Below are summaries of some of the invited addresses and other presentations at the 2013 Joint Mathematics Meetings, almost all of which were written by Claudia Clark, freelance science writer and former AMS-AAAS Media Fellow, and Evelyn Lamb, 2012 AMS-AAAS Media Fellow. Also see the JMM2013 blog written by former AMS- AAAS Media Fellow Adriana Salerno of Bates College.
"Green open access," "gold open access"---these buzzwords that seem to hail a new era for journal publishing were heard all over the San Diego Joint Meetings. Both terms refer to accessibility of research articles that have been accepted for publication in a journal. A "green open access" article is made freely available by the author, by posting it on his or her home page, or by depositing it in an open access web site such as the arXiv. A "gold open access" article is made freely available on the journal web site immediately upon publication. Some journals give authors the option of paying an Article Processing Charge (APC) in order to make their articles available on the gold open access basis. These fees range from US$500 to US$3,000.
*AMS Committee on Science Policy Panel Discussion: Who will pay for the papers we publish?
This session, organized by Kenneth Golden of the University of Utah, zeroed in on the issue of APCs. The first speaker was AMS Executive Director Don McClure, who described two recent actions, taken by large funding agencies, that point to the growing role of open access publishing. The first came last July 16th, when the Research Councils UK announced that, starting this spring, results of research supported by grants from the council must be made available immediately upon publication, with a preference for gold open access. The second action came the very next day, July 17th, when the European Commission announced its intention to adopt a very similar policy.
Sastry Pantula, Director of the Division of Mathematical Sciences of the National Science Foundation (NSF), also spoke in the session. He said that, in 2007, the National Institutes of Health (NIH) was mandated by law to require NIH-funded researchers to make an electronic version of their final peer-reviewed manuscripts available on a publicly accessible repository, such as PubMed Central, within 12 months of publication. PubMed Central now contains over 2.4 million articles and has about 700,000 users every weekday. Pantula said that, as early as this summer, or perhaps even sooner, the NSF would make an announcement about its policy regarding open access publishing.
David Goss of Ohio State University spoke about his experiences as the editor-in-chief of the Journal of Number Theory (published by Elsevier) during the first few months of the Cost of Knowledge protest against Elsevier. He cited several reasons for his decision to continue his work on the journal. For one thing, "our job is really to publish articles and do mathematics." The journal was started by colleagues at his institution, and he felt a duty to protect their creation. "I also felt very responsible for the young mathematicians who were trying to publish papers with us," he said. As a result of this experience, Goss realized that "we are just at the beginning of the impact of electronics on our life... In terms of how it's going to affect us as mathematicians and professors, it's just beginning... Open access doesn't just mean open access publishing. It can also mean open access education [such as massive open online courses (MOOCs)], and that could have a vast impact on us."
Robion Kirby of UC Berkeley heads the nonprofit Mathematical Sciences Publishers and is managing editor of the recently launched Cambridge University Press journals Forum of Mathematics, Pi, and Forum of Mathematics, Sigma. Both journals are gold open-access; CUP is covering all costs for three years, at which time the journals will begin charging APCs. Kirby answered the question of who shall pay with the following scenario. An author publishes in the best quality electronic, open access journal that he or she can. If the paper is published in a journal with APCs, the author takes the bill to the librarian at his or her institution, and the librarian pays the bill out of the library journals budget. If necessary, a subscription to a different journal---presumably a high-priced one---would be cut to free up funds. If the high-priced journal is part of a bundle, there will be pressure on the university to get out of the bundle; if the journal is not part of a bundle, there will be pressure on the journal to become more efficient.
*AMS Special Session on Topics and Issues in Electronic Publishing
Kirby also spoke in a separate session devoted to publication issues and organized by Steven G. Krantz of Washington University in St. Louis. Kirby asked the question of why, in publishing math papers, money has to change hands at all. He noted that half of all math journals are run without money. Some of these free journals are very respectable, but they rely on the enthusiasm of volunteers, which can flag over time. He said that high-quality papers that will have an impact in the field should undergo copyediting, they should be well-produced, and they should be well archived. And such services cost money. "So I accept that high-end papers cost something to publish," he said. He then went on to discuss the two new CUP journals and said he expected that, when the APCs are instituted, they would be around US$750 for a 25-page article.
