MCMP – Philosophy of Mathematics LudwigMaximiliansUniversität München

 Filosofia
Mathematical Philosophy  the application of logical and mathematical methods in philosophy  is about to experience a tremendous boom in various areas of philosophy. At the new Munich Center for Mathematical Philosophy, which is funded mostly by the German Alexander von Humboldt Foundation, philosophical research will be carried out mathematically, that is, by means of methods that are very close to those used by the scientists.
The purpose of doing philosophy in this way is not to reduce philosophy to mathematics or to natural science in any sense; rather mathematics is applied in order to derive philosophical conclusions from philosophical assumptions, just as in physics mathematical methods are used to derive physical predictions from physical laws.
Nor is the idea of mathematical philosophy to dismiss any of the ancient questions of philosophy as irrelevant or senseless: although modern mathematical philosophy owes a lot to the heritage of the Vienna and Berlin Circles of Logical Empiricism, unlike the Logical Empiricists most mathematical philosophers today are driven by the same traditional questions about truth, knowledge, rationality, the nature of objects, morality, and the like, which were driving the classical philosophers, and no area of traditional philosophy is taken to be intrinsically misguided or confused anymore. It is just that some of the traditional questions of philosophy can be made much clearer and much more precise in logicalmathematical terms, for some of these questions answers can be given by means of mathematical proofs or models, and on this basis new and more concrete philosophical questions emerge. This may then lead to philosophical progress, and ultimately that is the goal of the Center.

 video
A Hypothetical Conception of Mathematics in Practice
Colloquium Mathematical Philosophy, José Ferreirós (Sevilla) gives a talk at the MCMP Colloquium (11 June, 2015) titled "A Hypothetical Conception of Mathematics in Practice". Abstract: The aim of the talk will be to present some of the basic aspects of my approach to mathematical epistemology, developed in the forthcoming book Mathematical Knowledge and the Interplay of Practices (Princeton UP). The approach is agentbased, considering mathematical systems as frameworks that emerge in connection with practices of different kinds, giving rise to new practices. In particular, we shall consider the effects of placing the rather traditional thesis that advanced mathematics is hypothetical – based on 'constitutive,' not representational, hypotheses – in the setting of a web of interrelated practices. Insistence on the coexistence of a plurality of practices, I claim, modifies substantially that thesis and allows for the development of a novel epistemology.

 video
On the Contingency of Predicativism
Colloquium Mathematical Philosophy, Sam Sanders (MCMP) gives a talk at the MCMP Colloquium (16 April, 2015) titled "On the Contingency of Predicativism". Abstract: Following his discovery of the paradoxes present in naive set theory, Russell proposed to ban the vicious circle principle, nowadays called impredicative definition, by which a set may be defined by referring to the totality of sets it belongs to. Russell's proposal was taken up by Weyl and Feferman in their development of the foundational program predicativist mathematics. The fifth `Big Five' system from Reverse Mathematics (resp. arithmetical comprehension, the third Big Five systen) is a textbook example of impredicative (resp. predicative) mathematics. In this talk, we show that the fifth Big Five system can be viewed as an instance of nonstandard arithmetical comprehension. We similarly prove that the impredicative notion of bar recursion can be viewed as the predicative notion primitive recursion with nonstandard numbers. In other words, predicativism seems to be contingent on whether the framework at hand accommodates Nonstandard Analysis, arguably an undesirable feature for a foundational philosophy.

 video
Geometrical Roots of Model Theory: Duality and Relative Consistency
Colloquium Mathematical Philosophy, Georg Schiemer (Vienna/MCMP) gives a talk at the MCMP Colloquium (9 July, 2015) titled "Geometrical Roots of Model Theory: Duality and Relative Consistency". Abstract: Axiomatic geometry in Hilbert's Grundlagen der Geometrie (1899) is usually described as modeltheoretic in character: theories are understood as theory schemata that implicitly define a number of primitive terms and that can be interpreted in different models. Moreover, starting with Hilbert's work, metatheoretic results concerning the relative consistency of axiom systems and the independence of particular axioms have come into the focus of geometric research. These results are also established in a modeltheoretic way, i.e. by the construction of structures with the relevant geometrical properties. The present talk wants to investigate the conceptual roots of this metatheoretic approach in modern axiomatics by looking at an important methodological development in projective geometry between 1810 and 1900. This is the systematic use of the "principle of duality", i.e. the fact that all theorems of projective geometry can be dualized.The aim here will be twofold: First, to assess whether the early contributions to duality (by Gergonne, Poncelet, Chasles, and Pasch among others) can already be described as modeltheoretic in character. The discussion of this will be based on a closer examination of two existing justifications of the general principle, namely a transformationbased account and a (proto)prooftheoretic account based on the axiomatic presentation of projective space. The second aim will be to see in what ways Hilbert's metatheoretic results in Grundlagen, in particular his relative consistency proofs, were influenced by the previous uses of duality in projective geometry.

