Oscar E. Fernandez on The Calculus of Happiness

FernandezIf you think math has little to do with finding a soulmate or any other “real world” preoccupations, Oscar Fernandez says guess again. According to his new book, The Calculus of Happiness, math offers powerful insights into health, wealth, and love, from choosing the best diet, to finding simple “all weather” investment portfolios with great returns. Using only high-school-level math (precalculus with a dash of calculus), Fernandez guides readers through the surprising results. He recently took the time to answer a few questions about the book and how empowering mathematics can be.

The title is intriguing. Can you tell us what calculus has to do with happiness?

Sure. The title is actually a play on words. While there is a sprinkling of calculus in the book (the vast majority of the math is precalculus-level), the title was more meant to convey the main idea of the book: happiness can be calculated, and therefore optimized.

How do you optimize happiness?

Good question. First you have to quantify happiness. We know from a variety of research that good health, healthy finances, and meaningful social relationships are the top contributors to happiness. So, a simplistic “happiness equation” is: health + wealth + love = happiness. This book then does what any good applied mathematician would do (I’m am applied mathematician): quantify each of the “happiness components” on the left-hand side of the equation (health, wealth, and love), and then use math to extract valuable insights and results, like how to optimize each component.

This process sounds very much like the subtitle, how a mathematical approach to life adds up to health, wealth, and love. But just to be sure, can you elaborate on the subtitle?

That’s exactly right. Often we feel like various aspects of our lives are beyond our control. But in fact, many aspects of our lives, including some of the most important ones (like health, wealth, and love), follow mathematical rules. And by studying the equations that emerge from these rules you can quickly learn how to manipulate those equations in your favor. That’s what I do in the book for health, wealth, and love.

Can you give us some examples/applications?

I can actually give you about 30 of them, roughly the number discussed in the book. But let me focus on my three favorite ones. The first is what I called the “rational food choice” function (Chapter 2). It’s a simple formula: divide 100 calories by the weight (say, in grams) of a particular food. This yields a number whose units are calories per gram, the units of “energy density.” Something remarkable then happens when you plot the energy densities of various foods on a graph: the energy densities of nearly all the healthy foods (like fruits and vegetables) are at most about 2 calories per gram. This simple mathematical insight, therefore, helps you instantly make healthier food choices. And following its advice, as I discuss at length in the book, eventually translates to lower risk for developing heart disease and diabetes, weight loss, and even an increase in your life span! The second example comes from Chapter 3; it’s a formula for calculating how many more years you have to work for before you can retire. Among the formula’s many insights is that, in the simplest case, this magic number depends entirely on the ratio of how much you save each year to how much you spend. And the formula, being a formula, tells you exactly how changing that ratio affects your time until retirement. The last example is based on astronomer Frank Drake’s equation for estimating the number of intelligent civilizations in our galaxy (Chapter 5). It turns out that this alien-searching equation can also be used to estimate the number of possible compatible partners that live near you! That sort of equates a good date with an intelligent alien, and I suppose I can see some similarities (like how rare they are to find).

The examples you’ve mentioned have direct relevance to our lives. Is that a feature of the other examples too?

Absolutely. And it’s more than just relevance—the examples and applications I chose are all meant to highlight how empowering mathematics can be. Indeed, the entire book is designed to empower the reader—via math—with concrete, math-backed and science-backed strategies for improving their health, wealth, and love life. This is a sampling of the broader principle embodied in the subtitle: taking a mathematical approach to life can help you optimize nearly every aspect of your life.

Will I need to know calculus to enjoy the book?

Not at all. Most of the math discussed is precalculus-level. Therefore, I expect that nearly every reader will have studied the math used in the book at some point in their K-12 education. Nonetheless, I guide the reader through the math as each chapter progresses. And once we get to an important equation, you’ll see a little computer icon next to it in the margin. These indicate that there are online interactive demonstrations and calculators I created that go along with the formula. The online calculators make it possible to customize the most important formulas in the book, so even if the math leading up to them gets tough, you can still use the online resources to help you optimize various aspects of health, wealth, and love.

Finally, you mention a few other features of the book in the preface. Can you tell us about some of those?

Sure, I’ll mention two particular important ones. Firstly, at least 1/3 of the book is dedicated to personal finance. I wrote that part of the book to explicitly address the low financial literacy in this country. You’ll find understandable discussions of everything from taxes to investing to retirement (in addition to the various formulas derived that will help you optimize those aspects of your financial life). Finally, I organized the book to follow the sequence of math topics covered in a typical precalculus textbook. So if you’re a precalculus student, or giving this book to someone who is, this book will complement their course well. (I also included the mathematical derivations of the equations presented in the chapter appendixes.) This way the youngest readers among us can read about how empowering and applicable mathematics can be. It’s my hope that this will encourage them to continue studying math beyond high school.

Oscar E. Fernandez is assistant professor of mathematics at Wellesley College and the author of Everyday Calculus: Discovering the Hidden Math All around Us and The Calculus of Happiness: How a Mathematical Approach to Life Adds Up to Health, Wealth, and Love.

J. Richard Gott: What’s the Value of Pi in Your Universe?

Carl Sagan’s sci-fi novel Contact famously introduced wormholes for rapid transit between the stars. Carl had asked his friend Kip Thorne to tell him if the physics of wormholes was tenable and this led Thorne and his colleagues to investigate their properties. They found that traversable wormholes required exotic matter to prop them open and that, by moving the wormhole mouths one could find general relativity solutions allowing time travel to the past. A quantum state called the Casimir vacuum whose effects have been observed experimentally, could provide the exotic matter. To learn whether such time machines could be constructible in principle, we may have to master the laws of quantum gravity, which govern how gravity behaves on microscopic scales. It’s one of the reasons physicists find these solutions so interesting.

But in Contact there is lurking yet another fantastic sci-fi idea, which gets less publicity because it was not included in the movie version. In the book, the protagonist finds out from the extraterrestrials that the system of wormholes throughout the galaxy was not built by them, but by the long gone “old ones” who could manipulate not only the laws of physics but also the laws of mathematics! And they left a secret message in the digits of pi. In his movie Pi, Darren Aronofsky showed a man driven crazy by his search for hidden meanings in the digits of pi.

This opens the question: could pi have been something else? And if so, does pi depend on the laws of physics? Galileo said: “Philosophy is written in this grand book…. I mean the universe … which stands continually open to our gaze…. It is written in the language of mathematics.” The universe is written in the language of mathematics. Nobel laureate Eugene Wigner famously spoke of the “unreasonable effectiveness of mathematics” in explaining physics. Many philosophers take the Platonic view that mathematics would exist even the universe did not. And cosmologist Max Tegmark goes so far as to say that the universe actually is mathematics.

