My All 2020 Mathematics A to Z: John von Neumann


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.

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.

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.


Thank you all for reading. This grew a bit more serious than I had anticipated. This and all the other 2020 A-to-Z essays should appear at this link. Both the 2020 and all past A-to-Z essays should be at this link.

I am hosting the Playful Math Education Blog Carnival at the end of September, so appreciate any educational or recreational or fun mathematics material you know about. I’m hoping to publish next week and so hope that you can help me this week.

And, finally, I am open for mathematics topics starting with P, Q, and R to write about next month. I should be writing about them this month and getting ahead of deadline, but that seems not to be happening.

I’m looking for P, Q, and R topics for the All 2020 A-to-Z


And now I am at the actual halfway point in the year’s A-to-Z. I’m still not as far ahead of deadline as I want to be, but I am getting at least a little better.

As I continue to try to build any kind of publication buffer, I’d like to know of any mathematical terms starting with the letters P, Q, or R that you’d like me to try writing. I might write about anything, of course; my criteria is what topic I think I could write something interesting about. But that’s a pretty broad set of things. Part of the fun of an A-to-Z series is learning enough about a subject I haven’t thought about much, in time to write a thousand-or-more words about it.

So please leave a comment with any topics you’d like to see discussed. Also please leave a mention of your own blog or YouTube channel or Twitter account or anything else you do that’s worth some attention. I’m happy giving readers new things to pay attention to, even when it’s not me.

It hasn’t happened yet, but I am open to revisiting a topic I’ve written about before, in case I think I can do better. My list of past topics may let you know if something satisfactory’s already been written about, say, quaternions. But if you don’t like what I already have about something, make a suggestion. I might do better.

Topics I’ve already covered, starting with the letter ‘P’, are:


Topics I’ve already covered, starting with the letter ‘Q’, are:


Topics I’ve already covered, starting with the letter ‘R’, are:

Thanks for reading and thanks for your thoughts.

My All 2020 Mathematics A to Z: Möbius Strip


Jacob Siehler suggested this topic. I had to check several times that I hadn’t written an essay about the Möbius strip already. While I have talked about it some, mostly in comic strip essays, this is a chance to specialize on the shape in a way I haven’t before.

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.

Möbius Strip.

I have ridden at least 252 different roller coasters. These represent nearly every type of roller coaster made today, and most of the types that were ever made. One type, common in the 1920s and again since the 70s, is the racing coaster. This is two roller coasters, dispatched at the same time, following tracks that are as symmetric as the terrain allows. Want to win the race? Be in the train with the heavier passenger load. The difference in the time each train takes amounts to losses from friction, and the lighter train will lose a bit more of its speed.

There are three special wooden racing coasters. These are Racer at Kennywood Amusement Park (Pittsburgh), Grand National at Blackpool Pleasure Beach (Blackpool, England), and Montaña Rusa at La Feria Chapultepec Magico (Mexico City). I’ve been able to ride them all. When you get into the train going up, say, the left lift hill, you return to the station in the train that will go up the right lift hill. These racing roller coasters have only one track. The track twists around itself and becomes a Möbius strip.

Picture of some of the Grand National track, at Blackpool Pleasure Beach, showing off particularly a sign with an illustration of the launch station and 'What The Papers Say', showing the British national papers' 'reviews' of the roller coaster.
I’m not surprised I don’t have good pictures of Grand National. I’ve only been able to visit Blackpool Pleasure Beach the one time. And a few days before my camera had just been thoroughly soaked in heavy rain. I could only take a picture by turning it on; the camera would power up, take a picture as soon as it was ready, and power down again. It’s the other parks I’m surprised I lack better pictures for.

This is a fun use of the Möbius strip. The shape is one of the few bits of advanced mathematics to escape into pop culture. Maybe dominates it, in a way nothing but the blackboard full of calculus equations does. In 1958 the public intellectual and game show host Clifton Fadiman published the anthology Fantasia Mathematica. It’s all essays and stories and poems with some mathematical element. I no longer remember how many of the pieces were about the Möbius strip one way or another. The collection does include A J Deutschs’s classic A Subway Named Möbius. In this story the Boston subway system achieves hyperdimensional complexity. It does not become a Möbius strip, though, in that story. It might be one in reality anyway.

The Möbius strip we name for August Ferdinand Möbius, who in 1858 was the second person known to have noticed the shape’s curious properties. The first — to notice, in 1858, and to publish, in 1862 — was Johann Benedict Listing. Listing seems to have coined the term “topology” for the field that the Möbius strip would be emblem for. He wrote one of the first texts on the field. He also seems to have coined terms like “entrophic phenomena” and “nodal points” and “geoid” and “micron”, for a millionth of a meter. It’s hard to say why we don’t talk about Listing strips instead. Mathematical fame is a strange, unpredictable creature. There is a topological invariant, the Listing Number, named for him. And he’s known to ophthalmologists for Listing’s Law, which describes how human eyes orient themselves.

The Möbius strip is an easy thing to construct. Loop a ribbon back to itself, with an odd number of half-twist before you fasten the ends together. Anyone could do it. So it seems curious that for all recorded history nobody thought to try. Not until 1858 when Lister and then Möbius hit on the same idea.

Kennywood Racer blue roller coaster train, leaving the station on the (camera) left track.
It’s Kennywood’s Racer I’m surprised I don’t have a picture that shows what I really want. We get to Kennywood basically every year, 2020 excepted for obvious reasons. But there also really aren’t good observation tower-type attractions that let you get a photograph of Racer as a whole, I’ll tell myself is the problem.

An irresistible thing, while riding these roller coasters, is to try to find the spot where you “switch”, where you go from being on the left track to the right. You can’t. The track is — well, the track is a series of metal straps bolted to a base of wood. (The base the straps are bolted to is what makes it a wooden roller coaster. The great lattice holding the tracks above ground have nothing to do with it.) But the path of the tracks is a continuous whole. To split it requires the same arbitrariness with which mapmakers pick a prime meridian. It’s obvious that the “longitude” of a cylinder or a rubber ball is arbitrary. It’s not obvious that roller coaster tracks should have the same property. Until you draw the shape in that ∞-loop figure we always see. Then you can get lost imagining a walk along the surface.

And it’s not true that nobody thought to try this shape before 1858. Julyan H E Cartwright and Diego L González wrote a paper searching for pre-Möbius strips. They find some examples. To my eye not enough examples to support their abstract’s claim of “lots of them”, but I trust they did not list every example. One example is a Roman mosaic showing Aion, the God of Time, Eternity, and the Zodiac. He holds a zodiac ring that is either a Möbius strip or cylinder with artistic errors. Cartwright and González are convinced. I’m reminded of a Looks Good On Paper comic strip that forgot to include the needed half-twist.

Mobius Trip. A car, loaded for vacation, with someone in it asking 'Are we there yet? Are we there yet? Are we there yet?' The road is along a Mobius strip, with roadside bits like deer or road signs or opossums crossing the road on the margins.
Dan Collins’s Looks Good on Paper for the 10th of September, 2018. It originally ran the 10th of September, 2016. It falls a half-twist short of being the Möbius strip it intends to be. Essays in which I discuss Looks Good On Paper are at this link.

Islamic science gives us a more compelling example. We have a book by Ismail al-Jazari dated 1206, The Book of Knowledge of Ingenious Mechanical Devices. Some manuscripts of it illustrate a chain pump, with the chain arranged as a Möbius strip. Cartwright and González also note discussions in Scientific American, and other engineering publications in the United States, about drive and conveyor belts with the Möbius strip topology. None of those predate Lister or Möbius, or apparently credit either. And they do come quite soon after. It’s surprising something might leap from abstract mathematics to Yankee ingenuity that fast.

If it did. It’s not hard to explain why mechanical belts didn’t consider Möbius strip shapes before the late 19th century. Their advantage is that the wear of the belt distributes over twice the surface area, the “inside” and “outside”. A leather belt has a smooth and a rough side. Many other things you might make a belt from have a similar asymmetry. By the late 19th century you could make a belt of rubber. Its grip and flexibility and smoothness is uniform on all sides. “Balancing” the use suddenly could have a point.

I still find it curious almost no one drew or speculated about or played with these shapes until, practically, yesterday. The shape doesn’t seem far away from a trefoil knot. The recycling symbol, three folded-over arrows, suggests a Möbius strip. The strip evokes the ∞ symbol, although that symbol was not attached to the concept of “infinity” until John Wallis put it forth in 1655.

Kennywood Racer blue roller coaster train arriving at the station on the (camera) right track; the red train is approaching the (camera) left track.
Also Racer is usually quite packed, and we’d rather put time into Jackrabbit (100 years old this year) and Thunderbolt (originally built in 1924 as Pippin and extensively rebuilt in 1967). Still, people attending as part of special events can get more revealing photographs.

Even with the shape now familiar, and loved, there are curious gaps. Consider game design. If you play on a board that represents space you need to do something with the boundaries. The easiest is to make the boundaries the edges of playable space. The game designer has choices, though. If a piece moves off the board to the right, why not have it reappear on the left? (And, going off to the left, reappear on the right.) This is fine. It gives the game board, a finite rectangle, the topology of a cylinder. If this isn’t enough? Have pieces that go off the top edge reappear at the bottom, and vice-versa. Doing this, along with matching the left to the right boundaries, makes the game board a torus, a doughnut shape.

A Möbius strip is easy enough to code. Make the top and bottom impenetrable borders. And match the left to the right edges this way: a piece going off the board at the upper half of the right edge reappears at the lower half of the left edge. Going off the lower half of the right edge brings the piece to the upper half of the left edge. And so on. It isn’t hard, but I’m not aware of any game — board or computer — that uses this space. Maybe there’s a backgammon variant which does.

Still, the strip defies our intuition. It has one face and one edge. To reflect a shape across the width of the strip is the same as sliding a shape along its length. Cutting the strip down the center unfurls it into a cylinder. Cutting the strip down, one-third of the way from the edge, divides it into two pieces, a skinnier Möbius strip plus a cylinder. If we could extract the edge we could tug and stretch it until it was a circle.

And it primes our intuition. Once we understand there can be shapes lacking sides we can look for more. Anyone likely to read a pop mathematics blog about the Möbius strip has heard of the Klein bottle. This is a three-dimensional surface that folds back on itself in the fourth dimension of space. The shape is a jug with no inside, or with nothing but inside. Three-dimensional renditions of this get suggested as gifts to mathematicians. This for your mathematician friend who’s already got a Möbius scarf.

Log flume car splashing down at the bottom of its hill. In the background is the rust-red track and structure of the Racer roller coaster.
Anyway the roller coaster in background is Racer. This is a view from the Log Jammer, a log flume ride that Kennywood removed in 2018.

Though a Möbius strip looks — at any one spot — like a plane, the four-color map theorem doesn’t hold for it. Even the five-color theorem won’t do. You need six colors to cover maps on such a strip. A checkerboard drawn on a Möbius strip can be completely covered by T-shape pentominoes or Tetris pieces. You can’t do this for a checkerboard on the plane. In the mathematics of music theory the organization of dyads — two-tone “chords” — has the structure of a Möbius strip. I do not know music theory or the history of music theory. I’m curious whether Möbius strips might have been recognized by musicians before the mathematicians caught on.

And they inspire some practical inventions. Mechanical belts are obvious, although I don’t know how often they’re used. More clever are designs for resistors that have no self-inductance. They can resist electric flow without causing magnetic interference. I can look up the patents; I can’t swear to how often these are actually used. There exist — there are made — Möbius aromatic compounds. These are organic compounds with rings of carbon and hydrogen. I do not know a use for these. That they’ve only been synthesized this century, rather than found in nature, suggests they are more neat than practical.

Perhaps this shape is most useful as a path into a particular type of topology, and for its considerable artistry. And, with its “late” discovery, a reminder that we do not yet know all that is obvious. That is enough for anything.

Photograph of a model of the Montana Rusa roller coaster, under glass, and watched over by a Alebrije-styled dragon-ish figure. It sits on the platform for a roller coaster.
Montaña Rusa, sad to say, when we visited La Feria Chapultepec Magico in 2018 was running only one train at a time. It did have this gorgeous model of the roller coaster, although under plexiglass and roped off so that it was difficult to photograph.

There are three steel roller coasters with a Möbius strip track. That is, the metal rail on which the coaster runs is itself braced directly by metal. One of these is in France, one in Italy, and one in Iran. One in Liaoning, China has been under construction for five years. I can’t say when it might open. I have yet to ride any of them.


