## What is Calculus?: The Two Pillars

By Oscar Fernandez

This is the third in a series of short articles exploring calculus. The first article explored the origins of calculus, including the three “big problems” that drove calculus’ development. The second article explored limits, the foundation of calculus. This article discusses how limits help us solve the three “big problems” and introduces two of calculus’ pillars: derivatives and integrals.

In the first article in this series I discussed three Big Problems that drove the development of calculus: the instantaneous speed problem, the tangent line problem, and the area problem. I illustrated these via the figure below.

Reprinted, with permission, from Calculus Simplified (Princeton University Press). Click to expand.

These problems stumped mathematicians for millennia. (We briefly talked about why in the first article.) But their inability to solve these problems—echoing Morpheus in the movie The Matrix—was not due to the techniques they were using; it was due to their mindset.

How a Dynamics Mindset Solves the Three Big Problems

If you’ve read the second article in this series, you’ll remember my first characterization of calculus: calculus is a dynamics mindset. Yet nothing about the figure above says “dynamics.” Every image is a static snapshot of something (e.g., an area). So let’s calculus the figure. (Yep, I’m encouraging you to think of calculus as a verb.)

The figure below takes each Big Problem from the figure above and adds in the dynamics.

Reprinted, with permission, from Calculus Simplified (Princeton University Press). Click to expand.

These images show apples falling, gray lines approaching a blue tangent line, and areas being swept out. Lots of movement (dynamics)! Moreover, notice that as the central change in each row of the figure gets closer to zero —the quantity ∆t in the first row and ∆x in the second and third rows—the resulting diagram approaches the respective diagram in the first figure in this article. We’ve met this “as ∆t  approaches zero” language before—it’s the language of limits we discussed in the second article! Adding this new revelation to the figure above produces…

Reprinted, with permission, from Calculus Simplified (Princeton University Press. Click to expand.

Finally, expressing our result in terms of equations involving limits yields the final piece of the puzzle…

Notice how each row employs a dynamics mindset to recast the Big Problem (contained in the “limiting picture” column) as the limit of a sequence of similar quantities (e.g., speeds) involving finite changes, changes which pre-calculus mathematics can handle. Specifically:

• Row #1: The instantaneous speed of the falling apple is realized as the limit of its average speeds  ∆d / ∆t (ratios of changes in distance to changes in time) as ∆t —> 0.
• Row #2: The slope of the tangent line is realized as the limit of the secant line slopes ∆y / ∆x (the gray lines in the figure) as ∆x —> 0.
• Row #3: The area under the curve is realized as the limit as ∆x —> 0 of the area swept out from x = a up to ∆x  past b.

Introducing…Derivatives and Integrals

The limit obtained in the second row of the last figure is called the derivative of f(x) at x = a, the x-value of point P. The limit obtained in the third row of the Figure is called the definite integral of f(x) between x = a and x = b. Derivatives and integrals round out the three most important concepts in calculus (limits are the third).

You now have a working understanding of what derivatives and definite integrals are, what they measure, and how they arise from the application of a dynamics mindset to pre-calculus mathematics. The next post in this series will explore the derivative in greater details. We’ll discover that it has a nice geometric interpretation and a powerful real-world interpretation. (The last figure above hints to what these are.) Near the end of this series we will return to these interpretations to illustrate the power of derivatives, using them to help us understand phenomena as diverse as the fate of the Universe and, more pragmatically, how to find the best seat in a movie theater. Stay tuned!

Calculus Simplified
By Oscar E. Fernandez

Calculus is a beautiful subject that most of us learn from professors, textbooks, or supplementary texts. Each of these resources has strengths but also weaknesses. In Calculus Simplified, Oscar Fernandez combines the strengths and omits the weaknesses, resulting in a “Goldilocks approach” to learning calculus: just the right level of detail, the right depth of insights, and the flexibility to customize your calculus adventure.

Fernandez begins by offering an intuitive introduction to the three key ideas in calculus—limits, derivatives, and integrals. The mathematical details of each of these pillars of calculus are then covered in subsequent chapters, which are organized into mini-lessons on topics found in a college-level calculus course. Each mini-lesson focuses first on developing the intuition behind calculus and then on conceptual and computational mastery. Nearly 200 solved examples and more than 300 exercises allow for ample opportunities to practice calculus. And additional resources—including video tutorials and interactive graphs—are available on the book’s website.

Calculus Simplified also gives you the option of personalizing your calculus journey. For example, you can learn all of calculus with zero knowledge of exponential, logarithmic, and trigonometric functions—these are discussed at the end of each mini-lesson. You can also opt for a more in-depth understanding of topics—chapter appendices provide additional insights and detail. Finally, an additional appendix explores more in-depth real-world applications of calculus.

Learning calculus should be an exciting voyage, not a daunting task. Calculus Simplified gives you the freedom to choose your calculus experience, and the right support to help you conquer the subject with confidence.

• An accessible, intuitive introduction to first-semester calculus
• Nearly 200 solved problems and more than 300 exercises (all with answers)
• No prior knowledge of exponential, logarithmic, or trigonometric functions required
• Additional online resources—video tutorials and supplementary exercises—provided

## A Celebration of Mathematics Editor Vickie Kearn

This month, across the world, we have celebrated the enduring contributions of all women. For those of us at PUP, it is a chance as well to focus on a particularly generous, intelligent, and dynamic publisher, Vickie Kearn. In April, Vickie will retire from the Press after 18 years of synergistic and inspiring collaborations in math and computer science publishing, leaving us with a library of books that have educated and entertained millions, billions, and zillions of readers (borrowing from the title of one of her recent acquisitions).

Vickie has also been a powerful role model for women in STEM publishing, and one who empowered a population of publishers, myself included, and our new math editor Susannah Shoemaker as another. Vickie’s strength as a competitive publisher set the bar dauntingly high, but in that competition was also always an admirable collaboration, knowing that a cohort of us were changing the face of scholarly STEM publishing. It has been such a great privilege to be a colleague of Vickie’s since 2017, to travel to a math meeting with her, to meet incredibly creative authors with whom she has worked, and to learn from her at weekly project meetings. The PUP math list, particularly the popular math list, has grown exponentially and in multiple dimensions under Vickie’s leadership. If there are theorems or rules in math publishing, I would attribute these to Vickie’s rule: be smart, be curious, be generous, and be strong.

–Christie Henry

CH: Some say math is its own language. How did you learn to speak it?

