The Pythagorean Number Theory

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13 standing ol nature and the world beyond. But the technical following passage of the scholium to the Charmides, wherein interest of Pythagoras in mathematics was probably conditioned by his intuitive realization that number is the essence of things. In the absence of any relevant records, we can only the object of the former is explained: Logistic is the science dealing with numbered objects, not nue bers; it does not consider number in its essence, but it presupposes surmise the origin of this remarkable conception, which scems I as the unit, and the numbered object as number; that is, > regards 3 as u triad, 10 as a decad, and applics the SS to have inspired Plato in shaping his own philosophy, arithmetic to particular cases, Thus, logistic lunvestigaces. what Archimedes called the catthe-problem and also “melite” and plis. The partial sofutions of fonian thinkers concerning the nature of the primordial substance must have urged the Pythagoreans to find a more fundamental cause of things, discover a principle underlying the four elements, and give a lie” numbers, the lauer relating to bowls, the former to flocks, In other aspects, too, it investigates the numbers of material a: deeper explanation ol the Milesian systems. Furthermore, since nature treating them as absolute, lis subject-matter is everything that is is only a part of human experience, a more universal essence numbered? was needed to explain nature, reason, and religion as well, If the empirical aspects ol knowledge could somehow be satisfied with the operations of the natural elements, its rational and mystical aspects required a diflerent principle of explanat ion, Because of its rationality and permanence, mathematics could provide such a principle readily. The universal value of mathemátics suggested by the naturalistic account of knowledge was confirmed by the religious requirements of action. IL the emo- _— A comparative would have shown that analysis mathematical, of Eastern traditions especially numerical, relations were essential in mystical speculations and interpretation ol the world. in the Numbers were also indispensable in many practical fields, such as commerce and everyday social intercourse. At the time of Pythagoras, numbers were not the object of a separate science. They were considered to be almost as material as the ultimate principles of things—earth, air, fire, water-of the Milesians, In fact, they were merely used for practical purposes without being considered as purely rational entities, This distinction is illustrated by Plato when he says that “logistic and arithmetic are wholly concerned with numbers," and by the 2 Republic 5254, ber of stars composing it and the geometrical figure they form. Patient observation shows how to distinguish the various cone stellations in that way. Thus Plato wrote in Timacus that the vision of the day and night and of months and circling years has created the art of number, It has given us not only the notion of time, but also the means of studying the nature ol the universe, from which has emerged philosophy in all its tive Orphic rituals were to silisly the positive mind of the Pythagoreans, some rational basis was needed for their religious implications, Pythagoras would have remembered the Babylonian view that each constellation had two chief characteristics—the muro: —ecc: pt Pythagoras would have observed that the art of bi ra 94 music, so steadily practiced in the Brotherhood, was ruled si rhythm and number. This may have led him to dicon i e fundamental harmonic relations of a vibrating string stretc luc over a resounding board. By means of a movable bridge, he would divide the string into different lengths and produce various high and low totes, Though unable to determine the vibrations on which the separate sounds depended, he could measure the length of the vibrating string which was the mate: rial cause of the sound and determine the ratios corresponding to the various tones. ‘Through number, music was thus connected with astronomy. 3 Scholiun to Plato Charmides 165E. 4 Timucus 474.