This and all the other 2020 A-to-Z essays should be at this link. Both the 2020 and all past A-to-Z essays should be at this link. I am hosting the Playful Math Education Blog Carnival at the end of September, so appreciate any educational or recreational or simply fun mathematics material you know about. And, goodness, I’m actually overdue to ask for topics for the latters P through R; I’ll have a post for that tomorrow, I hope. Thank you for your reading and your help.

My All 2020 Mathematics A to Z: Leibniz


Today’s topic suggestion was suggested by bunnydoe. I know of a project bunnydoe runs, but not whether it should be publicized. It is another biographical piece. Biographies and complex numbers, that seems to be the theme of this year.

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.

Gottfried Wilhelm Leibniz.

The exact suggestion I got for L was “Leibniz, the inventor of Calculus”. I can’t in good conscience offer that. This isn’t to deny Leibniz’s critical role in calculus. We rely on many of the ideas he’d had for it. We especially use his notation. But there are few great big ideas that can be truly credited to an inventor, or even a team of inventors. Put aside the sorry and embarrassing priority dispute with Isaac Newton. Many mathematicians in the 16th and 17th century were working on how to improve the Archimedean “method of exhaustion”. This would find the areas inside select curves, integral calculus. Johannes Kepler worked out the areas of ellipse slices, albeit with considerable luck. Gilles Roberval tried working out the area inside a curve as the area of infinitely many narrow rectangular strips. We still learn integration from this. Pierre de Fermat recognized how tangents to a curve could find maximums and minimums of functions. This is a critical piece of differential calculus. Isaac Barrow, Evangelista Torricelli (of barometer fame), Pietro Mengoli, and Stephano Angeli all pushed mathematics towards calculus. James Gregory proved, in geometric form, the relationship between differentiation and integration. That relationship is the Fundamental Theorem of Calculus.

This is not to denigrate Leibniz. We don’t dismiss the Wright Brothers though we know that without them, Alberto Santos-Dumont or Glenn Curtiss or Samuel Langley would have built a workable airplane anyway. We have Leibniz’s note, dated the 29th of October, 1675 (says Florian Cajori), writing out \int l to mean the sum of all l’s. By mid-November he was integrating functions, and writing out his work as \int f(x) dx . Any mathematics or physics or chemistry or engineering major today would recognize that. A year later he was writing things like d(x^n) = n x^{n - 1} dx , which we’d also understand if not quite care to put that way.

Though we use his notation and his basic tools we don’t exactly use Leibniz’s particular ideas of what calculus means. It’s been over three centuries since he published. It would be remarkable if he had gotten the concepts exactly and in the best of all possible forms. Much of Leibniz’s calculus builds on the idea of a differential. This is a quantity that’s smaller than any positive number but also larger than zero. How does that make sense? George Berkeley argued it made not a lick of sense. Mathematicians frowned, but conceded Berkeley was right. By the mid-19th century they had a rationale for differentials that avoided this weird sort of number.

It’s hard to avoid the differential’s lure. The intuitive appeal of “imagine moving this thing a tiny bit” is always there. In science or engineering applications it’s almost mandatory. Few things we encounter in the real world have the kinds of discontinuity that create logic problems for differentials. Even in pure mathematics, we will look at a differential equation like \frac{dy}{dx} = x and rewrite it as dy = x dx . Leibniz’s notation gives us the idea that taking derivatives is some kind of fraction. It isn’t, but in many problems we act as though it were. It works out often enough we forget that it might not.

Better, though. From the 1960s Abraham Robinson and others worked out a different idea of what real numbers are. In that, differentials have a rigorous logical definition. We call the mathematics which uses this “non-standard analysis”. The name tells something of its use. This is not to call it wrong. It’s merely not what we learn first, or necessarily at all. And it is Leibniz’s differentials. 304 years after his death there is still a lot of mathematics he could plausibly recognize.

There is still a lot of still-vital mathematics that he touched directly. Leibniz appears to be the first person to use the term “function”, for example, to describe that thing we’re plotting with a curve. He worked on systems of linear equations, and methods to find solutions if they exist. This technique is now called Gaussian elimination. We see the bundling of the equations’ coefficients he did as building a matrix and finding its determinant. We know that technique, today, as Cramer’s Rule, after Gabriel Cramer. The Japanese mathematician Seki Takakazu had discovered determinants before Leibniz, though.

Leibniz tried to study a thing he called “analysis situs”, which two centuries on would be a name for topology. My reading tells me you can get a good fight going among mathematics historians by asking whether he was a pioneer in topology. So I’ll decline to take a side in that.

In the 1680s he tried to create an algebra of thought, to turn reasoning into something like arithmetic. His goal was good: we see these ideas today as Boolean algebra, and concepts like conjunction and disjunction and negation and the empty set. Anyone studying logic knows these today. He’d also worked in something we can see as symbolic logic. Unfortunately for his reputation, the papers he wrote about that went unpublished until late in the 19th century. By then other mathematicians, like Gottlob Frege and Charles Sanders Peirce, had independently published the same ideas.

We give Leibniz’ name to a particular series that tells us the value of π:

1 - \frac13 + \frac15 - \frac17 + \frac19 - \frac{1}{11} + \cdots = \frac{\pi}{4}

(The Indian mathematician Madhava of Sangamagrama knew the formula this comes from by the 14th century. I don’t know whether Western Europe had gotten the news by the 17th century. I suspect it hadn’t.)

The drawback to using this to figure out digits of π is that it takes forever to use. Taking ten decimal digits of π demands evaluating about five billion terms. That’s not hyperbole; it just takes like forever to get its work done.

Which is something of a theme in Leibniz’s biography. He had a great many projects. Some of them even reached a conclusion. Many did not, and instead sprawled out with great ambition and sometimes insight before getting lost. Consider a practical one: he believed that the use of wind-driven propellers and water pumps could drain flooded mines. (Mines are always flooding.) In principle, he was right. But they all failed. Leibniz blamed deliberate obstruction by administrators and technicians. He even blamed workers afraid that new technologies would replace their jobs. Yet even in this failure he observed and had bracing new thoughts. The geology he learned in the mines project made him hypothesize that the Earth had been molten. I do not know the history of geology well enough to say whether this was significant to that field. It may have been another frustrating moment of insight (lucky or otherwise) ahead of its time but not connected to the mainstream of thought.

Another project, tantalizing yet incomplete: the “stepped reckoner”, a mechanical arithmetic machine. The design was to do addition and subtraction, multiplication and division. It’s a breathtaking idea. It earned him election into the (British) Royal Society in 1673. But it never was quite complete, never getting carries to work fully automatically. He never did finish it, and lost friends with the Royal Society when he moved on to other projects. He had a note describing a machine that could do some algebraic operations. In the 1690s he had some designs for a machine that might, in theory, integrate differential equations. It’s a fantastic idea. At some point he also devised a cipher machine. I do not know if this is one that was ever used in its time.

His greatest and longest-lasting unfinished project was for his employer, the House of Brunswick. Three successive Brunswick rulers were content to let Leibniz work on his many side projects. The one that Ernest Augustus wanted was a history of the Guelf family, in the House of Brunswick. One that went back to the time of Charlemagne or earlier if possible. The goal was to burnish the reputation of the house, which had just become a hereditary Elector of the Holy Roman Empire. (That is, they had just gotten to a new level of fun political intriguing. But they were at the bottom of that level.) Starting from 1687 Leibniz did good diligent work. He travelled throughout central Europe to find archival materials. He studied their context and meaning and relevance. He organized it. What he did not do, by his death in 1716, was write the thing.

It is always difficult to understand another person. Moreso someone you know only through biography. And especially someone who lived in very different times. But I do see a particular an modern personality type here. We all know someone who will work so very hard getting prepared to do a project Right that it never gets done. You might be reading the words of one right now.

Leibniz was a compulsive Society-organizer. He promoted ones in Brandenberg and Berlin and Dresden and Vienna and Saint Petersburg. None succeeded. It’s not obvious why. Leibniz was well-connected enough; he’s known to have over six hundred correspondents. Even for a time of great letter-writing, that’s a lot.

But it does seem like something about him offended others. Failing to complete big projects, like the stepped reckoner or the History of the Guelf family, seems like some of that. Anyone who knows of calculus knows of the dispute about the Newton-versus-Leibniz priority dispute. Grant that Leibniz seems not to have much fueled the quarrel. (And that modern historians agree Leibniz did not steal calculus from Newton.) Just being at the center of Drama causes people to rate you poorly.

There seems like there’s more, though. He was liked, for example, by the Electress Sophia of Hanover and her daughter Sophia Charlotte. These were the mother and the sister of Britain’s King George I. When George I ascended to the British throne he forbade Leibniz coming to London until at least one volume of the history was written. (The restriction seems fair, considering Leibniz was 27 years into the project by then.)

There are pieces in his biography that suggest a person a bit too clever for his own good. His first salaried position, for example, was as secretary to a Nuremberg alchemical society. He did not know alchemy. He passed himself off as deeply learned, though. I don’t blame him. Nobody would ever pass a job interview if they didn’t pretend to have expertise. Here it seems to have worked.

But consider, for example, his peace mission to Paris. Leibniz was born in the last years of the Thirty Years War. In that, the Great Powers of Europe battled each other in the German states. They destroyed Germany with a thoroughness not matched until World War II. Leibniz reasonably feared France’s King Louis XIV had designs on what was left of Germany. So his plan was to sell the French government on a plan of attacking Egypt and, from there, the Dutch East Indies. This falls short of an early-Enlightenment idea of rational world peace and a congress of nations. But anyone who plays grand strategy games recognizes the “let’s you and him fight” scheming. (The plan became irrelevant when France went to war with the Netherlands. The war did rope Brandenberg-Prussia, Cologne, Münster, and the Holy Roman Empire into the mess.)

God: 'T-Rex remember the other day when you said you wanted to enhance the timeline?' T-Rex: 'Absolutely!' God: 'Well why enhance it only once?' T-Rex: 'Holy cow! Why indeed? I enhance the past so there's holodecks in the present. And then I teach cavepeeps to invent those, and then return to the future and find new entertainment technology so amazing I can't even imagine it right now! I could enhance the timeline over and over until me and all the other time travellers conclude it can't possibly be enhanced any more!!' Utahraptor: 'Which leaves us with two possibilities.' T-Rex: 'Oh?' Utahraptor: 'One: time travel isn't possible and we're stuck with this timeline.' T-Rex: 'Boo! Let's ignore that one!' Utahraptor: 'Two: time travel is possible, and this timeline is absolutely the best one anyone could come up with' T-Rex: 'Boo! That one --- that one gave me the sad feelings.'
Ryan North’s Dinosaur Comics for the 20th of August, 2020. (Spoiler: time travel isn’t possible.) And while I am still just reading the comics for fun, I have a number of essays discussing aspects of Dinosaur Comics at this link.

And I have not discussed Leibniz’s work in philosophy, outside his logic. He’s respected for the theory of monads, part of the long history of trying to explain how things can have qualities. Like many he tried to find a deductive-logic argument about whether God must exist. And he proposed the notion that the world that exists is the most nearly perfect that can possibly be. Everyone has been dragging him for that ever since he said it, and they don’t look ready to stop. It’s an unfair rap, even if it makes for funny spoofs of his writing.

The optimal world may need to be badly defective in some ways. And this recognition inspires a question in me. Obviously Leibniz could come to this realization from thinking carefully about the world. But anyone working on optimization problems knows the more constraints you must satisfy, the less optimal your best-fit can be. Some things you might like may end up being lousy, because the overall maximum is more important. I have not seen anything to suggest Leibniz studied the mathematics of optimization theory. Is it possible he was working in things we now recognize as such, though? That he has notes in the things we would call Lagrange multipliers or such? I don’t know, and would like to know if anyone does.

Leibniz’s funeral was unattended by any dignitary or courtier besides his personal secretary. The Royal Academy and the Berlin Academy of Sciences did not honor their member’s death. His grave was unmarked for a half-century. And yet historians of mathematics, philosophy, physics, engineering, psychology, social science, philology, and more keep finding his work, and finding it more advanced than one would expect. Leibniz’s legacy seems to be one always rising and emerging from shade, but never being quite where it should.


And that’s enough for one day. All of the 2020 A-to-Z essays should be at this link. Both 2020 and all past A-to-Z essays should be at this link. And, as I am hosting the Playful Math Education Blog Carnival at the end of September, I am looking for any blogs, videos, books, anything educational or recreational or just interesting to read about. Thank you for your reading and your help.

