Back in 2020, an exceptionally favorable critique was published regarding Benjamin Wardhaugh’s book, *The Book of Wonder*: *The Many Lives of Euclid’s Elements*. This ignited further interest in Wardhaugh’s other writings, including a commendatory review for *Reading Mathematics in Early Modern Europe*, a project he both edited and contributed to. This path led to the investigation of the research project, *Reading Euclid*, from which these books emerged.
Recently, Wardhaugh sent out a request to review his newly released work, *Counting*: *Humans, History* and *the Infinite Lives of Numbers*. Given the deep fascination with numerical history and the earlier brilliance displayed in Wardhaugh’s work on Euclid, accepting this invitation was unavoidable, and the material certainly met expectations.
The charm of numerical history enthralls readers, as Wardhaugh expresses that while numbers aid in counting, they are not solely essential, since counting can occur independently of numbers. Upon skimming through the book and capturing its essence, it became clear that it appeared almost anthropological in character rather than strictly historical, underscored by a quote from The Times on its cover: “An anthropological sweep through mathematical history from the Stone Age to the cyber age via six continents.”
Histories of science typically adhere to a systematic, chronological format, striving for thoroughness within certain bounds, but Wardhaugh’s exploration of counting history enchants by deviating from these conventions. His introduction doesn’t convey a seamless narrative; rather, it provides disjointed quotes that depict various dimensions of counting. Wardhaugh highlights the variety in counting methodologies, comparing it to “an unruly grab-bag of almost totally unrelated actions.”
Defining counting is complex. Wardhaugh cites a definition ascribed to the 17th-century German philosopher Gottfried Leibniz: “Counting is repeated attention… Counting occurs when you think ‘this…this…this…this’, with some method of tracking.” Additionally, Wardhaugh contends that counting lacks a singular history, comparing it to a tree with roots and many branches, symbolizing its intricacy and integration into human culture and history.
Wardhaugh steers readers through the historical origins, branches, and twigs of counting, moving across time and geography, unpacking intricacies with vivid exploration. The book is not a continuous story but is segmented into eight sections, with two sections focused on roots and six on branches. These sections contain self-contained essays that delve into various examples or dimensions of the main topic. Each section is meticulous about who counted, what was counted, the methods involved, and importantly, the motivations behind counting. Through this format, Wardhaugh underscores that counting is indeed a varied “grab-bag.”
The opening Roots section does not directly engage with counting. Instead, it investigates “Number sense before counting” through essays on estimating group sizes without counting and whether this ability is unique to humans or shared with animals. The following section, “Counting before writing,” presents essays on artifacts from prehistoric Africa that may have been utilized for counting. These assertions are speculative according to Wardhaugh, yet reinforced by robust arguments. Further essays investigate the intriguing but speculative origins of counting words.
Between the Roots and Branches lies an interlude that discusses the nature of numbers.
The exploration transitions into counting with “Counting with words and symbols in the Fertile Crescent”. Here, the earliest numeral system from the Sumerians is examined. The narrative remains in Mesopotamia to investigate figures like the monarch Tiglath-Pileser I, incorporating personal anecdotes to depict counting within cultures. This section shifts from specific narratives to broader cultural insights, revealing numeral systems and their utilization in various societies.
Delving into the next Branch labeled “Counters,” the journey commences in Ancient Greece, featuring insights into Athenian juror Philokleon using counters in court. The section broadens to encompass Greek and later Roman counting practices. Essays illustrate the Roman method of representing numbers with hands and fingers and explore counting boards, prevalent in early civilizations and exemplified by Blanche of Castile’s thirteenth-century account oversight.
With each Branch, Wardhaugh traverses historical avenues and scrutinizes distinct facets of counting, guiding readers through the progression of numerals and symbols, cultural interactions, and technological developments impacting counting.
Wardhaugh pauses with an interlude regarding Number symbols, offering additional insights.
Subsequent sections investigate East Asia’s counting devices, including Chinese rods, the *suanpan*, Japanese *soroban*, and Korean *jupan*, famously referenced in competitions like that of Kiyoshi Matsuzaki utilizing a *soroban* against a calculating machine, showcasing mental arithmetic without physical instruments.
Essays extend from Japan to America, covering early electric tabulating machines by Kawaguchi Ichitaro and Herman Hollerith, key figures in the annals of tabulation technology and precursors to IBM.
Wardhaugh even explores online counting spheres, such as social media “likes,”