The Pythagorean Number Theory With every constellation and every musical note character- Yet, in another passage discussing assimilation and resemized by a number, the study of the heave ns and of sound would suggest by analogy the establishment of a number theory extending to ethics and religion. From trade to liturgy through astronomy and acoustics, man's interests would together by the power of number. Merg be blance, Aristotle attributes to the Pythagoreans the opinion that numbers are altections or relations rather than substances. linked ed into the things perceived by the senses as well as into the higher values of life and destiny, numbers must have appea red to Pythagoras as more universal than any other huma n conception. Stripped of the accidents identifying them with specifi c objects, with the astrological chart of any person, numbe or even rs could be taken as the real constituents—the very nature of the world hus Pythagoras was led to declare that numbe r is the cuendo of things. | According to Aristotle, the Pythaporea ns do not place the objects of mathematics between the ideas and material things as Plato does, they say “that things themselves are aber and that “number is the matter of things as well as the form of their modifications and permanent states. ”® As the principles of mathematics, numbers are “the principles of all existing things.” Therefore, numbers canno t be attributes of something else; they are the substance of all existing things" and also “the causes of the reality of other things ." This follows from their participation in the One, which is a substance and not a predicate of something else. But are numbers transcendent or immanent? Whereas Plato maintained their transcendence, the Pythag oreans held that all things possess number and are numbers. Hence numbers are not separable from things: ay all existin g things ne made up of number s, the whole heaven is number, and even abstractions and immaterial things are numbers. The immanence ol number could not be expressed in stronger terms. 6 Met. 987» 27. 6 Met, 987“ 15, T Met. 985» 25, 8 Met. 9874 18, Y Met. 987b 24. 15 +nzo ‘They seemed to see in numbers many resemblances to the things that exist and come into being, more than in fire, in carth and in water. Such and such à modification of numbers was justice, another was soul and reason, another was opportunity, and similarly almost all other things were expressible numerically, Again, they saw that che attributes and the ratios of the musical scale were expressible in numbers. Since all things seemed in their whole nature to be modelled on numbers, and numbers seemed to be the first things in the whole of nature, they supposed the elements of numbers to be the clements of all things, and the whole heavens to be a musical scale and a number, ‘They collected and fitted into their scheme all the properties of numbers and scales they could show to agree with the attributes, the parts, and the whole arrangement of the heayens.10 Aristotle also asserts that the Pythagoreans say things exist by imitation of numbers, while Plato says they exist by participation. But he does not elaborate on this point, because “what the participation or the imitation of the forms could be they left an open question.”!! This is perhaps a reference to the views of Philolaus, for whom numbers are the paradigms or the substantial patterns of things—a conception apparently current among the younger Pythagoreans, which could have inspired Plato. Though the two views might have coexisted among the early Pythagoreans, Aristotle must have considered the complete identification of numbers and things to be their fundamental doctrine, as he devoted to it most ot his criticism. On the other hand, when Aristotle says the Pythagoreans “supposed real things to be numbers”! and “did not regard number as separable from the objects of sense,"!5 he surely 10 Met. 9856 27. 11 Met. 987b 13. 12 Met. 10904 20, 13 Phys. 2034 6.

means they mate reality, air. The lonian monism excluded a multiplication must have studied numbers as extern al objects, not as mere auxiliaries to ordinary computa tion. This view is emphasized by his reference to Eurytus, a disciple of Philolaus, who expressed the nature of objects by means of pebbles or counters. “Eurytus decided what was the number of an object (for example of a man or a horse) by imitating the figures of living things with pebbles, as some people bring number s into the forms of the triangle and square. "Theophrastus reports the same story, which probably goes back to Archytas ,!9 A more detailed account of the method of Eurytus is given by Alexander: ol causes. "pr QBveSAn=cTe> man, and 360 that which defines plant. Having laid this down, he took 250 counters, some green and some black, others red and counters