My All 2020 Mathematics A to Z: K-Theory


I should have gone with Vayuputrii’s proposal that I talk about the Kronecker Delta. But both Jacob Siehler and Mr Wu proposed K-Theory as a topic. It’s a big and an important one. That was compelling. It’s also a challenging one. This essay will not teach you K-Theory, or even get you very far in an introduction. It may at least give some idea of what the field is about.

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.

K-Theory.

This is a difficult topic to discuss. It’s an important theory. It’s an abstract one. The concrete examples are either too common to look interesting or are already deep into things like “tangent bundles of Sn-1”. There are people who find tangent bundles quite familiar concepts. My blog will not be read by a thousand of them this month. Those who are familiar with the legends grown around Alexander Grothendieck will nod on hearing he was a key person in the field. Grothendieck was of great genius, and also spectacular indifference to practical mathematics. Allegedly he once, pressed to apply something to a particular prime number for an example, proposed 57, which is not prime. (One does not need to be a genius to make a mistake like that. If I proposed 447 or 449 as prime numbers, how long would you need to notice I was wrong?)

K-Theory predates Grothendieck. Now that we know it’s a coherent mathematical idea we can find elements leading to it going back to the 19th century. One important theorem has Bernhard Riemann’s name attached. Henri Poincaré contributed early work too. Grothendieck did much to give the field a particular identity. Also a name, the K coming from the German Klasse. Grothendieck pioneered what we now call Algebraic K-Theory, working on the topic as a field of abstract algebra. There is also a Topological K-Theory, early work on which we thank Michael Atiyah and Friedrick Hirzebruch for. Topology is, popularly, thought of as the mathematics of flexible shapes. It is, but we get there from thinking about relationships between sets, and these are the topologies of K-Theory. We understand these now as different ways of understandings structures.

Still, one text I found described (topological) K-Theory as “the first generalized cohomology theory to be studied thoroughly”. I remember how much handwaving I had to do to explain what a cohomology is. The subject looks intimidating because of the depth of technical terms. Every field is deep in technical terms, though. These look more rarefied because we haven’t talked much, or deeply, into the right kinds of algebra and topology.

You find at the center of K-Theory either “coherent sheaves” or “vector bundles”. Which alternative depends on whether you prefer Algebraic or Topological K-Theory. Both alternatives are ways to encode information about the space around a shape. Let me talk about vector bundles because I find that easier to describe. Take a shape, anything you like. A closed ribbon. A torus. A Möbius strip. Draw a curve on it. Every point on that curve has a tangent plane, the plane that just touches your original shape, and that’s guaranteed to touch your curve at one point. What are the directions you can go in that plane? That collection of directions is a fiber bundle — a tangent bundle — at that point. (As ever, do not use this at your thesis defense for algebraic topology.)

Now: what are all the tangent bundles for all the points along that curve? Does their relationship tell you anything about the original curve? The question is leading. If their relationship told us nothing, this would not be a subject anyone studies. If you pick a point on the curve and look at its tangent bundle, and you move that point some, how does the tangent bundle change?

If we start with the right sorts of topological spaces, then we can get some interesting sets of bundles. What makes them interesting is that we can form them into a ring. A ring means that we have a set of things, and an operation like addition, and an operation like multiplication. That is, the collection of things works somewhat like the integers do. This is a comfortable familiar behavior after pondering too much abstraction.

Why create such a thing? The usual reasons. Often it turns out calculating something is easier on the associated ring than it is on the original space. What are we looking to calculate? Typically, we’re looking for invariants. Things that are true about the original shape whatever ways it might be rotated or stretched or twisted around. Invariants can be things as basic as “the number of holes through the solid object”. Or they can be as ethereal as “the total energy in a physics problem”. Unfortunately if we’re looking at invariants that familiar, K-Theory is probably too much overhead for the problem. I confess to feeling overwhelmed by trying to learn enough to say what it is for.

There are some big things which it seems well-suited to do. K-Theory describes, in its way, how the structure of a set of items affects the functions it can have. This links it to modern physics. The great attention-drawing topics of 20th century physics were quantum mechanics and relativity. They still are. The great discovery of 20th century physics has been learning how much of it is geometry. How the shape of space affects what physics can be. (Relativity is the accessible reflection of this.)

And so K-Theory comes to our help in string theory. String theory exists in that grand unification where mathematics and physics and philosophy merge into one. I don’t toss philosophy into this as an insult to philosophers or to string theoreticians. Right now it is very hard to think of ways to test whether a particular string theory model is true. We instead ponder what kinds of string theory could be true, and how we might someday tell whether they are. When we ask what things could possibly be true, and how to tell, we are working for the philosophy department.

My reading tells me that K-Theory has been useful in condensed matter physics. That is, when you have a lot of particles and they interact strongly. When they act like liquids or solids. I can’t speak from experience, either on the mathematics or the physics side.

I can talk about an interesting mathematical application. It’s described in detail in section 2.3 of Allen Hatcher’s text Vector Bundles and K-Theory, here. It comes about from consideration of the Hopf invariant, named for Heinz Hopf for what I trust are good reasons. It also comes from consideration of homomorphisms. A homomorphism is a matching between two sets of things that preserves their structure. This has a precise definition, but I can make it casual. If you have noticed that, every (American, hourlong) late-night chat show is basically the same? The host at his desk, the jovial band leader, the monologue, the show rundown? Two guests and a band? (At least in normal times.) Then you have noticed the homomorphism between these shows. A mathematical homomorphism is more about preserving the products of multiplication. Or it preserves the existence of a thing called the kernel. That is, you can match up elements and how the elements interact.

What’s important is Adams’ Theorem of the Hopf Invariant. I’ll write this out (quoting Hatcher) to give some taste of K-Theory:

The following statements are true only for n = 1, 2, 4, and 8:
a. R^n is a division algebra.
b. S^{n - 1} is parallelizable, ie, there exist n – 1 tangent vector fields to S^{n - 1} which are linearly independent at each point, or in other words, the tangent bundle to S^{n - 1} is trivial.

This is, I promise, low on jargon. “Division algebra” is familiar to anyone who did well in abstract algebra. It means a ring where every element, except for zero, has a multiplicative inverse. That is, division exists. “Linearly independent” is also a familiar term, to the mathematician. Almost every subject in mathematics has a concept of “linearly independent”. The exact definition varies but it amounts to the set of things having neither redundant nor missing elements.

The proof from there sprawls out over a bunch of ideas. Many of them I don’t know. Some of them are simple. The conditions on the Hopf invariant all that S^{n - 1} stuff eventually turns into finding values of n for for which 2^n divides 3^n - 1 . There are only three values of ‘n’ that do that. For example.

What all that tells us is that if you want to do something like division on ordered sets of real numbers you have only a few choices. You can have a single real number, R^1 . Or you can have an ordered pair, R^2 . Or an ordered quadruple, R^4 . Or you can have an ordered octuple, R^8 . And that’s it. Not that other ordered sets can’t be interesting. They will all diverge far enough from the way real numbers work that you can’t do something that looks like division.

And now we come back to the running theme of this year’s A-to-Z. Real numbers are real numbers, fine. Complex numbers? We have some ways to understand them. One of them is to match each complex number with an ordered pair of real numbers. We have to define a more complicated multiplication rule than “first times first, second times second”. This rule is the rule implied if we come to R^2 through this avenue of K-Theory. We get this matching between real numbers and the first great expansion on real numbers.

The next great expansion of complex numbers is the quaternions. We can understand them as ordered quartets of real numbers. That is, as R^4 . We need to make our multiplication rule a bit fussier yet to do this coherently. Guess what fuss we’d expect coming through K-Theory?

R^8 seems the odd one out; who does anything with that? There is a set of numbers that neatly matches this ordered set of octuples. It’s called the octonions, sometimes called the Cayley Numbers. We don’t work with them much. We barely work with quaternions, as they’re a lot of fuss. Multiplication on them doesn’t even commute. (They’re very good for understanding rotations in three-dimensional space. You can also also use them as vectors. You’ll do that if your programming language supports quaternions already.) Octonions are more challenging. Not only does their multiplication not commute, it’s not even associative. That is, if you have three octonions — call them p, q, and r — you can expect that p times the product of q-and-r would be different from the product of p-and-q times r. Real numbers don’t work like that. Complex numbers or quaternions don’t either.

Octonions let us have a meaningful division, so we could write out p \div q and know what it meant. We won’t see that for any bigger ordered set of R^n . And K-Theory is one of the tools which tells us we may stop looking.

This is hardly the last word in the field. It’s barely the first. It is at least an understandable one. The abstractness of the field works against me here. It does offer some compensations. Broad applicability, for example; a theorem tied to few specific properties will work in many places. And pure aesthetics too. Much work, in statements of theorems and their proofs, involve lovely diagrams. You’ll see great lattices of sets relating to one another. They’re linked by chains of homomorphisms. And, in further aesthetics, beautiful words strung into lovely sentences. You may not know what it means to say “Pontryagin classes also detect the nontorsion in \pi_k(SO(n)) outside the stable range”. I know I don’t. I do know when I hear a beautiful string of syllables and that is a joy of mathematics never appreciated enough.


Thank you for reading. The All 2020 A-to-Z essays should be available at this link. The essays from all A-to-Z sequence, 2015 to present, should be at this link. And I am still open for M, N, and O essay topics. Thanks for your attention.

My All 2020 Mathematics A to Z: Jacobi Polynomials


Mr Wu, author of the Singapore Maths Tuition blog, gave me a good nomination for this week’s topic: the j-function of number theory. Unfortunately I concluded I didn’t understand the function well enough to write about it. So I went to a topic of my own choosing instead.

The Jacobi Polynomials discussed here are named for Carl Gustav Jacob Jacobi. Jacobi lived in Prussia in the first half of the 19th century. Though his career was short, it was influential. I’ve already discussed the Jacobian, which describes how changes of variables change volume. He has a host of other things named for him, most of them in matrices or mathematical physics. He was also a pioneer in those elliptic curves you hear so much about these days.

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.

Jacobi Polynomials.

Jacobi Polynomials are a family of functions. Polynomials, it happens; this is a happy case where the name makes sense. “Family” is the name mathematicians give to a bunch of functions that have some similarity. This often means there’s a parameter, and each possible value of the parameter describes a different function in the family. For example, we talk about the family of sine functions, S_n(z) . For every integer n we have the function S_n(z) = \sin(n z) where z is a real number between -π and π.

We like a family because every function in it gives us some nice property. Often, the functions play nice together, too. This is often something like mutual orthogonality. This means two different representatives of the family are orthogonal to one another. “Orthogonal” means “perpendicular”. We can talk about functions being perpendicular to one another through a neat mechanism. It comes from vectors. It’s easy to use vectors to represent how to get from one point in space to another. From vectors we define a dot product, a way of multiplying them together. A dot product has to meet a couple rules that are pretty easy to do. And if you don’t do anything weird? Then the dot product between two vectors is the cosine of the angle made by the end of the first vector, the origin, and the end of the second vector.

Functions, it turns out, meet all the rules for a vector space. (There are not many rules to make a vector space.) And we can define something that works like a dot product for two functions. Take the integral, over the whole domain, of the first function times the second. This meets all the rules for a dot product. (There are not many rules to make a dot product.) Did you notice me palm that card? When I did not say “the dot product is take the integral …”? That card will come back. That’s for later. For now: we have a vector space, we have a dot product, we can take arc-cosines, so why not define the angle between functions?

Mostly we don’t because we don’t care. Where we do care? We do like functions that are at right angles to one another. As with most things mathematicians do, it’s because it makes life easier. We’ll often want to describe properties of a function we don’t yet know. We can describe the function we don’t yet know as the sum of coefficients — some fixed real number — times basis functions that we do know. And then our problem of finding the function changes to one of finding the coefficients. If we picked a set of basis functions that are all orthogonal to one another, the finding of these coefficients gets easier. Analytically and numerically: we can often turn each coefficient into its own separate problem. Let a different computer, or at least computer process, work on each coefficient and get the full answer much faster.

The Jacobi Polynomials have three coefficients. I see them most often labelled α, β, and n. Likely you imagine this means it’s a huge family. It is huger than that. A zoologist would call this a superfamily, at least. Probably an order, possibly a class.