Yet maybe it is the other way around. The laws of physics are just the laws by which matter behaves. They determine the nature of our universe. Maybe humans have simply developed the mathematics appropriate for describing our universe, and so of course it fits with what we see. The mathematician Leopold Kronecker said, “God created the integers, all the rest is the work of man.” Are the laws of mathematics discovered by us in the same way as we discover the laws of physics? And are the laws of mathematics we discover just those which would have occurred to creatures living in a universe with physics like ours? In our universe, physics produces individual identical particles: all electrons are the same for example. We know about integers because there are things that look the same (like apples) for us to count. If you were some strange creature in a fractal universe containing only one object—yourself—and you thought only recursively, you might not ever think of counting anything and would never discover integers.

What about π = 3.14159265.…? Might it have a different value in a different universe? In our universe we have a fundamental physical dimensionless constant, the fine structure constant α which is related to the square of the value of the electric charge of the proton in natural geometrical Planck units (where the speed of light is 1 and the reduced Planck constant is 1 and Newton’s gravitational constant is 1). Now 1/α = 137.035999… Some physicists hope that one day we may have a mathematical formula for 1/α using mathematical constants such as π and e. If a theory for the fine structure constant could be developed giving a value in agreement with observations but allowing it to be calculated uniquely from pure mathematics, and if more and more digits of the constant were discovered experimentally fulfilling its prediction, it would certainly merit a Nobel Prize. But many physicists feel that no such magic formula will ever be discovered. Inflation may produce an infinite number of bubble universes, each with different laws of physics. Different universes bubbling out of an original inflating sea could have different values of 1/α. As Martin Rees has said, the laws of physics we know may be just local bylaws in an infinite multiverse of universes. String theory, if correct, may eventually give us a probability distribution for 1/α and we may find that our universe is just somewhere in the predicted middle 95% of the distribution, for example. Maybe there could be different universes with different values of π.

Let’s consider one possible example: taxicab geometry. This was invented by Hermann Minkowski. Now this brilliant mathematician also invented the geometrical interpretation of time as a fourth dimension based on Einstein’s theory of special relativity, so his taxicab geometry merits a serious look. Imagine a city with a checkerboard pattern of equal-sized square blocks. Suppose you wanted to take a taxicab to a location 3 blocks east, and 1 block north of your location, the shortest total distance you would have to travel to get there is 4 blocks. Your taxi has to travel along the streets, it does not get to travel as the crow flies. You could go 1 block east, then 1 block north then 2 blocks east, and still get to your destination, but the total distance you traveled would also be 4 blocks. The distance to your destination would be ds = |dx| + |dy|, where |dx| is the absolute value of the difference in x coordinates and |dy| is the absolute value of the difference in y coordinates. This is not the Euclidean formula. We are not in Kansas anymore! The set of points equidistant from the origin is a set of dots in a diamond shape. See diagram.

Gott

Image showing an intuitive explanation of why circles in taxicab geometry look like diamonds. Wikipedia.

Now if the blocks were smaller, there would be more dots, still in a diamond shape. In the limit where the size of the blocks had shrunk to zero, one would have a smooth diamond shape as shown in the bottom section of the diagram. The set of points equidistant from the origin has a name—a “circle!” If the circle has a radius of 1 unit, the distance along one side of its diamond shape is 2 units: going from the East vertex of the diamond to the North vertex of the diamond along the diagonal requires you to change the x coordinate by 1 unit and the y coordinate by 1 unit, making the distance along one side of the diagonal equal to 2 units (ds = |dx| + |dy| = 1 + 1 units = 2 units). The diamond shape has 4 sides so the circumference of the diamond is 8 units. The diameter of the circle is twice the radius, and therefore 2 units. In the taxicab universe π = C/d = C/2r = 8/2 = 4. If different laws of physics dictate different laws of geometry, you can change the value of π.

This taxicab geometry applies in the classic etch-a-sketch toy (Look it up on google, if you have never seen one). It has a white screen, and an internal stylus that draws a black line, directed by horizontal and vertical control knobs. If you want to draw a vertical line, you turn the vertical knob. If you want to draw a horizontal line you turn the horizontal knob. If you want to draw a diagonal line, you must simultaneously turn both knobs smoothly. If the distance between two points is defined by the minimal amount of total turning of the two knobs required to get from one point to the other, then that is the “taxicab” distance between the two points. In Euclidean geometry there is one shortest line between two points: a straight line between them. In taxicab geometry there can be many different, equally short, broken lines (taxicab routes) connecting two points. Taxicab geometry does not obey the axioms of Euclidean geometry and therefore does not have the same theorems as Euclidean geometry. And π is 4.

Mathematician and computer scientist John von Neumann invented a cellular automaton universe that obeys taxicab geometry. It starts with an infinite checkerboard of pixels. Pixels can be either black or white. The state of a pixel at time step t = n + 1 depends only on the state of its 4 neighbors (with which it shares a side: north, south, east, west of it) on the previous time step t = n. Causal, physical effects move like a taxicab. If the pixels are microscopic, we get a taxicab geometry. Here is a simple law of physics for this universe: a pixel stays in the same state, unless it is surrounded by an odd number of black pixels, in which case it switches to the opposite state on the next time step. Start with a white universe with only 1 black pixel at the origin. In the next time step it remains black while its 4 neighbors also become black. There is now a black cross of 5 pixels at the center. It has given birth to 4 black pixels like itself. Come back later and there will be 25 black pixels in a cross-shaped pattern of 5 cross-shaped patterns.

Come back still later and you can find 125 black pixels in 5 cross-shaped patterns (of 5 cross-shaped patterns). All these new black pixels lie inside a diamond-shaped region whose radius grows larger by one pixel per time step. In our universe, drop a rock in a pond, and a circular ripple spreads out. In the von Neumann universe, causal effects spread out in a diamond-shaped pattern.

If by “life” you mean a pattern able to reproduce itself, then this universe is luxuriant with life. Draw any pattern (say a drawing of a bicycle) in black pixels and at a later time you will find 5 bicycles, and then 25 bicycles, and 125 bicycles, etc. The laws of physics in this universe cause any object to copy itself. If you object that this is just a video game, I must tell you that some physicists seriously entertain the idea that we are living in an elaborate video game right now with quantum fuzziness at small scales providing the proof of microscopic “pixelization” at small scales.

Mathematicians in the von Neumann universe would know π = 4 (Or, if we had a taxicab universe with triangular pixels filling the plane, causal effects could spread out along three axes instead of two and a circle would look like a hexagon, giving π = 3.). In 1932, Stanislaw Golab showed that if we were clever enough in the way distances were measured in different directions, we could design laws of physics so that π might be anything we wanted from a low of 3 to a high of 4.

