Aristotle was not primarily a mathematician but made important
contributions by systematising deductive logic. He wrote on physical subjects:
some parts of his Analytica posteriora show an unusual grasp of the
mathematical method. Primarily, however, he is important in the development of
all knowledge for, as the authors of write:-
Aristotle, more than any other thinker, determined the
orientation and the content of Western intellectual history. He was the author
of a philosophical and scientific system that through the centuries became the
support and vehicle for both medieval Christian and Islamic scholastic thought:
until the end of the 17th century, Western culture was Aristotelian. And, even
after the intellectual revolutions of centuries to follow, Aristotelian
concepts and ideas remained embedded in Western thinking.
Aristotle was born in Stagirus, or Stagira, or Stageirus, on the
Chalcidic peninsula of northern Greece. His father was Nicomachus, a medical
doctor, while his mother was named Phaestis. Nicomachus was certainly living in
Chalcidice when Aristotle was born and he had probably been born in that
region. Aristotle's mother, Phaestis, came from Chalcis in Euboea and her
family owned property there.
There is little doubt that Nicomachus would have intended
Aristotle to become a doctor, for the tradition was that medical skills were
kept secret and handed down from father to son. It was not a society where
people visited a doctor but rather it was the doctors who travelled round the
country tending to the sick. Although we know nothing of Aristotle's early
years it is highly likely that he would have accompanied his father in his
travels. We do know that Nicomachus found the conditions in Chalcidice less
satisfactory than in the neighbouring state of Macedonia and he began to work
there with so much success that he was soon appointed as the personal physician
to Amyntas III, king of Macedonia.
There is no record to indicate whether Aristotle lived with his
father in Pella, the capital of Macedonia, while Nicomachus attended to king
Amyntas at the court there. However, Aristotle was certainly friendly with
Philip, king Amyntas's son, some years later and it seems reasonable to assume
that the two, who were almost exactly the same age, had become friendly in
Pella as young children.
When Aristotle was about ten years old his father died. This
certainly meant that Aristotle could not now follow in his father's profession
of doctor and, since his mother seems also to have died young, Aristotle was
brought up by a guardian, Proxenus of Atarneus, who was his uncle (or possibly
a family friend as is suggested by some authors). Proxenus taught Aristotle
Greek, rhetoric, and poetry which complemented the biological teachings that
Nicomachus had given Aristotle as part of training his son in medicine. Since
in latter life Aristotle wrote fine Greek prose, this too must have been part
of his early education.
In 367 BC Aristotle, at the age of seventeen, became a student
at Plato's Academy in Athens. At the time that Aristotle joined the Academy it
had been operating for twenty years. Plato was not in Athens, but rather he was
on his first visit to Syracuse. We should not think of Plato's Academy as a
non-political organisation only interested in abstract ideas. The Academy was
highly involved in the politics of the time, in fact Plato's visit to Sicily
was for political reasons, and the politics of the Academy and of the whole
region would play a major role in influencing the course of Aristotle's life.
When Aristotle arrived in Athens, the Academy was being run by
Eudoxus of Cnidos in Plato's absence. Speusippus, Plato's nephew, was also
teaching at the Academy as was Xenocrates of Chalcedon. After being a student,
Aristotle soon became a teacher at the Academy and he was to remain there for
twenty years. We know little regarding what Aristotle taught at the Academy. In
 Diogenes Laertius, writing in the second century AD, says that Aristotle
taught rhetoric and dialectic. Certainly Aristotle wrote on rhetoric at this
time, issuing Gryllus which attacked the views on rhetoric of Isocrates, who
ran another major educational establishment in Athens. All Aristotle's writings
of this time strongly support Plato's views and those of the Academy.
Towards the end of Aristotle's twenty years at the Academy his
position became difficult due to the political events of the time. Amyntas, the
king of Macedonia, died around 369 BC, a couple of years before Aristotle went
to Athens to join the Academy. Two of Amyntas's sons, Alexander II and
Perdiccas III, each reigned Macedonia for a time but the kingdom suffered from
both internal disputes and external wars. In 359 BC Amyntas's third son, Philip
II came to the throne when Perdiccas was killed fighting off an Illyrian
invasion. Philip used skilful tactics, both military and political, to allow
Macedonia a period of internal peace in which they expanded by victories over
the surrounding areas.