I grew up in Venezuela and the English school only went through the 9th grade, so when I was 15, I went away to boarding school in North Carolina. There were only 125 girls in the whole school and there were two math teachers. One taught the girls who liked math and another taught those who did not like math. My class was very small since fewer of us liked math. Elsie Nunn was my teacher for three years and she made me fall in love with math. Before she taught anything new, she taught us about the person responsible for what we were about to learn. There was always a face behind the numbers, a person who had a family and hobbies. I found I could connect with these people. We had math club every day after school and she always had wonderful stories to tell. When I went to the University of Richmond, I knew I was going to major in math. This led to an unexpected benefit and a bit of a surprise. In the late 1960’s, University of Richmond was a Baptist school, and the classes for the men and women were held on separate sides of a lake. The one exception was that the upper level math classes were on the men’s side. Men and women were only allowed to talk with one another on Wednesday, Saturday, and Sunday, but I was able to talk with them every day because we had math class together. The surprise for me was that I was the only female math major. This felt strange at times, but Ms. Nunn had prepared me well and I got along fine with my classmates. The classes were small and we stuck together because unlike many people at UR we were more interested in math and less interested in parties.

CH: How can we continue to empower girls and women in STEM- as authors and publishers?

Based on conversations I have had with other women my age, I have had a very easy time in my career. This could be because I only have an undergraduate degree and did not experience the problems that arise in graduate school and a career as a mathematician. However, I would advise young women to join an organization that focuses on confidence building, like the Girl Scouts. I would also recommend finding a mentor—someone to look up to who can advise about a field that has long been male dominated. After I got my undergraduate degree, I taught school for 8 years, five of them in elementary school and 3 teaching math in junior high school. Most of the elementary teachers were female and the math teachers were both men and women. Although all of my college classmates in my math courses were male, it wasn’t until I went into publishing and attended my first mathematics meeting that I realized how gender specific math was.  I believe that as more women with math PhD degrees publish books and give plenary talks at conferences, the more visible they will be, and in turn, young women majoring in math will feel more a part of the mathematics community. It is critical for publishers to encourage female mathematicians to write scholarly books and ask them to review books under consideration for publication. We need more women who are advising publishers on the decisions we are making about the books we are publishing and not rely only on male scholars to help us make these decisions. Publishers need to ask female scholars to blurb books and endorse scholarly publications. There are many terrific female mathematicians and we need to increase their visibility in the book publishing community.

CH: You have published textbooks, popular math books, graphic works, works of magic, and monographs, all successful. What are the 5 essentials of a great math book?

A great book is not always measured by the number of copies it sells. It is sometimes measured by the impact it makes on a small community of scholars. Did it provide that one missing piece of information that led to the solution of an unsolved problem? Did it inspire a high school student to major in math? Did it turn a “math hater” into someone curious about math? Nevertheless, they all can benefit from some essential advice.

First, I feel that the most essential thing is that the author writes on something that she or he is passionate about. If this is the case, the reader will be engaged and love reading the book. Second, the author needs to clearly define the audience. No book can be for everyone. If the author defines the audience that way, then the book will be for no one. Third, the author needs to write for the audience and keep the mathematical level consistent throughout the book. One problem I have had with authors writing for audiences without an advanced math degree is over and under explaining math concepts. Fourth on my list is authors often introduce terms without defining them or define them by introducing other terms that need elaboration but instead lead to further confusion. Always provide examples that clarify definitions. Finally, if you have included any jokes or explanation marks in your manuscript, please delete them before sending the manuscript to your editor.

CH: What are the 5 math books you would gift to every aspiring female mathematician to learn about the art and science of math?

Before I reveal my suggestions, I would like to say that I think that the books I have suggested would make anyone want to learn about the art and science of math. They are particularly important to me because they point out the personal relationships that can develop out of the love of a subject. It is so hard for me to select only five because each book one selects to publish is special. Each one has a backstory. Most of my choices are, OF COURSE, Princeton University Press books because they are the ones I know the best and ones I have the time to read.

My first suggestion is not a book but a wonderful website, MacTutor History of Mathematics. I have spent many hours there and there is a link to Female Mathematicians, which is updated regularly.

The Calculus of Friendship: What a Teacher and a Student Learned about Life while Corresponding about Math by Steven Strogatz (Princeton University Press) is a book about a teacher and a student and their love of calculus as chronicled over thirty years through their letters. As you know by now, my love of math came from my high school math teacher. This author tried to help me find her. Unfortunately, we were unsuccessful. Later, at my 50th high school reunion I found out that she had passed away but it was the act of trying to find her that is illustrative of how tightly knit and wonderful I find the math community to be.

The Housekeeper and the Professor by Yoko Ogawa (Picador) was translated from Japanese. This is a novel about a math professor whose memory, due to an accident, is reset every 80 minutes, his housekeeper, and her young son. It is a wonderful story about how mathematics can bind three very different people.

Mathematics and Art: A Cultural History by Lynn Gamwell (Princeton University Press) covers the history of mathematics through exquisite works of art from antiquity to the present. I believe that learning about the history of mathematics is as important as the mathematics itself because you understand the time and place in which it is set and the math takes on more meaning.

The Seduction of Curves: The Lines of Beauty that Connect Mathematics, Art and the Nude by Allan McRobie (Princeton University Press) connects mathematics with art and engineering. This book focuses on the seven curves that are the basis of the catastrophe theory of mathematician René Thom. It is an accessible discussion of their role in nature, science, engineering, architecture, art, and other areas. Also included are their use in the work of David Hockney, Henry Moore, Anish Kapoor, and the delicate sculptures of Naum Gabo. The final two chapters focus on the collaborative work and friendship of Thom and Salvador Dalí. I searched for a book that could explain the work of René Thom for over twenty years before I found this one so it is pretty special.

CH: If you could invite five historic women mathematicians to join you at a dinner, who would they be, and why?

There are so many wonderful women mathematicians, historical and modern, that it is hard to choose just five. There are also many women who have made terrific contributions to mathematics who do not have advanced math degrees. See the references at the end of this post for additional resources.

At the top of my list would be Olga Taussky-Todd. Early in my career, I had the privilege of working with her on a book and got to know her a bit. I would love to spend more time with her. Not only was she smart, she had a great sense of humor. She made many contributions to the field of linear algebra, as did her husband, John, and we spent many hours talking about results in which, at the time, was one of my favorite topics in math. After Olga died, John gave me the poster from which the photo here was taken.

