A Notion of μηδέv in the Philosophy of Aristotle
The origin and development of mathematical symbols was closely connected with the development of mathematics itself and development of philosophy. It resulted from the fact that philosophy provided the motivation for investigations and creation of adequate and good mathematical symbols. Moreover, being one of the cultural factors, (1) it played a significant role in the process of accepting or rejecting certain notions.
This article aims at producing evidence that particular ideas of Hellenic philosophy made it impossible for Hellenic thinkers to accept notion of a zero. The following considerations will be preceded by brief information on the ancient notations.
The ancient numeric systems aimed at ascribing to a singular whole number or written symbol (up to a point determined by practical needs). This symbol was a combination of a limited number of signs, produced on the basis of more or less regular laws. (2) Three ancient groups of people: the Babylonians, the Chinese and the Mayas discovered a position principle, that is one of the prerequisites leading to discovering a zero and considering it a number. (3) The first appeared in the Babylonian numeration in the 3rd century BC as a result of overcoming ambiguity in the notation of numbers. The sign for a zero that is the so-called diagonally drafted double nail ( ) indicated, first of all, a lack of units of some "sixty" order. It was also treated as kind of an arithmetic operator, since adding it at the end meant multiplication by "sixty". But neither the Babilonian mathematicians nor astronomers treated zero as a number. A diagonally drafted double nail was conceived of as an empty place, that is a lack of unites of a respective order.
Hellenes people used two systems of denoting numbers. The Athenian system was mathematically equal to the Roman system, whereas the Ionic system, just like the Hebrew system, was a system of an alphabetic type. In both systems, just like in the Egyptian hieroglyphic system or the Hebrew numeration, numbers had their established values regardless of the place they were put in. (4) None of the Hellenic system was based on a position principle, none of them used a symbol of zero, either. However, there appeared an empty space in the Hellenic way of calculating, just as in the case of the Babylonians. The Pythagoreans constructed an instrument for calculating, known as Pythagorean table or abacus. It was a kind of tables made with pebbles. Vertical lines of abacus were to separate respective decimal orders. A lack of pebbles in a given column meant a lack of units in a given order. The construction of an abacus could have given the Hellenic scientists an idea of the position notation of numbers and, as a consequence of it, resulted in adopting a symbol of zero. But it did not happen in this way. Hellenes, who were overcome with their affection towards solving antinomies could not accept the idea of introducing "something" to indicate "nothing". Already the very thought of this contained some sort of contradiction. The construction of an abacus allowed them to avoid introducing a symbol of an empty space and therefore it was a guarantee for them to avoid antinomy. This fact, I believe, remains closely connected to the principal theses of hellenic philosophy.
A fundamental thesis of Parmenides' philosophy declared, that being is and it cannot not be, non-being is not and it cannot be. A little later Aristotle wrote: "to say of what is that it is not, or what is not it is, is false...". (5) The greatest Hellenic philosophers, Socrates, Plato and Aristotle were convinced that one could reach an absolute truth only by means of general notions, whereas contemporary mathematicians, Teodor, Teaitet and Eudoksos, believed that they proved "truths" or "true theorems". This conviction was of a metaphysical character. Its justification or attributing to it a sense which would not make it a tautology, was not possible on the basis of mathematics. According to his metaphysics, Plato presented mathematics as a way leading to "truth in itself", whereas mathematical objects that is numbers, values, figures,...., as having their independent being in the world of ideas. (6) Therefore mathematics was an introduction to philosophy.
A question of mathematics was presented in a different way by Aristotle, Plato's pupil. According to fundamental statements of his metaphysics, he rejected ontic independence of mathematical objects, while stressing that a definition (naturally meaning nominal definitions since the Hellenic mathematicians did not explain "primary notions") did not entail the existence of a defined thing which required evidence or a postulate. The second degree of the Aristotelian abstraction, that is, mathematical abstraction, concerned the foundations of mathematics. It consists in the fact that in a material being we pay attention to one property only, which is very significant or just essential - quantity. All other properties, particular as well as general ones, are left out. In abstraction of this kind, intellect creates a mathematical being by isolating quantity from the nature of a given substance. Since in Aristotle's philosophy quantity is quality which organises the matter derivatively, arranging its parts outside each other, a notion of quantity includes the relation of these parts to each other and to the whole. A set of these relations is connected inseparably with a basic function of quantity, that is, with organising the matter. Thanks to a quantitative (along with relations) organisation, the matter becomes readable for the intellect. That is why a mathematical being, superstructured on the expression of quantity, isolated in mathematical abstraction, is an instrument of a deeper insight into structure of the matter.
