How Mathematics is Redefining the Humanities

There should be no denying the importance of the humanities. And yet, many do deny its importance within dimly lit meeting rooms devoted to secretive budget discussions. What can we cut to save money? The arts and humanities seem like ripe targets for cuts. Do we need them? Search for “humanities funding” or “humanities cut” and you will find a treasure trove of results. Here is something from May 2017 where a part of SUNY Stony Brook was under attack in terms of its arts budget. Look at the above chart from Humanities Indicators (go to to NEH budget Request versus Final Appropriation document at the top of that page’s list of documents). The web is rife with the ongoing obsession with defunding the humanities. But what are the reasons offered for doing the cuts? We do love to read and listen to music — neither of these would be possible without the humanities. The very idea of the Western university is built upon the Greek tradition of inquiry — the gem of philosophy.

The humanists have been active in defending themselves. There is a recent report from Harvard where they observe that “Faced with evidence of failing concentrator numbers, Humanities faculty tend to blame someone else: the philistines who do not understand what we do…” (From Mapping the Future, p. 29). The report’s five conclusions are listed on page 51. Bullet 3 in the list: “We should reaffirm the critical, yet generalist and interdisciplinary tradition of undergraduate teaching,” and Bullet 4: “we should enlarge what we are doing by focusing on the interface between the Humanities and other divisions (notably some of the Social Sciences) or even other schools.”

I like Bullet 4: Or even other schools. I recommend humanists head directly to departments that have a strong base in mathematics. These departments represent the vast majority of most universities, and they often are grouped under the acronym STEM, which stands for Science, Technology, Engineering and Mathematics.

Consider curricular programs that arts and humanities colleges have created over the past few decades to help drive up numbers of students, and to reenergize interest in the humanities. Typically, these programs contain the word “digital” in them: digital humanities, digital arts. Sometimes, the words contain the phrase “technical” or “computational” (“techno” and “technology” are from the Greek “techne,” which means “craft”). This is a step in the right direction, but it is only a step. The real benefit of using digital tools is the effect that these tools, and their respective technologies, have on defining new ways of thinking. Framed in this way, the digital tool is merely a catalyst for a new way of thinking. The deep thing that makes the tool possible is mathematics. Let’s take two examples.

The first example is from a wonderful book by Franco Moretti entitled Graphs, Maps, Trees: Abstract Models for Literary History. When I first came across this book, and started reading, I couldn’t withhold my joy. Moretti is a literary scholar — but any mathematician or computer scientist (i.e., applied mathematician) would immediately recognize words like graphs, trees, and maps as abstract discrete structures. Undergraduates learn discrete structures in the first year of undergraduate study. Here is a diagram of a tree:

From Wikipedia Tree (graph theory)

The tree is beautiful because it elegantly interprets the world through an abstract lens. That is what Moretti was suggesting in his book — that things like abstract trees are useful in literature studies within the humanities. This observation needs the strongest clarification: any software tools or computers that allow us to understand literature are merely technological contrivances that surface underlying mathematics. The “digital”, the “technological”, and the “computational” are irrelevant in any deep sense. What is relevant is that beneath the multiple layers of this onion, we find a mathematical way of thinking. That is what the digital humanities is really about. It is not about computers or software any more than astronomy is about telescopes. It is about understanding the humanities mathematically. The same goes for digital arts. Forget the software and think closely, as Moretti has done, on what new underlying concepts are being presented, used, and learned when running the software. At the heart of the software lies a revolutionary new approach to interpreting culture and its creative products.

The Mad Hatter’s Tea Party (Illustration by John Tenniel) inside Peter Zelchenko’s digital edition of Alice’s Adventures in Wonderland.

On to the second example. In my Modeling class this semester, I asked students to choose their favorite work of literature or reading. Their goal was to translate this reading into mathematical logic or diagrammatic semantic equivalents (such as mind maps, concept maps, and semantic networks). That was project # 1. The semester’s project # 2 has focused on Lewis Carroll’s Alice’s Adventures in Wonderland. The students are producing really innovative results, and having cross-disciplinary team meetings. This is an excerpt from Chapter 7, “A Mad Tea Party”: “Alice looked all round the table, but there was nothing on it but tea. “I don’t see any wine, she remarked” “. That Alice did not see any wine but also saw tea can be translated into the languages of logic: propositional and predicate.

What are the goals in this logical translation? Through the recent emphasis on the digital, the computational, and the technological, the humanities has ventured into the land of digital tools. But as for Alice’s adventure, all is not as it seems. The tools eventually vanish. It takes time, but they disappear into a cloud of smoke bits. The tools are useful instruments for speedy information processing but that is it. In the wake of the vanishing tools, we find mathematics. It is has been there all along — right in front of us. But now we recognize it.

I don’t think we do a very good job in mathematics or computer science (CS) education in making these types of creative connections. When CS students learn logic, it is most often done in a dry fashion. If this then that, lots of Ps and Qs and excessive symbology and proofs. Theory, but mechanically delivered without feeling or true understanding. As I’ve emphasized before in other manuscripts — mathematical notation has nothing to do with mathematical thinking at any deep level. The notation merely allows specialists to communicate. Appreciating Alice’s adventures requires a bridging of disciplines starting with English words and phrases and putting aside the reams of logical expressions. Diagrams are essential. Even mathematics is largely misunderstood as a series of boring, mechanical exercises where the human student is forced into the role of machine. That is not mathematics. It is drudgery.