# Bertrand Russell’s Birthday and the Barber Paradox: A Brief Exploration into Analytical Philosophy

This week, we’re stepping away from the physical sciences to where mathematics and philosophy intersect to celebrate the life of Bertrand Russell.

Born on May 18, 1872, Bertrand Russell could be considered one of the most well-rounded contributors to modern thought that ever existed. Although Russell is most recognized as a pioneering logician, his work as a philosopher, mathematician, educator, social critic, and political activist is evidenced by his over 70 books and thousands of essays and letters.

When we think of intellectual accomplishment over the decades, one view centers scientists as those who have gone out into the world and done the hard work of discovering the physical properties and basic natural structures around which our most successful scientific theories have been built. A different perspective, however, attributes the success of science to the work of philosophers. Russell, credited with being one of the founders of what is now known as analytic philosophy, bridges the gap between these perspectives in that the endurance and profundity of his work is largely due to the importance of both empirical observation and logical analysis.

And only a man as brilliant as Russell could have come up with a logical paradox so confounding that it caused some of the leading thinkers of his time to simply avoid it. In the following, we’ll explore just one of Russell’s contributions to the field of mathematics as the field worked to develop a logical foundation for its theories—what has become known as Russell’s paradox.

## Set Theory: A Foundation for Paradoxical Reasoning?

The modern study of set theory was initiated by the German mathematicians Richard Dedekind and Georg Cantor in the 1870s. Simply put, a set is a collection of objects with something in common. A set might be, for example, prime numbers, birds that live in your backyard, or people with whom you have gone out for drinks in the past five years. The elements of a set are the things within it, such as prime numbers, birds, or people as in the examples above. They are also called the members of a set.

Seems straightforward enough, right? It was . . . for a short time. Then, in the next exciting developments in set theory threw a wrench in these mathematicians’ work around 1900, it was discovered that some interpretations of Dedekind and Cantor’s set theory gave rise to several contradictions, or paradoxes.

Around the same time that Dedekind and Cantor were working on set theory, Gottlob Frege, a German philosopher, logician, and mathematician, was developing mathematical logic. Frege spent many years of his life developing a mathematical/philosophical theory drawing on elements of set theory. Unfortunately for Frege, Russell took an interest in his work and discovered that Frege’s approach to set theory presented a conundrum.

Russell’s paradox describes the contradiction that occurs when considering that the set of all sets that are not members of themselves cannot exist. Such a set would be a member of itself if and only if it were not a member of itself. This paradox is based on the fact that some sets are members of themselves and some are not.

Russell's paradox is most easily understood when described as the barber paradox, which goes like this: A barber states that he shaves all who do not shave themselves. So who shaves the barber? Any answer to this question contradicts the barber's statement. (Check out this video for an animated description of the paradox.)

Why is this important? In drawing attention to this paradox, Russell highlighted flaws in the foundation of mathematics as it had existed up to that point, causing mathematicians to make several attempts to refine set theory. Russell's discovery led immediately to much research in set theory and logic to define the nature of sets, classes, and membership more accurately.

## Russell’s Legacy

Apart from his contributions to analytic philosophy, Russel’s advocacy for peace and social reform brought him most recognizably into the global limelight with his opposition to the First World War, during which he was imprisoned for his activism. In 1950, he was awarded the Nobel Prize in Literature for his prolific writings championing human rights and freedom of thought.

Now that you’ve been bit by the math bug, check out hBARSCI’s collection of hands-on kits, including abacuses, geometric shapes, puzzles, and more, that allow you to explore math fundamentals at home!