To understand the philosophy of Zeno one needs to make some
comments about the philosopher Epicurus who founded the Epicurean
School to which Zeno later belonged. Epicurus, who lived from
341 BC to 270 BC, founded his own School of philosophy based
on his teachings. These teachings were designed to indicate a
means of living ones life, and they aimed both to guarantee happiness
and to provide a means to find it. Epicurus had no interest in
science for its own sake and he was a severe critic of mathematics.
On science he wrote:
If we were not troubled by our suspicions of the phenomena
of the sky and about death, and also by our failure to grasp
the limits of pain and desires, we should have no need of natural
His criticisms of mathematics were very superficial of little
importance since he clearly had very little understanding of
the subject. In 306 BC he founded his School in Athens in the
garden of his house. Reasonably enough the School became known
as The Garden.
Apollodorus, the writer of more than 400 books, was a prominent
follower of Epicurus who lived in the 2nd century BC. Zeno of
Sidon was a student of Apollodorus and he studied, and later
taught, in the Garden in Athens. Cicero heard him teaching there
in 79 BC.
Zeno was a man of great learning who wrote on a very wide
range of topics. It is believed that, among the areas he studied,
he contributed to logic, atomic theory, biology, ethics, literary
style, oratory, poetry, the theory of knowledge, and to mathematics.
Except for the last mentioned two topics, one knows very little
about the contributions that he made. Here is a discussion of
the only two areas to which Zeno contributed and where details
of his contributions are quite well known, namely the theory
of knowledge and to mathematics.
Although Epicurus, the founder of the School to which Zeno
belonged, had no real mathematical abilities and criticized the
subject from a position of ignorance, this is far from true of
Zeno who had a deep understanding of the subject. Zeno made deep
criticisms of the axioms that Euclid set out in The Elements.
For example he claimed that Euclid's first proposition assumes
that two straight lines can intersect in at most one point but
Euclid does not have this as an axiom, nor can it be deduced
from the other axioms.
Zeno also attacked Euclid's proof of the equality of right
angles on the grounds that it presupposes the existence of a
right angle. Proclus also says that an Epicurean (almost certainly
Zeno but Proclus does not name him) claimed that Euclid assumes
that every curve is infinitely divisible, but again this cannot
be deduced from the axioms.
Some modern authors have suggested that these claims give
Zeno of Sidon some justification to be considered as having been
the first person to consider the possibility of non-Euclidean
geometry. This is a little far fetched particularly since Zeno's
aim was certainly not this. Rather his aim was to give substantial
arguments against mathematics supporting the anti-mathematical
beliefs of Epicurus.
Heath writes regarding comments by Proclus concerning Zeno:
Zeno argued generally that, even if we admit the fundamental
principles of geometry, the deductions from them cannot be proved
without the admission of something else as well which has not
been included in the said principles, and he intended by means
of these criticisms to destroy the whole of geometry.
Mathematicians of course, came to the defense of their subject,
rather than to try to understand the deep and justified comments
of Zeno. As von Fritz writes:
Zeno's criticisms of Euclid are pertinent, however,
and if any of the ancient philosophers and mathematicians who
tried to refute them had been able to grasp their full implications,
the development of mathematics might have taken a different turn.
Many people gain an important position in history, or fail
to gain such a position, as a result of luck. Had there been
a mathematician following Zeno who could have continued to develop
his ideas then one might know Zeno today as a major figure whose
flash of mathematical genius changed the course of mathematics.
This was not to be, however, and the brilliance of Zeno's ideas
was not appreciated for many centuries.
One knows more of Zeno through one of his students Philodemus
of Gadara. Now Philodemus studied under Zeno in Athens and then
moved to Rome in 75 BC to work for the Roman aristocrat Lucius
Calpurnius Piso. Philodemus then went to live in Lucius's villa
at Herculaneum, near Naples, taking with him his considerable
library of papyri.
When Vesuvius erupted in 79 AD, Herculaneum together with
Pompeii and Stabiae, was destroyed. Herculaneum was buried by
a compact mass of material about 16 m deep that preserved the
city until excavations began in the 18th century. Special conditions
of humidity of the ground conserved wood, cloth, food, and in
particular Philodemus's papyri.
The papyri contain remarkable information written by Philodemus
describing the arguments of his teacher Zeno with the Stoics.
Although Zeno's Epicurean philosophy of the desire for pleasure
seems the direct opposite of the Stoic's ethic of duty, the consequences
on how they lived their lives were quite similar. The arguments
described by Philodemus concerned the foundations of knowledge.
Von Fritz writes:
In this dispute Zeno defended the old Epicurean doctrine
that all human knowledge is derived exclusively from experience.
What make it interesting, however, is that he bases his defense
on a theory ... that is essentially an anticipation of John Stuart
Mill's theory of induction. ... Zeno insisted that all knowledge
is fundamentally derived by inference to all cases from a great
many cases without observed counter-instance.