Back to the inhabitants of the von Neumann universe who think π = 4. Might they be familiar with number we know and love, 3.14159265…? They might:

3.14159265… = 4 {(1/1) – (1/3) + (1/5) – (1/7) + (1/9) + …} (Leibnitz)

If they were familiar with integers, they might be able to discover 3.14159265… But maybe the only integers they know are 1, 5, 25, 125, … and 4 of course. They would know that 5 = SQRT(25), so they would know what a square root was. In this case they could still find a formula for

3.14159265. . . =
SQRT(4) {SQRT(4)/SQRT(SQRT(4))}{SQRT(4)/SQRT(SQRT(4) + SQRT(SQRT(4)))}{SQRT(4)/ SQRT(SQRT(4) + SQRT(SQRT(4) + SQRT(SQRT(4))))} …

This infinite product involving only the integer 4 derives from one found by Vieta in 1594.

There are indeed many formulas equal to our old friend 3.14159265… including a spectacular one found by the renowned mathematician Ramanujan. Though every real number can be represented by such infinite series, products and continued fractions, these are particularly simple. So 3.14159265… does seem to have a special intimate relationship with integers, independent of geometry. If physics creates individual objects that can be counted, it seems difficult to avoid learning about 3.14159265… eventually—“If God made the integers,” as Kronecker suggested. So 3.14159265… appears not to be a random real number and we are still left with the mystery of the unreasonable effectiveness of mathematics in explaining the physics we see in our universe. We are also left with the mystery of why the universe is as comprehensible as it is. Why should we lowly carbon life forms be capable of finding out as much about how the universe works as we have done? Having the ability as intelligent observers to ask questions about the universe seems to come with the ability to actually answer some of them. That’s remarkable.

UniverseGottJ. Richard Gott is professor of astrophysics at Princeton University. His books include The Cosmic Web: Mysterious Architecture of the Universe. He is the coauthor of Welcome to the Universe: An Astrophysical Tour with Neil DeGrasse Tyson and Michael A. Strauss.

Cipher challenge #3 from Joshua Holden: Binary ciphers

The Mathematics of Secrets by Joshua Holden takes readers on a tour of the mathematics behind cryptography. Most books about cryptography are organized historically, or around how codes and ciphers have been used in government and military intelligence or bank transactions. Holden instead focuses on how mathematical principles underpin the ways that different codes and ciphers operate. Discussing the majority of ancient and modern ciphers currently known, The Mathematics of Secrets sheds light on both code making and code breaking. Over the next few weeks, we’ll be running a series of cipher challenges from Joshua Holden. The last post was on subliminal channels. Today’s is on binary ciphers:

Binary numerals, as most people know, represent numbers using only the digits 0 and 1.  They are very common in modern ciphers due to their use in computers, and they frequently represent letters of the alphabet.  A numeral like 10010 could represent the (1 · 24 + 0 · 23 + 0 · 22 + 1 · 2 + 0)th = 18th letter of the alphabet, or r.  So the entire alphabet would be:

 plaintext:   a     b     c     d     e     f     g     h     i     j
ciphertext: 00001 00010 00011 00100 00101 00110 00111 01000 01001 01010

 plaintext:   k     l     m     n     o     p     q     r     s     t
ciphertext: 01011 01100 01101 01110 01111 10000 10001 10010 10011 10100

 plaintext:   u     v     w     x     y     z
ciphertext: 10101 10110 10111 11000 11001 11010

The first use of a binary numeral system in cryptography, however, was well before the advent of digital computers. Sir Francis Bacon alluded to this cipher in 1605 in his work Of the Proficience and Advancement of Learning, Divine and Humane and published it in 1623 in the enlarged Latin version De Augmentis Scientarum. In this system not only the meaning but the very existence of the message is hidden in an innocuous “covertext.” We will give a modern English example.

Suppose we want to encrypt the word “not” into the covertext “I wrote Shakespeare.” First convert the plaintext into binary numerals:

   plaintext:   n      o     t
  ciphertext: 01110  01111 10100

Then stick the digits together into a string:

    011100111110100

Now we need what Bacon called a “biformed alphabet,” that is, one where each letter can have a “0-form” and a “1-form.”We will use roman letters for our 0-form and italic for our 1-form. Then for each letter of the covertext, if the corresponding digit in the ciphertext is 0, use the 0-form, and if the digit is 1 use the 1-form:

    0 11100 111110100xx
    I wrote Shakespeare.

Any leftover letters can be ignored, and we leave in spaces and punctuation to make the covertext look more realistic. Of course, it still looks odd with two different typefaces—Bacon’s examples were more subtle, although it’s a tricky business to get two alphabets that are similar enough to fool the casual observer but distinct enough to allow for accurate decryption.

Ciphers with binary numerals were reinvented many years later for use with the telegraph and then the printing telegraph, or teletypewriter. The first of these were technically not cryptographic since they were intended for convenience rather than secrecy. We could call them nonsecret ciphers, although for historical reasons they are usually called codes or sometimes encodings. The most well-known nonsecret encoding is probably the Morse code used for telegraphs and early radio, although Morse code does not use binary numerals. In 1833, Gauss, whom we met in Chapter 1, and the physicist Wilhelm Weber invented probably the first telegraph code, using essentially the same system of 5 binary digits as Bacon. Jean-Maurice-Émile Baudot used the same idea for his Baudot code when he invented his teletypewriter system in 1874. And the Baudot code is the one that Gilbert S. Vernam had in front of him in 1917 when his team at AT&T was asked to investigate the security of teletypewriter communications.

Vernam realized that he could take the string of binary digits produced by the Baudot code and encrypt it by combining each digit from the plaintext with a corresponding digit from the key according to the rules:

0 ⊕ 0 = 0
0 ⊕ 1 = 1
1 ⊕ 0 = 1
1 ⊕ 1 = 0

For example, the digits 10010, which ordinarily represent 18, and the digits 01110, which ordinarily represent 14, would be combined to get:

1 0 0 1 0
0 1 1 1 0


1 1 1 0 0

This gives 11100, which ordinarily represents 28—not the usual sum of 18 and 14.

Some of the systems that AT&T was using were equipped to automatically send messages using a paper tape, which could be punched with holes in 5 columns—a hole indicated a 1 in the Baudot code and no hole indicated a 0. Vernam configured the teletypewriter to combine each digit represented by the plaintext tape to the corresponding digit from a second tape punched with key characters. The resulting ciphertext is sent over the telegraph lines as usual.

At the other end, Bob feeds an identical copy of the tape through the same circuitry. Notice that doing the same operation twice gives you back the original value for each rule:

(0 ⊕ 0) ⊕ 0 = 0 ⊕ 0 = 0
(0 ⊕ 1) ⊕ 1 = 1 ⊕ 1 = 0
(1 ⊕ 0) ⊕ 0 = 1 ⊕ 0 = 1
(1 ⊕ 1) ⊕ 1 = 0 ⊕ 1 = 1

Thus the same operation at Bob’s end cancels out the key, and the teletypewriter can print the plaintext. Vernam’s invention and its further developments became extremely important in modern ciphers such as the ones in Sections 4.3 and 5.2 of The Mathematics of Secrets.

