Chrysippus of Soli

Born: 280 BC in Soli, Cilicia, Asia Minor (now Soloi, Turkey)
Died: 206 BC in Athens, Greece

Chrysippus was of Phoenician roots. He came to Athens to study philosophy at the Academy with Arcesilaus. After a while he left the Academy and moved to the Stoa Poikile Academy in Athens where he was a pupil at the School founded by Zeno of Citium.

By the time Chrysippus joined the Stoa Poikile Academy, Cleanthes of Assos had become the second head following the death of Zeno. Chrysippus studied under Cleanthes, but he had also been influenced by the teachings of Plato. In 232 BC Chrysippus became the third head of the Stoa Poikile following the death of Cleanthes. He was to continue to hold this position until his own death.

There is evidence from the writings of Chrysippus that he was poor throughout his life. Certainly he stated that for a philosopher to become wealthy he might serve a king (or even, said Chrysippus, become a king himself). It is clear however that he did not adopt this route to a steady income. Otherwise, Chrysippus writes, the philosopher must rely on his friends and on teaching in order to live, and it would appear that this is the means by which he made his small income.

Another piece of information, which again is not surprising, is that Chrysippus wrote Greek with very poor style. This seems to have been a characteristic of people from Soli, and this is preserved today in the expression 'solecism'. Despite his Greek prose being awkward, he was a prolific writer who is said to have written 705 rolls of papyri, none of which are remains today.

Together with Zeno of Citium, Chrysippus is considered the cofounder of Stoicism. Stoicism takes its name from the Stoa Poikile Academy which in turn means "Painted Colonnade", the place where the founder of the academy usually lectured.

Chrysippus was one of the first to organise propositional logic as an intellectual discipline. Unanalysed propositions joined by connectives were studied. This allowed the Stoics to make major advances in mathematics and science. The logical term "disjunction" is certainly due to the Stoics and it is thought to have originated with Chrysippus. Diogenes Laertius in [3] lists 118 works on logic by Chrysippus, and of these 118 there are seven books occupying 15 rolls of papyri concerning the Liar Paradox.

One claim which Chrysippus made in the area of logic was to reject that the impossible does not follow from the possible. His arguments regarding this are presented in [8] which also examines more generally his views on modal logic.

In physics Chrysippus made distinctions between "whole" and "all" or "universe". He argued that the "whole" is the world while the "all" is the external void together with the world. He believed that logic and physics are necessary to differentiate between good and evil. For Chrysippus a knowledge of physics is necessary before ethics can be formulated. To him the value of physics and logic is mainly for this purpose.

Russell in [4] says:-

Chrysippus ... had an elaborate theory of knowledge, in the main empirical and based on perception, though [he] allowed certain ideas and principles, which were held to be established by ... the agreement of mankind.
One of the contributions of Chrysippus to mathematics is his claim that 'one' is a number. It may seem strange to us to realise that 'one' was ever not considered as a number but of course a little thought makes it clear that indeed there is no need for a number to describe a single object. In fact 'one' was considered as that by which things are measured. Aristotle, in Metaphysics writes (see for example [5]):-
... a measure is not the things measured, but the measure or the One is the beginning of number.
Chrysippus said 'one' was 'multitude one' and should be regarded as a number but this was not immediately accepted, Iamblichus writing that 'multitude one' was a contradiction in terms.

Plutarch in Common notions against the Stoics reports on a dilemma proposed by Democritus as reported by Chrysippus about a cone cut by a plane parallel to its base. Chrysippus argues against the interpretation of the dilemma claiming that it is based on the assumption that mathematical lines have an atomic structure and are not therefore infinitely divisible. However Heath [5] does not believe, as Chrysippus does, that Democritus regards mathematical lines as having an atomic structure. This aspect of Chrysippus's ideas are discussed in the interesting article [7].

There is a very old saying says of Chrysippus, that:-

... he alone is the sage, the others only act as shadows.
There is another saying (see for example [6]):-
If there had been no Chrysippus, there would have been no Stoicism
which certainly does not greatly overstate his importance.

There are a number of versions of how Chrysippus died, one of which says he drank some over proof wine while another says that he died of laughter. Consult [3] for more (probably fictitious) details of his life.

Article by: J J O'Connor and E F Robertson

April 1999
MacTutor History of Mathematics