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Gottlob Frege: Language Gottlob Frege (1848-1925) is most celebrated today for his contributions to mathematical logic and the philosophy of language. The first section below considers why a philosophical investigation of language mattered at all for Frege, the mathematician, and why it should have mattered to him. At the same time, the considerations may serve to illustrate some general motivations that were behind the development of
Bertrand Russell: Logic For Russell, Aristotelian syllogistic inference does not do justice to the subject of logic. This is surely not surprising. It may well be something of a surprise, however, to learn that in Russell’s view neither Boolean algebra nor modern quantification theory do justice to the subject. For Russell, logic is a synthetic a priori science studying all the kinds of structures there.
Inconsistent Mathematics Inconsistent mathematics is the study of commonplace mathematical objects, like sets, numbers, and functions, where some contradictions are allowed. Tools from formal logic are used to make sure any contradictions are contained and that the overall theories remain coherent. Inconsistent mathematics began as a response to the set theoretic and semantic paradoxes such as Russell’s Paradox and the Liar Paradox—the response being that
Skeptical Theism Skeptical theism is the view that God exists but that we should be skeptical of our ability to discern God’s reasons for acting or refraining from acting in any particular instance. In particular, says the skeptical theist, we should not grant that our inability to think of a good reason for doing or allowing something is indicative of whether or not God might
Friedrich Nietzsche: Philosophy of History Nietzsche was well-steeped in his contemporary methods and debates in the philosophy of history, which carried over into his philosophy in essential ways. Once a prodigy in classical philology, Nietzsche’s philosophy is everywhere concerned with traditions, historical shifts in custom and meaning, and, to adapt his key expression, “how things become what they are”. Beyond these, Nietzsche was closely concerned
Jules Henri Poincaré (1854—1912) Poincaré was an influential French philosopher of science and mathematics, as well as a distinguished scientist and mathematician. In the foundations of mathematics he argued for conventionalism, against formalism, against logicism, and against Cantor’s treating his new infinite sets as being independent of human thinking. Poincaré stressed the essential role of intuition in a proper constructive foundation for mathematics. He believed