In a Tortoiseshell: In the introduction to his junior paper in mathematics, Dylan Galt presents the motivating questions underlying his exposition of Stiefel-Whitney classes and cobordism, preparing the reader to appreciate the significance of the rich mathematics which will follow. Continue reading
As a Writing Center Fellow, I believe that good writing is necessary in all fields. However, it can be easy to conceive of writing (as I’m sure most people do) as an inherently humanistic act or practice. Writing in STEM fields is only a necessary way of communicating ideas, not intrinsically part of the discipline.
However, as I read G.H. Hardy’s essay “A Mathematician’s Apology” and Karl Popper’s lecture “Science: Conjectures and Refutations” for ENG 401 Literature and Science, I discovered that both Hardy and Popper describe “good” mathematical and scientific ideas in ways strikingly similar to how we at the Writing Center describe good theses. The foundation of a good argument, it seems, is consistent across disciplines, and we can use the standards provided by Hardy and Popper to inform our writing as much as our scientific or mathematical research.
In “A Mathematician’s Apology,” Hardy discusses what makes a mathematical idea “significant.” Hardy writes: “We can say, roughly, that a mathematical idea is ‘significant’ if it can be connected, in a natural and illuminating way, with a large complex of other mathematical ideas” (89). While we can quibble with exactly what Hardy finds significant or not in his essay, this basic definition of significant — “connected, in a natural and illuminating way, with a large complex of other mathematical ideas” — can be useful when thinking about a motivated thesis. Ask yourself: Does your thesis connect to a larger conversation of ideas? What exactly does it illuminate in that conversation?
In “Science: Conjectures and Refutations,” Popper articulates what makes a theory or idea “scientific” (versus “pseudo-scientific”) and, like Hardy, describes a good thesis statement in the process. Popper summarzies his conclusions in one line: “the criterion of the scientific status of a theory is its falsifiability, or refutability, or testability” (37). Here, Popper describes an essential element to a strong thesis: arguability. For a thesis to be good, someone must be able to argue against it; it cannot describe a factual state of being. Theses which rely heavily on plot summary or observable facts tend to veer into inarguable territory. Check yourself by asking: is there a counterargument to my thesis? If I had to write another paper disagreeing with myself, what might I say?
Hardy’s definition of a “significant” mathematical idea and Popper’s conception of a “scientific” theory can be used to understand what makes a good thesis. These criteria relate to Keith Shaw’s four-step thesis test:
- Is the thesis arguable? Can a reasonable person argue against it? Popper uses this standard for determining whether a theory is scientific.
- Is the thesis manageable? Is it responsive to the evidence at hand and suitable for the size/length of the paper?
- Is the thesis interesting? Does it address a question/puzzle/contradiction and go beyond the obvious?
- Is the thesis important? How is the claim significant in the context of the field? Hardy uses the term “significant” to describe an important mathematical idea.
The questions we ask at the Writing Center about what makes a good thesis statement are the same questions mathematicians and scientists ask about what makes a good argument in their fields. Rather than simply a form of communication, argumentative writing is in the same category as scientific hypotheses and mathematical theories, another form of the effort to argue and prove a new way of thinking about the world.
— Paige Allen ’21
Hardy, G. H.. A Mathematician’s Apology, Cambridge University Press, 2012. ProQuest
Popper, Karl R. “Science: Conjectures and Refutations.” Conjectures and Refutations: The Growth of Scientific Knowledge, Routledge, 2002, pp. 33–41.
It sometimes happens during the research process that I come to a point where I think I know what I should propose or argue, but I can’t see how to prove or disprove it. This happens fairly often when I’m writing papers in the humanities, but even more so when I am trying to solve a problem in mathematics. In fact, I ran into this issue while trying to prove one of the claims on my most recent math assignment. The goal was to prove the following claim:
(I include the claim for completeness, but it’s not terribly important.) My first step was to reduce the claim to a statement that seemed easier:
which I was able to do just by using the information given in the problem. After that, I was able to get nearly three-quarters of the way through the proof just using the given information. Then I ran into something I knew should work, but wasn’t able to verify:
Because I already knew what I was trying to prove, I was able to use the assumptions I described above to complete the problem (more or less). There was still a hole in my argument, but I was able to construct a solid “scholarly narrative” for the problem (that is, finish the proof) by carefully delineating what I understood and where my understanding went off a cliff. Writing out exactly what I thought was wrong also told me what I needed to learn the next time I worked on the topic — effectively, it told me what further research I needed to do. So, even though I wasn’t quite sure what was going on, I was still able to decide what I would need before I made a second pass over the problem.
There are, of course, several caveats to what I have said thus far. First, in this case, I knew that my overall claim was correct, so I was confident I should be able to jump over the hole in my reasoning and complete my argument. In general scholarly writing, we don’t know with certainty that our thesis is correct, so a question like the one above could throw the entire premise of the argument into doubt. The second caveat is that clearly, my argument couldn’t have been called “complete” while containing the excerpt above. Since there was a hole in my argument, I hadn’t really proved anything yet. Indeed, the kind of serious gap in understanding displayed above was only permissible because the homework assignment was the equivalent of an early draft.
