MCMP LudwigMaximiliansUniversität München

 Society & Culture

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 logicalmathematical 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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Mathematical Empiricism. A Methodological Proposal
Hannes Leitgeb (LMU/MCMP) gives a talk at the Workshop on Five Years MCMP: Quo Vadis, Mathematical Philosophy? (24 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, decisionmaking, 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 mathematicalempirical 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 mathematicalempirical frameworks to choose from; even when ultimately each of them needs to answer to mathematicalempirical 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 logicalmathematical empiricism 2.0.

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Notations and Diagrams in Algebra
Silvia de Toffoli (Stanford University) gives a talk at the Workshop on Five Years MCMP: Quo Vadis, Mathematical Philosophy? (24 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.

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Ethics and Morality in the Vienna Circle
Anne Siegetsleitner (Innsbruck) gives a talk at the Workshop on Five Years MCMP: Quo Vadis, Mathematical Philosophy? (24 June, 2016) titled "Ethics and Morality in the Vienna Circle". Abstract: In my talk I will present key aspects of a longoverdue 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 noncognitivsm. 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.

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Degrees of Truth Explained Away
Rossella Marrano (Scuola Normale Superiore Pisa) gives a talk at the Workshop on Five Years MCMP: Quo Vadis, Mathematical Philosophy? (24 June, 2016) titled "Degrees of Truth Explained Away". Abstract: The notion of degrees of truth arising in infinitevalued logics has been the object of longstanding 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.

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What Are NoGo Theorems Good for?
Radin Dardashti (LMU/MCMP) gives a talk at the Workshop on Five Years MCMP: Quo Vadis, Mathematical Philosophy? (24 June, 2016) titled "What Are NoGo Theorems Good for?". Abstract: Nogo 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 nogo 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 nogo theorems and try to address the question how nogo 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 nogo 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.

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Mathematical Philosophy and Leitgeb’s Carnapian Big Tent: Past, Present, Future
André W. Carus (LMU) gives a talk at the Workshop on Five Years MCMP: Quo Vadis, Mathematical Philosophy? (24 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 scientificallyoriented philosophers, such as Ladyman and Ross, embrace to varying degrees. So what, if anything, do these new, postVienna 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 counterintuitive 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.