Happy 100th Anniversary to Einstein’s General Theory of Relativity!

relativity jacketToday is the final day of our popular #ThanksEinstein series, in which an array of prominent scholars and scientists have shared their insights and reflections on relativity, Einstein, and how his work inspired their own careers. Scroll through this week’s blog posts to read pieces by Daniel Kennefick, Katherine Freese, Hanoch Gutfreund, Jürgen Renn, Alice Calaprice, Jimena Canales, J.P. Ostriker, and many more special features, including this piece on Einstein’s final days.

Einstein’s General Theory of Relativity celebrates its 100 year anniversary today. November 25, 1915, during a particularly strenuous time in his life, is when Einstein submitted his final version of the general theory of relativity to the Prussian Royal Academy, complete with the field equations that define how the force of gravity arises from the curvature of space and time by matter and energy. The theory, which is the current theory of gravitation in modern physics, has implications for everything from black holes to the idea of universe expansion. It gained rapid popularity after its conception in 1915, and in the early 1920s alone, it was translated into ten languages. Fifteen editions in the original German appeared over the course of Einstein’s lifetime.

Princeton University Press has released a special edition of Relativity: The Special and the General Theory to commemorate the anniversary, including commentary from Hanoch Gutfreund and Jürgen Renn, Einstein experts, as well as additional content such as title pages from several language translations. You can browse through them in the slideshow below. Happy 100th to the general theory of relativity! Science wouldn’t be the same without you.

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Relativity Title Page, English Edition

Relativity Title Page, Chinese Edition

Relativity Cover, Chinese 1921 Edition

Relativity Cover, Czech Republic 1923 Edition

Relativity Cover, German Edition

Relativity Title Page, Japanese Edition

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Was Einstein the First to Discover General Relativity?

Today the world celebrates the day 100 years ago that Albert Einstein submitted his final version of the general theory of relativity to the Prussian Royal Academy. A theory of gravitation with critical consequences, it completely transformed the field of theoretical physics and astronomy. Einstein has long been celebrated and popularized for his contribution, but some have continued to ask whether he was, in fact, the first to discover general relativity. Daniel Kennefick, co-author of An Einstein Encyclopedia, looks at the debate:

Einstein’s Race

By Daniel Kennefick

On November 25, 1915 Einstein submitted one of the most remarkable scientific papers of the twentieth century to the Prussian Academy of Sciences in Berlin. The paper presented the final form of what are called the Einstein Equations, the field equations of gravity which underpin Einstein’s General Theory of Relativity. Thus this year marks the centenary of that theory. Within a few years this paper had supplanted Newton’s Universal Theory of Gravitation as our explanation of the phenomenon of gravitation, as well as overthrown Newton’s understanding of such fundamental concepts as space, time and motion. As a result Einstein became, and has remained, the most famous and celebrated scientist since Newton himself.

EinsteinBut what if Einstein was not the first scientist to publish these famous equations? Should they be called, not the Einstein equations, but the Einstein-Hilbert equations, honoring also the German mathematician David Hilbert? In 1915, Einstein visited Hilbert in Gottingen, and Hilbert convinced him that the goal of a fully general relativistic theory was achievable, something Einstein had nearly convinced himself could not be done. Einstein returned to work, and by November, he had found the field equations which give General Relativity its final form. However, Hilbert also worked on the ideas Einstein had discussed with him and published a paper discussing how Einstein’s theory fitted in with his own ideas on the role of mathematics in physics.

The argument for honoring Hilbert lies in a paper written by him which included the Einstein equations, derived from fundamental principles. This paper, while appearing several months after Einstein’s, was submitted on November 20, and Hilbert even sent Einstein a copy which probably reached Einstein before he submitted his own paper. In fact, a few people have even gone so far as to propose that Einstein might have stolen the final form of his equations from Hilbert.

Of course even if that were true, we are talking only about one final term in the equations (Einstein had published a close to correct version earlier in the month) and to Einstein would still belong sole credit for the enormous amount of work which went into the argument by which equations with these unique properties were singled out in the first place. We would still recognize Einstein for the critical physical thinking, while acknowledging Hilbert’s superior mathematical ability in more quickly finding the final correct form of the equations. Still, perhaps Hilbert would deserve a share of the credit for that final step. Why then do the centenary celebrations mention Einstein only and omit Hilbert almost completely?

One reason is that in the late 1990s a historian working on Hilbert named Leo Corry made a remarkable discovery. He found a copy of the proofs of Hilbert’s paper, with a printers stamp dating it to December 6, 1915. These proofs show that Hilbert made significant changes to the paper after this date. In addition, the proofs do not contain the Einstein equations. The proofs have been cut up here and there (probably by the printers themselves as they worked), so it is possible that the equations would be there if we had the missing pieces. But it is also quite possible that amidst the changes Hilbert made to the paper, he took the opportunity to include the final form of the equations from Einstein’s paper. Indeed some of the changes he made after December 6 were to update his argument from earlier versions of Einstein’s theory to the later version.

Certainly it was Einstein who felt himself to be the injured party in this short-lived priority dispute (arguably the only occasion in his life when Einstein found himself in such a dispute). He complained to a friend that Hilbert was trying to “nostrify” his theory, to claim a share of the credit. Einstein complained to Hilbert himself indeed, and some of the changes made in proofs by Hilbert included the addition of remarks giving credit for the basic ideas behind the theory to Einstein. At any rate, Einstein tried not to let proprietary feelings color his feelings of gratitude for Hilbert. He recalled well that Hilbert had played an important role in encouraging Einstein to return to his theory at a time when Einstein had, to some extent, given up on his original goals. On December 20, 1915, he wrote to Hilbert:

“There has been a certain resentment between us, the cause of which I do not want analyze any further. I have fought against the feeling of bitterness associated with it, and with complete success. I again think of you with undiminished kindness and I ask you to attempt the same with me. It is objectively a pity if two guys that have somewhat liberated themselves from this shabby world are not giving pleasure to each other.” (translated and quoted in Corry, Renn and Stachel, 1997).

So if Einstein was becoming the new Newton, as the man who solved the riddle of gravity, he was far from being a new Newton in another sense; of being the sort of man who carries on scientific grudges to the detriment of his friendship with the other great thinkers of his day.

Daniel Kennefick is associate professor of physics at the University of Arkansas, an editor of the Collected Papers of Albert Einstein, and the author of An Einstein Encyclopedia and Traveling at the Speed of Thought: Einstein and the Quest for Gravitational Waves (Princeton).

For more on Einstein’s field equations, check out this article by Dennis Lehmkuhl at Caltech.

The Final Days of Albert Einstein


Albert Einstein’s time on earth ended on April 18, 1955, at the Princeton Hospital.