But let’s finish this post by going back to Bacon’s cipher.  I’ve changed it up a little — the covertext below is made up of two different kinds of words, not two different kinds of letters.  Can you figure out the two different kinds and decipher the hidden message?

It’s very important always to understand that students and examiners of cryptography are often confused in considering our Francis Bacon and another Bacon: esteemed Roger. It is easy to address even issues as evidently confusing as one of this nature. It becomes clear when you observe they lived different eras.

Answer to Cipher Challenge #2: Subliminal Channels

Given the hints, a good first assumption is that the ciphertext numbers have to be combined in such a way as to get rid of all of the fractions and give a whole number between 1 and 52.  If you look carefully, you’ll see that 1/5 is always paired with 3/5, 2/5 with 1/5, 3/5 with 4/5, and 4/5 with 2/5.  In each case, twice the first one plus the second one gives you a whole number:

2 × (1/5) + 3/5 = 5/5 = 1
2 × (2/5) + 1/5 = 5/5 = 1
2 × (3/5) + 4/5 = 10/5 = 2
2 × (4/5) + 2/5 = 10/5 = 2

Also, twice the second one minus the first one gives you a whole number:

2 × (3/5) – 1/5 = 5/5 = 1
2 × (1/5) – 2/5 = 0/5 = 0
2 × (4/5) – 3/5 = 5/5 = 1
2 × (2/5) – 4/5 = 0/5 = 0

Applying

to the ciphertext gives the first plaintext:

39 31 45 45 27 33 31 40 47 39 28 31 44 41
 m  e  s  s  a  g  e  n  u  m  b  e  r  o
40 31 35 45 46 34 31 39 31 30 35 47 39
 n  e  i  s  t  h  e  m  e  d  i  u  m

And applying

to the ciphertext gives the second plaintext:

20  8  5 19  5  3 15 14  4 16 12  1  9 14 
 t  h  e  s  e  c  o  n  d  p  l  a  i  n
20  5 24 20  9 19  1 20 12  1 18  7  5
 t  e  x  t  i  s  a  t  l  a  r  g  e

To deduce the encryption process, we have to solve our two equations for C1 and C2.  Subtracting the second equation from twice the first gives:


so

Adding the first equation to twice the second gives:


so

Joshua Holden is professor of mathematics at the Rose-Hulman Institute of Technology.

Browse Our Mathematics 2017 Catalog

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Fibonacci helped to revive the West as the cradle of science, technology, and commerce, yet he vanished from the pages of history. Finding Fibonacci is Keith Devlin’s compelling firsthand account of his ten-year quest to tell Fibonacci’s story.

Devlin Fibonacci cover

This annual anthology brings together the year’s finest mathematics writing from around the world. Featuring promising new voices alongside some of the foremost names in the field, The Best Writing on Mathematics 2016 makes available to a wide audience many articles not easily found anywhere else—and you don’t need to be a mathematician to enjoy them.

Pitici Best writing on Maths

In The Calculus of Happiness, Oscar Fernandez shows us that math yields powerful insights into health, wealth, and love. Using only high-school-level math, he guides us through several of the surprising results, including an easy rule of thumb for choosing foods that lower our risk for developing diabetes, simple “all-weather” investment portfolios with great returns, and math-backed strategies for achieving financial independence and searching for our soul mate.

Fernandez Calculus of Happiness

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Joshua Holden: The secrets behind secret messages

“Cryptography is all about secrets, and throughout most of its history the whole field has been shrouded in secrecy.  The result has been that just knowing about cryptography seems dangerous and even mystical.”

In The Mathematics of Secrets: Cryptography from Caesar Ciphers to Digital EncryptionJoshua Holden provides the mathematical principles behind ancient and modern cryptic codes and ciphers. Using famous ciphers such as the Caesar Cipher, Holden reveals the key mathematical idea behind each, revealing how such ciphers are made, and how they are broken.  Holden recently took the time to answer questions about his book and cryptography.


There are lots of interesting things related to secret messages to talk abouthistory, sociology, politics, military studies, technology. Why should people be interested in the mathematics of cryptography? 
 
JH: Modern cryptography is a science, and like all modern science it relies on mathematics.  If you want to really understand what modern cryptography can and can’t do you need to know something about that mathematical foundation. Otherwise you’re just taking someone’s word for whether messages are secure, and because of all those sociological and political factors that might not be a wise thing to do. Besides that, I think the particular kinds of mathematics used in cryptography are really pretty. 
 
What kinds of mathematics are used in modern cryptography? Do you have to have a Ph.D. in mathematics to understand it? 
 
JH: I once taught a class on cryptography in which I said that the prerequisite was high school algebra.  Probably I should have said that the prerequisite was high school algebra and a willingness to think hard about it.  Most (but not all) of the mathematics is of the sort often called “discrete.”  That means it deals with things you can count, like whole numbers and squares in a grid, and not with things like irrational numbers and curves in a plane.  There’s also a fair amount of statistics, especially in the codebreaking aspects of cryptography.  All of the mathematics in this book is accessible to college undergraduates and most of it is understandable by moderately advanced high school students who are willing to put in some time with it. 
 
What is one myth about cryptography that you would like to address? 
 
JH: Cryptography is all about secrets, and throughout most of its history the whole field has been shrouded in secrecy.  The result has been that just knowing about cryptography seems dangerous and even mystical. In the Renaissance it was associated with black magic and a famous book on cryptography was banned by the Catholic Church. At the same time, the Church was using cryptography to keep its own messages secret while revealing as little about its techniques as possible. Through most of history, in fact, cryptography was used largely by militaries and governments who felt that their methods should be hidden from the world at large. That began to be challenged in the 19th century when Auguste Kerckhoffs declared that a good cryptographic system should be secure with only the bare minimum of information kept secret. 
 
Nowadays we can relate this idea to the open-source software movement. When more people are allowed to hunt for “bugs” (that is, security failures) the quality of the overall system is likely to go up. Even governments are beginning to get on board with some of the systems they use, although most still keep their highest-level systems tightly classified. Some professional cryptographers still claim that the public can’t possibly understand enough modern cryptography to be useful. Instead of keeping their writings secret they deliberately make it hard for anyone outside the field to understand them. It’s true that a deep understanding of the field takes years of study, but I don’t believe that people should be discouraged from trying to understand the basics. 
 
I invented a secret code once that none of my friends could break. Is it worth any money? 
 
JH: Like many sorts of inventing, coming up with a cryptographic system looks easy at first.  Unlike most inventions, however, it’s not always obvious if a secret code doesn’t “work.” It’s easy to get into the mindset that there’s only one way to break a system so all you have to do is test that way.  Professional codebreakers know that on the contrary, there are no rules for what’s allowed in breaking codes. Often the methods for codebreaking with are totally unsuspected by the codemakers. My favorite involves putting a chip card, such as a credit card with a microchip, into a microwave oven and turning it on. Looking at the output of the card when bombarded 
by radiation could reveal information about the encrypted information on the card! 
 
