Workshop on Five Years MCMP: Quo Vadis, Mathematical Philosophy?, 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.

MCMP

Mathematical Empiricism. A Methodological Proposal

Workshop on Five Years MCMP: Quo Vadis, Mathematical Philosophy?, 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.

Notations and Diagrams in Algebra

Workshop on Five Years MCMP: Quo Vadis, Mathematical Philosophy?, 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.

Ethics and Morality in the Vienna Circle

Workshop on Five Years MCMP: Quo Vadis, Mathematical Philosophy?, 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.

Degrees of Truth Explained Away

Workshop on Five Years MCMP: Quo Vadis, Mathematical Philosophy?, 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.

What Are No-Go Theorems Good for?

Workshop on Five Years MCMP: Quo Vadis, Mathematical Philosophy?, 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.

Mathematical Philosophy and Leitgeb’s Carnapian Big Tent: Past, Present, Future

Workshop on Five Years MCMP: Quo Vadis, Mathematical Philosophy?, 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.

Valuing Questions

Workshop on Five Years MCMP: Quo Vadis, Mathematical Philosophy?, 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).

Relating Theories of Intensional Semantics: Established Methods and Surprising Results

Workshop on Five Years MCMP: Quo Vadis, Mathematical Philosophy?, 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.

Inductive Reasoning with Conceptual Spaces: A Proposal for Analogy

Workshop on Five Years MCMP: Quo Vadis, Mathematical Philosophy?, 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.

Five Years MCMP: Looking Back

Colloquium Mathematical Philosophy, Dolf Rami (Göttingen) gives a talk at the MCMP Colloquium (21 January, 2016) titled "On Some Puzzling Features of Existential Discourse". Abstract: Existence is a very puzzling notion that bewitched philosophers since the beginning of Western Philosophy. In this talk, I will compare the three most popular general views on existence and I will point out their main advantages and weaknesses. These are (a) the often so-called second-level view of existence, (b) the Meinongian view of existence and (c) the Parmendian view of existence. I will try to show that the best overall view of existence is a version of the Parmendian view of existence that makes use of negative free logic.

On Some Puzzling Features of Existential Discourse

Colloquium Mathematical Philosophy, Aaron Cotnoir (St. Andrews) gives a talk at the MCMP Colloquium (4 February, 2016) titled "On the Role of Supplementation Principles in Mereology". Abstract: Mereology is the formal theory of parts and wholes. Despite the frequent claim that a certain class of `supplementation' principles are analytically true of the concept of parthood, students of mereology often find such principles tricky to understand. This is made more complicated by the supposed relation between supplementation and mereological extensionality. In this paper, I outline the algebraic role of supplementation and argue that, contrary to received opinion, extensionalists and non-extensionalist alike should accept them.

On the Role of Supplementation Principles in Mereology

Colloquium Mathematical Philosophy, Matt Farr (Queensland) gives a talk at the MCMP Colloquium (20 January, 2016) titled "Causation & Time Reversal". Abstract: What would it be for a process to happen ‘backwards’ in time? Would such a process involve different causal relations? On a standard interpretation of time reversal, time reversal symmetric theories radically underdetermine causal relations between events. This has led many to imply that time reversal symmetry motivates eliminativism about causation. This paper assesses the compatibility of time reversal symmetry with causation by asking whether causal relations ought to invert under the action of time reversal or remain invariant. I show that in neither case is there an incompatibility between time reversal symmetry and causation and hence time reversal symmetric theories pose no special problem for causality. I argue for a ‘non-causal’ interpretation of time reversal, whereby time reversal does not invert causal relations, and assess the consequences of this interpretation for the epistemology and metaphysics of causation.

Causation & Time Reversal

Colloquium Mathematical Philosophy, Neil Barton (Birkbeck) gives a talk at the MCMP Colloquium (14 January, 2016) titled "On the Relationship Between Intrinsic and Extrinsic Justifications". Abstract: Recent discussions of the justification of new axioms for set theory have often focussed on a distinction between two different kinds of justification. Intrinsic justifications argue that putative axioms are implied by an underlying mathematical conception, whereas extrinsic justifications concern the consequences of said principle. In this paper, we argue that intrinsic and extrinsic justification as it has been explained in the literature is unsatisfactory. In its stead we propose a new account of intrinsic and extrinsic justification, one which develops a harmony between the two notions and avoids the problems we see for extant accounts.

On the Relationship Between Intrinsic and Extrinsic Justifications

Colloquium Mathematical Philosophy, Margaret Morrison (Toronto) gives a talk at the MCMP Colloquium (3 February, 2016) titled "Turbulence, Universality and Emergence". Abstract: Turbulent flows are paradigm cases of complex systems where multi-scale modelling is required. The fundamental problems in the field are strong fluctuations and couplings – problems that are also present in condensed matter physics (CMP) and field theory. Like the latter two areas of physics, renormalization group methods have been used to treat some of the theoretical difficulties with turbulent flows. However, unlike CMP where universality and emergence is, in some sense, reasonably understood, it is less than straightforward in cases of turbulence. I examine some of these issues, in particular the relation between multi-scale modelling and emergence, in an attempt to clarify how or even whether a notion of emergence might be applicable in the context of turbulent flows.

Turbulence, Universality and Emergence

Mathematical Philosophy - the application of logical and mathematical methods in philosophy - is about to experience a tremendous boom in various areas of philosophy. At the new Munich Center for Mathematical Philosophy, which is funded mostly by the German Alexander von Humboldt Foundation, philosophical research will be carried out mathematically, that is, by means of methods that are very close to those used by the scientists. The purpose of doing philosophy in this way is not to reduce philosophy to mathematics or to natural science in any sense; rather mathematics is applied in order to derive philosophical conclusions from philosophical assumptions, just as in physics mathematical methods are used to derive physical predictions from physical laws. Nor is the idea of mathematical philosophy to dismiss any of the ancient questions of philosophy as irrelevant or senseless: although modern mathematical philosophy owes a lot to the heritage of the Vienna and Berlin Circles of Logical Empiricism, unlike the Logical Empiricists most mathematical philosophers today are driven by the same traditional questions about truth, knowledge, rationality, the nature of objects, morality, and the like, which were driving the classical philosophers, and no area of traditional philosophy is taken to be intrinsically misguided or confused anymore. It is just that some of the traditional questions of philosophy can be made much clearer and much more precise in logical-mathematical terms, for some of these questions answers can be given by means of mathematical proofs or models, and on this basis new and more concrete philosophical questions emerge. This may then lead to philosophical progress, and ultimately that is the goal of the Center.

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