It turns out different relationships of these coefficients give you families of functions. Many of these families are noteworthy enough to have their own names. For example, if α and β are both zero, then the Jacobi functions are a family also known as the Legendre Polynomials. This is a great set of orthogonal polynomials. And the roots of the Legendre Polynomials give you information needed for Gaussian quadrature. Gaussian quadrature is a neat trick for numerically integrating a function. Take a weighted sum of the function you’re integrating evaluated at a set of points. This can get a very good — maybe even perfect — numerical estimate of the integral. The points to use, and the weights to use, come from a Legendre polynomial.

If α and β are both -\frac{1}{2} then the Jacobi Polynomials are the Chebyshev Polynomials of the first kind. (There’s also a second kind.) These are handy in approximation theory, describing ways to better interpolate a polynomial from a set of data. They also have a neat, peculiar relationship to the multiple-cosine formulas. Like, \cos(2\theta) = 2\cos^2(\theta) - 1 . And the second Chebyshev polynomial is T_2(x) = 2x^2 - 1 . Imagine sliding between x and cos(\theta) and you see the relationship. cos(3\theta) = 4 \cos^3(\theta) - 3\cos(\theta) and T_3(x) = 4x^3 - 3x . And so on.

Chebyshev Polynomials have some superpowers. One that’s most amazing is accelerating convergence. Often a numerical process, such as finding the solution of an equation, is an iterative process. You can’t find the answer all at once. You instead find an approximation and do something that improves it. Each time you do the process, you get a little closer to the true answer. This can be fine. But, if the problem you’re working on allows it, you can use the first couple iterations of the solution to figure out where this is going. The result is that you can get very good answers using the same amount of computer time you needed to just get decent answers. The trade, of course, is that you need to understand Chebyshev Polynomials and accelerated convergence. We always have to make trades like that.

Back to the Jacobi Polynomials family. If α and β are the same number, then the Jacobi functions are a family called the Gegenbauer Polynomials. These are great in mathematical physics, in potential theory. You can turn the gravitational or electrical potential function — that one-over-the-distance-squared force — into a sum of better-behaved functions. And they also describe zonal spherical harmonics. These let you represent functions on the surface of a sphere as the sum of coefficients times basis functions. They work in much the way the terms of a Fourier series do.

If β is zero and there’s a particular relationship between α and n that I don’t want to get into? The Jacobi Polynomials become the Zernike Polynomials, which I never heard of before this paragraph either. I read they are the tools you need to understand optics, and particularly how lenses will alter the light passing through.

Since the Jacobi Polynomials have a greater variety of form than even poison ivy has, you’ll forgive me not trying to list them. Or even listing a representative sample. You might also ask how they’re related at all.

Well, they all solve the same differential equation, for one. Not literally a single differential equation. A family of differential equations, where α and β and n turn up in the coefficients. The formula using these coefficients is the same in all these differential equations. That’s a good reason to see a relationship. Or we can write the Jacobi Polynomials as a series, a function made up of the sum of terms. The coefficients for each of the terms depends on α and β and n, always in the same way. I’ll give you that formula. You won’t like it and won’t ever use it. The Jacobi Polynomial for a particular α, β, and n is the polynomial

P_n^{(\alpha, \beta)}(z) = (n+\alpha)!(n + \beta)!\sum_{s=0}^n \frac{1}{s!(n + \alpha - s)!(\beta + s)!(n - s)!}\left(\frac{z-1}{2}\right)^{n-s}\left(\frac{z + 1}{2}\right)^s

Its domain, by the way, is the real numbers from -1 to 1. We need something for the domain. It turns out there’s nothing you can do on the real numbers that you can’t fit into the domain from -1 to 1 anyway. (If you have to do something on, say, the interval from 10 to 54? Do a change of variable, scaling things down and moving them, and use -1 to 1. Then undo that change when you’re done.) The range is the real numbers, as you’d expect.

(You maybe noticed I used ‘z’ for the independent variable there, rather than ‘x’. Usually using ‘z’ means we expect this to be a complex number. But ‘z’ here is definitely a real number. This is because we can also get to the Jacobi Polynomials through the hypergeometric series, a function I don’t want to get into. But for the hypergeometric series we are open to the variable being a complex number. So many references carry that ‘z’ back into Jacobi Polynomials.)

Another thing which links these many functions is recurrence. If you know the Jacobi Polynomial for one set of parameters — and you do; P_0^{(\alpha, \beta)}(z) = 1 — you can find others. You do this in a way rather like how you find new terms in the Fibonacci series by adding together terms you already know. These formulas can be long. Still, if you know P_{n-1}^{(\alpha, \beta)} and P_{n-2}^{(\alpha, \beta)} for the same α and β? Then you can calculate P_n^{(\alpha, \beta)} with nothing more than pen, paper, and determination. If it helps,

P_1^{(\alpha, \beta)}(z) = (\alpha + 1) + (\alpha + \beta + 2)\frac{z - 1}{2}

and this is true for any α and β. You’ll never do anything with that. This is fine.

There is another way that all these many polynomials are related. It goes back to their being orthogonal. We measured orthogonality by a dot product. Back when I palmed that card I told you was the integral of the two functions multiplied together. This is indeed a dot product. We can define others. We make those others by taking a weighted integral of the product of these two functions. That is, integrate the two functions times a third, a weight function. Of course there’s reasons to do this; they amount to deciding that some parts of the domain are more important than others. The weight function can be anything that meets a few rules. If you want to get the Jacobi Polynomials out of them, you start with the function P_0^{(\alpha, \beta)}(z) = 1 and the weight function

w_n(z) = (1 - z)^{\alpha} (1 + z)^{\beta}

As I say, though, you’ll never use that. If you’re eager and ready to leap into this work you can use this to build a couple Legendre Polynomials. Or Chebyshev Polynomials. For the full Jacobi Polynomials, though? Use, like, the command JacobiP[n, a, b, z] in Mathematica, or jacobiP(n, a, b, z) in Matlab. Other people have programmed this for you. Enjoy their labor.

In my work I have not used the full set of Jacobi Polynomials much. There’s more of them than I need. I do rely on the Legendre Polynomials, and the Chebyshev Polynomials. Other mathematicians use other slices regularly. It is stunning to sometimes look and realize that these many functions, different as they look, are reflections of one another, though. Mathematicians like to generalize, and find one case that covers as many things as possible. It’s rare that we are this successful.


I thank you for reading this. All of this year’s A-to-Z essays should be available at this link. The essays from every A-to-Z sequence going back to 2015 should be at this link. And I’m already looking ahead to the M, N, and O essays that I’ll be writing the day before publication instead of the week before like I want! I appreciate any nominations you have, even ones I can’t cover fairly.

I’m looking for M, N, and O topics for the All 2020 A-to-Z


I have reached the halfway point in this year’s A-to-Z! Not in the number of essays written — this week I should hit the 10th — but in preparing for topics? We are almost halfway done.

So for this, as with any A-to-Z essay, I’d like to know some mathematical term starting with the letters M, N, or O that you would like to see me write about. While I reserve the right to talk about anything I do care, I usually will pick the nominated topic I think I can be most interesting about. Or that I want to hurriedly learn something about. Please put in a comment with whatever you’d like me to discuss. And, please, if you do suggest something let me know how to credit you, and of any project that you do that I can mention. This project may be a way for me to show off, but I’d like everybody to have a bit more attention.

Topics I’ve already covered, starting with the letter ‘M’, are:


Topics I’ve already covered, starting with the letter ‘N’, are:


Topics I’ve already covered, starting with the letter ‘O’, are:


Thank you all for what you nominate. Also for thinking about nominations; I appreciate the work you do for me.

All of this year’s A-to-Z essays should be at this link. And all of the A-to-Z essays, from all years, should be at this link. Thanks for reading.

My All 2020 Mathematics A to Z: Imaginary Numbers


I have another topic today suggested by Beth, of the I Didn’t Have My Glasses On …. inspiration blog. It overlaps a bit with other essays I’ve posted this A-to-Z sequence, but that’s all right. We get a better understanding of things by considering them from several perspectives. This one will be a bit more historical.

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.

Imaginary Numbers.

Pop science writer Isaac Asimov told a story he was proud of about his undergraduate days. A friend’s philosophy professor held court after class. One day he declared mathematicians were mystics, believing in things they even admit are “imaginary numbers”. Young Asimov, taking offense, offered to prove the reality of the square root of minus one, if the professor gave him one-half pieces of chalk. The professor snapped a piece of chalk in half and gave one piece to him. Asimov said this is one piece of chalk. The professor answered it was half the length of a piece of chalk and Asimov said that’s not what he asked for. Even if we accept “half the length” is okay, how do we know this isn’t 48 percent the length of a standard piece of chalk? If the professor was that bad on “one-half” how could he have opinions on “imaginary numbers”?

This story is another “STEM undergraduates outwitting the philosophy expert” legend. (Even if it did happen. What we know is the story Asimov spun it into, in which a plucky young science fiction fan out-argued someone whose job is forming arguments.) Richard Feynman tells a similar story, befuddling a philosophy class with the question of how we can prove a brick has a interior. It helps young mathematicians and science majors feel better about their knowledge. But Asimov’s story does get at a couple points. First, that “imaginary” is a terrible name for a class of numbers. The square root of minus one is as “real” as one-half is. Second, we’ve decided that one-half is “real” in some way. What the philosophy professor would have baffled Asimov to explain is: in what way is one-half real? Or minus one?

We’re introduced to imaginary numbers through polynomials. I mean in education. It’s usually right after getting into quadratics, looking for solutions to equations like x^2 - 5x + 4 = 0 . That quadratic has two solutions, but it’s possible to have a quadratic with only one, such as x^2 + 6x + 9 = 0 . Or to have a quadratic with no solutions, such as, iconically, x^2 + 1 = 0 . We might underscore that by plotting the curve whose x- and y-coordinates makes true the equation y = x^2 + 1 . There’s no point on the curve with a y-coordinate of zero, so, there we go.

Having established that x^2 + 1 = 0 has no solutions, the course then asks “what if we go ahead and say there was one”? Two solutions, in fact, \imath and -\imath . This is all right for introducing the idea that mathematics is a tool. If it doesn’t do something we need, we can alter it.

But I see trouble in teaching someone how you can’t take square roots of negative numbers and then teaching them how to take square roots of negative numbers. It’s confusing at least. It needs some explanation about what changed. We might do better introducing them in a more historical method.

Historically, imaginary numbers (in the West) come from polynomials, yes. Different polynomials. Cubics, and quartics. Mathematicians still liked finding roots of them. Mathematicians would challenge one another to solve sets of polynomials. This seems hard to believe, but many sources agree on this. I hope we’re not all copying Eric Temple Bell here. (Bell’s Men of Mathematics is an inspiring collection of biographical sketches. But it’s not careful differentiating legends from documented facts.) And there are enough nerd challenges today that I can accept people daring one another to find solutions of x^3 - 15x - 4 = 0 .

Charlie Brown: 'You really need to work on your times tables, Sally, I can see that. Let's try the threes. How much is three times zero?' Sally: 'Four thousand? Six? Eleventy-twelve? Fifty-quillion? Overly-eight? Twiddely-two? Well? Am I getting closer?' Charlie Brown: 'Actually, it's kind of hard to say!'
Charles Schulz’s Peanuts for the 14th of October, 1967. You appreciate Schulz’s talent as a writer when you realize what a good nonsense word “Quillion” is. It sounds so plausible it’s easier to believe it is a number. “Overly-Eight” is another first-rate nonsense word and it’s just a shame that it’s so close to “Quillion” that it gets overshadowed. Reading the Comics essays with some mention of Peanuts are at this link.

Quadratics, equations we can write as ax^2 + bx + c = 0 for some real numbers a, b, and c, we’ve known about forever. Euclid solved these kinds of equations using geometric reasoning. Chinese mathematicians 2200 years ago described rules for how to find roots. The Indian mathematician Brahmagupta, by the early 7th century, described the quadratic formula to find at least one root. Both possible roots were known to Indian mathematicians a thousand years ago. We’ve reduced the formula today to

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

With that filtering into Western Europe, the search was on for similar formulas for other polynomials. This turns into several interesting threads. One is a tale of intrigue and treachery involving Gerolamo Cardano, Niccolò Tartaglia, and Ludovico Ferrari. I’ll save that for another essay because I have to cut something out, so of course I skip the dramatic thing. Another thread is the search for quadratic-like formulas for other polynomials. They exist for third-power and fourth-power polynomials. Not (generally) for the fifth- or higher-powers. That is, there are individual polynomials you can solve by formulas, like, x^6 - 5x^3 + 4 = 0 . But stare at it and you can see where that’s “really” a quadratic pretending to be sixth-power. Finding there was no formula to find, though, lead people to develop group theory. And group theory underlies much of mathematics and modern physics.

