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About Mathematics at Symposium
About Mathematics at Symposium
Studying mathematics with Symposium is unlike any conventional math learning you may have had.
Our term "mathematics" derives from the Greek mathemata, that which can be understood, communicated, taught, or learned on its own terms. The classical branches of mathematics are arithmetic, geometry, music, and astronomy (four liberal arts, or arts of things). This mathemata is the origin of liberal arts and Western philosophy, just as liberal arts lead into and presupposes philosophy.
By way of example, one excellent starting point is Euclid's Elements of Geometry. Euclid's book and Apollonius of Perga's On Conic Sections are a compendium of the rudimentary proofs and demonstrations of geometric truths and constructions. They are not properly understood as formulas or technical methodologies to be memorized and applied to problem sets, as is common in schools.
Euclid, Apollonius, Ptolemy investigate the nature of figures, two and three dimensional space, commensurability, relations, analogy, curves, the appearances of the heavenly motions and their explanations. Euclid opens with definitions: what is a point, a line, a surface, an angle, a circle, etc. Then he proposes five postulates, given presuppositions necessary to construct figures. And finally, the common notions, that which virtually all persons accept ("axiom" was introduced by Aristotle and amplified by Proclus). Book I comprises 48 propositions. The first shows how we can construct an equilateral triangle on any given finite straight line. The 47th is a geometric proof of the Pythagorean theorem. Step by step the inquiry progresses, building up. As we go, we must ask what kinds of figures are these, are they "eidetic" or merely idealized representations of actual figures? Can everything be counted or measured? And the questions multiply, and deepen our understanding.
This kind of study of mathematics is appropriate for two groups of people: (1) those who think they are not "math people" and who may have disliked and suffered school and college math courses; and (2) those who have a technical background in modern mathematics, physics, engineering. Both groups may be surprised and delighted, and each group will learn from the other.
Music at Symposium
Music at Symposium
Music with Symposium is not merely music appreciation, nor music history, certainly not a performance practicum, nor a seminar in music theory.
For ancients, medievals, and moderns, music has been understood as one of the four ways of the liberal arts.
It is a branch of mathematics (the learnables/teachables/knowables), built on ratios, proportions, dynamic intervals, the relating and ordering of tonal space, the rhythmic ordering of time, the resulting melody, harmony, counterpoint in polyphony. We measure magnitudes and their relations. Indeed, many mathematicians, scientists, and physicians become seriously interested in music as an expression of the orders they see in the world.
Music is also bound to language, structures of common usage or formal syntaxis, logic, rhetoric, and poetics. For the Hellenes music (musike) was the complete liberal art, and named after friendship with the divine muses. The educated person is the musical person, and vice-versa. Like the Pythagoreans, Plato, Aristotle, Boethius we inquire into the elements of music—back to the origins.
Beginners and non-musicians are most welcome, and the study is carried on in an amateur, philosophic spirit, not as professional musicians but as human beings stretching out toward understanding. To see how deep this inquiry plumbs consider a brief passage from Aristotle's Physics:
“In one way, then cause is said as that from which, being present in it, something comes to be, like the bronze of the statue and the silver of the bowl and the genera of these. In another way, [cause is said as] the species and the pattern; this is the account of the ‘what it was to be’ and the genera of this (as of the octave, [the ratio] two to one, and, generally, number) and the parts which are in the account.” II, 3, Becker lines 194b23-29, trans. Glenn Coughlin, Aristotle Physics: or Natural Hearing, South Bend IN: St. Augustine’s Press, 2005, p. 29.
As Jacob Klein points out, “The Logos makes us understand, if we follow Heraclitus, what the things themselves are saying, brightly and darkly, in tune and out of tune.” This is perhaps the origin of the metaphor “the book of nature” that has reverberated through the millennia (“Speech, Its Strength and Its Weaknesses”, lecture first delivered at St. John’s College, Annapolis, 23 February 1973, in Jacob Klein: Lectures and Essays, ed. Robert B. Williamson and Elliott Zuckerman, Annapolis MD: St. John’s College Press, 1985, pp. 361-374.).
We look then at great works such as JS Bach’s St. Matthew Passion, to see the elements and their relations at work, to learn about the possibilities of the Logos, and what we can thereby hear about nature, our own natures included, about the things themselves. What moves us and how? What is the power of music? What is the place of beauty?
