In this introduction we discuss the motivation behind the workshop “Towards a New Epistemology of Mathematics” of which this special issue constitutes the proceedings. We elaborate on historical and empirical aspects of the desired new epistemology, connect it to the public image of mathematics, and give a summary and an introduction to the contributions to this issue.
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. (shrink)
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. (shrink)
A brief presentation on occasion of Carnap’s visit to a meeting of the Indiana Philosophical Association at the same place exactly 75 ago; based on research in the archives of the IPA, the University of Pittsburgh, and the University of Konstanz. The lecture provides some hitherto unknown biographical background, a summary of Carnap's main arguments, and assigns it a place in Carnap's oeuvre.
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.  Jaffe, Arthur, Quinn, Frank. "Theoretical Mathematics: Towards a synthesis of mathematics and theoretical physics," Bulletin of the American Mathematical Society NS 29:1, 113.  Quinn, Frank. Contributions to a Science of Mathematics, manuscript, 98pp.  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. (shrink)