Wednesday, March 8, 2017

Finding Fibonacci


Devlin makes a pilgrimage to Pisa to see the
statue of Leonardo Fibonacci in 2002.
In 1983, I did something that would turn out to have a significant influence on the direction my career would take. Frustrated by the lack of coverage of mathematics in the weekly science section of my newspaper of choice, The Guardian, I wrote a short article about mathematics and sent it to the science editor. A few days later, the editor phoned me to explain why he could not to publish it. “But,” he said, “I like your style. You seem to have a real knack for explaining difficult ideas in a way ordinary people can understand.” He encouraged me to try again, and my second attempt was published in the newspaper on May 12, 1983. Several more pieces also made it into print over the next few months, eliciting some appreciative letters to the editor. As a result, when The Guardian launched a weekly, personal computing page later that year, it included my new, twice-monthly column MicroMaths. The column ran without interruption until 1989, when my two-year visit to Stanford University in California turned into a permanent move to the US.

Before long, a major publisher contracted me to publish a collection of my MicroMaths articles, which I did, and following that Penguin asked me to write a more substantial book on mathematics for a general audience. That book, Mathematics: The NewGolden Age, was first published in 1987, the year I moved to America.

In addition to writing for a general audience, I began to give lectures to lay audiences, and started to make occasional appearances on radio and television. From 1991 to 1997, I edited MAA FOCUS, the monthly magazine of the Mathematical Association of America, and since January 1996 I have written this monthly Devlin’s Angle column. In 1994, I also became the NPR Math Guy, as I describe in my latest article in the Huffington Post.

Each new step I took into the world of “science outreach” brought me further pleasure, as more and more people came up to me after a talk or wrote or emailed me after reading an article I had written or hearing me on the radio. They would tell me they found my words inspiring, challenging, thought-provoking, or enjoyable. Parents, teachers, housewives, business people, and retired people would thank me for awakening in them an interest and a new appreciation of a subject they had long ago given up as being either dull and boring or else beyond their understanding. I came to realize that I was touching people’s lives, opening their eyes to the marvelous world of mathematics.

None of this was planned. I had become a “mathematics expositor” by accident. Only after I realized I had been born with a talent that others appreciated—and which by all appearances is fairly rare—did I start to work on developing and improving my “gift.”

In taking mathematical ideas developed by others and explaining them in a way that the layperson can understand, I was following in the footsteps of others who had also made efforts to organize and communicate mathematical ideas to people outside the discipline. Among that very tiny subgroup of mathematics communicators, the two who I regarded as the greatest and most influential mathematical expositors of all time are Euclid and Leonardo Fibonacci. Each wrote a mammoth book that influenced the way mathematics developed, and with it society as a whole.

Euclid’s classic work Elements presented ancient Greek geometry and number theory in such a well-organized and understandable way that even today some instructors use it as a textbook. It is not known if any of the results or proofs Euclid describes in the book are his, although it is reasonable to assume that some are, maybe even many. What makes Elements such a great and hugely influential work, however, is the way Euclid organized and presented the material. He made such a good job of it that his text has formed the basis of school geometry teaching ever since. Present day high school geometry texts still follow Elements fairly closely, and translations of the original remain in print.

With geometry being an obligatory part of the school mathematics curriculum until a few years ago, most people have been exposed to Euclid’s teaching during their childhood, and many recognize his name and that of his great book. In contrast, Leonardo of Pisa (aka Fibonnaci) and his book Liber abbaci are much less well known. Yet their impact on present-day life is far greater. Liber abbaci was the first comprehensive book on modern practical arithmetic in the western world. While few of us ever use geometry, people all over the world make daily use of the methods of arithmetic that Leonardo described in Liber abbaci.

In contrast to the widespread availability of the original Euclid’s Elements, the only version of Leonardo’s Liber abbaci we can read today is a second edition he completed in 1228, not his original 1202 text. Moreover, there is just one translation from the original Latin, in English, published as recently as 2002.

But for all its rarity, Liber abbaci is an impressive work. Although its great fame rests on its treatment of Hindu-Arabic arithmetic, it is a mathematically solid book that covers not just arithmetic, but the beginnings of algebra and some applied mathematics, all firmly based on the theoretical foundations of Euclid’s mathematics.

After completing the first edition of Liber abbaci, Leonardo wrote several other mathematics books, his writing making him something of a celebrity throughout Italy—on one occasion he was summonsed to an audience with the Emperor Frederick II. Yet very little was written about his life.

In 2001, I decided to embark on a quest to try to collect together what little was known about him and bring his story to a wider audience. My motivation? I saw in Leonardo someone who, like me, devoted a lot of time and effort trying to make the mathematics of the day accessible to the world at large. (Known today as “mathematical outreach,” very few mathematicians engage in that activity.) He was the giant whose footsteps I had been following.

I was not at all sure I could succeed. Over the years, I had built up a good reputation as an expositor of mathematics, but a book on Leonardo would be something new. I would have to become something of an archival scholar, trying to make sense of Thirteenth Century Latin manuscripts. I was definitely stepping outside my comfort zone.

