By Keith Devlin @KeithDevlin@fediscience.org, @profkeithdevlin.bsky.social

Though my mathematical research since the mid-1980s has focused almost exclusively on the application of mathematical thinking to real-world problems, my PhD and almost all my research in the decade or more that followed it were in the decidedly pure areas of axiomatic set theory, in particular a study of the axioms themselves, undecidability proofs, and the mathematics of large infinities.

Those two  periods of my career might seem from the outside to be totally separate. Yet the reality is they are inescapably linked in myriad ways. The distinction between “pure” and “applied” that so appeals to many of those we elect to public office when they are deciding what to support, is short-sighted beyond belief, as nuclear power (and weapons) and modern internet security both illustrate dramatically.

The world of the mathematical infinite had fascinated me since, as a teenager, I had read popular books about the arithmetic of infinite numbers. My Devlin’s Angle post of December 1, 2011, Christmas Trees from the Land of Santa Claus, touched on some of that wonder. Two years later, in June 2013, I wrote a follow-up piece where I wondered whether all that work on infinities would ever find practical application: Will Cantor’s Paradise Ever Be of Practical Use?

My answer to that question then was nuanced. It remains so now. The content I worked on has not found application (in the way that, say, Newton’s use of infinity in calculus proved very useful). But the thinking I put in certainly did.

How useful? Well, the National Security Agency, the US Navy, and the US Army all funded my research in the years following the September 11, 2001 terrorist attack, as part of a national initiative to improve intelligence analysis. And that work grew out of my earlier “pure” research. As I point out in that 2013 post, that one example shows the foolishness of governments trying to judge the potential (future) value to society of academic research when deciding to allocate funding. But here we are, over a decade later, with the same short-sighted idiocy, this time in spades.

The standard axiomatic foundation of set theory, called the Zermelo-Fraenkel axioms after the two German mathematicians who formulated them at the start of the twentieth century, marked the end of a struggle that began in the seventeenth century, when the development of algebra and the invention of the calculus put demands on numbers that the number system in use at the time (which stretched back two millennia) could not support. By which I mean, it was not possible to provide rigorous proofs of the many uses those numbers were put to. Mathematicians were erecting elaborate buildings that everyone used every day, but they didn’t know what the bricks were made of.

I touched on that issue with regards to algebra in my essay of April 1, 2023, titled Monuments to Algebra. In the case of calculus, Newton himself was aware that he was making use of numbers well beyond their “safety limits,” and his unpublished notebooks contain a number of attempts to formulate a precise definition.

The question of “what a number is” was, in the end, not so much answered; rather it was officially banished by a change in perspective towards mathematics. The early twentieth-century axiomatization of mathematics (of which the Zermelo-Fraenkel axioms were one part) stipulated the properties the various kinds of numbers have, and the relationships between them, after which it no longer mattered (in terms of doing and applying mathematics) what exactly a number is.

If you want definitions, there are widely accepted candidates to choose from. For instance, John von Neumann provided a definition/construction of the natural numbers in set theory (i.e., as abstract sets). The rational numbers can be defined from the natural numbers in terms of ordered pairs. And late nineteenth century work by Cantor, Dedekind and others provided constructions of the real numbers from the rationals. But knowing those definitions doesn’t affect how you do or apply mathematics.

In fact, I doubt your intuitive sense of what various numbers are bears much similarity with any of those constructions. You probably have a reassuring mental image of points on a line (or in a plane in the case of complex numbers), and that’s enough. And that image provides college instructors of introductory courses in modern mathematics an opportunity to show you mind-boggling examples that highlight that there is a lot more to numbers than meets the eye.

At the end of the day, if you want to play the math game, you have to accept that it’s the rules that count, not the objects. This is much like chess, where chess pieces can come in all shapes and sizes, and you can play the game with slips of paper with marks on them.

Though algebra began, several thousand years ago, as an arithmetic tool to support commerce, its modern descendent depends (fundamentally) on our modern conception of numbers. And today’s algebra is a grounding pillar of modern society. It can rightly be described as the language of science and engineering, which is why it’s an obligatory school subject all over the world.

Even if the last time you “solved for x” was in your final school algebra test, you make use of it every day. It drives the Internet and it’s buried in all our technological devices. You solve equations all the time. Not by scribbling marks on paper, for the most part, but by moving your fingers on the screen of your mobile phone. Every time you use Google or Facebook, make a phone call, buy something online from Amazon or Walmart, ship a package by UPS or FedEx, or book a flight, the company’s computer converts your input into an equation—your equation, with your name attached to it—and solves it for you, to execute your request.

That’s the world we have created through many centuries of largely curiosity-driven “pure research” into “esoteric questions” about “abstract concepts,” seemingly having nothing to do with the everyday world.

The early development of algebra was largely a product of the medieval Muslim Empire. (Hence the original Arabic name al-jabr.) The Caliphs understood that you don’t judge in advance the importance of intellectual pursuit. You support it all. That’s why, in the ninth century, the Caliph asked a mathematician in Baghdad by the name of al-Khwārizmī to collect what was known about the method and write a book that preserved and explained it. That was the world’s first algebra textbook.

For all my life, the United States followed the same principle, and as a result became the world leader in the advancement of science, technology, and medicine. It became a magnet for researchers all over the world, and I was one of a great many who were invited to cross the ocean and do our part, with most of us staying.

Sadly, we have now elected to abandon that belief—with a vengeance. The US no longer seeks the world’s best to immigrate here. The pressure is for us to leave. Funding for research into science, technology, and medicine has been slashed. And the government demands fine-grained control of what research is done—an approach that led to the collapse of the Soviet Union after endless crop failures and then the Chernobyl nuclear disaster.

After our collapse, it will then be up to one or both of Europe and China (and possibly India down the line) to take on that leadership role. The steady (and growing) migration of scholars from the US to Europe, and a European-leaning Canada, we are now witnessing means the change of world STEMM leadership (the second M is for medicine) is going to be fast. Indeed, Europe is inviting talent every bit as aggressively as the US did throughout my lifetime.

This is surely going to be a much faster transfer of global leadership. The speed of massive change possible today is another consequence of the word we have built out of abstract ideas. Ideas come from, and reside in, human heads. Given the current pressure, a steady stream of talented individuals on airplanes will do it in months, not the years of the twentieth century and the centuries before then. The power shift of today’s global societal revolution is going to be rapid. If you blink, you’ll miss it.