The Book
Math History: A Long-Form Mathematics Textbook shows readers how people throughout history thought about math. It discusses what problems they cared about, how they expressed those problems, and how they solved them. It also aims to show the growth of math over time. It discusses the problems that gave rise to new fields, the ideas that allowed those fields to flourish, and the theorems and conjectures that propelled them to their modern form. Finally, it communicates the human side of math—the people who impacted it and the world in which they lived, how various cultures differed in their approach, and how the mathematical project has spanned the globe.
The first four chapters of this book cover the development of number systems, ancient mathematical methods, the transition to proofs, and axiomatic mathematics with a focus on Euclidean geometry. The final six chapters each track the history of a specific topic. These topics are number theory, algebra, early calculus, modern analysis, topology and combinatorics. It surveys math pretty broadly, but its focus is on the math topics that an undergraduate math major will encounter. The book contains many theorems with their historical proof. Each chapter also includes exercises.
Following each chapter is a People’s History section which expands the telling of math history to include surveyors, navigators, artists, merchants and student collectives, who developed practical math which anticipated future research.
The first appendix is on the history of applied math; the second is a mathematician pronunciation guide; the third is a collection of mathematical quotes; the fourth is an etymological dictionary of math words; the fifth is a history of math symbols; and final appendix is a collection 37 biographical sketches of the many fascinating lives of history’s premier mathematicians.
Primary Sources
To explore some primary sources from math history, click here.
If you know of a good primary source that I do not include, please let me know!
Bibliography
At the end of each chapter I include some readable sources for those interested in exploring the material further. To see a fuller list of sources, including more technical papers, click here.
I may also post other teaching resources. If you have some to share, please send them to me!
Solutions to Select Exercises
Some solutions to the exercises can be found here.
Errata
For a list of errors in the book, click here.
If you have found an error in the book, please send me an email at LongFormMath@gmail.com to let me know. When errors are found, I will post them here.
Use at Universities
If you know of a school which has used this book in their courses, please let me know!
A New Style of Textbook
Traditionally, math classes were taught in the “sage on the stage” style, in which a professor lectures at students, keeping tightly to the material with little to no discussion. This is slowly changing, and research shows that it is changing for the better.
Proof-based textbooks are traditionally a dry list of definitions and results, with terse proofs of those results and not much else; “sage on the page,” if you will. These textbooks are typically written by the foremost research experts in the field. I have great respect for these experts, and one of the most challenging tasks in writing a book is the determination of what to include and in what order to present it. These experts did a tremendous job on this, and without their work authors like myself would be unable to write the books we do.
That said, I believe a shift is beginning to occur in the textbook market. As prices continue to climb and mega-corporations are consolidating larger and larger slices of the industry, something has to give. Moreover, upper division math textbooks seem to be written for only the strongest and most self-motivated students. There are about 20,000 bachelor degrees given each year in mathematics and statistics at US institutions. And according to the AMS, there are only about 2,000 PhDs given in these fields per year. So even if one argues that reading terse math is proper because math research articles are (regrettably) written tersely, that argument ignores 90% of undergraduates for whom we want to understand and appreciate the material. Furthermore, it is always worthwhile to remind ourselves that only about half of the 10% that do go on to graduate school end up working in academia. So, as pointed out by Stan Yoshinobu, professors are peculiar, and we shouldn’t fall into the trap of only teaching only to the students we were.
I have argued that there is a need for a new style of textbook, and fortunately I came to this realization just as the gatekeepers of publishing were critically weakened by the advent of self-publishing and online marketplaces. I am not an analyst. I really like analysis, have taken half a dozen semesters of analysis and have self-studied on top of that, but I have never published a single paper in the field nor do I claim to be an expert. A decade ago there is no way that Springer would approach someone like me to write an analysis textbook, however today Amazon’s self-publishing arm couldn’t be happier to print my book. A decade ago I would have had enormous difficulty bringing my book to the attention of others, but with nearly a hundred million Amazon users in the US and even more worldwide, the only gatekeepers are my readers (please rate and review!). I expect this reality to change the publishing market in the next couple decades, as more educators self-publish their own takes on the classic textbooks.
Table of Contents
Scanned Sample of the Book
Will add soon.