in the outline of the face, some in that of the hands, some in that of other parts. Thus he completed and the outline of the man he was imagining by a number of counters equal in number to the units which he said defined the man. This doctrine of numbers and things prompted the Pythagoreans to collect and fit into their scheme all the properties of numbers they could discover 10 agree with particular experiences. Another reason for the Pythagorean doctrine that number is immanent may be found in the general charact er of philosophical speculations at the time. The lonians did not inquire the “likeness” of things, but about the “nature” of about things, their ulumate essence. The objects of sense perception were not explained through their participation with water 01 air, but through their ultimate identification with those elements. For example, Anaximenes considered the phenomena of the external world as the various modes of one single ulti- If things imitate numbers, the unanswered question of their ultimate cause might involve a possible multiplicity ol causes. Imitation explains neither essence nor existence, Mere assertion of an analogy between numbers and the objects of experience ollers no account of their nature. ‘The Pythagoreans could scarcely fail to perceive the weakness of the argument for simple imitation, The world of experience, therefore, could not be like numbers; it had to be numbers. Instead of being mere paradigms or archetypes of things, numbers had to be the things themselves, or at least the stuff out of which things are made. By word of mouth, this intuition was transmitted from one generation to another, until Philolaus aunounced Let us assume for example that 250 is the number which dchnes of many other colors; then smearing the wall with plaster and sketching on it 4 man and a plant, he proceeded to fix some of the 17 MeOp—Rrn+tIe — openly that “all things which can be known have number; for it is impossible for a thing to be conceived or known without number.” ? With this fundamental principle, the Pythagoreans developed their views on the nature ol particular things by a more intimate study of numbers. If number is the essence of things, why turn to experience for an explanation? In tracing the cause of things, the a posteriori method was of litue help to men already convinced that they possessed the most important clue to the solution of the mysteries of the world. Lf number is the basis of all knowledge and if its various transformations cause the nature of everything, it should sullice to organize and analyze a variety of numbers in order to understand rationally how the world is built. Hence Pythagoras “attached supreme importance to the study of arithmetic, which he advanced and removed from the region of commercial utility." This testimony is corroborated by Aristotle in his statement that “the Pythagoreans devoted themselves to the study of mathematics and were the first to advance this knowledge." 1Y What could be the stages of such a study? Establishment of 17 This fragment is preserved by Stobaeus; cf. Diels, Vors., 44 B 4, 111, 14 Met. LU92L La, 15 Theophrastus /n Met p Ga 19, 16 Alexander In Met. 827-829. Is Aristoxenus us quoted by Stobacus; cf. Diels, Vors., 58 B 4, UL p. 451. 14 Met. 9850 23.

an adequate number theory requires first the defin ition of number, Following the view of Tháles*Y that number 15 a collection of units, the Pythagorca ns “made number out of One." But they could scarcely regard one as a number, since a measure is not the thing measured and since one beginning of number." Euclid impli 19 ing always to the two dillerent kinds respectively.” This statement contains a relerence to the original conception of the dyad as being not a number but the beginning of the even, just as One is not a number but the starting-point of number— is “the a conception which must be very old, as Plato already speaks es this view in defining of two as even. Nicomachus gives also this other definition a unit as that by which an exist ing thing is calle d one, while without mentioning the dyad: “An even number is that which a number is che multitude made Up of unas. Among later admits of being divided by one and the same operation into as à “limiting quantity”; Chrysippus (third century B.C.) calls nuinber (i.e., into two halves), while an odd number is that Pythagoreans, Thymaridas (four th century ».c.) defines a unit it “multitude one.""2 Among neo-Pythag oreans, Moderatus (ca. 60 A.p.) considers number as “a progr ession of multitude beginning from a unit, and a regre ssion ending in Nicomachus (ca. 100 a...) defines it as “a the greatest and the least parts, greatest in size but least in which cannot be so divided, but is only divisible into two unequal parts." it,’ while As regards the term odd-even, Aristotle says that “the eleflow of Quantity made ments of number are the even and the odd” and that “the one proceeds from both of these for it is both even and odd." up of units."