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HoM Toolbox, or Historiography and Methodology for Mathematicians: Introduction – How Do We Know About the Past?

Amy Ackerberg-Hastings (MAA Convergence)


Sources of information about the past fall within three categories. Historians chiefly utilize primary sources, written, audiovisual, and other records created at or near the time of the events under study by the participants in the events. Several examples are shown below; note that primary sources can be created from a variety of materials—they are not all pieces of paper, and physical artifacts may be as informative as the words in written documents.

As historians process and interpret the evidence from primary sources, they create secondary sources, interpretations or reconstructions of the past. If they cannot access all of the primary sources relevant to their topic, historians may consult other secondary sources that discuss those materials. As a matter of courtesy and as the main means of furthering the conversations among historians that drive the discipline of history forward, authors will also acknowledge, extend, and critique the interpretations made by other historians in other secondary sources. We will revisit these conversations when we discuss approaches to historical interpretation. Another essential characteristic of a secondary source is that it will contain a thorough bibliographic trail (through footnotes and a bibliography) that allows other scholars to verify and evaluate the author's conclusions—so, historians “show their work” just like math students, and they invite their readers to join the conversations among historians.

Researchers may also come across tertiary sources, which are accounts of past events based entirely upon secondary sources without consultation of primary sources. These authors are simply sharing information that is already known without developing original interpretations of their own. Textbooks and encyclopedias are standard examples of tertiary sources, although popular history, documentaries, and historical fiction may also fall within this category.

The book, Newton's 1687 Principia Mathematica, is a primary source.The academic journal, American Journal of Mathematics, is a primary source.This letter by Florence Nightingale is a primary source.This manuscript with diagrams by a medieval student is a primary source.
This map drawn by Oronce Fine is a primary source.This newspaper clipping from the 18th century is a primary source.This geometric model from the late 19th century is a primary source.This photograph of the mathematician Li Shanlan with students is a primary source.
Figure 6. Some of the many types of primary sources in the history of mathematics,
as collected in Convergence’s Mathematical Treasures.

Just as you may have learned to do in a library skills class, historians assess the quality and reliability of all three categories of sources. What are the physical characteristics of a source, and what do those characteristics tell us about how and when it was made? How do we decide whether a source is authentic? (Feel free to take a break to check out this forgery in Mathematical Treasures.) What can we ascertain from the content of a source: are authors factual, free from error, explicit about their motives and biases, and so on? How does the source compare to other sources on the same topic?

History educators have developed a plethora of tools for collecting information from primary sources; two of my favorites are the (US-history-focused) websites History Matters by George Mason University and Getting Started with Primary Sources by the Library of Congress. I modified a worksheet from [Drake and Drake Brown 2003] for distribution to my own students as this Document Analysis Form and Object Analysis Form. (See [Ackerberg-Hastings 2019] for further discussion of primary-source analysis in the mathematics classroom.) The Convergence article, Online Museum Collections in the Mathematics Classroom [2014], offers a number of suggestions for teaching with historical mathematical objects, while [Wardhaugh 2010] guides readers through the unique challenges posed by making sense of historical mathematical writings.

An excellent case study from the history of mathematics for the analysis of primary sources—as well as the accidental way many primary sources survive to be discovered by researchers—is the Archimedes Palimpsest. It was copied in the 9th century, erased in the 13th century by thrifty scribes, re-discovered by Johan Ludwig Heiberg in the early 20th century, vanished (again) around the outbreak of World War I, returned to the public eye by an unidentified seller and buyer in 1998, and read through a variety of X-ray and digital techniques by a team assembled by the Walters Art Museum in Baltimore. The primary sources you work with may not have such a dramatic backstory, but they will still play an important role in helping you and your readers better understand the history of mathematics. On the next page, we will talk about taking the insights you gained from a set of primary sources and shaping them into a historical interpretation.

A portion of the Archimedes Palimpsest before imaging to uncover the scientific text.The same portion of the Archimedes Palimpsest after imaging, revealing a mathematical diagram.
Figure 7. Imaging techniques helped reveal the mathematics text and diagrams hidden beneath religious texts.
An introduction to the Archimedes Palimpsest is available in Convergence’s Mathematical Treasures.

Amy Ackerberg-Hastings (MAA Convergence), "HoM Toolbox, or Historiography and Methodology for Mathematicians: Introduction – How Do We Know About the Past?," Convergence (December 2022)