Mr Wu, author of the Singapore Maths Tuition blog, suggested another biographical sketch for this year of biographies. Once again it’s of a person too complicated to capture in full in one piece, even at the length I’ve been writing. So I take a slice out of John von Neumann’s life here.
John von Neumann.
In March 1919 the Hungarian People’s Republic, strained by Austria-Hungary’s loss in the Great War, collapsed. The Hungarian Soviet Republic, the world’s second Communist state, replaced it. It was a bad time to be a wealthy family in Budapest. The Hungarian Soviet lasted only a few months. It was crushed by the internal tension between city and countryside. By poorly-fought wars to restore the country’s pre-1914 borders. By the hostility of the Allied Powers. After the Communist leadership fled came a new Republic, and a pogrom. Europeans are never shy about finding reasons to persecute Jewish people. It was a bad time to be a Jewish family in Budapest.
Von Neumann was born to a wealthy, (non-observant) Jewish family in Budapest, in 1903. He acquired the honorific “von” in 1913. His father Max Neumann was honored for service to the Austro-Hungarian Empire and paid for a hereditary appellation.
It is, once again, difficult to encompass von Neumann’s work, and genius, in one piece. He was recognized as genius early. By 1923 he published a logical construction for the counting numbers that’s still the modern default. His 1926 doctoral thesis was in set theory. He was invited to lecture on quantum theory at Princeton by 1929. He was one of the initial six mathematics professors at the Institute for Advanced Study. We have a thing called von Neumann algebras after his work. He gave the first rigorous proof of an ergodic theorem. He partly solved one of Hilbert’s problems. He studied non-linear partial differential equations. He was one of the inventors of the electronic computer as we know it, both the theoretical and the practical ideas.
And, the sliver I choose to focus on today, he made game theory into a coherent field.
The term “game theory” makes it sound like a trifle. We don’t call “genius” anyone who comes up with a better way to play tic-tac-toe. The utility of the subject appears when we notice what von Neumann thought he was writing about. Von Neumann’s first paper on this came in 1928. In 1944 he with Oskar Morgenstern published the textbook Theory Of Games And Economic Behavior. In Chapter 1, Section 1, they set their goals:
The purpose of this book is to present a discussion of some fundamental questions of economic theory which require a treatment different from that which they have found thus far in the literature. The analysis is concerned with some basic problems arising from a study of economic behavior which have been the center of attention of economists for a long time. They have their origin in the attempts to find an exact description of the endeavor of the individual to obtain a maximum of utility, or in the case of the entrepreneur, a maximum of profit.
Somewhere along the line von Neumann became interested in how economics worked. Perhaps because his family had money. Perhaps because he saw how one could model an “ideal” growing economy — matching price and production and demand — as a linear programming question. Perhaps because economics is a big, complicated field with many unanswered questions. There was, for example, little good idea of how attendees at an auction should behave. What is the rational way to bid, to get the best chances of getting the things one wants at the cheapest price?
In 1928, von Neumann abstracted all sorts of economic questions into a basic model. The model has almost no features, so very many games look like it. In this, you have a goal, and a set of options for what to do, and an opponent, who also has options of what to do. Also some rounds to achieve your goal. You see how this abstract a structure describes many things one could do, from playing Risk to playing the stock market.
And von Neumann discovered that, in the right circumstances, you can find a rational way to bid at an auction. Or, at least, to get your best possible outcome whatever the other person does. The proof has the in-retrospect obviousness of brilliance. von Neumann used a fixed-point theorem. Fixed point theorems came to mathematics from thinking of functions as mappings. Functions match elements in a set called the domain to those in a set called the range. The function maps the domain into the range. If the range is also the domain? Then we can do an iterated mapping. Under the right circumstances, there’s at least one point that maps to itself.
In the light of game theory, a function is the taking of a turn. The domain and the range are the states of whatever’s in play. In this type of game, you know all the options everyone has. You know the state of the game. You know what the past moves have all been. You know what you and your opponent hope to achieve. So you can predict your opponent’s strategy. And therefore pick a strategy that gets you the best option available given your opponent is trying to do the same. So will your opponent. So you both end up with the best attainable outcome for the both of you; this is the minimax theorem.
It may strike you that, given this, the game doesn’t need to be played anymore. Just pick your strategy, let your opponent pick one, and the winner is determined. So it would, if we played our strategies perfectly, and if we didn’t change strategies mid-game. I would chuckle at the mathematical view that we study a game to relieve ourselves of the burden of playing. But I know how many grand strategy video games I have that I never have time to play.
