MCMP – Philosophy of Mathematics

Andrey Bovykin (Bristol) gives a talk at the MCMP Colloquium (16 January, 2013) titled "Recent metamathematical wonders and the question of arithmetical realism". Abstract: Metamathematics is the study of what is possible or impossible in mathematics, the study of unprovability, limitations of methods, algorithmic undecidability and "truth". I would like to make my talk very historical and educational and will start with pre-Godelean metamathematics and the first few metamathematical scenarios that emerged in the 19th century ("Parallel Worlds", "Insufficient Instruments", "Absence of a Uniform Solution", and "Needed Objects Are Yet Absent"). I will also mention some metamathematical premonitions from the era before Gauss that are the early historical hints that a mathematical question may lead to a metamathematical answer, rather than a "yes" or "no" answer. (These historical quotations have recently been sent to me by a historian Davide Crippa.) Then I am planning to sketch the history and evolution of post-Godelean metamathematical ideas, or, rather, generations of ideas. I split post-Godelean history of ideas into five generations and explain what people of each generation believed in (what are the right objects of study? what are the right questions? what are the right methods?). The most spectacular, and so far, the most important completed period in the history of metamathematics started in 1976 with the introduction of the Indicator Method by Jeff Paris and the working formulation of the Reverse Mathematics Programme by Harvey Friedman. I will describe this period (what people believed in that era) in some detail. We now live in a new period in metamathematics, quite distinct from the ideas of 1980s. I will speak about the question of universality of Friedman's machinery, the theory of templates, meaninglessnessisation (with an entertaining example of eaninglessnessisation), Weiermann's threshold theory and some new thoughts about the space of all possible strong theories. I will concentrate in most detail on the question of the Search for Arithmetical Splitting (the search for "equally good" arithmetical theories that contradict each other). and sketch some possible ways that may lead to the discovery of Arithmetical Splitting. There are people, who strongly deny the possibility of Arithmetical Splitting, most often under the auspices of arithmetical realism: "there exists the true standard model of arithmetic". They usually cite Godel's theorem as existence of ***true*** but unprovable assertions and concede only to some epistemological difficulties in reaching the "arithmetical truth". I will mention seven common arguments usually quoted by anti-Arithmetical-Splitting people and explain why it seems that all of these arguments don't stand. Historically and very tentatively, I would like to suggest that anti-Arithmetical-Splitting views should perhaps be considered not a viable anti-Pluralist philosophy of mathematics, but rather a conservative resistance to the inevitable new generation of metamathematical wonders. So anti-Arithmetical-Splitting views are more in line with such historical movements as "rejection of complex numbers", "resistance to non-Euclidean geometries in 1830s-1850s", and "refusal to accept concepts of abstract mathematics in the second half of the 19th century" and other conservative movements of the past. Arithmetical Splitting has not been reached yet, but we are well equipped to discuss it before its arrival.

Steve Awodey (CMU) gives a talk at the MCMP Colloquium (16 July, 2014) titled "The Univalence Axiom". Abstract: In homotopy type theory, the Univalence Axiom is a new principle of reasoning which implies that isomorphic structures can be identified. I will explain this axiom and consider its background and consequences, both mathematical and philosophical.