In that same session, Vindra Dass, a representative of Springer, discussed Springer's efforts to provide gold open access options for its mathematics journals. Some of those journals are "hybrid," meaning that they offer gold open access to authors in exchange for an APC of US$3000 per article, while still charging libraries to subscribe to the journals. Dass said that Springer has lowered the subscription prices of some journals that offer gold open access.
An AMS ad hoc committee has been working intensively for the last month to formulate a plan for how the Society can offer open-access options for some of its journals. At its meeting just before the JMM, the AMS Council discussed a proposal put forth by the Society. Don McClure has written an article describing the proposal. The article will appear in the March 2013 issue of the AMS Notices; that issue will be posted on the web on February 12, 2013. The proposal will receive further consideration from the AMS Board of Trustees and Council this spring.
--- Claudia Clark and Allyn Jackson
On the evening of January 9, Cédric Villani, director of the Institut Henri Poincaré and 2010 Fields Medalist, delivered the Gibbs Lecture, touching on several topics in statistical mechanics, which Villani described as the study of large time behavior. Depending on the system, this can mean billions of years or a few seconds.
Villani started by addressing the classical question of whether the solar system is stable. He discussed the different conclusions of Laplace, Poincaré and others, noting that mathematics is enriched by the interplay of ideas, even though some prove to be incorrect or in conflict with the theory.
He introduced the Kolmogorov-Arnold-Moser (KAM) theory of small perturbations of dynamical systems. In short, KAM says that small perturbations of systems should be quasi-stable. KAM theory can be used to show that catastrophic behaviors, even if not precluded by conservation laws, almost never happen. However, the perturbations required in KAM theory are so small that they basically never apply to real physical systems. But Villani says that even though the theory technically doesn't apply to real systems, it has been revolutionary in classical mechanics for mathematicians and physicists. "Maybe there is a paradox, maybe not. It depends on how you view mathematics," Villani said.
He went on to discuss what he called the "many-body revolution" of the nineteenth century, led by Maxwell and Boltzmann. Instead of using Newtonian mechanics to analyze a small number of particles or planets, probability is used to analyze large collections of objects, such as particles of gas in a container or stars in a galaxy. Some of the outcomes of this revolution were Einstein's and Smoluchowski's explanation of Brownian motion and evidence for atoms at the beginning of the 20th century.
The Boltzmann equation, famously inscribed on Boltzmann's tomb, describes the behavior of a fluid as it moves toward thermodynamic equilibrium. Villani discussed attempts to derive the Boltzmann equation mathematically, contrasting its utility in physics with its mysteriousness in mathematics. He also contrasted the Boltzmann equation to the Vlasov equation, which describes the behavior of plasma. Both have constant energy, but the Vlasov equation is reversible and has constant entropy, while Boltzmann's is irreversible and has increasing entropy.
Villani noted that Boltzmann's definition of entropy is distinct from some other mathematical uses of the term, but that it coincides with Claude Shannon's use of the term in information theory. He quoted von Neumann's suggestion to Shannon, "You should call it entropy, for two reasons. In the first place your uncertainty has been used in statistical mechanics under that name, so it already has a name. In the second place, and more important, no one really knows what entropy really is, so in a debate you will always have the advantage."
At the end of the lecture, Villani described connections between three of the most famous paradoxes of modern physics: the KAM theorem, plasma echoes, and Landau damping. In essence, Landau damping describes a situation in which perturbations damp away spontaneously. In 1946, Landau developed a linearization of the Vlasov equation, but Villani notes that the neglected term can go to infinity, which casts doubts on the validity of the linearization. Since then, efforts have been made to understand the nonlinear version of Landau damping. Villani closed by describing a dictionary between KAM and nonlinear Landau damping, two seemingly disparate ideas that may have a deep connection.
Ken Golden (University of Utah), who has traveled to Antarctica several times to study sea ice, gave the public Porter Lecture on the last afternoon of the meeting. His talk was one of several at the Joint Meetings in conjunction with the 2013 initiative Mathematics of Planet Earth.
Golden said that sea ice is the source of a great deal of uncertainty in climate change models. Far from an inert chunk of cold, white stuff, sea ice mediates the exchange of heat, gases, and momentum between the ocean and atmosphere, and the brine inclusions in sea ice create an ecosystem for algae and other microorganisms. These processes occur on both large and small scales, and climate change models can't take the small-scale effects into account. Golden said that even the best climate change models dramatically underestimate the extent of sea ice melt, and he believes this is because the small-scale processes, specifically related to brine inclusion in sea ice, are so poorly understood. That's where the mathematics comes in, said Golden. He uses the background of composite materials and statistical physics to understand fluid inclusion in sea ice with the aim of improving climate models.