After this 1928 paper von Neumann went on to other topics for about a dozen years. Why create a field of mathematics and then do nothing with it? For one, we see it as a gap only because we are extracting, after the fact, this thread of his life. He had other work, particularly in quantum mechanics, operators, measure theory, and lattice theory. He surely did not see himself abandoning a new field. He saw, having found an interesting result, new interesting questions..
But Philip Mirowski’s 1992 paper What Were von Neumann and Morgenstern Trying to Accomplish? points out some context. In September 1930 Kurt Gödel announced his incompleteness proof. Any logical system complex enough has things which are true and can’t be proven. The system doesn’t have to be that complex. Mathematical rigor must depend on something outside mathematics. This shook von Neumann. He would say that after Gödel published, von Neumann never bothered reading another paper on symbolic logic. Mirowski believes this drove von Neumann into what we now call artificial intelligence. At least, into mathematics that draws from empirical phenomena. von Neumann needed time to recover from the shock. And needed the prodding of Morgenstern to return to economics.
After publishing Theory Of Games And Economic Behavior the book … well, Mirowski calls it more “cited in reverence than actually read”. But game theory, as a concept? That took off. It seemed to offer a way to rationalize the world.
von Neumann would become a powerful public intellectual. He would join the Manhattan Project. He showed that the atomic bomb would be more destructive if it exploded kilometers above the ground, rather than at ground level. He was on the target selection committee which, ultimately, slated Hiroshima and Nagasaki for mass murder. He would become a consultant for the Weapons System Evaluation Group. They advised the United States Joint Chiefs of Staff on developing and using new war technology. He described himself, to a Senate committee, as “violently anti-communist and much more militaristic than the norm”. He is quoted in 1950 as remarking, “if you say why not bomb [ the Soviets ] tomorrow, I say, why not today? If you say today at five o’clock, I say why not one o’clock?”
The quote sounds horrifying. It makes game-theory sense, though. If war is inevitable, it is better fought when your opponent is weaker. And while the Soviet Union had won World War II, it was also ruined in the effort.
There is another game-theory-inspired horror for which we credit von Neumann. This is Mutual Assured Destruction. If any use of an atomic, or nuclear, weapon would destroy the instigator in retaliation, then no one would instigate war. So the nuclear powers need, not just nuclear arsenals. They need such vast arsenals that the remnant which survives the first strike can destroy the other powers in the second strike.
Perhaps the reasoning holds together. We did reach the destruction of the Soviet Union without using another atomic weapon in anger. But it is hard to say that was rationally accomplished. There were at least two points, in 1962 and in 1983, when a world-ruining war could too easily have happened, by people following the “obvious” strategy.
Which brings a flaw of game theory, at least as applied to something as complicated as grand strategy. Game theory demands the rules be known, and agreed on. (At least that there is a way of settling rule disputes.) It demands we have the relevant information known truthfully. It demands we know what our actual goals are. It demands that we act rationally, and that our opponent acts rationally. It demands that we agree on what rational is. (Think of, in Doctor Strangelove, the Soviet choice to delay announcing its doomsday machine’s completion.) Few of these conditions obtain in grand strategy. They barely obtain in grand strategy games. von Neumann was aware of at least some of these limitations, though he did not live long enough to address them. He died of either bone, pancreatic, or prostate cancer, likely caused by radiation exposure working at Los Alamos.
Game theory has been, and is, a great tool in many fields. It gives us insight into human interactions. It does good work in economics, in biology, in computer science, in management. But we can come to very bad conditions when we forget the difference between the game we play and the game we modelled. And if we forget that the game is value-indifferent. The theory makes no judgements about the ethical nature of the goal. It can’t, any more than the quadratic equation can tell us whether ‘x’ is which fielder will catch the fly ball or which person will be killed by a cannonball.
It makes an interesting parallel to the 19th century’s greatest fusion of mathematics and economics. This was utilitarianism, the attempt to bring scientific inquiry to the study of how society should be set up. Utilitarianism offers exciting insights into, say, how to allocate public services. But it struggles to explain why we should refrain from murdering someone whose death would be convenient. We need a reason besides the maximizing of utility.
