Commonwealth School teachers bring an infectious intellectual energy to their classrooms, fueled, in part, by their own innate curiosity. What happens when that curiosity is unleashed? The Hughes/Wharton Fund for Teachers aims to do just that, ensuring faculty can pursue their academic passions, access fulfilling professional development opportunities, and have the latitude to create new courses and reinvigorate existing ones. 

"I am a pure mathematician, not a historian," demurs Al Letarte. "It is the history of deductive thought as a basis for truth that I find most compelling." Even so, if you enjoy thinking about the origins and underpinnings of math (in addition to crunching the numbers), then you've found a compatriot in Mr. Letarte. For his most recent Hughes/Wharton project, he developed a new Mathematical Logic course, which he began by broadening his understanding of the global history of math, breaking from the largely Western voices that dominate the field. By following those diverse pathways, Mr. Letarte uncovered new ways of thinking about the philosophy of math—as well as his own identity as a teacher.

I ended up spending all of this summer’s book budget on the history of mathematics. I had not known, going into the project, that this was the direction my work would take, but when I began searching for readings, I ran across some unfamiliar sources that pulled me in unexpectedly exciting directions.

The readings on my list fell into three categories. First were well-known works that tell the story of mathematics from a largely Western point of view: an updated edition of Carl Boyer and Uta Merzbach’s A History of Mathematics in which the short chapters on ancient African, Chinese, Indian, and Islamic mathematics felt as though they had been added later; two volumes of essays for a general audience by Howard Eves under the title "Great Moments in Mathematics"; and a new book by Snezana Lawrence entitled A Little History of Mathematics, again pitched for a general audience and from a mostly Western perspective.

Next was a two-volume work about ancient Greek science by G. E. R. Lloyd that I acquired on the suggestion of Allison Allen [teacher of mythology for Dive In Commonwealth]. I am interested in this topic in part because of the obvious overlap between mathematics and science—but more importantly because I suspect that the Greeks’ thinking on the relationship between mathematical truth and every-day reality must have been tied to their views on the nature of science.

The third reading riveted my attention completely, drawing me away from the other two and consuming my summer. In search of interesting sources, I had come across the Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences, a massive two-volume work comprising 176 articles by historians of mathematics around the world. Collectively spanning ancient civilizations from Babylonian, Egyptian, Greek, Indian, and Islamic to African, Mayan, Chinese, Tibetan, Korean, and Hebrew, the articles proved concise, yet scholarly. Each came with an extensive bibliography—enough to occupy me for many future summers (and perhaps guide students in Medieval World History who have chosen the history of mathematics as a research topic). I at first tried reading them in order but soon found myself jumping around. I am a pure mathematician, not a historian: I am guided in my historical pursuits by my mathematical appetites. Not every past mathematical tradition has appreciated or embraced the deductive approach that I was brought up to love. Ancient Egyptian mathematics, for instance, was procedural and computational in nature, and it appears that the same can be said of mathematics in ancient China. The practice of wasan in seventeenth-century Japan focused on elegant and elaborate applications of computational procedures, but seemed to contain little in the way of deductive reasoning. The medieval Islamic mathematical tradition, on the other hand, built directly on the deductive foundation laid by the ancient Greeks and carried it further. It is the history of deductive thought as a basis for truth that I find most compelling.

Later in the summer, I came upon a trove of articles that traced the historical and philosophical development of logics, set theories, and the foundations of mathematics from George Boole and Giuseppe Peano to Alfred North Whitehead, Bertrand Russell, Alonzo Church, Kurt Gödel, Stephen Kleene, and Alfred Tarski. I have always found, perhaps because mathematical logic is such a young field (my own life overlaps those of most of the mathematicians on this list!), that the subject is difficult to grasp properly if one does not understand how logicians’ philosophical thinking developed and evolved between about 1840 and 1960. From the moment one embarks on a study of logic, the bewilderingly complicated interplay among the different branches of the subject becomes evident. For example, in the mathematical logic elective this fall, we were faced immediately with the following chicken-and-egg dilemma: all of mathematics can be expressed in the language of set theory. When we do any sort of mathematics, we are working in what is called a model of set theory. In particular, the development of model theory (the main focus of the course) takes place formally within a model of set theory. But we need model theory in order to understand what a model of set theory is. So where do we begin? In the course, we will address this dilemma by assuming that we already inhabit a world in which sets exist as “collections of objects” and behave in the way our intuition tells us they should. Later on, once we have established the necessary model-theoretic machinery, we can go back and discuss set theory more formally. (But what if there is no model of set theory? Then the entire course is a cruel joke!) Over the past 150 years, logicians have adopted sometimes competing perspectives on the philosophical status of mathematical truth. Tracing the development of their thinking through these articles is helping me better understand the subject as a whole. I look forward to sharing some of my insights with students.

I have noticed a shift in myself lately: It used to be that I found my identity as a faculty member at this school in what I did: designing newfangled courses, teaching advanced electives, and so on. Increasingly these past few years, it is instead about who I am: a person who thinks, studies, asks questions, examines interesting ideas, and questions old assumptions. I am very fortunate to work at a school that values these qualities in its students and faculty alike.

Explore Math at Commonwealth