Math Circle 2 -- Calculus, Meaning and Beyond

Of Stones, Summation, and Academic Sterility (Continued)

Calculus and Christian Foundations

EN: You argue that calculus is not just a mathematical achievement, but one deeply rooted in a Biblical worldview. How did Christian theology uniquely enable the development of calculus in ways that Greek or purely secular frameworks could not?

JDN: Allow me to quote some representative scholars on this point.

The Roman Catholic theologian/physicist Stanley L. Jaki (1924-2009), “The rise of science needed the broad and persistent sharing by the whole population, that is, an entire culture, of a very specific body of doctrines relating the universe to a universal and absolute intelligibility embodied in the tenet about a personal God, the Creator of all.” [The Road of Science and the Ways to God, (Scottish Academic Press,1978), p. 33.]

Alvin Plantinga (1932-), the John A. O’Brien Professor of Philosophy Emeritus at the University Of Notre Dame, argued that theistic belief provided the foundation of the very possibility of science and mathematics: ”the mediaeval concept [from Thomas Aquinas] of adaequatio intellectus ad rem [the correspondence, the “adequation,” of our thinking to the objectivities of the world outside of our thinking], which is based on the doctrine of the imago dei, well explains a host of human capacities, not least our ability to do science, something that naturalism has a much harder time doing [Where the Conflict Really Lies: Science, Religion, and Naturalism (Oxford University Press, 2011), p. 269-270.]

[Note: Naturalism is the view that we can understand the whole of reality on a solely and purely natural basis, a view that excludes discussion of or need for the supernatural or spiritual dimensions.]

Mathematician and philosopher Alfred North Whitehead (1861-1947) stunned his hearers in a mid-1920s lecture by affirming that the “greatest contribution of medievalism to the formation of the scientific movement” as “the inexpugnable belief that every detailed occurrence can be correlated with its antecedents in a perfectly definite manner, exemplifying general principles. Without this belief, the incredible labours of these men would have been without hope.” To him, this conviction “must come from the medieval insistence on the rationality of God.” [Science and the Modern World (Free Association Books, [1926] 1985), p. 15.]

Jaki notes the importance of this observation: “Half a century has passed since these words startled a distinguished audience at Harvard University and indeed the whole intellectual world. The magnitude of the shock merely corresponded to the impenetrable density of a climate of opinion for which the alleged darkness of the Dark Ages represented one of the forever established pivotal truths of the ‘truly scientific’ interpretation of Western intellectual tradition.” [Science and Creation: From Eternal Cycles to an Oscillating Universe (Scottish Academic Press,1974), p. 146.]

Comparing the medieval insistence on the rationality of God with the Greeks of old, Jaki remarks, “His [Heraclitus] thinking is indeed a classic example of the sad unbalance which is ultimately imposed on one’s thinking by the acceptance of the endless, cyclic recurrence as the basic pattern of existence. Within that framework one was ultimately left with no consistency in reasoning and observation … Clearly, in a philosophy of nature steeped in the idea of perennial cycles there remained ultimately no room except for disconnected sense perceptions.” [Science and Creation, p. 107.]

The failure of the Greeks was in their faulty theology. Their “gods” were inadequate; no ultimate purpose can be found in endless cycles of history. The Greek mind could discover proximate patterns, but there was no infinite-personal Mind who’s faithful and sustaining word of power upholds the patterns so discovered, no design of wisdom, no goal of the universe.

Recovering Meaning in Mathematics

EN: You cite concerns that calculus is often taught mechanically, stripped of its historical and philosophical depth. What do students lose when calculus is taught this way, and how would you ideally restore its meaning and wonder in education?

JDN: When we teach Calculus merely as mechanical procedures, e.g., we calculate the derivative of this function by using this formula, with no reference to the relationship of Calculus to history, to the nuances that required a certain way of looking at the world for it to blossom, to the derivation of said formula, to the meaning of this function, we mis-educate. We are not giving the student a broad range perspective, one that envelops the whole person into the fascinating realm of the interpenetration of ideas with history, personalities, and the workings of the physical world.

One of the better Calculus texts that ably presents the shalom of this wholeness is Calculus in Context (Johns Hopkins, 2017), written by the University of Notre Dame professor Alexander J. Hahn. See also his fascinating Mathematical Excursions to the World’s Great Buildings Princeton University Press, 2012) and Basic Calculus of Planetary Orbits and Interplanetary Flight: The Missions of the Voyagers, Cassini, and Juno (Springer, 2021).

Infinity, Limits, and Faith

EN: Concepts like infinity, infinitesimals, and limits are central to calculus and also touch on profound philosophical questions. How do you see these mathematical ideas reflecting—or even pointing toward—larger theological truths?

