Jonathan Borwein and Keith Devlin are well-known mathematicians who have a strong appreciation of, and expertise in, experimental mathematics. In this book they provide us with a concise, inviting introduction to the field.

The first chapter tries to succinctly explain what experimental mathematics is and why it’s a fundamental tool for the modern mathematician. The following is their definition:

Experimental mathematics is the use of a computer to run computations—sometimes no more than trial-and-error tests—to look for patterns, to identify particular numbers and sequences, to gather evidence in support of specific mathematical assertions that may themselves arise by computational means, including search. Like contemporary chemists—and before them the alchemists of old—who mix various substances together in a crucible and heat them to a high temperature to see what happens, today’s experimental mathematician puts a hopefully potent mix of numbers, formulas, and algorithms into a computer in the hope that something of interest emerges.

They immediately address some of the possible objections and illustrate how an approach that doesn’t focus on formal proof, but rather on exploration and experimentation, ultimately leads to hypotheses which can then be, in many cases, proved analytically. The authors argue that in this sense, thanks to the aid of advanced computers, mathematics is becoming more and more similar to other natural sciences.

They also make a case for how great mathematicians like Euler, Gauss, and Reimann were doing experimental mathematics well before calculators where available. Their calculations on paper were far more limited than what computers afford us these days, yet they served them well when it came to sharpening and verifying their intuitions.

The rest of the book is a continuous series of examples that show the advantages of this approach in practice. The examples are highly interesting (some of them stunning) and tend to focus on calculus, analysis and analytical number theory.

Each chapter is accompanied by a section called “Explorations”. I found this section to be particularly valuable. Within it you’ll find exercises, and further examples and considerations. The answers/solutions to the actual problems are provided in the second to last chapter, just before the brief epilogue.

Chapter 2 discusses how to calculate an arbitrary digit for irrational numbers like , in certain bases. They illustrate how the so called BBP Formula (Bailey-Borwein-Plouffe formula, co-discovered by Jonathan Borwein’s brother) came to be.

The use of a program which implements the PSQL integer relation algorithm in high-precision, floating-point arithmetic was key to its discovery. The BBP Formula in turn allowed the calculation of the quadrillionth binary digit of back in 2000.

Chapter 3 focuses on identifying numbers, digits patterns, and sequences once you obtain a numeric result through your calculations and experimentation. They introduce the subject with relatively obvious values like the approximations of or , but the chapter quickly escalates to an example where a closed form for a seemingly random sequence needs to be found.

Chapter 4 analyzes the Reimann Zeta function from the eyes of an experimental mathematician, and shows us what kind of insight we can gain from this unique perspective.

In chapter 5 we learn how by numerically evaluating definite integrals, it is sometimes possible to identify the resulting value which will help us to analytically resolve those particular integrals. The examples presented in this chapter originate for the most part from physics and are very challenging if attempted without the aid of experimental methods. To better grasp the kind of integrals discussed in this chapter, here is an example:

The explorations section provides a few more interesting integrals, including some for which a closed form is not known. The authors even include an integral that intentionally stumps Mathematica 6 and Maple 11.

Chapter 6 is dedicated to serendipitous discoveries (“proof by serendipity”) with a few interesting examples of how “luck” met preparation, ultimately enriching the body of mathematical knowledge almost by chance.

In chapter 7 the authors go back to talk about , this time in base 10, to calculate its digits with efficient, fast converging formulas and methods. The chapter wraps up with a discussion about the normality of , which hasn’t been proved of course, but appears to be empirically supported by the statistical analysis of the first trillion digits. In the explorations section there is a nice discussion about the implementation of fast arithmetic through the Karatsuba multiplication, and the subject of Montecarlo simulations (a very inefficient method of calculating , but a great way to show the idea behind Montecarlo simulations).

Chapter 8 has a bold title, “The computer knows more math than you do”. This provocative title is quickly diminished to put it in context though. The authors start by approaching a tough problem posed by Donald Knuth (of TeX and The Art of Computer Programming fame) to the readers of the American Mathematical Monthly:

In an attempt to solve this the authors invite us to go on a journey involving the Lambert W function, the Pochhammer function, and Abel’s limit theorem. The rest of the chapter illustrates another difficult problem whose solution obtained through the aid of Maple has important implications not only for mathematics, but also for quantum field theory and statistical mechanics.

In chapter 9 a few infinite series are calculated in order to show how CAS systems and experimental methodology can still be useful when dealing with problems that involve infinite sequences, series, and products.

Chapter 10 is dedicated to the limits and the dangers of this approach. Several examples showcase how one can be misled into making assumptions, and how to avoid this from happening. The ad hoc example below is correct to over half a billion digits:

After having calculated a few hundred digits, it would be natural to assume that the series converges to a natural number, when in reality it’s an irrational and transcendental number.

In chapter 11, conscious of the selective focus on analysis and analytical number theory throughout the book, Borwein and Devlin introduce other examples such as a topology problem whose proof was reached thanks to a deeper insight gained through computer visualization of a surface, a knot theory problem, the Four Color Theorem, the Robbins Conjecture, the computation of , and so on.

In truth, I feel that such a thin book could have used more examples like the ones in chapter 11, in order to make a stronger case for the applicability of experimental mathematics to areas outside of analysis.

The book is well written and the tone is never heavy, despite the advanced mathematical examples within it. The authors include historical background and anecdotes which makes for a more interesting read and provides a human perspective behind the formulas presented. The (at times) funny illustrations and occasional jokes are definitely a pleasant addition.

This book is relatively tool agnostic; Maple and Mathematica are referenced throughout, and so are a few online tools to identify number sequences and known numeric values. Overall though, the emphasis in on the methodology rather than a particular CAS (Computer Algebra System) or programming language. In fact, with the exception of a snippet of Maple code in one of the explorations in the first chapter, the book describe the examples from a mathematical and algorithmic standpoint. You won’t find source code for the examples illustrated.

The ideal target audience for The Computer as Crucible is graduate students and researchers. A bright, motivated high-school student will get the gist of this book, but a more mature mathematical audience will actually be able to follow the steps within the examples and fully appreciate the insight on how an experimental approach can aid their research.

Despite the numerous examples employed to make their case, the authors start the book by explaining that it is not intended to be comprehensive. It’s meant to be thought provoking and to whet your appetite as to what is now possible in mathematical research thanks to computers.