The first great breakthrough solving the general cubic, ax^3 + bx^2 + cx + d = 0 , came near the end of the 14th century in some manuscripts out of Florence. It’s built on a transformation. Transformations are key to mathematics. The point of a transformation is to turn a problem you don’t know how to do into one you do. As I write this, MathWorld lists 543 pages as matching “transformation”. That’s about half what “polynomial” matches (1,199) and about three times “trigonometric” (184). So that can help you judge importance.

Here, the transformation to make is to write a related polynomial in terms of a new variable. You can call that new variable x’ if you like, or z. I’ll use z so as to not have too many superscript marks flying around. This will be a “depressed polynomial”. “Depressed” here means that at least one of the coefficients in the new polynomial is zero. (Here, for this problem, it means we won’t have a squared term in the new polynomial.) I suspect the term is old-fashioned.

Let z be the new variable, related to x by the equation z = x - \frac{b}{3a} . And then figure out what z^2 and z^3 are. Using all that, and the knowledge that ax^3 + bx^2 + cx + d = 0 , and a lot of arithmetic, you get to one of these three equations:

z^3 + pz = q \\ z^3 = pz + q \\ z^3 + q = pz

where p and q are some new coefficients. They’re positive numbers, or possibly zeros. They’re both derived from a, b, c, and d. And so in the 15th Century the search was on to solve one or more of these equations.

From our perspective in the 21st century, our first question is: what three equations? How are these not all the same equation? And today, yes, we would write this as one depressed equation, most likely z^3 + pz = q . We would allow that p or q or both might be negative numbers.

And there is part of the great mysterious historical development. These days we generally learn about negative numbers. Once we are comfortable, our teachers hope, with those we get imaginary numbers. But in the Western tradition mathematicians noticed both, and approached both, at roughly the same time. With roughly similar doubts, too. It’s easy to point to three apples; who can point to “minus three” apples? We can arrange nine apples into a neat square. How big a square can we set “minus nine” apples in?

Hesitation and uncertainty about negative numbers would continue quite a long while. At least among Western mathematicians. Indian mathematicians seem to have been more comfortable with them sooner. And merchants, who could model a negative number as a debt, seem to have gotten the idea better.

But even seemingly simple questions could be challenging. John Wallis, in the 17th century, postulated that negative numbers were larger than infinity. Leonhard Euler seems to have agreed. (The notion may seem odd. It has echoes today, though. Computers store numbers as bit patterns. The normal scheme represents negative numbers by making the first bit in a pattern 1. These bit patterns make the negative numbers look bigger than the biggest positive numbers. And thermodynamics gives us a temperature defined by the relationship of energy to entropy. That definition implies there can be negative temperatures. Those are “hotter” — higher-energy, at least — than infinitely-high positive temperatures.) In the 18th century we see temperature scales designed so that the weather won’t give negative numbers too often. Augustus De Morgan wrote in 1831 that a negative number “occurring as the solution of a problem indicates some inconsistency or absurdity”. De Morgan was not an amateur. He coded the rules for deductive logic so well we still call them De Morgan’s laws. He put induction on a logical footing. And he found negative numbers (and imaginary numbers) a sign of defective work. In 1831. 1831!

Calvin: 'Here's another math problem I can't figure out. What's 9 + 4?' Hobbes: 'Ooh, that's a tricky one. You have to use calculus and imaginary numbers for this.' Calvin: 'IMAGINARY NUMBERS?' Hobbes: 'You know, eleventeen, thirty-twelve, and all those. It's a little confusing at first.' Calvin: 'How did YOU learn all this? You've never even gone to school!' Hobbes: 'Instinct. Tigers are born with it.'
Bill Watterson’s Calvin and Hobbes for the 6th of January, 1988. I could not in good conscience post this piece without including this comic. I discuss fewer Calvin and Hobbes strips than I thought, but I discuss them in Reading the Comics posts at this link.

But back to cubic equations. Allow that we’ve gotten comfortable enough with negative numbers we only want to solve the one depressed equation of z^3 + pz = q . How to do it? … Another transformation, then. There are a couple you can do. Modern mathematicians would likely define a new variable w, set so that z = w - \frac{p}{3w} . This turns the depressed equation into

w^3 - \frac{p^3}{27 w^3} - q = 0

And this, believe it or not, is a disguised quadratic. Multiply everything in it by w^3 and move things around a little. You get

(w^3)^2 - q(w^3) - \frac{1}{27}p^3 = 0

From there, quadratic formula to solve for w^3 . Then from that, take cube roots and you get three values of z. From that, you get your three values of x.

You see why nobody has taught this in high school algebra since 1959. Also why I am not touching the quartic formula, the equivalent of this for polynomials of degree four.

There are other approaches. And they can work out easier for particular problems. Take, for example, x^3 - 15x - 4 = 0 which I introduced in the first act. It’s past the time we set it off.

Rafael Bombelli, in the 1570s, pondered this particular equation. Notice it’s already depressed. A formula developed by Cardano addressed this, in the form x^3 = 15x + 4 . Notice that’s the second of the three sorts of depressed polynomial. Cardano’s formula says that one of the roots will be at

x = \sqrt[3]{\frac{q}{2} + r} + \sqrt[3]{\frac{q}{2} - r}

where

r = \sqrt{\left(\frac{q}{2}\right)^2 - \left(\frac{p}{3}\right)^3}

Put to this problem, we get something that looks like a compelling reason to stop:

x = \sqrt[3]{2 + \sqrt{-121}} + \sqrt[3]{2 - \sqrt{-121}}

Bombelli did not stop with that, though. He carried on as though these expressions of the square root of -121 made sense. And, if he did that he found these terms added up. You get an x of 4.

Which is true. It’s easy to check that it’s right. And here is the great surprising thing. Start from the respectable enough x^3 = 15x + 4 equation. It has nothing suspicious in it, not even negative numbers. Follow it through and you need to use negative numbers. Worse, you need to use the square roots of negative numbers. But keep going, as though you were confident in this, and you get a correct answer. And a real number.

We can get the other roots. Divide (x - 4) out of x^3 - 15x - 4 . What’s left is x^2 + 4x + 1 . You can use the quadratic formula for this. The other two roots are x = -2 + \frac{1}{2} \sqrt{12} , about -0.268, and x = -2 - \frac{1}{2} \sqrt{12} , about -3.732.

So here we have good reasons to work with negative numbers, and with imaginary numbers. We may not trust them. But they get us to correct answers. And this brings up another little secret of mathematics. If all you care about is an answer, then it’s all right to use a dubious method to get there.

There is a logical rigor missing in “we got away with it, I guess”. The name “imaginary numbers” tells of the disapproval of its users. We get the name from René Descartes, who was more generally discussing complex numbers. He wrote something like “in many cases no quantity exists which corresponds to what one imagines”.

John Wallis, taking a break from negative numbers and his other projects and quarrels, thought of how to represent imaginary numbers as branches off a number line. It’s a good scheme that nobody noticed at the time. Leonhard Euler envisioned matching complex numbers with points on the plane, but didn’t work out a logical basis for this. In 1797 Caspar Wessel presented a paper that described using vectors to represent complex numbers. It’s a good approach. Unfortunately that paper too sank without a trace, undiscovered for a century.

In 1806 Jean-Robert Argand wrote an “Essay on the Geometrical Interpretation of Imaginary Quantities”. Jacques Français got a copy, and published a paper describing the basics of complex numbers. He credited the essay, but noted that there was no author on the title page and asked the author to identify himself. Argand did. We started to get some good rigor behind the concept.

In 1831 William Rowan Hamilton, of Hamiltonian fame, described complex numbers using ordered pairs. Once we can define their arithmetic using the arithmetic of real numbers we have a second solid basis. More reason to trust them. Augustin-Louis Cauchy, who proved about four billion theorems of complex analysis, published a new construction of them. This used a group theory approach, a polynomial ring we denote as R[x]/(x^2 + 1) . I don’t have the strength to explain all that today. Matrices give us another approach. This matches complex numbers with particular two-row, two-column matrices. This turns the addition and multiplication of numbers into what Hamilton described.

And here we have some idea why mathematicians use negative numbers, and trust imaginary numbers. We are pushed toward them by convenience. Negative numbers let us work with one equation, x^3 + px + q = 0 , rather than three. (Or more than three equations, if we have to work with an x we know to be negative.) Imaginary numbers we can start with, and find answers we know to be true. And this encourages us to find reasons to trust the results. Having one line of reasoning is good. Having several lines — Argand’s geometric, Hamilton’s coordinates, Cauchy’s rings — is reassuring. We may not be able to point to an imaginary number of anything. But if we can trust our arithmetic on real numbers we can trust our arithmetic on imaginary numbers.

As I mentioned Descartes gave the name “imaginary number” to all of what we would now call “complex numbers”. Gauss published a geometric interpretation of complex numbers in 1831. And gave us the term “complex number”. Along the way he complained about the terminology, though. He noted “had +1, -1, and \sqrt{-1} , instead of being called positive, negative, and imaginary (or worse still, impossible) unity, been given the names say, of direct, inverse, and lateral unity, there would hardly have been any scope for such obscurity”. I’ve never heard them term “impossible numbers”, except as an adjective.

The name of a thing doesn’t affect what it is. It can affect how we think about it, though. We can ask whether Asimov’s professor would dismiss “lateral numbers” as mysticism. Or at least as more mystical than “three” is. We can, in context, understand why Descartes thought of these as “imaginary numbers”. He saw them as something to use for the length of a calculation, and that would disappear once its use was done. We still have such concepts, things like “dummy variables” in a calculus problem. We can’t think of a use for dummy variables except to let a calculation proceed. But perhaps we’ll see things differently in four hundred years. Shall have to come back and check.


Thank you for reading through all that. Once again a topic I figured would be a tight 1200 words spilled into twice that. This and the other A-to-Z topics for 2020 should be at this link. And all my A-to-Z essays, this year and past years, should be at this link.

I’m still looking for J, K, and L topics for coming weeks. I’m grateful for any subject nominations you’d care to offer.

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.

I’m looking for J, K, and L topics for the All 2020 A-to-Z


As the subject line says, I’m looking at what the next couple of letters should be for my 2020 A-to-Z. Please put in a comment here with something you think it’d be interesting to see me explain. I’m up for most any topic with some mathematical connection, including biographies.

Please if you suggest something, let me know of any project that you have going on. I’m happy to share links to other blogs, teaching projects, YouTube channels, or whatever else you have going on that’s worth sharing.

I am open to revisiting a subject from past years, if I think I could do a better piece on it. Topics I’ve already covered, starting with the letter ‘J’, are:


Topics I’ve already covered, starting with the letter ‘K’, are:


Topics I’ve already covered, starting with the letter ‘L’, are:


The essays for my All 2020 Mathematics A to Z are at this link. Posts from all of the A-to-Z posts, this year and previous years, are at this link.

My All 2020 Mathematics A to Z: J Willard Gibbs


Charles Merritt sugested a biographical subject for G. (There are often running themes in an A-to-Z and this year’s seems to be “biography”.) I don’t know of a web site or other project that Merritt has that’s worth sharing, but if I learn of it, I’ll pass it along.

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.

J Willard Gibbs.

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.

In the 1880s Gibbs worked on something which exploded through physics and mathematics. This was vectors. He didn’t create them from nothing. Hermann Günter Grassmann — whose fascinating and frustrating career I hadn’t known of before this — laid much of the foundation. Building on Grassman and W K Clifford, though, let Gibbs present vectors as we now use them in physics. How to define dot products and cross products. How to use them to simplify physics problems. How they’re less work than quaternions are. Gibbs was not the only person to recast physics in vector form. Oliver Heaviside is another important mathematical physicist of the time who did. But Gibbs identified the tools extremely well. You can read his Elements of Vector Analysis. It’s not very different from what a modern author would write on the subject. It’s terser than I would write, but terse is also respectful of someone’s time and ability to reason out explanations of small points.

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.


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. I’ll soon be opening f or topics for J, K, and L, essays also. Thanks for reading.

My All 2020 Mathematics A to Z: Fibonacci


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.

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.

Fibonacci.

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.

