42 The Lectures and Essays of Jacob Klein

application of the decimal system to all kinds of calculation and measurement. He demanded that all measures and weights be expressed in decimal units, a demand which was to be fulfilled in France during the French Revolution and which was later followed by practically all the world, except for England and the United States. Strangely enough, Stevin linked his symbolism of the decimal system with that of his Algebra. He writes what we express today as the unknown quantities x, x2

, x3, x4

••• as follows: CD.®.@.® ... , whereas@means not- as we may think-the unit, but any known number.

I cannot speak about his Algebra any further, because my time is up. But I should like to emphasize that Stevin's idea of an "age of wisdom"- that is to say, a golden age of science or, more exactly, an algebraic age of science- is still leading the modern conception of Science in general. The only difference between the idea of Stevin and the modern outlook is that we place that golden age not in the past but in the future. It is a question, whether we are right.

3 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 doubt­ful 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 sym­bolic mathematics. Physics, as we know it today, is not con­ceivable 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, becauc;e the existence of sym­bols 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, Pythagorean<>. These in a quite different sense from

today. Maeru:.ta is something that can be learned and once learned is known. The idea of

(ihno--r'll~-tll) is intimately connected with that con-Thus is the model for all Greek

and science. And this is especially true for Plato as welL main steps are: Theodorus (420 B.C.),

ucac:Lclu" (400 B.C.), Archytas (390 B.C.), Eudoxus (370 B.C.), (300 B.C.), Archimedes (250 B.C.), Apollonius (220 B.C.).

should also mention a later compiler, Pappus (300 B.C.). 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 re­of Apollonius, Diophantus, Pappus, and Proclus.

We are not going to deal with that f,lTeat mathematical tradi­tion. task will be to describe the Greek concept of number and the problems which arise in connection with that concept. We must 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 mean-

of is very complicated. It is a mediaeval term which translates an Aristotelian term. The words "things" and "numbers" 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 remarkable, 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

45

always comes to a rest when we pronounce a word like "hundred;' etc. Each of those words a

"number" (in Greek: an apt9~-t6c;). Thus, apt9~-t6c; means a definite number of definite things. And this meaning of the apt9~-t6c; change throughout all stages of mathematics and philosophy. It is also the meaning of the "numerus" until the sixteenth century.

concept of number involves two problems, two damental problems of Greek mathematics and philosophy. (1) What is the things in so as they are 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 concc:rr:ted first question. Their chief concern was is it possible that many things should be as We say chairs, seven people, ten cows. In every case number (five ... , seven ... , ten . . many things another one and at same time we comprehend

is not merely a sense of this word) but also and

a cosrnology, the science 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 the list of contraries as recorded it is worth while mentioning that these "EVEN," are listed in a peculiar way. list two columns: on the one side the terms rcn"\rt:>,C.OT't

positive and on the other, things of a nature. Thus GOOD is opposed to EVIL, LIGHT to DARKNESS,

• • • •

• •

The Lectum~ and .I acob Klein

MANY, MALE to FEMALE, etc. of the words ODD and EVEN,

the "BAD" things. For n&ptoa6<; .. ,..u .. ,,,, something which is rather

superfluous. But the Pytha-What to be superfluous

than a One. We can

ais1nnc~no:us are EVEN-ODD, express those words means of

+ + +1)

• • • • • • • • • • • • • • • •

• • • • • • • • • • • • • • • •

n , etc.

can obtained by adding a to which is later called the "limiting quantity"

noa6-rn<;). A "gnomon" is a configuration of dots which added to a figure of dots (or lines) produces

The of Number in Greek Mathematics and Philosophy 47

a method for obtaining all those numbers consists in substituting for n and m in the respective formulas the series of beginning with one.)

other distinctions of numbers in Pythagorean prime (or linear) numbers,

"''Jc•""'""'" numbers, etc. We are not going to deal must rather ask, What is the reason for

What is the purpose of this Pythagorean "'"'''"''""v of numbers? I have already it tries to a solu-

problem the unity of any number. of the an &tooc;, a Form of

EVEN-ODD, Square, them something which the unity of any number po&<>i-

six things can conceived as Form "triangle;· which

six things to be one. All numbers u"'''v"'l'. to that Form as all trees

The different "'"''~'"'"~·' what we call the

numeration, universe as a whole is

Every visible thing '·'"'·•v••J;;:.·> things and therefore to a certain

sense the "nature" of every visible Form of numbers .

lLuun:"''"' can expanded further, if possible relations t>P1ruu•Pn

the numbers Thus we can and things are

nor audible but are conveyed to us by means of audible to ratios, proportions, and properties. science of ratios (and

Logistic (from A.6yoc;). It is the basis '-'<u.~u.lG•c•v"'"• since calculating things is nothing but

numbers of things in relation to

tTr.raon Arithmetic (and Logistic), especially the figures of numbers, are probably the origin of the whole system of Greek mathematics in its later "geometrical" form. It seems

Lectures and Klein The of Number in Mathematics and Philosophy 49

a doctrine relations between pure, indivisible units hav­existence 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, in­divisible 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 prac­tical art, the subject of which is sensible things. This remains true within the entire Platonic, Nco-Platonic, and Nco­Pythagorean tradition. Their term Logistic becomes ambiguous, meaning either the pure doctrine of ratios and harmonics or- to a much extent the practical art of computation.

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 unify­ing Pythagorean "Forms" are partly alien to the numbers themselves. The "Forms" don't explain the real differences be­tween 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 in 51

nine this kind, first "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 apt8~-t6<; is a "number of things" not a number of things.) ONE is beyond

beyond any structure at all, beyond ouoiac:;: Republic 509b) it is the Idea of

numbers have unity they are im-Numbers." In that sense Aristotle is perfectly

contention (Metaph. A6, 987 b 10-13) that Plato only a2"10r€~an term J!tl-tflot<; into ~-t€8E~t<;. "ideal

are analogous to the "root-numbers" of the the Pythagoreans did with to tries to do with respect to the "true;'

The arithmological structure of the ideas allows also a solu­tion of the Platonic problem of "participation:' The real prob­lem of participation is the problem of the community among

Lectures and Klein

is a very radical is any unity in a number of A

,,.;;:.uuu;;;, many things and one at units a number doesn't mean the only possible a is subjected to is "apple." Platonic position with rt>cn.nL~r

it seems as if Aristotle didn't see the still awaits a solution.

41 Modern Rationalisn1

L adies and Gentlemen: I am sorry that I must ture of speaking without any but I L-'''"''"'" very welL However, this has one advantage, that

me speak without any ambiguity: a who <U!Jlul<:u with the language is, as a matter of human nature,

inclined to eloquent and therefore often I am a difficult and diffuse subject, the utmost

and precision is necessary in words and expressions. of all, I must state my premises so that we will find a common

I am to speak about the relationship between Ra-and Capitalism. I am not an economist, so I

the from the point of view of economics. my field is the History of Science and what people generally call Philosophy, naturally I approach the subject from this angle. But I should say here that the delimitation of different sci4Em<~es. however necessary it seems to be, is somewhat dangerous. We are to filing the truth in a number drawers, such as Religion, Politics, Economics, Science, Art.

subdivisions of the truth are apt to depend not on the

at which this lecture was delivered are not known, at some time in HJ38-l940 as a guest speaker in a class on Rationalism

and