Emmy Noether is very important to me as I published a biography of her in my first position as an acquiring editor. I learned a lot about her work and would like to know more about her as a person. She has been described by many as the most important woman in the history of math. She developed the theories of rings, fields, and algebras.

Sophie Germain and I share a birthday, so of course I have to have dinner with her. Due to the great opposition against women in mathematics Sophie was not able to have a career in mathematics. Even her parents opposed her. She learned from books in her father’s library, often secretly after everyone was asleep. In spite of this she made many contributions to math such as her work on Fermat’s Last Theorem.

CH: What are five of your favorite mathematical puzzles?

Instead of listing single puzzles, I’ve chosen my favorite puzzles as types or groups. The following are some illustrations.

Word logic puzzles are fascinating and can also drive you crazy. Here is an example from Brain Food:

At a family reunion were the following people: one grandfather, one grandmother, two fathers, two mothers, four children, three grandchildren, one brother, two sisters, two sons, two daughters, one father-in-law, one mother-in-law, and one daughter-in-law. But not as many people attended as it sounds. How many were there, and who were they? Go to Rinkworks.com for more excellent puzzles and the answer to this one. However, you should try to solve it first.

Kakuro is like a crossword puzzle with numbers. Each word” must add up to the number provided in the clue above it or to the left. Words can only use the numbers 1 through 9, and a given number can only be used once in a word. Every kakuro puzzle has one and only one solution and can be solved through logic alone.

Martin Gardner was a master puzzler. If you don’t know who he is, or his puzzles (like cutting the pie, twiddled bolts, and the mutilated chessboard) head over to martin-gardner.org You will be glad you did.

I love playing Yahtzee which is more a game of logic, luck, and chance but always a lot of fun.
Jenga also does not strictly fall into the category of math but a lot of my math friends love playing it and it often appears at math meetings.

CH: how should we best compute the impact of mathematical publishing on the world?

From teaching in rural and inner-city schools for 8 years, I learned that there were so many students and adults who knew nothing about surviving in an increasingly complicated world that depends on a mastery of basic math skills. Over the past 42 years, I have seen the publication of numerous wonderful books for this very audience. These are books coming from university presses, commercial presses and society presses. These are books that have been published for the “math haters” and those who think math is hard. They present math through music and art and in graphic novels, detective stories, and puzzle books. There are ancillary materials posted on websites where readers can manipulate equations and discover new math of their own invention. As the number of books being published continues to increase, more people are clearly reading them. I am finding that there is much more enthusiasm for mathematics than there was four decades ago. There has been an increase in math clubs, math circles are very active, and the Girl Scouts announce many new STEM badges each year. I believe that publishers will continue to produce high quality books from mathematical writers around the world. This includes books that are being translated from one language into another, fostering an understanding of cultural differences through books about mathematics. I take every opportunity I can to tell people about the cool factor of math. If you are reading this post and have not discovered the wonder and empowerment of math, I’d advise you to go find a mathematician or anyone who has and ask them to let you in on the secret.

Additional Resources for inspiring information on women in STEM
MacTutor
Grandma Got STEM
A Mighty Girl

## In Dialogue: Christopher Phillips and Tim Chartier on Sports & Statistics

Question: How would you describe the intersection between statistics and sports? How does one inform the other?

Christopher Phillips, author of Scouting and Scoring: Sports have undoubtedly become one of the most visible and important sites for the rise of data analytics and statistics. In some respects, sports seem to be an easy, even inevitable place to apply new statistical tools: most sports produce a lot of data across teams and seasons; games have fixed rules and clear measures of success (e.g., wins or points); players and teams have incentives to adjust in order to gain a competitive edge.

But as I discuss in my new book Scouting and Scoring: How We Know What We Know About Baseball, it is also easy to fall prey to myths about the use of statistics in sports. Though these myths apply across many sports, it is easiest to hone in on baseball, as that has been one of the most consequential areas for statistics.

Perhaps the most persistent and pernicious myth is that data emerge naturally from sporting events. There is no doubt that new video-, Doppler-, and radar-based technologies, especially when combined with increasingly cheap computing power and storage capability, have dramatically expanded the amount of data that can be collected. But it takes a huge about of labor to create, collect, clean, and curate data, even before anyone tries to analyze them. Moreover, some data, like errors in baseball, are inescapably the product of individual judgment which has to be standardized and monitored.

The second myth is that sport statistics emerged only recently, particularly after the rise of the electronic computer. In fact, statistical analysis in sports goes back decades: in baseball, playing statistics were being used to evaluate players for year-end awards and negotiate contracts for as long as professional baseball has existed. (And statistics were collected and published for cricket decades before baseball’s rules were formalized.) As new methods of statistical analysis emerged in the early twentieth century in fields like psychology and physiology, some observers immediately tried to apply them to sports. In the 1910 book Touching Second, the authors promoted the use of data for shifting around fielders and for scouting prospects, two of the most important uses of statistical data in the modern era as well. There’s certainly been a flurry of new statistics over the last twenty years, but the general idea isn’t new—consider that Allen Guttmann’s half-century-old book From Ritual to Record, highlights the “numeration of achievement” and the “quantification of the aesthetic” as defining features of modern sport.

Finally, it’s a myth that there is a fundamental divide between those who look at performance statistics (i.e., scorers) and those who evaluate bodies (i.e., scouts). The usual gloss is that scouts are holistic, subjective judges of quality whereas scorers are precise, objective measurers. In reality, baseball scouts have long used methods of quantification, whether for the pricing of amateur prospects, or for the grading of skills, or the creation of single metrics like the Overall Future Potential that reduce a player to a single number. There’s a fairly good case to be made that scouts and other evaluators of talent are even more audacious quantifiers than scorers in that the latter mainly analyze things that can be easily counted.

Tim Chartier, author of Math Bytes: Data surrounds us. The rate at which data is produced can make us seem like specks in the cavernous expanse of digital information.  Each day 3 billion photos and videos are shared on Snapchat.  In the last minute, 300 hours of video were uploaded to YouTube.  Data is offering new possibilities for insight. Sports is an area where data has a traditional role and newfound possibilities, in part, due to the enlarging datasets.

For years, there are a number of constants in baseball that include the ball, bat, bases, and statistics like balls, strikes, hits and outs.  Statistics are and have simply been a part of the game.  You can find from the 1920 box score that Babe Ruth got 2 hits in 4 at-bats in his first game as a Yankee. While new metrics have emerged with analytical advances, the game has been well studied for some time. As Ford C. Frick stated in Games, Asterisks and People,

“Baseball is probably the world’s best documented sport.”