Photograph of a Californian rabbit --- a small white rabbit with dark ears and musty grey snout --- sitting up in a cage as far from the camera as possible.
Penelope the rabbit is very happy to meet us!

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.)

He had other books. The Liber Quadratorum, for example, a book about algebra. Wikipedia seems to say we have it through a single manuscript, copied in the 15th century. Practica Geometriae, translated from Latin in 1442 at least. A couple other now-lost manuscripts. A couple pieces about specific problems.

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.

We could trace those hundred readers by the first thing anyone knows Fibonacci for. His rabbits, breeding in ways that rabbits do not, and the sequence of whole numbers those provide. Fibonacci did not discover this sequence. You knew that. Nothing in mathematics gets named for its discoverer. Virahanka, an Indian mathematician who lived somewhere between the sixth and eighth centuries, described the sequence exactly. Gopala, writing sometime in the 1130s, expanded on this.

Photograph of a Californian rabbit --- white, with black ears and a skeptical-looking red eye --- lying on the other side of the pen cage from a Flemish giant --- a large yellow-brown rabbit. They are not actually nose-to-nose, but look close to it, and we see the profile of both rabbits.
Penelope and Sunshine have a moment of togetherness before they decide they want less togetherness.

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 x^4 + y^4 = 1 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 2^{127} - 1 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.


And now I can try to get ahead of deadline for next week. This and all my other A-to-Z topics for the year should be at this link. All my essays for this and past A-to-Z sequences are at this link. And I am still taking topics to discuss in the coming weeks. Thank you for reading and please take care.

My All 2020 Mathematics A to Z: Exponential


GoldenOj suggested the exponential as a topic. It seemed like a good important topic, but one that was already well-explored by other people. Then I realized I could spend time thinking about something which had bothered me.

In here I write about “the” exponential, which is a bit like writing about “the” multiplication. We can talk about 2^3 and 10^2 and many other such exponential functions. One secret of algebra, not appreciated until calculus (or later), is that all these different functions are a single family. Understanding one exponential function lets you understand them all. Mathematicians pick one, the exponential with base e, because we find that convenient. e itself isn’t a convenient number — it’s a bit over 2.718 — but it has some wonderful properties. When I write “the exponential” here, I am looking at this function where we look at e^{t} .

This piece will have a bit more mathematics, as in equations, than usual. If you like me writing about mathematics more than reading equations, you’re hardly alone. I recommend letting your eyes drop to the next sentence, or at least the next sentence that makes sense. You should be fine.

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.

Exponential.

My professor for real analysis, in grad school, gave us one of those brilliant projects. Starting from the definition of the logarithm, as an integral, prove at least thirty things. They could be as trivial as “the log of 1 is 0”. They could be as subtle as how to calculate the log of one number in a different base. It was a great project for testing what we knew about why calculus works.

And it gives me the structure to write about the exponential function. Anyone reading a pop-mathematics blog about exponentials knows them. They’re these functions that, as the independent variable grows, grow ever-faster. Or that decay asymptotically to zero. Some readers know that, if the independent variable is an imaginary number, the exponential is a complex number too. As the independent variable grows, becoming a bigger imaginary number, the exponential doesn’t grow. It oscillates, a sine wave.

That’s weird. I’d like to see why that makes sense.

To say “why” this makes sense is doomed. It’s like explaining “why” 36 is divisible by three and six and nine but not eight. It follows from what the words we have mean. The “why” I’ll offer is reasons why this strange behavior is plausible. It’ll be a mix of deductive reasoning and heuristics. This is a common blend when trying to understand why a result happens, or why we should accept it.

I’ll start with the definition of the logarithm, as used in real analysis. The natural logarithm, if you’re curious. It has a lot of nice properties. You can use this to prove over thirty things. Here it is:

log\left(x\right) = \int_{1}^{x} \frac{1}{s} ds

The “s” is a dummy variable. You’ll never see it in actual use.

So now let me summon into existence a new function. I want to call it g. This is because I’ve worked this out before and I want to label something else as f. There is something coming ahead that’s a bit of a syntactic mess. This is the best way around it that I can find.

g(x) = \frac{1}{c} \int_{1}^{x} \frac{1}{s} ds

Here, ‘c’ is a constant. It might be real. It might be imaginary. It might be complex. I’m using ‘c’ rather than ‘a’ or ‘b’ so that I can later on play with possibilities.

So the alert reader noticed that g(x) here means “take the logarithm of x, and divide it by a constant”. So it does. I’ll need two things built off of g(x), though. The first is its derivative. That’s taken with respect to x, the only variable. Finding the derivative of an integral sounds intimidating but, happy to say, we have a theorem to make this easy. It’s the Fundamental Theorem of Calculus, and it tells us:

g'(x) = \frac{1}{c}\cdot\frac{1}{x}

We can use the ‘ to denote “first derivative” if a function has only one variable. Saves time to write and is easier to type.

The other thing that I need, and the thing I really want, is the inverse of g. I’m going to call this function f(t). A more common notation would be to write g^{-1}(t) but we already have g'(x) in the works here. There is a limit to how many little one-stroke superscripts we need above g. This is the tradeoff to using ‘ for first derivatives. But here’s the important thing:

x = f(t) = g^{-1}(t)

Here, we have some extratextual information. We know the inverse of a logarithm is an exponential. We even have a standard notation for that. We’d write

x = f(t) = e^{ct}

in any context besides this essay as I’ve set it up.

What I would like to know next is: what is the derivative of f(t)? This sounds impossible to know, if we’re thinking of “the inverse of this integration”. It’s not. We have the Inverse Function Theorem to come to our aid. We encounter the Inverse Function Theorem briefly, in freshman calculus. There we use it to do as many as two problems and then hide away forever from the Inverse Function Theorem. (This is why it’s not mentioned in my quick little guide to how to take derivatives.) It reappears in real analysis for this sort of contingency. The inverse function theorem tells us, if f the inverse of g, that:

f'(t) = \frac{1}{g'(f(t))}

That g'(f(t)) means, use the rule for g'(x), with f(t) substituted in place of ‘x’. And now we see something magic:

f'(t) = \frac{1}{\frac{1}{c}\cdot\frac{1}{f(t)}}

f'(t) = c\cdot f(t)

And that is the wonderful thing about the exponential. Its derivative is a constant times its original value. That alone would make the exponential one of mathematics’ favorite functions. It allows us, for example, to transform differential equations into polynomials. (If you want everlasting fame, albeit among mathematicians, invent a new way to turn differential equations into polynomials.) Because we could turn, say,

f'''(t) - 3f''(t) + 3f'(t) -  f(t) = 0

into

c^3 e^{ct} - 3c^2 e^{ct} + 3c e^{ct} -  e^{ct} = 0

and then

\left(c^3 - 3c^2 + 3c - 1\right) e^{ct} = 0

by supposing that f(t) has to be e^{ct} for the correct value of c. Then all you need do is find a value of ‘c’ that makes that last equation true.

Supposing that the answer has this convenient form may remind you of searching for the lost keys over here where the light is better. But we find so many keys in this good light. If you carry on in mathematics you will never stop seeing this trick, although it may be disguised.

In part because it’s so easy to work with. In part because exponentials like this cover so much of what we might like to do. Let’s go back to looking at the derivative of the exponential function.

f'(t) = c\cdot f(t)

There are many ways to understand what a derivative is. One compelling way is to think of it as the rate of change. If you make a tiny change in t, how big is the change in f(t)? So what is the rate of change here?

We can pose this as a pretend-physics problem. This lets us use our physical intuition to understand things. This also is the transition between careful reasoning and ad-hoc arguments. Imagine a particle that, at time ‘t’, is at the position x = f(t) . What is its velocity? That’s the first derivative of its position, so, x' = f'(t) = c\cdot f(t) .

If we are using our physics intuition to understand this it helps to go all the way. Where is the particle? Can we plot that? … Sure. We’re used to matching real numbers with points on a number line. Go ahead and do that. Not to give away spoilers, but we will want to think about complex numbers too. Mathematicians are used to matching complex numbers with points on the Cartesian plane, though. The real part of the complex number matches the horizontal coordinate. The imaginary part matches the vertical coordinate.

So how is this particle moving?

To say for sure we need some value of t. All right. Pick your favorite number. That’s our t. f(t) follows from whatever your t was. What’s interesting is that the change also depends on c. There’s a couple possibilities. Let me go through them.

First, what if c is zero? Well, then the definition of g(t) was gibberish and we can’t have that. All right.

What if c is a positive real number? Well, then, f'(t) is some positive multiple of whatever f(t) was. The change is “away from zero”. The particle will push away from the origin. As t increases, f(t) increases, so it pushes away faster and faster. This is exponential growth.

What if c is a negative real number? Well, then, f'(t) is some negative multiple of whatever f(t) was. The change is “towards zero”. The particle pulls toward the origin. But the closer it gets the more slowly it approaches. If t is large enough, f(t) will be so tiny that c\cdot f(t) is too small to notice. The motion declines into imperceptibility.

What if c is an imaginary number, though?

So let’s suppose that c is equal to some real number b times \imath , where \imath^2 = -1 .

I need some way to describe what value f(t) has, for whatever your pick of t was. Let me say it’s equal to \alpha + \beta\imath , where \alpha and \beta are some real numbers whose value I don’t care about. What’s important here is that f(t) = \alpha + \beta\imath .

And, then, what’s the first derivative? The magnitude and direction of motion? That’s easy to calculate; it’ll be \imath b f(t) = -\beta + \alpha\imath . This is an interesting complex number. Do you see what’s interesting about it? I’ll get there next paragraph.

So f(t) matches some point on the Cartesian plane. But f'(t), the direction our particle moves with a small change in t, is another poiat whatever complex number f'(t) is as another point on the plane. The line segment connecting the origin to f(t) is perpendicular to the one connecting the origin to f'(t). The ‘motion’ of this particle is perpendicular to its position. And it always is. There’s several ways to show this. An easy one is to just pick some values for \alpha and \beta and b and try it out. This proof is not rigorous, but it is quick and convincing.

If your direction of motion is always perpendicular to your position, then what you’re doing is moving in a circle around the origin. This we pick up in physics, but it applies to the pretend-particle moving here. The exponentials of \imath t and 2 \imath t and -40 \imath t will all be points on a locus that’s a circle centered on the origin. The values will look like the cosine of an angle plus \imath times the sine of an angle.

And there, I think, we finally get some justification for the exponential of an imaginary number being a complex number. And for why exponentials might have anything to do with cosines and sines.

You might ask what if c is a complex number, if it’s equal to a + b\imath for some real numbers a and b. In this case, you get spirals as t changes. If a is positive, you get points spiralling outward as t increases. If a is negative, you get points spiralling inward toward zero as t increases. If b is positive the spirals go counterclockwise. If b is negative the spirals go clockwise. e^{(a + \imath b) t} is the same as e^{at} \cdot e^{\imath b t} .

This does depend on knowing the exponential of a sum of terms, such as of a + \imath b , is equal to the product of the exponential of those terms. This is a good thing to have in your portfolio. If I remember right, it comes in around the 25th thing. It’s an easy result to have if you already showed something about the logarithms of products.


Thank you for reading. I have this and all my A-to-Z topics for the year at this link. All my essays for this and past A-to-Z sequences are at this link. And I am still interested in topics to discuss in the coming weeks. Take care, please.

I’m looking for G, H, and I topics for the All 2020 A-to-Z


When I look at how much I’m not getting ahead of deadline for these essays I’m amazed to think I should be getting topics for as much as five weeks out. Still, I should.

What I’d like is suggestions of things to write about. Any topics that have a name starting with the letters ‘G’, ‘H’, or ‘I’, that might be used for a topic. People with mathematical significance count, too. Please, with any nominations, let me know how to credit you for the topic. Also please mention any projects that you’re working on that could use attention. I try to credit and support people where I can.

These are the topics I’ve covered in past essays. I’m willing to revisit one if I realize I have fresh thoughts about it, too. I haven’t done so yet, but I’ll tell you, I was thinking hard about doing a rewrite on “dual”.


Topics I’ve already covered, starting with the letter ‘G’, are:


Topics I’ve already covered, starting with the letter ‘H’, are:


Topics I’ve already covered, starting with the letter ‘I’, are:


Thank you all for your thoughts, and for reading.

My All 2020 Mathematics A to Z: Delta


I have Dina Yagodich to thank for my inspiration this week. As will happen with these topics about something fundamental, this proved to be a hard topic to think about. I don’t know of any creative or professional projects Yagodich would like me to mention. I’ll pass them on if I learn of any.