No war is inevitable. One comes about only after many choices. Some are grand choices, such as a head of government issuing an ultimatum. Some are petty choices, such as the many people who enlist as the sergeants that make an army exist. We like to think we choose rationally. Psychological experiments, and experience, and introspection tell us we more often choose and then rationalize.
von Neumann was a young man, not yet in college, during the short life of the Hungarian Soviet Republic, and the White Terror that followed. I do not know his biography well enough to say how that experience motivated his life’s reasoning. I would not want to say that 1919 explained it all. The logic of a life is messier than that. I bring it up in part to fight the tendency of online biographic sketches to write as though he popped into existence, calculated a while, inspired a few jokes, and vanished. And to reiterate that even mathematics never exists without context. Even what seem to be pure questions on an abstract idea of a game is often inspired by a practical question. And that work is always done in a context that affects how we evaluate it.
My love and I, like many people, tried last week to see the comet NEOWISE. It took several attempts. When finally we had binoculars and dark enough sky we still had the challenge of where to look. Finally determined searching and peripheral vision (which is more sensitive to faint objects) found the comet. But how to guide the other to a thing barely visible except with binoculars? Between the silhouettes of trees and a convenient pair of guide stars we were able to put the comet’s approximate location in words. Soon we were experts at finding it. We could turn a head, hold up the binoculars, and see a blue-ish puff of something.
To perceive a thing is not to see it. Astronomy is full of things seen but not recognized as important. There is a great need for people who can describe to us how to see a thing. And this is part of the significance of J Willard Gibbs.
American science, in the 19th century, had an inferiority complex compared to European science. Fairly, to an extent: what great thinkers did the United States have to compare to William Thompson or Joseph Fourier or James Clerk Maxwell? The United States tried to argue that its thinkers were more practical minded, with Joseph Henry as example. Without downplaying Henry’s work, though? The stories of his meeting the great minds of Europe are about how he could fix gear that Michael Faraday could not. There is a genius in this, yes. But we are more impressed by magnetic fields than by any electromagnet.
Gibbs is the era’s exception, a mathematical physicist of rare insight and creativity. In his ability to understand problems, yes. But also in organizing ways to look at problems so others can understand them better. A good comparison is to Richard Feynman, who understood a great variety of problems, and organized them for other people to understand. No one, then or now, doubted Gibbs compared well to the best European minds.
Gibbs’s life story is almost the type case for a quiet academic life. He was born into an academic/ministerial family. Attended Yale. Earned what appears to be the first PhD in engineering granted in the United States, and only the fifth non-honorary PhD in the country. Went to Europe for three years, then came back home, got a position teaching at Yale, and never left again. He was appointed Professor of Mathematical Physics, the first such in the country, at age 32 and before he had even published anything. This speaks of how well-connected his family was. Also that he was well-off enough not to need a salary. He wouldn’t take one until 1880, when Yale offered him two thousand per year against Johns Hopkins’s three.
Between taking his job and taking his salary, Gibbs took time to remake physics. This was in thermodynamics, possibly the most vibrant field of 19th century physics. The wonder and excitement we see in quantum mechanics resided in thermodynamics back then. Though with the difference that people with a lot of money were quite interested in the field’s results. These were people who owned railroads, or factories, or traction companies. Extremely practical fields.
What Gibbs offered was space, particularly, phase space. Phase space describes the state of a system as a point in … space. The evolution of a system is typically a path winding through space. Constraints, like the conservation of energy, we can usually understand as fixing the system to a surface in phase space. Phase space can be as simple as “the positions and momentums of every particle”, and that often is what we use. It doesn’t need to be, though. Gibbs put out diagrams where the coordinates were things like temperature or pressure or entropy or energy. Looking at these can let one understand a thermodynamic system. They use our geometric sense much the same way that charts of high- and low-pressure fronts let one understand the weather. James Clerk Maxwell, famous for electromagnetism, was so taken by this he created plaster models of the described surface.
This is, you might imagine, pretty serious, heady stuff. So you get why Gibbs published it in the Transactions of the Connecticut Academy: his brother-in-law was the editor. It did not give the journal lasting fame. It gave his brother-in-law a heightened typesetting bill, and Yale faculty and New Haven businessmen donated funds.
Which gets to the less-happy parts of Gibbs’s career. (I started out with ‘less pleasant’ but it’s hard to spot an actually unpleasant part of his career.) This work sank without a trace, despite Maxwell’s enthusiasm. It emerged only in the middle of the 20th century, as physicists came to understand their field as an expression of geometry.
That’s all right. Chemists understood the value of Gibbs’s thermodynamics work. He introduced the enthalpy, an important thing that nobody with less than a Master’s degree in Physics feels they understand. Changes of enthalpy describe how heat transfers. And the Gibbs Free Energy, which measures how much reversible work a system can do if the temperature and pressure stay constant. A chemical reaction where the Gibbs free energy is negative will happen spontaneously. If the system’s in equilibrium, the Gibbs free energy won’t change. (I need to say the Gibbs free energy as there’s a different quantity, the Helmholtz free energy, that’s also important but not the same thing.) And, from this, the phase rule. That describes how many independently-controllable variables you can see in mixing substances.
