After tutoring the heir of King Phillip II, the future
Alexander the Great, for a number of years, Aristotle
returned to Athens in 335
B
.
C
.
E
. to found his own
school, the Lyceum. He intended the school to be as
broad-based as possible, exploring a wide range of sub-
jects but with prominence given to the study of the nat-
ural world. While at the Lyceum, Aristotle wrote 22
texts covering an astonishing range of topics: logic,
physics, astronomy, meteorology, theology, meta-
physics, ethics, rhetoric, poetics, and more. He founded
a theory of kinematics, a study of space, time, and
motion, and he established principles of physics that
remained unchallenged for two millennia.
With regard to mathematics, Aristotle is remem-
bered for his writings in logic, a subject he identified as
the basis of all scientific thought. He invented the syllo-
gism, a form of argument that comes in three parts: a
major premise, a minor premise, and a conclusion.
Although a straightforward notion for us today, this
work represented a first fundamental step toward under-
standing the structure of reasoning. He presented the fol-
lowing line of thought as an example of a syllogism:
Every Greek is a person.
Every person is mortal.
Therefore every Greek is mortal.
Aristotle recognized that any line of reasoning follow-
ing this form is logically valid by virtue of its structure,
not its content. Thus the argument:
Every planet is made of cheese.
Every automobile is a planet.
Therefore every automobile is made of cheese.
for example, is a valid argument, even though the valid-
ity of the premises may be in question. Removing con-
tent from structure was a sophisticated accomplishment.
Aristotle called his field of logic “analytics” and
described his work on the subject in his book Prior and
Posterior Analytics. He wanted to demonstrate the effec-
tiveness of logical reasoning in understanding science.
Aristotle also discussed topics in the philosophy of
mathematics. He argued, for instance, that an unknow-
able such as “infinity” exists only as a potentiality, and
never as a completed form. Although, for example,
from any finite set of prime numbers one can always
construct one more, speaking of the set of prime num-
bers as a single concept, he argued, is meaningless.
(Today we say that Aristotle accepted the “potentially
infinite” but rejected the “actual infinite.”)
It is recorded that Aristotle would often walk
through the gardens of the Lyceum while lecturing,
forcing his pupils to follow. His students became
known as the peripatetics, the word peripatetic mean-
ing “given to walking.” Copies of Aristotle’s lecture
notes taken by the peripatetics were regarded as valu-
able scholarly documents in their own right and have
been translated, copied, and distributed across the
globe throughout the centuries.
Political unrest forced Aristotle to leave Athens
again in 322
B
.
C
.
E
. He died soon afterward at the age
of 62 of an unidentified stomach complaint.
Aristotle’s analysis of critical thinking literally
shaped and defined the nature of logical thought we
exercise today in any academic pursuit. One cannot
exaggerate the profundity of Aristotle’s influence. By
identifying valid modes of thought and clarifying the
principles of logical reasoning, Aristotle provided the
tools necessary for sensible reasoning and astute sys-
tematic thinking. These are skills today deemed funda-
mental to basic goals of all levels of education.
See also
CARDINALITY
;
DEDUCTIVE
/
INDUCTIVE REA
-
SONING
;
PARADOX
.
arithmetic The branch of mathematics concerned with
computations using numbers is called arithmetic. This
can involve a number of specific topics—the study of
operations on numbers, such as
ADDITION
,
MULTIPLICA
-
TION
,
SUBTRACTION
,
DIVISION
, and
SQUARE ROOT
s,
needed to solve numerical problems; the methods needed
to change numbers from one form to another (such as
the conversion of fractions to decimals and vice versa);
or the abstract study of the
NUMBER SYSTEMS
,
NUMBER
THEORY
, and general operations on sets as defined by
GROUP THEORY
and
MODULAR ARITHMETIC
, for instance.
The word arithmetic comes from the Greek work
arithmetiké, constructed from arithmós meaning “num-
ber” and techné meaning “science.” In the time of
ancient Greece, the term arithmetic referred only to the
theoretical work about numbers, with the word logistic
used to describe the practical everyday computations
used in business. Today the term arithmetic is used in
both contexts. (The word logistics is today a predomi-
nantly military term.)
See also
BASE OF A NUMBER SYSTEM
;
FUNDAMENTAL
THEOREM OF ARITHMETIC
;
ORDER OF OPERATION
.
arithmetic 27