While Saunders Mac Lane studied for his D.Phil in Göttingen, he heard David Hilbert's weekly lectures on philosophy, talked philosophy with Hermann Weyl, and studied it with Moritz Geiger. Their philosophies and Emmy Noether's algebra all influenced his conception of category theory, which has become the working structure theory of mathematics. His practice has constantly affirmed that a proper large-scale organization for mathematics is the most efficient path to valuable specific results—while he sees that the question of which results are (...) valuable has an ineliminable philosophic aspect. His philosophy relies on the ideas of truth and existence he studied in Göttingen. His career is a case study relating naturalism in philosophy of mathematics to philosophy as it naturally arises in mathematics. Introduction Structures and Morphisms Varieties of Structuralism Göttingen Logic: Mac Lane's Dissertation Emmy Noether Natural Transformations Grothendieck: Toposes and Universes Lawvere and Foundations Truth and Existence Naturalism Austere Forms of Beauty. (shrink)

According to the familiar Quinean understanding of ontological commitment, (1) one undertakes ontological commitments only via theoretical regimentations, and (2) ontological commitments are to be identified with the domain of a theory’s quantifiers. Jody Azzouni accepts (1), but rejects (2). Azzouni accepts (1) because he believes that no vernacular expression carries ontological commitments. He rejects (2) by locating a theory’s commitments with the extension of an existence predicate. I argue that Azzouni’s two theses undermine each other. If ontological commitments follow (...) from predications of existence, then ontological commitments can be expressed in the vernacular via negative existential sentences. (shrink)

One of the complications of a clear evaluation of different positions called “realism” in metaphysics and ontology is discrepancy in terminology and variance of basic intuitions inherent this notion. I recommend a taxonomy that distinguishes three kinds of variance: variance in the sets of conditions for a philosophical position to be acknowledged as actual realism, variance in strength and variance in conceptual frameworks. Within this last case, I propose to differentiate realism in epistemology, in ontology, and in metaphysics, intersecting these (...) three variants with the local/global realism dichotomy. The taxonomy here provided aims at being meta-philosophical; expressive enough for a fine-grained articulation of different philosophical positions in the recent debate, including the question of metaphysical grounding; and impartial, with no need to be connected to the defense of one form or another of the philosophical positions called “realism.”. (shrink)