Archytas of Tarentum (c. 425 BCE—c. 350 BCE)

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ARCHYTAS OF TARENTUM
(C. 425 BCE—C. 350 BCE)

Archytas of Tarentum was active in the first half of the fourth century BCE as a mathematician and a philosopher in the Pythagorean tradition. He is famous for having sent a ship in 361 BCE to rescue Plato from Dionysius II, tyrant of Syracuse. Archytas is unique among ancient philosophers for his success in the political sphere—he was elected general seven consecutive times in a democratically governed Tarentum (at the time one of the most important Greek city-states in southern Italy).

More texts have been preserved in Archytas's name than in that of any other Pythagorean, but the majority of these texts are spurious. The pseudo-Pythagorean treatises of the first century BCE and later were often written in his name, considering him the latest of the three great early Pythagoreans (following Pythagoras himself and Philolaus). The spurious works on categories in Archytas's name were regarded as genuine by the commentators on Aristotle's Categories and were frequently cited. Four fragments survive from Archytas's genuine works, of which Harmonics was the most important, and there is a relatively rich set of testimonia.

Archytas provided the first solution to one of the most celebrated problems in ancient mathematics, the duplication of the cube. One romantic version of the problem reports that the inhabitants of the island of Delos were commanded by the god to build an altar double the size of the current altar, which had the shape of a cube. The problem was thus to determine the length of the side on which to build a cube of double the volume. Archytas's solution is a masterpiece of mathematical imagination, requiring one to envision the intersection of two lines drawn on the surface of a semicylinder—one by a rotating semicircle and one by a rotating triangle. In later antiquity, a story arose that Plato was critical of Archytas for using mechanical instruments to find the solution and thus perverting the true function of mathematics—that is, to direct the soul to the intelligible realm. This story was probably invented to explain the separation of the science of mechanics from philosophy. No physical instruments are employed in Archytas's solution, and it was criticized by some ancient authors as too abstract and of little practical application. Although Plato's complaints about the state of solid geometry in his day (Rep. 528b–d) may be directed at Archytas, they focus not on the use of instruments but rather on the failure of its practitioners to develop a coherent science of solid geometry.

Fr. 1, the beginning of Archytas's Harmonics, is the earliest text to identify a quadrivium of four sciences (the science of number, geometry, astronomy, and music). Archytas praises the sciences for beginning by distinguishing the universal concepts relevant to the specific science, but he regards their ultimate goal as an account of individual things in the world in terms of number, thus building on Philolaus's insight that all things are known through number. Archytas's own Harmonics begins by distinguishing important general conceptions in acoustics. His mistaken view that pitch depends on the speed with which a sound travels—it depends, in fact, on the frequency of impacts in a given period—was adopted with modifications by both Plato and Aristotle and was the most common view in antiquity. Archytas provided an important proof that the basic musical intervals such as the octave, which correspond to ratios of the form (n+1)/1, cannot be divided in half.

The goal of Archytas's harmonics, however, was the description of a particular set of phenomena—in this case the musical scales in use in his day—in terms of specific numerical ratios. Plato complained that such a science of harmonics sought numbers in the sensible world rather than ascending to more abstract problems, which were independent of the phenomena (Rep. 531c). For Archytas, however, there was no split between the intelligible and sensible world. Logistic, the science of number and proportion, was the master science for Archytas, because all other sciences ultimately rely on number to provide knowledge of individual things (Fr. 4). Just as his science aimed at mathematical description of concrete phenomena, so Archytas also developed a theory of definition that earned Aristotle's praise (Metaph. 1043a14–26) for taking into account not just the limiting (formal) aspect of the definiendum but also the unlimited (material) aspect.

Archytas argued that number was crucial in the political and ethical sphere as well. The stability of the state is based on the widely held human ability to calculate, which convinces the rich and the poor that they have their fair share (Fr. 3). Archytas regarded bodily pleasure as inimical to the rational calculation that should guide one's life, because, he believed, someone in the throes of the most intense pleasure (e.g. sexual orgasm) is manifestly unable to calculate.

There is little evidence for Archytas's cosmology, but he developed the most powerful ancient argument for the infinity of the universe. Archytas assumes that, if the universe is limited, it has an edge (modern science would question this assumption) and asks whether or not someone standing at the edge would be able to extend his or her hand beyond the edge. Normal assumptions about space suggest that it would be paradoxical if the person could not extend a hand beyond the edge. Archytas can ask the same question about any supposed limit, and hence the universe will not have a limit and will extend indefinitely. Versions of this argument were adopted by the Epicureans, Stoics, Locke, and Newton—although both Plato and Aristotle rejected it. Aristotle wrote three books—now lost—on Archytas and presents him favorably. Plato was impressed with Archytas's work in mathematics, but the two philosophers disagreed sharply on important philosophical issues.