44 3. MATHEMATICAL CULTURES IIPlato is famous for his theory of ideas, which had both metaphysical and epis-
temological aspects. The metaphysical aspect was a response to two of his pre-
decessors, Heraclitus of Ephesus (ca. 535-475 BCE), who asserted that everything
is in constant flux, and Parmenides (born around 515 BCE), who asserted that
knowledge is possible only in regard to things that do not change. One can see
the obvious implication: Everything changes (Heraclitus). Knowledge is possible
only about things that do not change (Parmenides). Therefore.... To avoid the
implication that no knowledge is possible, Plato restricted the meaning of Heracli-
tus' "everything" to objects of sense and invented eternal, unchanging Forms that
could be objects of knowledge.
The epistemological aspect of Plato's philosophy involves universal proposi-
tions, statements such as "Lions are carnivorous" (our example, not Plato's), mean-
ing "All lions are carnivorous." This sentence is grammatically inconsistent with
its meaning, in that the grammatical subject is the set of all lions, while the asser-
tion is not about this set but about its individual members. It asserts that each
of them is a carnivore, and therein lies the epistemological problem. What is the
real subject of this sentence? It is not any particular lion. Plato tried to solve this
problem by inventing the Form or Idea of a lion and saying that the sentence really
asserts a relation perceived in the mind between the Form of a lion and the Form
of a carnivore. Mathematics, because it dealt with objects and relations perceived
by the mind, appeared to Plato to be the bridge between the world of sense and
the world of Forms. Nevertheless, mathematical objects were not the same thing as
the Forms. Each Form, Plato claimed, was unique. Otherwise, the interpretation
of sentences by use of Forms would be ambiguous. But mathematical objects such
as lines are not unique. There must be at least three lines, for example, in order for
a triangle to exist. Hence, as a sort of hybrid of sense experience and pure mental
creation, mathematical objects offered a way for the human soul to ascend to the
height of understanding, by perceiving the Forms themselves. Incorporating math-
ematics into education so as to realize this program was Plato's goal, and his pupils
studied mathematics in order to achieve it. Although the philosophical goal was
not reached, the effort expended on mathematics was not wasted; certain geometric
problems were solved by people associated with Plato, providing the foundation of
Euclid's famous work, known as the Elements.
Within half a century of Plato's death, Euclid was writing that treatise, which
is quite free of all the metaphysical accoutrements that Plato's pupils had experi-
mented with. However, later neo-Platonic philosophers such as Proclus attempted
to reintroduce philosophical ideas into their commentary on Euclid's work. The
historian and mathematician Otto Neugebauer (1975, p. 572) described the philo-
sophical aspects of Proclus' introduction as "gibberish," and expressed relief that
scientific methodology survived despite the prevalent dogmatic philosophy.
According to Diels (1951, 44A5), Plato met the Pythagorean Philolaus in Sicily
in 390. In any case, Plato must certainly have known the work of Philolaus, since
in the Phaedo Socrates says that both Cebes and Simmias are familiar with the
work of Philolaus and implies that he himself knows of it at second hand. It
seems likely, then, that Plato's interest in mathematics began some time after the
death of Socrates and continued for the rest of his life, that mathematics played
an important role in the curriculum of his Academy and in the research conducted
there, and that Plato himself played a leading role in directing that research. We
do not, however, have any theorems that can with confidence be attributed to Plato