There are more pieces. They don’t all fit in a neat linear timeline; nobody’s life really does. Gibbs’s thermodynamics work, leading into statistical mechanics, foreshadows much of quantum mechanics. He’s famous for the Gibbs Paradox, which concerns the entropy of mixing together two different kinds of gas. Why is this different from mixing together two containers of the same kind of gas? And the answer is that we have to think more carefully about what we mean by entropy, and about the differences between containers.
There is a Gibbs phenomenon, known to anyone studying Fourier series. The Fourier series is a sum of sine and cosine functions. It approximates an arbitrary original function. The series is a continuous function; you could draw it without lifting your pen. If the original function has a jump, though? A spot where you have to lift your pen? The Fourier series for that represents the jump with a region where its quite-good approximation suddenly turns bad. It wobbles around the ‘correct’ values near the jump. Using more terms in the series doesn’t make the wobbling shrink. Gibbs described it, in studying sawtooth waves. As it happens, Henry Wilbraham first noticed and described this in 1848. But Wilbraham’s work went unnoticed until after Gibbs’s rediscovery.
And then there was a bit in which Gibbs was intrigued by a comet that prolific comet-spotter Lewis Swift observed in 1880. Finding the orbit of a thing from a handful of observations is one of the great problems of astronomical mathematics. Karl Friedrich Gauss started the 19th century with his work projecting the orbit of the newly-discovered and rapidly-lost asteroid Ceres. Gibbs put his vector notation to the work of calculating orbits. His technique, I am told by people who seem to know, is less difficult and more numerically stable than was earlier used.
Swift’s comet of 1880, it turns out, was spotted in 1869 by Wilhelm Tempel. It was lost after its 1908 perihelion. Comets have a nasty habit of changing their orbits on us. But it was rediscovered in 2001 by the Lincoln Near-Earth Asteroid Research program. It’s next to reach perihelion the 26th of November, 2020. You might get to see this, another thing touched by J Willard Gibbs.
Dina Yagodich suggested today’s A-to-Z topic. I thought a quick little biography piece would be a nice change of pace. I discovered things were more interesting than that.
I realized preparing for this that I have never read a biography of Fibonacci. This is hardly unique to Fibonacci. Mathematicians buy into the legend that mathematics is independent of human creation. So the people who describe it are of lower importance. They learn a handful of romantic tales or good stories. In this way they are much like humans. I know at least a loose sketch of many mathematicians. But Fibonacci is a hard one for biography. Here, I draw heavily on the book Fibonacci, his numbers and his rabbits, by Andriy Drozdyuk and Denys Drozdyuk.
We know, for example, that Fibonacci lived until at least 1240. This because in 1240 Pisa awarded him an annual salary in recognition of his public service. We think he was born around 1170, and died … sometime after 1240. This seems like a dismal historical record. But, for the time, for a person of slight political or military importance? That’s about as good as we could hope for. It is hard to appreciate how much documentation we have of lives now, and how recent a phenomenon that is.
Even a fact like “he was alive in the year 1240” evaporates under study. Italian cities, then as now, based the year on the time since the notional birth of Christ. Pisa, as was common, used the notional conception of Christ, on the 25th of March, as the new year. But we have a problem of standards. Should we count the year as the number of full years since the notional conception of Christ? Or as the number of full and partial years since that important 25th of March?
If the question seems confusing and perhaps angering let me try to clarify. Would you say that the notional birth of Christ that first 25th of December of the Christian Era happened in the year zero or in the year one? (Pretend there was a year zero. You already pretend there was a year one AD.) Pisa of Leonardo’s time would have said the year one. Florence would have said the year zero, if they knew of “zero”. Florence matters because when Florence took over Pisa, they changed Pisa’s dating system. Sometime later Pisa changed back. And back again. Historians writing, aware of the Pisan 1240 on the document, may have corrected it to the Florence-style 1241. Or, aware of the change of the calendar and not aware that their source already accounted for it, redated it 1242. Or tried to re-correct it back and made things worse.
This is not a problem unique to Leonardo. Different parts of Europe, at the time, had different notions for the year count. Some also had different notions for what New Year’s Day would be. There were many challenges to long-distance travel and commerce in the time. Not the least is that the same sun might shine on at least three different years at once.
