An Introduction to Bertrand Russell's Philosophy of Mathematics


Early Interest In Mathematics
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 'was one of the greatest events of my life, as dazzling as first love. I had not imagined that there was anything so delicious in the world' Even at this very early age he was very questioning of the grounds upon which Mathematics was based 'I had been told Euclid proved things, and was much disappointed that he started with axioms. At first I refused to accept them unless my brother could offer me some reason for doing so, but he said 'If you don't accept them we cannot go on' and as I wished to go on, I reluctantly accepted them.' This lead Russell to spend much of his early adult life in search of a way to put Mathematics on secure grounds, a lot has been made of his turbulent life as a child as what filled him with such drive to find a secure source of knowledge, but as I only intend this to be a relatively short introduction into his Philosophy of Mathematics there is not the time to go into his complex family life. (Though it should be noted it is covered in detail in Ray Monk's Bertrand Russell: 1872-1920 The Spirit of Solitude v. 1 & there is also a reasonable length chapter on his life in A.C Graylings Russell : A Very Short Introduction, good information about his life can also be found in many other texts to numerable to mention)

The Start of Russell's Journey
Before Russell arrived at Cambridge he had a Platonic conception of Mathematics seeing Maths 'According to Plato's doctrine, which I accepted only in a watered down form, there is an unchanging timeless world of ideas of which the world presented to our senses is an imperfect copy. Mathematics, according to this doctrine, deals with the world of ideas and has in consequence an exactness and perfection which is absent from the everyday world'. This lead to Russell being quite outraged with some of the teaching he received in Mathematics when he arrived at Cambridge 'The 'proofs' that were of Mathematical theorems were an insult to the logical intelligence. Indeed the whole of mathematics was taught as a set of clever tricks'

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 The Foundations of Geometry dealt with the problems geometry in a rather Kantian manner. The problem is that the arguments that are contained within Russell's The Foundations of Geometry is that they are one of the very few Philosophical arguments that can be successfully and certainly disproved by Scientific observation. Russell condemned his early work later in his life as 'foolish'


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.

The Road to Principia Mathematica
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 The Principles of Mathematics was released outlining the basic theory of Russell's system of classes which was to fulfil Russell's goal of reducing Mathematics to logic. In the appendix of the book Russell outlined a problem that arised from the German logician & mathematician Frege & Russells definition for 0, successor, and number. With Russell offering his first attempt to resolve what is now known as Russell's paradox and can be informally illustrated as such :


Suppose there is a town with just one male barber; and that every man in the town keeps himself clean-shaven: some by shaving themselves, some by attending the barber. It seems reasonable to imagine that the barber obeys the following rule: He shaves all and only those men in town who do not shave themselves.Under this scenario, we can ask the following question: Does the barber shave himself? Asking this, however, we discover that the situation presented is in fact impossible. If the barber does not shave himself, he must abide by the rule and shave himself and If he does shave himself, according to the rule he will not shave himself.


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 Principia Mathematica ever seeing the light of day and Russell never felt up to improving the original works.


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 'tautologies. I fear that, to a mind of sufficient intellectual power, the whole of mathematics would appear trivial, as trivial as the statement that a four footed animal is an animal.' Godel's Incompleteness theorem was to put the final nail in the coffin of Bertrand Russell's work in Philosophy of Mathematics, showing through a strict formal proof that what Russell and Whitehead had attempted to realise in the epic Principia Mathematica was not possible, as there would always be an axiom within a system of Mathematics that could not be proved. Russell never fully responded to Godel having mainly put his interest in the Philosophy of Mathematics behind him feeling that the work involved in Principia Mathematica had intellectually drained him.