That being said, many cryptographic systems throughout history have indeed been invented by amateurs, and many systems invented by professionals turned out to be insecure, sometimes laughably so. The moral is, don’t rely on your own judgment, anymore than you should in medical or legal matters. Get a second opinion from a professional you trustyour local university is a good place to start.   
 
A lot of news reports lately are saying that new kinds of computers are about to break all of the cryptography used on the Internet. Other reports say that criminals and terrorists using unbreakable cryptography are about to take over the Internet. Are we in big trouble? 
 
JH: Probably not. As you might expect, both of these claims have an element of truth to them, and both of them are frequently blown way out of proportion. A lot of experts do expect that a new type of computer that uses quantum mechanics will “soon” become a reality, although there is some disagreement about what “soon” means. In August 2015 the U.S. National Security Agency announced that it was planning to introduce a new list of cryptography methods that would resist quantum computers but it has not announced a timetable for the introduction. Government agencies are concerned about protecting data that might have to remain secure for decades into the future, so the NSA is trying to prepare now for computers that could still be 10 or 20 years into the future. 
 
In the meantime, should we worry about bad guys with unbreakable cryptography? It’s true that pretty much anyone in the world can now get a hold of software that, when used properly, is secure against any publicly known attacks. The key here is “when used properly. In addition to the things I mentioned above, professional codebreakers know that hardly any system is always used properly. And when a system is used improperly even once, that can give an experienced codebreaker the information they need to read all the messages sent with that system.  Law enforcement and national security personnel can put that together with information gathered in other waysurveillance, confidential informants, analysis of metadata and transmission characteristics, etc.and still have a potent tool against wrongdoers. 
 
There are a lot of difficult political questions about whether we should try to restrict the availability of strong encryption. On the flip side, there are questions about how much information law enforcement and security agencies should be able to gather. My book doesn’t directly address those questions, but I hope that it gives readers the tools to understand the capabilities of codemakers and codebreakers. Without that you really do the best job of answering those political questions.

Joshua Holden is professor of mathematics at the Rose-Hulman Institute of Technology in Terre Haute, IN. His most recent book is The Mathematics of Secrets: Cryptography from Caesar Ciphers to Digital Encryption.

This Halloween, a few books that won’t (shouldn’t!) die

If Halloween has you looking for a way to combine your love (or terror) of zombies and academic books, you’re in luck: Princeton University Press has quite a distinguished publishing history when it comes to the undead.

 

As you noticed if you follow us on Instagram, a few of our favorites have come back to haunt us this October morning. What is this motley crew of titles doing in a pile of withered leaves? Well, The Origins of Monsters offers a peek at the reasons behind the spread of monstrous imagery in ancient empires; Zombies and Calculus  features a veritable course on how to use higher math skills to survive the zombie apocalypse, and International Politics and Zombies invites you to ponder how well-known theories from international relations might be applied to a war with zombies. Is neuroscience your thing? Do Zombies Dream of Undead Sheep? shows how zombism can be understood in terms of current knowledge regarding how the brain works. Or of course, you can take a trip to the graveyard of economic ideology with Zombie Economics, which was probably off marauding when this photo was snapped.

If you’re feeling more ascetic, Black: The History of a Color tells the social history of the color black—archetypal color of darkness and death—but also, Michel Pastoureau tells us, monastic virtue. A strikingly designed choice:

In the beginning was black, Michel Pastoureau tells us in Black: A History of a Color

A post shared by Princeton University Press (@princetonupress) on

 

Happy Halloween, bookworms.

Peter Dougherty & Al Bertrand: On Being Einstein’s Publisher

by Peter Dougherty and Al Bertrand

So many people today—and even professional scientists—seem to me like somebody who has seen thousands of trees but has never seen a forest. (Albert Einstein to Robert A Thornton, 7 December 1944, EA 61-574)

For all of the scholarly influences that have defined Princeton University Press over its 111-year history, no single personality has shaped the Press’s identity as powerfully, both directly and indirectly, as Albert Einstein. The 2015 centenary of the publication of Einstein’s “Theory of General Relativity” as well as the affirmation this past February and again in June of the discovery of gravitational waves has encouraged us to reflect on this legacy and how it has informed our identity as a publisher.

The bright light cast by Einstein the scientist and by Einstein the humanist has shaped Princeton University Press in profound and far-reaching ways. It expresses itself in the Press’s standard of scholarly excellence, its emphasis on the breadth and connectedness of liberal learning across all fields, and in our mission of framing scholarly arguments to shape contemporary knowledge. All the while, Einstein’s role as a citizen of the world inspires our vision to be a truly global university press.

PUBLISHING EINSTEIN: A BRIEF HISTORY

Albert Einstein is not only Princeton University Press’s most illustrious author; he was our first best-selling author. Following his public lectures in Princeton in 1921, the Press—itself less than 20 years old at the time—published the text of those lectures, titled “The Meaning of Relativity”, in 1922. Publication followed the agitated exhortation of the Press’s then-manager, Frank Tomlinson, urging Professor Einstein to get his manuscript finished. Tomlinson wrote:

My dear Professor Einstein—

On July 6 I wrote you inquiring when we might expect to receive the manuscript of your lectures. I have had no reply to this letter. A number of people have been inquiring when the book will be ready, and we are considerably alarmed at the long delay in the receipt of your manuscript, which we were led to believe would be in our hands within a month after the lectures were delivered. The importance of the book will undoubtedly be seriously affected unless we are able to publish it within a reasonable time and I strongly urge upon you the necessity of sending us the copy at your earliest convenience. I should appreciate also the favor of a reply from you stating when we may expect to receive it.

the meaning of relativity jacketMr. Tomlinson’s letter marks something of a high point in the history of publishers’ anxiety, but far from failing, The Meaning of Relativity was a hit. It would go on to numerous successive editions, and remains very much alive today as both a print and digital book, as well as in numerous translated editions.

For all its glorious publishing history, The Meaning of Relativity can be thought of as a mere appetizer to the bounteous publishing banquet embodied in THE COLLECTED PAPERS OF ALBERT EINSTEIN, surely PUP’s most ambitious continuing publication and one of the most important editorial projects in all of scholarly publishing.

The Collected Papers of Albert Einstein

Authorized by the Einstein Estate and the PUP Board of Trustees in 1970, and supported by a generous grant from the late Harold W. McGraw, Jr., chairman of the McGraw-Hill Book Company, THE EINSTEIN PAPERS, as it evolves, is providing the first complete and authoritative account of a written legacy that ranges from Einstein’s work on the special and general theories of relativity and the origins of quantum theory, to expressions of his profound concern with civil liberties, education, Zionism, pacifism, and disarmament.

einstein old letterAn old saying has it that “good things come to those to wait,” words that ring resoundingly true regarding the EINSTEIN PAPERS. Having survived multiple obstacles in the long journey from its inception through the publication of its first volume in 1987, the Einstein Papers Project hit its stride in 2000 when Princeton University Press engaged Professor Diana Buchwald as its sixth editor, and moved the Project to Pasadena with the generous support of its new host institution, the California Institute of Technology.

