This past week, I began reading The Maniac, a fictionalized biography of John von Neumann. While von Neumann is perhaps lesser known than some of his contemporaries — Albert Einstein, J. Robert Oppenheimer, Bertrand Russell, and David Hilbert, to name only a few — his singular and relentless genius has profoundly affected the modern world. From his work on quantum mechanics, through the development of the atomic bomb, to the advent of modern computation and artificial intelligence, von Neumann’s contributions are legion.
This article is about someone else, though; the only person to have ever outsmarted von Neumann, according to The Maniac. This person is the logician Kurt Gödel, a name possibly even more obscure than von Neumann’s nowadays. Gödel, however, arguably shook the foundations of mathematics as profoundly as quantum physics, nuclear technology, and AI in their respective fields.
Beyond its impact on mathematics, I will also argue how Gödel’s work can provide some insights into the current political moment.
In the early 20th century, mathematicians like von Neumann, Russell, and Hilbert hoped to establish the consistency of mathematics. In particular, they were searching for a set of axioms — self-evident statements serving as starting points to a mathematical theory — that could be used to prove or disprove all other statements. In other words, these axioms would serve as a set of unimpeachable adjudicators, against which mathematicians could argue the truth or falsehood of any claim or conjecture.
Gödel’s incompleteness theorems, published in 1931, dashed these hopes. Gödel showed that any consistent set of axioms is incapable of proving or disproving every further statement within that system. In a certain sense, the axiomatic system would lead to a never-ending generation of new axioms — statements that seem true, whose truth cannot be proven with the other axioms at hand. In this way, the axioms are incomplete.
But what if we had a complete system, where no new axioms need to arise? Gödel demonstrated that this is possible, but with a key catch: such a system necessarily includes statements that are paradoxical or contradictory. In short, such complete systems will always be inconsistent, allowing for the provability of seemingly impossible concepts.
Gödel’s work found that the mathematical bedrock many of his contemporaries sought was in fact a chimera. Despite this shocking fact, we can still make mathematical progress, as we have seen time and again in the decades since the incompleteness theorems. We just need to be more mindful of the axioms we use, and whether those axioms may lead to an unacceptable incompleteness or inconsistency down the line.
In all this, I see several connections to U.S. politics, currently facing a similar foundational quandary. With the brazenness of the second Trump administration, and the range of responses to its actions (backlash and protest in some cases, acquiescence and enabling in others), the U.S. citizenry and the world have been confronted with deep questions. What does it mean to have, or defend, free speech? How much authority can a leader possess before it becomes authoritarian? How do countries respond to apparent lawlessness or the implementation of zero-sum politics?
Gödel’s work by no means answers any of these questions. Yet it does provide a possible perspective to finding answers. Our political system — with huge national parties, an intricate network of institutions, and myriad influences — is sufficiently complex to entail several paradoxes and contradictions, or otherwise feel grossly incomplete if there are self-evident “truths” on which we cannot agree.
It is perilous to ignore this reality, but I don’t think it’s impossible to accept it. Politicians have nevertheless built powerful coalitions over the years or worked with activists to stymie movements of repression and despair. We can all share our perspectives to argue for change and reach a compromise when disagreements persist. This is our toolbox in the absence of absolute provability. Rather than looking to destroy things down to a nonexistent bare truth, why not ask instead: what can we build?