We call him Fibonacci. Did he? The question defies a quick answer. His given name was Leonardo, and he came from Pisa, so a reliable way to address him would have “Leonardo of Pisa”, albeit in Italian. He was born into the Bonacci family. He did in some manuscripts describe himself as “Leonardo filio Bonacci Pisano”, give or take a few letters. My understanding is you can get a good fun quarrel going among scholars of this era by asking whether “Filio Bonacci” would mean “the son of Bonacci” or “of the family Bonacci”. Either is as good for us. It’s tempting to imagine the “Filio” being shrunk to “Fi” and the two words smashed together. But that doesn’t quite say that Leonardo did that smashing together.
Nor, exactly, when it did happen. We see “Fibonacci” used in mathematical works in the 19th century, followed shortly by attempts to explain what it means. We know of a 1506 manuscript identifying Leonardo as Fibonacci. But there remains a lot of unexplored territory.
If one knows one thing about Fibonacci though, one knows about the rabbits. They give birth to more rabbits and to the Fibonacci Sequence. More on that to come. If one knows two things about Fibonacci, the other is about his introducing Arabic numerals to western mathematics. I’ve written of this before. And the subject is … more ambiguous, again.
Most of what we “know” of Fibonacci’s life is some words he wrote to explain why he was writing his bigger works. If we trust he was not creating a pleasant story for the sake of engaging readers, then we can finally say something. (If one knows three things about Fibonacci, and then five things, and then eight, one is making a joke.)
Fibonacci’s father was, in the 1290s, posted to Bejaia, a port city on the Algerian coast. The father did something for Pisa’s duana there. And what is a duana? … Again, certainty evaporates. We have settled on saying it’s a customs house, and suppose our readers know what goes on in a customs house. The duana had something to do with clearing trade through the port. His father’s post was as a scribe. He was likely responsible for collecting duties and registering accounts and keeping books and all that. We don’t know how long Fibonacci spent there. “Some days”, during which he alleges he learned the digits 1 through 9. And after that, travelling around the Mediterranean, he saw why this system was good, and useful. He wrote books to explain it all and convince Europe that while Roman numerals were great, Arabic numerals were more practical.
It is always dangerous to write about “the first” person to do anything. Except for Yuri Gagarin, Alexei Leonov, and Neil Armstrong, “the first” to do anything dissolves into ambiguity. Gerbert, who would become Pope Sylvester II, described Arabic numerals (other than zero) by the end of the 10th century. He added in how this system along with the abacus made computation easier. Arabic numerals appear in the Codex Conciliorum Albeldensis seu Vigilanus, written in 976 AD in Spain. And it is not as though Fibonacci was the first European to travel to a land with Arabic numerals, or the first perceptive enough to see their value.
Allow that, though. Every invention has precursors, some so close that it takes great thinking to come up with a reason to ignore them. There must be some credit given to the person who gathers an idea into a coherent, well-explained whole. And with Fibonacci, and his famous manuscript of 1202, the Liber Abaci, we have … more frustration.
It’s not that Liber Abaci does not exist, or that it does not have what we credit it for having. We don’t have any copies of the 1202 edition, but we do have a 1228 manuscript, at least, and that starts out with the Arabic numeral system. And why this system is so good, and how to use it. It should convince anyone who reads it.
If anyone read it. We know of about fifteen manuscripts of Liber Abaci, only two of them reasonably complete. This seems sparse even for manuscripts in the days they had to be hand-copied. This until you learn that Baldassarre Boncompagni published the first known printed version in 1857. In print, in Italian, it took up 459 pages of text. Its first English translation, published by Laurence E Sigler in 2002(!) takes up 636 pages (!!). Suddenly it’s amazing that as many as two complete manuscripts survive. (Wikipedia claims three complete versions from the 13th and 14th centuries exist. And says there are about nineteen partial manuscripts with another nine incomplete copies. I do not explain this discrepancy.)
So perhaps only a handful of people read Fibonacci. Ah, but if they were the right people? He could have been a mathematical Velvet Underground, read by a hundred people, each of whom founded a new mathematics.
This is not to say Fibonacci copied any of these (and more) Indian mathematicians. The world is large and manuscripts are hard to read. The sequence can be re-invented by anyone bored in the right way. Ah, but think of those who learned of the sequence and used it later on, following Fibonacci’s lead. For example, in 1611 Johannes Kepler wrote a piece that described Fibonacci’s sequence. But that does not name Fibonacci. He mentions other mathematicians, ancient and contemporary. The easiest supposition is he did not know he was writing something already seen. In 1844, Gabriel Lamé used Fibonacci numbers in studying algorithm complexity. He did not name Fibonacci either, though. (Lamé is famous today for making some progress on Fermat’s last theorem. He’s renowned for work in differential equations and on ellipse-like curves. If you have thought what a neat weird shape the equation can describe you have tread in Lamé’s path.)