The dearth of hard information about Leonardo in the historical record meant that a traditional biography was impossible—which is probably why no medieval historian had written one. To tell my story, I would have to rely heavily on the mathematical thread that connects today’s world to that of Leonardo—an approach unique to mathematics, made possible by the timeless nature of the discipline. Even so, it would be a stretch.

In the end, I got lucky. Very lucky. And not just once, but several times. As a result of all that good fortune, when my historical account The Man of Numbers: Fibonacci’s Arithmetic Revolution was published in 2011, I was able to compensate for the unavoidable paucity of information about Leonardo’s life with the first-ever account of the seminal discovery showing that my medieval role-model expositor had indeed played the pivotal role in creating the modern world that most historians had hypothesized.

With my Leonardo project such a new and unfamiliar genre, I decided from the start to keep a diary of my progress. Not just my findings, but also my experiences, the project's highs and lows, the false starts and disappointments, the tragedies and unexpected turns, the immense thrill of holding in my hands seminal manuscripts written in the thirteenth and fourteenth centuries, and one or two truly hilarious episodes. I also encountered, and made diary entries capturing my interactions with, a number of remarkable individuals who, each for their own reasons, had become fascinated by Fibonacci—the Yale professor who traced modern finance back to Fibonacci, the Italian historian who made the crucial archival discovery that brought together all the threads of Fibonacci's astonishing story, and the remarkable widow of the man who died shortly after completing the world’s first, and only, modern language translation of Liber abbaci, who went to heroic lengths to rescue his manuscript and see it safely into print.

After I had finished the Man of Numbers, I decided that one day I would take my diary and turn it into a book, telling the story of that small group of people (myself included) who had turned an interest in Leonardo into a passion, and worked long and hard to ensure that Leonardo Fibonacci of Pisa will forever be regarded as among the very greatest people to have ever lived. Just as The Man of Numbers was an account of the writing of Liber abbaci, so too Finding Fibonacci is an account of the writing of The Man of Numbers. [So it is a book about a book about a book. As Andrew Wiles once famously said, “I’ll stop there.”]

This post is adapted from the introduction of Keith Devlin’s new book Finding Fibonacci: The Quest to Rediscover the Forgotten Mathematical Genius Who Changed the World, published this month by Princeton University Press.

Friday, January 6, 2017

So THAT’s what it means? Visualizing the Riemann Hypothesis

Two years ago, there was a sudden, viral spike in online discussion of the Ramanujan identity

1 + 2 + 3 + 4 + 5 + . . . = –1/12

This identity had been lying around in the mathematical literature since the famous Indian mathematician Srinivasa Ramanujan included it in one of his books in the early Twentieth Century, a curiosity to be tossed out to undergraduate mathematics students in their first course on complex analysis (which was my first exposure to it), and apparently a result that physicists made actual (and reliable) use of.

The sudden explosion of interest was the result of a video posted online by Australian video journalist Brady Haran on his excellent Numberphile YouTube channel. In it, British mathematician and mathematical outreach activist James Grime moderates as his physicist countrymen Tony Padilla and Ed Copeland of the University of Nottingham explain their “physicists’ proof” of the identity.

In the video, Padilla and Copeland manipulate infinite series with the gay abandon physicists are wont to do (their intuitions about physics tends to keep them out of trouble), eventually coming up with the sum of the natural numbers on the left of the equality sign and –1/12 on the right.

Euler was good at doing that kind of thing too, so mathematicians are hesitant to trash it, rather noting that it “lacks rigor” and warning that it would be dangerous in the hands of a lesser mortal than Euler.

In any event, when it went live on January 9, 2014, the video and the result (which to most people was new) exploded into the mathematically-curious public consciousness, rapidly garnering hundreds of thousands of hits. (It is currently approaching 5 million in total.) By February 3, interest was high enough for The New York Times to run a substantial story about the “result”, taking advantage of the presence in town of Berkeley mathematician Ed Frenkel, who was there to promote his new book Love and Math, to fill in the details.

Before long, mathematicians whose careers depended on the powerful mathematical technique known as analytic continuation were weighing in, castigating the two Nottingham academics for misleading the public with their symbolic sleight-of- hand, and trying to set the record straight. One of the best of those corrective attempts was another Numberphile video, published on March 18, 2014, in which Frenkel give a superb summary of what is really going on.

A year after the initial flair-up, on January 11, 2015, Haran published a blogpost summarizing the entire episode, with hyperlinks to the main posts. It was quite a story.

[[ASIDE: The next few paragraphs may become a bit too much for casual readers, but my discussion culminates with a link to a really cool video, so keep going. Of course, you could just jump straight to the video, now you know it’s coming, but without some preparation, you will soon get lost in that as well! The video is my reason for writing this essay.]]

For readers unfamiliar with the mathematical background to what does, on the face of it, seem like a completely nonsensical result, which is the MAA audience I am aiming this essay at (principally, undergraduate readers and those not steeped in university-level math), it should be said that, as expressed, Ramanujan’s identity is nonsense. But not because of the -1/12 on the right of the equals sign. Rather, the issue lies in those three dots on the left. Not even a mathematician can add up infinitely many numbers.

What you can do is, under certain circumstances, assign a meaning to an expression such as

X1 + X2 + X3 + X4 + …

where the XN are numbers and the dots indicate that the pattern continues for ever. Such expressions are called infinite series.

