02/09/2020

In Folha, Viana discusses the limits of the human mind.

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Reproduction of Marcelo Viana's column in Folha de S.Paulo.

At the end of the 19th century, science breathed optimism regarding the capacity of the human intellect to penetrate the mysteries of the universe. In physics, there was a conviction that the great discoveries had already been made, and what was lacking was little more than improving the precision of measurements. This confidence in the power of the mind was even greater in mathematics, with its spectacular list of advances.

In a famous lecture at the 1900 International Congress of Mathematicians in Paris, the German David Hilbert (1862–1943) listed 23 problems to be solved in the new century, stating: “The conviction that every mathematical problem can be solved is a powerful incentive for the researcher. We hear within ourselves the perpetual call: Here is the problem. Seek the solution. You can find it through pure reason, for in mathematics there is no 'I don't know'.”

However, the development of science throughout the 20th century would show that there are, indeed, certain insurmountable limits to the power of reasoning. The first was discovered in physics: the uncertainty principle, formulated in 1927 by the German physicist Werner Heisenberg (1901–1976), states that it is not possible to know simultaneously the position and velocity of a subatomic particle, such as the electron: the more precise the measurement of one of these quantities, the coarser the estimate of the other will necessarily be. I will return to this topic shortly.

Hilbert's major project was to formulate mathematics on rigorous foundations, so as to free it once and for all from the paradoxes of set theory. In fact, this was precisely the gist of the second problem on his list. But the incompleteness theorems proven in 1931 by the Austrian mathematician and philosopher Kurt Gödel (1906–1978) showed that this is not possible: Gödel showed that the fact that mathematics contains no contradictions can never be rigorously proven.

In his 1951 doctoral thesis, the American economist and mathematician Kenneth Arrow (1921–2017), Nobel laureate in economics in 1972, discovered another intriguing problem with important practical implications, the solution to which is beyond our reach.

Consider a selection process with three or more candidates (people, things, ideas, etc.). Each 'voter' votes by listing the candidates in order of preference. The problem is to determine, from the individual votes, an ordered list that reflects the overall preference of the electorate. There are three rules. If all voters prefer X to Y, then X must be ahead of Y on the final list. The relative position of any two candidates (who is ahead of whom) on the final list should depend only on their relative positions in the votes, and not on the voters' opinions about the other candidates.

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