As Emily Shuckburgh had done on Wednesday, Golden described the albedo feedback loop. When ice melts, albedo, the amount of solar radiation reflected by the earth, goes down. This then allows the earth to heat up more, further decreasing its albedo. This feedback loop makes sea ice not only an indicator of climate change but also an active participant. He mentioned that September 13, 2012 set a record for the lowest extent of Arctic sea ice, at just under half of the long-term average for that time of year.
A large part of the lecture was spent discussing attempts to understand and model the flow of brine in sea ice. Golden related the "rule of fives" relating sea ice permeability to temperature. Research has found that there is a critical phase transition around -5 degrees Celsius and 5% salinity: below these levels, sea ice is virtually impenetrable and does not allow brine flow in sea ice, which nearly halts algal growth in the ice. This critical point can be modeled in lattices, and the fractal geometry of melt ponds in the ice can be studied using Fourier surfaces.
Golden said that field experiments are essential and showed slides of x-ray tomography of sea ice, describing the differences between first-year ice and multi-year ice, and between the granular ice of the Antarctic and the columnar ice that dominates the Arctic.
Several times during the lecture, Golden mentioned the students he works with--not only the PhD candidates he advises but also high school students and undergraduates who have worked on theoretical models and even traveled to Antarctica with him on research vessels. During the questions at the end, Golden said that he has had good experiences working with younger students in what he called a "junior grad student" model instead of just during REUs. He also drew attention to efforts to get more young mathematics students into climate science.
The end of Golden's lecture consisted of a video of some notable moments in his Antarctic trips. In addition to adorable penguins and seals, there was fascinating footage of his vessel, Aurora Australis, stuck in sea ice. The scientists and crew were in no danger and continued to do experiments (and have a Halloween party) while they were stuck. Finally, without even having to dip into emergency rations, they were able to ram their way out of the ice and start their voyage back.
--- Evelyn Lamb
On Friday morning, Paul Zorn, professor of mathematics at St. Olaf College, and recently retired president of the MAA, spoke about various forms of mathematical communication. He began his lecture with a discussion of mathematics and art, as well as mathematics and literature, "and their symbiotic relationship in service of communication broadly understood" with examples including the still life paintings of 17th century painter Juan Sanches Cotan, and the writings of Jonathan Swift and Jorge Luis Borges.
The majority of Zorn’s lecture, however, was devoted to describing the challenge, and the importance, of communicating mathematics effectively, and how this requires everyone’s "best efforts, not just mathematical, but also linguistic, artistic, visual, pedagogical, sociological, psychological." Zorn starts by referencing William Thurston’s paper "On Proof and Progress in Mathematics," published in the Bulletin of the American Mathematical Society in April of 1994. To Zorn, Thurston’s definition of mathematics "implies that mathematical communication is not just descriptive of something else called mathematics, but is rather a valid form of mathematics in its own right." Thurston also points out that people understand mathematics in a variety of ways, and he urges mathematicians to "pay much more attention to communicating…our ways of thinking."
For students, Zorn says, part of the difficulty of math class is the special language of mathematicians, and the importance of word order, especially for students first seriously encountering theorems and proofs. Some of the words that mathematicians use--and, or, each, every, limit, derivative, measure, space--have meanings that differ from the meanings of these words in everyday speech. Other times mathematicians use language "ambiguously if not wrongly," especially with respect to functions, equations, and uses of the equal sign. (Zorn refers to some examples of each of these, as well as two essays--both of which appear in The Best Writing on Mathematics 2012--that address these issues, one written by Bonnie Gold, the other by Suzanna Epp.) "In the real world," Zorn says, "authors and teachers and speakers have continually…or should I say continously…to compromise between friendly but possibly vague informality on the one hand, and meticulous but possibly pedantic formality on the other hand." It is important to guide math students carefully and over time from the former to the latter style.
Two other issues arise for writers of mathematics textbooks. The first Zorn calls the "rulebook question": when teaching a child to play soccer, do you hand her a World Cup rule book to memorize or send her out to the back yard with a soccer ball? Analogously, for example, at what point do analysis and argument belong in a single-variable calculus text? Zorn has found that most students "will benefit from an expository approach that blends the rule book with the math analog of throwing the football out the back door." The second question is the "tour guide question": what information should be provided in a text and how much? Zorn thinks that trying to dodge this problem by trying to do too much is a mistake.