For instance, undergraduate mathematics students (and many high school students) learn that, provided X is a real number whose absolute value is less than 1, the infinite series

1 + X + X+ X3 + X+ …

can be assigned the value 1/(1 – X). Yes, I meant to write “can be assigned”. Since the rules of real arithmetic do not extend to the vague notion of an “infinite sum”, this has to be defined. Since we are into the realm of definition here, in a sense you can define it to be whatever you want. But if you want the result to be meaningful and useful (useful in, say, engineering or physics, to say nothing of the rest of mathematics), you had better define it in a way that is consistent with that “rest of mathematics.” In this case, you have only one option for your definition. A simple mathematical argument (but not the one you can find all over the web that involves multiplying the terms in the series by X, shifting along, and subtracting—the rigorous argument is a bit more complicated than that, and a whole lot deeper conceptually) shows that the value has to be 1/(1 – X).

So now we have the identity

(*) 1 + X +X+ X3 + X+ … = 1/(1 – X)

which is valid (by definition) whenever X has absolute value less than 1. (That absolute value requirement comes in because of that “bit more complicated” aspect of the rigorous argument to derive the identity that I just mentioned.)

“What happens if you put in a value of X that does not have absolute value less than 1?” you might ask. Clearly, you cannot put X = 1, since then the right-hand side becomes 1/0, which is totally and absolutely forbidden (except when it isn’t, which happens a lot in physics). But apart from that one case, it is a fair question. For instance, if you put X = 2, the identity (*) becomes

1 + 2 + 4 + 8 + 16 + … = 1/(1 – 2) = 1/(–1) = –1

So you could, if you wanted, make the identity (*) the definition for what the infinite sum

1 + X + X+ X3 + X4 + …

means for any X other than X = 1. Your definition would be consistent with the value you get whenever you use the rigorous argument to compute the value of the infinite series for any X with absolute value less than 1, but would have the “benefit” of being defined for all values of X apart from one, let us call it a “pole”, at X = 1.

This is the idea of analytic continuation, the concept that lies behind Ramanujan’s identity. But to get that concept, you need to go from the real numbers to the complex numbers.

In particular, there is a fundamental theorem about differentiable functions (the accurate term in this context is analytic functions) of a single complex variable that says that if any such function has value zero everywhere on a nonempty disk in the complex plane, no matter how small the diameter of that disk, then the function is zero everywhere. In other words, there can be no smooth “hills” sitting in the middle of flat plains, or even one small flat clearing in the middle of a “hilly” landscape—the quotes are because we are beyond simple visualization here.

An immediate consequence of this theorem is that if you pull the same continuation stunt as I just did for the series of integer powers, where I extended the valid formula (*) for the sum when X is in the open unit interval to the entire real line apart from one pole at 1, but this time do it for analytic functions of a complex variable, then if you get an answer at all (i.e., a formula), it will be unique. (Well, no, the formula you get need not be unique, rather the function it describes will be.)

In other words, if you can find a formula that describes how to compute the values of a certain expression for a disk of complex numbers (the equivalent of an interval of the real line), and if you can find another formula that works for all complex numbers and agrees with your original formula on that disk, then your new formula tells you the right way to calculate your function for any complex number. All this subject to the requirement that the functions have to be analytic. Hence the term “analytic continuation.'

For a bit more detail on this, check out the Wikipedia explanation or the one on Wolfram Mathworld. If you find those explanations are beyond you right now, just remember that this is not magic and it is not a mystery. It is mathematics. The thing you need to bear in mind is that the complex numbers are very, very regular. Their two-dimensional structure ties everything down as far as analytic functions are concerned. This is why results about the integers such as Fermat’s Last Theorem are frequently solved using methods of Analytic Number Theory, which views the integers as just special kinds of complex numbers, and makes use of the techniques of complex analysis.

Now we are coming to that video. When I was a student, way, way back in the 1960s, my knowledge of analytic continuation followed the general path I just outlined. I was able to follow all the technical steps, and I convinced myself the results were true. But I never was able to visualize, in any remotely useful sense, what was going on.

In particular, when our class came to study the (famous) Riemann zeta function, which begins with the following definition for real numbers S bigger than 1:

(**) Zeta(S) = 1 + 1/2S + 1/3S + 1/4S + 1/5S + …

I had no reliable mental image to help me understand what was going on. For integers S greater than 1, I knew what the series meant, I knew that it summed (converged) to a finite answer, and I could follow the computation of some answers, such as Euler’s

Zeta(2) = π2/6

(You get another expression involving π for S = 4, namely π4/90.)

It turns out that the above definition (**) will give you an analytic function if you plug in any complex number for S for which the real part is bigger than 1. That means you have an analytic function that is rigorously defined everywhere on the complex plane to the right of the line x = 1.

By some deft manipulation of formulas, it’s possible to come up with an analytic continuation of the function defined above to one defined for all complex numbers except for a pole at S = 1. By that basic fact I mentioned above, that continuation is unique. Any value it gives you can be taken as the right answer.

In particular, if you plug in S = –1, you get

Zeta(–1) = –1/12

That equation is totally rigorous, meaningful, and accurate.

