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Philosophy of Mathematics: Logic
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Mathematics and logic have been central topics of concern since the dawn of philosophy. Since logic is the study of correct reasoning, it is a fundamental branch of epistemology and a priority in any philosophical system. Philosophers have focused on mathematics as a case study for general philosophical issues and for its role in overall knowledge- gathering. Today, philosophy of mathematics and logic remain central disciplines in contemporary philosophy, as evidenced by the regular appearance of articles on these topics in the best mainstream philosophical journals; in fact, the last decade has seen an explosion of scholarly work in these areas.
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The terms philosophy of mathematics and mathematical philosophy are frequently used as synonyms.[1] The latter... may be used to mean at least three other things. One sense refers to a project of formalizing a philosophical subject matter, say, aesthetics, ethics, logic, metaphysics, or theology, in a purportedly more exact and rigorous form, as for example the labors of Scholastic theologians, or the systematic aims of Leibniz and Spinoza. Another sense refers to the working philosophy of an individual practitioner or a like-minded community of practicing mathematicians. Additionally, some understand the term mathematical philosophy to be an allusion to the approach taken by Bertrand Russell in his book Introduction to Mathematical Philosophy.
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It is necessary at the outset to distinguish the philosophy of mathematics from the philosophy of mathematicians. Philosophy of mathematics has its source in any moment that a person reflects on mathematical practice, whether it is another person's practice or that person's own. Having made that distinction between the more generic reflection on mathematics and its more specialized reflexive application, it is possible to see yet another distinction, analogous to what medieval logicians call logica docens, logic as taught, and logica utens, logic as used. C.S. Peirce, as a logician, mathematician, and philosopher who found it useful to study the past on the way to creating the future of mathematics, is a useful source here:
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Philosophy of mathematics today has transformed into a very complex network of diverse ideas, viewpoints, and theories. Sometimes the emphasis is on the "classical" foundational work (often connected with the use of formal logical methods), sometimes on the sociological dimension of the mathematical research community and the "products" it produces, then again on the education of future mathematicians and the problem of how knowledge is or should be transmitted from one generation to the next.
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The Department is particularly strong in logic, the philosophy of logic, and the philosophy of mathematics. William Hanson, and Geoffrey Hellman and specialize in these areas, while other faculty members sometimes teach courses in them.
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Dale Jacquette is Professor of Philosophy at The Pennsylvania State University. He is the author of Philosophy of Mind (1994), Meinongian Logic: The Semantics of Existence and Nonexistence (1996), Wittgenstein's Thought in Transition (1998), Symbolic Logic (2001), David Hume's Critique of Infinity (2001) and On Boole: Logic as Algebra (2001) as well as numerous articles on logic, metaphysics, philosophy of mind, and Wittgenstein.
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