From the Director:
Read Dr. Andrew Seeley's essay on the difference between the classical and modern approaches to the study of mathematics and the value of each. Where does each approach lead and how can the study of mathematics best accord with both truth and the human spirit? Read More
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A Lover of Latin and Math
This month, Dr. Seeley interviewed William Carey, teacher at Ad Fontes Academy in Virginia. Mr. Carey's belief in the value of the classical model of education shapes his method of teaching mathematics and logic as essential in a classical education and integrated with the other liberal arts. Read More
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Events of Interest
(Past) Portsmouth Institute 2022 Humanitas Summer Symposium -- June 10-11 -- Portsmouth, Rhode Island--Humanitas Summer Symposium on the blessings of liberty.
Zephyr Institute Defense of Icons Seminar -- June 29 -- Palo Alta, CA -- This seminar explores the use of icons through the work On the Divine Images by St. John of Damascus.
Collegium Institute Introduction to Aquinas on Happiness -- June 30-July 21 -- University City, PA and Remote -- A 4-part introductory survey of the Summa Theologica.
Witherspoon Institute Summer Seminar--Natural Law and Public Affairs -- July 5-9 -- Princeton, NJ -- A discussion of natural law theory and the application of the theory to moral and political issues.
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Further Enrichment
Teachers-True and False-The Catholic Thing In today's world, it is more important than ever to separate the role of the parent from that of the teacher.
Episode Six: Aristotle's "Nicomachean Ethics" -- Drs. Seeley and Lehman discuss Aristotle's presentation of the emotions in his Ethics, Macbeth, virtue, and Aquinas' later presentation of the emotions.
Episode Seven: Aquinas' Commentary on Aristotle's Posterior Analytics -- Drs. Seeley and Lehman consider logic through the lens of Aquinas and Aristotle.
Episode Eight: Anselm's Proslogion -- Drs. Seeley and Lehman are joined by Dr. Walz in a discussion of the Proslogion, the relation of our intellectual study with our search for God, and how these play out in the classroom.
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The Spirit of Mathematics
We accordingly thought it up to us so to train our actions even in the application of the imagination as not to forget in whatever things we happen upon the consideration of their beautiful and well ordered disposition, and to indulge in meditation mostly for the exposition of many beautiful theorems and especially of those specifically called mathematical. -Ptolemy, Almagest
Claudius Ptolemy, introducing his classic work on astronomy, beautifully expressed the spirit of ancient mathematics. Many readers of this article, reflecting on their own experience of mathematics, might find it unbelievable that anyone would connect imagination, beauty, and meditation with it. Imagination and beauty are found in literature and the fine arts, perhaps history, philosophy, and theology, but not mathematics. They would more readily identify with the biting indictment of math education expressed by Paul Lockhart, a New York high school teacher learned in the highest levels of mathematics, in his brilliant and beautiful essay, “A Mathematician's Lament”:
...If I had to design a mechanism for the express purpose of destroying a child’s natural curiosity and love of pattern-making, I couldn’t possibly do as good a job as is currently being done— I simply wouldn’t have the imagination to come up with the kind of senseless, soul-crushing ideas that constitute contemporary mathematics education.
How was Ptolemy’s spirit lost to the modern era? How can we get it back? My own experience suggests that a recovery of the ancient quadrivium – arithmetic, geometry, music, and astronomy – in some form is necessary, but it also makes me aware of the great divide between ancient and modern approaches to mathematics and the mathematical sciences, which led to a centuries-long battle for control of education.
I attended high school in the early 80’s, and was present at the beginning of the final push to replace and then subsume the classical liberal arts under an overall emphasis on science. There really was a fight at my school, culminating in my one and only participation in the sort of student-led protests made glamorous by the rebels of the late sixties. It seemed like an arcane issue. The administration wanted to increase the number of class periods from seven to eight. Student organizers claimed this was a move to undermine the core of the school’s curriculum – allowing the introduction of many more science and math courses – to compete with the school’s strong classical liberal arts focus: Latin (three years required, five years available), modern foreign language (two years), English (required each year, consisting of Greek mythology, English and American classics, and challenging modern readings), and History (required every year, including a year devoted to ancient and medieval history). Though I was a naturally apt math student, I found the math and science courses – compared to the arts courses – were spirit-less, and thought-less, drill and kill with little time for thoughtful questions. Geometry, which consisted of daily proof presentations based on assumed axioms, was the one exception.
