometry was first a constructive theory and that the ‘Platonic’ concepts: infinite space, straights, planes, etc., were used relatively late. Previous research does not consider the change of this ‘setting’ for the practice of geometry, so their detection makes it possible to describe and analyse the fundamental differences between ancient and more modern (including contemporary) mathematics. The description of these differences – so far not fully explored – is a prerequisite for the correctness of further reflections on the patterns of development of scientific knowledge and especially of mathematics.

The problems mentioned have not been described in more detail so far because they have yet to be noticed (or have been analysed from a different point of view). The mechanisms detected have relevance for contemporary science and may stimulate the emergence of new directions in the foundations of mathematics.

Expected end results: providing an alternative to the existing analysis of the very widely analysed and discussed in the contemporary philosophy of science and history of science origin of modern science by reconstructing the intuitive foundations of mathematics for ancient, modern and contemporary mathematics, providing a new model for the development of geometry and indicating the substantive usefulness of the notion of an intuitive model in the foundations of modern mathematics.

Adres: Instytut Filozofii i Socjologii PAN,

ul. Nowy Świat 72, pokój 104

00-330 Warszawa

e-mail: zbigkrol@wp.pl

Tel. (22) 657 27 65

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