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.

Delta.

In May 1962 Mercury astronaut Deke Slayton did not orbit the Earth. He had been grounded for (of course) a rare medical condition. Before his grounding he had selected his flight’s callsign and capsule name: Delta 7. His backup, Wally Schirra, who did not fly in Slayton’s place, named his capsule the Sigma 7. Schirra chose sigma for its mathematical and scientific meaning, representing the sum of (in principle) many parts. Slayton said he chose Delta only because he would have been the fourth American into space and Δ is the fourth letter of the Greek alphabet. I believe it, but do notice how D is so prominent a letter in Slayton’s name. And S, Σ, prominent in both Slayton and Schirra’s.

Δ is also a prominent mathematics and engineering symbol. It has several meanings, with several of the most useful ones escaping mathematics and becoming vaguely known things. They blur together, as ideas that are useful and related and not identical will do.

If “Δ” evokes anything mathematical to a person it is “change”. This probably owes to space in the popular imagination. Astronauts talking about the delta-vee needed to return to Earth is some of the most accessible technical talk of Apollo 13, to pick one movie. After that it’s easy to think of pumping the car’s breaks as shedding some delta-vee. It secondarily owes to school, high school algebra classes testing people on their ability to tell how steep a line is. This gets described as the change-in-y over the change-in-x, or the delta-y over delta-x.

Δ prepended to a variable like x or y or v we read as “the change in”. It fits the astronaut and the algebra uses well. The letter Δ by itself means as much as the words “the change in” do. It describes what we’re thinking about, but waits for a noun to complete. We say “the” rather than “a”, I’ve noticed. The change in velocity needed to reach Earth may be one thing. But “the” change in x and y coordinates to find the slope of a line? We can use infinitely many possible changes and get a good result. We must say “the” because we consider one at a time.

Used like this Δ acts like an operator. It means something like “a difference between two values of the variable ” and lets us fill in the blank. How to pick those two values? Sometimes there’s a compelling choice. We often want to study data sampled at some schedule. The Δ then is between one sample’s value and the next. Or between the last sample value and the current one. Which is correct? Ask someone who specializes in difference equations. These are the usually numeric approximations to differential equations. They turn up often in signal processing or in understanding the flows of fluids or the interactions of particles. We like those because computers can solve them.

Δ, as this operator, can even be applied to itself. You read ΔΔ x as “the change in the change in x”. The prose is stilted, but we can understand it. It’s how the change in x has itself changed. We can imagine being interested in this Δ2 x. We can see this as a numerical approximation to the second derivative of x, and this gets us back to differential equations. There are similar results for ΔΔΔ x even if we don’t wish to read it all out.

In principle, Δ x can be any number. In practice, at least for an independent variable, it’s a small number, usually real. Often we’re lured into thinking of it as positive, because a phrase like “x + Δ x” looks like we’re making a number a little bigger than x. When you’re a mathematician or a quality-control tester you remember to consider “what if Δ x is negative”. From testing that learn you wrote your computer code wrong. We’re less likely to assume this positive-ness for the dependent variable. By the time we do enough mathematics to have opinions we’ve seen too many decreasing functions to overlook that Δ y might be negative.

Notice that in that last paragraph I faithfully wrote Δ x and Δ y. Never Δ bare, unless I forgot and cannot find it in copy-editing. I’ve said that Δ means “the change in”; to write it without some variable is like writing √ by itself. We can understand wishing to talk about “the square root of”, as a concept. Still it means something else than √ x does.

We do write Δ by itself. Even professionals do. Written like this we don’t mean “the change in [ something ]”. We instead mean “a number”. In this role the symbol means the same thing as x or y or t might, a way to refer to a number whose value we might not know. We might not care about. The implication is that it’s small, at least if it’s something to add to the independent variable. We use it when we ponder how things would be different if there were a small change in something.

Small but not tiny. Here we step into mathematics as a language, which can be as quirky and ambiguous as English. Because sometimes we use the lower-case δ. And this also means “a small number”. It connotes a smaller number than Δ. Is 0.01 a suitable value for Δ? Or for δ? Maybe. My inclination would be to think of that as Δ, reserving δ for “a small number of value we don’t care to specify”. This may be my quirk. Others might see it different.

We will use this lowercase δ as an operator too, thinking of things like “x + δ x”. As you’d guess, δ x connotes a small change in x. Smaller than would earn the title Δ x. There is no declaring how much smaller. It’s contextual. As with δ bare, my tendency is to think that Δ x might be a specific number but that δ x is “a perturbation”, the general idea of a small number. We can understand many interesting problems as a small change from something we already understand. That small change often earns such a δ operator.

There are smaller changes than δ x. There are infinitesimal differences. This is our attempt to make sense of “a number as close to zero as you can get without being zero”. We forego the Greek letters for this and revert to Roman letters: dx and dy and dt and the other marks of differential calculus. These are difficult numbers to discuss. It took more than a century of mathematicians’ work to find a way our experience with Δ x could inform us about dx. (We do not use ‘d’ alone to mean an even smaller change than δ. Sometimes we will in analysis write d with a space beside it, waiting for a variable to have its differential taken. I feel unsettled when I see it.)

Much of the completion of work we can credit to Augustin Cauchy, who’s credited with about 800 publications. It’s an intimidating record, even before considering its importance. Cauchy is, per Florian Cajori’s History Mathematical Notations, one of the persons we can credit with the use of Δ as symbol for “the change in”. (Section 610.) He’s not the only one. Leonhardt Euler and Johann Bernoulli (section 640) used Δ to represent a finite difference, the difference between two values.

I’m not aware of an explicit statement why Δ got the pick, as opposed to other letters. It’s hard to imagine a reason besides “difference starts with d”. That an etymology seems obvious does not make it so. It does seem to have a more compelling explanation than the use of “m” for the slope of a line, or \frac{\Delta y}{\Delta x} , though.

Slayton’s Mercury flight, performed by Scott Carpenter, did not involve any appreciable changes in orbit, a Δ v. No crewed spacecraft would until Gemini III. The Mercury flight did involve tests in orienting the spacecraft, in Δ θ and Δ φ on the angles of the spacecraft’s direction. These might have been in Slayton’s mind. He eventually flew into space on the Apollo-Soyuz Test Project, when an accident during landing exposed the crew to toxic gases. The investigation discovered a lesion on Slayton’s lung. A tiny thing, ultimately benign, which discovered earlier could have kicked him off the mission and altered his life so.


Thank you all for reading. I’m gathering all my 2020 A-to-Z essays at this link, and have all my A-to-Z essays of any kind at this link. Here is hoping there’s a good week ahead.

My All 2020 Mathematics A to Z: Complex Numbers


Mr Wu, author of the Singapore Maths Tuition blog, suggested complex numbers for a theme. I wrote long ago a bit about what complex numbers are and how to work with them. But that hardly exhausts the subject, and I’m happy revisiting it.

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.

Complex Numbers.

A throwaway joke somewhere in The Hitchhiker’s Guide To The Galaxy has Marvin The Paranoid Android grumble that he’s invented a square root for minus one. Marvin’s gone and rejiggered all of mathematics while waiting for something better to do. Nobody cares. It reminds us while Douglas Adams established much of a particular generation of nerd humor, he was not himself a nerd. The nerds who read The Hitchhiker’s Guide To The Galaxy obsessively know we already did that, centuries ago. Marvin’s creation was as novel as inventing “one-half”. (It may be that Adams knew, and intended Marvin working so hard on the already known as the joke.)

Anyone who’d read a pop mathematics blog like this likely knows the rough story of complex numbers in Western mathematics. The desire to find roots of polynomials. The discovery of formulas to find roots. Polynomials with numbers whose formulas demanded the square roots of negative numbers. And the discovery that sometimes, if you carried on as if the square root of a negative number made sense, the ugly terms vanished. And you got correct answers in the end. And, eventually, mathematicians relented. These things were unsettling enough to get unflattering names. To call a number “imaginary” may be more pejorative than even “negative”. It hints at the treatment of these numbers as falsework, never to be shown in the end. To call the sum of a “real” number and an “imaginary” “complex” is to warn. An expert might use these numbers only with care and deliberation. But we can count them as numbers.

I mentioned when writing about quaternions how when I learned of complex numbers I wanted to do the same trick again. My suspicion is many mathematicians do. The example of complex numbers teases us with the possibilites of other numbers. If we’ve defined \imath to be “a number that, squared, equals minus one”, what next? Could we define a \sqrt{\imath} ? How about a \log{\imath} ? Maybe something else? An arc-cosine of \imath ?

You can try any of these. They turn out to be redundant. The real numbers and \imath already let you describe any of those new numbers. You might have a flash of imagination: what if there were another number that, squared, equalled minus one, and that wasn’t equal to \imath ? Numbers that look like a + b\imath + c\jmath ? Here, and later on, a and b and c are some real numbers. b\imath means “multiply the real number b by whatever \imath is”, and we trust that this makes sense. There’s a similar setup for c and \jmath . And if you just try that, with a + b\imath + c\jmath , you get some interesting new mathematics. Then you get stuck on what the product of these two different square roots should be.

If you think of that. If all you think of is addition and subtraction and maybe multiplication by a real number? a + b\imath + c\jmath works fine. You only spot trouble if you happen to do multiplication. Granted, multiplication is to us not an exotic operation. Take that as a warning, though, of how trouble could develop. How do we know, say, that complex numbers are fine as long as you don’t try to take the log of the haversine of one of them, or some other obscurity? And that then they produce gibberish? Or worse, produce that most dread construct, a contradiction?

Here I am indebted to an essay that ten minutes ago I would have sworn was in one of the two books I still have out from the university library. I’m embarrassed to learn my error. It was about the philosophy of complex numbers and it gave me fresh perspectives. When the university library reopens for lending I will try to track back through my borrowing and find the original. I suspect, without confirming, that it may have been in Reuben Hersh’s What Is Mathematics, Really?.

The insight is that we can think of complex numbers in several ways. One fruitful way is to match complex numbers with points in a two-dimensional space. It’s common enough to pair, for example, the number 3 + 4\imath with the point at Cartesian coordinates (3, 4) . Mathematicians do this so often it can take a moment to remember that is just a convention. And there is a common matching between points in a Cartesian coordinate system and vectors. Chaining together matches like this can worry. Trust that we believe our matches are sound. Then we notice that adding two complex numbers does the same work as adding ordered coordinate pairs. If we trust that adding coordinate pairs makes sense, then we need to accept that adding complex numbers makes sense. Adding coordinate pairs is the same work as adding real numbers. It’s just a lot of them. So we’re lead to trust that if addition for real numbers works then addition for complex numbers does.

Multiplication looks like a mess. A different perspective helps us. A different way to look at where point are on the plane is to use polar coordinates. That is, the distance a point is from the origin, and the angle between the positive x-axis and the line segment connecting the origin to the point. In this format, multiplying two complex numbers is easy. Let the first complex number have polar coordinates (r_1, \theta_1) . Let the second have polar coordinates (r_2, \theta_2) . Their product, by the rules of complex numbers, is a point with polar coordinates (r_1\cdot r_2, \theta_1 + \theta_2) . These polar coordinates are real numbers again. If we trust addition and multiplication of real numbers, we can trust this for complex numbers.

If we’re confident in adding complex numbers, and confident in multiplying them, then … we’re in quite good shape. If we can add and multiply, we can do polynomials. And everything is polynomials.

We might feel suspicious yet. Going from complex numbers to points in space is calling on our geometric intuitions. That might be fooling ourselves. Can we find a different rationalization? The same result by several different lines of reasoning makes the result more believable. Is there a rationalization for complex numbers that never touches geometry?

We can. One approach is to use the mathematics of matrices. We can match the complex number a + b\imath to the sum of the matrices

a \left[\begin{tabular}{c c} 1 & 0 \\ 0 & 1 \end{tabular}\right] + b \left[\begin{tabular}{c c} 0 & 1 \\ -1 & 0  \end{tabular}\right]

Adding matrices is compelling. It’s the same work as adding ordered pairs of numbers. Multiplying matrices is tedious, though it’s not so bad for matrices this small. And it’s all done with real-number multiplication and addition. If we trust that the real numbers work, we can trust complex numbers do. If we can show that our new structure can be understood as a configuration of the old, we convince ourselves the new structure is meaningful.