My collegiate experience, on the other hand, made me wonder why there was a conflict between the arts and the sciences at all. The fully integrated program aimed to include all seven of the classical liberal arts, and so required four years of math and of science, along with Latin, logic, music theory, theology, philosophy, and literature. In the ancient view, the mathematical disciplines of the quadrivium – arithmetic, geometry, astronomy, and music – were seen to complete the education begun in the language arts of grammar, logic, and rhetoric. Plato thought of the mathematical disciplines as honing the mind’s ability to see clearly the beauty that musical education revealed imaginatively and almost instinctively. Those who excelled in mathematical reasoning – not only in constructing and following mathematical arguments, but also in being engrossed in puzzles aimed at universalizing and grounding mathematical principles – were the ones most apt to pursue philosophy fruitfully.
The works of Euclid, Apollonius, and Ptolemy are excellent guides for young souls hungry for truth and beauty. They train the mind in logical reasoning like no other authors. Because they proceed from simple, clear starting points, step-by-logical-step, I felt I was always learning what I could see must be true. I learned why the angles of a triangle equal 180o while its area equals one half the base times the height, and things more astonishing: that only one straight line can touch a circle on one point, how to fit regular figures like pentagons and dodecahedrons (12-sided solids like a wargamers die) in circles and spheres, that open-ended hyperbolae are inside out ellipses whose sides grow infinitely close to becoming straight lines. Ptolemy expressed a common view among ancient mathematicians, that not only should mathematics train the mind and imagination, but also contribute to forming beautiful souls:
And indeed this same discipline would more than any other prepare understanding persons with respect to nobleness of actions and character by means of the sameness, good order, due proportion, and simple directness contemplated in divine things, making its followers lovers of that divine beauty, and making habitual in them, and as it were natural, a like condition of the soul.
Over the years, I have had the delightful experience of introducing many humanities teachers to Euclid. The imaginative character of the objects and the clear arguments which convince of truth yet open doors to wonder and discussion were a completely new experience of mathematics for them. Often they would say, “If only math had been like this in high school, I might be a math lover.”
Yet when I have advocated Euclid to high school math teachers, I have encountered resistance. “You are advocating approaches that I had to train myself out of in order to do advanced mathematics.” Teachers like this are following in the footsteps of the great Enlightenment thinkers such as Francis Bacon and Rene Descartes, who were harshly critical of the old ways of coming to know. They rejected both the three-fold way of the language arts and the four-fold way of the mathematical arts. Bacon, the founder of the modern scientific method, believed that ordinary language was a source of confusion and error rather than of insight into the reality of things. He argued that since ordinary language is based upon naturally acquired, unreflective concepts ordered to human conversation, it was unsuitable to facilitating serious thought.
Words are imposed according to the apprehension of the vulgar. And therefore the ill and unfit choice of words wonderfully obstructs the understanding….But words plainly force and overrule the understanding, and throw all into confusion, and lead men away into numberless empty controversies and idle fancies.
Descartes, founder of the algebraic approach to mathematics, furthered the separation between the language arts and science by teaching the world to replace words with symbols, eliminating the need for using images and concepts proper to each thing. The letters in an equation could stand for areas or lengths or numbers or speeds or times. According to Descartes, when looking at the equation, it is better not to think of what it might stand for; just focus on the relationships between what is known and unknown that will allow you to completely solve the problem posed. Classical mathematics used words and letters to refer directly to imagined objects and so found it difficult to get to the roots of real world problems, like the proper curvature of lenses. If Euclid, Apollonius, Archimedes had understood the proper method of geometry:
...they would not have put so much labor into writing so many books in which the very sequence of the propositions shows that they did not have a sure method of finding all, but rather gathered together those propositions on which they had happened by accident.