# ‘The assimilati on of units with points will be discussed later. Heath explains this strange view by submitting that the unil, The simplest operations performed with units are duplicabeing the principle of even and odd numbers, cannot itself be ton and its reverse, bipartition or mediation. This immediodd and must therefore be called even-odd.32 According to a ately supplies a broad principle ol classification of numbers better explanation suggested by Archytas and attributed to into those which are divisible into halve s and those which are not As Philolaus says, “number is of two special kinds, odd and even, with a third even-odd arisin two; and there are many forms g from a mixture of the of each kind.” 7 Nicomachus, who represents well enoug h the Pythagorean tradition, gives this ancient defini tion: that which can be divided both into two unequal parts (except Aristotle by ‘Fheon of Smyrna, the unit added to an even number makes an odd number, but when added to an odd number it produces an even number, and therefore must partake of both species.# On the other hand, lamblichus uses this term for even numbers like 6 and 10, which yield an odd num- “An even number js ber after a first bipartition.## This conception points to Plato's wwe equal parts and into distinction among “even times even,” “odd times odd,” “odd times even,” and “even times odd,"® taken up later by Euclid.2° These are examples of the possible classification of the various odd and even numbers referred to by Philolaus. the fundamental dyad which can be divided only into two equal parts) ; but however it is divided, its two parts must be of the same kind without share in the other kind. An odd number is that which, however divided, must in any case fall into two unequal parts belong- 28 Introductio Arithmetica i. 7.4. su Parmenides 1430. 20 CE. lamblichus Introductio Arithm etica, p- 10. 30 Introductio Arithmetica i. 7.3. 21 Met. 985a 20, 22 Met. 10884 7, 24 Elements vii. Defs. 1, 2. 24 lamblichus Introductio Arithmetica, 25 Stobacus Eclogae i. Procm. 8, 20 Introductio Arithmetica i. 7.1. 21 Diels, Vors., 44 B 5, IH, p. 408. 31 Met, Ygba 17. é2 Thomas L. Heath, 4 History of Greek Mathematics 1, pp. 11-12. p.71 ha Expositio Rerum Mathematicarum, p. 22. 34 Introductio Arithmetica, p. 22. 35 Parmenides 143E. 30 Elements vii. Defs. 8-10. (Oxtord, 1921),

The reciprocal operations of duplication and bipartiion lead naturally to the study of numbers with regard to their divisibility in general. The Pythagoreans were not slow in discovering that certain numbers can be divided by no other number than the unit. A fragment of Speusippus based upon the writings of Philolaus®? distinguishes “prime” and "incomposite” numbers and “secondary” or “composite” numbers. Thymaridas calls a prime number “rectilinear,” and Theon of Smyrna, “euthymetric” or “linear.” Theon says further that even numbers are not measured by the unit done, except 2, which is therefore odd-like without being prime, Hence, the neo-Pythagoreans definitely exclude 2 from the prime über; for their predecessors, the dyad, or 2, was not a number at all, but the principle of the even, just as the one was the principle of number. Yet in defining numbers “prime to one another” as those. "measured by a unit alone as common measure,” 3 Euclid accepts 2 as a prime number. So did Aristotle before him, when he speaks of the dyad as “the only prime number among the even numbers," which shows that this change was due to the immediate successors of Pythagoras. There are no reports ol the early use ol prime numbers tor cosmical or ethical considerations. Another early principle of combination of numbers was that of addition, By combining the unit with itself several times in succession, the series of the natural integers is obtained, Still to be considered are the results obtained from combining addition with duplication and multiplication. Such a combination 21 law of the formation of these numbers is described by Theon ol Smyrna in this way: “If we take successive double numbers starting from the unit, add them until a prime and incomposite number is found, and then multiply the sum by the last of the added terms, the resulting number will be perfect.”4 The proof of this relation is given by Euclid,tt who established a general connection between prime and perfect numbers. LE che sum of any number of terms of the series: 1, 2, 22, ... 