Since then, Professor Buchwald and her Caltech-based editorial team, along with their international network of scholarly editors, have produced successive documentary and English translation volumes at the rate of one every eighteen months. To give you an idea of just how impressive a pace this is, the Galileo papers are still a work in progress, nearly four centuries after his death.

The EINSTEIN PAPERS, having reached and documented Einstein’s writings up to 1925, has fundamentally altered our understanding of the history of physics and of the development of general relativity, for example by destroying the myth of Einstein as a lone genius and revealing the extent to which this man, with his great gift for friendship and collegiality, was embedded in a network of extraordinary scientists in Zurich, Prague, and Berlin.

Along with the EINSTEIN PAPERS, the Press has grown a lively publishing program of books drawn from his work and about Einstein. Satellite projects include The Ultimate Quotable Einstein, as well as volumes on Einstein’s politics, his love letters, and the “miraculous year” of 1905.

Last year the Press published two new books drawn from Einstein’s writings, The Road to Relativity, and the 100th anniversary edition of Relativity: The Special and General Theory, both volumes edited by Jürgen Renn of the Max Planck Institute in Berlin, and Hanoch Gutfreund of the Hebrew University in Jerusalem.   These volumes celebrate the centenary of Einstein’s publication of the theory of general relativity in November 1915.

In this same centenary year, PUP published several other Einstein titles, including:

— Volume 14 of the Collected Papers, The Berlin Years, 1923-1925.

An Einstein Encyclopedia, edited by Alice Calaprice, Daniel Kennefick, and Robert Schulman;

Einstein: A Hundred Years of Relativity, by Andrew Robinson

Especially notable, in January 2015 the Press released THE DIGITAL EDITION OF THE COLLECTED PAPERS OF ALBERT EINSTEIN, a publishing event that has attracted extraordinary worldwide attention, scientific as well as public. This online edition is freely available to readers and researchers around the world, and represents the historic collaboration between the Press and its partners, the Einstein Papers Project at Caltech and the Albert Einstein Archive in the Hebrew University in Jerusalem.

Moreover, works by and about Einstein sit at the crossroads of two major components of the Princeton list: our science publishing program which comprises a host of fields from physics through mathematics, biology, earth science, computer science, and natural history, and our history of science program which connects PUP’s Einstein output to our humanities publishing, helping to bridge the intellectual gap between two major dimensions of our list.

Einstein’s dual legacy at Princeton University Press thus serves to bookend the conversation defined by the Press’s unusually wide-ranging array of works across and throughout the arts and sciences, from mathematics to poetry. C.P. Snow famously described the sciences and the humanities as “two cultures.” Einstein’s legacy informs our effort as a publisher to create an ongoing correspondence between those two cultures in the form of books, which uniquely serve to synthesize, connect, and nurture cross-disciplinary discourse.

EINSTEIN’S LARGER PUBLISHING INFLUENCE

Much as the living legacy of the EINSTEIN PAPERS and its related publications means to Princeton University Press as a publisher, it holds a broader meaning for us both as editors and as leaders of the institution with which we’ve long been affiliated.

Like most of our colleagues, we arrived at the Press as editors previously employed by other publishers, and having little professional interest in physics. Each of us specialized in different editorial fields, economics and classics, respectively.

Our initial disposition towards the field of physics, while full of awe, was perhaps best summed up by Woody Allen when he said: “I’m astounded by people who want to ‘know’ the universe when it’s hard enough to find your way around Chinatown.”  

But we soon discovered, as newcomers to PUP inevitably do, that the Princeton publishing legacy of Albert Einstein carried with it a set of implications beyond his specific scientific bounty that would help to shape our publishing activity, as well as that of our colleagues. We see the Einstein legacy operating in three distinct ways on PUP’s culture:

First, it reinforces the centrality of excellence as a standard: simply put, we strive to publish the core scholarly books by leading authors, senior as well as first-time. Einstein’s legacy stands as a giant-sized symbol of excellence, an invisible but constant reminder that our challenge as publishers at Princeton is not merely to be good, but to be great. As we seek greatness by publishing those books that help to define and unite the frontiers of modern scholarship, and connect our authors’ ideas with minds everywhere, we are upholding a standard embodied in the work of Albert Einstein.

The second implication of the bounty Albert Einstein is a commitment to seeing liberal knowledge defined broadly, encompassing its scientific articulation as well as its expression in the humanities and social sciences. PUP purposefully publishes an unusually wide portfolio of subject areas, encompassing not only standard university press fields such as literary criticism, art history, politics, sociology, and philosophy, but a full complement of technical fields, including biology, physics, neuroscience, mathematics, economics, and computer science. A rival publisher once half-jokingly described PUP as “the empirical knowledge capital of the world.” She was referring to our capacious cultivation of scientific and humanistic publishing, an ambitious menu for a publisher producing only around 250 books a year, but one we think gives the Press its distinctive identity.

It is no coincidence that Albert Einstein, PUP’s most celebrated author, cast his influence across many of these fields both as a scientist and as a humanist, engaged fully in the life of the mind and of the world. His legacy thus inspires us to concentrate our editorial energies on building a list that focuses on knowledge in its broadest and deepest sense—that puts into play the sometimes contentious, and even seemingly incongruous, methodologies of science and the humanities and articulates a broad yet rigorous, intellectual vision, elevating knowledge for its own sake, even as the issues change from decade to decade.

A third implication appears in Einstein’s challenge to us to be a great global publisher. Einstein, a self-professed “citizen of the world” was in many ways the first global citizen, a scholar whose scientific achievement and fame played out on a truly global scale in an age of parochial and often violent nationalist thinking.

Einstein’s cosmopolitanism has inspired the Press to pursue a path of becoming a truly global university Press. To do this, PUP has built lists in fields that are cosmopolitan in their readership, opened offices in Europe and China, expanded its author and reviewer base all over the world, and has licensed its content for translation in many languages. As we go forward, we intend to continue to build a network that allows us to connect many local publishing and academic cultures with the global scholarly conversation. This vision of the Press’s future echoes Einstein’s call for a science that transcends national boundaries.

THE FUTURE

It has been nearly a century since publication of The Meaning of Relativity and half that since the original agreement for the EINSTEIN PAPERS was authorized. We can only imagine that the originators of the latter project would be proud of what our collective effort has produced, grateful to the principals for the job they have done in bringing the PAPERS to their current status, and maybe above all, awed by the global exposure the PAPERS have achieved in their print and now digital formats.