In April of 1955, shortly after Einstein’s death, a pathologist removed his brain without the permission of his family, and stored it in formaldehyde until around 2007, shortly before dying himself. In that time, the brain of the man who has been credited with the some of the most beautiful and imaginative ideas in all of science was photographed, fragmented —small sections parceled to various researchers. His eyes were given to his ophthalmologist.

These indignities in the name of science netted several so-called findings—that the inferior parietal lobe, the part said to be responsible for mathematical reasoning was wider, that the unique makeup of the Sulvian fissure could have allowed more neurons to make connections. And yet, there remains the sense that no differences can truly account for the cognitive abilities that made his genius so striking.

Along with an exhaustive amount of information on  the personal, scientific, and public spheres of Einstein’s life, An Einstein Encyclopedia includes this well-known if macabre “brain in a jar” story. But there is a quieter one that is far more revealing of the man himself: The story in which Helen Dukas, Einstein’s longtime secretary and companion, recounts his last days. Dukas, the encyclopedia notes, was “well known for being intelligent, modest, shy, and passionately loyal to Einstein.” Her account is at once unsensational and unadorned.

One might expect a story of encroaching death, however restrained, to chronicle confusion and fear. Medically supported death was a regular occurrence by the middle of the 20th century, and Einstein died in his local hospital. But what is immediately striking from the account is the simplicity and calmness with which Einstein met his own passing, which he regarded as a natural event. The telling of this chapter is matter of fact, from his collapse at home, to his diagnosis with a hemorrhage, to his reluctant trip to the hospital and refusal of a famous heart surgeon. Dukas writes that he endured the pain from an internal hemorrhage (“the worst pain one can have”) with a smile, occasionally taking morphine. On his final day, during a respite from pain, he read the paper and talked about politics and scientific matters.

“You’re really hysterical—I have to pass on sometime, and it doesn’t really matter when.” he tells Dukas, when she rises in the night to check on him.

As Mary Talbot  writes in Aeon, “Apprehending the truth that all things arise and pass away might be the ultimate groundwork for dying.” And certainly, it would be difficult to dispute Einstein’s wholehearted dedication to the truth throughout his life and work. His manifesto, referenced here by Hanoch Gutfreund on the occasion of the opening of the Hebrew University, asserts, “Science and investigation recognize as their aim the truth only.” From passionate debates on the nature of reality with Bohr, to his historic clash on the nature of time with Bergson, Einstein’s quest for the truth was a constant in his life.  It would seem that it was equally so at the time of his death. What, then, did he believe at the end? We can’t know, but An Einstein Encyclopedia opens with his own words,

Strange is our situation here upon earth. Each of us comes for a short visit, not knowing why, yet sometimes seeming to divine a purpose….To ponder interminably over the reason for one’s own existence or the meaning of life in general seems to me, from an objective point of view, to be sheer folly. And yet everyone holds certain ideals by which he guides his aspiration and his judgment. The ideals which have always shone before me and filled me with the joy of living are goodness, beauty, and truth. To make a goal of comfort or happiness has never appealed to me; a system of ethics built on this basis would be sufficient only for a herd of cattle.

Read a sample chapter of An Einstein Encyclopedia, by Alice Calaprice, Daniel Kennefick, & Robert Schulmann here.

#ThanksEinstein: Alice Calaprice on the man behind the myth

Thanks Einstein Meme 4

Becoming an Einstein Author

By Alice Calaprice

Alice Calaprice is the editor of the hugely popular collection of Einstein quotations that has sold tens of thousands of copies worldwide and been translated into twenty-five languages. This is the story of how her knack for German and quest for full-time work in Princeton, New Jersey led her to a career she never imagined.

As a child I did not dream of someday becoming an author of books about Albert Einstein, nor did I contemplate the possibility even after graduating from UC Berkeley in the 1960s. Such an idea would not even have occurred to me. Along with my interest in science, languages, cultures, and history, it was eventually serendipity that took me there.

In the late 1970s, after my family had settled well into the routine of raising school-age children in Princeton, New Jersey, I assigned myself the task of finding full-time work. I had recently completed a course in the then relatively new field of computer technology, hoping it would help bolster a future career. One day in early 1978, a friend told me about a new venture being undertaken by Princeton University Press: the publication of the papers of Albert Einstein in a voluminous series that would span many years. An intriguing project, for sure, but I did not imagine myself being a part of it.

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Calaprice at an Einstein statue in Washington DC (“worshipping at Einstein’s feet”).

Soon after, however, the founding editor of the project, physicist John Stachel, and I met after he had started some preliminary work on the papers. It interested him that I was a native German speaker, had spent time around computers, and wasn’t averse to physics jargon and working with physicists, being married to one at the time. He had been looking for someone for a specialized task: helping him prepare three electronic indexes of the contents of the Einstein archive. He explained that the archive contained about 10,000 documents, consisting of Einstein’s writings, correspondence, and third-party materials. The indexes would give him an overview of the archive’s size and contents–information crucial to the planning stages of the enormous undertaking.

Although the Collected Papers of Albert Einstein would be administered and published by the university press, the archive and his office were located at the nearby Institute for Advanced Study, in the same building where Einstein himself had worked during the last two decades of his life. Stachel asked if I was interested in helping to jump-start this initial phase of the project. The timing turned out to be perfect, and I agreed. I had no inkling that I was about to jump-start a lifelong career as well.

Hello, Einstein

This assignment, which required perusing and often carefully reading each document in the archive’s files, gave me the chance to familiarize myself with the details of Einstein’s legacy and life, with which I was not particularly familiar. It was also an opportunity to revive my long-neglected German-language aptitude, which had waned over the years. Einstein wrote almost exclusively in his native language, even after he came to America from Germany in 1933; his correspondence and papers were generally translated by his secretary or assistants. I was surprised by some of the particulars about his life. He was not so saintly, after all, and besides transforming scientific thinking he had also done ordinary things like play the violin and love animals.

My curiosity was piqued. I quickly became an autodidact, reading supplementary articles and books so I could put the archival material into context. Names of Einstein’s family, friends, and colleagues became familiar, as did the terms for concepts in physics used by him and his cohorts. The prewar and wartime venues and events in Germany became clearer, alive, and more personal. Berlin, the city of my wartime birth, took on new meaning: I discovered that the Einstein family had lived in the same neighborhood as my family, but, unlike them, we did not have to flee persecution. We did flee the city during the Allied bombings of 1945, long after the Einsteins had already departed for America. After short stints in various villages, we coincidentally ended up in Bad Cannstatt in southwestern Germany, which I later learned was also the ancestral home of Einstein’s mother. And, finally, both of us had found our way to Princeton, if at different times, by different routes, and for different reasons. After I had oriented myself to my new surroundings, I loved coming to work. I had found a stimulating job that suited me well. Not only was the timing of my employment in the archive ideal for me personally, but the times were exciting, too. The centennial of Einstein’s birth took place at the Institute—among other worldwide venues—in 1979. Some of Einstein’s assistants and collaborators were still alive and gave firsthand accounts of their recollections in a symposium on the campus. I was able to attend these talks.