Clearly pedagogical writing is hard work, Zorn concludes, but it’s an "honorable profession" with a historical precedent (see the writings of Augustus de Morgan, or Abraham de Moivre’s work, The Doctrine of Chances). This work continues to be done well and appropriately valued by our mathematical societies. Zorn calls on departments, colleges, and universites to likewise recognize and reward good mathematical exposition, but in "academic currency," such as tenure, promotion, and other forms of professional advancement.
--- Claudia Clark
This Saturday morning session was in some ways a sequel to MAA president Paul Zorn's Friday morning address (summarized above). In it, several established math writers and educators discussed strategies and pitfalls related to communicating mathematics to the public.
In "Exposing and Expositing Math to the Public," freelance writer Dana Mackenzie discussed what he says are the two primary paths mathematicians can take to become popular writers. His path was to leave academia and become a full-time math and science journalist. After being denied tenure, he completed the science writing program at UC Santa Cruz and now focuses on books. The other path, which he feels is more viable, is to start writing after getting tenure, giving as examples Steven Strogatz, John Allen Paulos, and Keith Devlin. He said that writing for the public requires a major attitude adjustment from academia. He said that academics need to get used to deadlines, editing, and including the human element in mathematics stories. He also discussed his foray into television, which happened as a result of one of the books he wrote. At the end, Mackenzie had some observations about how the mathematics community can improve the level of mathematics writing available to readers, echoing Zorn's recommendation from Friday morning to include mathematics communication in tenure decisions.
Barry Cipra's talk "First Things First: The Art of the Opening" was, naturally, about the beginnings of math stories. He shared several examples of effective openings from his own and others' mathematics writing, especially Martin Gardner. He included some interesting examples of the back-and-forth editing process, showing his first attempts at ledes and the way they evolved in discussions with his editors. Cipra's re-imagination of the beginning of Moby Dick as written by a mathematician amused the audience. "Let my name be Ishmael, let the captain's name be Ahab, let the boat's name be Pequod, and let the whale's name be as in the title."
Frank Morgan (Williams College) discussed his Huffington Post blog, which he started in March 2012. He was approached by the Huffington Post to start the blog and is not paid. He has incorporated videos of students and other researchers into his post. He discussed the double-edged sword of the comments section a few times and what he has learned about what works and what does not.
In "Mathematics, Meaning, and Misunderstanding," Gerald B. Folland (University of Washington) discussed some pitfalls related to mathematicians' very specific use of language, a topic Zorn had also brought up on Friday morning. Specific words such as "complex" (and the idea of "simple complex functions") were mentioned, as well as the contrast between prescriptive and descriptive definitions.
Frank Farris (Santa Clara University) closed the session with his presentation "Can an Art Show Teach Mathematics?" He discussed his recent art exhibit at Carleton College, which featured printed fabric printed with mathematical designs. He walked the audience through the creation of these designs. Generally, he started with a picture he had taken (often of a peach, evidently one of his favorite foods) and manipulated it using photo software. The manipulations focused on making radially symmetric designs using complex roots and using Fourier transforms to give the designs more interesting features. Farris described different mathematical lessons viewers could take from the exhibit, showing that an art exhibit can be an effective way to reach people at all levels of mathematical sophistication.
--- Evelyn Lamb
A Visual Tour of Lesser Known Art and Architecture as Examples of Mathematical Developments of World Civilizations, Betty Rogers
The first of a series of Thursday morning talks related to mathematics and the arts consisted of a visual tour of the sites that the presenter, Betty Rogers, discusses in her history of mathematics course, Multicultural Mathematics. These include the Chimu adobe village of Chan Chan in Peru; the Tower of Homage in Gibraltar; the Alhambra in Granada, Spain (image above is from the Alhambra); the Museum of Mathematics at Sharjah, UAE (at left); and the Capital Museum in Beijing.
DaVinci Revisited, Susan McBurney
In another talk, Susan McBurney spoke about her ongoing study of two pages of geometric sketches from the notebooks of Leonardo DaVinci. While some have interpreted these pages as the "playful doodling of an aging artist," McBurney’s examination has revealed some "subtle geometric themes" that she shared during her presentation. She started with a proof that two rectangles have equal area, and a proof that two shapes with curved sides have equal areas. McBurney then showed how she was able to reconstruct many of DaVinci’s diagrams, and has so far determined that every half circle DaVinci drew has the same amount of area shaded! She finished her presentation by showing a series of artworks she was inspired to create using these designs.
