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HoM Toolbox, or Historiography and Methodology for Mathematicians: Introduction – What is History?

Amy Ackerberg-Hastings (MAA Convergence)


Just like mathematicians, historians strive to articulate precise definitions. Thus, words that may be used interchangeably in everyday life need to be clarified and distinguished. In particular, “history” and ”the past” are not the same thing. For historians, the past consists of everything that ever happened. It is humanly impossible to record and report every detail about any or all past events—imagine for a moment that you decide to write a history of your life by noting each thing you did and every word you said. But we all experience dozens of events in just a single day that are lost to our memories almost as soon as they happen . . . and even if you could keep track of every minute happening, your readers likely would be bored to death by your account, and documenting your life would take so long that you would no longer have time to actually live your life. The past, therefore, is not history.

In addition to being necessarily selective, history involves answering questions—most notably, the 5 Ws and 1 H: Who? What? When? Where? Why? How? The first four questions are mainly informational, so it may seem fairly straightforward to collect facts from the sources that were created during the time period a historian is studying. (We will see on the next two pages that this assumption is actually an oversimplification.) Indeed, the “how?” and “why?” questions are the ones that move us from the past to history, because people in the past rarely were self-aware enough to write down their true motives or to document all of the processes that influenced their actions. In other words, historians must make inferences from the sources in order to draw conclusions about the meaning and significance of past events. These conclusions, or interpretations, comprise history.

Infographic with the 5W1H question words used by historians of mathematics.
Figure 4. Historians of mathematics ask many, many questions.
Resource created for Teachers Pay Teachers by Mr. Dignan’s Desk.

The history of mathematics, then, is the effort to understand how and why mathematics developed over time through identifying a representative selection of primary-source evidence created in the past and relevant to the topic whose history is being written, putting the information documented by the evidence into context, identifying relationships between the evidence, and drawing conclusions about how and why mathematics unfolded as it did. These interpretations give us insights into how human beings think, act, and relate to each other. Doing history thus helps us discern not only how mathematics evolved into the forms we know today, but also why mathematics became the way it is—because human beings made choices about what to study, where to share their knowledge, who could be involved, how to connect abstractions with the physical world, why the subject matters, and so on. While historians of mathematics, like all historians, must rely on the memories of people from the past—memories that could be incomplete, faulty, or biased—they use critical reading and thinking skills to sort through and interpret the evidence. This is sober, thoughtful work grounded in the effort to understand; historians avoid making assumptions as much as possible.

Ivor Grattan-Guinness explaining the history of mathematics.One of Grattan-Guinness's famous articles on the difference between history and heritage in mathematics.
Figure 5. In obituaries such as the one posted by Bocconi University in Milan, Ivor Grattan-Guinness (1941–2014) was shown explaining the history of mathematics. He wrote numerous articles on the differences between “history of mathematics” and “heritage of mathematics,” including [Grattan-Guinness 2004].

Assumptions about the past are often connected to a shared sense of memory that is usually called “tradition” or “heritage”. Ivor Grattan-Guinness wrote several essays (such as [2004]) explaining the difference between history and heritage while challenging readers to create sound history; see also [Tosh 2021]. Heritage typically celebrates the perceived heroes and highlights of a culture's or nation's past, which has the positive effect of providing a community with a common identity. For instance, Americans like to remember that the entire United States was outraged by Japan's attack on Pearl Harbor and went on to fight a good and honorable war, in which any non-heroic or bad actions that happened before or during the conflict were forgotten (e.g., the antisemitism of Charles Lindbergh and Henry Ford). Selectively mining the past for heritage thus leads researchers into at least three pitfalls:

  1. Thinking of the past as we want it to have been (nostalgia) instead of asking what really happened, according to primary source evidence;
  2. Using assumptions about the past to guide behavior in the present (upholding supposed traditions);
  3. Critiquing past people for not thinking the way we do in the present (both a neglect of context and a belief in constant and inevitable progress).

In contrast, approaching the past with the mindset of a historian includes:

  1. Recognizing that what motivated people in the past may be different from what motivates people in the present;
  2. Considering people in their original political, social, economic, cultural, intellectual, and religious settings or context;
  3. Tracing patterns of change and continuity over time.

These activities begin during the collection and analysis of historical evidence, as we will see in the next section.

Amy Ackerberg-Hastings (MAA Convergence), "HoM Toolbox, or Historiography and Methodology for Mathematicians: Introduction – What is History?," Convergence (December 2022)