As a computer programmer who’s passionate about mathematics, experimental mathematics fascinates me greatly. As such, I hope to work my way through the actual textbooks that are generally suggested as a follow up to this book. Namely, I’ve already started reading Mathematics by Experiment: Plausible Reasoning in the 21st Century (Second Edition), which is co-authored by Jonathan Borwein himself. Other textbooks referenced in this introduction are Experimental Mathematics in Action and Experimentation in Mathematics: Computational Paths to Discovery.

In conclusion, The Computer as Crucible is a lovely little book which builds a strong case for experimental mathematics. Any practicing mathematician or serious amateur should consider checking out this introduction to a topic that will no doubt transform mathematics.

Full disclosure: We received this book for free from the publisher, but we’re under no obligation to review or endorse it. We routinely receive a fair number of books from several publishers that never make the cut for an actual review. The links have our Amazon referral id which gives us a tiny percentage if you buy a book. In turn this helps support this site.

The list is not exhaustive by any stretch, capping at a total of 52 books so far, but it covers recommendations for 20ish different areas of mathematics. The focus is on outstanding introductory texts, with a few must-have math references thrown in here and there. We’ve briefly reviewed each of these math books in order to provide you with a general idea of its style and content.

Of course, not everyone will agree with the way we opted to divide the field of mathematics, nor will they necessarily agree with every selection we made. But it’s impossible to please everyone or be exhaustive in every way. We are confident that most readers will agree with us and recognize at least a few classic and well regarded books.

A lot of work went into the making of this page, as such we hope that it will serve you well, help you discover new and interesting titles, as well as act as a wish list of sorts.

Finally, this list of book reviews has been published with my name of it, but it was a collaborative effort. A special thanks goes to my friends Jonathan, Mary, Marco, Paolo, Alex and David for their research and review contributions.

May this list inspire and further guide you in the pursuit of mathematics.

The reasonable man adapts himself to the world; the unreasonable one persists in trying to adapt the world to himself. Therefore all progress depends on the unreasonable man. — George Bernard Shaw (attributed)

On November 11, 2002, Grigory Perelman, a Russian mathematician known to his friends as “Grisha”, posted a research paper to the www.arXiv.org preprint server containing, amongst other things, the outline of a proof of the Poincaré Conjecture, a famous conjecture in topology first articulated in 1904 by the great mathematician Henri Poincaré. Dr. Perelman also e-mailed a few selected mathematicians directly, drawing attention to his somewhat curious paper. This rapidly created a stir as the mathematicians realized that he might well have proven the Poincaré Conjecture, an extremely difficult problem that had eluded the talents of many top mathematicians including Poincaré. Perelman went on to post two more papers to arXiv.org elaborating his proof. The Clay Institute, which had offered a prize of $1 million for the proof (or disproof) of the Poincaré Conjecture, funded two teams of mathematicians to verify Perelman’s proof. The National Science Foundation also funded efforts to verify and expand upon the proof. By 2006, the “consensus” in the mathematical community was that Dr. Perelman had proved the Poincaré Conjecture. Dr. Perelman was offered the prestigious Fields Medal, close to the Nobel Prize of mathematics. He became the first mathematician to decline the Fields for reasons that remain somewhat unclear.

Two recent books attempt to tell the story of Grigory Perelman and the Poincaré Conjecture. Masha Gessen’s Perfect Rigor is the first biography of the elusive and enigmatic Perelman. It gives a great deal of information about the world of Soviet mathematics in which Perelman grew up and Perelman’s life to date. The author was unable to interview Perelman who has declined nearly all interviews; he has given an interview to Sylvia Nasar and David Gruber for their New Yorker article “Manifold Destiny“, about which more later. The book suffers from an unremittingly hostile, perhaps jealous, view of the unusual Dr. Perelman, who is variously portrayed as extremely naive, weird, and possibly mentally ill.

Dr. Perelman’s father was an electrical engineer and his mother a mathematics teacher at a Soviet trade school. His mother apparently had a strong interest in mathematics and almost pursued a doctorate before marrying his father. Perelman appears to have been involved in mathematics at an early age and joined a competitive math club. He competed and won a gold medal at the International Math Olympiad in Budapest, Hungary in 1982 at the age of 16. He attended a special math and physics school, Leningrad Secondary School #239, usually identified as “School 239″ in Perfect Rigor. He then became a student at Leningrad State University. In 1987, he became a graduate student at the Leningrad (subsequently the St. Petersburg) branch of the Steklov Mathematical Institute, the mathematics division of the Soviet (now Russian) Academy of Sciences. The mathematician Yuri Burago was his adviser. Perelman defended his dissertation in 1990. He continued to work at the Steklov Institute until 1992, publishing a number of papers in Russian and American mathematical journals.

In the fall of 1992, Perelman came to the United States for a semester at the Courant Institute at New York University and then another semester at the State University of New York Stony Brook in early 1993. At New York University, he met and may have become friends with the mathematician Gang Tian. Perelman and Gang Tian traveled together from NYU to the Institute for Advanced Study at Princeton to listen to mathematics lectures. Then, Perelman became a prestigious Miller Fellow at Berkeley. During this time he proved the Soul Conjecture, a difficult problem in topology. His Miller Fellowship ended in 1995. He received several job offers from a number of top universities. However, he wanted a tenured position. His job offers appear to have been untenured, tenure-track positions. He returned to Russia and the Steklov Institute in 1995 where he was part of the Mathematical Physics group, dropping almost entirely out of sight, publishing nothing. He appears to have spent the next seven years working on the Poincaré conjecture. In 2002, he stunned the mathematical world by posting his proof to the Internet, flouting tradition by declining to submit the proof to a peer reviewed mathematics journal. The Clay Institute would fund mathematicians John Morgan and Gang Tian (Perelman’s friend or acquaintance at NYU) as well as a separate team at the University of Michigan to verify Perelman’s work in the form of a peer reviewed academic book.

In 2006, the prominent mathematician Shing-Tung Yau and two of his former students argued that Perelman had published an incomplete proof which they “fixed” in a lengthy paper published in the Asian Journal of Mathematics. At this point the elusive Dr. Perelman appears to have struck back with a vengeance, possibly exhibiting something other than the naivete imputed in Pefect Rigor. Perelman granted a rare interview to Sylvia Nasar, best known as author of A Beautiful Mind about the mathematician John Forbes Nash, and David Gruber for an article in the New Yorker magazine, “Manifold Destiny,” which all but openly accused Yau and his former students of blatant plagiarism.

The article quotes Perelman attributing his decision to decline the Fields medal and withdraw from the mathematics profession to the low ethical standards of the profession (in his opinion). The article also discusses the alleged rivalry between Yau and his former student Gang Tian, Perelman’s acquaintance from NYU and co-author with John Morgan of the book on Perelman’s proof. Yau threatened legal action against the New Yorker which stood by its story. Yau soon appears to have retreated under a storm of negative publicity and criticism within the mathematics “community”.