Now comes the tempting, but wrong, part that is not rigorous. If you plug in S = –1 in the original infinite series, you get

1 + 1/2-1 + 1/3-1 + 1/4-1 + 1/5-1 + …

which is just

1 + 2 + 3 + 4 + 5 + …

and it seems you have shown that

1 + 2 + 3 + 4 + 5 + . . . = –1/12

The point is, though, you can’t plug S = –1 into that infinite series formula (**). That formula is not valid (i.e., it has no meaning) unless S > 1.

So the only way to interpret Ramanujan’s identity is to say that there is a unique analytic function, Zeta(S), defined on the complex plane (apart from at the real number 1), which for all real numbers S greater than 1 has the same values as the infinite series (**), which for S = –1 gives the value Zeta(–1) = –1/12.

Or, to put it another way, more fanciful but less accurate, if the sum of all the natural numbers were to suddenly find it had a finite answer, that answer could only be –1/12.

As I said, when I learned all this stuff, I had no good mental images. But now, thanks to modern technology, and the creative talent of a young (recent) Stanford mathematics graduate called Grant Sanderson, I can finally see what for most of my career has been opaque. On December 9, he uploaded this video onto YouTube.



It is one of the most remarkable mathematics videos I have ever seen. Had it been available in the 1960s, my undergraduate experience in my complex analysis class would have been so much richer for it. Not easier, of that I am certain. But things that seemed so mysterious to me would have been far clearer. Not least, I would not have been so frustrated at being unable to understand how Riemann, based on hardly any numerical data, was able to formulate his famous hypothesis, finding a proof of which is agreed by most professional mathematicians to be the most important unsolved problem in the field.

When you see (in the video) what looks awfully like a gravitational field, pulling the zeros of the Zeta function towards the line x = 1/2, and you know that it is the only such gravitational field there is, and recognize its symmetry, you have to conclude that the universe could not tolerate anything other than all the zeros being on that line.

Having said that, it would, however, be really interesting if that turned out not to be the case. Nothing is certain in mathematics until we have a rigorous proof.

Meanwhile, do check out some of Grant’s other videos. There are some real gems!

Tuesday, December 13, 2016

You can find the secret to doing mathematics in a tubeless bicycle tire

The author climbing the locally-notorious Country View Road just south of San Jose, CA

As regular readers may know, one of my consuming passions in life besides mathematics is cycling. Living in California, where serious winters were wisely banned many years ago, on any weekend throughout the year you are likely to find me out on a road- or a mountain bike.

Being also a lover of well-designed technology, I long ago switched to using tubeless tires on my road bike. Actually, it’s bikes, in the plural—my road bikes number four, all with different riding conditions in mind, but all having in common the same kind of ultra- narrow saddle that non-cyclists think must be excruciatingly painful, but is in fact engineered to be the only thing comfortable enough to sit on for many hours at a stretch. [Keep going; I am working my way to making a mathematical point. In fact, I am heading towards THE most significant mathematical point of all: What is the secret to doing math?]

Road tubeless tires have several advantages over the more common type of tire, which requires an airtight innertube. One advantage is that you need inflate them only to 80 pounds per square inch, as opposed to the 110 psi or more for a tubed tire, which provides even more comfort over those many hours in the saddle.

You need tire pressures 3 or more times that of a car tire because of the extremely low volume in a road-bike tire, which sits on a 700 cm diameter wheel with a rim whose width is between 21 mm and 25 mm. It is that high pressure that made the manufacture of tubeless wheels and tires for bicycles such a significant challenge. How can you ensure an almost totally airtight fit when the tire is inflated, and it still be possible for an average person to remove and mount a deflated tire with their bare hands. (Tire levers can easily damage tubeless wheels and tires.) We are almost to the secret to doing math. Hang in there.

Clever design: Tubeless rims and tires on a road bike wheel.

The airtight fit is possible precisely because of that relatively high pressure inside the tire—80 psi is over five times the air pressure outside the tire. (An automobile tire is inflated to roughly twice atmospheric pressure, much lower.) The cross-sectional photo on the left shows how a tubeless tire has a squared-off ridge that fits into a matching notch in the rim. The more air pressure there is in the tire, the tighter that ridge binds to the rim, increasing the air seal.

The problem is, as I mentioned, getting the tire on and off the rim. The tire ridge that fits into the rim-notch has a steel wire running through it, and its squared-off shape is designed to make it difficult for the tire to separate from the rim—that, after all, is the point. To solve the mounting/removal problem, the wheel has a channel in the middle, as shown more clearly in the photo on the right.

To mount the tire, you push the two tire-rims into that channel, one after the other. By the formula for the circumference of a circle, when a tire rim is in that center channel, you have just over 3 times the depth of the channel of superfluous tire length to play with, roughly 12mm of tire looseness. The idea is to use that “looseness” to work your way around the wheel, pushing (actually rolling) first one tire edge over the wheel rim and into the channel, then the other. Once the tire is seated on the rim, inflating it with a hand pump forces the tire rims out of the channel into the notches. To remove the tire after it is deflated, you push the two tire rims into the channel and reverse the process.

That, at least, is the theory. Putting theory into practice turns out to be quite a challenge. When I first started to use road tubeless tires, several years ago, I read several online manuals and watched a number of YouTube videos demonstrating how to do it, and could never do it. I usually ended up taking the wheel and tire to my local bike shop, where the mechanic would do it for me with seeming ease before my eyes. “Fifteen dollars, please.”

