Bertrand Russell is one of the giants of modern logic. His work in attempting to clarify the underpinnings of mathematics was unprecedented and comprehensive. Russell was interested in mathematics and systems of logic from a young age. In his mind math was the one thing that produced absolutely certain answers in life, an antidote to the insanity that haunted his family line.
Unfortunately, it quickly became obvious that math was not a perfect system. Math relied in many cases on “Axioms”, things that were just true because they had to be true. Here’s one that bugged Russell as a child learning geometry: if I draw a line, and then I draw a point outside of that line, only one line can be drawn that is parallel to the original and also crosses through this point. It’s a useful principle for geometric thinking, but where is the proof? Russell’s tutor could give him no reason for the fact beyond “it is assumed”. Russell became obsessed with eliminating axioms and building mathematics on a solid foundation of absolute provable fact, and worked with this idea all through his education. In fact, this quest for truth dogged him most of his adult life.
Russell for a time thought he was the only person driven to recreate math, but he soon found like-minded contemporaries and even a few mentors. The three that are most important to understand are Georg Cantor, David Hilbert, and Gottlob Frege. Cantor created Set Theory, and then went mad. Russell hoped these were unrelated. Frege created the Begriffsschrift, a language for logical operations as daunting as its crazy German name that was designed to examine the underpinnings of mathematics and build them on a solid foundation. The Begriffsschrift was based on Cantor’s Set Theory. Hilbert set out the challenge to mathematicians at the International Congress of Mathematics that Arithmetic, as it underpins all other maths, must be built on total certainty in order for mathematics to become impregnable to doubt. Hilbert, like Russell, shared the dream of a mathematics where any problem stated rigorously could be absolutely and with complete clarity solved. Russell set out to achieve this dream by using the underpinning of Set Theory and a logical language not unlike that developed by Frege.
It was around this time that Russell accidentally undid the work of Cantor and Frege, and gave Hilbert a rather nasty shock. It all had to do with Set Theory. A Set is a collection of objects defined by a common property. For instance, “the Set of all Green Things” would include a blade of grass, a leaf, and the Incredible Hulk. Similarly, the number 3 could be defined as “the Set of all Sets with Three Elements”. Three hats, three cars, three dogs, all of these are included in the infinite set of things that can be defined by the number 3. Set theory was at the heart of logic, it was the method that Russell, Frege, and many others were hoping to use to create a truly unassailable underpinning for mathematics. Russell killed that dream with a stray thought: What about a “Set of all Sets that do not contain themselves”?
It may not be immediately obvious that this should shock you to your core. See, “the Set of all Sets that do not contain themselves” contains itself… unless it does. In which case it doesn’t. Do you see the issue? It’s like saying “I am now lying”. If you are, you aren’t and if you aren’t, you are. A simple paradox, but it was a paradox that ate away the very heart of modern logic. Set theory could not be said to be absolutely internally consistent, and therefore all the work based on Set Theory was itself no longer completely consistent. Russell, in his search for an unshakable foundation, had destroyed what little work was already complete.
Russell attempted to make up for this by writing the Principia Mathematica. It was to be a formal system that used a few clever tricks to get out of the trap that Russell’s own Paradox set for him. Unfortunately young Kurt Godel, a man Russell admitted was probably one of the few to read the Principia in its entirety, would definitively prove that there were some problems that were inherently un-answerable. He proved beyond a doubt that Arithmetic was inherently, inextricably incomplete. Thus, any system based on it was also necessarily incomplete. The work of decades by Frege, Hilbert, Cantor, and Russell was in one stroke proven to be 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.
Note: A lot of the information in this post comes from “Logicomix“, a very well-written and engaging account of the life and times of Bertrand Russell.