Math Circles 3 -- Quotable Quotes
"...to challenge, interest, humor, delight, and inspire the reader."
For many, if not most people, mathematics is associated with the dull, the difficult, and the uninteresting. Economics has oft been labelled “the dismal science, “ but math may be even higher on that scale.
As we move through the Math Circles of James Nickel, the third circle is a 30-page collection of quotes about mathematics in general and in relation to other disciplines. It’s an impressive and enlightening read.
Here is his introduction: “Mathematicians are labeled as odd, mad (to quote G. K. Chesterton), esoteric, absentminded, isolated, and unapproachable. One way to dispel these negative feelings towards mathematics in general and mathematicians in particular is to use mathematical aphorisms or quotations in their appropriate context. Quotations are concise packages of thought expressed in such a way as to communicate rich associations and connotations. The following quotations are meant to challenge, interest, humor, delight, and inspire the reader. I have included quotations that I do not agree with (philosophically), but they are still useful in that they will engage the reader to analyze their world view starting points.”
1. Beauty and the Transcendent
EN: Your collection brings together voices that describe mathematics as both beautiful and transcendent. What do you think these thinkers are seeing in mathematics that the average student—or even teacher—often misses?
JDN: Allow me first to talk about quotes and their relationship to pedagogy. In my many years of teaching high school math, I would regularly start a class by writing related quotes on the black/white board, quotes to trigger thought and discussion both that day and throughout the subsequent week.
One of my educational pet-peeves is the ever-reduced time allotments given to teachers to develop themes that invoke discussion. In many settings, teachers (especially math instructors) are forced to stick to the topics that will be later tested, as in standardized tests. Also, other school events (many extracurricular) either overrule or cut into teaching time. When I taught in Australia in the 1980s, the school year comprised 42 weeks of instruction. You got that right … 42 weeks with little or no interruptions. In the 2000s, when I was teaching math classes in the United States at a private Christian school, I averaged about 23 full weeks of instruction per year given a 36-week school year. What happened? Interruptions, special events, and other subjects, a bit here and a bit there, reduced the time required for full-orbed math instruction. I informed my headmaster that such a reduction was theft. My communique served as a shock treatment, revealing how much teaching time this leader had taken away from one of his teachers.
We need to sculpt discussion time to bring about a fullness of shalom in a math class. Discussion creates the framework for teaching subsequent concepts. I believe whenever math instruction is short-circuited there needs to be respectable shouting from the housetops in a way that engenders a change of priorities.
Thus ends my vociferation.
Back to the question.
I have written a recent book entitled Mathematics: Exploring Beauty—Trinitarian Perspectives (Sound Mind Press, 2024) that investigates the relationship between transcendent dynamics and its proximate reflection through mathematical beauty.
Why do some of these quotes highlight thinkers who see in mathematics a stunning beauty that the average student—or even teacher—often misses?
The answer comes from the English physicist Herbert H. Huntley, “… the classroom where science and mathematics are taught [there is] the feast of beauty … unlimited both in abundance and variety. Every discerning teacher knows that in the sphere of the created world, Beauty is an utterance of the divine voice, but scarcely one in ten thousand attempts to teach this language to the rising generation: ‘It’s not in the syllabus.’” [The Divine Proportion: A Study in Mathematical Beauty (Dover, 1970), pp. 104‑105.]
If math teachers have not been trained to see beauty in the wonderful world of mathematics, how can we expect students to see it?
A call for and implementation of re-orientation/re-training will resolve this conundrum.
Addendum: Learning in an independent atmosphere, a goal of the homeschooling experience, will give the student more time to reflect. If given robust reading materials, the student can engage in thought processes that build a way of life thinking, an outlook not just confined to the one-hour of math instruction/one hour of math homework per day that is typically experienced in the traditional day school setting. Food for thought!
2. Language Fabric
EN: Many of the quotes suggest that mathematics is not just a tool, but a kind of language for understanding creation. How should that idea reshape the way we think about math in relation to science and even theology?
JDN: While the language of science is mathematics, the language of mathematics is Algebra. Yes, Algebra, replete with its hieroglyphics of symbols, is not just a tool of mathematics; it is its language. Its symbols and the syntax surrounding them reveal the implicit order of its operations. Like learning any foreign language, it takes time to learn the way it speaks, time which requires, as language theorists say, “full immersion,” i.e., “indwelling.”
It took hundreds of years for mathematicians to develop and refine the symbols and the syntax through which they communicate structure. Did they invent this syntax merely out of the web of their minds, or is there a controlling factor?
Scottish theologian Thomas F. Torrance (1913-2007) gives a hint to that factor, “Everything we know in the created universe, macrocosmically or microcosmically, we learn from light signals, but their mathematical patterns have to be deciphered and coordinated with word in the formulation of scientific theory and the development of knowledge.” [Theological and Natural Science (Wipf and Stock, 2002), p. 15.]
New Testament scholar, Reformed theologian, and polymath Vern Poythress (1946-) echoes Torrance: “The created world, as result of God’s speech, bears within it from top to bottom a kind of quasilinguistic character … through God’s act of creation, things in the world themselves become wordless voices to the praise of God.” [“Science as Allegory—Mathematics as Rhyme,” in A Third Conference on Mathematics from a Christian Perspective (Wheaton College, 1981), Robert L. Brabenec, ed., p. 5.]