When scholars sought to restore the liberal arts in America, at places like Columbia University in the 1920s, University of Chicago in the 1930s, and St. John’s College in the 1940s for instance, music naturally rushed in to take its rightful place in their plans of study due to irrepressible student demand. Indeed, the “father of the great books seminar,” John Erskine, left Columbia University to be the founding president of the Juilliard School in 1927. His old student Jacque Barzun revived the great books seminars at Columbia in 1936.
We invite you join us in this unusual opportunity to include more music in your lifelong learning.
See a sampling of accessible books congenial to Symposium’s approach to music
See a sampling of accessible books congenial to Symposium’s approach to music
Aaron Copland, What to Listen For in Music, New York: McGraw-Hill Inc, 1939 and in print ever since, 307 pp.
Grosvenor W. Cooper, Learning to Listen: A Handbook for Music, Chicago and London: The University of Chicago Press, 1957, 165 pp.
Victor Zuckerkandl, The Sense of Music, Princeton: Princeton University Press, 1959, 255 pp.
__________ Sound and Symbol: Music and the External World, trans. from the German by Willard R. Trask, Bollingen Series XLIV, Princeton: Princeton University Press, 1956, 399 pp.
__________ Man the Magician, Bollingen Series XLIV.2, trans. from the German by Norbert Guterman, Princeton: Princeton University Press, 1973, 370 pp.
The Study of Counterpoint from Johann Joseph Fux’s Gradus ad parnassum, trans. and ed. by Alfred Mann, New York and London: W.W. Norton and Co., 1943, 1965, 1971, 156 pp.
Wye Jamison Allanbrook, Rhythmic Gesture in Mozart: Le nozze di Figaro & Don Giovanni, Chicago and London: The University of Chicago Press, 1983, 396 pp.
Peter Pesic “Hearing the Irrational: Music and the Development of the Modern Concept of Number” in Isis v. 101, pp. 501-530, 2010, PDF available online at Pesic, Hearing the Irrational
___ Music and the Making of Modern Science, Cambridge MA and London: The MIT Press, 2014, 347 pp.
___ Polyphonic Minds: Music of the Hemispheres, Cambridge MA and London: The MIT Press, 2017, 330 pp.
___ Sounding Bodies: Music and the Making of Biomedical Science, Cambridge MA and London: The MIT Press, 2022, 408 pp.
Peter Kalkavage, Music and the Idea of a World, Philadelphia: Paul Dry Books, 2024, 325 pp.
About Languages at Symposium
About Languages at Symposium
Beginners groups start in January and June each year.
Beginners groups start in January and June each year.
Language learning with Symposium is offered in an all-too-rare manner. Rather than merely learning ancient Greek and Latin (more languages to come hopefully), we use Greek and Latin to inquire into language itself, especially grammatical elements, but also logic, rhetoric, poetics, scansion, etymology. After all, how many of us know what precisely a past participle is, and how often do we use, misuse, and abuse past participles? We do this at a slow but steady, sustainable pace for working professionals, and semi-retired and retired persons.
Language learning happens practically in small tutorials, in which a group proceeds together with a shared text and guide. More experienced participants help the less experienced, and the burden is on each to learn what they can, at their own pace.
Ancient languages are particularly helpful to us in various ways: they are our parent languages; they are inflected languages; they are at once similar and different enough from English to illuminate the foundations of our own language and other Indo-European languages. They open worlds and enable us to read in the originals Homer, Sappho, Euclid, Ptolemy, Diotima, Plato, Aristotle, Hypatia, Herodotus, Thucydides, Vergil, Ovid, Lurcretius, the Greek Bible, Augustine, John Damascene, Macrina of Cappadocia, Aquinas, Campion, Herbert, Marvell, Copernicus, Gilbert, Kepler, Galileo, Descartes, Francis Bacon, Harvey, Hobbes, Spinoza, Locke, Huygens, Euler, Newton, Gauss, Bernoulli, Samuel Johnson, and others, the progenitors of Dante, Camoes, Cervantes, Shakespeare, Goethe, Moliere, Pushkin, Melville, Faulkner. Through them we can overcome our recent amnesia and orphandom into greater possession of most of our patrimony. Indeed the demise of the classical, that is to say the robust, liberal arts throughout the world has tended to go hand in hand with the demise of the teaching and learning of the classical languages.
How can we be present if we do not recollect our past-present? How can we stand upon the shoulders of giants if we don't speak Giant? Let us heed the words of old Lupus, the learned abbot of Ferrieres, "reviridicencia litterarum/the growing green again of letters."
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