Bernd Buldt

To discuss the developments of mathematics that have to do with the introduction of new objects, we distinguish between ‘Aristotelian’ and ‘non-Aristotelian’ accounts of abstraction and mathematical ‘top-down’ and ‘bottom-up’ approaches. The development of mathematics from the 19th to the 20th century is then characterized as a move from a ‘bottom-up’ to a ‘top-down’ approach. Since the latter also leads to more abstract objects for which the Aristotelian account of abstraction is not well-suited, this development has also lead to a decrease of visualizations in mathematical practice.

Philosophy of Mathematics has become a well-established field of philosophical inquiry. And while it is quite common to attribute philosophers before Kant, from Plato through Locke, a philosophy of mathematics, the term itself was not coined before 1800. A longish paper underlying this talk traces the history of the new term and investigates the reasons philosophers and mathematicians had for adopting a new discipline with that name. In terms of its underlying methodology, the paper combines an evaluation of bibliometric data with an analysis of the philosophic filiations among the various authors. Due to time constraints, the actual talk focuses primarily on the early history of the discussion as it took place in Germany, with an emphasis on Fries and Fichte; other authors, like Schelling, Hegel, Krause, and their students, receive mention as well. Earlier developments outside Germany and the development around 1900 find summative treatment only. At the end, two theses will be defended. First, it can be shown that the new term was introduced around 1800 in response to Kant’s philosophy and that it’s revival around 1900 is again caused by, broadly speaking, Neo-Kantian considerations. Second, some paradigmatic decisions on how to conceive of mathematics philosophically are tied to these earlier, Kantian reflections but have lost their justification in the meantime.

Frank Quinn of Jaffe-Quinn fame worked out the basics of his own account of how mathematical practice should be described and analyzed, partly by historical comparisons with 19th century mathematics, partly by an analysis of contemporary mathematics and its pedagogy. Despite his claim that for this task, "professional philosophers seem as irrelevant as Aristotle is to modern physics," this philosophy talk will provide a critical summary of his main observations and arguments. The goal is to inject some of Quinns remarks to the current conversation on mathematical practice. [1] Jaffe, Arthur, Quinn, Frank. "Theoretical Mathematics: Towards a synthesis of mathematics and theoretical physics," Bulletin of the American Mathematical Society NS 29:1, 113. [2] Quinn, Frank. Contributions to a Science of Mathematics, manuscript, 98pp. [3] Quinn, Frank. "A revolution in mathematics? What really happened a century ago and why it matters today," Notices American Mathematical Society 59:1 31-37.