Some things cannot be proven and will forever remain uncertain. To accept that dangerous idea, one may need to be mad or be willing to risk it. Janna Levin’s novel, A Madman Dreams of Turing Machines looks at the real lives of “two mad treasures,” the most important mathematicians of the last century, Kurt Gödel (1906-78) and Alan Turing (1912-54).
Einstein’s General and Special Theories of Relativity were not the only challenges to early twentieth-century Enlightenment certainty. Kurt Gödel’s Incompleteness Theorem, which demonstrated that there are some mathematical truths that can never be proven within a system, raised a firestorm. Until then, math was the one thing considered verifiable and certain by all. For Gödel, the numbers we study are independent of our thought, and in a throwback to Platonism, these are known only by intuition. We cannot prove everything.
Alan Turing, another mathematician and code breaker, built on Gödel’s mathematical discoveries, creating the Turing Machine, a prototype of the computer, arguing that the cognitive process of a computer—thinking and replying—makes it no different from a human mind. Levin’s novel weaves back and forth between the lives of these two men, mixing fact and fiction to create a story of two brilliant minds whose lives spiral out of control.
Both men demonstrate the inseparable dance of mathematics and philosophy.
Their deaths are hardly the secret of the book, rather, it is the inevitable destination. The trip there is fascinating. It is fraught with very human obstacles—illusion versus reality, free will versus determinism, God versus naturalism, heterosexuality versus homosexuality, love versus suspicion.
Levin’s fictional narrative incorporates facts from the lives of her characters, but the story itself is a unique interpretation with deliberate changes. Dialogue is sometimes drawn from letters. Kurt Gödel was in fact so paranoid of being poisoned he starved himself to death. Turing was indeed gay, a crime in England that led to his forced chemical castration. His anguish and loss of friends, family, and a lover led to a tragic suicide by an apple laced with cyanide.
While the story has a basis in history that is interesting in and of itself, two elements of this novel will immediately stand out to the reader. The first is the amazing attention to detail that Levin’s work provides. The book’s prose is a scientific analysis, full of sounds, colors, and tastes. For example, Kurt’s wife and caretaker, Adele, is keenly aware of his phobias.
She fills the air with harmless words and themes much as radio would and thereby creates for him a background landscape of white noise. Adele never demands of him any actual conversation. She seems to understand that she is there to keep time and drown out the alarm of more individuated noises. The clink of a pin drop that can fray his nerves, the grinding of gravel beneath the weight of a man on the street, the spike of a dulled conversation as a couple pass beneath his window over Langegasse, the mounting pitch of the conspiratorial exchange as they approach.
Levin’s attention to detail sets this novel apart, avoiding the formulaic or clichéd, and this makes sense given that she is a theoretical physicist. Dialogue is sparse, but the reader is not left wanting for it.
Secondly, the narrator claims to be the writer and inserts herself into the text, or, as one might say, the story’s system. Who the narrator is, however, is an unresolvable question. Is the narrator real? Is the story real? Is the narrator Levin or is the narrator the reader? The narrative and narrator are both incomplete. She is a personification of the Incompleteness Theorem, with her unprovable identity. The narrative remains incomplete, with the narrator even saying toward the end of the novel “There no ending. I’ve tried to invent one but it was a lie and I don’t want to be a liar.”
The narrator is introspective, deeply considering the implications of the lives and ideas of Gödel and Turing. “I am on an orbit through the universe that crosses the paths of some girls, a teenager, a dog, an old woman,” says the narrator. “Maybe I should list more significant events that shaped my relationship to truth but that story would be a lie too. I could have written this book entirely differently, but then again maybe this book is the only way it could be, and these are the only choices I could have made. This is me, an unreal composite, maybe part liar, maybe not free.”
Both of these points, the attention to detail and the embedded narrator, help bring the reader into a relationship with the story, both real and imagined, true and false, reality and dream.
The Incompleteness Theorem continues to stay with me, particularly in terms of its religious connotations. In the world of theological studies, many insist on the possibility of knowing everything. Systematic theologians often provide their detailed outlines, mapping God’s entire being and defining everything precisely. This helps the theologian decide who is in and who is out, and it can provide a sense of assurance and completeness.
Even when we do not know the answer, we offer the solution we call mystery. Mystery can be a way for theologians to say that something must be true, but they cannot prove it. The Trinity is often the prime example of this. God is three persons, but one God. Both his threeness and oneness seem to be indicated in Scripture and verified by tradition, the theologian will note, but seem impossible in our world. It is, therefore, a mystery.
The idea of mystery may also appear as a cop-out. When something is held as a belief, but nothing in this world seems to indicate that this belief has any basis in reality, we might say, “this is part of the great divine mystery.” There. Now we feel better and we do not have to prove a thing. It can be a way of feeling complete with our incompleteness.
Sometimes those things we believe are valuable because they help us cope with our narratives, sometimes they are actually mysterious. What the Incompleteness Theorem demonstrates is that even in what might be considered empirically verifiable there is incompleteness and mystery. As Olga says to Kurt:
We wanted to construct complete worldviews, complete and consistent theories and philosophies, perfect solutions where everything could find its place. But we cannot. The girls I hear playing in the park when I walk to the institute, our neighbor the old woman who will die soon, our own circle, we all prize a resolution, a gratifying ending, completeness and unity, but we are surrounded by incompleteness.
Aside from this, there are other nagging questions that come from the Incompleteness Theorem. I’ll put myself out there with the following. My mathematically gifted friends can enlighten me.
Theologians often discuss the relationship of God to mathematics. For example, is the mathematics of our universe eternal or did God create it? Answers from Christian theists might include the idea that mathematics is eternal and uncreated, or that God created the mathematics of our universe. In the latter case, the mathematics he created is considered a shadow of the higher, divine math that makes up the being of God. God is, therefore, outside of the system, and so he can prove 1+1=2.
But what is divine math?
In Christian theism that would be related to Trinitarianism, namely, God is three persons, but one God. So the theologian might say that there is no analogy for this kind of math, since nothing in our universe is equivalent. (One might object, however, that on the quantum level these numerical paradoxes are possible.) Just how different God’s math is, however, appears to be part of the problem. If God is three, he is not just two, and this implies in some form or another the reality of 1+1=2 even for God.
This leaves me with a question: Even if God has his own math—for example, that something can be both three and one—would he be able to prove that idea of threeness and oneness since he is part of that system? Would demonstrating his own mathematical nature be a truth for God that is unprovable even by him?
Is God his own mystery?
A Madman Dreams of Turing Machines, winner of the PEN/ Robert Bingham Fellowship for Writers, is a great, stimulating read at only 230 pages. If you are interesting in more, see the following resources below and the video above.