This post collects some thoughts on the graphic novel “Logicomix” written by Apostolos Doxiadis and Christos Papadimitriou, with drawing from Alecos Papadatos and Annie Di Donna. Lots of spoilers ahead.
I’d heard a long time ago that Papadimitriou had co-written a graphic novel, something unusual among the leading figures of theoretical computer science (or so I think!), and I was naturally interested. Removing some of the unusualness, the subject turns out - as the title suggest - to be logic!
It’s a fun book that subverts not one stereotype and raises some pretty general questions about the quest of science and its emotional side. At the same time, it’s biographically accurate, well, more or less. As usual, truth is stranger than fiction.
Logic can get pretty emotional
The book details the struggles of several influential mathematicians of the late 19th and early 20th century to establish firm foundations of mathematics. The story is centered around Bertrand Russell’s “epic quest for truth”, the search for rigorous foundations of mathematics through set theory - and it is indeed told in an epic way1.
But it’s also told in a personal, emotional and human way: yes, even though it’s about “logic”! One of the central themes is the connection between logic and madness, and the unusual prevalence of mental disorders among logicians. There are some inconvenient questions to be confronted: is logic so removed from reality that it distorts the thinking of those who dive deep into it? Or, is it an inclination to madness that pushes people to fanatically pursue absolute, logical certainty? You’re free to reach your own conclusions2, but the book delivers plenty of high-quality food for thought backed up by historical evidence. Throwing in Russell’s love life and the two world wars in the tangled mix of emotions influencing the “epic quest for truth”, the picture of the men of logic is far from an abstract, icy portrait. It is a clash of fiery temperaments that drive the characters to extreme actions, not just in mathematics, but in the painfully real world, for good or for bad3.
Thus, it’s interesting to see how a book about logicians - who should be pretty much the ultimate embodiment of the “stereotypical mathematician”4 - takes these same stereotypes, smashes them to pieces and replaces them not with their complete opposite, which would be playing by the rules of the stereotypes again, but with something uniquely sideways to that polarity. Still, it rings honest and original. Truth is stranger than fiction.
Another interesting aspect was the personal biography of Russell. The book convincingly depicts a single thread of profound feeling of failure running through his life - even though on the outside he was one of the leading thinkers of the 20th century5! Ah, the inevitable doubts accompanying the scientific quest for reason.
The story of the math
Long story short, around the turn of the 19th century, people were like ‘yeah, of course it works, it’s obvious, we don’t need to prove that’ way too much, and that was making some people worried. Russell was particularly worried that all of mathematical logic might be built on a foundation of sand, and thus unable to provide certain knowledge and/or prone to contradictions that could break everything any moment. It probably helped that one of his fondest childhood memories was being exposed to Euclid’s Elements.
Yet, when we went to Cambridge, the other students in the Cambridge math tripos weren’t worried at all about the precise definitions of infinitesimals and such. A fellow named Whitehead was the only one sympathetic to Russell’s logical concerns.
Soon Russell got to writing a book to save math from contradictions, and in the process, discovered a contradiction in the very axioms he (and lots of other people) was using. That was a bummer. Then he got together with Whitehead, and they wrote stuff for 10 doubt-filled permanent-rewriting middle-of-the-night-arguments years, until they got something they weren’t that satisfied with, but they published it anyway6, under the name “Principia mathematica”. That was around 1910-1915.
Some people were crazy enough to read it, among them Wittgenstein. He became Russell’s student for a while, and had some revolutionary theories that language cannot describe, so it makes no sense to try to do that in a blog post. Let all my philosophy friends hate me.
Then, after a while, Kurt Godel got to reading the “Principia”, and figured out what Russell and Whitehead had been trying to do was provably impossible! That was around 1930.
Roughly, Godel’s contribution was showing that:
- a consistent system of axioms cannot be able to prove all true statements about the natural numbers
- such a system cannot prove its own consistency
Without getting into the details, this intuitively destroys the hope of having a single “right” system of consistent axioms where everything true is provable, including the consistency of the system itself. This was the hope, the vision shared by Russell and his like-minded logician friends. “It’s all over”, in the words of John von Neumann after him hearing Godel’s proof.