Philip captured Olynthus and annexed Chalcidice in 348 BC.
Stagirus, the town of Aristotle's birth, held out for a while but was also
defeated by Philip. Athens worried about the powerful threatening forces of
Macedonia, and yet Aristotle had been brought up at the Court of Macedonia and
had probably retained his friendship with Philip. The actual order of events is
now a little uncertain. Plato died in 347 BC and Speusippus assumed the
leadership of the Academy. Aristotle was certainly opposed to the views of
Speusippus and he may have left the Academy following Plato's death for
academic reasons or because he failed to be named head of the Academy himself.
Some sources, however, suggest that he may have left for political reasons
before Plato died because of his unpopularity due to his Macedonian links.
Aristotle travelled from Athens to Assos which faces the island
of Lesbos. He was not alone in leaving the Academy for Xenocrates of Chalcedon
left with him. In Assos Aristotle was received by the ruler Hermias of Atarneus
with much acclaim. It is likely that Aristotle was acting as an ambassador for
Philip and he certainly was treated as such by Hermias. Aristotle married
Pythias, the niece and adopted daughter of Hermias, and they had one child, a
daughter also called Pythias. However, Aristotle's wife died about 10 years
after their marriage. It is thought that she was much younger than Atistotle,
being probably of age of about 18 when they married.
On Assos, Aristotle became the leader of the group of
philosophers which Hermias had gathered there. It is possible that Xenocrates
was also a member of the group for a time. Aristotle had a strong interest in
anatomy and the structure of living things in general, an interest which his
father had fostered in him in his early years, that helped him to develop a
remarkable talent for observation. Aristotle and the members of his group began
to collect observations while in Assos, in particular in zoology and biology.
Barnes writes in that Aristotle's:-
... studies on animals laid the foundations of the biological
sciences; and they were not superseded until more than two thousand years after
his death. The enquiries upon which those great works were based were probably
carried out largely in Assos and Lesbos.
Aristotle probably begun his work Politics on Assos as well as
On Kingship which is now lost. He began to develop a philosophy distinct from
that of Plato who had said the kings should be philosophers and philosophers
kings. In On Kingship Aristotle wrote that it is:-
... not merely unnecessary for a king to be a philosopher, but
even a disadvantage. Rather a king should take the advice of true philosophers.
Then he would fill his reign with good deeds, not with good words.
However, Aristotle's time in Assos was ended by political
events. The Persians attacked the town and Hermias was captured and executed.
Aristotle escaped and stopped on the island of Lesbos on his way to Macedonia.
It was more than a passing visit for he remained there for about a year and
must have had the group of scientists from Assos with him for they continued
their biological researches there.
Macedonia was now at peace with Athens, for Philip had made a
treaty in 346 BC. In 343 BC Aristotle reached the Court of Macedonia and he was
to remain there for seven years. The often quoted story that he became tutor to
the young Alexander the Great, the son of Philip, is almost certainly a later
invention as was pointed out by Jaeger, see . Grayeff in suggests that
Philip saw in Aristotle a future head of the Academy in Athens. Certainly this
would have suited Philip well for Speusippus, the then head of the Academy, was
strongly opposed to Philip and strongly encouraging Athens to oppose the rise
The treaty between Athens and Macedonia began to fall apart in
340 BC and preparations for war began. The following year Speusippus died but
Aristotle, although proposed as head of the Academy, was not elected. The
position went to Xenocrates and Philip lost interest in his support for
Aristotle. He moved back to his home in Stagirus and took with him to Stagirus
his circle of philosophers and scientists.
Aristotle did not marry again after the death of his wife but he
did form a relationship with Herpyllis, who came from his home town of
Stagirus. It is not clear when they first met but together they had a son,
Nicomachus, named after Aristotle's father.
Philip was now at the height of his power but, as so often
happens, that proved the time for internal disputes. Aristotle supported
Alexander, Philip's son who soon became king. Alexander decided on a policy
similar to his father in regard to Athens and sought to assert his power by
peaceful means. Alexander protected the Academy and encouraged it to continue
with its work. At the same time, however, he sent Aristotle to Athens to found
a rival establishment.
In 335 BC Aristotle founded his own school the Lyceum in Athens.
