My All 2020 Mathematics A to Z: Hilbert’s Problems

Beth, author of the popular inspiration blog I Didn’t Have My Glasses On …. proposed this topic. Hilbert’s problems are a famous set of questions. I couldn’t hope to summarize them all in an essay of reasonable length. I’d have trouble to do them justice in a short book. But there are still things to say about them.

Color cartoon illustration of a coati in a beret and neckerchief, holding up a director's megaphone and looking over the Hollywood hills. The megaphone has the symbols + x (division obelus) and = on it. The Hollywood sign is, instead, the letters MATHEMATICS. In the background are spotlights, with several of them crossing so as to make the letters A and Z; one leg of the spotlights has 'TO' in it, so the art reads out, subtly, 'Mathematics A to Z'.
Art by Thomas K Dye, creator of the web comics Projection Edge, Newshounds, Infinity Refugees, and Something Happens. He’s on Twitter as @projectionedge. You can get to read Projection Edge six months early by subscribing to his Patreon.

Hilbert’s Problems.

It’s easy to describe what Hilbert’s Problems are. David Hilbert, at the 1900 International Congress of Mathematicians, listed ten important problems of the field. In print he expanded this to 23 problems. They covered topics like number theory, group theory, physics, geometry, differential equations, and more. One of the problems was solved that year. Eight of them have been resolved fully. Another nine have been partially answered. Four remain unanswered. Two have generally been regarded as too vague to resolve.

Everyone in mathematics agrees they were big, important questions. Things that represented the things mathematicians of 1900 would most want to know. Things that guided mathematical research for, so far, 120 years.

It does present us with a dilemma. Were Hilbert’s problems listed because he understood what mathematicians would find important? Or did mathematicians find them important because Hilbert listed them? Sadly, mathematicians know of no professionals who have studied questions like this and could offer insight.

There is reason to say that Hilbert’s judgement was good. He listed, for example, the Riemann hypothesis. The hypothesis is still unanswered. Many interesting results would follow from it being proved true, or proved false, or proved unanswerable. Hilbert did not list Fermat’s Last Theorem, unresolved then. Any mathematician would have liked an answer. But nothing of consequence depends on it. But then he also listed making advances in the calculus of variations. A good goal, but not one that requires particular insight to want.

So here is a related problem. Why hasn’t anyone else made such a list? A concise summary of the problems that guides mathematical research?

It’s not because no one tried. At the 1912 International Conference of Mathematicians, Edmund Landau identified four problems in number theory worth solving. None of them have been solved yet. Yutaka Taniyama listed three dozen problems in 1955. William Thurston put forth 24 questions in 1982. Stephen Smale, famous for work in chaos theory, gathered a list of 18 questions in 1998. Barry Simon offered fifteen of them in 2000. Also in 2000 the Clay Mathematics Institute put up seven problems, with a million-dollar bounty on each. Jair Minoro Abe and Shotaro Tanaka gathered 22 questions for a list for 2001. The United States Defense Advanced Research Projects Agency put out a list of 23 of them in 2007.

Apart from Smale’s and the Clay Mathematics lists I never heard of any of them either. Why not? What was special about Hilbert’s list?

For one, he was David Hilbert. Hilbert was a great mathematician, held in high esteem then and now. Besides his list of problems he’s known for the axiomatization of geometry. This built not just logical rigor but a new, formalist, perspective. Also, he’s known for the formalist approach to mathematics. In this, for example, we give up the annoyingly hard task of saying exactly what we mean by a point and a line and a plane. We instead talk about how points and lines and planes relate to each other, definitions we can give. He’s also known for general relativity: Hilbert and Albert Einstein developed its field equations at the same time. We have Hilbert spaces and Hilbert curves and Hilbert metrics and Hilbert polynomials. Fans of pop mathematics speak of the Hilbert Hotel, a structure with infinitely many rooms and used to explore infinitely large sets.

So he was a great mind, well-versed in many fields. And he was in an enviable position, professor of mathematics at the University of Göttingen. At the time, German mathematics was held in particularly high renown. When you see, for example, mathematicians using ‘Z’ as shorthand for ‘integers’? You are seeing a thing that makes sense in German. (It’s for “Zahlen”, meaning the counting numbers.) Göttingen was at the top of German mathematics, and would be until the Nazi purges of academia. It would be hard to find a more renowned position.