*Inevitable conclusions (cover image of “Logicomix”) *
Given that not-that-happy story, the other main question posed by “Logicomix”, quite directly, is whether the whole quest was a tragedy. That might sound like a dumb question about definitions, but it’s really asking something fundamental about science: are endeavors like the “Principia Mathematica” inherently “tragic”?
To properly ask and answer the question in all its epic artsiness and stuff, our Greek authors just couldn’t resist making a parallel to their cultural heritage. So we get a brief subplot that parallels the main one (it’s mostly significant at the end of the book), about a staging of the “Oresteia” in modern-day Athens.
In parallel, there’s a second brief subplot developed, voiced through Papadimitriou’s character7, about the early days of the digital computer, and the ideas behind it as developed by Alan Turing and John von Neumann. Without Russell and Whitehead’s attempts, the book argues, the next generation of thinkers - Turing, von Neumann and Godel - would have had no frame of reference. And it seems to be true - in modern speak, while initial critical reviews were skeptical, the “Principia” became a cult classic later on. Go read about it on the Stanford Encyclopedia.
What makes classical Greek tragedies such a good fit for the topic is just how fundamentally and inevitably screwed from the very beginning the heroes are. The protagonist of your average tragedy is necessarily an inherently good person, who however entangles herself, as a result of her choices, in a web of events that ultimately destroys the foundations of her life. The sad thing is that she is heading straight for doom all along, yet she doesn’t realize it until she gets there.
This nicely parallels the inevitability of logical conclusions. The quest of the “Principia” was in a sense also doomed all along. Yet, the beautiful convergence of all the plot lines at the end of “Logicomix”, at the peformance of the final act of the “Oresteia”, makes the case that the seemingly endless vicious cycle of building yet another doomed theory on top of the previous one is ended, and our heroes are redeemed. Thanks to their efforts, we learned we can’t prove everything, focused on the small part that we can, and got the computer. Not a bad deal.
Perhaps there’s a way in which these questions are relevant right now, too. The ongoing crisis in many experimental sciences is approaching ‘epic’ proportions, and entire fields, such as social psychology, now reject foundational results that seemed ironclad only ten years ago.
The problem runs deep: it’s not something wrong with a particular machine to measure human reaction time to gummy bears, it’s the entire methodology that is bogus. The current statistical tests and publishing practices just don’t work, and even well-meaning scientists have all incentives to produce faulty science under this model. Tragic. Those who don’t mean well have a lot of other tools at their disposal to cause even more damage. It is a foundational problem, and there were the Russells and Whiteheads of this crisis, warning against the lack of rigor (even in the 60s!), but their voices were barely audible until only several years ago.
So, what did we learn from that? On paper, we’ve learned some stuff, but it’s not clear yet what we’ve actually learned. Has much changed for a hundred years, between the way the mathematicians at Cambridge ignored issues of rigour, and the way many scientific communities today resisted - and still resist8 - acknowledging the crisis?
Well, let’s just hope those on the wrong side of the crisis right now understand the conclusion of “Logicomix”: it’s not a tragedy (well… not a complete one). It’s all over, we had to go through it I guess, and now we need to deal with it.
being Greek probably helps here ↩
on average. The book doesn’t hold a strictly neutral position on this matter - rather, it seems to give you examples of both things causing one another. Depending on who you are, you’ll probably take some of these arguments more strongly than others ↩
as it turns out, it’s often for personal bad right now and common good in fifty years. That’s how science progresses, yay! ↩
a being somewhere between a computer and the socially awkward kid with glasses in second grade ↩
Nobel prize in literature and stuff like that ↩
nobody wanted to read it, so they had to pay for the publishing themselves, welcome to the world of abstract math. ↩
the creators of the book are also in the book, a play on the idea of self-reference key to many developments in logic ↩
to be fair, it’s one thing to abstractly think about Russell a hundred years ago, but I can imagine that when you’re directly involved in it and it happens all around you, it is much harder to understand what is really going on and separate truth from statistical shenanigans. ↩