By most accounts, Perelman is an unusual person. He left his job at the Steklov Institute and apparently resides with his aging mother in her apartment in St. Petersburg. He has reportedly indicated that he is no longer interested in mathematics and generally refuses interviews, prizes, and so forth. It is not unlikely that many prominent research universities and institutions would fall over themselves to offer him a tenured professorship or something similar if he expressed any interest. It remains to be seen whether he will decline the Clay Institute’s $1 million prize if offered. Without knowing more about Perelman and his adventures in mathematics than can be found in Perfect Rigor or other accounts to date, it is difficult to draw firm conclusions about the man or even his discovery.

Notwithstanding, a few thoughts come to mind. Perfect Rigor and some other accounts implicitly criticize Perelman for his decision to turn down the job offers in 1995 and return to the Steklov Institute, imputing arrogance or just plain nuttiness. Some mathematicians and scientists would kill for some of the offers that Perelman turned down. Most major breakthroughs take a long time, usually five years or more. Perelman spent at least seven years on the Poincaré conjecture and he probably was working on it while in the United States. Most tenure track positions involve a seven year period. The assistant professor is up for review typically in six years; he or she usually must produce allegedly ground breaking work within six years. If he or she is denied tenure, he or she has one year, the seventh year, to find another job. Most assistant professors have acquired a spouse and small children by this time. There is considerable pressure to produce research papers, write grant proposals and raise money. Perelman apparently published nothing from 1995 until 2002. He most likely would not have gotten tenure had he tried to do this at any of the jobs that he turned down in 1995.

There appears to be a long history of mathematicians developing serious psychological problems. The aforementioned John Forbes Nash succumbed to mental illness, diagnosed as paranoid schizophrenia, and was well known to Princeton students for wandering around campus scribbling incomprehensible formulas on blackboards. Kurt Gödel developed psychological problems and allegedly starved himself to death. Georg Cantor became increasingly erratic as he got older. There are many anecdotal accounts of high levels of concentration and mental efforts sustained over months or years resulting in a kind of mental exhaustion and other problems. Both the western and eastern literature of meditation, which often involves prolonged concentration, contain warnings about various adverse psychological effects including anxiety attacks and hallucinations. Disillusioned former adherents of various meditation movements or “cults” have alleged serious adverse effects of heavy meditation, meaning many hours per day every day, similar to those recounted in ancient traditional sources on meditation. Although computer programming can be exhilarating, many programmers appear to experience mental exhaustion and “burnout” after lengthy programming projects involving high levels of sustained concentration.

In engineering there is an adage: “if you are one step ahead, you are a genius; if you are two steps ahead, you are an idiot!” Perfect Rigor portrays Perelman as astonishingly naive, protected from the “real world” by the bizarre Soviet mathematical system. While this may have some truth, a number of Perelman’s actions may exhibit much foresight, like a champion chess player sacrificing a piece for subsequent gain. Is pretending not to notice the alleged anti-Semitism (Perelman is a Russian Jew) in the Soviet mathematical system naive or politically astute? Declining the Fields medal, as some have noted, attracted enormous attention to Perelman. He is now one of the best known recipients (or non-recipients in this case) of the Fields Medal. It also gave him a great deal of moral authority which he seems to have used effectively to fend off Shing-Tung Yau’s alleged attempt to steal credit for proving the Poincaré Conjecture. Refusing to grant interviews also means that Perelman probably has a great deal of leverage with journalists in the rare cases when he grants an interview, as he did with such great effect in The New Yorker in 2006.

Perelman was a math prodigy, returning home with a gold medal and a perfect score from the 1982 International Math Olympiad. Prodigies are often not as successful as one might expect. Math and physics prodigies often flame out, sometimes catastrophically. While prodigies are more common among people who make major inventions and scientific discoveries than in the general population, they are not nearly as common as most people probably think. Perfect Rigor portrays Perelman’s success in proving the Poincaré Conjecture as a logical consequence of his youthful training and competition in the sometimes bizarre Soviet mathematical system. Since Perelman has revealed little about the process of his discovery, this is difficult to evaluate.

Prodigies often run into problems and don’t realize their seeming potential later in life. This has been observed in math, physics, and other fields for many generations. There are probably several causes. Some prodigies are probably frauds, manufactured by ambitious parents; that such people fail to make major breakthroughs is not surprising. Some prodigies are probably the product of a hothouse environment, driven or manipulated by parents or others to practice heavily and perform at an unusually high level that is difficult to sustain. As they get older and establish their own lives, other interests or needs intervene. Some prodigies undoubtedly fall afoul of politics that they are ill-prepared to deal with.

Academic homework, exams, competitions like the International Math Olympiad, admissions exams such as the SAT or GRE exams in the United States, specialized exams and competitions such as the famous Putnam math examinations, and so forth do not necessarily either teach or measure some of the skills required in actual invention or discovery. Exams and homework in math and physics tend to test the ability to accurately and quickly perform certain calculations or apply certain known mathematical methods to a problem. Some people either through heavy practice or rare natural ability can learn to perform these calculations rapidly with negligible error. This does not translate directly into the ability to handle unsolved research problems which often seem to require large amounts of frustrating trial and error and often deeper understanding of concepts, mental visualization, and so forth.

Many topics taught at a high school, college, and even beginning graduate school level are quite mature. Logical and technical flaws that abound in original research papers have been cleaned up and eliminated. Teachers and textbook writers have learned how to present the material clearly so that a bright or highly motivated student may be able to easily master the material quickly. Prodigies can sometimes read a textbook and immediately start performing the methods described in the textbook very accurately. This becomes more difficult as one reaches the “bleeding edge” where the available learning materials are original research papers or badly written textbooks that may contain errors, impenetrable jargon, opaque language, and even deliberate obfuscation of logical or technical flaws. Prodigies may encounter a sudden drop off of their remarkable abilities which they may inaccurately attribute to a lack of the magic “ability” required for the field rather than the immature state of the bleeding edge knowledge. Perelman presumably navigated these difficulties as he progressed in mathematical research.

One is reminded of the old sayings “actions speak louder than words” and “talk is cheap”. If Perelman’s proof stands the test of time, he has done much. If he is sincere in declining prizes, honors, and adulation, he sets an example by his actions. In reading Perelman’s story, one also cannot shake the impression that he may have had some unhappy experiences during his stay in the United States and went home silently vowing “I’ll show them,” which he apparently has.

