Presenter: Ioannis M. Vandoulakis
In modern history and philosophy of mathematics, the concept of “style” of mathematical thinking is used to characterise certain meta-mathematical aspects of mathematical thinking, such as the manière of mathematical writing, preferred mathematical patterns, commitment to a rule or standard, mathematical practices, cognitive attitudes, modes of understanding, etc.
A mathematical style does not generate either a discovery or a demonstration; it is not easily definable and may characterise either an individual or a scholarly community, for instance, the “French-school style” or the “Gottingen’s style,” or the “Euclidean” or “Archimedean style”. It can also be a methodologically defined style (for instance, “experimental” style) or a style defined by a problem.
Alistair Cameron Crombie in his monument work Styles of Scientific Thinking in the European Tradition: The History of Argument and Explanation Especially in the Mathematical and Biomedical Sciences and Arts (London: Gerald Duckworth & Company, 1995) ascribes to the Greeks the achievement of the postulational (or axiomatic style) of mathematical thinking, as exemplified in Euclid’s Elements. In our talk, we will present two historical types of styles of Greek mathematical thinking that cannot be characterised as postulational.
Notably, the (Neo-) Pythagorean style of arithmetic thinking used in developing a proofless version of Greek number theory by genetic constructions (definitions), and the Euclidean style of arithmetic, which can be characterised as genetic, in the sense that number theory is developed from below by using effective procedures of proof. These studies enabled us to approach anew the age-old problem of relation between Greek mathematics and philosophy and reveal an unexpected relation between Greek arithmetic and Eleatic philosophy.
Ioannis VANDOULAKIS is a Senior Lecturer in the history and philosophy of science at the Hellenic Open University and the Open University of Cyprus. Since October 2016, he is a key expert in an EU-funded mainstream project for the reform of General and Secondary Education in Turkmenistan (2016-2020). He holds a Ph.D. in history of mathematics from Moscow M.V. Lomonosov State University. He has been research fellow at the Russian Academy of Science (Moscow) and CNRS (Paris). He has published on history and philosophy of Greek mathematics, history of mathematical logic, philosophy and the foundations of mathematics, general artificial intelligence.
His latest research interests concentrate around questions of mathematical proof, its history and styles of mathematical proving across various cultures. Among his recent works in this area are: (in cooperation with Dun Liu, Chinese Academy of Sciences, China) Navigating across Mathematical Cultures and Times: Exploring the Diversity of Discoveries and Proofs (World Scientific 2017), (in cooperation with P. Stefaneas, National Technical University of Athens, Greece) “Proofs as spatio-temporal processes” (Philosophia Scientiæ, 2014), “Proof-events in History of Mathematics” (Gaņita Bhāratī, 2013), and others.
He currently serves the Editorial Board of Gaņita Bhāratī, the Executive Committee of the International Society for the Interdisciplinary Study of Symmetry (ISIS) and its journal Symmetry: Art and Science.