PHILOSOPHY AND HERMENEUTICS OF MATHEMATICS RESEARCH TEAM

The team exists since 1 Jan. 2010.

Team leader is Prof. Zbigniew Król, Ph. D., D. Sc.

The Philosophy and Hermeneutics of Mathematics Research Team is doing interdisciplinary research work in philosophy of mathematics (ontology and epistemology of mathematics, factors in advancement of mathematical knowledge, foundations of mathematics) combining historical, hermeneutical and formal methods. Hermeneutics of mathematics strives to reconstruct and analyze hidden assumptions, background knowledge and tacit knowledge in order to show their role in creation of even the most extremely formalized theories of contemporary mathematics. Among such hidden determinants in creation of mathematics are broadly conceived platonic methods involved, for instance, in most applications of classical logic, or non predicative concepts, etc. Although considerable emphasis is put on historical context of mathematics, this is completely unrelated to “cultural and historical relativism”; cf. http://www.calculemus.org/zjazd-filoz2008/okr-stol/index.html (in Polish).

Research is concerned mainly with the inner workings of creation and development of mathematical knowledge. The analysis of old historical forms of mathematics, esp. those of Ancient Greece, early modern period, and at the outset of contemporary mathematics allows to pinpoint particular historical features of contemporary formalized mathematics. The upshot is an abundance of new and important information on classical philosophical issues concerning mathematics on one hand, and on the other an obvious necessity and possibility to analyze certain array of open mathematical problems, never analyzed before, in the foundations, the theory of mathematical truth, and set and category theories.

The team is engaged in multiannual research project called *Intuitive foundations of mathematics. Intuition and truth in mathematics*.

Research areas are:

(1) Reconstruction of intuitive basis for mathematical creations in various historical periods and at present showing involvement of intuitive foundations of mathematics in science and classical philosophy (ontology, epistemology, philosophy and general methodology of science, philosophical hermeneutics and phenomenology).

(2) Research in development and history of mathematics is only part of the program, another part being new systems of foundations, sometimes formalized, and investigation of properties of such systems.

(3) Inventing and improving new and mathematically promising concept of intuitive models in mathematics.

(4) A five-year project called *Changes in intuitive foundations of mathematics and historical variability of mathematical knowledge. Genesis of infinitary concepts in euclidean geometry and constitution process of mathematical platonism as a reason for modern science*.

The results can be generalized and applied to research of the problem of historical variability of mathematical knowledge and there is a possibility to propose rigorously justified and entirely new schemes of mathematical change, especially in geometry. The research shows that geometric theory was at first constructive, and platonic notions of infinite space, straight line and surface came to use much later. No research to date was done on this change of scene for producing geometry yet only after that one can analyze fundamental dissimilarities between ancient and modern mathematics, including early modern one. Correct description of these differences is a precondition for any further research on the schemes of development of scientific knowledge and, most of all, mathematics.

These problems were never described in any detail, since nobody saw them, or were analyzed from entirely different point of view. The workings discovered thus far are important clue for modern science and can stimulate new directions in the foundations of mathematics

Expected final results: new alternative analysis of origins of modern science by way of reconstruction of intuitive foundations of mathematics for ancient, modern and contemporary mathematics; new model for the development of geometry; demonstration of instrumental value of the concept of intuitive model for foundations of modern mathematics.

Address: Instytut Filozofii i Socjologii PAN,

72 Nowy Świat St. Room 105.

00-330 Warsaw

e-mail: zbigkrol@wp.pl

phn. (22) 657 27 65, fax. 826 78 23