Things picked up for Fibonacci’s reputation in 1876, thanks to Édouard Lucas. (Lucas is notable for other things. Normal people might find interesting that he proved by hand the number was prime. This seems to be the largest prime number ever proven by hand. He also created the Tower of Hanoi problem.) In January of 1876, Lucas wrote about the Fibonacci sequence, describing it as “the series of Lamé”. By May, though in writing about prime numbers, he has read Boncompagni’s publications. He says how this thing “commonly known as the sequence of Lamé was first presented by Fibonacci”.
And Fibonacci caught Lucas’s imagination. Lucas shared, particularly, the phrasing of this sequence as something in the reproduction of rabbits. This captured mathematicians’, and then people’s imaginations. It’s akin to Émile Borel’s room of a million typing monkeys. By the end of the 19th century Leonardo of Pisa had both a name and fame.
We can still ask why. The proximate cause is Édouard Lucas, impressed (I trust) by Boncompagni’s editions of Fibonacci’s work. Why did Baldassarre Boncompagni think it important to publish editions of Fibonacci? Well, he was interested in the history of science. He edited the first Italian journal dedicated to the history of mathematics. He may have understood that Fibonacci was, if not an important mathematician, at least one who had interesting things to write. Boncompagni’s edition of Liber Abaci came out in 1857. By 1859 the state of Tuscany voted to erect a statue.
So I speculate, without confirming that at least some of Fibonacci’s good name in the 19th century was a reflection of Italian unification. The search for great scholars whose intellectual achievements could reflect well on a nation trying to build itself.
And so we have bundles of ironies. Fibonacci did write impressive works of great mathematical insight. And he was recognized at the time for that work. The things he wrote about Arabic numerals were correct. His recommendation to use them was taken, but by people who did not read his advice. After centuries of obscurity he got some notice. And a problem he did not create nor particularly advance brought him a fame that’s lasted a century and a half now, and looks likely to continue.
I am always amazed to learn there are people not interested in history.
To start this year’s great glossary project Mr Wu, author of the MathTuition88.com blog, had a great suggestion: The Atiyah-Singer Index Theorem. It’s an important and spectacular piece of work. I’ll explain why I’m not doing that in a few sentences.
Mr Wu pointed out that a biography of Michael Atiyah, one of the authors of this theorem, might be worth doing. GoldenOj endorsed the biography idea, and the more I thought it over the more I liked it. I’m not able to do a true biography, something that goes to primary sources and finds a convincing story of a life. But I can sketch out a bit, exploring his work and why it’s of note.
Theodore Frankel’s The Geometry of Physics: An Introduction is a wonderful book. It’s 686 pages, including the index. It all explores how our modern understanding of physics is our modern understanding of geometry. On page 465 it offers this:
The Atiyah-Singer index theorem must be considered a high point of geometrical analysis of the twentieth century, but is far too complicated to be considered in this book.
I know when I’m licked. Let me attempt to look at one of the people behind this theorem instead.
The Riemann Hypothesis is about where to find the roots of a particular infinite series. It’s been out there, waiting for a solution, for a century and a half. There are many interesting results which we would know to be true if the Riemann Hypothesis is true. In 2018, Michael Atiyah declared that he had a proof. And, more, an amazing proof, a short proof. Albeit one that depended on a great deal of background work and careful definitions. The mathematical community was skeptical. It still is. But it did not dismiss outright the idea that he had a solution. It was plausible that Atiyah might solve one of the greatest problems of mathematics in something that fits on a few PowerPoint slides.
So think of a person who commands such respect.
His proof of the Riemann Hypothesis, as best I understand, is not generally accepted. For example, it includes the fine structure constant. This comes from physics. It describes how strongly electrons and photons interact. The most compelling (to us) consequence of the Riemann Hypothesis is in how prime numbers are distributed among the integers. It’s hard to think how photons and prime numbers could relate. But, then, if humans had done all of mathematics without noticing geometry, we would know there is something interesting about π. Differential equations, if nothing else, would turn up this number. We happened to discover π in the real world first too. If it were not familiar for so long, would we think there should be any commonality between differential equations and circles?