While this is true, the prevalence of data does not necessarily result in trusting the recommendations of those who study it.  For example, Manager Bobby Bragen stated, “Say you were standing with one foot in the oven and one foot in an ice bucket. According to the percentage people, you should be perfectly comfortable.”  This underscores an important aspect of data and analytics.  Data, inherently, can lead to insight but it becomes actionable when one trusts in how accurately it reflects our world.

Other sports, while not as statistically robust as baseball also have an influx of data.  In basketball, cameras positioned in the rafters report the (x,y) position of every player on the court and the (x,y,z) position of the ball throughout the entire game every fraction of a second.  As such, we can replay aspects of games via this data for years to come.  With such information comes new information.  For example, we know that Steph Curry, while averaging just over 34 minutes a game, runs, on average, just over 2.6 miles per game. He also runs almost a quarter of a mile more on offense than defense.

While such data can be stunning with its size and detail, it also comes with challenges. How do you recognize a pick and roll versus an isolation play simply from essentially dots moving in a plane?  Further, basketball, like football but unlike baseball, generally involves multiple players at a time.  How much credit do players get for a basket on offense?  A player’s position may open up possibilities for scoring, even if that player didn’t touch the ball.  As such, metrics have been and continued to be created in order to better understand the game.

Sports are played with a combination of analytics, gut and experience.  What combination depends on the sport, player, coach and context.  Nonetheless, data is here and will continue to give insight on the game.

## Pi: A Window into the World of Mathematics

Mathematicians have always been fascinated by Pi, the famous never-ending never-repeating decimal that rounds to 3.14. But why? What makes Pi such an interesting number? Every mathematician has their own answer to that question. For me, Pi’s allure is that it illustrates perfectly the arc of mathematics. Let me explain what I mean by taking you on a short mathematical adventure.

Picture yourself in a kitchen, rummaging the pantry for two cans of food. Let’s say you’ve found two that have circular bases of different diameters d1 and d2. Associated with each circle is a circumference value, the distance you’d measure if you walked all the way around the circle.

Were you to perfectly measure each circle’s circumference and diameter you would discover an intriguing relationship:

In other words, the ratio of each circle’s circumference to its diameter doesn’t change, even though one circle is bigger than the other. (This circumference-to-diameter number is  (“Pi”), the familiar 3.14-ish number.) This is the first stop along the arc of mathematics: the discovery of a relationship between two quantities.

Where this story gets very interesting is when, after grabbing even more cans and measuring the ratio of their circumferences to their diameters—you seem to have lots of free time on your hands—you keep finding the same ratio. Every. Time. This is the second stop along the arc of mathematics: the discovery of a pattern. Shortly after that, you begin to wonder: does every circle, no matter its size, have the same circumference-to-diameter ratio? You have reached the third stop along the arc of mathematics: conjecture. (Let’s call our circumference-to-diameter conjecture The Circle Conjecture.)

At first you consider proving The Circle Conjecture by measuring the ratio C/d for every circle. But you soon realize that this is impossible. And that’s the moment when you start truly thinking like a mathematician and begin to wonder: Can I prove The Circle Conjecture true using mathematics? You have now reached the most important stop along the arc of mathematics: the search for universal truth.

One of the first thinkers to make progress on The Circle Conjecture was the Greek mathematician Euclid of Alexandria. Euclid published a mammoth 13-book treatise text called Elements circa 300 BC in which he, among other accomplishments, derived all the geometry you learned in high school from just five postulates. One of Euclid’s results was that the ratio of a circle’s area A to the square of its diameter d2 is the same for all circles:

This is close to what we are trying to prove in The Circle Conjecture, but not the same. It would take another giant of mathematics—the Greek mathematician Archimedes of Syracuse—to move us onto what is often the last stop on the arc of mathematics: thinking outside the box.

Archimedes went back to Euclid’s five postulates, all but one of which dealt with lines, and extended some of Euclid’s postulates to handle curves. With these new postulates Archimedes was able to prove in his treatise Measurement of a Circle (circa 250 BC) that the area, circumference, and radius r of a circle are related by the equation:

(You may recognize this as the area of a triangle with base C and height r. Indeed, Archimedes’ proof of the formula effectively “unrolls” a circle to produce a triangle and then calculates its area.) Combining Archimedes’ formula with Euclid’s result, and using the fact that r = d/2, yields:

Et Voilà! The Circle Conjecture is proved! (To read more about the mathematical details involved in proving The Circle Conjecture, I recommend this excellent article.)

This little Pi adventure illustrated the core arc of mathematics: discovery of a relationship between to quantities; discovery of a more general pattern; statement of a conjecture; search for a proof of that conjecture; and thinking outside the box to help generate a proof. Let me end our mathematical adventure by encouraging you to embark on your own. Find things you experience in your life that are quantifiable and seem to be related (e.g., how much sleep you get and how awake you feel) and follow the stops along the arc of mathematics. You may soon afterward discover another universal truth: anyone can do mathematics! All it takes is curiosity, persistence, and creative thinking. Happy Pi Day!

Oscar E. Fernandez is associate professor of mathematics at Wellesley College. He is the author of Calculus Simplified, Everyday Calculus, and The Calculus of Happiness (all Princeton).

## Ken Steiglitz: Happy π Day!

As every grammar school student knows, π is the ratio of the circumference to the diameter of a circle. Its value is approximately 3.14…, and today is March 14th, so Happy π Day! The digits go on forever, and without a pattern. The number has many connections with computers, some obvious, some not so obvious, and I’ll mention a few.

The most obvious connection, I suppose, is that computers have allowed enthusiasts to ﬁnd the value of π to great accuracy. But how accurately do we really need to know its value? Well, if we knew the diameter of the Earth precisely, knowing π to 14 or 15 decimal places would enable us to compute the length of the equator to within the width of a virus. This accuracy was achieved by the Persian mathematician Jamshīd al-Kāshī in the early 15th century. Of course humans let loose with digital computers can be counted on to go crazy; the current record is more than 22 trillion digits. (For a delightful and oﬀ-center account of the history of π, see A History of Pi, third edition, by Petr Beckmann, St. Martin’s Press, New York, 1971. The anti-Roman rant in chapter 5 alone is worth the price of admission.)