But what would happen if I had a flat on one of the remote rides I regularly do in the mountains that surround Silicon Valley, where I live? One major advantage of tubeless tires is that, even if they puncture, usually the air leaks out only very slowly, and can generally be stopped by inflating the tire from a small pressure-can of air and liquid latex you carry in your back pocket, which seals the hole. Which is how I was always able to get to a bike shop where someone else could solve the problem for me. But a major puncture in the remote, with no cell phone access, could leave me dangerously stranded. Clearly, I had to learn how to do it myself.

From now on, when I say “change a tubeless tire”, you can interpret it as “do mathematics”. The secret is coming up. Moreover, it is coming with a moral that those of us in mathematics education ignore at our students’ peril.

What I find cool is that, for me I somehow stumbled on the secret to doing math fairly early in life, before math had become such a problem that I felt I could never do it. But taking up cycling later in life, when I had a fully developed set of metacognitive skills, I approached the problem of changing a tubeless tire in much the same way as many people—including, I suspect, the mechanics in my local bicycle shop—see math. Namely, people like me (and that smart kid sitting in the front row in the school math class) make doing math look effortless, but many people feel they could never master it in a million years.

Nothing, surely, can look less requiring of skill or expertise than putting a tire on a bicycle wheel. (This is why I think this is such a great example.) Surely, you just need to read an instruction manual, or perhaps have someone demonstrate to you. But no matter how many times I read the instructions, no matter how many times I viewed—and re-viewed—those how-to YouTube videos, and no matter how many times I stood alongside the bike shop mechanic and watched as he quickly and effortlessly put the tire onto the wheel, I could never do it.

Just think about that for a moment. For some tasks, instruction (on its own) just does not work. Not even for the seemingly simple task of changing a bicycle tire. And yet we think that forcing kids to sit in the math class while we force-feed instruction will result in their being able to do math! Dream on.

What does work, in fact what is absolutely necessary, both for changing tubeless tires and doing math, is that the learner has to learn to see things the way the expert does. And, since instruction does not work, that key step has to be made by the learner. All that a good teacher can do, then, is find a way to help the learner make that key leap. [That short initial word “all” belies the human expertise required to do this.]

Clearly, when I was, yet again, standing in the bicycle repair shop, watching the mechanic change my tire, what he was doing—more precisely, what he was experiencing—was very different from what I was doing and experiencing when I tried and failed. What was I not getting?

My big breakthrough finally came the one time when the mechanic, holding the wheel horizontally pressed to his stomach, while manipulating the tire with both hands, told me what he was really doing. “You have to think of the tire as alive,” he said. “It wants to be sitting firmly on the rim” [that, after all, is what it was—expensively—designed for], “but it is not very disciplined. It’s like a small child. It moves around and resists your attempts to force it. You have to understand it, and be aware, through your hands, of what it is doing. Work with it—be constantly aware of what it is trying to do—so you both get what you want: the tire gets onto the wheel, where it belongs, and you can inflate it and get back on your bike (where you want to be).”

Fanciful? Maybe. But it worked. And it continues to work. As a result, not only can I now change my tubeless tires, it has for me become “mindless and automatic,” as effortless (to me) as Picasso drawing a simple doodle on a restaurant napkin to pay the bill for his meal was to him. (I thought that if you got this far, you deserved a second example with greater cultural overtones.)

It took many years for Picasso to learn to draw the way he did (and for the marketplace to assign high value to his work), but that does not mean his work was not skillful; rather, he simply routinized part of it. When I watch a film of him at work, I see superficially how he created, and it looks routine and effortless, but I do not see his canvas as he did, and I could not draw as he did.

Likewise, my skill in fitting a tubeless tire, now effortless and automatic, is a result of my now seeing and understanding what earlier had been opaque.

I admit that it is far easier to learn to mount a tubeless tire on a road bike wheel than to draw like Picasso. But I am less sure the difference is so great between changing a tubeless tire and being able to solve any one particular kind of math problem. Still, no matter how great the difference in the degree of skill required, it is possible to learn from the analogy.

Given what I have said here, will reading this essay mean you can go out and immediately be able to change a tubeless tire? Have I just made a case for instruction working after all? It’s possible—for changing bicycle tires, but surely not for painting like Picasso. Instruction can and does work, and it is an important part of learning. But my guess is you would find my words are not enough. I think that the reason that one piece of bike-shop instruction was so instantly transformative for me was that I had spent an aggregate of many hours struggling to change my tire and failing. I had reached a stage where the effective key was to get inside the mind of an expert. But the ground had to be prepared for that simple revelation to work.

In education, as in so many parts of life, there are no silver bullets. But given enough of the right preparation—enough experience acquired through repeated trying and failing—an ordinary lead bullet will do the job.

----

This month’s column is loosely adapted from a passage of my forthcoming book Finding Fibonacci: The Quest to Rediscover the Forgotten Mathematical Genius Who Changed the World, due out in March.

