Truth
I just finished Logicomix: An epic search for truth and thoroughly enjoyed it.
Logicomix is a graphic novel that does a beautiful job of explaining the generations-long search for the foundations of mathematics through the life of mathematician and philosopher Bertrand Russell.
I picked it up because I’m making my way through Gödel’s Incompleteness Theorem with the help of another wonderful book that’s focused on the theorem itself: Gödel’s Proof by Ernest Nagel and James Newman.
Logicomix places that theorem in context, making it clear just what a breakthrough it was and why its impact was so immense.
What I loved was not only the story itself but how it was told. I was amazed at how much emotion a graphic novel can elicit. I felt I knew Bertrand Russell.
There are many ways to connect with a character, whether in a film, a fiction novel, or a biography, but I found the effect here was different than all of those. I realized it was a unique combination of factors that can only happen in a graphic novel:
You’re seeing the character interact with others, and you’re seeing the character’s facial expressions and language. That’s similar to a film.
You’re hearing their internal thoughts. That’s similar to a fiction novel.
There’s a meta commentary about the characters and concepts. That’s similar to a biography or non-fiction “explainer” book. Logicomix does this through a meta layer: the graphic novel’s creators talk about the novel’s characters and the approach they took to the story. It’s an effective device because it helps explain and add context to tough concepts, adds nuance and ambiguity to a complex story, and makes connections that help connect the narrative through time and concepts.
In other words, it combined the best of film, fiction novels, biographies, and non-fiction explainer books.
***
Briefly, here are the key concepts and characters:
Bertrand Russell is the main character, and we follow the course of his life.
A series of childhood events led Russell to have an unshakable belief in logic and reason. So much so that, as a student at Cambridge University, he challenged his professors, pointing out that mathematics didn’t rest on any foundations—that it’s truth wasn’t proven; rather, it was assumed.
He found himself trapped between mathematics and philosophy and aimed to combine the two worlds—the philosophers’ search for truth with the rigor of mathematics.
But he was frustrated:
Russell’s search for tools with which to establish that foundation led him to Gottleb Frege, a German mathematician, philosopher, and logician. Frege had created a concept script that had promise as a fully logical language.
Russel also built on the ideas of Georg Cantor, who pioneered set theory and more rigorous thinking about infinity.
Set theory pitted two giants of mathematics against each other at the time: Henri Poincaré and David Hilbert. Poincaré believed in the importance of human intuition. Hilbert believed in the rigorous exactness of logical proof.
Hilbert in particular inspired Russell to focus on providing a foundational basis for arithmetic. The ability to know for sure that arithmetic problems, properly defined, can be solved with certainty was inspiring to Russell and others at the time.
Russell started writing The Principles of Mathematics, which he hoped would be a new and greater Euclid, establishing new foundations for mathematics.
But in the process he ended up discovering something that turned logic upside down, breaking the foundations of set theory. Russell’s Paradox, as it came to be known, showed that there’s a paradox in set theory and, therefore, it is flawed. The issue is that set theory breaks down when it becomes self-referential—when you ask the question:
“Does the set of all sets which do not contain themselves contain itself?”
If it does, then it doesn’t. And if it doesn’t, then it does. Paradox.
Other popular forms of this paradox include:
The statement: I am lying
The conceptual book: A complete catalog of books that are not self-referential
The paradox destroyed the foundation on which Gottleb Frege had built his life’s work.
Everyone was dejected by the paradox and the setback due to it, including Russell. Bertrand Russell and Alfred North Whitehead decided to team up to build a new foundation for mathematics: Principia Mathematica.
But, by Russell and Whitehead’s own admission, they failed. Principia Mathematica was published, but it was incomplete and controversial.
Here, the graphic novel, already great, takes an even greater, quite beautiful turn.
The authors introduce the Greek tragedy Oresteia by Aeschylus, and the meta story they create adds a beautiful layer to the whole story—the concept of reality versus the map of reality:
Ludwig Wittgenstein makes a very entertaining appearance, questioning Russell with an intensity and passion even greater than that which Russell had brought to Cambridge as a young student.
And Wittgenstein turned all of logic on its head:
And, finally, Kurt Gödel demolished the entire endeavor with his Incompleteness Theorem, which proved that there will always be elements of arithmetic that can be unprovable. In other words, “There will always be unanswerable questions” in arithmetic. Arithmetic is, by necessity, incomplete.
The essence of the story is that the truth is tricky…
Again:
“…if even in Logic and Mathematics, the paragons of certainty, we cannot have perfect assurances of Reason, then even less can this be achieved in the messy business of human affairs—either private, or public.”
***
There is a positive side to all this, of course.
The path continued: Hilbert to John Von Neumann to Alan Turing.
Alan Turing extended the work done by Hilbert and Gödel, tackling Hilbert’s “decision problem,” which had survived Gödel’s analysis.
The decision problem asked: Given a logical system, is there an algorithm for deciding whether a proposition is provable within the system or not?
Turing answered: no.
And to show this, he rigorously defined the notion of an algorithm using a theoretical machine that had a central control and a tape for memory, input, and output.
The key property of this Turing machine was universality—the ability to carry out any computational task, provided it is supplied with an appropriate program for doing so.
The concept of the Turing machine became central to modern computers as well as crypto networks, such as Ethereum.