The Poincare Conjecture

Donal O’Shea’s The Poincare Conjecture is a more pleasant book to read than Perfect Rigor, lacking the hostile tone of Perfect Rigor and sugar coating a number of topics. Perelman is “eccentric”. Little is said about “Manifold Destiny” or the ugly priority dispute. O’Shea focuses on the history of geometry, the Poincaré Conjecture, mostly inspiring stories about great mathematicians, and tries to explain the mathematics of the Poincaré Conjecture to a general audience.

On the whole, The Poincare Conjecture is an enjoyable and informative book to read. O’Shea carefully debunks the myth that scholars in the Middle Ages and the ancient world believed the Earth was flat. He gives an interesting account of Columbus, the slow discovery of the exact shape and geography of the Earth, confirming the ancient theory of the spherical Earth. He slowly and deftly leads the reader through the history of mathematics and geometry to the Poincaré Conjecture, the many failed attempts to prove it, and the seeming final solution by Perelman.

Some of the illustrations leave a bit to be desired. In discussing mathematics in the ancient world, O’Shea uses modern CIA maps of the modern world to show the ancient Greek kingdom of Ionia where Pythagoras was born and to show the Middle East. One map, for example, shows modern Bulgaria which did not exist in the time of Pythagoras. Similarly, O’Shea is discussing ancient Babylonia and Persia but the associated map shows modern Iraq and Iran. Hopefully, this will be fixed in a future edition.

Some of the discussion of hyperbolic geometry and most of the chapter on Poincaré’s topology papers, which presents the actual Poincaré conjecture, could be improved. The diagrams and explanation on page 27 in the chapter “Possible Worlds” showing how the surface of a two-holed torus can be mapped to an octagon is hard to follow. O’Shea returns to the two-holed torus and the octagon in Chapter 10, “Poincaré’s Topological Papers”. Probably many readers will have forgotten the discussion on page 27 by then. The term “natural geometry” is used in this chapter but not defined clearly. A number of diagrams in this chapter are small and difficult to follow. Interested readers can find a better explanation of some of the relevant aspects of hyperbolic geometry in the second chapter of Roger Penrose’sThe Road to Reality which features some entertaining Escher prints showing the so-called “Poincaré disc model” of hyperbolic geometry (first discovered not by Poincaré, but by Eugenio Beltrami, Penrose carefully points out).

One can only go so far with analogies to rubber sheets or cloth fabric in describing topology and especially differential geometry. This is a problem many popular mathematics and science books encounter. If we had a better way of explaining and introducing differential calculus to a general audience, this would improve the general public’s ability to follow issues in mathematics and science and also improve our educational system.

Pure mathematics today suffers from a particularly opaque and confusing language. It now typically takes several months for a skilled person to master the arcane language of modern pure mathematics. Abstraction has been taken to an extreme. Words and phrases such as “algebra”, “ring”, “module”, “field”, and so forth have meanings in pure mathematics that differ both from common usage and the language of applied mathematics used in most engineering and also much physics. The Poincare Conjecture suffers in places from terms like “natural geometry” that have a special meaning in pure mathematics.

Conclusion

Both books focus on the genius of Perelman and famous mathematicians such as Gauss, Riemann, Poincaré, and others. Indeed, the subtitle of Perfect Rigor is “A Genius and the Mathematical Breakthrough of the Century”. This superman theory of scientific progress and a strong focus on extreme intelligence is common in popular science and math books and articles.

The story of the Poincaré Conjecture, at least until Perelman, is a story of large amounts of trial and error (lots of error) as both books allude to. Henri Poincaré formulated the conjecture in 1904 and published an incorrect proof. Almost every year has seen publication or presentation of attempts to prove the Poincaré Conjecture. Numerous mathematicians, including very top mathematicians, have published incorrect proofs. Many different approaches to the problem have been developed. Most failed. Richard Hamilton developed the basic approach that Perelman built upon but apparently stopped making progress in the 1980′s or early 1990′s. It is common to find large amounts of trial and error in the detailed history of inventions and discoveries, including discoveries in pure and applied mathematics.

It is clear that Perelman spent at least seven years on the Poincaré conjecture. We have no idea how much trial and error and how much failure took place during those seven years. Perelman reportedly fixed two minor errors in his first paper in the subsequent two papers posted to www.arXiv.org in 2002 and 2003. Other inventors and discovers have frequently gone through long periods of trial and error and repeated failure before their “breakthrough”. While respecting Perelman’s accomplishments, we should also be interested in the precise process used to reach the answer and avoid attributing it to magical genius alone.

Both Perfect Rigor and The Poincare Conjecture are interesting and informative books for general audiences. Even practicing mathematicians may gain some insights and new information from Perfect Rigor. Yet, Grigory Perelman remains an enigma. A definitive biography remains to be written. The world might learn a lot from more details on how he discovered his proof of the Poincaré Conjecture.

(C) Copyright 2010, John F. McGowan, Ph.D.

About the Author

John F. McGowan, Ph.D. is a software developer, research scientist, and consultant. He works primarily in the area of complex algorithms that embody advanced mathematical and logical concepts, including speech recognition and video compression technologies. He has extensive experience developing software in C, C++, Visual Basic, Mathematica, and many other programming languages. He is probably best known for his AVI Overview, an Internet FAQ (Frequently Asked Questions) on the Microsoft AVI (Audio Video Interleave) file format. He has worked as a contractor at NASA Ames Research Center involved in the research and development of image and video processing algorithms and technology. He has published articles on the origin and evolution of life, the exploration of Mars (anticipating the discovery of methane on Mars), and cheap access to space. He has a Ph.D. in physics from the University of Illinois at Urbana-Champaign and a B.S. in physics from the California Institute of Technology (Caltech). He can be reached at jmcgowan11@earthink.net

1. Some of our readers are likely familiar with your work, but could you tell us more about yourself and your mathematical background?

I majored in mathematics as an undergraduate at Yale, from which I graduated in 1976. I think I even won a couple of math prizes, but I have to confess that I wasn’t the top mathematician in my class. That distinction would surely have gone to Jonathan Rogawski, who last I knew was a professor of mathematics at UCLA. (Notice that I just created the impression that I was the second-best mathematician in my class. I don’t know whether that’s true, but I’ll take it.)

Anyway, I went on to get a Ph.D. in mathematics at M.I.T. and have remained in the Boston area ever since. I went into the investment business in the early 1980s, based on the assumption that quantitative expertise would be a good match. But the truth is that I got progressively more qualitative as time went by, going from securities analyst to investment writer. I don’t know whether that transition made complete sense, but it ultimately gave me the opportunity to write some books – first about investments and then about numbers, including several volumes of puzzle books.