Einstein’s Inner Circle

There and at other times, I met many people who had been associated with Einstein either directly or were now members of boards that were planning the eventual publication of his papers. Outstanding among these was Helen Dukas, Einstein’s longtime, modest, and intensely loyal secretary, who, after his death in 1955, had become the first archivist of his papers. Now in her early eighties, she still came to work almost daily. Her office was around the corner from mine on the third floor of Fuld Hall. She stopped by to chat every morning after exiting the elevator located across from my office, often inspecting the never-ending clutches of house finches nesting outside my window in spring and summer. She came to our house for dinner, and she invited my family to be her guests at the swimming pool in the Institute Woods.

At Helen’s crowded memorial service after her death in 1982, I heard her old friend Otto Nathan, the executor of Einstein’s estate, tearfully proclaim, “When Helen died, Einstein died a second time.” The Institute, a cosmopolitan place of world-renowned scholars, where foreign languages were heard more often than English, was a place where one could thrive professionally and personally.

We completed the indexes by the 1980 deadline. Because the 10,000 estimated documents had more than quadrupled to 42,000, we had hired a part-time assistant to help accomplish the task. I spent long hours working off-site in the evenings, when mainframe computers at the university’s Computer Center and, later, in my husband’s cyclotron laboratory in the physics department, were more readily available for use.

Herb Bailey, the well-regarded director of Princeton University Press who had long advocated for publication of the Collected Papers, was apparently pleased with my work. He now offered me a position in the editorial offices at the Press’s historic Scribner building on the university’s campus. My first day of work was on April Fool’s Day 1980, but I was assured my employment was not a joke. John Stachel continued his sole editorship of the papers at the Institute, and later at Scribner with a small staff. I was in touch with the group almost daily, grounding my interest in what came to be known as the Einstein Papers Project.

Fluent in Einstein

Five years later, after I had become a senior editor at PUP, I had the opportunity to again read the documents and letters that were about to be published in volume 1 of the Collected Papers. In 1985, the first manuscript in the series was turned over to the Press’s editorial office, and I was asked to take charge. I helped to set an editorial style for the series, copyedited the volumes as they arrived in-house, and became administrator and “principal investigator” of the concomitant National Science Foundation-funded English-translation project. Over a span of almost thirty years, I copyedited all fifteen of the volumes in the series—more recently as a freelancer—that have been published so far, including the translated volumes. Alas, so much reading, yet I never succeeded in understanding physics and relativity theory! Despite this shortfall, I became the liaison for nonscientific Einstein-related inquiries, book projects, film documentaries, and even the movie IQ in the early 1990s. I was a resource on matters dealing with Einstein, consistently learning something new in the process and having contact with an assortment of Einstein aficionados around the world. At the same time, I handled many other editing projects, mostly in the sciences. Surrounded by a group of wonderful, supportive, and good-humored colleagues and a continuously changing stream of engaging authors, I was having the time of my life. Those years set the stage for the twenty years ahead.

In 1995, I had an especially good year. First, it was the year I began mitigating my restlessness at home by taking annual trips to unlikely parts of the world, and I went to eastern Siberia with a small group of fellow nature lovers. Second, on my return, I received the news that I would receive the national Literary Market Place (LMP) Award for Individual Editorial Achievement in Scholarly Publishing, to be presented at the New York Public Library the following year. Third, Trevor Lipscombe, PUP’s acquisitions editor in physics at the time, discussed with me the prospect of publishing a book of quotations by Einstein. Like all those familiar with Einstein’s life, Trevor was aware that the physicist was multidimensional and fearless in expressing opinions on a variety of topics of interest to many: there was much more to him than relativity theory. Unbeknownst to Trevor, I had already collected many quotations while working on the indexes and copyediting the first few volumes of the Collected Papers—simply because they had struck a chord with me. When I showed him my blue box of index cards containing the quotations, he suggested I write the book myself rather than find someone else to do so. I was excited at the prospect of being on the other side of the author/editor relationship.

The Quotable Einstein is born

Soon after I returned from another adventure trip about a year later, this time into the Amazon Basin in northeastern Peru, the first edition of The Quotable Einstein was published. It contained four hundred quotations and their sources, arranged by topic, such as Einstein on religion, on his family, on Jews, on politics, on science and scientists, and so forth. The initial print run was modest, as there were doubts that the book would have wide appeal. The volume quickly sold out, however, and was reprinted six times. For a long time, it was at the top of PUP’s sales list, which I admired in disbelief and awe whenever one was posted on the bulletin board. Three more enlarged editions followed at approximately five-year intervals, and more than twenty-five foreign-language translations have been contracted, some in obscure languages I had never heard of. I believe these books were successful because they showed Einstein in all his guises, in his own uncensored words—a human being beyond the prevailing hagiographic and absent-minded-professor myths and falsely attributed quotations. The Ultimate Quotable Einstein, containing about 1,600 documented quotations and published in 2008, was my fourth and final contribution to this series of quotation books.

Because of the success of these volumes, I was now, to my surprise, perceived as an authority. I was asked to give Calaprice_Einstein_Encyclopediatalks for nonacademic audiences and participate in television shows and documentaries. I was invited to the German embassy to celebrate the special relativity centennial in 2005, and sat next to the German ambassador for lunch. I had book signings. I appeared on Ira Flatow’s “Science Friday” at the NPR studio in New York, along with Dennis Overbye of the New York Times. I have to confess that I found these new challenges difficult. I felt more comfortable doing research and writing, so I agreed to write three more books for other publishers who approached me.

Now, well into retirement in California, I am back with PUP for my swan song in the Einstein genre. Having often felt the need for a concise Einstein reference guide while doing research, I had submitted to the publisher an informal proposal to write An Einstein Encyclopedia. My expertise on specialized topics relating to Einstein is limited, so two Einstein scholars with broad experience on the Einstein Papers Project, historian Robert Schulmann and physicist Dan Kennefick, fortunately agreed to join me in this project as co-authors. Our final proposal was accepted, the three of us had a productive long-distance collaboration, and, best of all, we managed to stay friends throughout the process. As our reward, we are now the proud authors of a reference book that we expect will be of use and interest to an eclectic readership.

Alice Calaprice is a renowned authority on Albert Einstein and the author of several popular books on Einstein, including The Ultimate Quotable Einstein (Princeton).

#ThanksEinstein image courtesy of the official Albert Einstein Facebook page.

#ThanksEinstein: Jürgen Renn on popularizing Einstein

Thanks Einstein Meme 3

Einstein: Missionary of Science

By Jürgen Renn

Jürgen Renn is a director at the Max Planck Institute for the History of Science in Berlin. This is the story of how he came to play such a major role in popularizing Einstein.

