About St. John's College: Santa Fe
The Curriculum: The Mathematics Tutorial
Mathematics is a vital part of education, that this is true or ought to be is suggested by the word itself, for it is derived from a Greek word meaning "to learn." It is regrettable, then, that students should come to dislike mathematics or to think of themselves as unmathematical. It is equally regrettable that competent mathematicians are often unaware of the philosophical assumptions upon which mathematical equations and formulas are based. Mathematics at St. John's is studied as a liberal art, not artificially separated from what have come to be called the humanities. When mathematics is taught at an unhurried pace, in an atmosphere of reflective inquiry, and from treatises chosen not only for their matter but also for their elegance and imagination, as it is at St. John's, mathematics becomes not only the most readily learnable liberal art but also one that provides ready access to others and significant analogies with them.
There are two main reasons for studying mathematics. First, it pervades our modern world, perhaps even defines it. Therefore anyone who means to criticize or reform, to resist or cooperate with this world, not only must have some familiarity with the mathematical methods by which it is managed, but also must have thought about the assumptions that underlie their application. It is the task of the mathematics tutorial and the laboratory together to help students to think about what it means to count and measure things in the universe.
The second main reason for studying mathematics concerns the mathematics tutorial more specifically. Since mathematics has, as its name implies, a particularly close connection with the human capacity for learning, its study is especially useful in helping students to think about what it means to come to know something.
To prepare themselves for such reflection, students study artfully composed mathematical treatises, demonstrate propositions at the blackboard, and solve problems. By doing this over four years, they learn a good deal of mathematics and they gain noticeably in rigor of thought, nimbleness of imagination, and elegance of expression. But while they are practicing the art of mathematics in all its rigor, they are continually encouraged to reflect on their own activity.
Scores of questions, of which the following are examples, are raised during the four years: Why and how do mathematical proofs carry such conviction? What is a mathematical system and what are its proper beginnings and ends? What is the relation of logic to mathematics? What do "better" and "worse," "ugly" and "beautiful" signify in mathematics? Do mathematical symbols constitute a language? Are there "mathematical objects"? How might the discoverer of a particular theorem have come to see it? By means of such questions, which grow out of the daily work and which excite the intellect and the imagination at the same time, a discussion is initiated in the mathematics tutorial that is easily and often carried over into the larger sphere of the seminar.
The students begin with the Elements of Euclid. Using Euclid's organization of the mathematical discoveries of his predecessors, the students gain a notion of deductive science and of a mathematical system in general; they become acquainted with one view of mathematical objects - its central expression found in the theory of ratios - which is buried under the foundations of modern mathematics. After Euclid, they begin the study of Ptolemy's Almagest, centering their attention on the problem of "hypotheses" constructed to "save the appearances" in the heavens.
That the tutorial reads Ptolemy indicates the difference between the mathematics tutorial at St. John's and the ordinary course in mathematics. Ptolemy presents a mathematical theory of the heavenly motions, but he gives more than that: His work is both an example of mathematics applied to phenomena and a companion to the philosophical, poetic, and religious readings that are taken up in the first and second years.
In the second year, the students continue the study of Ptolemy, with emphasis upon those difficulties and complexities of the geocentric system that are brilliantly transformed by the Copernican revolution. They study Copernicus' transformation of the Ptolemaic theory into heliocentric form. They next take up the Conics of Apollonius to learn a synthetic presentation of the very objects whose analytical treatment by Descartes marks the beginning of modern mathematics. After this they study analytic geometry, which presents the conic sections in algebraic form. They thus gain an understanding of algebra as the "analytic art" in general.
In the third year, calculus is studied both analytically in its modern form and geometrically as Newton presented it in his Principia Mathematica. This is followed by an examination of Dedekind's theory of real numbers, the endeavor to provide a rigorous arithmetical foundation for the calculus. The students then return to Newton's Principia to take up its treatment of astronomy, in which Newton brings heavenly and earthly motions under one law and replaces a purely geometric astronomy with a "dynamic" theory in which orbits are determined by laws of force.
The mathematics tutorial is both an introduction to physics and a foundation for the study of the philosophical outlook of the modern world.
In the fourth year, the reading of Lobachevski's approach to non-Euclidean geometry invites reflection on the postulates of geometry, as well as on the nature of the geometric art as a whole. Seniors also study Einstein's special theory of relativity, which challenges our conventional understanding of the nature of time and space. In Santa Fe, the mathematics and language tutorials of the senior year are replaced for part of the second semester with a visual arts tutorial that includes a close study of classic paintings, beginning with Giotto's frescoes and ending with Picasso's Les Demoiselles d 'Avignon.