And he was speaking at a great moment. The transition from one century to another is a good one for ambitious projects and declarations to be remembered. But the International Congress of Mathematicians was of particular importance. This was only the second meeting of the International Congress of Mathematicians. International Congresses of anything were new in the late 19th century. Many fields — not only mathematics — were asserting their professionalism at the time. It’s when we start to see professional organizations for specific subjects, not just “Science”. It’s when (American) colleges begin offering elective majors for their undergraduates. When they begin offering PhD degrees.

So it was a field when mathematics, like many fields (and nations), hoped to define its institutional prestige. Having an ambitious goal is one way to define that.

It was also an era when mathematicians were thinking seriously about what the field was about. The results were mixed. In the last decades of the 19th century, mathematicians had put differential calculus on a sound logical footing. But then found strange things in, for example, mathematical physics. Boltzmann’s H-theorem (1872) tells us that entropy in a system of particles always increases. Poincaré’s recurrence theorem (1890) tells us a system of particles has to, eventually, return to its original condition. (Or to something close enough.) And therefore it returns to its original entropy, undoing any increase. Both are sound theorems; how can they not conflict?

Even ancient mathematics had new uncertainty. In 1882 Moritz Pasch discovered that Euclid, and everyone doing plane geometry since then, had been using an axiom no one had acknowledged. (If a line that doesn’t pass through any vertex of a triangle intersects one leg of the triangle, then it also meets one other leg of the triangle.) It’s a small and obvious thing. But if everyone had missed it for thousands of years, what else might be overlooked?

I wish now to share my interpretation of this background. And with it my speculations about why we care about Hilbert’s Problems and not about Thurston’s. And I wish to emphasize that, whatever my pretensions, I am not a professional historian of mathematics. I am an amateur and my training consists of “have read some books about a subject of interest”.

By 1900 mathematicians wanted the prestige and credibility and status of professional organizations. Who would not? But they were also aware the foundation of mathematics was not as rigorous as they had thought. It was not yet the “crisis of foundations” that would drive the philosophy of mathematics in the early 20th century. But the prelude to the crisis was there. And here was a universally respected figure, from the most prestigious mathematical institution. He spoke to all the best mathematicians in a way they could never have been addressed before. And presented a compelling list of tasks to do. These were good tasks, challenging tasks. Many of these tasks seemed doable. One was even done almost right away.

And they covered a broad spectrum of mathematics of the time. Everyone saw at least one problem relevant to their field, or to something close to their field. Landau’s problems, posed twelve years later, were all about number theory. Not even all number theory; about prime numbers. That’s nice, but it will only briefly stir the ambitions of the geometer or the mathematical physicist or the logician.

By the time of Taniyama, though? 1955? Times are changed. Taniyama is no inconsiderable figure. The Taniyama-Shimura theorem is a major piece of elliptic functions. It’s how we have a proof of Fermat’s last theorem. But by then, too, mathematics is not so insecure. We have several good ideas of what mathematics is and why it should work. It has prestige and institutional authority. It has enough Congresses and Associations and Meetings that no one can attend them all. It’s moreso by 1982, when William Thurston set up questions. I know that I’m aware of Stephen Smale’s list because I was a teenager during the great fractals boom of the 80s and knew Smale’s name. Also that he published his list near the time I finished my quals. Quals are an important step in pursuing a doctorate. After them you look for a specific thesis problem. I was primed to hear about great ambitious projects I could not possibly complete.

Only the Clay Mathematics Institute’s list has stood out, aided by its catchy name of Millennium Prizes and its offer of quite a lot of money. That’s a good memory aid. Any lay reader can understand that motivation. Two of the Millennium Prize problems were also Hilbert’s problems. One in whole (the Riemann hypothesis again). One in part (one about solutions to elliptic curves). And as the name states, it came out in 2000. It was a year when many organizations were trying to declare bold and fresh new starts for a century they hoped would be happier than the one before. This, too, helps the memory. Who has any strong associations with 1982 who wasn’t born or got their driver’s license that year?