A photo of a European wildcat, Felis silvestris silvestris. The original photo is on the left. On the right is a version where the compression ratio gradually increases from right to left, thereby decreasing the image quality. The original photograph is by Michael Ga¨bler; it was modiﬁed by AzaToth to illustrate the eﬀects of compression by JPEG. [Public domain, from Wikimedia Commons]

Don’t condemn the apparent absurdity of setting world records like this; the results can be useful. Running the programs on new hardware or software and comparing results is a good test for bugs. But more interesting is the question of just how the digits of π are distributed. Are they essentially random? Do any patterns appear? Is there a message from God hidden in this number that, after all, God created? Alas, so far no pattern has been found, and the digits appear to be “random” as far as statistical tests show. On the other hand, mathematicians have not been able to prove this one way or another.

Putting aside these more or less academic thoughts, the value of π is embedded deep in the code on your smartphone or computer and plays an important part in storing the images that people are constantly (it seems to me) scrolling through. Those images take up lots of space in memory, and they are often compressed by an algorithm like JPEG to economize on that storage. And that algorithm uses what are called “circular functions,” which, being based on the circle, depend for their very life on… π. The ﬁgure shows how the quality of an original image (left) degrades as it is compressed more and more, as shown on the right.

I’ll close with an example of an analog computer which we can use to ﬁnd the value of π. The computer consists of a piece of paper that is ruled with parallel lines 3 inches (say) apart, and a needle 3 inches long. Toss the needle so that it has an equal chance of landing anywhere on the paper, and an equal chance of being at any angle. Then it turns out that the chance of the needle intersecting a line on the piece of paper is 2/π, so that by repeatedly tossing the needle and counting the number of times it does hit a line we can estimate the value of π. Of course to ﬁnd the value of π to any decent accuracy we need to toss the needle an awfully large number of times. The problem of ﬁnding the probability of a needle tossed this way was posed and solved by Georges-Louis Leclerc, Comte de Buﬀon in 1777, and the setup is now called Buﬀon’s Needle. This is just one example of an analog computer, in contrast to our beloved digital computers, and you can ﬁnd much more about them in The Discrete Charm of the Machine.

Ken Steiglitz is professor emeritus of computer science and senior scholar at Princeton University. His books include The Discrete Charm of the MachineCombinatorial OptimizationA Digital Signal Processing Primer, and Snipers, Shills, and Sharks (Princeton). He lives in Princeton, New Jersey.

## Ken Steiglitz: It’s the Number of Zeroes that Counts

We present the third installment in a series by The Discrete Charm of the Machine author Ken Steiglitz. You can find the first post here and the second, here.

The scales of space and time in our universe; in everyday life we hang out very near the center of this picture: 1 meter and 1 second.

As we’ll see in The Discrete Charm the computer world is full of very big and very small numbers. For example, if your smartphone’s memory has a capacity of 32 GBytes, it means it can hold 32 billion bytes, or 32000000000 bytes. It’s awfully inconvenient and error-prone to count this many zeros, and it can get much worse, so scientists, who are used to dealing with very large and small numbers, just count the number of zeros. In this case the memory capacity is 3.2×1010 bytes. At the other extreme, pulses in an electronic circuit might occur at the rate of a billion per second, so the time between pulses is a billionth of a second, 0.000000001, a nanosecond, 1 × 10−9 seconds. In the time-honored scientific lingo, a factor of 10 is an “order of magnitude,” and back-of-the-envelope estimates often ignore factors of 2 or 3. What’s a factor of 2 or 3 between friends? What matters is the number of zeroes. In the last example, a nanosecond is 9 orders of magnitude smaller than a second.

Such big and small numbers also come up in discussing the size of transistors, the number of them that fit on a chip, the speed of communication on the internet in bits per second, and so on. The figure shows the range of magnitudes we’re ever likely to encounter when we discuss the sizes of things and the time that things take. At the low extremes I indicate the size of an electron and the time between the crests of gamma-ray waves, just about the highest frequency we ever encounter. The electron is about 6 orders of magnitude smaller than a typical virus (and a single transistor on a chip); the frequency of gamma rays is about 10 orders of magnitude faster than a gigahertz computer chip.

To this computer scientist a machine like an automobile is pretty boring. It runs only one program, or maybe two if you count forward and reverse gear. With few exceptions it has four wheels, one engine, one steering wheel—and all cars go about as fast as any other, if they can move in traffic at all. I could take my father’s 1941 Plymouth out for a spin today and hardly anyone would notice. It cost about \$845 in 1941 (for a four-door sedan), or about \$14,000 in today’s dollars. In other words, in our order-of-magnitude world, it is a product that is practically frozen in time. On the other hand, my obsolete and clumsy laptop beats the first computer I ever used by 5 orders of magnitude in speed and memory, and 4 orders of magnitude in weight and volume. If you want to talk money, I remember paying about 50¢ a byte for extra memory for a small laboratory computer in 1971—8 orders of magnitude more expensive than today, or maybe 9 if you take inflation into account.

The number of zeros is roughly the logarithm (base-10), and plots like the figure are said to have logarithmic scales. You can see them in the chapter on Moore’s law in The Discrete Charm, where I need them to get a manageable picture of just how much progress has been made in computer technology over the last few decades. The shrinkage in size and speedup has been, in fact, exponential with the years—which means constant-size hops in the figure, year by year. Anything less than exponential growth would slow to a crawl. This distinction between exponential and slower-than-exponential growth also plays a crucial role in studying the efficiency of computer algorithms, a favorite pursuit of theoretical computer scientists and a subject I take up towards the end of the book.

Counting zeroes lets us to fit the whole universe on a page.

Ken Steiglitz is professor emeritus of computer science and senior scholar at Princeton University. His books include The Discrete Charm of the MachineCombinatorial OptimizationA Digital Signal Processing Primer, and Snipers, Shills, and Sharks. He lives in Princeton, New Jersey.

## Ken Steiglitz: Garage Rock and the Unknowable

Here is the second post in a series by The Discrete Charm of the Machine author Ken Steiglitz. You can access the first post here

I sat down to draft The Discrete Charm of the Machine with the goal of explaining, without math, how we arrived at today’s digital world. It is a quasi-chronological story; I take what I need, when I need it, from the space of ideas. I start at the simplest point, describing why noise is a constant threat to information and how using discrete values (usually zeros and ones) affords protection and a permanence not possible with information in analog (continuous) form. From there I sketch the important ideas of digital signal processing (for sound and pictures), coding theory (for nearly error-free communication), complexity theory (for computation), and so on—a fine arc, I think, from the boomy and very analog console radios of my childhood to my elegant little internet radio.

