July 27, 2023

Canadian-American educator and writer Lawrence Peter in his book The Peter Principle observes: “Democracy is a process by which people freely choose a scapegoat.” The average citizen in a democracy receives a constant message that the situation in the country is getting worse and worse, dangerous and more dangerous, tragic and more tragic. It is not uncommon for him to need an embodiment of his anxiety in a particularly hated politician. The average citizen in a totalitarian state receives signals that the country’s situation is very good and that it can only get even better. However, the depressing pressure of democracy on the citizen must be compensated for in some way. This compensation is provided by utilitarianism. Utilitarianism is based on the “greatest happiness principle,” that is, it preaches the pursuit of the greatest happiness and the desire to avoid the unpleasant. The English philosopher John Stuart Mill was critical of the satisfactory feelings of his fellow citizens. He believed that “It is better to be a dissatisfied man than a life-satisfied pig; it is better to be a life-satisfied Socrates than a life-satisfied fool. And if a fool or a pig has a different point of view, it is only because they know exclusively their own point of view.” However, the desire for the ideal order of society reflects wishful thinking.

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In 1931, the Austrian-American logician and mathematician Kurt Gödel proved two theorems about incompleteness and inconsistency in mathematics. This was a true revolution in mathematics. Before Gödel, mathematicians believed that their science was complete, consistent, and that its theorems could be deduced logically unambiguously from axioms. Consider the example of Euclid’s geometry, which is based on axioms from which theorems follow. In it, it can be proved that the statement “the sum of the angles of a triangle is 180 degrees” is true and the statement “the sum of the angles of a triangle is 100 degrees” is false. This is how all mathematics was done before Gödel. It was considered to be logically rigorous, a perfect axiomatic science. Gödel proved that if formal arithmetic is non-contradictory, then it is impossible to derive a formula asserting the non-contradictory nature of arithmetic and that the non-contradictory nature of any axiomatic theory cannot be proved by means of this theory itself. If such a rigorous, slender, and seemingly logical science as mathematics is incomplete and contradictory, can democracy be a complete and consistent system?

This question was answered in the negative by the same Kurt Gödel at the time he obtained U.S. citizenship. In 1948, he appeared for the U.S. citizenship exam accompanied by his colleague at the Institute for Advanced Study in Princeton, Albert Einstein. From a mathematician’s point of view, a country’s constitution is a set of logical connected axioms. Knowing the incompleteness of axiomatic systems, Gödel found contradictions in the U.S. constitution that allowed him to establish a dictatorial regime under the guise of democracy and freedom. One of the members of the examination board asked Gödel:

“Until now you have been a German subject.” 

Gödel corrected the examiner:

“Not Germany, but Austria.”

“It doesn’t matter,” said the examiner. “In any case you have lived under a monstrous dictatorship, which is impossible in our country.”

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But Gödel contradicted him:

“On the contrary, I undertake to prove mathematically that a dictatorship in the United States is possible.”

Einstein managed to convince Gödel not to anger the examiners, otherwise he would not receive U.S. citizenship.

Three years later, in 1951, the American mathematician and economist Kenneth Joseph Arrow, a professor at the University of Chicago, Stanford, and Harvard University, proved in a general way the theorem proved by Gödel in mathematics. When applied to political structures, it is called the “impossibility of democracy” theorem as a “collective choice” or “dictator’s inevitability theorem.” This theorem, also called the Gödel-Arrow theorem, states that the equilibrium of a system depends on the preferences of voters who have no idea of the system’s margin of safety and are not inclined to trust it when deviations from the norm occur in it. The Gödel-Arrow theorem proves that no collective choice procedure can optimally reflect individual voter preferences. In particular, correct determination of the winner in democratic elections is not always possible. Arrow identified five conditions, now generally recognized as essential axioms of democracy, in which social decisions are made by revealing the preferences of individuals, i.e. by voting results. Using elementary mathematical apparatus, Arrow showed that these conditions are contradictory: it is impossible to create an electoral system that would not violate at least one of them. Such violation occurs not by someone’s evil will, but due to an inherent defect of the system, which is incomplete and contradictory. Arrow received the Nobel Prize in 1972 for proving the impossibility of simultaneous compliance with the requirements of reasonableness and equality and the impossibility of establishing the ranking of social priorities. According to Gödel-Arrow’s theorem, any electoral system is flawed. Arrow’s results dashed the hopes of many sociologists and mathematicians to find a perfect voting system.

Fierce struggle for democracy is one of the effective methods of its undermining and even destruction, i.e. establishment of dictatorship. In Soviet times, a joke was circulated ridiculing the “struggle for peace” waged by the USSR, organizing wars and awarding the people it needed, not necessarily peacemakers, with international Lenin prizes “for the struggle for peace”: “There was no war, but there was such a struggle for peace that there was nothing left of peace”. It is in the light of Arrow’s theorem that the struggle for and against judicial reform in Israel should be understood. Since the time of the pioneering philosophers of the doctrine of the necessity of separation of powers, the English philosopher John Locke and the French philosopher Charles Louis Montesquieu, it is clear that a struggle between the executive, legislative, and judicial powers is preferable to the harmony between them that exists under a dictatorship. However, the problem of a properly quantified separation of powers does not have a single solution, as democracy cannot be complete and consistent. Both sides fighting in Israel believe that they are fighting for true democracy. In reality, they are undermining the existing democracy, which, like any other democracy, is incomplete, contradictory, and certainly an imperfect system. To paraphrase the aforementioned Soviet joke, there is such a struggle for democracy that democracy may be established, but there may be nothing left of the state of Israel.  

Image: Public Domain