I encountered Albert Einstein at crucial turning points in my life, first studying his general theory of relativity while exploring quantum field theory on curved space-time backgrounds for my diploma thesis in physics at the Freie Universität in Berlin. I published my first papers on general relativity together with two postdocs I had the fortune to work with at the time: Tevian Dray and Don Salisbury. I would like to have pursued this topic for my PhD thesis as well but instead turned to quantum field theory and statistical physics. Meanwhile, I developed a passion for the history of science and began to prepare an edition of Galileo’s manuscripts. In 1985, working on my PhD in Rome, I was convinced that I could do physics and the history of science at the same time, and that I would stay in Italy for a long time to come. But things would soon change dramatically.

Kurt Sundermeyer, one of the people who taught me about general relativity, brought my attention to an advert looking for an assistant editor at the Collected Papers of Albert Einstein, then located at Boston University. I quickly applied and, after being interviewed by the founding editor, John Stachel, got the position. The work I did for the edition turned out to be a revelation and deeply shaped my future career. Arriving in Boston in 1986, the first volume was already underway and included the early letters between Albert and his fiancé Mileva Marić.

einstein old lettersThis newly discovered source gave key insights into Einstein’s early intellectual biography, leading up to his “miraculous year” 1905. Together with Robert Schulmann I published a special edition of these letters for Princeton University Press. Working on the scientific annotation of these letters, I was very fortunate to work with and learn from my senior colleagues John Stachel, Robert Schulmann, and David Cassidy. Later I also profited from encounters with other Einstein experts such as Fabio Bevilacqua, Diana Buchwald, Jean Eisenstaedt, Peter Galison, Hubert Goenner, Gerald Holton, Don Howard, David Kaiser, Martin Klein, Anne Kox, John Norton, Karin Reich, David Rowe, Robert Rynasiewicz, and many others, some of whom have meanwhile become close friends. John Stachel played a pivotal role in launching Einstein studies as a field of collaboration among physicists, historians, and philosophers of science and has always been my mentor in this field. He also pioneered broad-ranging studies in the history of general relativity, a field that I soon made my own, working in close collaboration with talented younger colleagues, in particular, Michel Janssen, Tilman Sauer, and Matthias Schemmel. Eventually, Michel, Tilman, the two Johns, several other younger colleagues, and I formed the team that would produce a four-volume study on The Genesis of General Relativity, published with Springer in 2007. But this is getting ahead of things.

In the late 1980s, commuting between Boston and Berlin, I also collaborated closely with the exceptional science historian and native Berliner Peter Damerow, who was always a great source of inspiration for my work. Together with an Italian colleague, Paolo Galluzzi, Peter and I developed a vision to create an electronic Galileo-Einstein Archive which would make all of Galileo’s and Einstein’s archival resources openly available in digital form. The idea was supported by the NSF and its program director Ron Overman, and we used the grant they subsequently awarded to explore our vision of an electronic archive in hypertext format. Like-minded colleagues all over the world were contacted, including the people who were just then creating the Web at CERN in Geneva. But our vision was evidently premature and the result was eventually limited to an electronic archive of Galileo’s manuscripts on mechanics. This was realized only after the foundation of the Max Planck Institute for the History of Science where I became a director in 1994. It took the persistence and courage of the current director of the Collected Papers of Albert Einstein, Diana Buchwald, supported by Prineinstein old letterceton University Press, to eventually realize over twenty years later the vision of a freely accessible Digital Einstein Archive.

The Genesis of General Relativity was the first major collaborative research project of the newly founded Max Planck Institute for the History of Science. Today, the collaboration endures as the new image to emerge from this study of Einstein’s most important achievement continues to be developed. The project has also been expanded by the work of younger colleagues at the institute such as Alex Blum and Roberto Lalli. Together with one of the founders of the project, Michel Janssen, another colleague, Christoph Lehner, recently published the Cambridge Companion to Einstein. The research undertaken in this field is not confined to the intellectual dimension of Einstein’s work, however, but also extends to the cultural and political contexts, as is illustrated by Milena Wazeck’s study Einstein’s Opponents, or by Giuseppe Castagnetti’s and Hubert Goenner’s studies on the institutional contexts. I plan to bring some of these perspectives together in my forthcoming book on Einstein, entitled On the Shoulders of Giants and Dwarfs.

Einstein’s engagement as a missionary and popularizer of science has made a deep impression on me and it is in this spirit that my collaborators and I became involved in the Einstein Year 2005, when the centenary of Einstein’s miraculous year was celebrated. The centerpiece of this celebration in Germany was the extensive exhibition “Albert Einstein — Chief Engineer of the Universe,” an online presentation of which can still be seen today.

relativity 100 yearsWorking with other scholars on Einstein’s life and work continues to be a great source of inspiration for me. I am particularly grateful for the friendships that have developed from my various collaborations. One striking example is my friendship with Hanoch Gutfreund, a great scholar, an interminable source of energy, and a wonderful human being. With Hanoch, I recently wrote two books for Princeton University Press, The Road to Relativity and Relativity: The Special and the General Theory. 100th Anniversary Edition. In preparing these books, we developed a common style of popularization without compromising on scientific rigor. Having met late in life, we are all the more determined to write many more books together.

Jürgen Renn is a director at the Max Planck Institute for the History of Science in Berlin. His books include The Road to Relativity.

Einstein graphic courtesy of the Albert Einstein Facebook page.

#ThanksEinstein: Katherine Freese on how relativity rejuvenated her career

Thanks Einstein Meme 3Under the Spell of Relativity

By Katherine Freese

Katherine Freese is director of Nordita, the Nordic Institute for Theoretical Physics, in Stockholm, and author of The Cosmic Cocktail, which tells of the epic quest to solve one of the most compelling enigmas of modern science—what is the universe made of? This is the story of how one of today’s foremost pioneers in the study of dark matter came back from the brink of burnout because of Relativity.

My career choice was hugely influenced by the work of Albert Einstein. I chose a career in physics precisely because I was inspired by his theories of relativity. My first exposure to physics was at Exeter Summer School in New Hampshire when I was fifteen years old. I went there after my junior year in high school because, frankly, I enjoyed learning and would otherwise have been bored over the long summer. I took an introductory course in physics and have to admit that, at first, I was a bit intimidated. But I got into it quickly and was gratified to discover that I did really well. The course was inspiring, and my teacher Mr. Dudley probably has no idea what an impact he had on me.

It was when the summer course turned to Special Relativity that I became really excited. What a bizarre and fascinating subject! To begin with, the idea that there is no absolute reference frame was an eye-opener. I later tried to explain this to friends, but they persisted in arguing that the Earth really does provide a special reference frame, world freeseat least for humans, so we should just compute everything from our own point of view.