 video
A Computational Perspective on Metamathematics
Colloquium Mathematical Philosophy, Vasco Brattka (UniBwM Munich) gives a talk at the MCMP Colloquium (29 January, 2015) titled "A Computational Perspective on Metamathematics". Abstract: By metamathematics we understand the study of mathematics itself using methods of mathematics in a broad sense (not necessarily based on any formal system of logic). In the evolution of mathematics certain steps of abstraction have led from numbers to sets of numbers, from sets to functions and eventually to function spaces. Another meaningful step in this line is the step to spaces of theorems. We present one such approach to a space of theorems that is based on a computational perspective. Theorems as individual points in this space are related to each other in an order theoretic sense that reflects the computational content of the related theorems. The entire space is called the Weihrauch lattice and carries the order theoretic structure of a lattice enriched by further algebraic operations. This space yields a mathematical framework that allows one to classify theorems according to their complexity and the results can be essentially seen as a uniform and somewhat more resource sensitive refinement of what is known as reverse mathematics. In addition to what reverse mathematics delivers, a Weihrauch degree of a theorem yields something like a full "spectrum" of a theorem that allows one to determine basically all types of computational properties of that theorem that one would typically be interested in. Moreover, the Weihrauch lattice is formally a refinement of the Borel hierarchy, which provides a wellknown topological complexity measure (and the relation of the Weihrauch lattice to the Borel hierarchy is very much like the relation between the manyone or Turing semilattice and the arithmetical hierarchy). Well known classes of functions that have been studied in algorithmic learning theory or theoretical computer science have meaningful and very succinct characterizations in the Weihrauch lattice, which underlines that this lattice yields a very natural model. Since the Weihrauch lattice is defined using a concrete model, the lattice itself and theorems as points in it can also be studied directly using methods of topology, descriptive set theory, computability theory and lattice theory. Hence, in a very true and direct sense the Weihrauch lattice provides a way to study metamathematics without any detour over formal systems and models of logic.

 video
Symmetry and Mathematicians' Aesthetic Preferences: a Case Study
Colloquium Mathematical Philosophy, Irina Starikova (Sao Paulo) gives a talk at the MCMP Colloquium (8 January, 2015) titled "Symmetry and Mathematicians' Aesthetic Preferences: a Case Study". Abstract: Symmetry plays an important role in some areas of mathematics and has traditionally been regarded as a factor of visual beauty. In this talk I explore the ways that symmetry contribute to mathematicians’ aesthetics judgments about mathematical entities and representations. I discuss an example from algebraic graph theory. Comparing two isomorphic drawings of the Petersen graph, I argue that we need to refine the question by distinguishing between perceptual and intellectual beauty and by noting that some mathematical symmetries are revealed to us in diagrams while others are hidden.

 video
An Aristotelian continuum
Colloquium Mathematical Philosophy, Stewart Shapiro (Ohio) gives a talk at the MCMP Colloquium (18 December, 2014) titled "An Aristotelian continuum". Abstract: Geoffrey Hellman and I are working on a pointfree account of the continuum. The current version is “gunky” in that it does not recognize points, as part of regions, but it does make essential use of actual infinity. The purpose of this paper is to produce a more Aristotelian theory, eschewing both the actual existence of points and infinite sets, pluralities, or properties. There are three parts to the talk. The first is to show how to modify the original gunky theory to avoid the use of (actual) infinity. It is interesting that there are a number of theorems in the original theory (such as the existence of bisections and differences, and the Archimedean property) that have to be added, as axioms. The second part of the talk is to take the “potential” nature of the usual operations seriously, by using a modal language. The idea is that each “world” is finite; the usual operations are understood as possibilities. This part builds on some recent work on set theory by Øystein Linnebo. The third part is an attempt to recapture points, but taking the notion of a potentially infinite sequence seriously.