As we continue our work with our colleagues at Caltech and the Hebrew University to extend the EINSTEIN PAPERS into the future, we are reminded of the significance of the great scientist’s legacy, especially as it bears on our identity as a global publisher, framing the pursuit of knowledge imaginatively across the arts and sciences.

The eminent Italian publisher Roberto Calasso, in his recent book, The Art of the Publisher, encourages readers to imagine a publishing house as,

“a single text formed not just by the totality of books that have been published there, but also by its other constituent elements, such as the front covers, cover flaps, publicity, the quantity of copies printed and sold, or the different editions in which the same text has been presented. Imagine a publishing house in this way and you will find yourself immersed in a very strange landscape, something that you might regard as a literary work in itself, belonging to a genre all its own.”

Now, at a time when the very definition of publishing is being undermined by technological and economic forces, it is striking to see each publisher as a “literary work unto itself.” So it is with Princeton University Press. In so far as PUP can claim a list having a diversified but well-integrated publishing vision, one that constantly strives for excellence and that stresses the forest for the trees, it is inescapably about the spirit and substance reflected in the legacy of Albert Einstein, and it is inseparable from it.

Einstein_blog (small)

 


 

Peter J. Dougherty is Director of Princeton University Press. This essay is based in part on comments he delivered at the Space-Time Theories conference at the Hebrew University in Jerusalem in January, 2015. Al Bertrand is Associate Publishing Director of Princeton University Press and Executive Editor of the Press’s history of science publishing program, including Einstein-related publications.

Happy Birthday, Alan Turing

Hodges_AlanTuring movie tie inThursday, June 23rd marks the birthday of Alan Turing, widely credited with being the father of the modern computer and artificial intelligence, as well as with leading the Bletchley Park codebreakers in cracking an encryption method used by the Nazi’s. PUP is proud to have published Alan Turing: The Enigma, a scientific biography of the famous cryptologist that went on to become a New York Times Bestseller, and was adapted in the 2014 historical drama/thriller The Imitation Game. The film was a commercial and critical success, grossing over $233 million worldwide. Turing is, for many, a modern day mathematical hero in the spirit of Albert Einstein or John Nash.

Yet despite his genius and groundbreaking accomplishments, Turing was hounded to his early death about his sexuality. After facing a 1952 charge of indecency over his relationship with another man, a criminal act in the UK at the time, he endured chemical castration and took his own life only two years later at age 41.

Archaic attitudes and inhumane treatment of LGBT people continued by the agency (and more broadly in society) for decades after Turing’s death. But in a historic move this past April, the GCHQ (UK Government Communications Headquarters) issued a formal apology, acknowledging that the treatment of Turing was “horrifying”.

You can read more about the apology here.

A man who changed the modern world while anticipating gay liberation by decades, Turing’s tragically brief four decades of life were unarguably well spent. Happy birthday, Alan Turing.

An interview with John Stillwell on Elements of Mathematics

elements of mathematics jacketNot all topics that are part of today’s elementary mathematics were always considered as such, and great mathematical advances and discoveries had to occur in order for certain subjects to become “elementary.” Elements of Mathematics: From Euclid to Gödel, by John Stillwell gives readers, from high school students to professional mathematicians, the highlights of elementary mathematics and glimpses of the parts of math beyond its boundaries.

You’ve been writing math books for a long time now. What do you think is special about this one?

JS: In some ways it is a synthesis of ideas that occur fleetingly in some of my previous books: the interplay between numbers, geometry, algebra, infinity, and logic. In all my books I try to show the interaction between different fields of mathematics, but this is one more unified than any of the others. It covers some fields I have not covered before, such as probability, but also makes many connections I have not made before. I would say that it is also more reflective and philosophical—it really sums up all my experience in mathematics.

Who do you expect will enjoy reading this book?

JS: Well I hope my previous readers will still be interested! But for anyone who has not read my previous work, this might be the best place to start. It should suit anyone who is broadly interested in math, from high school to professional level. For the high school students, the book is a guide to the math they will meet in the future—they may understand only parts of it, but I think it will plant seeds for their future mathematical development. For the professors—I believe there will be many parts that are new and enlightening, judging from the number of times I have often heard “I never knew that!” when speaking on parts of the book to academic audiences.

Does the “Elements” in the title indicate that this book is elementary?

JS: I have tried to make it as simple as possible but, as Einstein is supposed to have said, “not simpler”. So, even though it is mainly about elementary mathematics it is not entirely elementary. It can’t be, because I also want to describe the limits of elementary mathematics—where and why mathematics becomes difficult. To get a realistic appreciation of math, it helps to know that some difficulties are unavoidable. Of course, for mathematicians, the difficulty of math is a big attraction.

What is novel about your approach?

JS: It tries to say something precise and rigorous about the boundaries of elementary math. There is now a field called “reverse mathematics” which aims to find exactly the right axioms to prove important theorems. For example, it has been known for a long time—possibly since Euclid—that the parallel axiom is the “right” axiom to prove the Pythagorean theorem. Much more recently, reverse mathematics has found that certain assumptions about infinity are the right axioms to prove basic theorems of analysis. This research, which has only appeared in specialist publications until now, helps explain why infinity appears so often at the boundaries of elementary math.

Does your book have real world applications?

JS: Someone always asks that question. I would say that if even one person understands mathematics better because of my book, then that is a net benefit to the world. The modern world runs on mathematics, so understanding math is necessary for anyone who wants to understand the world.

John Stillwell is professor of mathematics at the University of San Francisco. His many books include Mathematics and Its History and Roads to Infinity. His most recent book is Elements of Mathematics: From Euclid to Gödel.

Nicholas J. Higham: The Top 10 Algorithms in Applied Mathematics

pcam-p346-newton.jpg

From “Computational Science” by David E. Keyes in Princeton Companion to Applied Mathematics

In the January/February 2000 issue of Computing in Science and Engineering, Jack Dongarra and Francis Sullivan chose the “10
algorithms with the greatest influence on the development and practice of science and engineering in the 20th century” and presented a group of articles on them that they had commissioned and edited. (A SIAM News article by Barry Cipra gives a summary for anyone who does not have access to the original articles). This top ten list has attracted a lot of interest.

Sixteen years later, I though it would be interesting to produce such a list in a different way and see how it compares with the original top ten. My unscientific—but well defined— way of doing so is to determine which algorithms have the most page locators in the index of The Princeton Companion to Applied Mathematics (PCAM). This is a flawed measure for several reasons. First, the book focuses on applied mathematics, so some algorithms included in the original list may be outside its scope, though the book takes a broad view of the subject and includes many articles about applications and about topics on the interface with other areas. Second, the content is selective and the book does not attempt to cover all of applied mathematics. Third, the number of page locators is not necessarily a good measure of importance. However, the index was prepared by a professional indexer, so it should reflect the content of the book fairly objectively.