JDN: To delve deeper into infinity and how the concepts play out in theology, metaphysics, and mathematics, I would recommend Infinity: New Research Frontiers (Cambridge University Press, 2011), ed. Michael Heller and W. Hugh Woodin. Be prepared for a challenge!

In theology, infinity is ascribed to God in terms of eternality, i.e., in the way we view time as created beings, God has neither beginning nor end; He exists (the “I AM”) before the universe came into being in “eternity past” and that existence will never cease.

The way modern mathematicians look at infinity is quite different. It involves the created world of number and space, specifically that which is quantifiable. They use infinite processes as a microscope to view the world of “instantaneous time” and actual areas and volumes of solids.

Regarding infinity, the Greek philosopher Aristotle (384-322 BC) posited two types: potential and actual. He denied the existence of the infinite (the actual) but affirmed that you could approximate it (potential).

The Greek astronomer/mathematician Eudoxus of Cnidus (ca. 408-355 BC) used Aristotle’s potential infinity idea in developing the “method of exhaustion” to approximate the volume of curved figures like the sphere. As a walking encyclopedia of his time, Archimedes of Syracuse (ca. 287-ca. 212 BC) also used this method of approximation to prove multiple geometric propositions.

The Greek mathematician/geometer Euclid (ca. 300 BC) also used the potential concept of the infinite to prove, using reductio ad absurdum (indirect proof), that there is a limitless number of prime numbers.

Despite these accomplishments, the theology and metaphysics of the Greeks stopped short of using the “limit of infinitesimal processes” that is the foundation of Calculus.

The ancient Greek world’s horror of the actual infinite (horror infiniti) limited its mathematical scope and operation.

The medieval world gave us the vision of the infinite God, building on the revelation of the Father through the Incarnate Son, i.e., their eternal relationship. When that world embraced this vision, it then created the dynamics of infinitesimal processes upon which differential and integral Calculus is built.

As quoted in my essay, mathematics historian Carl B. Boyer (1906-1976) summarizes, “The blending of theological, philosophical, mathematical, and scientific considerations which has so far been evident in Scholastic thought is seen to even better advantage in a study of what was perhaps the most significant contribution of the fourteenth century to the development of mathematical physics … a theoretical advance was made which was destined to be remarkably fruitful in both science and mathematics, and to lead in the end to the concept of the derivative.” [The History of the Calculus and its Conceptual Development (Dover, [1949] 1959), pp. 70-71].

According to Jaki, the derivative was an essential tool for understanding the physical science of dynamics: “The Greeks failed to develop science, by which I mean an intellectual enterprise in which one discovery generates another discovery and does so at an increasingly accelerated rate … the Greeks of old failed to make any breakthrough in the science of motion or dynamics which is the basis of all physics and which in turn is the basis of all modern exact science.” [The Absolute Beneath the Relative and Other Essays (The Intercollegiate Studies, Inc., 1988), pp. 61-62.]

Calculus as a Window into Reality

EN: You suggest that calculus reveals something fundamental about the nature of reality—especially the relationship between change and continuity. In what sense does calculus help us “see” the world more truthfully, and what does that reveal about the nature of creation?

JDN: What is fundamental about the nature of reality is that in its objectiveness it speaks to us of an order and complexity that we can, in part, understand as interrogate it. We do not come to objective reality outside of us with preconceived, i.e., a priori, notions. We come to it with our questions along with appropriate experimentation and with them a willingness to change or refine the way we think about the object under consideration.

The questions posed by the early founders of the Scientific Revolution required a willingness to go beyond the Greeks, to engage in the meaning of change and continuity as it relates to motion, to embrace the dynamism of the created order, the integration of form and matter, theory and the empirical.

We are free to create theoretical concepts and structures, but this freedom is not absolute. As we interrogate the created order, our minds come under the compelling demand of reality, to which we ourselves, the imago dei, belong in mind and body,

Adaequatio intellectus ad rem … there is an actual correspondence, indeed, a harmony, between reality, the form, the structured order of creation, and the way we think. We only realize this interpenetration between outward form and inward thinking when we conform our thinking to the mode of rationality given by reality. A realization of this harmony through subjection is what biblical doctrine of creation inculcates.

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For a deeper dive, read James Nickel’s A Case Study for Calculus. https://biblicalchristianworldview.net/documents/caseCalculus.pdf

James D. Nickel, B.A. (Mathematics), B.Th., B.Miss., M.A. (Education), Senior Fellow in Mathematics and Education at the Center for Cultural Leadership, taught high school mathematics in the late 1970s in Hawaii, in the 1980s in Australia, and from 2005 to 2012 in Washington State and online. He also has nearly 25 years of experience in Information Technology and is the author of numerous books including Mathematics: Is God Silent and a math curriculum The Dance of Number.

Read the Introduction to Math Circle 2: The One and the Many.