Hilbert’s Challenge to the Mathematicians of Tomorrow

Episode #1 of the course Great math problems for the 21st-century mind by John Robin

Welcome to the course.

My name is John Robin and I have a confession. Despite being an editor, writer, and all-around entrepreneur who helps self-publishing authors put out great books, in my spare time, I love to engage in recreational mathematics. In fact, before I chose the publishing path as my career, I worked in academia and during this time, fell in love with the mysteries that lurked in the margins of my textbooks.

Among the greatest of them were the problems that mathematicians even today are stuck on. In fact, in 2000, the Clay Institute put out a $1,000,000 reward on seven of them for anyone who could solve them. Because of this, they have become known as the Millennium Problems, but some of them date back to the 1800s. One of them, the Poincaré Conjecture, has already been solved—and the mathematician, Grigori Perelman, turned down the $1,000,000 reward!

The idea that we might someday solve all the problems in the field of mathematics dates back to the early 1900s. The mathematician David Hilbert listed off several problems that remained to be solved and in 1921, even went so far as to propose that all of mathematics might be unified under a system of theories that would allow us to deduce the truth or falsity of anything.

This came to be known as Hilbert’s Program, and throughout the 1920s, several brilliant theorists worked hard to realize it. Among them was John von Neumann, one of the early theoreticians whose work helped pave the way to modern computing. They all worked diligently on this problem. Then, in 1931, Kurt Gödel, a young and brilliant Austrian-born logician, found a solution, but it was not what Hilbert expected.

In fact, Gödel found through his research that such a system (at least, as Hilbert originally proposed it) could never exist. He conceptualized this system by breaking down every proof to the absolute fundamentals, called axioms. These are basic assumptions that are true without needing to be proven (for example, the sky is blue). Hilbert’s complete system would contain all the axioms with which every statement about reality could be deduced.

So, for example, let’s say I want to prove a very complicated problem about sums of prime numbers. Hilbert’s complete system of axioms would give me all the basic truths about reality that I could use, combine, and build upon to prove that problem. Complete, there would be no problem in the world that would be unsolvable.

What Gödel actually found was that many proofs gave rise to further axioms that were not part of the original system. He was able to prove that no matter how rigorously we build a fundamental system, there will always exist problems whose proofs give rise to new understandings about reality that become new axioms. In the case of our example, our proof about prime number sums might very well lead us to understand a new fundamental axiom about numbers and would be unsolvable with only the pre-existing set of axioms.

This has become known today as the Incompleteness Theorem and has been at the heart of many advances in mathematics over the last decades.

But Hilbert’s Program paved the way nonetheless for an awareness of many of the outstanding problems that mathematicians would need to solve in order to make sense of the mysteries they had unearthed since the time of the Ancient Greek thinkers. There might be no one grand unifying system for all of mathematics, but there certainly were many giants waiting to be tamed, and even now, in the 21st century, we still grapple with them.

Over the next nine days, we will explore some of the most popular—and most challenging!—of these problems. And if you’re daring, there’s $2,000,000 out there waiting for the mathematician who can tame two of the ones I’ll be covering.

Tomorrow, we will begin our tour with one problem that stumped mathematicians since the 1600s and that was only solved recently: Fermat’s Last Theorem.

Recommended book

Gödel, Escher, Bach: An Eternal Golden Braid by Douglas Hofstadter

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