Friday, November 4, 2016

Mathematical Milk and the U.S. Presidential Election

Keith Devlin mails his completed election ballot. What does math have to say about his act?
With the United States is the final throes of a presidential election, my mind naturally turned to the decidedly tricky matter of election math. Voting provides a great illustration of how mathematics – which rules supreme, yielding accurate and reliable answers to precise questions, in the natural sciences and engineering – can lead us astray when we try to apply it to human and social activities.

A classic example is how we count votes in an election, the topic of an earlier Devlin’sAngle post, in November, 2000. In that essay, I looked at how different ways to tally votes could affect the imminent Bush v. Gore election, at the time blissfully unaware of how chaotic would be the process of counting votes and declaring a winner on that particular occasion. The message there was, particularly in the kinds of tight race we typically see today, the different ways that votes can be tallied can lead to very different results.

Everything I said back then remains just as valid and pertinent today (mathematics is like that), so this time I’m going to look at another perplexing aspect of election math: why do we make the effort to vote? After all, while elections are sometimes decided by a small number of votes, it is unlikely in the extreme that an election on the scale of a presidential election will hang on the decision of a single voter. Even if it did, that would be well within the range of procedural error, so it makes no difference if any one individual votes or not.

To be sure, if a large number of people decide to opt out, that can affect the outcome. But there is no logical argument that takes you from that observation to it being important for a single individual to vote. This state of affairs is known as the Paradox of Voting, or sometimes Downs Paradox. It is so named after Anthony Downs, a political economist whose 1957 book An Economic Theory of Democracy examined the conditions under which (mathematical) economic theory could be applied to political decision-making.

On the face of it, Downs’ analysis does lead to a paradox. Economic theory tells us that rational beings make decisions based on expected benefit (a notion that can be made numerically precise). That approach works well for analyzing, say, why people buy insurance year after year, even though they may never submit a claim. The theory tells you that the expected benefit is greater than the cost; so it is rational to buy insurance. But when you adopt the same approach to an election, you find that, because the chance of exercising the pivotal vote in an election is minute compared to any realistic estimate of the private individual benefits of the different possible outcomes, the expected benefits of voting are less than the cost. So you should opt out. [The same observation had in fact been made much earlier, in 1793, by Nicolas de Condorcet, but without the theoretical backing that Downs brought to the issue.]

Yet, many otherwise sane, rational citizens do not opt out. Indeed, society as a whole tends to look down on those who do not vote, saying they are not "doing their part." (In fact, many countries make participation in a national election obligatory, but that is a separate, albeit related, issue.)

So why do we (or at least many of us) bother to vote? I can make the question even more stark, and personal. Suppose you have intended to "do your part" and vote. You wake up on election morning with a sore throat, and notice that it is raining heavily. Being numerically able (as all Devlin’s Angle readers must be), you say to yourself, "It cannot possibly affect the result if I just stay at home and nurse my throat. I was intending to vote, after all. Changing my mind about voting at the last minute cannot possibly influence anyone else. Especially if I don’t tell anyone." The math and the logic, surely, are rock solid. Yet, professional mathematician as I am, I would struggle out and cast my vote. And I am sure many Devlin’s Angle readers would too – most of them, I would suspect.

So what is going on? We can do the math. We are good logical thinkers. Why don’t we act according to that reasoning? Are we conceding that mathematics actually isn’t that useful? [SPOILER: Math is useful; but only when applied with a specific purpose in mind, and chosen/designed in a way that makes it appropriate for that purpose.]

Which brings me to my main point. To make it, let me switch for a moment from elections to the Golden Ratio. In April 2015, the magazine Fast Company Design published an article titled "The Golden Ratio: Design’s Biggest Myth," in which I was quoted at length. (The author also drew heavily on a Devlin’s Angle post of mine from May 2007.)

With a readership much wider than Devlin’s Angle, over the years the Fast Company Design piece has generated a fair amount of correspondence from people beyond mathematics academia, often designers who have not been able to overcome drinking Golden Ratio Kool-Aid during their design education. One recent email came, not from a designer but a high school math teacher, who objected to a statement the article quoted me (accurately) as saying, “Strictly speaking, it's impossible for anything in the real-world to fall into the golden ratio, because it’s an irrational number.” The teacher had, it was at once clear to me, drunk not just Golden Ratio Kool-Aid, but Math Kool-Aid as well.

In the interest of full disclosure, let me admit that, in the early part of my career as a mathematics expositor, I was as guilty as anyone of distributing both Golden Ratio Kool-Aid and Math Kool-Aid, to whoever would drink it. But, as a committed scientist, when presented with evidence to the contrary, I re-examined my thinking, admitted I had been wrong, and started to push better, more honest products, which I will call Golden Ratio Milk and Mathematical Milk. I described Golden Ratio Milk in my 2007 MAA post and peddled it more in that Fast Company Design interview. Here I want to talk about Mathematical Milk.

The reason why the Golden Ratio’s irrationality prevents its use in, say architecture, is that the issue at hand involves measurement. Measurement requires fixing a unit of measure – a scale. It doesn’t matter whether it is meters or feet or whatever, but once you have fixed it, that is what you use. When you measure things, you do so to an agreed degree of accuracy. Perhaps one or two decimal places. Almost never to more than maybe twenty decimal places, and that only in a few instances in subatomic physics. So it terms of actual, physical measurement, or manufacturing, or building, you never encounter objects to which a numerical measurement has more than a few decimal places. You simply do not need a number system that has fractions with denominator much greater than, say, 1,000,000, and generally much less than that.

