Science

Is There a Logical Inconsistency in the Constitution?

The mathematician who wrote the “Incompleteness Theorem” proof was apparently alarmed by our founding document. He just never explained why.

Illustration: A Donald Trump-like figure stands on a large copy of the Constitution laid across two tanks as other soldiers prepare.
Doris Liou

We hear a lot about American “exceptionalism”—the idea that the United States is different from all other nations, and that, by virtue of its democracy, any of its actions assumes a priori the moral high ground. This is in part grounded in our belief in our Constitution, a document, practically sacred, that we hold up as some kind of insurance policy to the world and ourselves, as a guarantee that our effort to “form a more perfect Union” and “establish Justice, insure domestic Tranquility, provide for the common defence, promote the general Welfare, and secure the Blessings of Liberty to ourselves and our Posterity,” will always be a moral guidepost for our actions and with that a benefit to all humankind. Fascism? It Can’t Happen Here, famously wrote novelist and social critic Sinclair Lewis in 1935—except it was as the title of a novel suggesting that, with the right combination of charismatic demagogue, dire economic conditions, and savvy use of the media, it might.*

Fast forward to today and Donald Trump’s relentless attack on the Constitution and seemingly complete misunderstanding of its basic tenets are forcing us to rethink that insurance policy. Many of us—myself included—held on to the hope that our institutions and the processes would be able to safeguard the democratic process. But the walls between the executive, legislative, and judicial branches have perhaps never felt less solid. This leaves more than a few of us wondering about the foundations of our “great experiment,” about whether our country could actually be otherwise, given the constitutional system of rules that we claim must legally bind us and guide us.

While the Constitution is hardly a mathematical document, the question of its necessary implications—and thus the objective and even irrefutable nature of our democratic principles—is a mathematical kind of question. Does democracy follow from the Constitution like a proof of Euclidean geometry? Can we check to make sure it really holds? This was effectively a question asked in 1946 by Kurt Gödel, the most famous mathematical logician of the 20th century.

Gödel was Austrian by birth and had come to the United States in 1940 to escape the fascist regime of his homeland to assume a professorship at Princeton’s Institute for Advanced Study. Following World War II, he was encouraged by many to apply for U.S. citizenship. Despite the assurances of his friends (including the great physicist Albert Einstein and Oskar Morgenstern, the father of game theory) that the process was a formality, Gödel addressed the task with diligence and seriousness, taking the time to study local and national law, and finally the Constitution.

One of Gödel’s most important (and best known) mathematical achievements is his proof of “The Incompleteness Theorem,” a result that reveals a fundamental incompatibility between the logical properties of “completeness” and “consistency.” “Completeness” means that all statements can either be proved true or false within a basic system of formal reasoning (like the axiom-based deductive proofs of high school geometry). “Consistency” is the idea that the system does not allow for any statement to be both true and false. Gödel’s Incompleteness Theorem says that if a system is sufficiently complicated, it cannot be both consistent and complete. (“Sufficiently complicated” means complex enough to encode basic arithmetic.) In most logical systems—and certainly in mathematics—we must have consistency. So that leaves us without completeness, which means we have statements that can neither be proved true or false. The great mystery is that we don’t know a priori which ones they are.

Gödel proved his theorem by using logical rules to produce the statement “This Theorem is False.” Try the mental exercise of chasing your tail by assuming that this is either true or false. … If the theorem is false, then the statement is true. If the theorem is true, the statement is false.

But here is the real kicker: If you did identify any such “independent” statement, it or its negation could be appended to the system of axioms (fundamental assumed truths), and leave it consistent, thereby producing alternative logical systems, both consistent, but each containing statements that contradict statements in the other. For example, some of you might recall that in good old high school Euclidean geometry there are five axioms, one of which is the “parallel axiom.” It says that given a line and a point not on the line, you can only draw one line parallel through the point that is parallel to the original line. (Try it out!). Seems obvious, right? Well, for centuries mathematicians tried to prove that this statement actually followed from the other four axioms of Euclidean geometry—but as it turned out, the parallel axiom is actually independent of them. Thus, there is the possibility (and as we now know, the reality) of consistent Euclidean geometries with unique parallel lines and non-Euclidean geometries without unique parallel lines. This was an earthshaking discovery that exploded traditional intuitions and assumptions of the certitude of mathematical work.

Gödel’s reading of the Constitution seems to have led him to an analogous finding about the United States. As the day of his naturalization interview drew near, Gödel horrified Morgenstern by telling him that it would be completely possible within the laws of the Constitution for a dictator to emerge and put a Fascist regime in place. In Gödel’s Incompleteness-primed mind, both democracy and anti-democracy were consistent with the Constitution. Morgenstern and Einstein did their best to steer Gödel away from this line of thought, worrying that it would undermine his citizenship application. As luck would have it, during his exam, the subject came up. Gödel tried to explain this constitutional conundrum to his examiner. The examiner quickly changed topic and Gödel became a U.S. citizen, presumably saved by finding yet another person who didn’t like to talk about math.

The exact argument that Gödel had in mind—effectively a proof of Gödel’s “Un-Democracy Theorem”—is unknown. But in recent months, I’ve found myself thinking about it more and more, especially as I think about the United States our children will inherit. From the fascist trappings of many a Trump rally, to the bullying and attempted muzzling of the press that is aligned with a consistent undermining of the very nature of truth and facts, and lastly, the ideological battles that drive the interpretation of the Constitution and affect our fundamental freedoms, I’ve found myself wondering what was the weak point that Gödel identified? Just what kind of government and country does “follow” from the Constitution?

As a mathematician, did Gödel immediately understand the implications of the weird non-Euclidean geometry of gerrymandering? Did he see the inherent arithmetic inconsistencies in the Electoral College process and understand how the system could be gamed to empower a minority? Lately, I wonder if he had zeroed in on the potential inconsistencies of the presidential pardon. A full-blown ability to pardon anyone—including himself (as Supreme Court nominee Brett Kavanaugh might support)—might so weaken the separation of powers as to enable a de facto dictatorship. Indeed, the notion of a “President who pardons all and only those who can’t pardon themselves” is strikingly close to the kind of self-referencing logical antinomy (“This Theorem is False”) that underlies Gödel’s Incompleteness Theorem proof. On the other hand, maybe it was something as simple as the inconsistency of a document about freedom being drafted—at least in part—by slaveholders.

We have yet to discover an actual text outlining Gödel’s theory that our Constitution could enable fascism, but the larger lesson is worth considering. Does our Constitution ensure the democratic ideals it espouses? Does the fact of our Constitution and the institutions that have evolved under those who have “guarded” it, guarantee the freedoms and rights we unconsciously assume? There are plenty of countries with high-minded and floridly written constitutions that would not make anyone’s top 10 list of homes for safe living and free thought. Since the time of the first modern national constitution (our own), constitutions have on average survived for 17 years before being rewritten.

Gödel is also known—if perhaps less so—for work in mathematical physics, where he discovered a solution to Einstein’s equations for the structure of the universe that describe a world very different from the one we live in. Among the possibilities of this world is time travel. I’m hoping that we don’t need to rely on that discovery in order to avoid the constitutional alternate reality Gödel may have foreseen.

*Correction, Aug. 6, 2018: This piece originally misidentified the author of It Can’t Happen Here. It was Sinclair Lewis, not Upton Sinclair.