Please note: These papers were prepared for the Greek Science course taught at Tufts University by Prof. Gregory Crane in the spring of 1995. The Perseus Project does not and has not edited these student papers. We assume no responsibility over the content of these papers: we present them as is as a part of the course, not as documents in the Perseus Digital Library. We do not have contact information for the authors. Please keep that in mind while reading these papers.

Chris Weinkopf

Greek Mathematics

April 19, 1995

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Table of Contents

    1. The Divisions
    2. The Pure and the Physical
    3. Order, Purpose, and Method

Greek mathematics was premised on inductive reasoning. Whereas ancient historians sough to deduce facts from observations, the mathematicians sought to explore and discover truths working from a factual foundation. Theoretical mathematics also provided ancient philosophers with the tools of logic, which were thus employed in the pursuit of practical ends. The classical interpretation of mathematics, as well as the subdivision of the discipline into specific categories, demonstrates the Greeks' approach to the subject.

The Divisions

Afragment of Archytus, a Pythagorean friend and contemporary of Plato, explains the Quadrivivium, the four fields into which the Pythagoreans divided mathematics: arithmetic, geometry, sphaeric (astronomy), and music. Plato classified mathematics into the same categories, but included as a separate division stereometry-- three-dimensional geometry-- between geometry and sphaeric. (The Pythagoreans considered stereometry as subfield of geometry).

The ordering of the Pythagorean and Platonic classifications of mathematics was based on the physical dimensions. Arithmetic, which deals exclusively with numbers, is one-dimensional, is consequently the most simplistic of the mathematics and therefor falls first in the order. It is followed by two-dimensional geometry, then stereometry, and next astronomy, which deals with three dimensional objects in motion. Music follows astronomy because both are governed by the laws of harmony-- the flow of celestial bodies is harmonious to the eye as music is to the ear.

Theon of Smyrna and Proclus ordered the divisions differently. Like the Pythagoreans they placed arithmetic first, but followed it with music, then geometry, and finally sphaeric. Whereas the Pythagorean and Platonic orderings were based on physical dimensions, the discriminating criteria for Theon and Proclus were magnitude and motion. Arithmetic and Music were first and second in the ordering, because unlike geometry and sphaeric, they do not have magnitude. Arithmetic preceded music, however, because the former deals just with numbers, the latter with numbers and their relation to something else. Sphaeric proceeded geometry, because its magnitude is in motion, whereas the magnitude of geometry is at rest. For more on this distinction, see Aristotle's Metaphysics 1078.

Despite the technical differences between the Pythagorean/Platonic and Smyrna/Proclus orderings of mathematical divisions, however, the two were alike in as much that both began with what was believed to be the "simplest" branch of the discipline, and became more complex with each subsequent level. The ordering of the systems was consistent with the nature of the discipline itself: it stressed working from the simplest of truths to the most complex, not through speculation, but through rigid application of the logical process.

Aristotle added optics and mechanics to the subdivisions of mathematics. He classified the branches into two categories: "pure" (theoretical math) and "physical" (applied). His order rests on the notion that all other fields of mathematics are derived ultimately from arithmetic and geometry. The division was also based on inductive reasoning, as Aristotle stressed that physical math required the pure for its proof.

The Pure and the Physical

The Greek mathematicians defined "arithmetic" differently than we do today. The word encompassed not only logistic, or what we commonly associate as arithmetic, but also the Theory of numbers. The latter falls into the Aristotelian division of "pure mathematics," and was used in connection with geometrical proofs. Logistic, however, an applied or physical form of mathematics, had many day to day applications.

In his Theaetetus, Plato portrays arithmetic as a function of logic when two characters determine that mathematics is the most rational way to solve a problem. Platonic characters use simple numbers as a way of expressing worth-- in this case that of individuals-- in Statesmen. And in The Republic, Socrates characterizes the art of calculation as a prerequisite for all other forms of though. The three Platonic quotes encompass Ancient Greek attitudes towards arithmetic: a logical process, a tool of everyday life, and the fundamental building block of all knowledge.

Geometry too, in the Hellenistic world, was classified according to its pure and physical elements. Theoretical geometry, such as the Pythagorean proofs, were used to derive practical applications and explain natural phenomena. Geodesy, or mensurnation, ]for example, was a field of geometry which provided Greeks with a practical measurement of volumes and surfaces, as illustrated in Aristotle's Metaphysics.

Geminus employed the laws of geometry to explain optics, specifically why placement and distance affect the appearance of an object's size and shape. He also used geometry to design better scenes for the theater and explore Catoprica, the theory of mirrors. Similarly, he applied theoretical notions of mechanics, such as centers of gravity and equilibrium, to the creation if weapons or theatrical stage equipment. He noted that mechanics could be used to design scale models of heavenly bodies, and Archimedes is credited for having made such a device.

Plato noted that theoretical astronomy had its application in phaenomena, heavenly observation. Geminus expanded on the applications of astronomy by including the use of gnomons or sundials to measure time, the art of measuring the heights of stars crossing the meridian, and determining the positions of sun, moon, and stars.

Order, Purpose, and Method

The Greek division and ordering of mathematics are emblematic of the discipline's methodology. Ancient mathematicians classified the simplest or purest forms of their fields first, as it was from these that other forms were derived. Greek mathematicians were dedicated to the discovery of truths, based on existing axioms, and then applying them to discover newer truths still.