Even if you go beyond physical measurement, to the theoretical realm where you imagine having an unlimited number decimal places available, you will still be in the domain of the rational numbers. Which means the Golden Ratio does not arise. Irrational numbers arise to meet mathematical needs, not the requirements of measurement. They live not in the physical world but in the human imagination. (Hence my Fast Company Design quote.) It is important to keep that distinction clear in our minds.

The point is, when we abstract from our experiences of the world around us, to create mathematical models, two important things happen. A huge amount of information is lost; and there is a significant gain in precision. The two are not independent. If we want to increase the precision, we lose more information, which means that our model has less in common with the real world it is intended to represent. Moreover, when we construct a mathematical model, we do so with a particular question, or set of questions in mind.

In astronomy and physics, and related domains such as engineering, all of this turns out to be not too problematic. For example, the simplistic model of the Solar System as a collection of point-masses orbiting around another, much heavier, point-mass, is extremely useful. We can formulate and solve equations in that model, and they turn out to be very useful. At least they turn out to be useful in terms of the goal questions, initially in this case predicting where the planets will be at different times of the year. The model is not very helpful in telling us what the color of each planet’s surface is, or even if it has a surface, both of which are certainly precise, scientific questions.

When we adopt a similar approach to model money supply or other economic phenomena, we can obtain results that are, mathematically, just as precise and accurate, but their connection to the real world is far more tenuous and unreliable – as has been demonstrated several times in recent years when those mathematical results have resulted in financial crises, and occasionally disasters.

So what of the paradox of voting? The paradox arises when you start by assuming that people vote to choose, say, a president. Yes, we all say that is what we do. But that’s just because we have drunk Election Kool-Aid. We don’t actually behave in accordance with that statement. If we did, then as rational beings we would indeed stay at home on election day.

Time to throw out the Kool-Aid and buy a gallon jug of far more beneficial Election Milk: (Presidential) elections are about a society choosing a president. Where that purpose impacts the individual voter is not who we vote for, but in providing social pressure to be an active member of that society.

That this is what is actually going on is illustrated by the fact that U.S. society created, and millions of people wear, "I have voted" badges on election day. The focus, and the personal reward, is not "Who I voted for" but "I participated in the process." [For an interesting perspective on this, see the recent article in the Smithsonian Magazine, "WhyWomen Bring Their “I Voted” Stickers to Susan B. Anthony’s Grave."]

To be sure, you can develop mathematical models of group activities, like elections, and they will tend to lead to fewer problems (and "paradoxes") than a single-individual model will, but they too will have limitations. All mathematical models do. Mathematics is not reality; it is just a model of reality (or rather, it is a whole, and constantly growing, collection of models).

When we develop and/or apply a mathematical model, we need to be clear what questions it is designed to help us answer. If we try to apply it to a different question, we may get lucky and get something useful, but we may also end up with nonsense, perhaps in the form of a "paradox."

With both measurement and the election, as is so often the case, one benefit we get from trying to apply mathematics to our world and to our lives is we gain insight into what is really going on.

Attempting to use the real numbers to model the acts of measuring physical objects leads us to recognize the dependency on the physical activity of measurement.

Likewise, grappling with Downs Paradox leads us to acknowledge what elections are really about – and to recognize that choosing a leader is a societal activity. In a democracy, who each one of us votes for is inconsequential; that we vote is crucial. That’s why I did not just spend a couple of hours yesterday making my choices and filling in my ballot and leaving it at that. I also went out earlier today – in light rain as it happens (and without a sore throat) – and put my ballot in the mailbox. Yesterday I acted as an individual, motivated by my felt societal obligation to participate in the election process. Today I acted as a member of society.

As a professional set theorist, I am familiar with the relationship between, and the distinction between, a set and its members. When we view a set in terms of its individual members, we say we are treating it extensionally. When we consider a set in terms of its properties as a single entity, we say we are treating in intensionally. In an election, we are acting intensionally (and intentionally) – at the set level, not as an element of a set.

* A shorter version of this article was published simultaneously in The Huffington Post.



Wednesday, October 12, 2016

It was Twenty Years Ago Today

The title of the famous Beatles song does not exactly apply to Devlin’s Angle. The online column (now run on a blog platform, but unlike most blogs, still subject to an editor’s guiding hand) is in its twentieth year, but it actually launched on January 1, 1996.

In last month’s column, I looked back at the very first post. It was a fascinating exercise to try to put myself back in the mindset of how the world looked back then, which was about the time when the World Wide Web was just starting to find its way onto university campuses, but had not yet penetrated the everyday lives of most of the world’s population.

That period of intense technological and societal change – looking back, it is clear it was just beginning, in the first half of the 1990s being more evolutionary rather than the revolutionary that was soon to follow – and the strong sensation of change both underway and pending, is reflected in some of the topics I chose to write about each month in that first year. Here is a list of those first twelve posts, with hyperlinks.


Along with essays you might find in a mathematics magazine for students (February, June, July, August, November, December), there are reflections on where mathematics and its role in the world might be heading in the next few years.

