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In this final chapter, titled "The Philosophy of Logical Analysis," Bertrand Russell reiterates the great divide in philosophy, reaching back to Pythagoras, between those inspired by mathematics and those by the empirical sciences. The mathematical party is more speculative and religious, while the empirical party is more practical and down to earth. Russell now seeks to locate his own school of logical analysis, which sets out to eliminate "Pythagoreanism" (metaphysics) from mathematics and "combine empiricism with an interest in the deductive parts of human knowledge." He explains how more recent mathematicians, such as Georg Cantor (1845–1918), have eliminated certain errors from analytical geometry and infinitesimal calculus. He mentions that philosopher and mathematician Gottlob Frege (1848–1925) more clearly has defined what numbers are. "From Frege's work it followed that arithmetic, and pure mathematics generally, is nothing but a prolongation of deductive logic." Frege's work has disproved Kant's theory about arithmetic propositions and led to the development of pure mathematics from logic, which is laid out in Russell's and Alfred North Whitehead's Principia Mathematica. Whitehead, a philosopher and mathematician, worked with Russell on the theory that mathematics could be reduced to a branch of logic. Philosopher Rudolf Carnap (1891–1970) has proposed that "philosophical problems are really syntactical," and "when errors in syntax are avoided, a philosophical problem is ... either solved or shown to be insoluble."
Physics has "supplied material for the philosophy of logical analysis ... especially through the theory of relativity and quantum mechanics." Space-time has replaced space and time. What previously was thought of as a particle will now have to be looked at as a series of events. The new physics has taught philosophers that "matter" is a "convenient way of collecting events into bundles." "Continuity of motion ... appears to have been a mere prejudice," according to quantum theory.
Modern analytical empiricism differs from the old empiricism because it incorporates mathematics and has developed "a powerful logical technique." As a result, it can address certain problems that seem to be more in the realm of science than philosophy. Nonetheless, while the new empiricism addresses the part of philosophy concerned with the nature of the world, there are still the ethical or political doctrines that address the question of how a person should live. Unfortunately, these two streams in philosophy have often been conflated, leading to a lot of confused thinking. Moreover, some philosophers have imposed their own moral prejudices on philosophical thinking in a desire to educate people or make them more virtuous. But Russell says, "Morally, a philosopher who uses his professional competence for anything except a disinterested search for truth is guilty of a kind of treachery." Philosophers of logical analysis "confess frankly that the human intellect" cannot find answers to many of the most profoundly important questions that lie at the heart of life. At the same time, these empiricists don't believe there is a "'higher' way of knowing ... hidden from science and the intellect." In Russell's view, even though philosophy (i.e., his philosophy) has let go of dogmatism, it still can "suggest and inspire a way of life."
In this chapter Bertrand Russell reiterates the argument between rationalism and empiricism that dates back to the Greek philosopher Pythagoras and then recaps the advances in mathematics and logic until the present day, mentioning his own volume written with Alfred North Whitehead, Principia Mathematica, which seeks to locate mathematics as a subset within logic. However, he fails to mention the mathematician who permanently disproved his most cherished theory, Kurt Gödel (1906–78). A mathematician, logician, and philosopher, Gödel formulated one of the most significant theories in the history of mathematics when he published his two incompleteness theorems in 1931.
The first incompleteness theorem says, according to the Stanford Encyclopedia of Philosophy, that within "any consistent formal system F within which a certain amount of arithmetic can be carried out, there are statements of the language of F which can neither be proved nor disproved in F." The second incompleteness theorem says that such a system cannot prove itself to be consistent. Thus Gödel's proofs establish a distinction between proof and truth, when previously their unity had been the unquestioned foundation of mathematics. His proofs expose the limits of logic in seeking truth. Gödel permanently blocks the road on which Whitehead and Russell set out, detailed in the three volumes of Principia Mathematica. This magnum opus describes a set of axioms and rules of inference in symbolic logic from which all mathematical truths can in principle be proven. While Whitehead and Russell's work is still considered to be a seminal text in the philosophy of mathematics, Gödel's work destroyed Russell's dream of building a grand edifice of logic. Gödel's theorems show that no system can ever be complete nor prove its own consistency. Alas there can never be one set of axioms sufficient for all mathematics. Gödel's incompleteness theorems have had a profound impact on mathematics and artificial intelligence (AI) development, since computers are powered by formal programs that use symbolic logic and rules of inference. Gödel was to mathematics what Copernicus was to astronomy and Albert Einstein to physics. They all heralded momentous and game-changing revolutions.
Despite railing against Platonic idealism, Russell took a Platonic view of logic, which he later described as a "kind of mathematical mysticism." As an old man he said of himself: "I disliked the real world and sought refuge in a timeless world, without change or decay or the will-o'-the-wisp of progress." Russell's God is mathematics, where he hopes to find immutable truths. He had believed this realm was accessible through logic, which is why he pursued this field with such single-minded love and purpose.
Russell ends his treatise, written during the madness of Hitler's rampage through Europe, by asserting that amid "the welter of conflicting fanaticisms" scientific truthfulness is a unifying force. Scientific truth is defined by Russell as belief based on observation and inference, "divested of local and temperamental bias, as [much as] is possible for human beings." He insists on this virtue in philosophy, and he says his school of analytical philosophy has "invented a powerful method by which [scientific truth] can be rendered fruitful." But perhaps he goes too far in categorically asserting that there is no "'higher' way of knowing ... hidden from science and the intellect." There may be multiple ways of knowing, after all, and they are not necessarily hierarchical. The truth-seeking philosopher ought never to close off any avenues of inquiry.