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

New decades—and even more so new centuries and (in our lifetime) a new millennium—inevitably prompt writers to find out and report on what was going on one or more units earlier. I am no exception, and have often written such pieces.

But a run-of-the-mill year like xx24 simply does not create the same sense of occasion. Which is precisely why I found myself wondering just what the big news was in mathematics in 1924, and before then 1824, and on back to 1624. After all, by focusing on the number-base years, we miss nine others in which interesting things surely happened. (Or we group discoveries by decades or centuries, but then why tag them by the first year thereof?) Anyway, here goes for the 24s.

1924

Having worked in axiomatic set theory for the first half of my mathematical career, I was delighted to see that 1924 saw two of the results that helped propel me into that part of mathematics.

First, that was the year the mind-bending Banach-Tarski paradox was proved; or, to express it more accurately, the Banach-Tarski Theorem. There was no paradox; rather, the result simply highlighted how our geometric intuitions fail us when we enter the world of infinite sets—in this case the infinite as it operates when the Axiom of Choice is available.

What the theorem says is that, given a solid ball in three-dimensional space, there exists a decomposition of the ball into a finite number of disjoint components (five is enough), which can then be put back together in a different way to yield two identical copies of the original ball. Indeed, the reassembly process involves only moving the pieces around and rotating them without changing their shape. The rub is that the pieces themselves are not “solids” in the usual sense, but infinite sets of scattered points. Stefan Banach and Alfred Tarski gave the proof in their paper Sur la décomposition des ensembles de points en parties respectivement congruentes, Fundamenta Mathematicae, 6: 244–277.

Results like this show that the correct answer to the question, “Is mathematics invented or discovered?” is “Both.” In this case, the (Zermelo-Fraenkel) axioms of set theory were invented, but consequences such as the Banach-Tarski Theorem were discovered.

The second notable result that year was Hilbert’s hotel, introduced by David Hilbert in a 1924 lecture “Über das Unendliche”.

Hilbert proposed a thought experiment to illustrate a counterintuitive property of infinite sets.

A simple argument shows that a fully occupied hotel with infinitely many rooms (i.e. rooms that can be numbered using all the natural numbers) may still accommodate additional guests.

In fact, it can accommodate infinitely many additional guests.

Hilbert’s idea was popularized by George Gamow in his 1947 book One Two Three... Infinity.

If you haven’t seen Hilbert’s Hotel before, it’s more fun to figure out for yourself how the desk clerk would accommodate the additional guest(s).

1824

Going back another hundred years, in 1824 Niels Henrik Abel showed that there was no general solution to a quintic polynomial equation, presenting his proof in the book Mémoire sur les équations algébriques où on démontre l'impossibilité de la résolution de l'équation générale du cinquième degré (“Memoir on algebraic equations, in which the impossibility of solving the general equation of the fifth degree is proven”).

1724

Going further back to 1724, in that year Daniel Bernoulli published his earliest mathematical work, Exercitationes quaedam mathematicae (“Mathematical Exercises”). One of the results he presented in that book expressed the numbers of the Fibonacci sequence in terms of the golden ratio (PHI). One way to write it is:

F(n) = [PHI^n – (–PHI)^(-n)]/SQRT(5)

where F(n) is the n’th Fibonacci number.

1624

In 1624, the English mathematician Henry Briggs (1561–1630) published his Arithmetica Logarithmica, a folio containing the logarithms of thirty-thousand natural numbers to fourteen decimal places (1-20,000 and 90,001 to 100,000). Briggs had changed the original logarithms invented by John Napier into the common (base 10) logarithms, that were still in use when I went to school in the 1960s (and were frequently referred to as “Briggsian logarithms”).

FOOTNOTE

It would have made a more pleasing story to go back 500 years, as far as 1524, but as far as I could ascertain, there’s nothing momentous from that year. The Italian Gerolamo Cardano (1501–1576), who would later develop the basic rules of modern probability calculations—a definite step towards what would be probability theory, an important technique of modern mathematics—was still a 23-year-old medical student at the time, so his influential mathematical work was several years in the future.

So too in the (near) future was the general solution to the cubic equation by Scipione del Ferro (1465–1526) and Niccolò Tartaglia (1500–1557). And nothing else came close to 1524.

If you go back any earlier than the 1500s, you are no longer in the world of mathematics as we understand it today—a world built on the modern concept of reified, abstract numbers, with basic tools such as abstract algebra, Cartesian geometry, and differential calculus. That world started to come onto the scene in the seventeenth century; the ensuing changes were so revolutionary that the new concepts rapidly came to be what we call “mathematics”, and pretty well obliterated the collection of concepts that had previously carried that name.

The fact that we can read works from as far back as antiquity and see—and understand—them as “what we do today” is because we automatically and implicitly impose a modern conceptual frame on what was in fact conceptually very different. (In fact, books on the history of mathematics typically present the work in modern terms.) But that’s a story for another day—or more precisely, another month. Stay tuned.

BREAKING: A shiny, upgraded version of the “Devlin’s Angle” archive on my profkeithdevlin.org site just went live. I had to modify the site because a recent upgrade to WordPress broke some of the old one, temporarily making all the Angle posts from January 1996 to July 2011 once again inaccessible. (So this is a “Breaking” announcement about breaking.)

Dr. Keith Devlin is an emeritus mathematician at Stanford University, a co-founder and Executive Director Emeritus of the Stanford H-STAR institute, a co-founder of the Stanford mediaX research network, and a Senior Researcher Emeritus at CSLI.