He arrived in the city with assistants to staff the school and a large range of
teaching materials he had gathered while in Macedonia; books, maps, and other
teaching material which may well have been intended at one stage to support
Aristotle in his bid to become head of the Academy. The Academy had always been
narrow in its interests but the Lyceum under Aristotle pursued a broader range
of subjects. Prominence was given by Aristotle to the detailed study of nature
and in this and all the other subjects he studied:-
His own researches were carried out in company, and he
communicated his thoughts to his friends and pupils, never thinking to retain
them as a private treasure-store. He thought, indeed, that a man could not
claim to know a subject unless he was capable of transmitting his knowledge to
others, and he regarded teaching as the proper manifestation of knowledge.
Whether the works that come down to us are due to Aristotle or
to later members of his school was questioned by a number of scholars towards
the end of the 19th century. The reasons are discussed by Jaeger , but in this
work Jaeger argues that the apparent differences in the approach by Aristotle
in different works can be explained by his ideas developing over a number of
years. Grayeff  examines certain texts in detail and again claims that they
represent developments in the ideas of Aristotle's school long after his death.
According to a tradition which arose about two hundred and fifty
years after his death, which then became dominant and even today is hardly
disputed, Aristotle in these same years lecturer - not once, but two or three
times, in almost every subject - on logic, physics, astronomy, meteorology,
zoology, metaphysics, theology, psychology, politics, economics, ethics,
rhetoric, poetics; and that he wrote down these lectures, expanding them and
amending them several times, until they reached the stage in which we read
them. However, still more astounding is the fact that the majority of these
subjects did not exist as such before him, so that he would have been the first
to conceive of and establish them, as systematic disciplines.
After the death of Alexander the Great in 323 BC,
anti-Macedonian feeling in Athens made Aristotle retire to Chalcis where he
lived in the house which had once belonged to his mother and was still retained
by the family. He died the following year from a stomach complaint at the age
It is virtually impossible to give an impression of Aristotle's
personality with any certainty but the authors of write:-
The anecdotes related of him reveal him as a kindly,
affectionate character, and they show barely any trace of the self-importance
that some scholars think they can detect in his works. His will, which has been
preserved, exhibits the same kindly traits; he makes references to his happy
family life and takes solicitous care of his children, as well as his servants.
He was a bit of a dandy, wearing rings on his fingers and
cutting his hair fashionably short. He suffered from poor digestion, and is
said to have been spindle-shanked. He was a good speaker, lucid in his
lectures, persuasive in conversation; and he had a mordant wit. His enemies,
who were numerous, made him out to be arrogant and overbearing. ... As a man he
was, I suspect, admirable rather than amiable.
We have commented above on the disputes among modern scholars as
to whether Aristotle wrote the treatises now assigned to him. We do know that
his work falls into two distinct parts, namely works which he published during
his lifetime and are now lost (although some fragments survive in quotations in
works by others), and the collection of writings which have come down to us and
were not published by Aristotle in his lifetime. We can say with certainty that
Aristotle never intended these 30 works which fill over 2000 printed pages to
be published. They are certainly lecture notes from the courses given at the
Lyceum either being, as most scholars believe, the work of Aristotle, or of
later lecturers. Of course it is distinctly possible that they are notes of
courses originally given by Aristotle but later added to by other lecturers
after Aristotle's death.
The works were first published in about 60 BC by Andronicus of
Rhodes, the last head of the Lyceum. Certainly:-
The form, titles, and order of Aristotle's texts that are
studied today were given to them by Andronicus almost three centuries after the
philosopher's death, and the long history of commentary upon them began at this
What do these works contain? There are important works on logic.
Aristotle believed that logic was not a science but rather had to be treated
before the study of every branch of knowledge. Aristotle's name for logic was
"analytics", the term logic being introduced by Xenocrates working at
the Academy. Aristotle believed that logic must be applied to the sciences:-
The sciences - at any rate the theoretical sciences - are to be
axiomatised. What, then, are their axioms to be? What conditions must a
proposition satisfy to count as an axiom? again, what form will the derivations
within each science take? By what rules will theorems be deduced from axioms?
Those are among the questions which Aristotle poses in his logical writings,
and in particular in the works known as Prior and Posterior Analytics.
