Zeno of Sidon
Born: about 150 BC in Sidon (now Saida in Lebanon)
Died: about 70 BC in Athens, Greece
Zeno of Sidon was born in the city of Sidon on the Mediterranean coast
of what today is Lebanon.
To understand the philosophy of Zeno we need to make some comments about
the philosopher Epicurus who founded the Epicurean School to which Zeno
later belonged. Epicurus, who lived from 341BC 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
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 science.
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.
us, 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 criticised 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.
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 in [2] 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.
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 [1]:-
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 defence 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
Of course there are many examples in mathematics where Zeno's
observations of many cases would actually suggest a false result.