These are my suppositions, though. I could be giving a too-complicated answer. It’s easy to remember that United States President John F Kennedy challenged the nation to land a man on the moon by the end of the decade. Space enthusiasts, wanting something they respect to happen in space, sometimes long for a president to make a similar strong declaration of an ambitious goal and specific deadline. President Ronald Reagan in 1984 declared there would be a United States space station by 1992. In 1986 he declared there would be by 2000 a National Aerospace Plane, capable of flying from Washington to Tokyo in two hours. President George H W Bush in 1989 declared there would be humans on the Moon “to stay” by 2010 and to Mars thereafter. President George W Bush in 2004 declared the Vision for Space Exploration, bringing humans to the moon again by 2020 and to Mars thereafter.

No one has cared about any of these plans. Possibly because the first time a thing is done, it has a power no repetition can claim. But also perhaps because the first attempt succeeded. Which was not due only to its being first, of course, but to the factors that made its goal important to a great number of people for long enough that it succeeded.

Which brings us back to the Euthyphro-like dilemma of Hilbert’s Problems. Are they influential because Hilbert chose well, or did Hlbert’s choosing them make them influential? I suspect this is a problem that cannot be resolved.

Thank you for reading. This and the other other A-to-Z topics for 2020 should be at this link. All my essays for this and past A-to-Z sequences are at this link. And I am taking nominations for J, K, and L topics. I’m grateful for anything you can offer me.

A Weird Kind Of Ruler

I ran across something neat. It’s something I’ve seen before, but the new element is that I have a name for it. This is the Golomb Ruler. It’s a ruler made with as few marks as possible. The marks are supposed to be arranged so that the greatest possible number of different distances can be made, by measuring between selected pairs of points.

So, like, in a regularly spaced ruler, you have a lot of ways to measure a distance of 1 unit of length. Only one fewer way to measure a distance of 2 units. One fewer still ways to measure a distance of 3 units and so on. Convenient but wasteful of marks. A Golomb ruler might, say, put marks only where the regularly spaced ruler has the units 1, 2, and 4. Then by choosing the correct pairs you can measure a distance of 1, 2, 3, or 4 units.

There’s applications of the Golomb ruler, stuff in information theory and sensor design and stuff. Also logistics. Never mind those. They present a neat little puzzle: can you find, for a given number of marks, the best possible arrangement of them into a ruler? That would be the arrangement that allows the greatest number of different lengths. Or perhaps the one that allows the longest string of whole-number differences. Your definition of best-possible determines what the answer is.

As a number theory problem it won’t surprise you to know there’s not a general answer. If I’m reading accurately most of the known best arrangements — the ones that allow the greatest number of differences — were proven by testing out cases. The 24-mark arrangement needed a test of 555,529,785,505,835,800 different rulers. MathWorld’s page on this tells me that optimal mark placement isn’t known for 25 or more marks. It also says that the 25-mark ruler’s optimal arrangement was published in 2008. So it isn’t just Wikipedia where someone will write an article, and then someone else will throw a new heap of words onto it, and nobody will read to see if the whole thing still makes sense. Wikipedia meanwhile lists optimal configurations for up to 27 points, demonstrated by 2014.

And as this suggests, you aren’t going to discover an optimal arrangement for some number of marks yourself. Unless you should be the first person to figure out an algorithm to do it. It’s not even known how complex an algorithm has to be. It’s suspected that it has to be NP-hard, though. But, while you won’t discover anything new to mathematics in pondering this, you can still have the fun of working out arrangements yourself, at least for a handful of points. There are numbers of points with more than one optimal arrangement.

(Golomb here is Solomon W Golomb, a mathematician and electrical engineer with a long history in information theory and also recreational mathematics problems. There are several parties who independently invented the problem. But Golomb actually did work with rulers, so at least they aren’t incorrectly named.)

Some Difficult Math Problems That You Understand

What the heck, it’s Friday, I might as well put up a little fun thing here. Over on the Maths In A Minute blog is a quartet of mathematics problems which anyone can understand on their first reading of them, but which are unsolved and which have been sitting there unsolved for generations.

There’s, realistically, no chance that you personally are going to solve them. While probably everyone has some mathematical talent, and anyone with the ability to reason is at least in principle endowed with the relevant tools, it’s not likely that you’ll find a path to solving them that hasn’t been tried and that’s failed before. But that doesn’t mean they’re not worth looking at and thinking about because you can nevertheless discover things you didn’t know before, and notice stuff about mathematics that you hadn’t realized. Even if your discovery only delights yourself, it’s still a delight to you.

(I didn’t run across this page myself, though could have; the Algebra Fact of the Day brought me to it.)