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**An Introduction to Bertrand Russell's Philosophy of Mathematics**

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Fri 14 Aug, 2009 06:44 am

From an early age the young Bertrand Russell had displayed a keen interest in Mathematics, remarking that the day his older brother Frank taught him Euclid's system of geometry

Before Russell arrived at Cambridge he had a Platonic conception of Mathematics seeing Maths

This lead to Russell seeking refugee in a form of Kantian transcendental idealism. Which aimed to solve the question of how the axioms of something such as Euclid geometry could be built by reason alone but then still hold true for the empirical world which surrounds us. Kant answered this by concluding that the world that Euclid geometry describes not the actual world but how the word appears to us or as how Kant would say that Euclid geometry describes our 'form of intuition' with regard to space. Russell's first philosophical book

Russell then briefly turned to a form of neo-Hegelianism under the influence of Mc Taggart a then very influential Hegelian professor of Philosophy based at Cambridge at the time, the majority of the work written by Russell in this period is still unpublished. Russell then began to catch up with some of the developments made in German mathematics work as such as that done by Cantor had provided mathematics with a much more sophisticated grounds, with several terms being redefined and certain notions banished. Much of this work was very counter to common sense and had be described as a form of 'cancerous growth' by Wittgenstein. But for Russell going against common sense was a very small price to pay, to reach his long term goal of being able to found the basics of mathematics on reason alone.

In Paris in 1900 at an academic conference Russell met the Italian mathematician Peano, who had managed to show he could base the whole mathematics of a system which only contained three basic notions and five initial axioms. Though hugely impressed with Peano's system Russell felt that it would be possible to take it further showing that Maths could be derived form logic alone through a more basic notion of class. In 1903 the book

Along with Whitehead, Russell set out to write Principia Mathematica in which he aimed to remove the paradox that plagued his system, through his solution the Theory of types though the Theory of types was not at all trouble free and involved the construction of a proper logical language which was a very challenging prospect and was never completed by Russell with only three of the four volume's of

By 1913 Wittgenstein (who at the time was Russell's brilliant student who he hoped would take over his work) had managed to convince Russell that there was no such thing as a logical object. Eventually Russell came to believe that Mathematics consisted of

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