--- Claudia Clark
This panel took place on the first day of the meeting and was in part a response to a recent Science paper about low retention rates of women faculty in all areas of science, which indicated that mathematics had a particularly low retention rate. In addition, recent research shows that both men and women tended to rate the exact same application as more competent when a male name was on the top than when it was a female name. To start the session, panelists were asked several questions to open up the discussion. The first question was what math departments can do to attract more women.
Suggestions included thinking about startup packages and childcare support, hiring couples, casting a broad net with respect to candidates' research areas, and not telling female candidates they are "diversity hires." (For obvious reasons, this can be off-putting, leading them to believe that others don't think them qualified or that they may not have good opportunities for collaboration.) Several panelists mentioned reaching out beyond the mathematics department in various ways. For example, a hiring committee can make sure women candidates meet other women in science beyond the math department during on-campus interviews.
Professor de Pillis said that stereotypes are a cognitive shortcut: when we stereotype people, we don't have to spend as much energy thinking about our reactions. This can be helpful in certain situations, but when this is part of a hiring decision, it can be harmful. She recommended that hiring committee members should have to explain their reasons for ranking one person above another. This can combat subconscious bias by making the process more conscious. She also emphasized that hiring decisions should not be rushed because that can add pressure and make it harder to avoid stereotyping.
The next question was how to support female faculty members after they had been hired.
Haynes and de Pillis both said that transparent, consistent review procedures are essential, as is transparency in the amount of service and teaching expected. Haynes suggested that these guidelines be written down so everyone will be on the same page and have a reference. Bertozzi said that not being pushed into service work as a junior faculty member was key to her success as a tenured but young faculty member. She mentioned that the chair of her department was particularly supportive of her staying off of unnecessary committees. His backing helped her stand up to people who tried to convince her to take on more service work.
Other suggestions included pushing for salary equity, mentoring, and providing assistance in finding grants and awards, as women are often reluctant to self-nominate. Several suggestions were made regarding department climate and caring for children while working. Gavosto mentioned that people can be unintentionally (or possibly intentionally) unwelcoming when they make comments about female mathematicians, for example calling it a "Girl Scout meeting" when two or three faculty members talking about mathematics all happen to be women.
A lively discussion followed the pre-set questions, with audience members sharing their personal experiences and questions. Topics included stigmas related to parenting (Bertozzi recommended mentioning the fact that Terence Tao met one of his collaborators at daycare), women's math groups, and data about family-friendly policies (men and women utilize these policies in similar numbers; they meet a need for everyone, not just women).
For more information, see the panel's website.
--- Evelyn Lamb
On January 9, Emily Shuckburgh (British Antarctic Survey) gave this AMS-MAA joint invited address that focused on the role of fluid dynamics of the atmosphere and oceans in understanding climate change. The talk was one of several at the Joint Meetings in conjunction with the 2013 initiative Mathematics of Planet Earth. The official US launch of MPE was later that evening.
Shuckburgh started by noting that in order to understand how mathematics can be used to model climate change, "you require a knowledge of mathematical techniques, but very importantly, you also require an intuition based on observations of the systems." The talk was roughly organized correspondingly: she began by introducing some of the mathematical underpinnings of her work and then showed how those theories inform experimental observation of global climate.
The talk avoided extensive forays into dense pages of equations, but Shuckburgh did show the nuts and bolts a few times. She explained a very simplistic model of earth's temperature, which assumes that the earth is a black body with a constant albedo (ratio of reflected to incident radiation), noting that the model yields surprisingly accurate results. By adding in the assumption that the atmosphere transmits a larger fraction of shortwave than longwave radiation, the estimate improves to within 2 degrees of the observed value, and Shuckburgh mentioned some other assumptions that can be added to improve the estimate even further.
Shuckburgh also described the feedback loop of Arctic sea ice melt due to changing albedo. Ice, being white ("albedo" comes from the Latin word for "white") has a much higher albedo than water, so sea ice reflects the Sun's energy much more effectively than the open ocean. When sea ice melts, the albedo of the Arctic Ocean is lowered, heating up the ocean. This in turn leads to more sea ice melt and even lower Arctic albedo.