Paolo Mancosu (UC Berkeley) gives a talk at the MCMP Colloquium (8 May, 2014) titled "In Good Company? On Hume's Principle and the assignment of numbers to infinite concepts.". Abstract: In a recent article (Review of Symbolic Logic 2009), I have explored the historical, mathematical, and philosophical issues related to the new theory of numerosities. The theory of numerosities provides a context in which to assign numerosities to infinite sets of natural numbers in such a way as to preserve the part-whole principle, namely if a set A is properly included in B then the numerosity of A is strictly less than the numerosity of B. Numerosities assignments differ from the standard assignment of size provided by Cantor’s cardinality assignments. In this talk, I generalize some specific worries emerging from the theory of numerosities to a line of thought resulting in what I call a ‘good company’ objection to Hume’s principle. The talk has four main parts. The first takes a historical look at nineteenth-century attributions of equality of numbers in terms of one-one correlations and argues that there was no agreement as to how to extend such determinations to infinite sets of objects. This leads to the second part where I show that there are countably infinite many abstraction principles that are ‘good’, in the sense that they share the same virtues of HP and from which we can derive the axioms of second order arithmetic. The third part connects this material to a debate on Finite Hume Principle between Heck and MacBride and states the ‘good company’ objection. Finally, the last part gives a tentative taxonomy of possible neo-logicist responses to the ‘good company’ objection and makes a foray into the relevance of this material for the issue of cross-sortal identifications for abstractions.

Simon Huttegger (UC Irvine) gives a talk at the MCMP Colloquium (8 May, 2014) titled "Learning Experiences, Expected Inaccuracy, and the Value of Knowledge". Abstract: I argue that van Fraassen's reflection principle is a principle of rational learning. First, I show that it follows if one wants to minimize expected inaccuracy. Second, the reflection principle is a consequence of a postulate describing genuine learning situations, which is related to the value of knowledge theorem in decision theory. Roughly speaking, this postulate says that a genuine learning experience cannot lead one to foreseeably make worse decisions after the learning experience than one could already have made before learning.

Eric Schliesser (Ghent) gives a talk at the MCMP Colloquium (25 June, 2014) titled "Anti-Mathematicism and Formal Philosophy". Abstract: Hannes Leitgeb rightly claims that "contemporary critics of mathematization of (parts of) philosophy do not so much put forward arguments as really express a feeling of uneasiness or insecurity vis-à-vis mathematical philosophy." (Leitgeb 2013: 271) This paper is designed to articulate arguments in the place of that feeling of uneasiness. The hope is that this will facilitate more informed discussion between partisans and critics of formal philosophy. In his (2013) paper Leitgeb articulates and refutes one argument from Kant against formal philosophy. This paper will show, first, that Kant's argument as part of a much wider 18th century debates of formal methods in philosophy (prompted by success of Newton and anxiety over Spinoza). Now, obviously 'philosophy' has a broader scope in the period, so I will confine my discussion of arguments about formal methods in aeas we still consider 'philosophical' (metaphysics, epistemology, moral philosophy, etc.) In order to facilitate discussion I offer the fllowing taxonomy of arguments: (i) the global strategy, by which we mean that that the epistemic authority and security of mathematical applications as such are challenged and de-privileged; (ii) the containment strategy, by which the successful application of mathematical technique is restricted only to some limited domain; (iii) non-epistemic theories, by which the apparent popularity of mathematics within some domain of application is explained away in virtue of some non-truth-tracking features. The non-epistemic theory is generally part of a debunking strategy. I will offer examples from Hume and Mandeville of all three strategies. The second main aim is to articulate arguments that call attention to potential limitations of formal philosophy. By 'limitation' I do not have in mind formal, intrinsic limits (e.g., Godel). I explore two: (A) firs, formal philosophers often assume a kind of topic neutrality or generality when justifying their methods (even if this idea has been challenged from within Dutilh Novaes 2011). But this means that formal methods are not self-justifying (outside logic and mathematics, perhaps) and unable to ground their own worth; it follows straightforwardly that for such grounding formal approaches require substantive (often normative) non-formal premises. (B) Second, Leitgeb (2013: 274-5) insightfully discusses the ways in which formal approaches like any other method, may be abused. But because abuses with esoteric techniques may be hard to detect by bystanders (philosophical and otherwise) absent other means of control and containment, there is a heavy responsibility on practitioners of formal philosophy to develop institutional and moral safeguards that are common in, say, engineering and medical sciences against such abuses. Absent safeguards, formal philosophers require a strong collective self-policing ethos; it is unlikely that current incentives promote such safeguards.

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.