The result appeared to conflict with Russell's assumption that there is a class which contains all other objects this class would be the greatest cardinal. When he applied this approach to Cantor's proof that there is no greatest largest cardinal number he stumbled on paradox. Russell employed the methods of Peano to give derivations of mathematical results based on logic alone. Read moreĬonrad Asmus, Greg Restall, in Handbook of the History of Logic, 2012 10 Russell Īt the beginning of the 20 th century, Russell was involved in a reductive project similar to Frege's. At the heart of this vision, in a reductionist form, is the idea that, once the appropriate logical vocabulary is supplied, and the concepts of a field made sufficiently clear, all knowledge can be reduced to empirical observation and data collection. The task of logical analysis, in Russell’s telling introduction to Wittgenstein’s work, is to show ‘how traditional philosophy and traditional solutions arise out of ignorance of the principles of Symbolism and out of misuse of language’ ( Wittgenstein 1921). The impetus behind Russell and Whitehead’s project remained that of Leibniz’ dream, but phrased now in tones less of wishful thinking and more of matter-of-factual inevitability. Principia Mathematica, more than any other work, was responsible for directing Anglo-American philosophy away from metaphysics, idealism and the naive empiricism of the nineteenth century, and towards an empiricism instead founded on the precise use of a language resolutely committed to describing facts-a language epitomised in the severe codes of symbolic logic. This work developed on Frege’s system, and was to prove instrumental in inspiring the development of the austere brand of philosophy by the Vienna circle in the 1920s, known as logical positivism ( van Heijenoort 1967). In Principia Mathematica, Whitehead and Russell (1910) endeavoured to refound the entirety of mathematics on the new ‘meta-mathematics’ of formal logic. Liam Magee, in Towards a Semantic Web, 2011 1910s-Mathematical principles An historical introduction to formal knowledge systems

Principia of mathematics