Yet the path through the book is not quite so breezy and trouble-free. In the final three chapters we encounter three mysteries, each progressively more fundamental and thorny. I hope your curiosity and sense of wonder will be piqued; there are ample references to further reading. Here are the problems in a nutshell:

1. Is it no harder to find a solution to a problem than to merely check a solution? (Does P = NP?) This question comes up in studying the relative difficulty of solving problems with a computing machine. It is a mathematical question, and is still unresolved after almost 40 years of attack by computer scientists.
As I discuss in the book, there are plenty of reasons to believe that P is not equal to NP and most computer scientists come down on that side. But … no one knows for sure.
2. Are the digital computers we use today as powerful—in a practical sense—as any we can build in this universe (the extended Church-Turing thesis)? This is a physics question, and for that reason is fundamentally different from the P=NP question. Its answer depends on how the universe works.
The thesis is intimately tied to the problem of building machines that are essentially more powerful than today’s digital computers—the human brain is one popular candidate. The question runs deep: some believe there is magic to found beyond the world of zeros and ones.
3. Can a machine be conscious? Philosopher David Chalmers calls this the hard problem, and considers it “the biggest mystery.” It is not a question of mathematics, nor of physics, but of philosophy and cognitive science.

I want to emphasize that this is not merely the modern equivalent of asking how many angels could dance on the point of a pin. The answer has most serious consequences for us humans: it determines how we should treat our android creations, the inevitable products of our present rush to artificial intelligence. If machines are capable of suffering we have a moral responsibility to treat them compassionately.

My first reaction to the third question is that it is unanswerable. How can we know about the subjective mental life of anyone (or any thing) but ourselves? Philosopher Owen Flanagan called those who take this position mysterians, after the proto-punk band ? and the Mysterians. Michael Shermer joins this camp in his Scientific American column of July 1, 2018. I discuss the difficulty in the final chapter and remain agnostic—although I am hard-pressed even to imagine what form an answer would take.

I suggest, however, a pragmatic way around the big third question: Rather than risk harm, give the machines the benefit of the doubt. It is after all what we do for our fellow humans.

Ken Steiglitz is professor emeritus of computer science and senior scholar at Princeton University. His books include The Discrete Charm of the MachineCombinatorial OptimizationA Digital Signal Processing Primer, and Snipers, Shills, and Sharks. He lives in Princeton, New Jersey.

## Ken Steiglitz: When Caruso’s Voice Became Immortal

We’re excited to introduce a new series from Ken Steiglitz, computer science professor at Princeton University and author of The Discrete Charm of the Machine, out now.

The first record to sell a million copies was Enrico Caruso’s 1904 recording of “Vesti la giubba.” There was nothing digital, or even electrical about it; it was a strictly mechanical affair. In those days musicians would huddle around a horn which collected their sound waves, and that energy was coupled mechanically to a diaphragm and then to a needle that traced the waveforms on a wax or metal-foil cylinder or disc. For many years even the playback was completely mechanical, with a spring-wound motor and a reverse acoustical system that sent the waveform from what was often a 78 rpm shellac disc to a needle, diaphragm, and horn. Caruso almost single-handedly started a cultural revolution as the first recording star and became a household name—and millionaire (in 1904 dollars)—in the process. All without the benefit of electricity, and certainly purely analog from start to finish. Digital sound recording for the masses was 80 years in the future.

Enrico Caruso drew this self portrait on April 11, 1902 to commemorate his first recordings for RCA Victor. The process was completely analog and mechanical. As you can see, Caruso sang into a horn; there were no microphones. [Public domain, from Wikimedia Commons]

The 1904 Caruso recording I mentioned is perhaps the most famous single side ever made and is readily available online. It was a sensation and music lovers who could afford it were happy to invest in the 78 rpm (or simply “78”) disc, not to mention the elaborate contraption that played it. In the early twentieth century a 78 cost about a dollar or so, but 1904 dollars were worth about 30 of today’s dollars, a steep price for 2 minutes and 28 seconds of sound full of hisses, pops, and crackles, and practically no bass or treble. In fact the disc surface noise in the versions you’re likely to hear today has been cleaned up and the sound quality greatly improved—by digital processing of course. But being able to hear Caruso in your living room was the sensation of the new century.The poor sound quality of early recordings was not the worst of it. That could be fixed, and eventually it was. The long-playing stereo record (now usually called just “vinyl”) made the 1960s and 70s the golden age of high fidelity, and the audiophile was born. I especially remember, for example, the remarkable sound of the Mercury Living Presence and Deutsche Grammophon labels. The market for high-quality home equipment boomed, and it was easy to spend thousands of dollars on the latest high-tech gear. But all was not well. The pressure of the stylus, usually diamond, on the vinyl disc wore both. There is about a half mile of groove on an LP record, and the stylus that tracks it has a very sharp, very hard tip; records wear out. Not as quickly as the shellac discs of the 20s and 30s, but they wear out.

The noise problem for analog recordings is exacerbated when many tracks are combined, a standard practice in studio work in the recording industry. Sound in analog form is just inherently fragile; its quality deteriorates every time it is copied or played back on a turntable or past a tape head.

Everything changed in 1982 with the introduction of the compact disc (CD), which was digital. Each CD holds about 400 million samples of a 74-minute stereo sound waveform, each sample represented by a 2-byte number (a byte is 8 bits). In this world those 800 million bytes, or 6.4 billion bits (zeros or ones) can be stored and copied forever, absolutely perfectly. Those 6.4 billion bits are quite safe for as long as our civilization endures.

There are 19th century tenors whose voices we will never hear. But Caruso, Corelli, Domingo, Pavarotti… their digital voices are truly immortal.

Ken Steiglitz is professor emeritus of computer science and senior scholar at Princeton University. His books include The Discrete Charm of the MachineCombinatorial OptimizationA Digital Signal Processing Primer, and Snipers, Shills, and Sharks. He lives in Princeton, New Jersey.

## Browse our 2019 Mathematics Catalog

Our new Mathematics catalog includes an exploration of mathematical style through 99 different proofs of the same theorem; an outrageous graphic novel that investigates key concepts in mathematics; and a remarkable journey through hundreds of years to tell the story of how our understanding of calculus has evolved, how this has shaped the way it is taught in the classroom, and why calculus pedagogy needs to change.

