The Beginnings & Evolution Of Algebra, by Isabella Bashmakova and Galina Smirnova

At first glance, one would not think that a book burdened with large amounts of equations and an extremely dry approach would contain particularly useful insights on areas outside of algebra, but one would be wrong. This book, it should be noted, is not an easily accessible one to the reader, and it assumes a great deal of familiarity with cutting-edge algebra research that is far beyond what most students of mathematics are likely ever to conceive of. My father once told me that he had done well at math until he got to algebra, and then he struggled because it required a level of abstraction that he just was not able to understand, and I do not think he was bragging about his ability to deal with more concrete forms of mathematics nor his difficult struggle with abstract algebra. It is not coincidental that the authors discuss (in one of the book’s appendices) the fact that wherever algebra appeared in the world, it appeared first in its geometric form simply because the human mind does not automatically or easily deal with abstraction. And if there is one thing most of this book is full of, it is increasing levels of abstraction as algebra has progressed, and where the ability to assimilate and handle that abstraction failed, algebra correspondingly failed to develop, especially because it was hard for mathematicians to understand cutting-edge work, whose proofs and insights they mistrusted, and because mathematicians themselves got caught up in all kinds of bloody wars and duels and squabbles over politics and religion, demonstrating that even mathematics is as full of human frailty as any other endeavor.

In terms of this book’s contents, the order is topical along a mostly chronological order. The book begins with a discussion of the origins of algebra in ancient Babylon [1] and then looks at algebra in ancient Greece and the first unsolvable problems the transformation of algebra into an abstract deductive science. Then there is a look at the birth of literal algebra with a focus on Diophantus. The next chapter examines Algebra in the Middle Ages in the Arabic realms and in Europe, followed by a chapter on the first achievements of algebra in Renaissance Europe, with a particular focus on Bombelli and Viete. The authors then turn their attention to algebra in the 17th and 18th centuries with a focus on increasing arithmetization and the work of Descartes and Gauss. At this point the book takes a more topical turn, first looking at the theory of algebraic equations in the 19th century with the work of Abel and Galois and the victorious “march” of group theory and then a look at problems in number theory and the birth of commutative algebra and ideal theory. After this the book turns its attention to linear and noncommuative algebra, remaining in the 19th and early 20th centuries, and then concludes and discusses a revisionist view of ancient geometric algebra.

Much of this book is written very dryly, and it is sometimes hard to appreciate the insight the authors have. In the midst of summarizing an important theorem, or the assumptions that were made about continuity and comments in praise of abstraction, the authors have a lot to say of great value concerning the fate of mathematics and mathematicians. The authors draw the sensible conclusion that the development of mathematics requires a lot of elements, including the preservation of writing about math, and the ability of insights and learning to be communicated in a way that can be understood by others. Additionally, the authors point out that development in mathematics requires a certain optimism and a certain educational infrastructure. Often in this book the insights of lone geniuses are lost because there is an absence of others who can understand it, or because they die young in absurd duels. This book is a very cerebral and abstract history, and it likely would be a difficult read for anyone who is not very interested in both math and history, but for those who are, this book offers some rich insights intermixed with its difficult prose and dense mathematical framework.

[1] https://edgeinducedcohesion.wordpress.com/2016/02/28/book-review-the-babylonian-theorem/