Strange paradoxes arise when one makes one simple postulate, that the speed of light is the same in every reference frame. Two observers moving with relativistic speeds (relative to one another) measure completely different things. Clocks measure different times, and rulers measure different lengths. The shortest time is measured in the reference frame where the event takes place, and in every other frame time appears dilated. So an astronaut, who goes off into space and eventually returns, ages more slowly than the rest of us. There can be time travel! In the sense that the astronaut can come back to the Earth at an arbitrarily distant point in the future…if she can tolerate traveling at those speeds. Recently I met quite a few astronauts in Stockholm at the Congress of the Association of Space Explorers. They are amazing people. I was invited to give a 20 minute talk on “What we know about the Universe today.” A tall order in front of these folks. Can you guess what I talked about? Cosmology, beginning with Einstein’s relativity, of course.

These exciting things I learned when I was 15 made me determined to learn more physics, and I ended up majoring in physics in college. I went very young, at 16, and graduated with a bachelor’s degree in physics from Princeton University at the age of 20. It was really hard, I was burning out quickly, and at that point I wasn’t sure I wanted to continue. Chapter One of The Cosmic Cocktail, the book that was published by Princeton University Press just over a year ago, describes what happened next. I decided to take some time off from school. With my best friend, I went off to Tokyo to teach English and ended up serving drinks in bars for a giant salary. (I finally surpassed it a few years ago as a Full Professor.) A year and a half later, I went to Korea to renew my visa. While I was traveling around Pusang, my stomach, or so I thought, started to hurt. When I returned to Tokyo I was walking around doubled over with pain. Indeed it turned out to be appendicitis. I went to the Catholic Hospital, run by English nuns, and had my appendix removed.

While I was lying in the hospital bed, I read the only book I had brought with me, Spacetime Physics by Taylor and Wheeler. It is a book about Einstein’s special relativity. The book is beautifully written and only requires simple knowledge of forces, energy, and so on, and I loved it. The minute I got out of the hospital, I flew back to the US, reinvigorated by the desire to study physics. I contacted Columbia University, which had previously accepted me, and they let me in at a moment’s notice. I was lucky they did.

Einstein’s influence persisted. Two years into my graduate program at Columbia University, I went to Fermilab, the particle physics accelerator outside of Chicago, to work in experimental high energy physics. However, I also took a class in cosmology at the University of Chicago twice a week, out of curiosity. Plus, it took me into the city of Chicago. Fermilab is on a farm an hour west and has buffalo roaming around. The professor who taught the course, David Schramm, was a giant both physically and mentally, and one of the founders of the field of astroparticle physics, where the smallest particles explain the properties of the largest galaxies. We nicknamed him “Schrammbo.” (If you want to know more about him, you’ll have to read my book.) In that course, Einstein’s equations were applied to the Universe as a whole. Wow. I stopped showing up in the lab and instead sat in my housing at Fermilab and read about general relativity, this time at a graduate level framed by far deeper mathematics. Again, it was a turning point. I transferred to the University of Chicago to get my PhD with David Schramm in the field of cosmology.

In human history, every culture has had creation myths. In the past 100 years we have developed our own, the Big Bang. The difference is that the Hot Big Bang is right! The achievements over the past century in the field of cosmology are breakthroughs for all of mankind. We understand everything about our observable Universe all the way out to the farthest distant that light could have traveled to us in the age of the Universe (anything farther out could not have impacted us because the information could not travel in excess of the speed of light).

Now I’m a professional. I work with Einstein’s equations or their immediate consequences every day. I’m a theorist. I invent things and hope they turn out to match reality. All my work lies within the framework of modern cosmology, which began with Einstein’s work in relativity in 1915. What a brilliant man he was! Ever since I learned about relativity I’ve been under its spell, and I still am.

Katherine Freese is director of Nordita, the Nordic Institute for Theoretical Physics, in Stockholm, and professor of physics at the University of Michigan. She is the author of The Cosmic Cocktail.

#ThanksEinstein image courtesy of the official Albert Einstein Facebook page.

#ThanksEinstein: Hanoch Gutfreund on the revelation of relativity

Einstein meme 2The Revelation of Relativity

By Hanoch Gutfreund

Hanoch Gutfreund is professor emeritus of theoretical physics at the Hebrew University of Jerusalem, where he is also the academic director of the Albert Einstein Archives. This is the story about how Einstein’s General Theory of Relativity revolutionized his teaching, understanding, and career.

My present day interest in Einstein evolved late in my academic life. It started when as Rector and then President of the Hebrew University, in the 1990’s, I became aware of the unique cultural asset possessed by the university – the Albert Einstein Archives. When I stepped down from the presidency, with einstein lightthe encouragement of my successor, I began to devote more and more time to promote the Einstein – H.U. connection, through public lectures on various Einstein topics and by organizing and helping to organize Einstein exhibitions in different places in the world.

As professor of theoretical physics, for many years I taught everything that Einstein did in his miraculous year – 1905. However, only in the late nineties did I read the original papers with commentaries by John Stachel. For me this was a revelation. Einstein’s way of thinking, his motivations, his introductions and conclusions – all this was very different from the way these topics were treated in ordinary textbooks. I believe that if I had known and understood what I know and understand today, my students would have appreciated and benefited from my lectures even more. Motivated by this revelation, I decided to fill a gap in my own physics education. As a student, I never had a course in general relativity. In the learning process, the historical context and Einstein’s intellectual struggle were for me at least as important as the scientific results.

Teinstein speed of lighto mark the 50th anniversary of the Israeli Academy of Science, we displayed the most important manuscript in the Einstein Archives, the manuscript of Einstein’s seminal paper on general relativity. Each one of the 46 pages of this manuscript was enclosed in a dimly illuminated box. People visited this exhibit as if they were entering a shrine.

Following this experience, I met with Jurgen Renn, director of the Max Planck Institute for the History of Science. We discussed an option to publish this manuscript as part of a comprehensive account of Einstein’s intellectual odyssey to general relativity.

Gutfreund_RoadtoRelativityThis meeting led to a fruitful collaboration, which has now produced The Road to Relativity: The History and Meaning of Einstein’s The Foundation of General Relativity. It attempts to make the essence of general relativity accessible to broader audiences. We have also initiated the recently published, 100th anniversary edition of Einstein’s popular booklet on the special and general theory of relativity, with extensive background material and a reading companion, intended to resent Einstein’s text in a historical and modern context. We are already considering other Einsteinian projects in the future. This year, as the world marks the 100th anniversary of general relativity, there are many requests addressed to the Albert Einstein Archives and to myself for assistance in organizing special exhibitions, for participation in scientific conferences and in public events, for interviews in the media and for help and advice in various other initiatives. It’s an exciting time, and I remain very grateful for this inspiring phase in my life.