The term of _ρι'μός (7) has two meanings in Aristotle's approach. The Philosopher defines _ρι'μός in a philosophical sense also with other terms: μovαδικός, (8) μα'ηματικός, (9) or _ρι'μητικός. (10) In the philosophical approach a number is a "plurality of units" (πλ_'oς μovάδωv), (11) a "synthesis of unites" (σύv'εσις μovάδωv), (12) a "discrete quantity" (πόσov διωρισμέvov). (13) Aristotle differentiated between an abstract number and a concrete number (τις _ρι'μός) included in particular groups of things: "Number, we must note, is used in two senses - both of what is counted or the countable and also of that with which we count" (_στιv _τερov _ _ρι'μo_μεv κα_ τ_ _ρι'μoύμεvov) (14) In Physics, book IV, Aristotle considered a possibility of adopting a zero number. "Now there is a ratio in which the void is exceeded by body, as there is no ratio of 0 to a number (τ_ μηδέv πρ_ς _ρι'μόv). For if 4 exceeds 3 by 1, and 2 by more than 1, and 1 by still more than it exceeds 2, still there is no ratio by which it exceeds 0; for that which exceeds must be divisible into the excess + that which is exceeded, so that 4 will be what it exceeds 4 + 0." (15) There is no doubt that the name of "zero" appearing in the above quotation from Physics means "a number zero" and is evidently connected with a notion of "nothing" or "nothingness". It is also indicated by the term of μηδέv used by Aristotle. This term does not belong to a dictionary of basic terms of Aristotle's system for obvious reasons. The Philosopher employed it when he analysed views of his predecessors while writing about beings coming to be from nothingness (16) and according to the ex nihilo nihil fit principle he rejected non-being conceived of as nothingness. Coming to be is a change "from non-subject to subject, the relation being that of contradiction (κατα _vτίφασιv) is "coming to be" - "unqualified coming to be" when the change takes place in an unqualified way, "particular coming to be" when the change is a change in a particular character: for instance, a change from not-white to white is coming to be of the particular thing, white, while change from unqualified non-being to being is coming to be in an unqualified way, in respect of which we say that a thing "comes to be" without qualification, not that it "comes to be" some particular thing". (17) Aristotle talked about absolute coming to be (γέvεσις _πλ_) and detailed coming to be (γέvεσίς τις). Coming to be in an absolute sense cannot be approached as coming to be from nothingness. A basic difference between γέvεσίς τις and γέvεσις _πλ_ consists in the fact that in the γέvεσίς τις process, the substance still exists in spite of change, it just acquires new qualities, while the γέvεσις _πλ_ process, there is no substance, there is only the matter as a component of a substance, unable to exist without a form. Bearing this in mind, one can say that a substance originated from the state in which it did not exist, that is from some kind of non-being, however not from nothingness. In Aristotle's system a notion of non-being (o_δέv, μ_ _v) is connected with a notion of being (τ_ _v) and is a negation of it. Hence, "being" has as many senses as "non-being". Analysing senses of "non-being" the Philosopher wrote: "For "non-being" also has many senses, since "being" has; and "not being a man" means not being a certain substance, "not being straight" not being of certain quality; "not being three cubits long" not being of certain quantity. [...] But since "non-being" taken in its various cases has as many senses as there are categories, and besides this the false is said not to be, and so is the potential, it is from this that generation proceeds, man from that which is not man but potentially man, and white from that which is not white but potentially white, and this whether it is some one thing that is generated or many". (18)
The first notion of an abstract zero, that is a number zero, in the history of human thought appeared in Aristotle's philosophy in the 4th c. BC, when the Babilonians elaborated a zero as a lack of units of some order. The Philosopher could not accepted it since it would lead him to contradiction. The basic principles of his metaphysics demanded the rejection of this notion just as they demanded the rejection of a notion of nothingness or actual infinity. The rejection of an abstract zero was demanded by the principles of his logic, demanding the rejection of an empty set.
(1) Another factors, in my opinion less significant, is a position occupied by a originator of symbols in the scientific world.
(2) The most frequent way of doing it was reducing whole numbers to sums of "successive units" b, b2,..., bn, bn+1,... where each number is a multiple of a preceeding one. Usually bn/bn+1 equals b that is a basis of the system. A befitting notation should show a number of bi "units" of each i order.
(3) A knowledge of position numeration made it easy to extend a basis to any number, for example 10.
(4) A notation of a number indicating multiplicity of k · bi "units", where k changes from 1 to (b(i+1)/bi ) - 1, included symbols that depended on k and i at the same time.
(5) Aristotle, Metaphysics, translated by W.D.Ross, 1011 b 26-27.
(6) Plato, The Republic, 510 c - e.
(7) The term of _ρι'μός is discussed by P.Pritchard in: Plato's Philosophy of Mathematics, Academia Verlag, Sankt Augustin, 1995, pp. 23 - 33.
(8) Aristotle, Nicomachean Ethics, 1131 a 30.
(9) Aristotle, Metaphysics, 1080 a 21.
(10) Aristotle, Metaphysics, 1083 b 16.
(11) Aristotle, Metaphysics, 1053 a 30, Physics, 207 b 7.
(12) Aristotle, Metaphysics, 1039 a 12.
(13) Aristotle, Categories, 4 b 22, 31.
(14) Aristotle, Physics, translated by R.P.Hardie and R.K.Gaye, 219 b 5 - 8.
(15) Aristotle, Physics, 215 b 12 - 17. Compare: Metaphysics, 1080 a.
(16) Aristotle, On Generation and Corruption, 316 a, 317 a; On Melissos, 974 a 1 - 8, 974 b 10 - 12, 975 a 21 - 23; On Gorgias, 979 a 35, 979 b 6, 31 - 37.
(17) Aristotle, Physics, 225 a 12 - 17. Compare 187 a.
(18) Aristotle, Metaphysics, 1089 a 16 - 32.