After explaining some theoretical foundations, Shuckburgh described one of the experimental parts of her work: drilling into the ice in Antarctica to obtain ice cores that date from as much as 800,000 years ago. By analyzing the air bubbles in the ice, scientists can measure the amounts of CO2 and other gases in the atmosphere at various points in time. The ratios of isotopes of oxygen in water depends on temperature, so analysis of the water in the ice reveals past temperatures.
Shuckburgh's presentation included sobering graphs depicting long-term trends of CO2 and temperature, showing the correlation between high atmospheric levels of CO2 and warm surface temperatures. One graph in particular illustrated the stark contrast between historic fluctuations of CO2 levels and the increasing trend in the past few decades. That graph made it very clear that the current rise in atmospheric levels of carbon dioxide is not a part of the natural cycle of atmospheric greenhouse gas fluctuation.
Shuckburgh said that the Equator emits less radiation than one would expect based on how much energy it receives. She explained how fluid dynamics models of the atmosphere based on the well-known Navier-Stokes equations in a rotating frame can describe global wind patterns. These wind patterns then facilitate heat transport between the Equator and poles.
It is fairly common knowledge that the Arctic and Antarctic regions are very different: the North Pole is in the middle of water surrounded by land, and the South Pole is in the middle of land surrounded by water. Shuckburgh explained that global warming affects these two regions differently; in particular, the Arctic is losing sea ice and warming rapidly, while in the Antarctic, the changes are slower, and a few isolated regions are even becoming slightly colder as a result. These differences are due in part to the efficiency with which the Arctic sea transfers heat from the Equator to the North Pole and in part to the wind jets created by rising temperatures.
One of the most engaging parts of the presentation was Shuckburgh's description of an experiment done by her and her team to better understand the role of ocean eddies in the transport of heat from the Equator to arctic regions. Using satellite data, they developed a dynamical systems model for this transport, including an estimate of stable and unstable manifolds. The team released small floats into the ocean to test their predictions. The floats, which were named after members of the research group, were released near the tip of South America, and their paths were tracked for two years. Shuckburgh proudly announced that the float named for her has nearly reached Africa.
Shuckburgh said that although the captain of the ship from which they released the floats was dubious, the experiment worked, and the floats did track unstable manifolds. Their models were not entirely accurate; small perturbations led to greater changes in trajectories of the floats than predicted by the model based on satellite data. This was to be expected due to the limitations in resolution of satellite pictures.
Shuckburgh ended by talking about the high risk of rapid, irreversible climate changes now facing the planet. She mentioned the AVOID program, which has quantified the risk of permanent climate change and has recommendations for how to avoid this. According to their calculations, if global greenhouse gas emissions peak in 2016, there is only a 50 percent chance of avoiding a 2 degree Celsius rise in global temperature. But she said this is one of the only ways to avoid catastrophic increases in global temperature.
--- Evelyn Lamb
After Wednesday evening’s SIGMAA business meeting, Keith Devlin (Stanford University) started by telling the standard story of how modern arithmetic spread, which includes the story of Leonardo Pisano--known to us as Leonardo Fibonacci--and his influence on the rise of commercial modern mathematics in Europe. As a young person, Leonardo traveled with his father, an international businessman, to Bugia in North Africa, where he was first introduced to what we call the Hindu-Arabic number symbols—0 through 9—by Muslim traders. On returning to Pisa, Leonardo wrote a book, known as Liber abbaci, showing how to use these numbers to perform calculations that would be useful to merchants and to mathematicians. The first edition was completed in 1202, and, although it no longer survives, a second edition was written in 1228. The first publication of Liber abbaci made Leonardo famous and it seemed very likely "that it was the appearance of Liber abbaci, if not the contents of Liber abbaci, that instigated the sudden growth in modern commercial methods and accounting methods that came out of Tuscany in the 13th century onwards."
Then came the discovery in the 1960s of hundreds of mostly unattributed books written in vernacular Italian between 1300 and the 1500s. These books, which came to be known as abbacus books or abbacus tracts, were used to teach people the basics of commercial arithmetic. However, no direct link was made between Leonardo’s known works and these abbacus books until the 2003 discovery in a Florence library of a shorter, simpler book for merchants that Leonardo wrote after writing Liber abbacci. To learn more about this fascinating story, read Devlin’s book, The Man of Numbers: Fibonacci’s Arithmetic Revolution.
--- Claudia Clark
Photos on this page, unless identified otherwise, are by Sandy Huffaker.