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Quine's and Putnam's arguments have also been influential outside philosophy of mathematics, inspiring indispensability arguments in other areas of philosophy. For example, David Lewis , who was a student of Quine, used an indispensability argument to argue for modal realism in his 1986 book On the Plurality of Worlds .
Philosophy of mathematics is the branch of philosophy that deals with the nature of mathematics and its relationship to other areas of philosophy, particularly epistemology and metaphysics. Central questions posed include whether or not mathematical objects are purely abstract entities or are in some way concrete, and in what the relationship ...
According to Richards, he mastered the language "in almost hardly over a week", [9] although other sources show he had taken one year of German in school. [10] Ramsey was then able, at the age of 19, to make the first draft of the translation of the German text of Ludwig Wittgenstein 's Tractatus Logico-Philosophicus .
During Middle Ages, Euclid's Elements stood as a perfectly solid foundation for mathematics, and philosophy of mathematics concentrated on the ontological status of mathematical concepts; the question was whether they exist independently of perception or within the mind only (conceptualism); or even whether they are simply names of collection ...
The role of mathematics in Western philosophy has grown and expanded from Pythagoras onwards. It is clear that numbers held a particular importance for the Pythagorean school , although it was the later work of Plato that attracts the label of mathematicism from modern philosophers.
In classical real analysis, one way to define a real number is as an equivalence class of Cauchy sequences of rational numbers.. In constructive mathematics, one way to construct a real number is as a function ƒ that takes a positive integer and outputs a rational ƒ(n), together with a function g that takes a positive integer n and outputs a positive integer g(n) such that
Paul Thagard writes that "the current philosophy of mathematics that fits best with what is known about minds and science is James Franklin's Aristotelian realism." [ 12 ] In the philosophy of probability, he argues for an objective Bayesian view according to which the relation of evidence to conclusion is strictly a matter of logic. [ 13 ]
Ludwig Wittgenstein considered his chief contribution to be in the philosophy of mathematics, a topic to which he devoted much of his work between 1929 and 1944. [1] As with his philosophy of language, Wittgenstein's views on mathematics evolved from the period of the Tractatus Logico-Philosophicus: with him changing from logicism (which was endorsed by his mentor Bertrand Russell) towards a ...