MCMP – Philosophy of Mathematics

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.

Manuel Rebuschi (Poincaré Archives, University of Lorraine, Nancy) gives a talk at the Workshop on Mathematics: Objectivity by Representation (11 November, 2014) titled "IF epistemic logic and mathematical knowledge". Abstract: Can epistemic logicstate anything interesting about the epistemology of mathematics? That's one of Jaakko Hintikka’s claims. Hintikka was not only the founder of modal epistemic logic (1962), since he also worked on the foundations of mathematics (1996). Using what he calls "second generation" epistemic logic (2003), i.e. independence-friendly (IF) epistemic logic, Hintikka revisits the epistemology of mathematics, and in particular the debate between classical and intuitionistic mathematics (2001). The aim of the talk is to show that Hintikka is right regarding IF epistemic logic, for such a logic enables us to account for interesting features of mathematical knowledge. However, the path is not as easy as that Hintikka suggests. I will show that the well-known issue of logical omniscience directly threatens the understanding of intuitionism offered by IF epistemic logic.

Stanislav O. Speranski (Sobolev Institute of Mathematics) gives a talk at the MCMP Colloquium (6 November, 2014) titled "A useful method for obtaining alternative formulations of the analytical hierarchy". Abstract: In mathematical philosophy one often employs various formal systems and structures for solving philosophical tasks. In particular, many important results in Kripke's theory of truth and the like rest on definability techniques from second-order arithmetic. With this in mind, I will present one useful method for obtaining alternative formulations of the analytical hierarchy. The latter plays a key role in foundations of mathematics and theory of computation, being the generally accepted classification of undecidable problems which capture the truth predicate for first-order arithmetic of natural numbers, and whose computational complexities are less than that of second-order true arithmetic. In the course of the presentation I will mention some relevant contributions of J. Robinson, H. Putnam, J.Y. Halpern, I. Korec and others. Further applications, including those dealing with probabilistic logics, will be discussed in the final part of the talk.

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.

Florian Steinberger (MCMP/LMU) gives a talk at the Workshop on Mathematics: Objectivity by Representation (11 November, 2014) titled "Three ways in which logic might be normative". Abstract: Logic, the tradition has it, is, in some sense, normative for reasoning. Famously, the tradition was challenged by Gilbert Harman who argued that there is no interesting general normative link between facts about logical consequence and norms of belief formation and revision. A number of authors (e.g. John MacFarlane and Hartry Field) have sought to rehabilitate the traditional view of the normative status of logic against Harman. In this paper, I argue that the debate as a whole is marred by a failure of the disputing parties to distinguish three different types of normative assessment, and hence three distinct ways in which logic might be said to be normative. I show that as a result of their failure to appreciate this three-fold distinction, authors have been talking past one another. Finally, time-permitting, I show how each of the three types of normative assessments relate to broader epistemological commitments, specifically commitments within the internalism/externalism debate.

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.

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.

Christopher Menzel (Texas A&M University) gives a talk at the MCMP Colloquium (18 June, 2014) titled "Haecceities and Mathematical Structuralism". Abstract: It is well-known that some earlier versions of mathematical structuralism (notably from Resnik and Shapiro) appeared to be committed to a rather strong form of the Identity of Indiscernibles (II) that is falsified by the existence of structures like the complex field that admit of non-trivial automorphisms, or symmetries. In light of more recent work (notably, by MacBride, Ketland, Shapiro, Ladyman, and Leitgeb and Ladyman), it is widely accepted that the mathematical structuralist is not committed to II and that, in fact, the principle can be rejected outright on robustly structuralist grounds. I accept a qualified form of this view but I don't think the issue is as cut and dried as it might appear. In a 2007 Analysis article, José Bermúdez suggests that a strong version of II is still in play for the structuralist that can meet the challenge of non-trivial symmetries. The key to the proposal (as I will interpret it) lies in allowing identity properties, or haecceities, like being identical to c (for an arbitrary complex number c, say) to count as structural properties. Typically, structuralists dismiss such properties as obviously non-structural. I will argue to the contrary that haecceities can be viewed as properly structural and, in some circumstances, can serve as legitimate properties for discerning otherwise indiscernible “positions” in structures. Drawing on the model theoretic concept of an expansion, I base my argument on a notion of discernibility rooted intuitively in “underlying structure”. This notion turns out to be equivalent to a notion of discernibility identified in some previous studies but proves useful in focusing when haecceities can legitimately be invoked and why Bermúdez's proposed version of II falls short of a fully satisfactory structuralist principle.

Christopher Menzel (Texas A&M University) gives a talk at the MCMP Colloquium (18 June, 2014) titled "Haecceities and Mathematical Structuralism". Abstract: It is well-known that some earlier versions of mathematical structuralism (notably from Resnik and Shapiro) appeared to be committed to a rather strong form of the Identity of Indiscernibles (II) that is falsified by the existence of structures like the complex field that admit of non-trivial automorphisms, or symmetries. In light of more recent work (notably, by MacBride, Ketland, Shapiro, Ladyman, and Leitgeb and Ladyman), it is widely accepted that the mathematical structuralist is not committed to II and that, in fact, the principle can be rejected outright on robustly structuralist grounds. I accept a qualified form of this view but I don't think the issue is as cut and dried as it might appear. In a 2007 Analysis article, José Bermúdez suggests that a strong version of II is still in play for the structuralist that can meet the challenge of non-trivial symmetries. The key to the proposal (as I will interpret it) lies in allowing identity properties, or haecceities, like being identical to c (for an arbitrary complex number c, say) to count as structural properties. Typically, structuralists dismiss such properties as obviously non-structural. I will argue to the contrary that haecceities can be viewed as properly structural and, in some circumstances, can serve as legitimate properties for discerning otherwise indiscernible “positions” in structures. Drawing on the model theoretic concept of an expansion, I base my argument on a notion of discernibility rooted intuitively in “underlying structure”. This notion turns out to be equivalent to a notion of discernibility identified in some previous studies but proves useful in focusing when haecceities can legitimately be invoked and why Bermúdez's proposed version of II falls short of a fully satisfactory structuralist principle.

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.