In his younger years, Russell was a dedicated mathematician, but continuously disturbed by what he considered the shaky foundations of mathematics. In college, for example, he strongly criticized a professor for teaching calculus using infinitesmals, infinitely small values such that an infinite number added up to finite values. And he did have a point; as we discussed in podcast 65, when taught without using the more rigorous concept of limits, calculus does have a quasi-mystical quality.
More importantly, Russell went on to discover an inherent paradox in set theory, which came to be known as Russell's Paradox. If you are allowed freedom in defining sets, define the set S as the set of all sets that do not contain themselves. Does set S contain itself? To put it more concretely, suppose I tell you that I listen to precisely those podcasts whose hosts don't listen to their own podcasts. Do I listen to Math Mutation? If I don't, then I do, and if I do, then I don't! Fellow mathematician Gottlob Frege thought Russell's paradox so important that he delayed publication of his new book on set theory, hastily adding an appendix explaining the paradox. In the comic, he angrily demanded that the printer destroy the printing plates and all copies of his book, though I couldn't find other references to this incident online.
Russell then went on to work with Alfred North Whitehead on the Principia Mathematica, a massive work that was supposed to finally establish solid logical foundations for mathematics from the ground up, based on set theory augmented with 'types' to solve Russell's Paradox. Numerous times over the decade they worked on it, Russell would find some basic flaw and insist on starting over from scratch, driving his partner nuts. After this project, they never collaborated again. Finally they reached an acceptable version of their first volume, which took over 300 pages just to prove 1+1=2, and had to pay to self-publish it because the publishers estimated that so few people would actually read it that it would result in a net loss financially. In the end, it became a highly influential work, found on the shelves of every respectable math library. Though I am still a bit skeptical that more people actually read the whole thing than the publishers had originally estimated.
Ironically, just as Russell had smugly dismantled the beliefs of his predecessors, successors soon arose who were just as much a pain in Russell's rear. As we discussed back in podcast 24, "A Mathematical Nuclear Bomb", Godel eventually proved that no matter how rigorous Russell was, he could never put together a perfect mathematical system from the ground up-- any sufficiently complex mathematical system is guaranteed to be either inconsistent, having internal contradictions, or incomplete, having true statements that are unprovable. His student Wittgenstein further challenged even the basic concepts of mathematics being applicable to life, claiming all existential propositions are meaningless. When Russell gave as an example the statement "There is no hippopotamus in the room at present", Wittgenstein held his ground, claiming he could not judge its truth or falsehood. I think Russell may have been in the right on that one. But regardless, Russell concluded that Wittgenstein was a genius, and after working with him, decided he could never again do foundational work in mathematics or philosophy.
The story in Logicomix is framed by Russell talking to a group of American college students on the eve of World War II, who have insisted that he use his logic to show conclusively that entering the war would be irrational. After narrating his autobiography in the flashbacks that take up most of the comic, he disappoints the students by refusing to give a definitive answer. Instead, he uses his experiences to show that you can't always get all the answers from mathematics and logic, and discusses the various propositions claimed by both sides of the issue. Needless to say, the students were a bit disappointed.
While quite informative and entertaining, the graphic novel does have its flaws. I was a little disappointed to read in the afterward that many of the amusing encounters depicted between Russell and other mathematicians and philosophers did not actually happen-- while Russell encountered their ideas, his interactions with the people were apparently not as colorful as required for a comic. I also was disappointed that the book does not talk at all about Russell's later life: it ends before World War II, though he was prominent in philosophy and politics until his death in 1970. For example, David Horowitz's memoir "Radical Son" includes some sad scenes of Russell being used and manipulated by young leftists in the 1960s, and I really would have liked to see some of those incidents discussed in the context of this story. I'm also not sure I fully buy the level of logical modesty and open-mindedness assumed by Russell at the end of the comic, since he did go on to spend the next three decades advocating strongly for political causes. But still, I really enjoyed this book, and would highly recommend it to others interested in math or philosophy.
And this has been your math mutation for today.
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