Monday, March 28, 2011

Introduction to the Pythagorean Mathematical Philosophy & the Quadrivium

The philosophical tradition surrounding the name of Pythagoras derives its mistique from how little we know of its origins, while at the same time can claim the mathematical development and scientific advancements as verification of its basis in wisdom and truth.

The Pythagoreans divided their teaching into a four-fold system called the Quadrivium, consisting of Arithmetic/Number, Harmonics/Music, Geometry, Cosmology/Astronomy. Nicomachus of Gerasa, well known as being the greatest Neo-Pythagorean of his time, can help us understand the meaning of this division of their teaching:

"Things, then, both those properly so called and those that simply have the name, are some of them unified and continuous, for example, an animal, the universe, a tree, and the like, which are properly and peculiarly called "magnitudes"; others are discontinuous, in a side-by-side arrangement, and, as it were, in heaps, which are called "multitudes", a flock, for instance, a people, a heap, a chorus, and the like.
"Wisdom, then, must be considered to be knowledge of these two forms.  Since, however, all multitude and magnitude are by their own nature of necessity infinite-- for multitude starts from a definite root and never ceases increasing; and magnitude, when division beginning with a limited whole is carried on, cannot bring the dividing process to an end, but proceeds therefore to infinity-- and since sciences are always sciences of limited things, and never of infinites, it is accordingly evident that a science dealing either with magnitude, per se, or with multitude, per se, could never be formulated, for each of them is limitless in itself, multitude in the direction of the more, and magnitude in the direction of the less.  A science, however, would arise to deal with something separated from each of them, with quantity, set of from multitude, and size, set off from magnitude.

Again, to start afresh, since of quantity one kind is viewed by itself, having no relation to anything else, as "even", "odd", "perfect", and the like, and the other is relative to something else and is conceived of together with its relationship to another thing, like "double", "greater", "smaller", [etc], it is clear that two scientific methods will lay hold of and deal with the whole investigation of quantity; arithmetic, absolute quantity, and music, relative quantity.

"And once more, inasmuch as part of "size" is in a state of rest and stability, and another part in motion and revolution, two other sciences in the same way will accurately treat of "size", geometry the part that abides and is at rest, astronomy that which moves and revolves."    
 -- Nicomachus of Gerasa: Introduction to Arithmetic I - ch. 2-3

Another clear description of the way the Pythagoreans divided learning into the Quadrivium is given by the late Neo-Platonist Proclus:

"The Pythagoreans considered all mathematical science to be divided into four parts: one half they marked off as concerned with quantity, the other half with magnitude; and each of these they posited as twofold.  A quantity can be considered in regard to its character by itself or in its relation to another quantity; magnitudes as either stationary or in motion.  Arithmetic, then, studies quantity as such; music the relations between quantities; geometry [studies] magnitude at rest, spherics [studies] magnitude inherently moving.  The Pythagoreans consider quantity and magnitude not in their generality, however, but only as finite in each case.  For they say that the sciences study the finite in abstraction from infinite quantities and magnitudes, since it is impossible to comprehend infinity in either of them.  Since this assertion is made by men who have reached the summit of wisdom, it is not for us to demand that we be taught about quantity in sense objects or magnitude that appears in bodies.  To examine these matters is, I think, the province of the science of nature, not that of mathematics itself."   
 -- Proclus: A Commentary on the First Book of Euclid's Elements - Prologue I ch. 7

 The study of arithmetic was given before anything else, so fundamental was the doctrine of Number for their philosophical school.  Nicomachus can help us see why arithmetic must be studied first:

"Which then of these four methods must we first learn?  Evidently, the one which naturally exists before them all, is superior and takes the place of origin and root and, as it were, of mother to the others.  And this is arithmetic, not solely because we said that it existed before all the others in the mind of the creating God like some universal and exemplary plan, relying upon which as a design and archetypal example the creator of the universe sets in order to their proper ends; but also because it is naturally prior in birth, inasmuch as it abolishes other sciences with itself, but is not abolished together with them.
"So it is with the foregoing sciences; if geometry exists, arithmetic must also needs be implied, for it is with the help of this latter that we can speak of triangle, quadrilateral, octahedron, icosahedron, double, eightfold, or one and one-half times, or anything else of the sort which is used as a term by geometry, and such things cannot be conceived of without the numbers that are implied with each one.  For how can "triple" exist, or be spoken of, unless, the number 3 exists beforehand, or "eightfold", without 8?  But on the contrary 3, 4, and the rest might be without the figures existing to which they give names.
"Hence arithmetic abolishes geometry along with itself, but is not abolished by it, and while it is implied by geometry, it does not itself imply geometry.
"And once more is this true in the case of music; not only because the absolute is prior to the relative, as "great" to "greater" and "rich" to "richer" and "man" to "father", but also because the musical harmonies, diatessaron, diapente, and diapason, are named for numbers; similiarly all of their harmonic ratios are arithmetical ones, for the diatessaron is the ratio 4:3, the diapente that of 3:2, and the diapason the double ratio; and the most perfect, the didiapason, is the quadruple ratio.
"More evidently still astronomy attains through arithmetic the investigations that pertain to it, not alone because it is later than geometry in origin-- for motion naturally comes after rest-- nor because the motions of the stars have a perfectly melodious harmony, but also because risings, settings, progressions, retrogressions, increases, and all sorts of phases are governed by numerical cycles and quantites.
"So then we have rightly undertaken first the systematic treatment of this, as the science naturally prior, more honorable, and more venerable, and as it were, mother and nurse of the rest."    
 -- Nicomachaus of Gerasa: Introduction to Arithmetic I - ch. 4-5

It is recommended that a deeper understanding of the philosophy of the Quadrivium is gained by meditation on these quotations from the mathematician-philosophers.

We shall be exploring the Quadrivium and related areas in future posts.

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