Week 5 & 6 & Lecture & Seminar 3

Seminar Paper

Russell's Work In Logic

Russell could be seen as one of the giants of modern logic. His work attempts to clarify the underpinning of mathematics was unprecedented and comprehensive. Rusell was interested in mathematics and systems of logic. According to him math was the one thing that had the answers in life. At the time he believed he was only person drive to recreate math, but soon found like minded contemporaries and even few mentors. The Three most important to understand were Georg Cantor, David Hilber and Gottlob Frege. Cantor created Set Theory, and Russell Hoped these were unrelated. Grege created the Begriffsschrift, a language for logical operation as daunting as its crazy German name that was designed to examine and underpin the mathematics and build them on a solid foundation. That work was based on Canto's Set Theory. Hilbert challenges mathematicians at the International Congress of Mathematics that Arithmetic, is a underpins of all other maths. Hilbert, like Russell, shared the dream of mathematics where any problem stated rigorously could be absolutely and with complete clarity solved.

Russell set to achieve his dream by the use of Set Theory and a logical logical language not unlike that developed by Frege. Russell accidentally undid Frege and Cantor work and gave Hilbert a shock. It was to do with Set Theory where a set is a collection of objects define by a common property. For instance, "the set of all green things" would inlcude a blade of grass, a leaf and the Incredible Hulk. Similarly, the number 3 could be defined as " set of all sets with three elements". Three hats, three cars, three dogs all of these can be grouped together to be defined by the number 3. Set theory was the the heart of logic and it was a method that was used by Russell and Frege and many others that were hoping to create a unassailable underpinning for mathematics. But Russell killed that dream with his thought. He stated what about a " Set of all Sets that do not contain themselves"? In which case it doesn't. Its like saying " I am now lying". If you are, you aren't and if your aren't you are. Its a simple paradox that ate away the very heard of modern logic. The set of Set Theory was itself no longer completely consistent. Russells work had destroyed what little work was already complete. Russell proved beyond that Arithmetic was inherently and inextricable incomplete. Any system based on it was also necessarily incomplete. The work of decades by Frege, Hilbert, Cantor, and Russell was in on stroke proven to an impossible quest.

This didn't despair Russell, however it merely clarified things for him, He had found, at last, irrefutable evidence that there was no perfect path to absolute truth, and in fact such a path could never exist. With grace, he acknowledged that perhaps some problems exist that cannot be solved with a simple logical calculus. And in fact, there was no reason to despair. His work in logic built the foundation for Godel’s work, who built the foundation for Turing, who invented computing, which paved the way for much of the modern world. Russell spent much of his life in a quest for truth that ultimately failed, but his passions enriched the world along the way.