Back in 2020, I penned an extremely favorable review of Benjamin Wardhaugh’s captivating work, The Book of Wonder: The Many Lives of Euclid’s Elements. This prompted me to also write a positive review of Reading Mathematics in Early Modern Europe, a volume that Wardhaugh contributed to and edited. This was succeeded by a brief blog entry about the research initiative, Reading Euclid, from which both books originated.
Recently, I was delighted to receive an email from Benjamin Wardhaugh inquiring if I would be interested in obtaining a copy of his latest book, Counting: Humans, History and the Infinite Lives of Numbers (William Collins, 2024). Given my significant interest in the history of numbers and the quality of Wardhaugh’s Euclid book, I naturally agreed. I have not been let down.
I previously mentioned my fascination with the history of numbers, yet as Wardhaugh highlights, while numbers have been and are used in counting, they are not always essential for such tasks, and there has historically been ample counting without the inclusion of numbers. At one point, I found myself chuckling, as I skimmed through the book to get a general sense of it; I perceived this as being much more about anthropology than history. That was when I first paused to read the quote from The Times on the front cover:
“An anthropological sweep through mathematical history from the Stone Age to the cyber age via six continents.”
Accounts of science or science-related subjects usually adhere to a linear chronological structure and strive to be thorough. One should not expect this from Wardhaugh’s engaging, intriguing, and at times provocative exploration of the history of counting. Why is that? Wardhaugh provides the answer in the introduction, which itself is not a continuous narrative but a collection of fragmentary quotes that reflect aspects of what counting is or could potentially be. In response to the question, “But what is it?” Wardhaugh informs the reader:
‘Counting’ can appear as a chaotic assortment of largely unrelated actions; a term that encompasses a vast array of very distinct cultural practices. The variety of activities grouped under counting seems far too broad for comfort, and at least on the surface, it is not obvious what they all share in common; or even if they share anything in common whatsoever.
Nearly any definition of counting proves to be problematic, but one of the most effective is credited to the seventeenth-century German philosopher Gottfried Leibniz. He posited that counting signifies repeated attention. Counting occurs when you think ‘this…this…this…this’ and have a method of keeping track.
Wardhaugh continues by telling us:
Counting does not possess a singular history.
[…]
The narrative of counting is instead shaped like a tree. It has multiple roots, numerous branches, and countless twigs and leaves. Counting has evolved and spread alongside the human species, branching out into nearly every culture, both past and present. Sometimes it is feasible to trace a single branch for a considerable distance: at other times, a branch appears to intersect, to touch (or nearly touch) other branches.
In this book, Wardhaugh guides his readers on a journey that initially dives into the root system, then ascends along some of the branches, traversing time and continents; pausing to investigate a twig, pluck a leaf, or taste some ripe fruit. The book is not structured as a continuous narrative but is organized into eight sections, two of which explore the roots, while the other six navigate through the diverse branches.
Each of the eight sections comprises a collection of self-contained essays, each addressing an example or an aspect of the topic indicated in the section title. In those sections that focus on counting examples, which constitute most but not all of them, Wardhaugh meticulously provides details about who is conducting the counting, what is being counted, how it is being counted, and importantly, why it is being counted. Through this method, he clearly illustrates that counting is, as he stated in his introduction, ‘an unruly grab-bag of almost totally unrelated actions’.
The initial Roots section does not address counting directly but focuses on Number sense prior to counting and contains three essays on how individuals estimate the size of groups of objects without performing counts and whether this skill is exclusive to humans or if animals share this intrinsic capability.
The second Roots section, Counting before writing, presents three essays regarding artifacts discovered in prehistoric Africa that may have played a role in the counting process. Wardhaugh provides compelling arguments for such utilization but also cautions that ultimately, the claims remain speculative. A fourth essay thoughtfully speculates on the potential origins of counting words.
Between the Roots sections and the Branches, there exists an interlude discussing numbers and their nature.
We now transition into the domain of counting in a section titled, Counting with words and symbols in the Fertile Crescent. The first essay in this section examines the Sumer