by John Allen Paulos
Kurt Gödel was a logician whose work in mathematical logic was seminal and fundamental. His famous incompleteness theorems, in particular, have changed our view of mathematics and computer science. He was born in Austria and lived through political turmoil there before fleeing the country after the Nazis annexed it in 1938. He came to America and settled for a time at the Institute for Advanced Study in Princeton, New Jersey, where after the war in 1947 he applied for US citizenship. While preparing for the test, the ever punctilious Gödel noted a logical inconsistency in the Constitution, a loophole that would allow American democracy to legally become a dictatorship. His friends at the Institute, including Einstein, counseled him not to express his misgivings when he appeared before the judge lest he not be granted citizenship. He did, but happily the judge ignored them.
It’s never been clear what Gödel’s logical objection was, but it’s likely, as F. E. Guerra-Pujol speculated in his 2012 paper, “Gödel’s Loophole,” that it centered around Article V of the U.S. Constitution, which described the procedure by which the Constitution might be amended. Given that Gödel’s proof of his first incompleteness theorem involves a sort of self-reference, it’s not surprising that his loophole arises from the observation that Article V’s procedures to amend the Constitution might be employed to amend itself. Article V could be modified to make it easier to amend Article V. Thus, although Article V makes the Constitution difficult to amend, an amended Article V could make it easier to do so. There could also be amendments to the amendments to make it easier still, allowing future politicians to do away with the constitutional safeguards of fundamental rights in the Constitution.
Gödel’s loophole argument could be dismissed as describing a quite unlikely situation, but it doesn’t seem quite so unrealistic as it did until recently. A topical scenario involving a very different sort of self-reference has arisen in Israel. What happens, for example, if a person, group, or party challenges the Netanyahu-backed bill severely limiting the power of the Israeli Supreme Court and this challenge goes up to the Israeli Supreme Court itself? Could the court support the challenge and scuttle the bill? The court has just agreed to hear such a case, and it’s unclear what will happen, especially since Israel doesn’t have a constitution and in deciding cases the court relies on the nebulous standard of what is “reasonable.” Whether a sort of unchecked legislative dictatorship develops is, of course, up to Israelis to determine.
The word “loophole” brings to mind Douglas Hofstadter’s notion of a strange loop, which he discusses in his magisterial books, Gödel, Escher, Bach, and I Am a Strange Loop. A strange loop can be thought of metaphorically as a circle that weaves its way up or down through different levels of a largely hierarchical system ending up where it began. Although he didn’t use the words, Gödel demonstrated that a mathematical system containing only basic arithmetic gave rise to a strange loop between two levels, the truth of mathematical statements in the system and the very symbols used to convey the truth of these statements. He exploited the loop to prove his incompleteness theorems.
The interplay between two or more levels can in general give rise to a paradox-like self-reference. Think of the Escher drawing where descending one staircase after another leads nevertheless to the top staircase. Paul Simon’s lyrics also come to mind: “You know the nearer your destination, the more you’re slip slidin’ away.” Hofstadter gives many other examples of strange loops including the one that results in human consciousness and the very notion of personal identity and “I-ness.”
To get back to legal matters, however, note that another example is the largely hierarchical nature of governments. In many countries, including the US and Israel, there is some analogue to our system of three governmental branches, the executive, legislative, and judicial. These branches (levels in a hierarchy of sorts) are often in tension, although theoretically they may be co-equal. The result in Israel is the anomalous and fraught relationship between the judicial and the other two branches. Other sorts of tangled legal hierarchies exist, say those between states and localities or between states and the federal government, that can also lead to strange loops and paradoxical-seeming results. Politics is complicated.
Indeed, strange loops are not all that strange.
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John Allen Paulos is a Professor of Mathematics at Temple University and the author of Innumeracy and A Mathematician Reads the Newspaper. His most recent book is Who’s Counting –Uniting Numbers and Narratives with Stories from Pop Culture, Puzzles, Politics, and More.