Thinking in the Language of Calculus

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By Tyson Schlect

Thinking in the Language of Calculus

By Tyson Schlect

Implicit in the pages of Calculus for Everyone is the idea that symbolic mathematical language is for everyone. In the curricular context, all students should study the symbolic mathematical language of calculus because all students must understand what calculus means. This is a distinct pedagogical goal from understanding how calculus works or what calculus does.

To illustrate the difference, consider education in Latin and Greek. One purpose (perhaps the key one) of ancient language education is to cultivate a student’s ability to think in Latin or to think in Greek (not to translate into English), and thereby to read classic works of literature and history in the original language without translating it in his head. C.S. Lewis, in recounting his own adventures in ancient language education, articulated this difference.

“Those in whom the Greek word lives only while they are hunting for it in the lexicon, and who then substitute the English word for it, are not reading the Greek at all; they are only solving a puzzle. The very formula, ‘Naus means a ship,’ is wrong. Naus and ship both mean a thing, they do not mean one another”
– C.S. Lewis, Surprised By Joy

In the same way, the language of calculus – its mathematical symbols, its collection of symbols into equations, and its modification of equations into differentials – refers to things which every student must understand in order to think in calculus. Things to which the language of calculus refers are philosophical objects of ancient Greek thought. Calculus for Everyone begins with discussions of Pythagorean mathematics precisely because the development of mathematics in the Newtonian paradigm depended on prior philosophical commitments in the history of western philosophy. If calculus developed out of need to assess objects in motion, then what precisely is an object? Certainly it might be made out of a material, but the material world is full of non-objects (mud, for example, is not properly considered an object, even though mud is the very material from which God created Man). Objects must therefore have the added feature that they may be counted using cardinal numbers (there is one of this thing, two of this thing). This is straightforward enough until one considers the oddity that a necessary feature of a material object is that it contains a purely philosophical notion of countability. 

Thus Calculus for Everyone delves into the significance of assigning number to the world. Critical structural similarities exist between physical space and time, mathematical objects, and numbers. Because calculus is the mathematics of change, this structural similarity is at the heart of the meaning of calculus in assessing the philosophical problem of change in the world. To adequately understand the meaning of calculus, every student must—as it were—see a mathematical object moving in space and time, just as Newton did. It is not sufficient to see only the mathematical symbols and rearrange them according to seemingly arbitrary rules. Why can’t we just divide on one side of the equation? Why do we have to divide on both sides? It is not sufficient to learn all the rules for shortcutting the Method of Increments, which is virtually the entirety of what students learn in AP Calculus. Instead, every student needs to develop the skill of thinking in the language of calculus and of knowing its meaning.

No other calculus textbook accomplishes this particular pedagogical goal, and therefore no thorough philosophical education is quite complete without it. Calculus for Everyone fills a unique curricular hole by introducing students to the objects to which calculus refers and drawing out the philosophical development of calculus—a vastly different pedagogical enterprise than working the mathematical manipulations of calculus which got hammered out long after Newton. Students grounded in the meaning of calculus may then progress into rigorous mathematics with broad ranging applications in the world, or into heightened philosophical reflection on the western tradition, or into a deeper reflection on the reality that we are embodied creatures who change while serving a Triune God who does not change. Whatever the future vocational calling of students, knowing the meaning of the symbolic mathematical language of calculus is a curricular and pedagogical necessity.    

Related Articles and Resources:

• Calculus for Everyone (curriculum), by Dr. Mitch Stokes
• What does Jesus have to do with STEM, by Dr. Mitch Stokes
• Re-Enchanting the Sciences, by Tyson Schlect (via Beza Institute)
• “Our 1-star Calculus for Everyone Review” • Introduction and Testimonials

Tyson Schlect is a water and environmental strategy consultant with a background in engineering. He is also a Lecturer in Science at New Saint Andrews College where he has taught calculus and the history and philosophy of science. He lives in Spokane, Washington with his wife and four daughters.