Hanoch Gutfreund is professor emeritus of theoretical physics at the Hebrew University of Jerusalem, where he is also the academic director of the Albert Einstein Archives.

Check out the earlier post in this series by Jimena Canales.

#ThanksEinstein image courtesy of the official Albert Einstein Facebook page.

#ThanksEinstein: Jimena Canales on the ideal figure of Einstein

Thanks Einstein meme 2

Me, Myself and Einstein

By Jimena Canales

Jimena Canales is the author of The Physicist and the Philosopher, which tells the remarkable story of how an explosive debate between two intellectual giants transformed our understanding of time and drove a rift between science and the humanities that persists today. This is the story of how she came to study the iconic physicist when she initially had no interest in “such a great man, or any great men.”

I arrived at Einstein after following a winding, circuitous road. Like so many others, I was acquainted with his life and works since my college years. I majored in Engineering Physics, taking the required relativity lessons as part of my Modern Physics courses. Like so many others, I struggled to understand the philosophical significance of the theory’s paradoxes (particularly those pertaining to simultaneity and time and length dilation). Comprehending that was a lot harder than the comparatively simple number-crunching that led me to the right answers in the final exam.

But as a historian of science, I was initially not interested in such a great man or in any great men, for that matter. In fact, early on in my career I was more interested in understanding broader social and historical transformations than those that could ever be brought about by single individuals, no matter how brilliant they were. No one person created modernity, and I was interested in understanding how it came about.

Yet once I came to know Einstein, my career and my views about the history of science changed dramatically. It all started when I found the transcript of a meeting at the Société française de philosophie that took place in April 6, 1922. I had been looking for that particular document because I was trying to find out everything I could about Henri Bergson, one of the most important philosophers of the century widely renowned for his insightful views about time. Bergson’s conception of time had been more famous than Einstein’s — it was only natural that I would focus on him. But what I found out upon reading the transcript shocked me beyond belief. I read about Henri Bergson debating, face-to-face, with Albert Einstein himself.

I tried to read as much as I could about this day, and—surprisingly—found almost nothing in the existing Einstein scholarship. But it seemed to me that everywhere else I looked I found references to that particular meeting. What is more, I found an astounding number of prominent intellectuals and scientists all discussing Einstein and Bergson together, and asking which of the two men was correct when it came to the prickly question of time.

Given my interest in Bergson, I was invariably led to study Einstein. I had thought that everything interesting about Einstein must have already been worked on to death by historians. I thought I would have nothing new to contribute. But the Einstein I was starting to get to know was quite different from the one I had read about.

I was at first a reluctant Einstein scholar, but as I read more and more I was hooked. My encounter with his work affected me in the most unexpected and wonderful ways imaginable. I was reminded how in even the most treaded upon topics there remain elements of surprise. Truth be told, the Einstein I got to know through my sources was not the mythical figure we all know—he is of this world. In my book, we find him saving a small piece of soap to give to his wife because he cannot afford to buy it; we find him desperately trying to combat the objections of Bergson to defend his theory; we see him speaking on the radio and reenacting for television some of the most important moments of his life; and, finally, we encounter him reflecting about Bergson in melancholic and personal letters written to his best friend in the years before his death. By reading his private correspondence, I got to know his sense of humor as much as his callousness; his sectarianism as much as his noble internationalist ideals; his pacifism as much as his uncompromising politics (I was surprised, for example, to see him take such a strong stance against the League of Nations). I was able to see his brilliance as well as his limitations (in almost everything that involved knowing the language and culture of France).

Einstein slowly appeared to me as much more than a great man—he became an ideal figure through which we could explore broader questions, such as the division between the science and the humanities, the role of expert knowledge versus lay wisdom, the relation of science to the media and to other areas of culture (including art). We could learn how cosmological and universal conceptions of time (in theory) are related to our use of time (in practice). We could explore processes that lead to the formation of our own subjectivities and psychological make-up and, most importantly, analyze our ever-changing place amongst things, machines and new technologies.

Years after my first scholarly encounter with Einstein, I cannot but be more than grateful to that great physicist, who a century after creating his General Theory of Relativity, continues to teach us so much.

Jimena Canales holds the Thomas M. Siebel Chair in the History of Science at the University of Illinois, Urbana-Champaign, and was previously associate professor of the history of science at Harvard University. She is the author of The Physicist and the Philosopher, and A Tenth of a Second: A History.

#ThanksEinstein image courtesy of the official Albert Einstein Facebook page.

PUP congratulates writers chosen for The Best Writing on Mathematics 2015

Highlighting the finest articles published throughout the entire year, The Best Writing on Mathematics 2015 shines the spotlight on math’s brightest, most creative minds. Edited by Mircea Pitici, the volume is inviting to experienced mathematicians and numbers novices alike.

The Best Writing on Mathematics, in its sixth edition, offers surprising and meaningful insights and perspectives into the highly influential world of mathematics. Colm Mulcahy and Dana Richards express their appreciation and reflections of the significant work of icon Martin Gardner, Toby Walsh creatively uses the popular game Candy Crush as a vehicle to analyze the hardships of solving computational problems, Benoît Rittaud and Albrecht Heeffer investigate and question the true derivation of the pigeonhole principle, Carlo Cellucci considers and defines beauty in mathematics — and that’s just the beginning.

Best Writing on Math 2015

Congratulations to those chosen to be included in The Best Writing in Mathematics 2015!

Interpreting mathematics is not about mathematical truth (or any other truth); it is a personal take on mathematical facts, and in that it can be true or untrue, or it can even be fiction; it is vision, or it is rigorous reasoning, or it is pure speculation, all occasioned by mathematics; it is imagination on a mathematical theme; it goes back several millennia and it is flourishing today, as I hope this series of books lays clear, (xiii)

