Hilbert's Power

29th of September 2025

"C'est avec les hochets que l'on mène les hommes." – Napoléon

David Hilbert was a giant of mathematics. He and Henri Poincaré are sometimes brought up as the last two mathematicians who deeply understood and contributed to all areas of mathematics, instead of nibbling away at one of the many increasingly specialized corners. Several of his important theorems are named after him, such as the Nullstellensatz. He also symbolizes the final chapter of the illustrious history of mathematics at the University of Göttingen, which had opened with Gauss 125 years earlier.

Still, if we're to consider Hilbert's influence on our world today, his true power did not lie in his ability to prove theorems, which is supposedly the hallmark of a good mathematician. In comparison to that special something, all his technical contributions look minor. That special power is a taste for open problems, and the charisma to get his colleagues to work on them.

In 1900, Hilbert presented a list of 23 open problems that he believed were most important to work on to progress mathematics as a field. For the decades since, and up until today, his colleagues are toiling away at them; when one is solved, a paper along the lines of About the seventh problem of D. Hilbert is written. Hilbert's problems are reincarnated today as the Millennium Prize Problems, which also include Hilbert's eighth problem, the Riemann Hypothesis.

Consider also Hilbert's program to show that mathematics itself does not contain a contradiction, published in 1921. It was destined to fail in an incredibly fruitful way: it led Gödel to discover his incompleteness theorem, and with it a whole branch of math. Perhaps even more influentially, Hilbert then formulated the Entscheidungsproblem (literally, decision problem), which asks for an algorithm to decide the truth value of any statement. Its resolution by Alan Turing marked the inception of computing.

Are the millennium problems the right questions to look at in this moment of mathematics? I have no clue. My knowledge of math is far too superficial to comment. But what I can say for sure is that everyone is completely obsessed with them, simply because they are so well known. The influence that a Hilbert-like figure can exert over their descendants—just by publishing a list of problems claimed to be important—is vast.

The opposite archetype is represented by Paul Erdős, the most productive mathematician ever. He put out surprising results by the thousands, contributed greatly to our understanding of finite sets, was extremely collaborative and unique in character. Yet he is less well known than Hilbert. This might be because he mostly solved conjectures set out by others, or published new problems and their solutions simultaneously, thus leaving little inspiration to influence the long-term trajectory of mathematics—at least relative to what he was capable of.

As proofs are being automated, mathematicians will spend more time thinking about the what instead of the how. The exact same can be said of many other professions. We'll need Hilbert's power to choose objective functions and evaluations when we have access to increasingly powerful optimizers. It is time to study the "ought" of our fields.