This looks like a job for Bertrand Russell!
In college, I took several courses on formal logic. I enjoyed those courses greatly, and wanted to continue my engagement with the subject. As the years passed, however, I grew reluctant to read anything that didn’t have a lot of pictures. So I was excited to hear that last week, Bloomsbury USA released an American edition of Logicomix: An Epic Search for Truth, a graphic novel based on Bertrand Russell’s role in the development of formal logic.
Apparently the book focuses on the intellectual crisis Russell suffered when he discovered Russell’s Paradox. The classic illustration of this paradox is the town with one barber. This barber is a man in town who shaves all those men in town who do not shave themselves, and none of the men who do shave themselves. If he does not shave himself, he would be a man in town who does not shave himself, and would therefore have to shave himself. But if he does shave himself, he would be a man who does shave himself, and could not therefore shave himself.
Whimsical as this paradox sounds, it did in fact have large consequences in Russell’s career and in the history of thought. Russell had hoped to reduce mathematics to logic, that is to say, to demonstrate that the truths of mathematics are a subset of the truths of logic. The basis for this project was a particular theory of sets. That theory of sets presupposed a set which included all and only those sets that are not members of themselves. That set would be a member of itself if and only if it were not a member of itself. When Russell found that the theory of sets he was using generated this paradox, he came to realize that he had defeated the theory. He then had to scrap that theory and adopt a more modest theory of sets, a theory which was in fact too modest to do what Russell needed it to do if he were to reduce mathematics to logic.
acilius
believer1
lefalcon
losthunderlads
vthunderlad
Panther Red 
cymast
/ October 15, 2009Russell’s Paradox, and the similar mathematical paradox, are assuming comparing self to self and comparing self to others share the same vantage, which they do not.
acilius
/ October 15, 2009You sound like Georg Cantor, whose theory of sets is the one mathematicians actually use. He specified as an axiom that every set is a member of itself. So Cantor would agree with you that the relation of self to self is not comparable to the relation of self to other. And of course Russell’s Paradox cannot arise in Cantor’s theory.
Russell’s goal had been to take the arbitrary postulates out of mathematics and reduce mathematics to a branch of logic. Since Cantor’s axiom that every set must include itself as a member was not a truth of logic, but an arbitrary postulate, the triumph of Cantor’s theory of sets spelled the doom of Russell’s project.
cymast
/ October 15, 2009Wow, that made my hour!
acilius
/ October 15, 2009Glad to hear it!
The book came in the mail the other day. It’s really a lot of fun. I don’t know if I’ll post a review of it, but I do recommend it highly.
cymast
/ October 15, 2009You’re saying a mathematics book is fun? Whenever I feel like laughing hysterically, I crack open an economic theory textbook.
acilius
/ October 16, 2009There’s math in it, for sure. But it’s mainly about Bertrand Russell’s life. It starts with some rather hideous stories about Russell’s childhood which led him to look for a world where everything made sense and nothing had to be taken on authority. That’s how they explain his desire to reduce math to logic. Little Bertie is thrilled by his first encounter with geometry, because it seems to be such a world. When he asks why Euclid’s axioms are true, his teacher tells him that we just have to accept them as a starting point. The boy crashes into disappointment, and spends the next few decades trying to build a kind of math that won’t need starting points, where everything can be proven. When the man Russell discovers that this is impossible, he suffers a defeat that overwhelms him and changes virtually everything about his life.
cymast
/ October 16, 2009Sound great except for the hideous childhood stories part! I hate those.
acilius
/ October 16, 2009They handle them with a light touch. Once they’ve made it clear that, for example, his grandparents kept it from Russell that his parents had died and that he found it out by stumbling upon their graves, they move on. They treat it as obvious that such an event would scar a person, they don’t dwell on it.