I do not mean to say Atiyah is right and his critics wrong. I’m no judge of the matter at all. What is interesting is that one could imagine a link between a pure number-theory matter like the Riemann hypothesis and a physical matter like the fine structure constant. It’s not surprising that mathematicians should be interested in physics, or vice-versa. Atiyah’s work was particularly important. Much of his work, from the late 70s through the 80s, was in gauge theory. This subject lies under much of modern quantum mechanics. It’s born of the recognition of symmetries, group operations that you can do on a field, such as the electromagnetic field.
In a sequence of papers Atiyah, with other authors, sorted out particular cases of how magnetic monopoles and instantons behave. Magnetic monopoles may sound familiar, even though no one has ever seen one. These are magnetic points, an isolated north or a south pole without its opposite partner. We can understand well how they would act without worrying about whether they exist. Instantons are more esoteric; I don’t remember encountering the term before starting my reading for this essay. I believe I did, encountering the technique as a way to describe the transitions between one quantum state and another. Perhaps the name failed to stick. I can see where there are few examples you could give an undergraduate physics major. And it turns out that monopoles appear as solutions to some problems involving instantons.
This was, for Atiyah, later work. It arose, in part, from bringing the tools of index theory to nonlinear partial differential equations. This index theory is the thing that got us the Atiyah-Singer Index Theorem too complicated to explain in 686 pages. Index theory, here, studies questions like “what can we know about a differential equation without solving it?” Solving a differential equation would tell us almost everything we’d like to know, yes. But it’s also quite hard. Index theory can tell us useful things like: is there a solution? Is there more than one? How many? And it does this through topological invariants. A topological invariant is a trait like, for example, the number of holes that go through a solid object. These things are indifferent to operations like moving the object, or rotating it, or reflecting it. In the language of group theory, they are invariant under a symmetry.
It’s startling to think a question like “is there a solution to this differential equation” has connections to what we know about shapes. This shows some of the power of recasting problems as geometry questions. From the late 50s through the mid-70s, Atiyah was a key person working in a topic that is about shapes. We know it as K-theory. The “K” from the German Klasse, here. It’s about groups, in the abstract-algebra sense; the things in the groups are themselves classes of isomorphisms. Michael Atiyah and Friedrich Hirzebruch defined this sort of group for a topological space in 1959. And this gave definition to topological K-theory. This is again abstract stuff. Frankel’s book doesn’t even mention it. It explores what we can know about shapes from the tangents to the shapes.
And it leads into cobordism, also called bordism. This is about what you can know about shapes which could be represented as cross-sections of a higher-dimension shape. The iconic, and delightfully named, shape here is the pair of pants. In three dimensions this shape is a simple cartoon of what it’s named. On the one end, it’s a circle. On the other end, it’s two circles. In between, it’s a continuous surface. Imagine the cross-sections, how on separate layers the two circles are closer together. How their shapes distort from a real circle. In one cross-section they come together. They appear as two circles joined at a point. In another, they’re a two-looped figure. In another, a smoother circle. Knowing that Atiyah came from these questions may make his future work seem more motivated.
But how does one come to think of the mathematics of imaginary pants? Many ways. Atiyah’s path came from his first research specialty, which was algebraic geometry. This was his work through much of the 1950s. Algebraic geometry is about the kinds of geometric problems you get from studying algebra problems. Algebra here means the abstract stuff, although it does touch on the algebra from high school. You might, for example, do work on the roots of a polynomial, or a comfortable enough equation like . Atiyah had started — as an undergraduate — working on projective geometries. This is what one curve looks like projected onto a different surface. This moved into elliptic curves and into particular kinds of transformations on surfaces. And algebraic geometry has proved important in number theory. You might remember that the Wiles-Taylor proof of Fermat’s Last Theorem depended on elliptic curves. Some work on the Riemann hypothesis is built on algebraic topology.
(I would like to trace things farther back. But the public record of Atiyah’s work doesn’t offer hints. I can find amusing notes like his father asserting he knew he’d be a mathematician. He was quite good at changing local currency into foreign currency, making a profit on the deal.)
It’s possible to imagine this clear line in Atiyah’s career, and why his last works might have been on the Riemann hypothesis. That’s too pat an assertion. The more interesting thing is that Atiyah had several recognizable phases and did iconic work in each of them. There is a cliche that mathematicians do their best work before they are 40 years old. And, it happens, Atiyah did earn a Fields Medal, given to mathematicians for the work done before they are 40 years old. But I believe this cliche represents a misreading of biographies. I suspect that first-rate work is done when a well-prepared mind looks fresh at a new problem. A mathematician is likely to have these traits line up early in the career. Grad school demands the deep focus on a particular problem. Getting out of grad school lets one bring this deep knowledge to fresh questions.