Veteran Portland, Oregon, math teacher Larry Zimmerman (1931-2016) speaks in the same tongue, “… we would expect the deepest scientific probes into the micro- or macro-cosmos to reveal a language fabric in which are woven the forces and relationships governing the tangible creation. This language fabric should itself be suggestive of an intellectual antecedent, an orderly, powerful, infinitude of thought, a ‘terra incognita of pure reasoning’ which ‘casts a chill on human glory.’” [The Biblical Educator (1980), David Chilton, ed., 2:2, 1]
We add to this perspective of University of Oxford mathematician John Lennox (1943-), “The idea that the universe did not come to be without the input of information and energy from an intelligent source … has been amply confirmed by scientific discovery … the language of mathematics has proved to be a powerful tool in describing how things work. Its codification of the laws of nature into short and elegant ‘words’ consisting of symbols surely reflect the greater Word that is ultimately responsible for the physical structures of the universe.” [Seven Days that Divide the World (Zondervan: 2011), p. 100.]
In Mathematics: Exploring Beauty (Sound Mind Press, 2024), I write, “Since the physical creation is held together by the Logos, we would expect to see revealed some sort of language fabric. The language of mathematics is ideally suited to unveil this underlying structure. It is a peerless tool by which we discover and investigate creational wonders, the beautiful rhythms of its multi-layered, interpenetrating structures. Number, therefore, is the intelligible order of things that makes it open to quantification, open to assessment by the symbolic representation of patterns. We develop number systems and mathematical symbols to encapsulate this rationality.” (p. 77.)
The language of mathematics takes on new life in terms of the revelation of the living Logos, the Incarnate Son of the Father. It is not just a hieroglyphic mix-mash of symbols; it is a language fabric suggestive of a deeper, transcendent order created and sustained in, through, and by the Logos of the Father. It behooves us to learn that language within the realm of this richer context.
3. Procedures and Meaning
EN: Several quotations emphasize that mathematics must be taught as something alive and meaningful, not just procedural. What would a classroom look like if teachers truly embraced that vision?
JDN: In mathematics pedagogy, meaning both precedes the procedural and, if taught correctly, meaning is in the teaching of procedure.
Let me equate procedure with poetry. Mathematics as poetry? Have I lost my mind?
Mathematics historian Morris Kline (1908-1992) sees this connection: “One has to be blind to beauty to not be able to see it in a mathematical proof (e.g., Euclid’s proof of the infinity of the primes) for an elegantly executed proof is a poem in all but the form in which it is written.” [Mathematics in Western Culture (1953), p. 470.]
Albert Einstein (1879-1955) once reflected that mathematical derivations are “… the poetry of logical ideas.” [Obituary written by Einstein after the death of the German mathematician Emmy Noether (1882-1935), The New York Times, May 4, 1935.]
In Mathematics: Exploring Beauty, after several pages of mathematical exposition where I find the velocity necessary for an object like a rocket to escape the gravitational pull of the Earth, I conclude, “The flow of this logic and analysis, to one mathematically attuned to it, borders on the sublime symmetry of poetry. The beauty revealed by such logical demonstrations is what attracts men and women to the discipline of mathematics, and why many engage it as their lifetime vocation, either as research mathematicians or teachers of mathematics.” [p. 230].
When we grasp the language fabric framework (The answer to Question 2.), a veil will be lifted from the eyes of our heart. Behind that veil is the world of mathematical beauty, the wonder of seeing the language of mathematics as poetry. Again, from my book Mathematics: Exploring Beauty, “Scientific breakthroughs are a form of kata-physical poetics. By kata-physical I mean the objective nature of the cosmos, its inner structure. Scientists dive deeply into these objectivities. By their intense study, they indwell its objective nature. They interrogate it. They let objective rationality speak to them. This indwelling/listening mode is what triggers their imagination and their intuitive perception creating the poetics of mathematical formulation. Einstein’s life and study exemplifies the nature of kata-physical poetics.” [p. 93].
A classroom imbued with wonder is what we will see with these new eyes.
4. Why this?
EN: You have also included quotes by atheists or math minds that don’t align with your Biblical perspective. Would you care to comment on that?
JDN: My mentor, history and sociology professor Dr. Glenn R. Martin (1935-2004) of Indiana Wesleyan University, often repeated this challenge, “Biblical Christians should not only know what we believe, but why we believe it (simultaneously, we should know what we do not believe and why). If we do not know the ‘what and why,’ we shall be led or victimized by those who do know the ‘what and why.’”
The reason I include quotes by atheists or math minds that do not align with my thinking is to invoke discussion. In the pedagogical world, we need to allow students the time to follow trains of thought, to know what assumptions undergird philosophical excursions and the ramifications. When we do this fairly, and when we present the alternative Christian viewpoint, our goal is to engender analysis, examination that involves point by point contrasting. And hopefully, we open eyes to see the power of grounding a view of life in terms of the revelation of the God the Father Almighty who created all things visible and invisible, all things that cohere in His Son in and through the living breath of the communion of the Holy Spirit.
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