Math Circles 7: A Glossary
"By words we learn thoughts, and by thoughts we learn life."—Jean Baptiste Girard
James Nickel’s “Math Circles” swirl in a variety of directions. Math Circles 7 is a glossary of mathematics terminology. As Erin McKean observes, “People say jargon is a bad thing, but it’s really a shortcut vocabulary professionals use to understand one another.”
On Definitions and Purpose, Mathematical Accuracy and Engagement
EN: What inspired you to create this glossary with its unique blend of precise mathematical definitions, humor/puns, and explicit Christian worldview elements (like references to God, sin, or Scripture)? Was there a particular audience or teaching goal in mind? How did you approach balancing mathematical accuracy with accessibility and engagement?
In the following discussion, I will answer all of these questions.
JDN: I prepared this glossary for teaching a Precalculus course to high school students (2005-2007). I wanted to give them a quick reference point for the topics discussed, plus a little flavor of humor to spice the learning experience.
I like to add the extra of humor to math instruction to help dispel the idea that learning math is dull drudgery. It is one way of “balancing mathematical accuracy with accessibility and engagement.”
For example, here are a few excerpts from a lesson entitled “The Mathematics of Change at an Instant.” I interweave mathematical exposition with worldview, literature, history, and humor. It would be fairly accurate to say that most people have never learned the basics of mathematical physics this way.
I start with Charles Dodgson, aka Lewis Carroll (Through the Looking Glass) and Winston Churchill (“Examinations,” My Early Life (1874-1904), p. 26.)
Lewis Carroll (1832-1898), English author, mathematician/logician, and Anglican clergyman:
Alice looked around her in great surprise. “Why I do believe we’ve been under this tree the whole time! Everything’s just as it was!”
“Of course it is,” said the Queen. “What would you have it?”
“Well, in our country, “said Alice, still panting a little, “you’d generally
get to somewhere else—if you ran very fast for a long time as we’ve been doing.”
“A very slow sort of country!” said the Queen. “Now, here, you see, it takes all the
running you can do, to keep in the same place.”
Winston Churchill (1874-1965):
Further dim chambers lighted by sullen, sulphurous fires were reputed to contain a dragon called the ‘Differential Calculus.’
To Galileo:
In the Assayer (1623), he wrote with convincing authority, linking the study of God’s works to the principles of mathematics:
Philosophy is written in this grand book, which stands continually open to our gaze. But the book cannot be understood unless one first learns to comprehend the language and read the letters in which it is composed. It is written in the language of mathematics, and its characters are triangles, circles, and other geometrical figures without which it is humanly impossible to understand a single word of it; without these, one wanders about in a dark labyrinth. (Galileo Galilei, Discoveries and Opinions of Galileo, trans. Stillman Drake, 1957, p. 196.)
Stanley L. Jaki summarizes Galileo’s foundations for scientific work:
The creative science of Galileo was anchored in his belief in the full rationality of the universe as the product of the fully rational Creator, whose finest product was the human mind, which shared in the rationality of its Creator. (The Road of Science and the Ways to God, 1978, p. 106.)
On Galileo’s method:
Galileo’s analytical skills incorporated three principles in his study of the “grand book of God’s creation.”
a. Obtain basic principles through observation and experimentation. Sometimes his experiments were “thought” experiments based upon what he observed.
b. Major on the major. Strip away incidental or minor effects. He tried to understand what was happening with falling objects by stripping away air resistance (i.e., he assumed they were falling in a vacuum).
c. Apply the derived principles and other mathematical demonstrations back to the real world with all of its limitations (i.e., air resistance).
Above all, Galileo proposed to seek quantitative descriptions in terms of mathematical equations and formulas. His commitment to quantitative analysis was the fruit of one thousand years of the leavening influence of the Christian Gospel in Europe (since the fall of Rome). His appreciation of an orderly and understandable creation presupposes the God “who created everything according measure, number and weight” (“sed omnia mensura et numero et pondere disposuisti,” Wisdom of Solomon 11:20-21), even in a world of motion.
[Note: re. “… but thou hast ordered all things in measure and number and weight. For it is always in thy power to show great strength, and who can withstand the might of thy arm?”—it is the most often quoted and alluded to phrase in Medieval Latin texts.]