— Mircea Pitici, Editor


Articles and authors selected in The Best Writing on Mathematics 2015

Articles Authors
A Dusty Discipline Michael J. Barany and Donald MacKenzie
How Puzzles Made Us Human Pradeep Mutalik
Let the Games Continue Colm Mulcahy and Dana Richards
Challenging Magic Squares for Magicians Arthur T. Benjamin and Ethan J. Brown
Candy Crush’s Puzzling Mathematics Toby Walsh
Chaos on the Billiard Table Marianne Freiberger
Juggling with Numbers Erik R. Tou
The Quest for Randomness Scott Aaronson
Synthetic Biology, Real Mathematics Dana Mackenzie
At the Far Ends of a New Universal Law Natalie Wolchover
Twisted Math and Beautiful Geometry Eli Maor and Eugen Jost
Kenichi Miura’s Water Wheel, or The Dance of the Shapes of Constant Width Burkard Polster
Dürer: Disguise, Distance, Disagreements, and Diagonals! Annalisa Crannell, Marc Frantz, and Fumiko Futamura
The Quaternion Group as a Symmetry Group Vi Hart and Henry Segerman
The Steiner-Lehmus Angle Bisector Theorem John Conway and Alex Ryba
Key Ideas and Memorability in Proof Gila Hanna and John Mason
The Future of High School Mathematics Jim Fey, Sol Garfunkel, Diane Briars, Andy Isaacs, Henry Pollak, Eric Robinson, Richard Scheaffer, Alan Schoenfeld, Cathy Seeley, Dan Teague, and Zalman Usiskin
Demystifying the Math Myth: Analyzing the Contributing Factors for the Achievement Gap between Chinese and U.S. Students Guili Zhang and Miguel A. Padilla
The Pigeonhole Principle, Two Centuries before Dirichlet Benoît Rittaud and Albrecht Heeffer
A Prehistory of Nim Lisa Rougetet
Gödel, Gentzen, Goodstein: The Magic Sound of a G-String Jan von Plato
Global and Local James Franklin
Mathematical Beauty, Understanding, and Discovery Carlo Cellucci
A Guide for the Perplexed: What Mathematicians Need to Know to Understand Philosophers of Mathematics Mark Balaguer
Writing about Math for the Perplexed and the Traumatized Steven Strogatz
Is Big Data Enough? A Reflection on the Changing Role of Mathematics in Applications Domenico Napoletani, Marco Panza, and Daniele C. Struppa
The Statistical Crisis in Science Andrew Gelman and Eric Loken
Statistics and the Ontario Lottery Retailer Scandal Jeffrey S. Rosenthal
Never Say Never David J. Hand

Mircea Pitici holds a PhD in mathematics education from Cornell University, where he teaches math and writing. He has edited The Best Writing on Mathematics since 2010.

Feynman on the historic debate between Einstein & Bohr

The golden age of quantum theory put many of the greatest minds of the 20th century in contact with some of the most significant scientific and philosophical questions of their era. But it also put these minds in contact with one another in ways that have themselves been a source of curiosity and ongoing scientific debate.

Richard Feynman and Albert Einstein, two towering geniuses of their time, were both as revered for their scientific contributions as they were beloved for their bursts of wisdom on a wide range of subjects. It’s hard not to wonder just what these men thought of one another. Princeton University Press, which published The Ultimate Quotable Einstein in 2010 publishes The Quotable Feynman this fall. The book includes reflections by Feynman on Einstein, from his memorable mannerisms to his contributions to some of the most heated debates in 20th century science.Feynman quote

Perhaps because of the gap between their career high points, (Einstein died in 1955; Feynman didn’t receive his Nobel Prize until 1965), there are no verified quotes where Einstein alludes to Feynman or his expansive body of work. But Feynman had made observations on the older physicist, several of which revolve around Einstein’s famous 1927 public debate with Niels Bohr on the correctness of  quantum mechanics. Central to the debate was this question: Were electrons, light, and similar entities waves or particles? In some experiments they behaved like the former, and in others, the latter.

In an attempt to resolve the contradictory observations, Einstein proposed a series of “thought experiments”, which Bohr responded to. Bohr essentially took the stance that the very act of measuring alters reality, whereas Einstein insisted that reality exists, independent of the act of measurement. Key to the philosophy of science, the dispute between the two giants is detailed by Bohr in “Discussions with Einstein on Epistemological Problems in Atomic Physics”. Richard Feynman is quoted as commenting on the debate:Feynman quote 2

An Einstein Encyclopedia contains a section on the Einstein-Bohr debates, as well as a wealth of other information on Einstein’s career, family, friends. There is an entire section dedicated to righting the various misconceptions that swirl around the man, and another on his romantic interests (actual, probable, and possible).

In spite of their differences, Bohr and Einstein were friends and shared great respect for each others’ work. Until Einstein’s death 3 decades later, they continued their debates, which became, in essence, a debate about the nature of reality itself.  feynman quote 3

Check out other new Einstein publications this fall, including:

An Einstein Encyclopedia
The Road to Relativity

Introducing the new video trailer for The Quotable Feynman

Nobel Prize-winning physicist Richard P. Feynman (1918–88) is widely known for his scientific genius. But during his life, he became as famous for the wit, wisdom, and lucidity of his popular lectures and writings as for his fundamental contributions to physics. We are pleased to present the new video trailer for The Quotable Feynman, including approximately 500 quotations carefully selected by his daughter, Michelle Feynman, from his spoken and written legacy:

Check out chapter one here.





An interview with Nicholas Higham on The Princeton Companion to Applied Mathematics

Higham jacket

We are excited to be running a series of posts on applied mathematics by Nicholas Higham over the next few weeks. Higham is editor of The Princeton Companion to Applied Mathematics, new this month. Recently he took the time to answer some questions about the book, and where the field is headed. Read his popular first post on color in mathematics here.

What is Applied Mathematics?

NH: There isn’t any generally agreed definition, but I rather like Lord Rayleigh’s comment that applied mathematics is about using mathematics to solve real-world problems “neither seeking nor avoiding mathematical difficulties”. This means that in applied mathematics we don’t go out of our way to consider special cases that will never arise in practice, but equally we do not sidestep genuine difficulties.

What is the purpose of The Companion?

NH: The Companion is intended to give an overview of the main areas of applied mathematics, to give examples of particular problems and connections with other areas, and to explain what applied mathematicians do—which includes writing, teaching, and promoting mathematics as well as studying the subject. The coverage of the book is not meant to be exhaustive, but it is certainly very broad and I hope that everyone from undergraduate students and mathematically interested lay readers to professionals in mathematics and related subjects will find it useful.

What is an example of something aspect of applied mathematics that you’ve learned while editing the book?

NH: Applied mathematics is a big subject and so there are many articles on topics outside my particular areas of expertise. A good example concerns applications of computational fluid dynamics in sport. An article by Nicola Parolini and Alfio Quarteroni describes the mathematical modeling of yachts for the America’s cup. The designer wishes to minimize water resistance on the hull and maximize the thrust produced by the sails. Numerical computations allow designs to be simulated without building and testing them. The article also describes mathematical modeling of the hi-tech swimsuits that are now banned from competition. The model enables the benefit of the suits on race times to be estimated.

The Companion is about 1000 pages. How would advise people to read the book.

NH: The book has a logical structure, with eight parts ranging from introductory material in Part I, the main areas of applied mathematics in Part IV (the longest part), through to broader essays in the final part. It is a good idea to start by reading some of the articles in Part I, especially if you are less familiar with the subject. But a perfectly sensible alternative approach is to select articles of interest from the table of contents, read them, and follow cross-references. Or, you can just choose a random article and start reading—or simply follow interesting index entries! We worked very hard on the cross-references and index so an unstructured approach to reading should lead you around the book and allow you to discover a lot of relevant material.

What was the hardest thing about editing The Companion?

