Category: Philosophical theories

Plato and mathematics

Plato becomes all the more favourably inclined toward the science of his time, especially mathematics and its method. In the educational programme for the philosopher-kings in the Republic, mathematical knowledge becomes a necessary stage preceding the initiation to true philosophy. However, Plato’s familiarity with mathematics does not imply an uncritical acceptance of mathematical practices.

Platonic philosophy and Mathematics

There is significant pre-existing knowledge in the theory of numbers and in geometry in the fourth century. The discovery of incommensurable magnitudes is dated around 430 BCE and is considered to be pivotal in the development of Greek mathematics. Greek mathematicians are familiar with concepts such as demonstration, analysis and composition, infinite regress, analogy. Hippocrates of Chios is said to have been the first to write geometrical “Elements” at the end of the fifth century, i.e. to have consciously taken steps toward setting axiomatic foundations in geometry, a project taken up with growing intensity during the entire fourth century and in which main figures were Archytas, Theaetetus, Eudoxos, friends and companions of Socrates and Plato.

Plato’s activity as an author spans the first fifty years of the fourth century BCE. During this long period his stance toward the sciences changes. In the early dialogues the Platonic Socrates is portrayed as indifferent to, or even suspicious of, the naturalistic and mathematical tradition of his contemporaries and predecessors. His own interest is limited to ethical issues, while scientific aptitude is often a boasting point of his Sophist rivals. During the middle period, the Meno and the Phaedo being significant milestones, one notices an impressive elevation of the import of mathematics. Mathematical knowledge is regarded as a model of precision and validity, so philosophy is called to imitate the method of mathematics -- the so-called “hypothetical method”, according to which a questionable result is proven based on a basic claim (υπόθεσις).

“Well, allow me to intervene and pose the question—whether virtue comes by teaching or some other way—to be examined by means of hypothesis. I mean by hypothesis what the geometricians often do in dealing with a question put to them [...] In the same way with regard to our question about virtue, since we do not know either what it is or what kind of thing it may be, we had best make use of a hypothesis in considering whether it can be taught or not, as thus: In the first place, if it is something dissimilar or similar to επιστήμη, and secondly if this means that it is teachable or not” [Meno 86e-87b]
“However, that is the way I began. In each case I assume some principle [υποθέμενος] which I consider strongest, and whatever seems to me to agree with this, whether relating to cause or to anything else, I regard as true [...] And if anyone held fast to the principle, you would pay him no attention and you would not reply to him until you had examined the consequences to see whether they agreed with one another or not; and when you had to give an explanation of the principle, you would give it in the same way by assuming some other principle which seemed to you the best of the higher ones, and so on until you reached one which was adequate. [Phaedo 100a]

In the educational programme for the philosopher-kings in the Republic, mathematical knowledge (in its five divisions: arithmetic, plane geometry, solid geometry, astronomy, harmony) becomes a necessary stage preceding the initiation to true philosophy (“dialectic”), in contrast to natural research, which assumes a marginal place. Plato’s justification for the thorough study of mathematics is political and cognitive. We seek, says Socrates, a useful “lesson (or educational course)” for future rulers, a “μάθημα that would draw the soul away from the world of becoming to the world of being” (Republic 521d). The strict deductive structure of mathematics prepares the mind for its becoming independent from the counterfeit world of phenomena, opinions and desires (from this “barbaric slough”, 533d) and for realising that true knowledge is pure intellection.

In the later dialogues Plato’s scientific interests are significantly expanded not only in the direction of the social arts (rhetoric, legislation, medicine), but also toward the entire spectrum of the natural sciences. In the Timaeus Plato develops a complete teleological world-picture, which serves simultaneously as an encyclopaedia of fourth-century natural sciences. In the so-called “unwritten doctrines” he also attempts to incorporate basic mathematical concepts (unit, infinity) in the heart of his metaphysics.

Plato’s criticism of the method of mathematics

Plato’s familiarity with the sciences, and especially with mathematics, does not imply an uncritical acceptance of mathematical practices. Whenever Plato deals with scientific or mathematical knowledge, he does not fail to stress that one should not conflate scientific knowledge with usefulness. What he appreciates in the sciences is their relation to truth.

Even pure science, free from any ties to usefulness, can never reach absolute cognitive validity. This role is assigned to philosophy, which is knowledge of the immutable and eternal Forms. By definition the cognitive range of the sciences is limited, since they retain their connection to the sensible world and its unstable and changing entities.

“And can we say that any of these things becomes certain, if tested by the touchstone of strictest truth, since none of them ever was, will be, or is in the same state? [...] That fixed and pure and true and what we call unalloyed knowledge has to do with the things which are eternally the same without change or mixture, or with that which is most akin to them” [Philebus 59a-c]
Even mathematics, the most abstract science, cannot rid itself completely from the sensible world, because it necessarily makes use of diagrams, i.e. visible representations (Republic 510 d-e). Moreover, its “hypotheses” (its axioms) stand without proof, in contrast to dialectic, which is capable of comprehending “that which requires no υπόθεσις and is the ανυπόθετος principle of all” (Repubic 511b).

Author: Vassilis Kalfas
  • Fowler, D.H. The Mathematics of Plato's Academy. A New Reconstruction. Oξφόρδη, 1987.
  • Menn, S. "Plato and the Method of Analysis." Phronesis 47 (2002)
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