A problem facing anyone who compiles such a list is to define what is meant by “algorithm”. Where does one draw the line between an algorithm and a technique? For a simple example, is putting a rational function in partial fraction form an algorithm? In compiling the following list I have erred on the side of inclusion. This top ten list is in decreasing order of the number of page locators.

  1. Newton and quasi-Newton methods
  2. Matrix factorizations (LU, Cholesky, QR)
  3. Singular value decomposition, QR and QZ algorithms
  4. Monte-Carlo methods
  5. Fast Fourier transform
  6. Krylov subspace methods (conjugate gradients, Lanczos, GMRES,
    minres)
  7. JPEG
  8. PageRank
  9. Simplex algorithm
  10. Kalman filter

Note that JPEG (1992) and PageRank (1998) were youngsters in 2000, but all the other algorithms date back at least to the 1960s.

By comparison, the 2000 list is, in chronological order (no other ordering was given)

  • Metropolis algorithm for Monte Carlo
  • Simplex method for linear programming
  • Krylov subspace iteration methods
  • The decompositional approach to matrix computations
  • The Fortran optimizing compiler
  • QR algorithm for computing eigenvalues
  • Quicksort algorithm for sorting
  • Fast Fourier transform
  • Integer relation detection
  • Fast multipole method

The two lists agree in 7 of their entries. The differences are:

PCAM list 2000 list
Newton and quasi-Newton methods The Fortran Optimizing Compiler
Jpeg Quicksort algorithm for sorting
PageRank Integer relation detection
Kalman filter Fast multipole method

Of those in the right-hand column, Fortran is in the index of PCAM and would have made the list, but so would C, MATLAB, etc., and I draw the line at including languages and compilers; the fast multipole method nearly made the PCAM table; and quicksort and integer relation detection both have one page locator in the PCAM index.

There is a remarkable agreement between the two lists! Dongarra and Sullivan say they knew that “whatever we came up with in the end, it would be controversial”. Their top ten has certainly stimulated some debate, but I don’t think it has been too controversial. This comparison suggests that Dongarra and Sullivan did a pretty good job, and one that has stood the test of time well.

Finally, I point readers to a talk Who invented the great numerical algorithms? by Nick Trefethen for a historical perspective on algorithms, including most of those mentioned above.

This post originally appeared on Higham’s popular website.

Higham jacketNicholas J. Higham is the Richardson Professor of Applied Mathematics at The University of Manchester. He most recently edited The Princeton Companion to Applied Mathematics.

Happy Birthday, Albert Einstein!

What a year. Einstein may have famously called his own birthday a natural disaster, but between the discovery of gravitational waves in February and the 100th anniversary of the general theory of relativity this past November, it’s been a big year for the renowned physicist and former Princeton resident. Throughout the day, PUP’s design blog will be celebrating with featured posts on our Einstein books and the stories behind them.

HappyBirthdayEinstein Graphic 3

Here are some of our favorite Einstein blog posts from the past year:

Was Einstein the First to Discover General Relativity? by Daniel Kennefick

Under the Spell of Relativity by Katherine Freese

Einstein: A Missionary of Science by Jürgen Renn

Me, Myself and Einstein by Jimena Canales

The Revelation of Relativity by Hanoch Gutfreund

A Mere Philosopher by Eoghan Barry

The Final Days of Albert Einstein by Debra Liese

 

Praeteritio and the quiet importance of Pi

pidayby James D. Stein

Somewhere along my somewhat convoluted educational journey I encountered Latin rhetorical devices. At least one has become part of common usage–oxymoron, the apparent paradox created by juxtaposed words which seem to contradict each other; a classic example being ‘awfully good’. For some reason, one of the devices that has stuck with me over the years is praeteritio, in which emphasis is placed on a topic by saying that one is omitting it. For instance, you could say that when one forgets about 9/11, the Iraq War, Hurricane Katrina, and the Meltdown, George W. Bush’s presidency was smooth sailing.

I’ve always wanted to invent a word, like John Allen Paulos did with ‘innumeracy’, and πraeteritio is my leading candidate–it’s the fact that we call attention to the overwhelming importance of the number π by deliberately excluding it from the conversation. We do that in one of the most important formulas encountered by intermediate algebra and trigonometry students; s = rθ, the formula for the arc length s subtended by a central angle θ in a circle of radius r.

You don’t see π in this formula because π is so important, so natural, that mathematicians use radians as a measure of angle, and π is naturally incorporated into radian measure. Most angle measurement that we see in the real world is described in terms of degrees. A full circle is 360 degrees, a straight angle 180 degrees, a right angle 90 degrees, and so on. But the circumference of a circle of radius 1 is 2π, and so it occurred to Roger Cotes (who is he? I’d never heard of him) that using an angular measure in which there were 2π angle units in a full circle would eliminate the need for a ‘fudge factor’ in the formula for the arc length of a circle subtended by a central angle. For instance, if one measured the angle D in degrees, the formula for the arc length of a circle of radius r subtended by a central angle would be s = (π/180)rD, and who wants to memorize that? The word ‘radian’ first appeared in an examination at Queen’s College in Belfast, Ireland, given by James Thomson, whose better-known brother William would later be known as Lord Kelvin.

The wisdom of this choice can be seen in its far-reaching consequences in the calculus of the trigonometric functions, and undoubtedly elsewhere. First semester calculus students learn that as long as one uses radian measure for angles, the derivative of sin x is cos x, and the derivative of cos x is – sin x. A standard problem in first-semester calculus, here left to the reader, is to compute what the derivative of sin x would be if the angle were measured in degrees rather than radians. Of course, the fudge factor π/180 would raise its ugly head, its square would appear in the formula for the second derivative of sin x, and instead of the elegant repeating pattern of the derivatives of sin x and cos x that are a highlight of the calculus of trigonometric functions, the ensuing formulas would be beyond ugly.

One of the simplest known formulas for the computation of π is via the infinite series ????4=1−13+15−17+⋯

This deliciously elegant formula arises from integrating the geometric series with ratio -x^2 in the equation 1/(1+????^2)=1−????2+????4−????6+⋯

The integral of the left side is the inverse tangent function tan-1 x, but only because we have been fortunate enough to emphasize the importance of π by utilizing an angle measurement system which is the essence of πraeteritio; the recognition of the importance of π by excluding it from the discussion.

So on π Day, let us take a moment to recognize not only the beauty of π when it makes all the memorable appearances which we know and love, but to acknowledge its supreme importance and value in those critical situations where, like a great character in a play, it exerts a profound dramatic influence even when offstage.

LA MathJames D. Stein is emeritus professor in the Department of Mathematics at California State University, Long Beach. His books include Cosmic Numbers (Basic) and How Math Explains the World (Smithsonian). His most recent book is L.A. Math: Romance, Crime, and Mathematics in the City of Angels.