2. What inspired you to write Number Freak?

I was asked by a publisher to come up with a concept that would do for mathematics what a slightly different concept did for the natural sciences. The idea I came up with was more of a coffee-table book than the sized-down version I now have in my hands, but that effort was considered too expensive. I subsequently cast a wider net for the project, and was fortunate enough to attract publishers in the U.S., the U.K., and Australia.

3. The book is chock-full of interesting facts about the first 200 natural numbers. What did you learn in the process of writing this book that you didn’t know before?

Well, I guess the pat answer is that I learned how little I actually knew. Some of the work on planar tilings was new to me, even though it probably shouldn’t have been – for example, the Archimedean and Laves tilings I discuss in #11 are quite beautiful but I hadn’t been aware of their categorization and duality. And I wasn’t familiar with the work of mathematicians such as Erich Friedman of Stetson University, somebody who surely could have pulled off a book like this: I was only too happy, for example, to include “Friedman numbers” such as 127.

In self-defense, I wasn’t a complete neophyte. One big advantage I had in writing the book – apart from doing it in the Internet age, which gave me an abundance of material – was that I have a good memory for mathematical and pop culture trivia. For example, I enjoyed reaching back and remembering that the ultra-high security “D” block at Alcatraz prison had precisely 42 individual cells, something that meshed quite nicely with the picture of the “magic cube” I displayed elsewhere in the discussion of #42.

4. Having read this book I feel that it’s accessible to virtually anyone. Who do you feel is the ideal target audience for the book?

Boy is that a good question. My answer is that it’s for absolutely anyone, but if that’s too mealy-mouthed a reply, I guess I would say that I’d be especially pleased if parents bought Number Freak to (successfully!) introduce their kids to the world of numbers in a way that maybe, just maybe, is friendlier than what those kids were getting elsewhere.

5. Was there anything that you wish you could have included in the book but didn’t?

Another good question, and I’m afraid a painful one. The book was originally slated to go from 1 to 300 — as in a perfect game in bowling, among other things — but the editorial powers-that-be eventually whittled that down to 200. Too bad, as my discussion of the infamous 256th level of Pac-Man was worth the price of admission. (Say, that’s a topic I didn’t know about when I started the book!) I also lost some precious photos, charts and diagrams along the way. And you can imagine how I felt when a friend berated me for not mentioning “77 Sunset Strip,” when of course my original manuscript mentioned the show – and I have a photograph of Efrem Zimbalist, Jr. to prove it! (Those of my vintage – I’m 54 – will remember the show’s catchy theme song, but not many are aware that 77 was a particular good choice for the street address because it is the smallest integer whose English pronunciation requires five syllables.)
Other than that, I deliberately went easy on the cult surrounding the number 23, for example, and left a bunch of numerology and religious interpretations for somebody else to ponder. That’s another book all by itself.

6. What’s the answer to life, the universe and everything?

Why it’s 42, of course. You know, I had already answered question #3 above before I saw this one!

7. What’s your favorite number and why?

When I started the book, 17 had the edge. First of all, “At 17” by Janis Ian is probably my favorite song of all time. It came out in 1975, which was my favorite music year of all time. (Perhaps I should have written it in 1975.) But 17 is famous in mathematics for Carl Friedrich Gauss’s famous straightedge-and-compass construction of a regular 17-gon, for the 17 “wallpaper” symmetries of the plane, and for the fact that if you connect 17 suitably spaced dots with a segment of red, blue, or green, you will automatically create a “monochromatic” triangle whose three vertices are among the original 17 dots. And nobody has yet created a solvable Sudoku puzzle with fewer than 17 original entries. How about that?

But by the time I finished Number Freak, my favorite number had become 36. What happened is that while doing research for the book I came across a conjecture from the 18th century called the 36 Officer Problem. I had never heard of it before (yet another example!), perhaps because the problem was resolved in the early 20th century and then ceased to be of interest. But there was a three-dimensional wrinkle to the problem that hadn’t been explored, and I used that wrinkle to design a puzzle with a gray base and 36 towers of various colors. I went to Toy Fair and showed the puzzle to ThinkFun, a great game and puzzle company out of Alexandria, Virginia. And guess what? They made me a deal for the puzzle and after a year tinkering with the basic model, they launched it as “36 Cube” in the fall of 2008—many months before Number Freak came out! I was thrilled that the lessons of the book came to life in such a tangible way, so I’d be lying if I didn’t admit that 36 holds a very special place in my heart.

Thank you very much, Derrick, for your insightful answers. And to our readers, if you haven’t already done so, check out his book.

Mathematicians: An Outer View of the Inner World is a hardcover book of photos that centers around 92 well-established mathematicians. It is a work of art that presents, through each of the glossy images (which are printed on excellent quality paper), an autobiographical note on the left, and a large, black and white photograph of a mathematician on the right hand side.

The beautiful portraits, along with the short essays, help the reader to establish a brief emotional connection with each mathematician, as you become privy to the “outer view of their inner world”. Some of the featured mathematicians are young, many are older, some are well-known worldwide, others are relatively unknown. Their stories are intense and each fascinating in its own right. Many of these brilliant minds lived through WW2, and found that doing so had a strong impact on their lives and career choices.

Each mathematician interpreted the assignment of writing their micro-bio in a different way. Some tell you about their childhood and how they came to discover they joy of mathematics, others about what their biggest accomplishments to date have been, and a few talk about the open challenges they see in their field. The common thread amongst this diverse group is the wondrous search for knowledge, beauty and truth that transpires from their words. As Mariana Cook (the book’s photographer) mentions in the foreword, though she has photographed scientists and many other unrelated groups of people, she’s never experienced so many people referring so often to beauty and truth. Indeed math is beauty, elegance and truth!

Stunning photography and intimate essays make for a great work of art. Mathematicians is a book that can be enjoyed by anyone, from people who are passionate about mathematics to those who have been adverse to the subject all their lives. The latter group will probably look at the subject of math in a new light after experiencing this book.

The affordable price and large format of this book make it a coffee table piece that every geek should have. The hardest part for most math aficionados will be treating it like a coffee table book though, and not reading it cover to cover right away, as I did. This title left me with a strange sense of harmony and peace, and a renewed awareness about the importance of mathematics, as well as a desire to join their efforts in furthering the discipline. Well above what I was expecting from a photography-based book.

Have you read a math book that you’d like to review for Math-Blog.com? Let us know.