In fact in Prior Analytics Aristotle proposed the now famous
Aristotelian syllogistic, a form of argument consisting of two premises and a
conclusion. His example is:-
(i) Every Greek is a person.
(ii) Every person is mortal.
(iii) Every Greek is mortal.
Aristotle was not the first to suggest axiom systems. Plato had
made the bold suggestion that there might be a single axiom system to embrace
all knowledge. Aristotle went for the somewhat more possible suggestion of an
axiom system for each science. Notice that Euclid and his axiom system for
geometry came after Aristotle.
Another topic to which Aristotle made major contributions was
natural philosophy or rather physics by today's terminology. (I [EFR] show my
age and the traditional nature of St Andrews University if I remark that in the
1960s a pass in 'General Natural Philosophy' formed part of my degree.)
Aristotle looks at matter, change, movement, space, position, and time. He also
made contributions to the study of astronomy where in particular he studied
comets, geography with an examination of features such as rivers), chemistry
where he was interested in processes such as burning, as well as meteorology
and the study of rainbows.
As well as important works on zoology and psychology, Aristotle
wrote his famous work on metaphysics. This, according to Aristotle, studies:-
... the most general or abstract features of reality and the
principles that have universal validity. ... metaphysics studies whatever must
be true of all existent things just insofar as they exist, [and] it studies the
general conditions which any existing thing must satisfy.
Although Aristotle does not appear to have made any new
discoveries in mathematics, he is important in the development of mathematics.
As Heath explains in :-
The importance of a proper understanding of the mathematics in
Aristotle lies principally in the fact that most of his illustrations of
scientific method are taken from mathematics.
Clearly Aristotle had a thorough grasp of elementary mathematics
and believed mathematics to have great importance as one of three theoretical
sciences. However, it is fair to say that he did not agree with Plato, who
elevated mathematics to such a prominent place of study that there was little
room for the range of sciences studied by Aristotle. The other two theoretical
sciences, Aristotle claimed, were (using modern terminology) philosophy and
Heath notes in the introduction to some of the
mathematics referred to by Aristotle in his works:-
... Aristotle was aware of the important discoveries of Eudoxus
which affected profoundly the exposition of the Elements by Euclid. One
allusion clearly shows that Aristotle knew of Eudoxus's great Theory of
Proportion which was expounded by Euclid in his Book V, and recognised the
importance of it. Another passage recalls the fundamental assumption on which
Eudoxus based his ' method of exhaustion' for measuring areas and volumes; and,
of course, Aristotle was familiar with the system of concentric spheres by
which Eudoxus and Callippus accounted theoretically for the independent motions
of the sun, moon, and planets. ...
The incommensurable is mentioned over and over again, but the
case mentioned is that of the diagonal of a square in relation to its side;
there is no allusion to the extension of the theory to other cases by Theodorus
Heath also mentions the mathematics which Aristotle,
perhaps surprising, does not refer to. There is:-
... no allusion to conic sections, to the doubling of the cube,
or to the trisection of an angle. The problem of squaring the circle is
mentioned in connection with the attempts of Antiphon, Bryson, and Hippocrates
to solve it; but there is nothing about the curve of Hippias ...
While Heath discusses the many mathematical references in
Aristotle, the book attempts to construct (or reconstruct) a work on
Aristotle's view of the philosophy of mathematics. As Apostle writes in:-
... numerous passages on mathematics are distributed throughout
the works we possess and indicate a definite philosophy of mathematics, so that
an attempt to construct or reconstruct that philosophy with a fairly high
degree of accuracy is possible.
We end our discussion with an illustration of Aristotle's ideas
of 'continuous' and 'infinite' in mathematics. Heath explains Aristotle's
idea that 'continuous':-
... could not be made up of indivisible parts; the continuous is
that in which the boundary or limit between two consecutive parts, where they
touch, is one and the same...
As to the infinite Aristotle believed that it did not actually
exist but only potentially exists. Aristotle writes in Physics (see for example
But my argument does not anyhow rob mathematicians of their
study, although it denies the existence of the infinite in the sense of actual
existence as something increased to such an extent that it cannot be gone
through; for, as it is, they do not need the infinite or use it, but only
require that the finite straight line shall be as long as they please. ...
Hence it will make no difference to them for the purpose of proofs.