Hannes Leitgeb (LMU/MCMP) gives a talk at the Workshop on Five Years MCMP: Quo Vadis, Mathematical Philosophy? (2-4 June, 2016) titled "Mathematical Empiricism. A Methodological Proposal". Abstract: I will propose a way of doing (mathematical) philosophy which I am calling 'mathematical empiricism'. It is the proposal to rationally reconstruct language, thought, ends, decision-making, communication, social interaction, norms, ideals, and so on, in conceptual frameworks. The core of each such framework will be a space of "possibilities", however, these "possibilities" will consist of nothing else than mathematical structures labeled by empirical entities. Mathematical empiricism suggests to carry out (many) rational reconstructions in such mathematical-empirical conceptual frameworks. When the goal is to rationally reconstruct a part of empirical science itself (which is but one philosophical goal amongst many others), it will be reconstructed as "taking place" within such frameworks, whereas the frameworks themselves may be used to rationally reconstruct some of the presuppositions of that part of empirical science. While logic and parts of philosophy of science study such frameworks from an external point of view, with a focus on their formal properties, metaphysics will be embraced as studying such frameworks from within, with a focus on what the world looks like if viewed through a framework. When mathematical empiricists carry out their investigations in these and in other areas of philosophy, no entities will be postulated over and above those of mathematics and the empirical sciences, and no sources of epistemic justification will be invoked beyond those of mathematics, the empirical sciences, and personal and social experience (if consistent with the sciences). And yet mathematical empiricism, with its aim of rational reconstruction, will not be reducible to mathematics or empirical science. When a fragment of science is reconstructed in a framework, the epistemic authority of science will be acknowledged within the boundaries of the framework, while as philosophers we are free to choose the framework for reconstruction and to discuss our choices on the metalevel, all of which goes beyond the part of empirical science that is reconstructed in the framework. There is a great plurality of mathematical-empirical frameworks to choose from; even when ultimately each of them needs to answer to mathematical-empirical truth, this will underdetermine how successfully they will serve rational reconstruction. In particular, certain metaphysical questions will be taken to be settled only by our decisions for or against conceptual frameworks, and these decisions may be practically expedient for one purpose and less so for another. The overall hope will be to take what was good and right about the distinctively Carnapian version of logical empiricism, and to extend and transform it into a more tolerant, less constrained, and conceptually enriched logical-mathematical empiricism 2.0.

Silvia de Toffoli (Stanford University) gives a talk at the Workshop on Five Years MCMP: Quo Vadis, Mathematical Philosophy? (2-4 June, 2016) titled "Notations and Diagrams in Algebra". Abstract: The aim of this talk is to investigate the roles of Commutative Diagrams (CDs) in a specific mathematical domain, and to unveil the reasons underlying their effectiveness as a mathematical notation; this will be done through a case study. It will be shown that, differently from other mathematical diagrams, CDs do not depict spatial relations, but represent mathematical structures. CDs will be interpreted as a hybrid notation, that goes beyond the traditional bipartition of mathematical representations into graphic and linguistic. It will be argued that one of the reasons why CDs form a good notation is that they are highly ‘mathematically tractable’: experts can obtain valid results by ‘calculating’ with CDs. These calculations, take the form of a ‘diagram chase’. In order to draw inferences, experts move algebraic elements around the diagrams. These diagrams present a dynamic nature. It is thanks to their dynamicity that CDs can externalize the relevant reasoning and allow experts to draw conclusions directly by manipulating them. Lastly, it will be shown that CDs play essential roles in the context of proof as well as in other phases of the mathematical enterprise, such as discovery and conjecture.

Anne Siegetsleitner (Innsbruck) gives a talk at the Workshop on Five Years MCMP: Quo Vadis, Mathematical Philosophy? (2-4 June, 2016) titled "Ethics and Morality in the Vienna Circle". Abstract: In my talk I will present key aspects of a long-overdue revision of the prevailing view on the role and conception of ethics and morality in the Vienna Circle. This view is rejected as being too partial and undifferentiated. Not all members supported the standard view of logical empiricist ethics, which is held to be characterized by the acceptance of descriptive empirical research, the rejection of normative and substantial ethics as well as an extreme non-cognitivsm. Some members applied formal methods, some did not. However, most members shared an enlightened and humanistic version of morality and ethics. I will show why these findings are still relevant today, not least for mathematical philosophers.

Rossella Marrano (Scuola Normale Superiore Pisa) gives a talk at the Workshop on Five Years MCMP: Quo Vadis, Mathematical Philosophy? (2-4 June, 2016) titled "Degrees of Truth Explained Away". Abstract: The notion of degrees of truth arising in infinite-valued logics has been the object of long-standing criticisms. In this paper I focus on the alleged intrinsic philosophical implausibility of degrees of truth, namely on objections concerning their very nature and their role, rather than on objections questioning the adequacy of degrees of truth as a model for vagueness. I suggest that interpretative problems encountered by the notion are due to a problem of formalisation. On the one hand, indeed, degrees of truth are artificial, to the extent that they are not present in the phenomenon they are meant to model, i.e. graded truth. On the other hand, however, they cannot be considered as artefacts of the standard model, contra what is sometimes argued in the literature. I thus propose an alternative formalisation for graded truth based on comparative judgements with respect to the truth. This model provides a philosophical underpinning for degrees of truth of structuralist flavour: they are possible numerical measures of a comparative notion of truth. As such, degrees of truth can be considered artefacts of the model, thus avoiding the aforementioned objections.