I bought some of these books, others were sent to me as evaluation copies for my consideration. I plan to devote detailed reviews to my very favorites in the coming months. Here, I’m simply going to list all my math-related reading up to mid-2009, for those of you who’d like to check it out.

Sacred Mathematics: Japanese Temple Geometry

Mathematicians: An Outer View of the Inner World

Is God a Mathematician?

Euler’s Gem: The Polyhedron Formula and the Birth of Topology

The Great Equations: Breakthroughs in Science from Pythagoras to Heisenberg

The Golden Ratio: The Story of PHI, the World’s Most Astonishing Number

A Mathematical Nature Walk

The Mathematical Mechanic: Using Physical Reasoning to Solve Problems

Divine Proportions: Rational Trigonometry to Universal Geometry

Modeling with Data: Tools and Techniques for Scientific Computing

Probability, Markov Chains, Queues, and Simulation: The Mathematical Basis of Performance Modeling

Prime-Detecting Sieves. (LMS-33) (London Mathematical Society Monographs)

Many parents face the challenge of helping their children with math homework, which for some stems in part to having developed a strong phobia or dislike of the subject themselves. Along with a psychological component, in many cases the challenge is augmented by a lack of basic skills (when it comes to knowing how to approach math problems and work their way through mathematical nomenclature). For some it’s like trying to help their child with French homework, when they don’t speak the language. Otherwise perfectly intelligent adults end up finding themselves worrying over problems that most math-savvy people would consider straightforward.

Math for Moms and Dads tries to solve this predicament by providing a vocabulary of essential terms, a very gentle introduction to problem solving and mathematical reasoning, fundamental concepts of elementary (primary) and middle school mathematics, and step-by-step solutions to basic exercises. It also stresses the importance of the parent-child and parent-teacher relationships when it comes to teaching and assisting with the learning of math. This book is very basic and relatively short, which means that it’s something most parents would be able to squeeze time into their schedule to read (which I feel is a positive element of this book). As someone with a passion for math, I’m biased and admit that I do not find this type of book terribly exciting myself, but I fully realize its usefulness for people who need a “less than scary” introduction (or refresher) to the subject.

The first chapter introduces the book and provides parents with a few pointers on how to use a calculator and when its usage is appropriate. The content on these pages will appear pretty obvious to a large number of readers, but this book tries not to make any assumptions, and as such it aims to cover concepts that many people might take for granted.

Chapter two details the mathematical vocabulary mentioned earlier in this article, and within this chapter parents will learn about fundamental math terminology, including terms such as absolute value, congruent, coordinate plane, diagonal, fraction, permutations and so on. The second part of the chapter provides the reader with more descriptive information about common, basic concepts like commutative and associative property, prime and composite numbers, rational and irrational numbers, union and intersection, linear and quadratic equations, etc.

Chapter three covers the basic rules necessary for resolving a variety of problems, including order of operations, exponents and their rules, properties of numbers, fraction and integer based arithmetic, expressions and equations, and so on.

Chapters four and five tackle the issue of solving homework exercises and preparing for math tests. Together these chapters help clarify how to approach mathematical problems, with examples that are solved in a step-by-step manner.

Chapter six is a pedagogical chapter about how to approach study, which covers topics such as how to create the right study conditions and find the ideal place in your house to turn into a homework area, as well as how to develop note taking and test preparation skills.

Chapter seven is entitled “When will I use this, anyway?”, and it attempts to convince both parents and their children that learning mathematics is an important and useful real world skill. I felt that this chapter (which is about a subject – the importance of math beyond the classroom – I believe strongly in) was on the weaker side, but it may still be useful to some.

Lastly chapter 8 deals with parent-teacher communication, a topic that I felt was important for this kind of book.

Should you feel that your own math skills are not your strongest suit or if you need a concise and easy to follow along with refresher course on numerous basic math topics, so that you can better assist your child with their studies, you will likely find this book right up your alley.

If you are a publisher and would like to have your books reviewed, please contact me at antonio@math-blog.com. As a policy, we will only publish reviews for book worth recommending, informing the publisher if a book doesn’t meet (in our opinion) the standard.

And what was he?
Forsooth, a great arithmetician.
-Shakespeare Othello, I.i

Perhaps it should not be surprising, considering the vast libraries of published works, that mathematics should appear topically in works of fiction, but anyone who has sat through an introductory course in algebra can understand the difficulty that an author might have in captivating a reader if the subject is predominantly mathematical. This is not to say that casual readers should hold Jean-Pierre Serre up to the same literary standards of William Shakespeare, but that fiction can involve mathematics in complex and beautiful ways.

Mathematics is a rich ground for metaphor: the first scene of Tom Stoppard’s Arcadia is an excellent example of the intertwining of such unusual topics of conversation. In the very first scene, our young protagonist Thomasina is trying get her tutor, Septimus, to explain sexuality to her, whereas Septimus would rather encourage her studies:

Septimus: Carnal embrace is sexual congress, which is the insertion of the male genital organ into the female genital organ for purposes of procreation and pleasure. Fermat’s Last Theorem, by contrast, asserts that when x,y and z are whole numbers each raised to power of n, the sum of the first two can never equal the third when n is greater than 2.
(Pause)
Thomasina: Eurghhh!
Septimus: Nevertheless, that is the theorem.
Thomasina: It is disgusting and incomprehensible. Now when I am grown to practice it myself I shall never do so without thinking of you. (3)

Thomasina grunts out of mathematical frustration or coital pleasure, or perhaps both. And what precisely will she be doing when thinking of Septimus? This early comparison of algebraic theory with sexual self-discovery alters the tone of every mathematical reference uttered through the rest of the play, and double-entendres arise from otherwise innocent mathematical statements. Arcadia is laced with math, most prominently geometry and chaos theory. From carnal embrace to architecture, math is among the primary metaphors for examining the impermanence of forms and the struggle to find meaning in a world of background static.

While sex and death are compelling, authors also use math not as a metaphor for other things but as itself: a philosophical tool with which to pry off the lid of the universe. Neal Stephenson has written a number of historical fiction (or arguably science fiction) novels whose primary or secondary characters are mathematicians. The Baroque Cycle is a compendium of three novels centered around the dispute between Newton and Leibniz. In Quicksilver, the first book in the cycle, a character muses on the significance of conic forms (ie tracing the intersection of a cone and a plane) and gravitation:

Comets passed freely through space, their trajectories shaped only by (still mysterious) interactions with the Sun. If they moved on conic sections, it was no accident. A comet following a precise hyperbolic trajectory through the æther was a completely different thing from Daniel’s just happening to trace a roughly hyperbolic course through the English countryside. If comets and planets moved along conic sections, it had to be some kind of necessary truth, an intrinsic feature of the universe. It did mean something. What exactly? (676)

Daniel ponders an old question: is nature written according to the rules of Euclidean Geometry, or is geometry just the illusion of patterns in the fog? Newton’s “On the Motion of Bodies in Orbit” appears to answer the question, and in so doing open a window to the mind of God. The influence of mathematics on metaphysics has a tremendous effect on religious, and therefore political, thinking at the time. In this way mathematics drives a thousand plot lines spinning off from its philosophical implications.

