The Lectures and Essays of Jacob Klein
3 I The Concept
of Nun1ber
in Greek
Mathen1atics
and Philosophy
T
he subject of my paper is the concept of number in Greek
mathematics and Greek philosophy. This subject is of some
importance, if we consider the role of mathematics not only in
Greek philosophy but also in modern science. Indeed it is doubt­
ful whether philosophy exists today, but certainly the existence
of mathematical physics is not doubtful. All our life and thoughts
are molded by it. In fact, mathematical physics, this immense
construction of our mind, is one of the most important things,
if not the most important, of our modern world. Now the
medium of mathematical physics, or rather its very nerve, is sym­
bolic mathematics. Physics, as we know it today, is not con­
ceivable without symbolic mathematics. We are used to this kind
of symbolic expression to the extent that we have no difficulty
in handling symbols and are not even aware of the fact that we
are dealing with symbols. A school of thought which calls itself
Logistic is trying to interpret this fact in its own way. I think,
however, they do not understand it, becaw,e the existence of sym­
bols appears to them to be self-evident. But symbols are in
themselves a great problem. They didn't exist for the Greeks,
at least not in the same way they exist for us. The great
A paper delivered before the Philosophy Club of the University of Virginia, March 6,
1939.
Lectures and
Jacob Klein
is Geometry. It began
people who were later
collective name, Pythagoreans. These
in a quite different sense from
today. Ma0riµa is something that can be learned
and once learned is known. The idea of
(emcr-r11 µ ri) is intimately connected with that conThus
,._,,.,, .. ,n,., is the model for all Greek
and science. And this is especially true for Plato
as well.
main steps are: Theodorus (420 B.C.) ,
1 nc;c:,c;cc;,u;, (400 B.C. ) , Archytas (390 B.C . ) , E udoxus (370 B.C.),
(300 B.C.), Archimedes (250 B.C.), Apollonius (220 B.C.) .
should also mention a later compiler, Pappus (300 B.C. ) .
is
another, non-geometrical tradition more directly
with the Pythagoreans represented by Nicomachu's
Smyrna (120 A.D. ) , and Domninos (fifth century
A.D.). Finally there are Diophantus (60 A.D.) and Proclus (fifth
century A.D.), one of the commentators of Euclid. I should like
to mention in passing that modern mathematics, as it arises in
sixteenth century, is the result of a rediscovery and re­
of Apollonius, Diophantus, Pappus, and Proclus.
We are not going to deal with that i:,rreat mathematical tradi­
tion.
task will be to describe the Greek concept of number
and the problems which arise in connection with that concept.
We m ust start with the "Pythagoreans." Modern books on the
of philosophy and mathematics usually state that the
main contention of the Pythagoreans was: the essences of things
are numbers. This statement in itself is without sense. The meanof ess,enc)e is very complicated. It is a mediaeval term which
translates an Aristotelian term . The words "things" and
are both ambiguous. It would be safer to render the
contention in the following way: everything that
we see or hear can be counted. This is a rem arkable, but unfor­
false, statement. But even its falsity is of the utmost
for the discovery of the falsity of that statement
means nothing less than the discovery of incommensurables.
What were the things counted by the Pythagoreans and what
the very process of counting mean? The answer to the first
question is: all things which are perceivable by our senses,
all visible things. As to the process of counting it
The
of Number in Greek Mathematics and
45
always comes to a rest when we pronoun ce a word like
a
"hundred;' etc. E ach of those words
a
means
"numbe r" (in Greek: an apt0µ6c; ) . Thus, api0µ6c;
definite number of definite things. And this meaning of the
change through out all stages of
apt0µ6c;
mathem atics and philosophy. It is also the meaning of the
"numerus" until the sixteenth century.
concept of number involves two problem s, two
damenta l problems of Greek m athematics and
as they are
What is the n ,- ... �.. ,,v things in so
In what sense are they "units" submitted to
sense is the number of those things or
unity'? Is the number expressed by one word a
The Pythagoreans were not very much c(}::l:c :�r· n ,,,rt
first question. Their chief concern was
is it possible that many things should
We say
ch airs, seven people, ten cows.
number (five . . . , seven . . . , ten . .
things
another one and
at
same time we comprehend
is not merely a
sense of this word) but also and
the science
a cosrnology,
of this universe.
The books of Nicomachus,
the main
Arithmetic. They
a classification of
tion which we partly find also in the so-called
of Euclid (VII, VIII, IX) . The
and even numbers. "ODD" and "EVEN" can
r ,rt>Gtrl list of contraries as recorded
the
it is worth while mentioning that these
"EVEN," are listed in a peculiar way.
list
two columns: on the one side the terms ""'"r"-""'1 t
and on the other, things of a
nature.
positive
Thus GOOD is opposed to EVIL, LIGHT to DARKNESS,
u
The Lectures and
Jacob Klein
MANY, MALE to FEMALE, etc.
of the words ODD and EVEN,
the "BAD" things. For 1t&p1006<;
u e:cu,,,.,. something which is rather
superfluous. But the PythaWhat
to be superfluous
than a One. We can
EVEN-ODD,
those words
means of
+
•
•
•
• •
• • • •
• • • •
• • • •
• • • •
• • • •
+ l)
•
The
of Number in Greek Mathematics and Philosophy
47
a x,.u,.u,,.,.
