Figure 1.

Three years ago, I wrote the “Word from …” column about book publishing at the American Mathematical Society (see Notices of the AMS, vol. 67, issue 7, August 2020). Today I want to address a broader question, which can be asked as follows: why does publishing mathematics books continue to exist as an important and valuable activity? In this column, I’ll try to answer this question. I should add that what I am writing here is coming from my personal experience of working for the Publication Program at the AMS, as well as from talking with my colleagues at the AMS and with many mathematicians and publishers of mathematics. Other people may give different answers to this question.

Why mathematicians write books

Mathematics is and always was a collective endeavor. Mathematicians, probably more often than representatives of other sciences, want to share these discoveries with their immediate colleagues and with other mathematicians. Over many years several tools to do this have been invented. In antiquity and the early Middle Ages mathematics was communicated through personal contacts, such as conversations with pupils and, later, in mathematical letters, which were quite common in Europe in the 17th and 18th centuries (e.g., Euler, members of the Bernoulli family). But there was also an urge to communicate with people beyond their immediate colleagues, and mathematicians used other available methods including clay tablets; writing on papyrus scrolls, on parchment, and on paper; and, finally, printing. As the number of mathematicians grew and technology progressed, printed materials superseded everything else, and, in mathematics, led to printed journals and books. In the 1980s, the internet appeared, bringing with it various new methods of communicating science (electronic books and journals, blogs, videos of talks, online seminars, and much more), and there seems to be no end to new inventions in this area.

Nowadays, whenever a mathematician proves new results, an article (or several) is written and submitted for publication to a journal and/or posted on the arXiv or a similar repository. In many cases the authors view this as “the end of the story” and move to the next challenge. However, from time to time a mathematician (or a group of mathematicians) feels a need to step back, to think about a broader picture, and summarize what was achieved and where the subject should go next. This is when they may decide to write a monograph presenting classical and recent developments in an important area of mathematics. This may also happen when a mathematician believes that the research has reached a peak and wants to organize what was achieved in the area. Oftentimes, writing a book allows the author to discover hidden patterns and simple ideas behind sometimes very long and complicated formal arguments.

For many mathematicians teaching is a significant part of their professional activity. Developing a new or innovative course involves a lot of effort, and a textbook based on such a course offers an avenue for sharing their insight and resources with others. The result might be a textbook on a standard undergraduate or beginning graduate topic (analysis, abstract algebra, topology, probability, etc.) or on a more advanced topic (such as Sobolev spaces, hyperbolic dynamics, analytic number theory, to name a few). The author’s hope when writing a textbook might be that it can help bring more students into mathematics and invite instructors to teach a new important topic. Of course, there is a wide gray zone: a research monograph can be used as text for a topics course, and advanced graduate textbook may contain an exposition of recent research.

Recently, a new format of sharing knowledge, which can be called a “collective online initiative,” emerged. This is an endeavor of a group of experts in a certain area of mathematics, where a lot of important information is scattered over hundreds or, sometimes, thousands of research articles in dozens of journals. The goal of such an endeavor is to collect and organize new results and approaches in one place to help others learn them and to facilitate further progress. Often, such collective projects are presented (published) online and may require well developed database-type coding that allows the user to find a particular topic within the area. Examples of such online projects are

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The Stacks Project: https://stacks.math.columbia.edu/

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The L-functions and Modular Forms Database (LMFDB): https://www.lmfdb.org/

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The On-Line Encyclopedia of Integer Sequences® (OEIS®): https://oeis.org/

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NIST Digital Library of Mathematical Functions: https://dlmf.nist.gov/

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Atlas of Lie Groups and Representations: http://www.liegroups.org/

To summarize, I would say that mathematicians write books because they want to share their findings and their experience both in mathematical research and in teaching mathematics.

Next, there is a question of why the American Mathematical Society publishes books. The answer to this question is simple: part of the mission of the AMS, as a society of professional mathematicians, is to advance research. Publishing books that present results of mathematical research or aim to help more people enter and master the profession or use mathematics in other areas (e.g., industry, business, public service, etc.,) is among the most direct and important ways to achieve this goal. The books the AMS publishes are chosen with this in mind, and this choice is made by professionals (both the AMS acquisition editors and the members of its book editorial committees). This is one of the main aspects of AMS publishing that distinguishes it from many other publishers of mathematics. I hope that publishing can be viewed as an important part of the AMS’s contribution to advancing research and teaching.

To conclude, I want to repeat what I said in my 2020 column. If you, as a mathematician and an author of a book, share the goal of advancing research and teaching mathematics, then publishing your book with the AMS is a step toward achieving these goals. Many books published by the AMS have a strong positive impact on the profession and on its appreciation by the society. We welcome any author who wants to contribute to our collective vision of mathematics and its role in science, technology, and beyond.