But the subjects need not always be as weighty as the Universal Law of Gravitation. Math can appear within a sense of whimsy and joy in the beauty of solving puzzles and word games. Lewis Carroll’s Alice’s Adventures in Wonderland is full of subtle math jokes woven into the fabric of his fantastic tale. At the Mad Tea Party, when the Hatter wants another cup of tea, he has the whole party move around the table in a kind of infinite sequence. When Alice asks what happens when all the places are used up, the March Hare asks for a change of subject. In Through the Looking Glass, the chess queens grill Alice on her arithmetical skill:

“And you do Addition?” the White Queen asked. “What’s one and one and one and one and one and one and one and one and one and one?”
“I don’t know,” said Alice. “I lost count.”
“She can’t do Addition,” the Red Queen interrupted. “Can you do Subtraction? Take nine from eight.”
“Nine from eight I can’t, you know,” Alice replied very readily: “but – “
“She can’t do Subtraction,” said the White Queen. (222)

These are the joyous little jokes that make the Alice’s Adventures in Wonderland and Through the Looking Glass so endearing. In addition to writing children’s stories, Lewis Carroll was a mathematician; word games and clever exchanges such as these are the result of a mathematical attention to the precise meanings of words. Many excellent articles have been written on the logical puzzles hidden in his works.

Thus, contrary to the common perception that mathematics and literature occupy opposite ends on the spectrum of human thought, one can see how well the two disciplines may interweave. This is but a sampling, there is much more math to be found in fiction than just the few examples above. Whether heavy-handed or light-hearted, explicit or metaphorical, math appears in all sorts of ways throughout literary works. If you have but the patience to look into it and see, mathematics can be a light to brighten the many worlds you might visit.

References

1. Carroll, Lewis. Alice’s Adventures in Wonderland & Through the Looking Glass. New York: Penguin Putnam, 2000.
2. Stephenson, Neal. Quicksilver. New York: HarperCollins, 2003.
3. Stoppard, Tom. Arcadia. New York: Faber and Faber, 1993.

I’d like to claim that I’ve read it cover to cover. But I haven’t. At 1014 pages (and a hefty five and a half pounds), this recently published hardcover tome will probably take me a while to properly read through in its entirety. It can serve as a reference that can be randomly accessed if you so chose, but from what I’ve read so far, I’m extremely impressed and am eager to pursue my way through it systematically. It will be time really well spent, and dare I say, a life changing experience. This is the book to bring on a proverbial desert island, if you were allowed only one title.

The Princeton Companion to Mathematics is not only a beautiful book from an aesthetic standpoint, with its heavy, high quality pages and sturdy binding, but above all it’s a monumental piece of work. I have never seen a book like this before. It rigorously illustrates the (pure) mathematical field while remaining as accessible as possible to the general reader. There is no mathematician in the world who, upon reading this book cover to cover, would not have learned a great deal from it. And I’m sure this includes Timothy Gowers himself, who was the book’s Chief Editor and who brilliantly managed to coordinate a team of world class experts to cover (again in an accessible manner) their respective fields of expertise. Such experts not only are the best mathematicians alive today in their respective areas of expertise, but are also absolutely wonderful teachers who have the uncanny ability to divulge information in a understandable manner, under the editorial guidance of Professor Gowers. Timothy Gowers, by the way, also has a blog, which contains discussions about the book and a helpful errata.

This book is what I now consider to be the bible of mathematics, and unlike a typical reference or encyclopedia, The Princeton Companion to Mathematics never fails to provide a sense of unity and cohesion, both of which are essential if you want to truly provide an (nearly) complete panorama of a subject. While all the basics are well explained with the clarity and simplicity of really good popular science, this tome doesn’t skimp on details or theorems when it comes to highly advanced topics that few people are familiar with. The style remains geared towards providing a good introduction to each subject, as opposed to a PhD thesis, and as such it will prove useful to the ambitious high school student, as well as professional mathematicians or graduate students. And as if all this wasn’t enough, they managed to squeeze in a biographical overview of the most important historical mathematicians from Pythagoras to Bourbaki, as well as a respectable (yet not overly comprehensive section) about applied mathematics, and math’s influence on other disciplines.

I believe this is the kind of book that will still be is use a hundred years from now, even though by then it will be slightly outdated. This title is destined to be fully revered as a classic and monumental review of the subject of pure mathematics. I salivate like Pavlov’s dogs at the idea of the amount of fun I will have exploring this book, which will no doubt expose my ignorance about several key areas of math, and yet at the same time help me to remedy such things.

This Christmas, give yourself a great gift and get this book. If you are looking for the perfect gift for people who’re interested in mathematics, this is the right book. If you are a parent, I especially encourage you to pick it up for your son or daughter, it could change their lives. Ladies, your geeky boyfriend will likely propose to you if you put a copy under the tree. Jokes aside, The Princeton Companion to Mathematics makes for a great read to start 2009 off with.

Hypatia of Alexandria (AD 350 to 370 – 415): Born nearly 17 centuries ago, Hypatia of Alexandria was a brazen, highly intelligent woman who excelled in the fields of science, math and philosophy, which at the time (and for hundreds upon hundreds of years further) were seen squarely as the domain of men. Hypatia’s foremost teacher was her father, Theon Alexandricus, a mathematician and philosopher, who she would later go on to contribute to several mathematical works with. Hypatia herself was a teacher, as well as being the inventor of the hydrometer. Though she forged ahead in a time when women were all but ignored in the realm of mathematics, this bright Greek woman eventually met with a tragic death when her chariot was attacked and she was brutally murdered by a gang of Christians. Though her life was cut short, while she was alive, through her accomplishments, Hypatia was able lay the groundwork for future female pioneers of mathematics.