The process by which we learn to trust them as numbers, guides us to learning how to trust any new mathematical structure. So here is a new thing that complex numbers can teach us, years after we have learned how to divide them. Do not attempt to divide complex numbers. That’s too much work.

My All 2020 Mathematics A to Z: Butterfly Effect


It’s a fun topic today, one suggested by Jacob Siehler, who I think is one of the people I met through Mathstodon. Mathstodon is a mathematics-themed instance of Mastodon, an open-source microblogging system. You can read its public messages here.

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.

Butterfly Effect.

I take the short walk from my home to the Red Cedar River, and I pour a cup of water in. What happens next? To the water, anyway. Me, I think about walking all the way back home with this empty cup.

Let me have some simplifying assumptions. Pretend the cup of water remains somehow identifiable. That it doesn’t evaporate or dissolve into the riverbed. That it isn’t scooped up by a city or factory, drunk by an animal, or absorbed into a plant’s roots. That it doesn’t meet any interesting ions that turn it into other chemicals. It just goes as the river flows dictate. The Red Cedar River merges into the Grand River. This then moves west, emptying into Lake Michigan. Water from that eventually passes the Straits of Mackinac into Lake Huron. Through the St Clair River it goes to Lake Saint Clair, the Detroit River, Lake Erie, the Niagara River, the Niagara Falls, and Lake Ontario. Then into the Saint Lawrence River, then the Gulf of Saint Lawrence, before joining finally the North Atlantic.

Photograph of a small, tree-lined riverbed from a wooden bridge over it.
To the right: East Lansing and the Michigan State University campus. To the left, in a sense: the Atlantic Ocean.

If I pour in a second cup of water, somewhere else on the Red Cedar River, it has a similar journey. The details are different, but the course does not change. Grand River to Lake Michigan to three more Great Lakes to the Saint Lawrence to the North Atlantic Ocean. If I wish to know when my water passes the Mackinac Bridge I have a difficult problem. If I just wish to know what its future is, the problem is easy.

So now you understand dynamical systems. There’s some details to learn before you get a job, yes. But this is a perspective that explains what people in the field do, and why that. Dynamical systems are, largely, physics problems. They are about collections of things that interact according to some known potential energy. They may interact with each other. They may interact with the environment. We expect that where these things are changes in time. These changes are determined by the potential energies; there’s nothing random in it. Start a system from the same point twice and it will do the exact same thing twice.

We can describe the system as a set of coordinates. For a normal physics system the coordinates are the positions and momentums of everything that can move. If the potential energy’s rule changes with time, we probably have to include the time and the energy of the system as more coordinates. This collection of coordinates, describing the system at any moment, is a point. The point is somewhere inside phase space, which is an abstract idea, yes. But the geometry we know from the space we walk around in tells us things about phase space, too.

Imagine tracking my cup of water through its journey in the Red Cedar River. It draws out a thread, running from somewhere near my house into the Grand River and Lake Michigan and on. This great thin thread that I finally lose interest in when it flows into the Atlantic Ocean.

Dynamical systems drops in phase space act much the same. As the system changes in time, the coordinates of its parts change, or we expect them to. So “the point representing the system” moves. Where it moves depends on the potentials around it, the same way my cup of water moves according to the flow around it. “The point representing the system” traces out a thread, called a trajectory. The whole history of the system is somewhere on that thread.

Harvey, sulking: 'What a horrible year! How did it come to this?' Penny: 'I blame chaos theory.' Harvey: 'If it's chaos theory, I know EXACTLY who to blame! Some stupid butterfly flapped its wings and here we are.'
Stephen Beals’s Adult Children for the 21st of June, 2020. There were at least two “chaos theory”/“butterfly effect” comic strips in my feed just last week, and I wasn’t even looking. You can find comics essays where I talk about Adult Children at this link, and comic strips in general here.

Phase space, like a map, has regions. For my cup of water there’s a region that represents “is in Lake Michigan”. There’s another that represents “is going over Niagara Falls”. There’s one that represents “is stuck in Sandusky Bay a while”. When we study dynamical systems we are often interested in what these regions are, and what the boundaries between them are. Then a glance at where the point representing a system is tells us what it is doing. If the system represents a satellite orbiting a planet, we can tell whether it’s in a stable orbit, about to crash into a moon, or about to escape to interplanetary space. If the system represents weather, we can say it’s calm or stormy. If the system is a rigid pendulum — a favorite system to study, because we can draw its phase space on the blackboard — we can say whether the pendulum rocks back and forth or spins wildly.

Come back to my second cup of water, the one with a different history. It has a different thread from the first. So, too, a dynamical system started from a different point traces out a different trajectory. To find a trajectory is, normally, to solve differential equations. This is often useful to do. But from the dynamical systems perspective we’re usually interested in other issues.

For example: when I pour my cup of water in, does it stay together? The cup of water started all quite close together. But the different drops of water inside the cup? They’ve all had their own slightly different trajectories. So if I went with a bucket, one second later, trying to scoop it all up, likely I’d succeed. A minute later? … Possibly. An hour later? A day later?

By then I can’t gather it back up, practically speaking, because the water’s gotten all spread out across the Grand River. Possibly Lake Michigan. If I knew the flow of the river perfectly and knew well enough where I dropped the water in? I could predict where each goes, and catch each molecule of water right before it falls over Niagara. This is tedious but, after all, if you start from different spots — as the first and the last drop of my cup do — you expect to, eventually, go different places. They all end up in the North Atlantic anyway.

Photograph of Niagara Falls, showing the American Falls and the Bridal Veil, with a faint rainbow visible to the left of image and a boat sailing to the right.
Me, screaming to the pilot of the boat at center-right: “There’s my water drop! No, to the left! The left — your other left!”

Except … well, there is the Chicago Sanitary and Ship Canal. It connects the Chicago River to the Des Plaines River. The result is that some of Lake Michigan drains to the Ohio River, and from there the Mississippi River, and the Gulf of Mexico. There are also some canals in Ohio which connect Lake Erie to the Ohio River. I don’t know offhand of ones in Indiana or Wisconsin bringing Great Lakes water to the Mississippi. I assume there are, though.

Then, too, there is the Erie Canal, and the other canals of the New York State Canal System. These link the Niagara River and Lake Erie and Lake Ontario to the Hudson River. The Pennsylvania Canal System, too, links Lake Erie to the Delaware River. The Delaware and the Hudson may bring my water to the mid-Atlantic. I don’t know the canal systems of Ontario well enough to say whether some water goes to Hudson Bay; I’d grant that’s possible, though.

Think of my poor cups of water, now. I had been sure their fate was the North Atlantic. But if they happen to be in the right spot? They visit my old home off the Jersey Shore. Or they flow through Louisiana and warmer weather. What is their fate?

I will have butterflies in here soon.

Imagine two adjacent drops of water, one about to be pulled into the Chicago River and one with Lake Huron in its future. There is almost no difference in their current states. Their destinies are wildly separate, though. It’s surprising that so small a difference matters. Thinking through the surprise, it’s fair that this can happen, even for a deterministic system. It happens that there is a border, separating those bound for the Gulf and those for the North Atlantic, between these drops.

But how did those water drops get there? Where were they an hour before? … Somewhere else, yes. But still, on opposite sides of the border between “Gulf of Mexico water” and “North Atlantic water”. A day before, the drops were somewhere else yet, and the border was still between them. This separation goes back to, even, if the two drops came from my cup of water. Within the Red Cedar River is a border between a destiny of flowing past Quebec and of flowing past Saint Louis. And between flowing past Quebec and flowing past Syracuse. Between Syracuse and Philadelphia.

How far apart are those borders in the Red Cedar River? If you’ll go along with my assumptions, smaller than my cup of water. Not that I have the cup in a special location. The borders between all these fates are, probably, a complicated spaghetti-tangle. Anywhere along the river would be as fortunate. But what happens if the borders are separated by a space smaller than a drop? Well, a “drop” is a vague size. What if the borders are separated by a width smaller than a water molecule? There’s surely no subtleties in defining the “size” of a molecule.

That these borders are so close does not make the system random. It is still deterministic. Put a drop of water on this side of the border and it will go to this fate. But how do we know which side of the line the drop is on? If I toss this new cup out to the left rather than the right, does that matter? If my pinky twitches during the toss? If I am breathing in rather than out? What if a change too small to measure puts the drop on the other side?

And here we have the butterfly effect. It is about how a difference too small to observe has an effect too large to ignore. It is not about a system being random. It is about how we cannot know the system well enough for its predictability to tell us anything.

The term comes from the modern study of chaotic systems. One of the first topics in which the chaos was noticed, numerically, was weather simulations. The difference between a number’s representation in the computer’s memory and its rounded-off printout was noticeable. Edward Lorenz posed it aptly in 1963, saying that “one flap of a sea gull’s wings would be enough to alter the course of the weather forever”. Over the next few years this changed to a butterfly. In 1972 Philip Merrilees titled a talk Does the flap of a butterfly’s wings in Brazil set off a tornado in Texas? My impression is that these days the butterflies may be anywhere, and they alter hurricanes.

Comic strip Chaos Butterfly Man: 'Bitten by a radioactive chaos butterfly, Mike Mason gained the powers of a chaos butterfly!' [ Outside a bank ] Mason flaps his arm at an escaping robber. Robber: 'Ha-ha! What harm can THAT do me?' [ Nine days later ] Robber: 'Where did this thunderstorm finally come fr ... YOW!' (He's struck by lightning.)
RubenBolling’s Super-Fun-Pak Comix for the 23rd of May, 2020. Ruben Bolling uses Chaos Butterfly a good bit in both Super-Fun-Pak Comix and in the main strip, Tom the Dancing Bug. I have fewer essays exploring these Chaos Butterfly strips than you might imagine from that because I ran out of different things to say about the joke. Bolling’s is a great strip, though, and I recommend you consider it.

That we settle on butterflies as agents of chaos we can likely credit to their image. They seem to be innocent things so slight they barely exist. Hummingbirds probably move with too much obvious determination to fit the role. The Big Bad Wolf huffing and puffing would realistically be almost as nothing as a butterfly. But he has the power of myth to make him seem mightier than the storms. There are other happy accidents supporting butterflies, though. Edward Lorenz’s 1960s weather model makes trajectories that, plotted, create two great ellipsoids. The figures look like butterflies, all different but part of the same family. And there is Ray Bradbury’s classic short story, A Sound Of Thunder. If you don’t remember 7th grade English class, in the story time-travelling idiots change history, putting a fascist with terrible spelling in charge of a dystopian world, by stepping on a butterfly.

The butterfly then is metonymy for all the things too small to notice. Butterflies, sea gulls, turning the ceiling fan on in the wrong direction, prying open the living room window so there’s now a cross-breeze. They can matter, we learn.

I’m looking for D, E, and F topics for the All 2020 A-to-Z


It does seem like I only just began the All 2020 Mathematics A-to-Z and already I need some fresh topics. This is because I really want to get to having articles done a week before publication, and while I’m not there yet, I can imagine getting there.

I’d like your nominations for subjects. Please comment here with some topic, name starting ‘D’, ‘E’, or ‘F’, that I might try writing about. This can include people, too. And also please let me know how to credit you for a topic suggestion, including any projects you’re working on that could do with attention.

These are the topics I’ve covered in past essays. I’m open to revisiting ones that I think I have new ideas about. Thanks all for your ideas.

Topics I’ve already covered, starting with the letter D, are:


Topics I’ve already covered, starting with the letter E, are:


Topics I’ve already covered, starting with the letter F, are:


Thank you for reading, and for your thoughts.

My All 2020 Mathematics A to Z: Michael Atiyah


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.

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.

Michael Atiyah.

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 x^2 + y^2 = 1 . 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.

I turn out also to need B and C topics for the All 2020 A-to-Z


In looking over what I want my deadlines to be, and what the calendar looks like, I learn I should also be asking for topics that start with the letters B and C … yesterday, really. Today is probably close enough. And I am still interested in letter-A topics.

It all remains straightforward. Please comment with some mathematical topic, with a name that starts ‘B’ or ‘C’, that you’d like to see me try to write about. This can include people, too. I’m not likely to add to the depth of people’s biographical understandings. But I can share what I know, or have learned. Please also let me know how to credit you for a topic, and any project you’re working on that I can give some attention to.

I’m not afraid of re-visiting a topic, if I think I can improve it. But topics I have already covered, starting with the letter B, are:

And here are topics I’ve written about for the letter C before:

My actual deadline will remain “about three minutes before publication”.