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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.