The Science of Plato

Sources

Impact. Plato is now best known as the main representative of idealist philosophy in the West and the founder of the Academy, the first Western university. He is far less often recognized, however, as a mathematical philosopher (or philosophical mathematician), even though that is how he might have preferred to describe himself.

Forms. His purely philosophical activity also influenced Greek scientific research, although that influence was certainly a mixed and not always encouraging one. Plato’s idealism on the whole tended to devalue the world of the senses as an unstable realm of continual “coming-to-be” and “passing-away.” It is but a pale shadow of the real world of the Forms. If the world we perceive is unreal, however, then research into natural phenomena is correspondingly of less value and importance than efforts to comprehend the ideal Forms on which things in our world are modeled. Truth is to be found not in the visible but instead in the intelligible world and must be accessed by reason and not by sensory experience. Why waste time using the imperfect senses to learn more about a world that is ultimately an illusion?

Parmenides. In his epistemological position, Plato clearly owed much to Parmenides, who likewise rejected the senses in favor of pure, abstract logic. On these terms, Plato judged the value of a particular line of scientific research by how much it uncovered patterns in the natural world that pointed the mind toward the eternal, ideal realm of changeless Being. For the most part, only geometry and astronomy, both highly mathematical, seemed to live up to this high standard. The emphasis here on mathematics also reveals a Pythagorean influence on Plato.

Cosmology. Central to an understanding of Platonic science is the late dialogue known as the Timaeus (circa 355 to 347 b.c.e.). There, Plato gives what he calls a “plausible story”—the best that can be offered, given the fact that it aims to explain the dim world of the senses. Specifically, he provides an account of the arrangement of all things in the world by a divine artisan or Demiurge (demiourgos). Unlike the God of the Judeo-Christian and Islamic traditions, the Demiurge is not a creator in the strict sense of the word, since he works with preexistent matter. Further, he is not even all-powerful, since matter itself can stubbornly resist his efforts to form it in one way or another. He is instead a god who shapes, and who does so by keeping one eye fixed on the eternal Forms, which provide him with patterns for the things in this world. From this starting point, and in the form of a mythic tale that perhaps should not be not taken too literally, Plato goes on to construct an entire cosmology. It intends it to be as comprehensive as possible, combining mathematics, physics, biology, and physiology into a unique and complex system.

Geometrical Shapes. His debts to earlier Greek thinkers are fairly easy to identify. From Empedocles he borrowed the notion that all substances are compounded from the four basic “roots,” earth, water, air, and fire. In a much more radical move, however, and certainly with Pythagorean inspiration, he identified each of these four with a solid geometrical form. Fire is a four-sided figure, or tetrahedron, and resembles a pyramid. Earth is a cube and thus has six sides. Air is an octahedron, or eight-sided polygon. The last and most complex element, finally, is the twenty-sided icosahedron of water. Greek geometry, whose exact details in the century before Plato are hazy at best,

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had already determined that these four shapes—plus a fifth, the dodecahedron (twelve sides)—are the only regular geometrical solids.

Triangles. Plato then went on to analyze each of these shapes even further, namely into a combination of one or the other of two basic types of triangles: the right and the equilateral. Two right isosceles triangles, for instance, or else four equilateral ones, can be joined to form one face of a cube; the whole solid itself, then, would equal 6 x 4 or 24 triangles. The element air, identified with an octahedron, can be broken down into eight such triangles; water, the icosahedron, is made up of twenty.

Problem of Change. Bizarre as this theory might seem to modern eyes, it makes sense and even has some recognizable strengths within its own. To begin with, it marks an advance over Empedocles and the other Pluralists, since it reduces their four elements to a single, fundamental reality. Moreover, by analyzing fire, air, water, and earth into four, eight, twenty, and twenty-four triangles, respectively, it takes care of a major problem that the Empedoclean system leaves unresolved. According to Empedocles, each of the four roots is fundamental and basic, in the sense that it is incapable of any further analysis. Air, for instance, cannot be broken down into smaller, constituent parts, nor can it ever change into anything else—despite the fact that direct observation offers plenty of evidence that air indeed condenses into water, and water evaporates into air. How can these simple changes be explained if each of the four roots permanently keeps its own form?

Transformation. Plato’s composite elements, by contrast, can be easily transformed into one another by the simple addition or subtraction of the triangles out of which they are made. Remove two triangles from earth, for instance, and the result will be fire. This results in a kind of chemistry of combinations and proportions that is similar to the one that Empedocles proposed, but which operates on a much more fundamental level. Further, Empedocles’ vague talk of “four parts fire, two parts water, two parts earth” as a recipe for the production of “bone,” for example, could now be replaced by seemingly more precise mathematical formulas, such as the one Plato offers to account for how (one icosahedron of) water changes in the process of boiling into (two octahedra of) air and (one tetrahedron of) fire: 20 = [2 x 8} + 4.

Math and Nature. What is perhaps most important of all about the cosmological account in the Timaeus is the fact that it completes the project begun by the Pythagoreans. In their belief that numbers are the basic reality, the Pythagoreans had sought (both mystically and mathematically) to uncover the hidden numerical structure of the world. In this respect, they mathematized nature. Plato’s own work in the Timaeus carries this process to its logical conclusion by identifying ultimate reality with geometrical forms. The earlier Greek thinkers were materialists: the fundamental “stuff” of the world was always seen by them as precisely that— “stuff”— that is, as a physical entity. Although Plato’s four elements (earth, air, fire, and water) are material, however, his system reduced them to the far more abstract, nonmaterial shapes of the triangles. The physical world could now be constructed, broken down, and remade again by combining geometrical parts. Once this had been accomplished, the old questions of what reality is made of and how it undergoes all its innumerable changes seemed that much closer to being resolved. Moreover, the preferred language in which acceptable answers to these questions were to be cast was from then on decidedly the language of mathematics.

Sources

D. R. Dicks, Early Greek Astronomy to Aristotle (Ithaca, N.Y.: Cornell University Press, 1970).

David C. Lindberg, The Beginnings of Western Science: The European Scientific Tradition in Philosophical, Religious, and Institutional Context, 600 b.c.e. to c.e. 1450 (Chicago: University of Chicago Press, 1992).

Samuel Sambursky, The Physical World of the Greeks (New York: Macmillan, 1956).