But despite these two cautionary notes, annotating incomplete arguments as I do above has often proved helpful (and necessary) for me in both mathematics and the humanities. By whittling down the “unknown” part of the problem to a single nugget that we are not equipped to attack with our current tools, we mark the bounds of our own knowledge, and in doing so, lay the groundwork needed to push those bounds even further than before.
–Isabella Khan ’21
In a Tortoiseshell: In the introduction to his interdisciplinary senior thesis merging Game Theory and Latin American Studies, José L. Pabón effectively orients his readers to the structure and motive of his paper. By first providing a succinct outline, which he expands on in the following paragraphs, he prepares the reader for the content of his thesis. Then, he pivots smoothly into a discussion of his underlying motive in writing this thesis, introducing the reader to the perspective he will adopt in his argument, and deftly presenting the material in such a way as to capture the reader’s attention and make him or her immediately sympathetic to the arguments and analysis presented in the rest of the essay.
Let’s face it: there is a divide between the humanities and STEM fields. As an undergraduate at Johns Hopkins, I felt like I was on the losing side of a battle. I was a triple major in some very different fields: mathematics, creative writing, and French literature. However, this combination was not always well received by professors, and much less by administration. I still recall how the registrar’s office spent an hour telling me how it was not possible to complete a degree with three majors, even though I had finished almost all the coursework and had the appropriate signatures!
As a doctoral candidate, I have been able to combine these interests into a doctoral dissertation about the mathematical methods of an experimental group of French writers, the Oulipo. While it’s been a joy for me to investigate the subtle channels that lead from mathematics to literature and back again, for some hardcore literature scholars, using the word “mathematics” evokes fear and anxiety, which brings me to a big question: how do you write about one discipline to people who have been trained almost exclusively in another? More specifically, how was I to write about mathematics for people who took their last mathematics class in high school?
Below are some of the strategies I’ve found that help me deal with these challenges. Hopefully my experience can be beneficial to others who intend to fly in the face of disciplinary boundaries at some point in their studies.
In my project, I have found that defining a clear thesis early on was key, and that the thesis had to be motivated not just by the interdisciplinary nature of the project but grounded in my own discipline. To use the magic thesis statement (MTS):
By looking at the mathematical methods of the Oulipo, we can see
- The historical reasons behind these choices and how the Oulipo situates itself within a literary tradition that precedes it as well as within the larger history of mathematics,
- The way the mathematics changes the compositional methods of these authors, and
- How this changes the reading experience, which most people don’t see.
This is important because it helps us understand:
- The Oulipo’s singularity among other 20th century French experimental groups,
- The Oulipo’s influence on mathematics and computer science, and
- How mathematics and literature are complementary, and how one can influence the other.
As you can see from how I divided my MTS, I found it necessary to break down my argument and motive into their constituent parts, not only because a doctoral dissertation is extremely long and therefore requires a more complex thesis, but to facilitate the reading experience for my audience. To that end, I have three main parts of the thesis:
- History (both literary history and history of science/mathematics)
- Compositional methods (rhetorical analysis, the heart and soul of literary scholarship)
- And finally the reader (situating the Oulipo within a theoretical context)
In the last part of my MTS, I broke the motive down into three parts, again for my intended audience, which bridges disciplinary boundaries:
- Why this matters for literary scholarship of 20th-century France (or Europe in general)
- Why this matters for historians of mathematics or science
- Why this matters for larger disciplinary questions
This brings me to orienting. The primary reader of my dissertation is my advisor, who is unlike any other literary scholar in that he is not afraid of mathematics! As a translator, he had to learn a great deal about mathematics and the Oulipo to translate the works of Georges Perec, and he has continued to do a great deal of work on the group and mathematics. That said, my other readers are not well acquainted with mathematics, so I felt that I needed to assume a novice reader. Now the question was, how can I orient such a reader without talking down to him or her? Paradoxically, I came to the answer in preparation for my prospectus defense, during which I had to give a brief presentation about my work and answer questions from the French Department faculty. To get them thinking mathematically, I realized, they needed to do mathematics!
So I walked them through a proof, a simple one to say the least. Beginning with a mathematical anecdote to hook them in, I explained the mathematical reasoning that Gauss used as a child and how to abstract that into a theorem about sums of positive, consecutive integers. Then, I linked these notions of formal language and axioms to literature through the Oulipo and a few key texts. Mathematics is the study of abstract patterns, and literary sensibility as well depends upon a human capacity for pattern recognition. In that sense, the reader of my dissertation—regardless of his or her mathematical background—already possesses all the tools needed to decipher the Oulipo’s use of mathematics! It just requires a different kind of reading and—as I have learned—writing.