My initial comments expounding Galileo’s theory of motion:
Let’s envision Mr. Delta as he drops a stone from the top of a building. As Professor Galileo watches, he first notices that the stone’s velocity is not constant; i.e., it increases with time. Stop the motion for a second! Freeze the frame!
At this juncture, Galileo’s theorizing (now called the theory of inertia) departs from the ancient Greek theory of motion, popularized by Aristotle (384-322 BC). Among other dicta, Aristotle postulated that an object, like a thrown ball, keeps moving only as long as something was actually in contact with it, imparting motion to it all the time. Whatever this “something” was (Aristotle thought it was “air” closing behind the ball), he said that it continually pushed the object along. (Herbert Butterfield, The Origins of Modern Science,1961, pp. 3-4.)
[Note: Inertia, from Latin, means “idleness.” The theory of impetus is also called the theory of momentum, Latin for “movement.”]
The Alexandrian theologian John Philoponus (ca. 500), based on Christian convictions, first challenged this idea in the 6th century. Contrary to Aristotelian dogma and amazingly similar to Galileo’s findings, Philoponus resolved that (Cited in Stanley Jaki, Science and Creation: From Eternal Cycles to an Oscillating Universe,1974, pp. 185-187):
a. All bodies would move in a vacuum with the same speed regardless of their weight (or mass).
b. Bodies of differing weights would, falling from the same height, hit the ground at the same time. This is easy to validate experimentally, something Aristotle never tried.
c. Projectiles move across the air not because the air keeps closing behind them, but because they were imparted with a “quantity of motion” (or momentum).
The ideas of Philoponus were transmitted into the thinking of some key medieval theologians, particularly the French philosopher-theologians Jean Buridan (ca. 1295-1358) and Nicole Oresme (ca. 1323‑1382), via the work of the Arabic thinkers and translators. These two medieval scholars refined the thoughts of Philoponus, especially the elementary impetus [impetus, in Latin, means “to attack against.”] theory of motion (point c.) thus building the foundation for Galileo’s work in momentum and inertia and Isaac Newton’s (1642‑1727) formulation of the first law of motion, i.e., every object persists in a state of rest or a state of uniform motion in a straight line unless acted upon by an external force that changes that initial state.
Mr. Delta interrupts, “Excuse me, this history lesson is illuminating, but I’m still waiting for the stone to hit the ground.”
“Hold on for a few more minutes,” replies Professor Galileo. “Let’s flush out some mathematics before we start the motion again.”
I then expound the mathematics of free-fall motion. After several pages of algebraic work, I let Galileo have the last word:
“Let the stone continue its fall!” cries Professor Galileo.
“Thank you, Professor; it’s about time,” replies Mr. Delta.
On Mathematics and Humor
EN: As I read these definitions I find myself smiling, even laughing a times, at the way humorous remarks and notations you’ve sprinkled throughout. It’s almost hilarious. Perhaps it pops out because it’s so unexpected. Would you mind commenting on this?”
JDN: To repeat, I use humor to spice the learning. I want students to not only know the geometric definition of an arc but also have Noah floating on the ocean of their thinking as they memorize it.
A few one-liners from Bob Hope (1903-2003):
“When she started to play, Steinway came down personally and rubbed his name off the piano.”
“A bank is a place that will lend you money if you can prove that you don’t need it.”
“I grew up with six brothers. That’s how I learned how to dance—waiting for the bathroom.”
Great comedians like Hope have often said something like this: “Humor occurs when your audience least expects it.”
Although the text of the New Testament does not say that Jesus had a sense of humor, I am sure that there was a twinkle in His eye and a chuckle from the audience at many of His sayings.
To realize that God manifested in the flesh had a sense of humor puts a different perspective on one’s view of God. God with skin laughs!
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Related Links from our Math Circles Series
Whither Mathematics Education in the 21st Century?
Of Stones, Summation, and Academic Sterility
Calculus, Meaning and Beyond
Quotable Quotes
Jaki Musings
Wonders of the Fibonacci Sequence
On Book Reviews