The Seven Liberal Arts, Hands-On: Logic, Proof and the Quadrivium

The seven liberal arts were the core curriculum of the ancient and medieval West: three arts of language and argument, then four arts of number. They are old, but they are not museum pieces. Each one can be practiced this week, and each one can leave behind something you made. This page gives you a practice and a product for every art, then a short path through the three great teachers of method: Socrates, Aristotle and Euclid.

The seven arts and their order

The set of seven is usually traced to Martianus Capella (5th century CE), and the word quadrivium is usually credited to Boethius (early 6th century). The names trivium and quadrivium for the two groups became standard later in the medieval period. Established as history; the exact origin of each term is less certain than the arts themselves.

The order carries an argument. You first learn to read precisely (grammar), reason validly (logic) and speak persuasively and honestly (rhetoric). Then you apply those tools to quantity: number at rest (arithmetic), number in space (geometry), number in time (music) and number in motion (astronomy). The quadrivium is where reasoning meets things that can be checked. If you skip the trivium, you cannot tell a proof from a plausible story. If you skip the quadrivium, the trivium never meets resistance.

The trivium in practice

Grammar here means more than syntax. It means reading slowly enough to find out what a text actually says. The same habit helps with contracts and technical documentation. For logic, see Proof and Precise Reasoning: From Arguments to Theorems, which covers validity, soundness and counterexamples. For rhetoric, English Expression Mastery: Words, Sentences, Paragraphs and Speech gives a twelve-week routine. The test of rhetoric is honesty under pressure: your essay should state the strongest objection fairly before it answers it.

The quadrivium in practice

Do one quadrivium art at a time; do the trivium weekly.

Arithmetic. Read the number-theory books of Euclid (VII to IX) or Nicomachus. The made thing is a short program that tests a conjecture before you try to prove it. Example: Euler's polynomial n squared plus n plus 41 gives a prime for every n from 0 to 39, which tempts you to believe it always does. Run the code and n = 40 gives 1681, which is 41 squared. Testing refutes quickly; it never proves. Demonstrated by the calculation. Euclid's proof that there are infinitely many primes (Elements IX.20) shows what proof adds.

Geometry. Work through Euclid's Book I one proposition at a time. Write the proof yourself, then draw the construction in code. See Geometry, Euclid and Trigonometry and Euclid's postulates.

Music. Music is the art of ratios. The Pythagorean tradition is credited with the whole-number ratios of consonant intervals: 2:1 for the octave, 3:2 for the fifth. The comparison to make is between three tunings. A just major third (5:4) is about 386 cents; a Pythagorean major third (81:64) is about 408 cents; the equal-tempered third is exactly 400 cents. Stacking twelve pure fifths overshoots seven octaves by the Pythagorean comma, about 23.5 cents. Generate the intervals in code, listen to them, and try the monochord. Demonstrated (it is arithmetic you can hear).

Astronomy. Compare Ptolemy's epicycles with Kepler's ellipses. A caution: epicycles are not simply "wrong". With enough of them you can approximate almost any periodic motion, a mathematical fact later formalized in Fourier analysis. Kepler's gain was a compact, physically meaningful description. Fit his third law to real data: Mars has a semi-major axis of about 1.524 AU and a period of about 1.881 years, and 1.524 cubed is about 3.54 while 1.881 squared is also about 3.54. Established

Engineering as a seventh studio

The medieval seven left out the arts of making. A modern learner can add a studio. After every real fix, write a five-line root-cause note: symptom, evidence, cause, fix, verification. Once a month, write about the principle behind a design decision. Read George Polya's How to Solve It, Herbert Simon's The Sciences of the Artificial and Henry Petroski's To Engineer Is Human, which is about learning from failure. See Evidence-Based Troubleshooting: Separating Explanations for drills that sharpen the habit.

Three teachers of method

Socrates: definition, humility, examination

Plato's early dialogues show a method, the elenchus, in which a claim is tested by asking for definitions and drawing out consequences (see the elenchus). In the Euthyphro, Socrates asks what piety is and finds fault with each answer. In the Meno, he asks whether virtue can be taught and leads a slave boy to see that doubling a square's area means building on its diagonal. In the Apology, he says that the unexamined life is not worth living (38a). Take from these a habit: ask what the word means before arguing about the thing.

Aristotle: the Organon

The Organon is a later editors' grouping of six logical works. The Categories classifies kinds of things. On Interpretation treats propositions. The Prior Analytics founds the syllogism, for example: all mammals are animals; all dogs are mammals; so all dogs are animals. The Posterior Analytics describes demonstration from first principles. The Topics covers dialectical argument and the Sophistical Refutations catalogues fallacies. Established as history; Aristotle's syllogistic is a valid but limited fragment of modern logic.

Euclid: definition, axiom, construction, proof

Euclid begins with definitions, postulates and common notions, then builds 48 propositions in Book I. A good exercise is to rewrite an argument from daily life in Euclidean form: define terms, state assumptions, derive the conclusion step by step. Note that Euclid's proofs are not gap-free by modern standards. For example, I.1 assumes without proof that two circles intersect, a gap that Hilbert's axioms (1899) later closed. Even so, the model remains one of the clearest pictures of deductive order.

Telling kinds of claim apart

A graduate of this path can distinguish proof (the conclusion follows necessarily from the premises), evidence (observation bearing on a claim), inference (a step from one to the other), analogy (a similarity, which suggests but does not establish), rhetoric (persuasion, which can be honest) and propaganda (persuasion that works by bypassing your judgment). Label which one you are reading, and which one you are writing. Analogy deserves special care: the parallels this page draws between music, astronomy and argument are analogies, not demonstrations.

Pacing and where this leads

Do one trivium exercise each week, and give each quadrivium art a month. When the path is solid, you are ready for Boole (The Laws of Thought, 1854), Russell and Whitehead (Principia Mathematica, 1910 to 1913), Goedel (1931), Jaynes (Probability Theory: The Logic of Science, 2003) and Pearl (Causality, 2000). Judgment Under Uncertainty: Probability, Causation and Forecasting covers the last two. A Map of Mathematics: Twenty-Five Areas and What Depends on What shows how the branches depend on each other, and How to Read a Great Book: Levels, Modes and the Completion Standard explains how to read the primary texts named here.

Try this

1. Pick a short paragraph from a text you admire. Write its premises and conclusion as a numbered list, and mark the weakest premise.
2. Write a short program that tests "n squared plus n plus 41 is prime" for n from 0 to 50. Note where it fails, and write one sentence on what the test did and did not show.
3. Compute the three major thirds (5:4, 81:64, and 2 to the power 1/3) in cents and listen to them in order.

Further reading

• Euclid, Elements, Book I, in Heath's translation or the Green Lion Press edition.
• Aristotle, Prior Analytics I.1 to I.7 and Sophistical Refutations.
• Plato, Meno 80d to 86c.
• Sister Miriam Joseph, The Trivium.
• J. M. Barbour, Tuning and Temperament: A Historical Survey (Michigan State College Press, 1951).
• G. Polya, How to Solve It.