Radin Dardashti (LMU/MCMP) gives a talk at the Workshop on Five Years MCMP: Quo Vadis, Mathematical Philosophy? (2-4 June, 2016) titled "What Are No-Go Theorems Good for?". Abstract: No-go Theorems in physics have often been construed as impossibility results with respect to some goal. These results usually have had two effects on the field. Either, the no-go result effectively stopped that research programme or one or more of the assumptions involved in the derivation were questioned. In this talk I address some general features of no-go theorems and try to address the question how no-go results should be interpreted. The way they should be interpreted differs significantly from how they have been interpreted in the history of physics. More specifically, I will argue that no-go theorems should not be understood as implying the impossibility of a desired result, and therefore do not play the methodological role they purportedly do, but that they should be understood as a rigorous way to outline the methodological pathways in obtaining the desired result.

André W. Carus (LMU) gives a talk at the Workshop on Five Years MCMP: Quo Vadis, Mathematical Philosophy? (2-4 June, 2016) titled "Mathematical Philosophy and Leitgeb’s Carnapian Big Tent: Past, Present, Future". Abstract: Hannes Leitgeb’s conception of mathematical philosophy, reflected in the success of the MCMP, is characterized by a pluralism — a Big Tent program — that shows remarkable continuity with the Vienna Circle, as now understood. But logical empiricism was notoriously opposed to metaphysics, which Leitgeb and other recent scientifically-oriented philosophers, such as Ladyman and Ross, embrace to varying degrees. So what, if anything, do these new, post-Vienna scientific philosophies exclude? Ladyman and Ross explicitly exclude much of recent analytic metaphysics, decrying it — very much in the logical empiricist spirit of critical Enlightenment — as vernacular “domestication” of counter-intuitive science. But it turns out, in the light of recent research on Carnap’s later thought, that Leitgeb’s Big Tent conception, though it excludes less than Ladyman and Ross, adheres more closely to Carnap’s Enlightenment ideal.

Liam Kofi Bright (CMU Pittsburgh) gives a talk at the Workshop on Five Years MCMP: Quo Vadis, Mathematical Philosophy? (2-4 June, 2016) titled "Valuing Questions". Abstract: If all scientists seek the truth, will they agree on how this search should be carried out? Social epistemologists have alleged that were scientists to be truth seekers they would display an unwelcome homogeneity in their choice of what projects to pursue. However, philosophers of science have argued that the injunction to seek the truth is incapable of providing any guidance to scientific project selection. Drawing on theories of the semantics of questions to construct a model of project selection, I argue that the injunction to seek the truth can guide choice through a philosophcially well motivated decision theory, but may indeed discourage division of cognitive labour. I end by discussing methods of maintaining heterogeneity among a community of inquirers, even veritistic ones, in light of my results.

Kristina Liefke (LMU/MCMP) gives a talk at the Workshop on Five Years MCMP: Quo Vadis, Mathematical Philosophy? (2-4 June, 2016) titled "Relating Theories of Intensional Semantics: Established Methods and Surprising Results". Abstract: Formal semantics comprises a plethora of ‘intensional’ theories which model propositional attitudes through the use of different ontological primitives (e.g. possible/impossible worlds, partial situations, unanalyzable propositions). The ontological relations between these theories are, today, still largely unexplored. In particular, it remains unclear whether the basic objects of some of these theories can be reduced to objects from other theories (s.t. phenomena which are modeled by one theory can also be modeled by the other theories), or whether some of these theories can even be reduced to ontologically ‘poor’ theories (e.g. extensional semantics) which do not contain intensional objects like possible worlds.

This talk surveys my recent work on ontological reduction relations between the above theories. This work has shown that – more than preserving the modeling success of the reduced theory – some reductions even improve upon the theory’s modeling adequacy or widen the theory’s modeling scope. Our talk illustrates this observation by two examples: (i) the relation between Montague-/possible world-style intensional semantics and extensional semantics, and (ii) the relation between intensional semantics and situation-based single-type semantics. The relations between these theories are established through the use of associates from higher-order recursion theory (cf. (i)) and of type-coercion from programming language theory (cf. (ii)).

Part of this work is joined with Markus Werning (RUB Bochum) and Sam Sanders (LMU Munich/MCMP).

Marta Sznajder (University of Groningen/MCMP) gives a talk at the Workshop on Five Years MCMP: Quo Vadis, Mathematical Philosophy? (2-4 June, 2016) titled "Inductive Reasoning with Conceptual Spaces: A Proposal for Analogy". Abstract: In his late work on inductive logic Carnap introduced the conceptual level of representations – i.e. conceptual spaces – into his system. Traditional inductive logic (e.g. Carnap 1950) is a study of inductive reasoning that belongs to the symbolic level of cognitive representation (in the three-level view of representations presented by Gärdenfors (2000)). In the standard, symbolic approach the confirmation functions are functions applied to propositions defined with respect to a particular formal language. In my project I investigate alternative approach that is a step towards modelling inductive reasoning directly on the conceptual spaces: considering probability densities (or distributions) over the set of points in a conceptual space rather than traditional credences over propositions.

I will present one way in which analogical effects can enter inductive reasoning, using the tools of Bayesian statistics and building up from Carnap’s idea that analogical dependencies between predicates can be read off conceptual spaces via the distances that encode similarity relations between predicates. I consider a quasi-hierarchical Bayesian model in which the different hypotheses considered by the agent are probability distributions over a one-dimensional conceptual space, representing possible distributions of the particular qualities among a studied population.

Roland Poellinger (LMU/MCMP) gives a talk at the Workshop on Five Years MCMP: Quo Vadis, Mathematical Philosophy? (2-4 June, 2016) titled "Five Years MCMP: Looking Back". Abstract: In this presentation I will speak about the MCMP's outreach and line up some of the center's achievements in the last five years. I will put special emphasis on our media output since many of our activities are mirrored in our media-related efforts such as our video channels on iTunes U, our Coursera online courses, and our publication database on the MCMP's web portal.