NH: The hardest aspect of the project was ensuring that it was completed in a reasonable time-frame. With 165 authors it’s hard to keep track of everything and to to ensure that drafts, revisions, and corrected proofs are delivered on time.

How much of the book did you write?

NH: I wrote about 100 of the 1000 pages. This was great fun, but it was some of the hardest writing I’ve done. The reason is partly that I was sometimes writing about topics that I don’t normally write about. But it was also because Companion articles are quite different from the papers I’m used to writing: they should have a minimal number of equations and formal statements of theorems, lots of diagrams and illustrations, and no citations (just Further Reading at the end of the article).

How did you choose the cover?

NH: We considered many different ideas. But after a lot of thought we settled on the motor boat picture, which captures the important topics of fluid mechanics, waves, and ocean, all of which are covered in the book in a number of articles.

What is the most unexpected article?

NH: Perhaps the article Mediated Mathematics: Representations of Mathematics in Popular Culture and Why These Matter by sociologist of education Heather Mendick. She discusses the way mathematics is represented in numerous TV shows and films.

What would you be doing if you hadn’t become a mathematician?

NH: I’d be playing a Hammond B3 organ in a jazz or blues band. I’m a keen musician and played keyboards semi-professionally for many years, starting in my teens.

How did you go about organizing the book?

NH: I recruited five Associate Editors with expertise in different areas and we met and planned out the eight parts of the book and the articles, along with a list of authors to invite. We looked for authors who are leading international experts in their field and are at the same time excellent expositors. Signing up the 165 authors was quite a long process. We were able to find authors for almost every article, so just a very small number had to be dropped. In some cases the authors suggested changes of content or emphasis that we were happy to agree with.

What range of readers is The Companion aimed at?

NH: The target audience for The Companion is very broad. It includes mathematicians at undergraduate level or above, students, researchers, and professionals in other subjects who use mathematics, and mathematically interested lay readers. Some articles will also be accessible to students studying mathematics at pre-university level.

Why not just seek information online? Why is there a need for a book?

NH: When Princeton University Press asked me to edit The Companion they told me that reference books still have great value. Many people have trouble navigating the vast amount of information available online and so the need for carefully curated thematic reference works, written by high calibre authors, is as great as ever. So PUP’s experience is that print is definitely not dead, and indeed my own experience is that I have many books in PDF form on my computer, but if I want to read them seriously I use a hard copy.

How have you ensured that the book will not go out of date quickly?

NH: This was a major consideration. This was a five and a half year project and we wanted to make sure that the book will still be relevant 10, 20, or 50 years from now. To do that we were careful to choose articles on topics that have proven long-term value and are not likely to be of short-term interest. This is not to say that we don’t cover some relatively new, hot topics. For example, there are articles on compressed sensing (recovering sparse, high-dimensional data from a small number of indirect measurements) and on cloaking (hiding an object from an observer who is using electromagnetic, or other, forms of imaging, as in Harry Potter or Romulan space ships in Star Trek), both of which are areas that have grown tremendously in the last decade.

What sort of overview of applied mathematics does the book give?

NH: Applied mathematics is a huge subject, so we cannot cover everything in 1000 pages. We have tried to include the main areas of research as well as key underlying concepts, key equations, function and laws, as well as lots of example of applied mathematics problems. The examples range from the flight of a golf ball, to robotics, to ranking web pages. We also cover the use of applied mathematics in other disciplines such as engineering, biology, computer science, and physics. Indeed the book also has a significant mathematical physics component.

Where is the field going?

NH: Prior to the 20th century, applied mathematics was driven by problems in astronomy and mechanics. In the 20th century physics became the main driver, with other areas such as biology, chemistry, economics, engineering, and medicine also providing many challenging mathematical problems from the 1950s onwards. With the massive and still growing amounts of data available to us in today’s digital society information, in its many guises, will be an increasingly important influence on applied mathematics in the 21st century.

To what extent does The Companion discuss the history of applied mathematics?

NH: We have an excellent 25-page article in Part I titled The History of Applied Mathematics by historians of mathematics June Barrow-Green and Reinhard Siegmund-Schultze. Many articles contain historical information and anecdotes. So while The Companion looks to the future it also gives an appreciation of the history of the subject.

How do you see the connections between applied mathematics and other disciplines developing?

NH: Applied mathematics is becoming ever more interdisciplinary. Many articles in The Companion illustrate this. For example,

  • various facets of imaging feature in several articles, including those on compressed sensing, medical imaging, radar, and airport baggage screening,
  • the article on max-plus algebras shows how what may seem like an esoteric area of pure mathematics has applications to all kinds of scheduling processes,
  • the article on the spread of infectious diseases shows the value of mathematical models in epidemiology,
  • several articles show how mathematics can be used to help understand the earth’s weather and climate, focusing on topics such as weather prediction, tsunamis, and sea ice.

What are you thoughts on the role of computing in applied mathematics?

NH: Computation has been a growing aspect of applied mathematics ever since the first stored program computer was invented here in Manchester. More and more it is the case that numerical computations and simulations are used in tandem with, or even in place of, the classical analysis that relies just on pen and paper. What I find particularly interesting is that while the needs of mathematics and of science in general have, naturally, influenced the development of computers and programming languages, there have been influences in the other direction. For example, the notation for the ceiling and floor functions that map a real number to the next larger or smaller integer, respectively, was first introduced in the programming language APL.

Of course numerical computations are expressed in terms of algorithms, and algorithms are ubiquitous in applied mathematics, and strongly represented in the book.

Do you have any views on ensuring the correctness of work in applied mathematics?

NH: As the problems we solve become every more complicated, and the computations we perform run for longer and longer, questions about the correctness of our results become more important. Applied mathematicians have always been good at estimating answers, perhaps by an asymptotic analysis, so we usually know roughly what the answer should look like and we may be able to spot gross errors. Several particular aspects of treating correctness are covered in The Companion.

Uncertainty quantification is about understanding how uncertainties in the data of a problem affect the solution. It’s particularly important because often we don’t know the problem data exactly—for example, in analyzing groundwater flow we don’t know the exact structure of what lies under the ground and so have to make statistical assumptions, and we want to know how these impact the computed flows.

A different aspect of correctness concerns the reproducibility of our computations and treats issues such as whether another scientist can reproduce our results and whether a computation on a high-performance computer will produce exactly the same answer when the computation is repeated.

All of these issues are covered in multiple articles in the book.

Nicholas J. Higham is the Richardson Professor of Applied Mathematics at The University of Manchester. Mark R. Dennis is professor of theoretical physics at the University of Bristol. Paul Glendinning is professor of applied mathematics at The University of Manchester. Paul A. Martin is professor of applied mathematics at the Colorado School of Mines. Fadil Santosa is professor of mathematics at the University of Minnesota. Jared Tanner is professor of the mathematics of information at the University of Oxford.