9n—3 be prime, and said sum be multiplied by the last term, the product will be a perfect number—that is, equal to the sum of all its factors; in modern notation: Sy + 27! is a perfect number. When a number is smaller than the sum of its aliquot parts, it is called an over-perfect number; a deficient number is greater than that sum. The first four perfect numbers are 6, 28, 496, and 8128; for 6 = (1 + 2 + 3) and 28 = (1 + 244+7 ++ 14). The numbers 12 and 20 are over-perfect because the sum of the aliquot parts (14-2-+3 +44 6) ot 12 is greater than 12, and the sum of the aliquot parts (1 4 2 4-4 4-5 + 10) of 20 is greater than 20. The numbers 8 and 14 are deficient; for 8 is greater than the sum of its aliquot parts (1 + 2 + 4) and 14 is greater than the sum (1 + 2 4-7) ot its aliquot parts. No reference to perfect, over-perlect, and deficient numbers is found in the fragments of Philolaus or anywhere before Euclid. We are told by lamblichus*? chat Speusippus had written about perfect numbers in “a neat little book entitled On the Pythagyielded the perfect and friendly numbers, which result from orean Numbers” based on the writings of Philolaus; the book the relations between a given number and its component parts has been lost. with respect to multiplication and addition. By taking together the factors of a given number, including one but exclud- Yet perfect numbers were probably known to Pythagoras if we accept a statement by lamblichust* attributing to him the ing that number, it may happen that their sum is equal to the given number, which is then called a “perfect” number, The 47 Kathleen Freeman, 1, The Ihe Pre-socratic Pre-: MP 7 aa Philosophers 86 Elements vii. Def. 12. 39 Topica 1574 35. “i i (Cambridge, Mass., du Ex positio Kerum Mathematiwarun, P- 45. in 41 Elements ix. 96, The algebraic proof la given by Thomas L, Heath424 p. ll, 1908), ge, (Cambrid Elements tuctid’s of Books Thirteen The (repub. in New York, 1956). 42 Theologumena Aritlimelicae, p. 82. 43 Introductio Arithmetica, p. 35.

23 discovery of friendly numbers. These are pairs of numbers such that each is the sum of all the aliquot parts of the other, as by the lonians sufficiently rational to account for the er 284 and 220: the aliquot parts of 284 (1 + 244 + 71 + 142 together equal 220, and the sum of the aliquot parts of nique of deduction probably bad 10 work out whatever rational explanation je could in order to connect the general propositions OF pu 220 (1 +42 4 4 4 5 4 10 4 11 + 20 + 22 + 44 + 55 4- 110) equals 284. Pythagoras investigated 11 numbers reciprocally equal to the sums ol their aliquot parts, he must also have considered the simpler class of numbers which are equal to the sum of their own aliquot parts. There is a story that Pythagoras defined a friend as “One who is the other J, such as 220 and 284." Discovery of such couples of numbers, which appear to have been known to the Hindus before the sixth century 1.6, wis a problem ol considerable difficulty for the Greeks; because verification of such numbers usually requires the handling of very high digits. General intgrest in the friendly numbers seems 10 result from the belief that a good and geometrical figures. ur A prin properties of numbers proper was not yet established, x yt Fa e with their practical applications. Such un Le ss sé meen be suggested by number, If number is rational and male: we at the same time, then extension, which is also a manifestation of matter, must be number. That is why the Pythagorcans con nected the unit in arithmetic and the point in geometrvy, by defining the unit as a “point without position, and the ae as "a unit having position.” The development ol a ae ret theory should therefore embrace geometry, which hn È in turn account for astronomy and the general study of nature, omen was attached to them. The Pythagoreans considered development of the purely arithmetical properties of numbers as uot only mathematically important, but also as preparation for wider application of iN. Greer MEZ AS PI ACT numbers to the sensible world through their connection with figures or forms. To be sure, figures are the most natural tink between purely numerical relations and presentations of the external world. From the pebbles of Eurytus to the simultaneous consideration of the numbers of stars of a particula r con: stellation and the geometrical figure they trace out in space, many Observations suggest an intimate connection between numbers and figures. H everything is number, chen die figure of everything must be a number. These and similar considera ions may have suggested to Pythagoras an essential relation between numbers and geometry, On the other hand, the practical mensuration of the Egyptians and the mathematical generalizations of Thales, with which Pythagoras was presumably acquainted, could have fed him to investigate the significance of his theories. For the Egyptians had not given any reason for their utilitarian geometry, nor were the material principles of things proposed PHAILCGSDPAY VCR L