Gabrielle Émilie Le Tonnelier de Breteuil, marquise du Châtelet (December 17, 1706 – September 10, 1749): A woman of many intellectual interests, Émilie was a mathematician, author, and physicist who hailed from France. Born into a well-to-do family, Châtelet was a gifted child with a natural penchant for linguistics. Given her family’s high social status, Émilie was able to receive a degree of education far above the vast majority of French women at the time. Her place in society also put her in a position wherein she was able to mingle with some of the leading minds of her time (such as Voltarie, who would go onto become one of her lovers). In 1740, Châtelet published a book entitled Institutions de Physique, which put forth some of her knowledge regarding both science and philosophy. In her last year of life, Émilie translated Newton’s well-known Principia Mathematica. In her early forties she became pregnant, and though she initially survived the pregnancy, a few days later both she and her newborn child passed away. Émilie was an independent, articulate and highly intelligent woman, who was somehow able to hold down both her role as a leading lady in French high society and as a mathematician, an equation which deserves respect in its own right.

Maria Gaetana Agnesi (May 16, 1718 – January 9, 1799): A woman of many skills, Agnesi was an Italian mathematician, linguist, and philosopher whose profound intelligence was evident from an early age. Born into a wealthy and large family (due in part to siblings which sprang from her father’s two subsequent marriages after Maria’s mother passed away), Agnesi was a devoted and studious woman who would go onto publish the first book that dealt with both integral and differential calculus. In 1750, Maria was appointed as chair of mathematics and natural philosophy at the Bologna Academy of Sciences, an incredible accomplishment for any woman in the mid eighteenth century, when exceptionally few universities in Europe allowed women to study, let alone hold teaching positions. Later in life, Agnesi, a deeply religious woman, joined a nunnery and ended her days tending to the less fortunate.

Marie-Sophie Germain (April 1, 1776 – June 27, 1831): Parisian born Germain was a passionate mathematician with a love of number theory and differential geometry. During her lifetime (which, in the context of both France and Europe in general, was a highly tumultuous era) Germain often corresponded under a pseudonym (Monsieur Le Blanc) as a means of hiding her gender when writing to leading male mathematicians of the time such as Lagrange and Gauss. In 1816 Sophie won a contest that was held by the French Academy of Science which dealt with the area of vibrations on elastic surfaces, that in turn lead her to become the first woman (short of some of the staffs’ wives) to attend classes at the Academy. In 1831, the University of Gottengen bestowed an honorary degree to Germain, however she died as a result of breast cancer before she was able to receive the degree. A self-taught mathematician who came of age during a truly unstable period in French history, Sophie will long be remembered for her mathematical contributions in the field of number theory.

Augusta Ada Byron King, Countess of Lovelace (December 10, 1815 – November 27, 1852): English born Ada was the daughter of famed poet Lord Byron, though he was not active in his daughter’s life. Aside from her famous father, Ada is primarily known for her programming work regarding Charles Babbage’s invention of the analytical engine, a very early mechanical general-purpose computer. Lovelace was ahead of her time in this field, as she believed that computers held the capacity to do more than just simply act as calculators. Like many of the women in this list, Ada met with an early death; she was only 36 when she died due to uterine cancer. Today Lovelace is remembered fondly as the first female computer programmer (in era before the modern computer came into existence), and the programming language Ada was named in her honor.

Sofia Vasilyevna Kovalevskaya (January 15, 1850 – February 10, 1891): Generally acknowledged as the first well-known Russian female mathematician, Kovalevskaya (portrayed above) began teaching herself advanced mathematics as a young teen, before going on to leave Russia so that she could attend university in mainland Europe (something that women were not allowed to do in Russia at the time). A very bright, quite and gentle person, Sofia loved to learn and was eager to share this passion with others by teaching math, though this proved to be very challenging for a woman in nineteenth century Russian and Kovalevskaya would again have to leave her homeland so as to take up a position lecturing at the University of Stockholm. Prior to her relatively young passing due to pneumonia, Kovalevskaya published numerous papers on topics pertaining to mathematics and mathematical physics, and won a prestigious award (the Prix Bordin) from the French Academy of Sciences. (Here you can find a mathematical book about her work.)

Amalie Emmy Noether (March 23, 1882 – April 14, 1935): Considered by Einstein to be most important woman in history of mathematics, Emmy (as she generally went by) was an early twentieth century German mathematician with a passion for such areas as theoretical physics and abstract algebra. Noether was both an accomplished university professor and a prolific writer of mathematical papers, as well as someone with a profound ability to grasp abstract thought. As the Nazi stronghold grew in Germany during the 1930s, Emmy found herself, like so many other Jewish professors, barred from teaching. Towards the end of 1933, Noether was able to escape Germany and take up a position at the American college of Bryn Mawr. However, sadly, two years later Emmy’s life was cut short when she died just days after undergoing surgery. To this day Noether’s many contributions towards mathematics and theoretical physics are highly revered, and many remain relevant to the math of the twenty-first century.

Dame Mary Lucy Cartwright (December 17, 1900 – April 3, 1998): An accomplished British mathematician, Cartwright led a long and distinguished career that focused on function theory. In her lifetime, Mary published in excess of 100 papers and was the first female mathematician to be elected as a Fellow of the Royal Society of England; a theorem regarding analytical function that she put forth, Cartwright’s theorem, shares her name. Cartwright received numerous awards and recognitions throughout her life including, the De Morgan Medal of the London Mathematical Society and the Sylvester Medal of the Royal Society.

Julia Hall Bowman Robinson (December 8, 1919 – July 30, 1985): An American mathematician who was born in St. Louis, Robinson is known for her work regarding Hilbert’s tenth problem and the field of decision problems. Though plagued by health problems for most of her life, Julia didn’t let this stand in the way of her love of math and the pursuit of knowledge. She taught as a professor at Berkley and was the first female mathematician to be elected to the National Academy of Sciences. An historical first in her career included becoming president of the American Mathematical Society. She would also go on to become elected to the American Academy of Arts and Sciences in the mid 1980s, just a few short years before she passed away from leukemia.

Shafi Goldwasser (b.1958 –): A native of New York (and the only living mathematician on our list), Goldwasser is both a professor of mathematics (at the Weizmann Institute of Science) and of computer science (at MIT, where she was the first person to hold an RSA Professorship). Shafi’s research focuses on areas such as cryptography, complexity theory and computation number theory, and she is well-known for her work with zero-knowledge proofs. For her work in the field of complexity theory, Goldwasser was awarded the Gödel Prize in theoretical computer science twice (1993 and 2001, respectively).

In this article we’ve taken a gander at ten well known and highly esteemed female mathematicians, but the list doesn’t stop here. Throughout history there have been numerous other women whose contributions to the field of mathematics have made significant impacts. In 1971 the Association for Women in Mathematics was formed with the intent of helping to establish and promote equal opportunities and treatment for girls and women in all areas of mathematics, while at the same time helping to encourage more to get involved with math.