The Music Tutorial
One of the aims of the St. John's program has been to restore music as a liberal art to the curriculum. The study of music at St. John's is not directed toward performance, but toward an understanding of the phenomena of music. The ancients accorded music a place among the liberal arts because they understood it as one of the essential functions of the mind, associated with the mind's power to grasp number and measure. The liberal art of music was based, for them, on the ratios among whole numbers.
In particular, the music program at St. John's aims at the understanding of music through close study of musical theory and analysis of works of musical literature. In the freshman year, students meet once a week to study the fundamentals of melody and its notation. Demonstration takes place primarily by singing, and by the second semester the students perform some of the great choral works. In the sophomore year, a tutorial meets three times a week. Besides continuing the singing, the music tutorial reflects two different but complementary aspects of music. On the one hand, music is intimately related to language, rhetoric, and poetry. On the other, it is a unique and self-sufficient art, which has its roots deep in nature.
The work of the tutorial includes an investigation of rhythm in words as well as in notes, a thorough investigation of the diatonic system, a study of the ratios of musical intervals, and a consideration of melody, counterpoint, and harmony. None of these are done apart from the sounding reality of good music. The inventions of Bach, the songs of Schubert, the masses of Palestrina, the operas of Mozart, and the instrumental works of Beethoven are the real textbooks.
In the second semester, at least one major work is analyzed closely. Seminars on great works of music are included as part of the regular seminar schedule. Instead of reading a text, students listen to recordings of a composition and familiarize themselves with its score before the seminar meets. Group discussion of a work of music, as of a book, facilitates and enriches the understanding of it.
Three hundred years ago, algebra and the arts of analytic geometry were introduced into European thought, mainly by René Descartes. This was one of the great intellectual revolutions in recorded history, paralleling and in part determining the other great revolutions in industry, politics, morals, and religion. It has redefined and transformed our whole natural and cultural world. It is a focal point of the St. John's program and one that the college takes special care to emphasize. There is scarcely an item in the curriculum that does not bear upon it. The last two years of the program exhibit the far-reaching changes that flow from it, and these could not be appreciated without the first two years, which cover the period from Homer to Descartes.
Modern mathematics has made possible the exploration of natural phenomena on an immense scale and has provided the basis for what is known to us as the laboratory. The intellectual tools of the laboratory are the consequence of the vast project of study conceived by the great thinkers of the seventeenth century. They are based on a mathematical interpretation of the universe, which transforms the universe into a great book written in mathematical characters. Liberal learning is concerned with the artifices of the human mind and hand that help us to relate our experiences to our understanding. For this purpose, St. John's has set up a three-year laboratory in the natural sciences, wherein characteristic and related topics of physics, biology, and chemistry are pursued. There is the art of measurement, which involves the analytical study of the instruments of observation and measurement; crucial experiments are reproduced; the interplay of hypothesis, theory, and fact has to be carefully scrutinized. All of this is supported by the mathematics tutorials, which provide the necessary understanding of mathematical techniques.
The task, however, is not to cover exhaustively the various scientific disciplines, to bring the student up to date in them, or to engage in specialized research. It is rather to make the student experience and understand the significance of science as a human enterprise involving fundamental assumptions and a variety of skills. The college does not subscribe to the sharp separation of scientific studies from the humanities, as if they were distinct and autonomous domains of learning. There need not be "two cultures." Different fields of exploration require different methods and techniques, but the integrity of scientific pursuits stems from sources common to all intellectual life.
The Organization of the Laboratory Work
The laboratory program is largely determined by three considerations relevant to the liberalization of the study of science: (1) The formally scheduled experimental work must be combined with a full and free discussion of the instruments and principles involved in it. (2) The content of the work should be so chosen as to enable the students to trace a scientific discipline to its roots in principle, assumption and observation. Thus certain integrated wholes of subject matters are to be selected as problems in which the roles of theory and experimentation can be distinguished through critical study. (3) The schedule of laboratory work shouldgive opportunity for leisurely but intensive experimentation. The students must have time to satisfy themselves as to the degree of accuracy their instruments permit, to analyze procedures for sources of error, to consider alternative methods, and on occasion to repeat an entire experiment. Only thus can they come to a mature understanding of the sciences called "exact.".
A laboratory section consists of fourteen to sixteen students working under the guidance of a tutor, with the help of more advanced students serving as assistants. Sections meet two or three times a week. A laboratory session may be used for exposition and discussion of theory, for experimentation, or for both, as the progress of the work requires. Occasionally, a laboratory meeting is reserved for the discussion of a classic paper or other text directly related to the topic at hand; writings of Aristotle, Galen, Harvey, Huygens, Newton, Lavoisier, Maxwell, Thomson, Rutherford, and Bohr are among those regularly used in this way. In all the work of the laboratory and in the laboratory manuals written at the college, the purpose is to achieve an intimate mixture of critical discussion and empirical inquiry.