If you’re attending the Joint Mathematics Meetings in Baltimore this week, you can stop by Booth 500 to check out our mathematics titles!

Integers and permutations—two of the most basic mathematical objects—are born of different fields and analyzed with different techniques. Yet when the Mathematical Sciences Investigation team of crack forensic mathematicians, led by Professor Gauss, begins its autopsies of the victims of two seemingly unrelated homicides, Arnie Integer and Daisy Permutation, they discover the most extraordinary similarities between the structures of each body. Prime Suspects is a graphic novel that takes you on a voyage of forensic discovery, exploring some of the most fundamental ideas in mathematics. Beautifully drawn and wittily and exquisitely detailed, it is a once-in-a-lifetime opportunity to experience mathematics like never before.

99 Variations on a Proof offers a multifaceted perspective on mathematics by demonstrating 99 different proofs of the same theorem. Each chapter solves an otherwise unremarkable equation in distinct historical, formal, and imaginative styles that range from Medieval, Topological, and Doggerel to Chromatic, Electrostatic, and Psychedelic. With a rare blend of humor and scholarly aplomb, Philip Ording weaves these variations into an accessible and wide-ranging narrative on the nature and practice of mathematics. Readers, no matter their level of expertise, will discover in these proofs and accompanying commentary surprising new aspects of the mathematical landscape.

Exploring the motivations behind calculus’s discovery, Calculus Reordered highlights how this essential tool of mathematics came to be. David Bressoud explains why calculus is credited to Isaac Newton and Gottfried Leibniz in the seventeenth century, and how its current structure is based on developments that arose in the nineteenth century. Bressoud argues that a pedagogy informed by the historical development of calculus presents a sounder way for students to learn this fascinating area of mathematics.

## Ken Steiglitz on The Discrete Charm of the Machine

A few short decades ago, we were informed by the smooth signals of analog television and radio; we communicated using our analog telephones; and we even computed with analog computers. Today our world is digital, built with zeros and ones. Why did this revolution occur? The Discrete Charm of the Machine explains, in an engaging and accessible manner, the varied physical and logical reasons behind this radical transformation. Ken Steiglitz examines why our information technology, the lifeblood of our civilization, became digital, and challenges us to think about where its future trajectory may lead.

What is the aim of the book?

The subtitle: To explain why the world became digital. Barely two generations ago our information machines—radio, TV, computers, telephones, phonographs, cameras—were analog. Information was represented by smoothly varying waves. Today all these devices are digital. Information is represented by bits, zeros and ones. We trace the reasons for this radical change, some based on fundamental physical principles, others on ideas from communication theory and computer science. At the end we arrive at the present age of the internet, dominated by digital communication, and finally greet the arrival of androids—the logical end of our current pursuit of artificial intelligence.

What role did war play in this transformation?

Sadly, World War II was a major impetus to many of the developments leading to the digital world, mainly because of the need for better methods for decrypting intercepted secret messages and more powerful computation for building the atomic bomb. The following Cold War just increased the pressure. Business applications of computers and then, of course, the personal computer opened the floodgates for the machines that are today never far from our fingertips.

How did you come to study this subject?

I lived it. As an electrical engineering undergraduate I used both analog and digital computers. My first summer job was programming one of the few digital computers in Manhattan at the time, the IBM 704. In graduate school I wrote my dissertation on the relationship between analog and digital signal processing and my research for the next twenty years or so concentrated on digital signal processing: using computers to process sound and images in digital form.

What physical theory played—and continues to play—a key role in the revolution?

Quantum mechanics, without a doubt. The theory explains the essential nature of noise, which is the natural enemy of analog information; it makes possible the shrinkage and speedup of our electronics (Moore’s law); and it introduces the possibility of an entirely new kind of computer, the quantum computer, which can transcend the power of today’s conventional machines. Quantum mechanics shows that many aspects of the world are essentially discrete in nature, and the change from the classical physics of the nineteenth century to the quantum mechanics of the twentieth is mirrored in the development of our digital information machines.

What mathematical theory plays a key role in understanding the limitations of computers?

Complexity theory and the idea of an intractable problem, as developed by computer scientists. This theme is explored in Part III, first in terms of analog computers, then using Alan Turing’s abstraction of digital computation, which we now call the Turing machine. This leads to the formulation of the most important open question of computer science, does P equal NP? If P equals NP it would mean that any problem where solutions can just be checked fast can be solved fast. This seems like asking a lot and, in fact, most computer scientists believe that P does not equal NP. Problems as hard as any in NP are called NP-complete. The point is that NP-complete problems, like the famous traveling problem, seem to be intrinsically difficult, and cracking any one of them cracks them all.  Their essential difficulty manifests itself, mysteriously, in many different ways in the analog and digital worlds, suggesting, perhaps, that there is an underlying physical law at work.

What important open question about physics (not mathematics) speaks to the relative power of digital and analog computers?

The extended Church-Turing thesis states that any reasonable computer can be simulated efficiently by a Turing machine. Informally, it means that no computer, even if analog, is more powerful (in an appropriately defined way) than the bare-boned, step-by-step, one-tape Turing machine. The question is open, but many computer scientists believe it to be true. This line of reasoning leads to an important conclusion: if the extended Church-Turing thesis is true, and if P is not equal to NP (which is widely believed), then the digital computer is all we need—Nature is not hiding any computational magic in the analog world.

What does all this have to do with artificial intelligence (AI)?

The brain uses information in both analog and digital form, and some have even suggested that it uses quantum computing. So, the argument goes, perhaps the brain has some special powers that cannot be captured by ordinary computers.

What does philosopher David Chalmers call the hard problem?

We finally reach—in the last chapter—the question of whether the androids we are building will ultimately be conscious. Chalmers calls this the hard problem, and some, including myself, think it unanswerable. An affirmative answer would have real and important consequences, despite the seemingly esoteric nature of the question. If machines can be conscious, and presumably also capable of suffering, then we have a moral responsibility to protect them, and—to put it in human terms—bring them up right. I propose that we must give the coming androids the benefit of the doubt; we owe them the same loving care that we as parents bestow on our biological offspring.

Where do we go from here?