The revolution instigated by thinkers such as Bacon and Descartes has proven remarkably successful in opening the natural world and placing it in service to humanity. It has developed whole new fields of beautiful mathematics undreamed of in ancient times. But with it has come a spirit antithetical to the most important elements of human formation. A new recovery of the quadrivial disciplines and approaches can restore the place of mathematics in forming complete human beings, while guiding us in a healthy appropriation of the power and promise of modern mathematics.
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An Interview with William Carey
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A Lover of Latin and Math
When I met William Carey this past spring, I immediately liked him because he was almost envious of the fact that I had been able to engage my college students in a semester-long study of the epitome of ancient astronomy, Ptolemy’s Almagest. Then I learned he is an honored Latinist who also loves mathematics both ancient and modern, one of the few people I know qualified to read, understand, and translate mathematical texts originally written in Latin, a practice kept up well into the nineteenth century. He is currently enjoying a few years’ sabbatical from “the tyranny of the urgent that is teaching”, throwing himself into reading T.L. Heath’s classic work on Greek mathematics and Andre Beaufre’s An Introduction to Strategy, while helping 2500 Afghan refugees find housing and furniture.
William was accidentally introduced to the study of Latin by his father, a trial attorney who enjoyed reading books he was interested in to his 10 year-old son. In the midst of reading Caesar’s Gallic Wars, he mentioned that the work was not originally English but Latin. “Why aren’t we reading it in Latin?” asked his young son. Having no answer but ignorance, they worked through Jenney’s First Year Latin together, and then went through it again. This path eventually led his father to a Master’s in Classics, an adjunct position at George Mason University, and founding The Latin Library, an online resource of original texts. It led William to excel in Latin in high school and college, winning the prestigious Marian W. Stocker Prize for the best undergraduate Latinist at the University of Virginia.
But his grandfather, a chemical engineer, gave him a love of engineering and mathematics, allowing him to work on various instruments in his garage and teaching him to code Apple IIe computers. He began college as an engineering major, but hated the culture and pedagogy: tons of work ordered to crush the spirit and a grading system designed to foster intense competition (only a small percentage of students could get A’s). “When complaints were raised about engineering students regularly pulling all-nighters in the hallways of the building, the department supplied cots.” William quickly changed majors to Classics, though his love of history (“we read great and interesting texts”) led him to the verge of a double major.
He returned to mathematics as a teacher at the classical Christian school Ad Fontes Academy. Hired to teach Latin, he was asked to fill needs in calculus and logic. It took him a few years to feel comfortable teaching calculus, but he was struck by how disconnected it was from the spirit of the rest of the curriculum. “Our school prepares students to lead flourishing, essentially human lives. But contemporary high school mathematics seems ordered to the technological needs of the 1920s, producing human computers adept at transcendental calculus. It’s like ordering your entire history curriculum to give an intense understanding of nothing but World War I.”
But William also realized that his students had a whole toolbox that he didn’t, which they learned through their 9th grade study of Euclid. Learning to present cogent demonstrations, to field questions, to imagine alternative proofs made them excellent pre-calculus students. “I worked to bring those strengths to the other math courses. Less drill, more proof. I do need to drill them in some things, but this is so that they can engage fruitfully in interesting discussions and texts.” William believes that too many classical educators exempt mathematics from classical pedagogy. “Learning to read great texts, discussion, discovery, clear and persuasive reasoning should be as much a part of mathematics as it is in English and History.”
His students respond well to his methods. His best students become delighted when they learn that mathematics is not based on arbitrary authority; they become excited to discover, they insist on knowing rather than believing. “It becomes addictive, like a drug. In one discussion, a student asked with almost disbelief, ‘Mr. Carey, is this leading to the quadratic formula?’ That completely changed his expectations for mathematics.” His students never ask, “What use is this?” “I never pretend it’s useful, and they just enjoy playing with truth. Very few high school students are impressed by career-oriented learning.”
For further information on William Carey, click here.
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