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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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.
The 'fitting problem' for logical semantic systems
MCMP – Mathematical Philosophy (Archive 2011/12)
1 hour 8 minutes 55 seconds
6 years ago
The 'fitting problem' for logical semantic systems
Catarina Duthil-Novaes (ILLC/Amsterdam) gives a talk at the MCMP Colloquium titled "The 'fitting problem' for logical semantic systems". Abstract: When applying logical tools to study a given extra-theoretical, informal phenomenon, it is now customary to design a deductive system, and a semantic system based on a class of mathematical structures. The assumption seems to be that they would each capture specific aspects of the target phenomenon. Kreisel has famously offered an argument on how, if there is a proof of completeness for the deductive system with respect to the semantic system, the target phenomenon becomes „squeezed“ between the extension of the two, thus ensuring the extensional adequacy of the technical apparatuses with respect to the target phenomenon: the so-called squeezing argument. However, besides a proof of completeness, for the squeezing argument to go through, two premises must obtain (for a fact e occurring within the range of the target phenomenon):
(1) If e is the case according to the deductive system, then e is the case according to the target phenomenon.
(2) If e is the case according to the target phenomenon, then e is the case according to the semantic system.
In other words, the semantic system would provide the necessary conditions for e to be the case according to the target phenomenon, while the deductive system would provide the relevant sufficient conditions. But clearly, both (1) and (2) rely crucially on the intuitive adequacy of the deductive and the semantic systems for the target phenomenon.
In my talk, I focus on the (in)plausibility of instances of (2), and argue That the adequacy of a semantic system for a given target phenomenon must not be taken for granted. In particular, I discuss the results presented in (Andrade-Lotero & Dutilh Novaes forthcoming) on multiple semantic systems for Aristotelian syllogistic, which are all sound and complete with respect to a reasonable deductive system for syllogistic (Corcoran˙s system D), but which are not extensionally equivalent; indeed, as soon as the language is enriched, they start disagreeing with each other as to which syllogistic arguments (in the enriched language) are valid. A plurality of apparently adequate semantic systems for a given target phenomenon brings to the fore what I describe as the „fitting problem“ for logical semantic systems: what is to guarantee that these technical apparatuses adequately capture significant aspects of the target phenomenon? If the different candidates have strikingly different properties (as is the case here), then they cannot all be adequate semantic systems for the target phenomenon. More generally, the analysis illustrates the need for criteria of adequacy for semantic systems based on mathematical structures. Moreover, taking Aristotelian syllogistic as a case study illustrates the fruitfulness but also the complexity of employing logical tools in historical analyses.
MCMP – Mathematical Philosophy (Archive 2011/12)
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.
