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)

We clarify the respective roles fixed points, diagonalization, and self- reference play in proofs of Gödel’s first incompleteness theorem.

Guided by questions of scope, this paper provides an overview of what is known about both the scope and, consequently, the limits of Gödel’s famous first incompleteness theorem.

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.

Remarks on the emergence of the concept “Spielraum” as a foundation for probability theory.

Slides for the first tutorial on Gödel's incompleteness theorems, held at UniLog 5 Summer School, Istanbul, June 24, 2015.

Slides for the second tutorial on Gödel's incompleteness theorems, held at UniLog 5 Summer School, Istanbul, June 24, 2015.

Slides for the third tutorial on Gödel's incompleteness theorems, held at UniLog 5 Summer School, Istanbul, June 24, 2015.

A short biography of Johannes von Kries, followed by a bibliography of his works, containing more than 70 previously unknown works.

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. (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)

While the Gödel centenary year 2006 triggered a lot of conference and workshop activity on Gödel, the years leading to it stand out by exhibiting several excellent publications on Gödel's life and work, most notably the completion of the Kurt Gödel Collected Works series . The two volumes of Kurt Gödel. Wahrheit & Beweisbarkeit, written in German and edited by E. Köhler et al., constitute something like the ‘German-Austrian contribution’ to this renewal of interest in Gödel's legacy, even though not (...) all of the articles in the volumes actually have German or Austrian authors.2 Indeed, as P. Weibel explains in the preface of the first volume, one of the intentions of the editors was to present Gödel as part of the cultural history of Austria. At the same time, the volumes highlight the contrast between Gödel's national reception—or lack thereof—and the way in which he was perceived and appreciated internationally, and although this discrepancy is now fortunately a thing of the past, its history is worth documenting. In this respect the two volumes will be of lasting value and importance, and the editors must be given credit not just for recording Gödel's delayed acceptance by Austrian politicians and university headquarters, but also for promoting Gödel's case. This being said, there is something extravagant and slightly queer about the two Kurt Gödel. Wahrheit & Beweisbarkeit volumes. Parts of them seem to be written for a generally educated audience, while other parts will be understandable only to logicians and philosophers who specialize in the topics covered. The volumes consist partly of biographical essays, interviews about Gödel, photos of him and his family, and letters by or to him, but they also include several non-historical articles which use …. (shrink)

Immanuel Kants "Grundlegung" ist ohne Zweifel einer der bedeutendsten Texte der abendlandischen Moralphilosophie. Sie gilt mitunter allerdings auch als einer der "dunkelsten" Texte Kants - wenn nicht sogar der abendlandischen Philosophie uberhaupt. Der Kommentar macht deutlich, dass die vorgebliche Dunkelheit im Wesentlichen der Tatsache geschuldet ist, dass der Text seit langem im Schatten der falschen Annahme gelesen wird, Kant wolle eine Begrundung (oder Rechtfertigung) des Sittengesetzes liefern, um damit den moralischen Skeptizismus abzuwehren. Liest man ihn stattdessen unter der (auch historisch (...) angemesseneren) Perspektive, dass Kant mit dem kategorischen Imperativ 'nur' eine neue Formel fur das Prinzip der allgemein im Schwange gehenden Sittlichkeit 'angeben und rechtfertigen' will, um so den moralischen Eudaimonismus durch eine Moral der Autonomie zu ersetzen - dann zeigt sich, dass der Text nicht bloss ein philosophisches, sondern zugleich ein schriftstellerisches Meisterwerk ist. (shrink)