MCMP – Philosophy of Mathematics

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.

Kate Hodesdon (Nancy) gives a talk at the Workshop on Mathematics: Objectivity by Representation (11 November, 2014) titled "Discernibility from a countable perspective". Abstract: In this talk I discuss formal methods for discerning between uncountably many objects with a countable language, building on recent work of James Ladyman, Øystein Linnebo and Richard Pettigrew. In particular, I show how stability theory provides the resources to characterize theories in which this is possible, and discuss the limitations of the stability theoretic approach.

Marco Panza (Paris I) gives a talk at the Workshop on Mathematics: Objectivity by Representation (11 November, 2014) titled "What are the challenges of Benacerrafs Dilemma? A Reinterpretation". Abstract: Despite its enormous influence, Benacerraf's dilemma admits no standard, unanimously accepted, version. This mainly depends on Benacerraf's having originally presented it in a quite colloquial way, by avoiding any compact, somehow codified, but purportedly comprehensive formulation. But it also depends on Benacerraf's appealing, while expounding the dilemma, to so many conceptual ingredients so as to spontaneously generate the feeling that most of them are in fact inessential for stating it. It is almost unanimously admitted that the dilemma is, as such, independent of the adoption of a causal conception of knowledge, though Benacerraf appealed to it. This apart, there have not been, however, and still there is no agreement about which of these ingredients have to be conserved so as to get a sort of minimal version of the dilemma, and which others can, rather, be left aside (or should be so, in agreement with an Okkamist policy). My purpose is to come back to the discussion on this matter, with a particular attention to Field's reformulation of the problem, so as to identify two converging and quite basic challenges, addressed by Benacerraf's dilemma to a platonist and to a combinatorialist (in Benacerraf's own sense) philosophy of mathematics, respectively. What I mean by dubbing these challenges 'converging' is both that they share a common kernel, which encompasses a challenge for any plausible philosophy of mathematics, and that they suggest (at least to me) a way-out along similar lines. Roughing these lines out is the purpose of the two last part of the talk.

Helmut Schwichtenberg (LMU) gives a talk at the MCMP Colloquium (5 December, 2013) titled "Remarks on the foundations of mathematics". Abstract: We consider minimal logic with implication and universal quantification over (typed) object variables. Free type and predicate parameters may occur. For mathematics we need (i) data (the Scott - Ershov partial continuous functionals) and (ii) predicates (defined inductively or coinductively). In this setting we can define (Leibniz) equality, falsity and the missing logical connectives (negation, disjunction, existential quantification, conjunction). Ex-falso-quodlibet can be proved. Using Kreisel's (modified) realizability we can (even practically) extract computational content from proofs, and (internally) prove soundness.

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.