On Mathematical Structuralism. A Theory of Unlabeled Graphs as Ante Rem Structures Video

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 model-theoretic 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 model-theoretic 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 model-theoretic in character. The discussion of this will be based on a closer examination of two existing justifications of the general principle, namely a transformation-based account and a (proto-)proof-theoretic 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.

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 agent-based, 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.

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.

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 well-known topological complexity measure (and the relation of the Weihrauch lattice to the Borel hierarchy is very much like the relation between the many-one or Turing semi-lattice 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.

Stanislav O. Speranski (Sobolev Institute of Mathematics) gives a talk at the MCMP Colloquium (4 December, 2014) titled "Quantified probability logics: how Boolean algebras met real-closed fields". Abstract: This talk is devoted to one interesting probability logic with quantifiers over events — henceforth denoted by QPL. That is to say, the quantifiers in QPL are intended to range over all events of the probability space at hand. Here I will be concerned with fundamental questions about the expressive power and algorithmic content of QPL. Note that these two aspects are closely connected, and in fact there exists a trade-off between them: whenever a logic has a high computational complexity, we expect some natural higher-order properties to be definable in the corresponding language. For each class of spaces, we introduce its theory, viz. the set of probabilistic sentences true in every space from the class. Then the theory of the class of all spaces turns out to be of a huge complexity, being computably equivalent to the true second-order arithmetic of natural numbers. On the other hand, many important properties of probability spaces — like those of being finite, discrete and atomless — are definable by suitable formulas of our language. Further, for any class of atomless spaces, there exists an algorithm for deciding whether a given probabilistic formula belongs to its theory or not; moreover, this continues to hold even if we extend our formalism by adding quantifiers over reals. To sum up, the proposed logic and its variations, being attractive from the viewpoint of probability theory and having nice algebraic features, prove to be very useful for investigating connections between probability and logic. In the final part of the talk I will compare QPL with other probabilistic languages (including Halpern's first-order logics of probability and Keisler's model-theoretic logics with probability quantifiers), discuss analogies with research in Boolean algebras and show, in particular, how QPL can serve for classification of probability spaces.