A funny thing happens on the way from chapter 1 to 12. I begin with the modest plan of describing, in the simplest way I can, the ideas behind the analog-to-digital revolution.  We visit along the way some surprising tourist spots: the Antikythera mechanism, a 2000-year old analog computer built by the ancient Greeks; Jacquard’s embroidery machine with its breakthrough stored program; Ada Lovelace’s program for Babbage’s hypothetical computer, predating Alan Turing by a century; and B. F. Skinner’s pigeons trained in the manner of AI to be living smart bombs. We arrive at a collection of deep conjectures about the way the universe works and some challenging moral questions.

Ken Steiglitz is professor emeritus of computer science and senior scholar at Princeton University. His books include Combinatorial OptimizationA Digital Signal Processing Primer, and Snipers, Shills, and Sharks (Princeton). He lives in Princeton, New Jersey.

## Browse our 2019 Computer Science Catalog

Our new Computer Science catalog includes an introduction to computational complexity theory and its connections and interactions with mathematics; a book about the genesis of the digital idea and why it transformed civilization; and an intuitive approach to the mathematical foundation of computer science.

If you’re attending the Information Theory and Applications workshop in San Diego this week, you can stop by the PUP table to check out our computer science titles!

Mathematics and Computation provides a broad, conceptual overview of computational complexity theory—the mathematical study of efficient computation. Avi Wigderson illustrates the immense breadth of the field, its beauty and richness, and its diverse and growing interactions with other areas of mathematics. With important practical applications to computer science and industry, computational complexity theory has evolved into a highly interdisciplinary field that has shaped and will further shape science, technology, and society.

A few short decades ago, we were informed by the smooth signals of analog television and radio; we communicated using our analog telephones; and we even computed with analog computers. Today our world is digital, built with zeros and ones. Why did this revolution occur? The Discrete Charm of the Machine explains, in an engaging and accessible manner, the varied physical and logical reasons behind this radical transformation, and challenges us to think about where its future trajectory may lead.

Discrete mathematics is the basis of much of computer science, from algorithms and automata theory to combinatorics and graph theory. This textbook covers the discrete mathematics that every computer science student needs to learn. Guiding students quickly through thirty-one short chapters that discuss one major topic each, Essential Discrete Mathematics for Computer Science can be tailored to fit the syllabi for a variety of courses. Fully illustrated in color, it aims to teach mathematical reasoning as well as concepts and skills by stressing the art of proof.

## Gift Guide: Biographies and Memoirs!

Not sure what to give the reader who’s read it all? Biographies, with their fascinating protagonists, historical analyses, and stranger-than-fiction narratives, make great gifts for lovers of nonfiction and fiction alike! These biographies and memoirs provide glimpses into the lives of people both famous and forgotten:

Walatta-Petros was an Ethiopian saint who lived from 1592 to 1642 and led a successful nonviolent movement to preserve African Christian beliefs in the face of European protocolonialism. Written by her disciple Galawdewos in 1672, after Walatta-Petros’s death, and translated and edited by Wendy Laura Belcher and Michael Kleiner, The Life of Walatta-Petros praises her as a friend of women, a devoted reader, a skilled preacher, and a radical leader, providing a rare picture of the experiences and thoughts of Africans—especially women—before the modern era.

This is the oldest-known book-length biography of an African woman written by Africans before the nineteenth century, and one of the earliest stories of African resistance to European influence. This concise edition, which omits the notes and scholarly apparatus of the hardcover, features a new introduction aimed at students and general readers.

The forgotten mathematician: Fibonacci

The medieval mathematician Leonardo of Pisa, popularly known as Fibonacci, is most famous for the Fibonacci numbers—which, it so happens, he didn’t invent. But Fibonacci’s greatest contribution was as an expositor of mathematical ideas at a level ordinary people could understand. In 1202, his book Liber abbaci—the “Book of Calculation”—introduced modern arithmetic to the Western world. Yet Fibonacci was long forgotten after his death.

Finding Fibonacci is Keith Devlin’s compelling firsthand account of his ten-year quest to tell Fibonacci’s story. Devlin, a math expositor himself, kept a diary of the undertaking, which he draws on here to describe the project’s highs and lows, its false starts and disappointments, the tragedies and unexpected turns, some hilarious episodes, and the occasional lucky breaks.

The college president: Hanna Gray

Hanna Holborn Gray has lived her entire life in the world of higher education. The daughter of academics, she fled Hitler’s Germany with her parents in the 1930s, emigrating to New Haven, where her father was a professor at Yale University. She has studied and taught at some of the world’s most prestigious universities. She was the first woman to serve as provost of Yale. In 1978, she became the first woman president of a major research university when she was appointed to lead the University of Chicago, a position she held for fifteen years. In 1991, Gray was awarded the Presidential Medal of Freedom, the nation’s highest civilian honor, in recognition of her extraordinary contributions to education.

Gray’s memoir An Academic Life is a candid self-portrait by one of academia’s most respected trailblazers.

The medieval historian: Ibn Khaldun

Ibn Khaldun (1332–1406) is generally regarded as the greatest intellectual ever to have appeared in the Arab world—a genius who ranks as one of the world’s great minds. Yet the author of the Muqaddima, the most important study of history ever produced in the Islamic world, is not as well known as he should be, and his ideas are widely misunderstood. In this groundbreaking intellectual biography, Robert Irwin presents an Ibn Khaldun who was a creature of his time—a devout Sufi mystic who was obsessed with the occult and futurology and who lived in a world decimated by the Black Death.

Ibn Khaldun was a major political player in the tumultuous Islamic courts of North Africa and Muslim Spain, as well as a teacher and writer. Irwin shows how Ibn Khaldun’s life and thought fit into historical and intellectual context, including medieval Islamic theology, philosophy, politics, literature, economics, law, and tribal life.

The novelist and philosopher: Iris Murdoch

Iris Murdoch was an acclaimed novelist and groundbreaking philosopher whose life reflected her unconventional beliefs and values. Living on Paper—the first major collection of Murdoch’s most compelling and interesting personal letters—gives, for the first time, a rounded self-portrait of one of the twentieth century’s greatest writers and thinkers. With more than 760 letters, fewer than forty of which have been published before, the book provides a unique chronicle of Murdoch’s life from her days as a schoolgirl to her last years.

The letters show a great mind at work—struggling with philosophical problems, trying to bring a difficult novel together, exploring spirituality, and responding pointedly to world events. We witness Murdoch’s emotional hunger, her tendency to live on the edge of what was socially acceptable, and her irreverence and sharp sense of humor. Direct and intimate, these letters bring us closer than ever before to Iris Murdoch as a person.