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 Intrinsic Hamilton-Jacobi Dynamics of General Relativity and its Implications for the Semi-Classical Emergence of Time
MCMP – Philosophy of Physics
48 minutes 49 seconds
10 years ago
The Intrinsic Hamilton-Jacobi Dynamics of General Relativity and its Implications for the Semi-Classical Emergence of Time
Donald Salisbury (Austin) gives a talk at the Workshop on the Problem of Time in Perspective (3-4 July, 2015) titled "The Intrinsic Hamilton-Jacobi Dynamics of General Relativity and its Implications for the Semi-Classical Emergence of Time". Abstract: The quantization of the general theory of relativity is notoriously difficult, in particular on account of the underlying general covariance and the consequent appearance of constraints in the classical Hamiltonian theory. The notion of time in the quantum theory is especially troubling since differing ideas of time suggest themselves depending on the quantum rules that are employed and the interpretations given to time in the classical theory itself. I will address the problem of time from a perspective in which constraints are implemented in a Hamilton-Jacobi framework through the use of intrinsic coordinates. The canonical approach is especially suited for this task. The decisive result is that the problem of time is even greater than one might have expected; there are arbitrarily many equally valid and possibly inequivalent time choices that one can introduce in this manner, all involving the use dynamical variables that are invariant under the action of the four-dimensional diffeomorphism-induced group as described in Pons, Salisbury, and Sundermeyer, Phys. Rev. (2009), 084015. I will review a Kuchař-inspired, but fully diffeomorphism covariant, classical Hamiltonian approach to general relativity in which spacetime scalar phase space variables are introduced that can serve as intrinsic coordinates. There corresponds to each choice a constraint which can be converted (as originally proposed by Asher Peres for conventional variables) into an Einstein-Hamilton-Jacobi (EHJ) equation. The choice of intrinsic coordinates is rendered simple in terms of these new variables, as are the choices in the new intrinsic EHJ equation. Indeed, the resulting intrinsic dynamics follows immediately from the EHJ equation. No Lagrangian is obtained, and this might be expected given that spacetime scalars must depend on time derivatives of the metric and their introduction into the Einstein action would result in the appearance higher derivative contributions. To each EHJ equation there corresponds a Wheeler-DeWitt quantum equation with its own emergent time. I will begin to examine possible quantum implications of the existence of distinct emergent intrinsic times.
MCMP – Philosophy of Physics
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.