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.

While Gödel’s first theorem remains valid under substitution of various provability predicates, Gödel’s second theorem does not. This is one reason to label G1 as “extensional” but to call G2 “intensional.” Although this asymmetry between G1 and G2 is known for long, no satisfying account of G2’s intensionality has been put forward. After briefly reviewing the discussion so far, the paper presents a new analysis based on two observations. First, the underestimated role of provable closure under modus. Provable closure under modus ponens—usually listed as the second derivability condition—or similar requirements, do not get much attention since their proof isn’t perceived as much of challenge. Closer inspection shows, however, that the requirement of provable closure under modus ponens interacts differently with different formalized proof predicates and has thus a direct bearing on the discussion of G2’s intensionality. Second, in order to establish the co-extensionality of various proof predicates and hence the extensionality of G1, the role informal consistency assumption play goes largely unnoticed and unanalyzed. Closer inspection shows, however, that if such informal consistency assumptions are being made precise, the former distinction between G1 and G2 becomes blurry and G2’s intensionality can be made go away. Both observation then support the thesis that the traditional extensional/intensional distinction is not as robust as the received view believed it to be.