It is easy, in a career, to keep studying problems one has already had great success in, for good reason and with good results. It tends not to keep producing revolutionary results. Atiyah was able — by chance or design I can’t tell — to several times venture into a new field. The new field was one that his earlier work prepared him for, yes. But it posed new questions about novel topics. And this creative, well-trained mind focusing on new questions produced great work. And this is one way to be credible when one announces a proof of the Riemann hypothesis.
Here is something I could not find a clear way to fit into this essay. Atiyah recorded some comments about his life for the Web of Stories site. These are biographical and do not get into his mathematics at all. Much of it is about his life as child of British and Lebanese parents and how that affected his schooling. One that stood out to me was about his peers at Manchester Grammar School, several of whom he rated as better students than he was. Being a good student is not tightly related to being a successful academic. Particularly as so much of a career depends on chance, on opportunities happening to be open when one is ready to take them. It would be remarkable if there wre three people of greater talent than Atiyah who happened to be in the same school at the same time. It’s not unthinkable, though, and we may wonder what we can do to give people the chance to do what they are good in. (I admit this assumes that one finds doing what one is good in particularly satisfying or fulfilling.) In looking at any remarkable talent it’s fair to ask how much of their exceptional nature is that they had a chance to excel.
It’s a collection of short autobiographical essays from mathematicians. The focus is on the hard part. Getting to an advanced degree implies some a lot of work, much of it intellectually challenging. I felt a great relief every essay I found where someone else struggled with real analysis, or the great obstacle of qualifying exams. And I can take some comfort, thinking back at all the ways I was a bad student, to know how many other people were also bad students in the same ways. (Is there anyone who goes on to a doctoral program in mathematics who learns how to study before about two years into the program?)
There are other challenges. It’s not a surprise that American life, even academic life, is harder if you can’t pass as a heterosexual white male. A number of the writers are women, or black people, or non-heterosexual people. And they’re generous enough to share infuriating experiences. One can admire, for example, Robin Blankenship’s social engineering that convinced her algebraic topology instructor that she did, indeed, understand the material as well as her male peers did. One can feel the horror of the professor saying he had been giving her lousy scores because his impression was that she didn’t know what she was doing even as she gave correct answers. And she wasn’t the only person who struggled against a professor grading from the impression of his students rather than their actual work. There is always subjectivity and judgement in grading, even in mathematics, even in something as logically “pure” as a proof. But that …
I do not remember having a professor grading me in a way that seemed out of line with my work. The grade to expect if the grading were done single-blind, with no information about the identity of the exam-taker. But, then, I’m white and male and anyone looking at me would see someone who looks like he should be a mathematician. I think my presentation is, to be precise, “high school physics teacher who thinks this class just might be ready to see an amazing demonstration about something we call the Conservation of Angular Momentum”. That’s near enough to “mathematician” for most needs. This makes many of these essays, to me, embarrassing eye-openings.
So I think the book’s worth reading. And it is a great number of essays, most of them two or three pages. So it’s one you can pick up and read when you just have a few free minutes.
I haven’t forgotten about writing original material here — actually I’ve been trying to think of why something I’ve not thought about a long while is true, which is embarrassing and hard to do — but in the meanwhile I’d like to remember Leonhard Euler’s 306th birthday and point to Richard Elwes’s essay here about Euler’s totient function. “Totient” is, as best I can determine, a word that exists only for this mathematical concept — it’s the count of how many numbers are relatively prime to a given number — but even if the word comes only from the mildly esoteric world of prime number studies, it’s still one of my favorite mathematical terms. It feels like a word that ought to be more successful. Someday I’ll probably get in a nasty argument with other people playing Boggle about it.
Apparently, though, Euler didn’t dub this quantity the “totient”, and the word is a neologism coined by James Joseph Sylvester (1814 – 1897). That’s pretty respectable company, though: Sylvester — whose name you probably brush up against if you study mathematical matrices — is widely praised for his skill in naming things, although the only terms I know offhand that he gave us were “totient” and “discriminant”. That term in the quadratic formula which tells you whether a quadratic equation has two real, one real, or two imaginary solutions, was a name (not a concept) given by him, and he named (and extended) the similar concept for cubic equations. I do believe there are more such Sylvester-dubbed terms, just, that we need a Wikipedia category to gather them together.