January’s post, about the growth of computer viruses in the digital domain, was clearly in that Brave New World vein, as I noted last month, and in February I focused on another aspect of the rapid growth of the digital world, with a look at the ongoing debate about the future of Artificial Intelligence. Though that field has undoubtedly made many advances in the ensuing two decades, the core argument I summarized there seems as valid today as it did then. Digital devices still do not “think” in anything like a human fashion (though these days it can sometimes be harder to tell the difference).

The posts for April, May, and October looked at different aspects of the “Where is mathematics heading?” question. Of course, I was not claiming then, nor am I suggesting now, that the core of pure mathematics is going to change. (Though the growth of Experimental Mathematics in the New Millennium was a new direction, one I addressed in a Devlin’s Angle post in March 2009.) Rather, I was taking a much broader view of mathematics, stepping outside the mathematics department of colleges and universities and looking at the way mathematics is used in the world.

The October post, in particular, turned out to be highly prophetic for my own career. Shortly after the terrorist attack on the World Trade Center on September 11, 2001, I was contacted by a large defense contractor, asking if I would join a large team they were putting together to bid for a Defense Department contract to find ways to improve intelligence analysis. I accepted the offer, and worked on that project for the next several years. (From my perspective, that project and the work that followed did not end uniformly well, as I lamented in an AMS Notices opinion piece in 2014.) When that project ended, I did similar work for a large contractor to the US Navy and another project for the US Army. In all three projects, I was living in the kind of world I portrayed in that October, 1996 column.

In fact, my professional life as a mathematician for the entire life of Devlin’s Angle has been in that world – a way of using mathematics I started to refer to as “mathematical thinking.” In a Devlin’s Angle post in 2012, I tried to articulate what I mean by that term. (The term is used by others, sometimes with different meanings, though I see strong overlaps and general agreements among them all.) That same year, I launched the world’s first mathematical MOOC on the newly established online course platform Coursera, with the title “Introduction to Mathematical Thinking”, and published a book with the same title.

With the world as it is today, in particular the pervasive (though largely hidden) role played by mathematics and mathematical ideas in almost every aspect of our lives, I would hazard a guess that there are far more people using “mathematical thinking” than there are people doing mathematics in the traditional sense.

If so, that would make the professions of mathematician and mathematics educator two of the most secure careers in the world. For there is one thing in particular you need in order to engage in (effective) mathematical thinking about a real world problem: an adequate knowledge of, and conceptual understanding of, mathematics. In fact, that need was ever so, but it often tended to be overlooked in the pre-digital eras, when doing mathematics meant engaging in a lot of paper-and- pencil, symbolic computations, which meant that the bulk of mathematics instruction focused on computation, with wide ranging knowledge and conceptual understanding often getting short shrift.

But those days are gone. Today, we carry around in our pockets devices that give us instant access to pretty well all of the world’s mathematical information and computational procedures we might need to use. (Check out Wolfram Alpha.) But the thinking still has to be done where it always has: in our heads.


Tuesday, September 13, 2016

Then and Now: Devlin’s Angle Turned Twenty This Year

Devlin’s Angle turned 20 this year. The first post appeared on January 1, 1996, as part of the MAA’s move from print to online. I was the editor of the MAA’s regular print magazine MAA FOCUS at the time, continuing to act in that capacity until December 1997. (See the last edition of MAA FOCUS that I edited here.)

Keith Devlin at a mathematical exposition summit in Oregon in 1997. L to R: Ralph Abraham (Univ of California at Santa Cruz), Devlin, Roger Penrose (Univ of Oxford, UK), and Ivars Peterson (past MAA Director of Publications for Journals and Communications).

One of the innovations I made when I took over as MAA FOCUS editor in September 1991 was the inclusion of an editorial (written by me) in each issue. Though my ten-times-a-year essays were very much my own personal opinion, they were subject to editorial control by the organization's Executive Director, supported by an MAA oversight committee, both of which had approved my suggestion to do this. Over the years, the editorials generated no small amount of controversy, sometimes based on a particular editorial content, and other times on the more general principle of whether an editor’s personal opinion had a proper place in a professional organization's newsletter.

As to the latter issue, I am not sure anyone’s views changed over the years of my editorial reign, but the consensus at MAA Headquarters was that it did result in many more MAA members actually picking up MAA FOCUS when it arrived in the mail and reading it. That was why I was asked to write a regular essay for the new MAA Online. Though blogs and more generally social media were still in the future, the MAA leadership clearly had it right in thinking that an online newsletter was very much an organ in which informed opinion had a place.

And so Devlin’s Angle was born. When I realized recently that the column turned twenty this year — in its early days we thought of it very much an online “column”, with all that entailed in the world of print journalism — I was curious to remind myself what topic I chose to write about in my very first post.

Back then, I would have needed to explain to my readers that they could click on the highlighted text in that last sentence to bring up that original post. For the World Wide Web was a new resource that people were still discovering, with 1995-96 seeing its growth in academia. Today, of course, I can assume you have already looked at that first post. The words I wrote then (when I might have used the term “penned”, even though I typed them at a computer keyboard) provide an instant snapshot of how the present-day digital world we take for granted looked back then.

A mere twenty years ago.