MCMP – Philosophy of Mathematics
22 episodes - English - Latest episode: about 5 years ago - ★★★★★ - 2 ratingsMathematical 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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Episodes
Recent metamathematical wonders and the question of arithmetical realism
April 18, 2019 19:09 - 1 hour - 948 MB VideoAndrey 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 scenario...
The Univalence Axiom
April 18, 2019 18:55 - 56 minutes - 864 MB VideoSteve 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.
In Good Company? On Hume's Principle and the assignment of numbers to infinite concepts.
April 18, 2019 18:52 - 1 hour - 1 GB VideoPaolo 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 ...
Learning Experiences, Expected Inaccuracy, and the Value of Knowledge
April 18, 2019 18:50 - 56 minutes - 866 MB VideoSimon 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...
Anti-Mathematicism and Formal Philosophy
April 18, 2019 18:47 - 49 minutes - 759 MB VideoEric 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 th...
Geometrical Roots of Model Theory: Duality and Relative Consistency
July 14, 2015 04:00 - 1 hour - 1.04 GB VideoGeorg 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 concer...
A Hypothetical Conception of Mathematics in Practice
June 30, 2015 07:00 - 57 minutes - 872 MB VideoJosé 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 kin...
On the Contingency of Predicativism
May 11, 2015 01:51 - 49 minutes - 749 MB VideoSam 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 math...
A Computational Perspective on Metamathematics
February 10, 2015 07:17 - 1 hour - 962 MB VideoVasco 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...
Quantified Probability Logics: How Boolean Algebras Met Real-Closed Fields
February 10, 2015 04:59 - 51 minutes - 787 MB VideoStanislav 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 ...
Symmetry and Mathematicians' Aesthetic Preferences: a Case Study
January 16, 2015 03:12 - 44 minutes - 676 MB VideoIrina Starikova (Sao Paulo) gives a talk at the MCMP Colloquium (8 January, 2015) titled "Symmetry and Mathematicians' Aesthetic Preferences: a Case Study". Abstract: Symmetry plays an important role in some areas of mathematics and has traditionally been regarded as a factor of visual beauty. In this talk I explore the ways that symmetry contribute to mathematicians’ aesthetics judgments about mathematical entities and representations. I discuss an example from algebraic graph theory. Compar...
An Aristotelian continuum
December 31, 2014 12:07 - 46 minutes - 720 MB VideoStewart Shapiro (Ohio) gives a talk at the MCMP Colloquium (18 December, 2014) titled "An Aristotelian continuum". Abstract: Geoffrey Hellman and I are working on a point-free account of the continuum. The current version is “gunky” in that it does not recognize points, as part of regions, but it does make essential use of actual infinity. The purpose of this paper is to produce a more Aristotelian theory, eschewing both the actual existence of points and infinite sets, pluralities, or prop...
Neuropsychology of numbers
December 20, 2014 02:32 - 29 minutes - 456 MB VideoHourya Benis-Sinaceur (Paris I) gives a talk at the Workshop on Mathematics: Objectivity by Representation (11 November, 2014) titled "Neuropsychology of numbers". Abstract: How do we extract numbers from our perceiving the surrounding world? Neurosciences and cognitive sciences provide us with a myriad of empirical findings that shed light on hypothesized primitive numerical processes in the brain and in the mind. Yet, the hypotheses based on which the experiments are conducted, hence the re...
IF epistemic logic and mathematical knowledge
December 18, 2014 13:27 - 1 hour - 134 MB VideoManuel 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 ...
Natural numbers in philosophy of mathematics and in cognitive science
December 18, 2014 11:54 - 57 minutes - 870 MB VideoPaula Quinon (Lund) gives a talk at the MCMP Colloquium (27 November, 2014) titled "Natural numbers in philosophy of mathematics and in cognitive science". Abstract: Natural numbers are the object of studies in various disciplines. Two such disciplines are the philosophy of mathematics and research in developmental cognitive sciences. My current endeavor consists in studying the borders and possible mutual influences between these two. In my talk I compare the conceptual frameworks of the two...
On Mathematical Structuralism. A Theory of Unlabeled Graphs as Ante Rem Structures
December 18, 2014 06:29 - 1 hour - 1.09 GB VideoHannes Leitgeb (MCMP/LMU) gives a talk at the Workshop on Mathematics: Objectivity by Representation (11 November, 2014) titled "On Mathematical Structuralism. A Theory of Unlabeled Graphs as Ante Rem Structures". Abstract: There are different versions of structuralism in present-day philosophy of mathematics which all take as their starting point the structural turn that mathematics took in the last two centuries. In this talk, I will make one variant of structuralism—ante rem structuralism—...
What are the challenges of Benacerrafs Dilemma? A Reinterpretation
December 18, 2014 03:22 - 56 minutes - 864 MB VideoMarco 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 al...
Discernibility from a countable perspective
December 18, 2014 02:20 - 32 minutes - 498 MB VideoKate 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 ...
Three ways in which logic might be normative
December 18, 2014 01:18 - 1 hour - 983 MB VideoFlorian 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. Joh...
A useful method for obtaining alternative formulations of the analytical hierarchy
December 12, 2014 00:45 - 1 hour - 1.11 GB VideoStanislav 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 wil...
Haecceities and Mathematical Structuralism
June 19, 2014 00:00 - 54 minutes - 830 MB VideoChristopher 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...
Remarks on the foundations of mathematics
February 21, 2014 04:16 - 1 hour - 1.35 GB VideoHelmut Schwichtenberg (LMU) gives a talk at the MCMP Colloquium (5 December, 2013) titled "Remarks on the foundations of mathematics". Abstract: We consider minimal logic with implication and universal quantification over (typed) object variables. Free type and predicate parameters may occur. For mathematics we need (i) data (the Scott - Ershov partial continuous functionals) and (ii) predicates (defined inductively or coinductively). In this setting we can define (Leibniz) equality, falsity ...