certainty—or the lack thereof--7/16/25

16 Jul 2025

certainty—or the lack thereof--7/16/25

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Today's selection-- from Proof by Adam Kucharski. Can mathematics be both complete and noncontradictory?:

“In the early twentieth century, some mathematicians tried to tackle the growing number of paradoxes by building a collection of fixed axioms from which all mathematical statements could be derived. Where Euclid had tried—and failed—they would build something taller, sturdier, and more monster—proof. Crucially, the collection would need to be complete, so that any true statement could be proved from these axioms, as well as noncontradictory, so that it wasn't possible to prove two conflicting statements. David Hilbert at the University of Gottingen led the efforts, and the search would become known as ‘Hilbert's program.’ The aim was to move away from intuition-based mathematics—which [German mathematician Karl] Weierstrass and others had shown to be unreliable—while avoiding the logical quagmire that mathematicians would encounter if they had to prove theorems without being able to take anything as ‘self-evident.’

“One person who hoped to contribute to Hilbert's program was a PhD student at the University of Vienna named Kurt Godel. Unfortunately, he soon hit upon some paradoxes that suggested they were facing, a much larger problem. It appeared there were true statements about arithmetic that could not be proved using arithmetic. The following year, Godel published his findings in the form of two ‘incompleteness theorems.’ News of the theorems-and their implications for Hilbert's program-spread rapidly. No matter how detailed the mathematical axioms, Godel had shown there were always some situations that they would not cover.

“He had achieved his proof by focusing on self-referential statements. The best-known example of this is the ‘liar's paradox.’ Imagine someone says to you: ‘I am a liar.’ If they're telling the truth, then their statement implies they are a liar, which is a contradiction. But if they're lying about being a liar, then it implies they are truthful, resulting in another contradiction. Godel used an analogous approach, analyzing the mathematical equivalent of statements such as: ‘This statement cannot be proved.’ If this statement can be proved, then what it says is false, and we have an inconsistency. Alternatively, if it cannot be proved, then our system of logic is incomplete, because it includes a true statement that we cannot prove.

“Thanks to Cantor's ‘youth corrupting’ work on infinite sets, Godel was able to extend his proof to cover all arithmetic. In doing so, he demonstrated the limitations of axiomatic systems. Mathematics could not be both complete and noncontradictory. Axioms alone were not enough.

Godel's theorems explain why Frederick and Napoleon ran into problems while developing their legal codes. They also show why some modern bureaucratic systems remain so frustrating, and why software engineers sometimes struggle to develop successful decision-making algorithms. As Godel found, it's remarkably easy to hit upon a set of rules that is either incomplete or contradictory. In which case, there will be situations that either no rule covers, or we have two conflicting rules that we—or an algorithm—must somehow follow.