method for obtaining all those numbers
consists in substituting for n and m in the respective formulas
beginning with one. )
the series of
other distinctions of numbers in Pythagorean
prime (or linear) numbers,
a u uuu aJ, a numbers, etc. We are not going to deal
must rather ask, What is
reason
What is the purpose of this Pythagorean
""'"1"-c of numbers? I have already said it tries to
a soluof
the
unity
of
any
number.
problem
the
Lu c,l,!.u·, v,u,., an &torn;, a Form of
EVEN-ODD, ,au"'"''· Square,
something which
them
the unity of any number possisix things can
conceived as
Form "triangle;' which
six
to be one. All numbers
as all trees
to that Form
The different s 111.,c 11:::s
what we call
numeration,
universe as a whole is
Every visible thing u1:::., u1 nt:.
things and therefore to a certain
sense the "nature" of every visible
Form of numbers .
nu.u11:;u'-' can
expanded further, if
possible relations
the numbers
Thus we can
are
and furthermore
things
nor audible but are conveyed to us by means of
audible "'"·"'''" to ratios, proportions,
and properties.
science of ratios (and
Logistic (from l6yo<;). It is the basis
'-'<tl.\.: u un,v,,,,,. since calculating things is nothing
but
numbers of things in relation to
UvJ, v u, ,..
•
•
•
•
• •
•
•
•
•
•
, etc .
can
obtained by adding a
to
which is later called the "limiting quantity"
1too61"T1 <;) . A "gnomon" is a configuration of dots
which added to a figure of dots (or lines) produces
Arithmetic (and Logistic) , especially
figures of numbers, are probably the origin of the whole system
of Greek m athem atics in its later " geometrical" form. It seems
,..,,... ·r a ,.,, »
Lectures and
Klein
The
Number in
Mathematics and Philosophy
49
a doctrine relations between pure, indivisible units hav­
existence in themselves can no longer form the "theoretical"
basis of our "practical" calculations. For, in our calculations,
we continually make use of fractions, in other words, we divide
units which we compute. The relations between pure, in­
divisible units don't allow a computation of those units in­
the use of fractions. The art of calculation our
Arithmetic is, therefore, relegated to the
of a merely prac­
tical art, the subject of which is sensible things. This remains
true within the entire Platonic, Neo-Platonic, and Neo­
Pythagorean tradition. Their term Logistic becomes ambiguous,
meaning
the pure doctrine of ratios and harmonics or - to
extent the practical art of computation.
a much
The new point of view from which Plato approaches the
problem of numbers leads him to a further step in answering
the second question connected with
concepts of numbers.
The question is: How can many pure units form one number.
The answer to this question given by the ("purified")
Pythagorean Arithmetic is not entirely satisfactory. The unify­
ing Pythagorean "Forms" are partly alien to the numbers
themselves. The "Forms" don't explain the real differences be­
tween numbers under the same Form . According to Plato,
Arithmetic cannot be sufficiently explained by itself, which is
true also for the whole system of mathematics in the restricted
sense of the word. The true "principles" of the unity of any
number can only be found in Ideas of Numbers. And those ideas
of numbers may solve at the same time, as we shall see, the great
Platonic problem of "participation." Let me state the problem
in Plato's own terms. In the Phaedo, Socrates wonders how one
thing brought to another one thing produces two things. Neither
of the things is two. Is the "two" something apart from the single
things, so to speak, outside of them? Where is the "two"? (We
must not forget that our symbol "2" doesn't mean anything in
itself.) In the Greater Hippias Socrates asks the sophist Hippias
whether he thinks that something which is common to two
things may belong to neither of them. Hippias contemptuously
rejects this suggestion . He argues this way: If we, Socrates and
Hippias, are both just or healthy or wounded, and so on, then
Socrates is just, healthy, wounded, and Hippias is just, healthy,
The Lectures and
Klein
The
51
of Aristotle
first
nine
this kind,
"TWO," which is identical with the idea
"absolute"
which is unique and not
other units, is not a "number" at all. (One in
order is not a number either; the first
is "two." This is valid for all
because an apt0µ6<; is a "number of things"
not a number of things.)
ONE is beyond
b eyond any structure at all, beyond
ouoiac:;: Republic 509b) it is the Idea of
numbers have unity '""''" a.''· " c they are im­
Numbers." In that sense Aristotle is perfectly
conten1:ion (Metaph. A6, 987 b 10-13) that Plato only
t!UJLQ"orea.n term µiµ11mc:; into µt0e�t<;.
"ideal
are analogous to the "root-numbers" of the
the Pythagoreans did with respect to
tries to do with respect to the "true;'
The arithmological structure of the ideas allows also a solu­
tion of the Platonic problem of "participation:' The real prob­
lem of participation is the problem of the community among
Lectures and
Klein
is a very radical
is any unity in a number of
A
"""'· " "-''""' many things and
one at
units a number doesn't mean the
only possible
a
is subjected to
is
Pl atonic position with rocnc.,M
it seems as if Aristotle didn't see the
still awaits a solution.