'Mathema' means ''study subject'' and in the Pythagorean-Platonic tradition the term ''mathematics'' comprehended geometry, arithmetic, music and astronomy, the so called ''quadrivium'', that in the Platonic Academia was not an introductory discipline, but a sort of 'final training' for the aristocratic Guardians described in the Platonic Respublica. They appear organised as two pairs: geometry/astronomy and arithmetic/music, each pair including both the abstract and the concrete aspects, and describing respectively the being and signs worlds.
In the Greek mathematics it is noteworthy the distinction between arithmetic and logistics.
The former is a ''number theory'' and concerns with the numbers 'in themselves', by distinguishing even and odd, linear, plane and solid numbers, with a likely evident genetic reference to their ancient representation as pebbles on a surface, according to the ancient Pythagorean tradition. Following definitions are those of prime, square, cubic, oblong numbers. Proofs are geometric too, representing the numbers as figures, squares, rectangles, triangles, gnomons, and so on, according to the so called ''dot-algebra'' ((KNORR 1975)), and are the bases of that ''geometric algebra'' which we are going to analyse in the following sections.
The latter copes with the practical aspects in measurement and trade, and deals with ''specific numbers of perceived and counted things'', using also special number-names depending on the numbered things: as, for example, 'phialites' ''number of bowls'', from 'phiale', ''bowl''. From this point of view, it is worthwhile to remind the adjectival nature of the Greek numbers (in Greek 1, 2, 3, 4, hundreds, thousands, ten thousands are inflected for genders and cases). Logistics comprehends elementary arithmetic operations and simple equations solution too.
However, in the earliest Greek arithmetic, as well as in logistics, the number is always a ''number of things'', and the word 'number' (arithmos) is used only for integers. Thus we can recognise in this Pythagorean limitation of the idea of number, despite of the practical restrictions it entails, a crucial feature of the cultural role of mathematics and its theoretical development direction ((BOYER 1968)). And this cultural change, with respect to the Babylonian and Egyptian roots, fits into a brand new intellectual social class, thoroughly different from both the oriental Scribes and the traditional indo-european priesthood.
This aspect must be taken into account in tracing the differences with the Babylonian mathematics. Many authors have pointed out the existence of two different styles in Greek mathematics:
- the first rooted in earlier oriental systems, more practical and popular, based on problems and numerical solutions, using fractions, without distinctions between continuous and discrete magnitudes,
- the second, product of the Greek civilisation, theoretical and logic, based on postulates and proofs, using only pure integers, with sharp cut between arithmetic and geometry. (see (NEUGEBAUER 1957), (WILDER 1968), (van der WAERDEN 1983)).
However, it is sure that Greek mathematics begins by 'importing' the former tradition, and it is everything but obvious the process leading from the former to the latter style. Practical computations, which followed those earlier traditions, were performed by the abacus, with moveable or fixed pebbles in the different columns. A classic abacus were split in two fields: the first, for the integers, represented in any column a multiple of ten. It had in the upper part only one pebble (representing 5) and in the lower one four pebbles. In the second field there was just one difference: five instead of four pebbles in the lower part, for the duodecimal, instead of decimal, character of the practically used Greek measure-unity submultiples (see (HEATH 1921) I,21). We shall return on this point in the conclusions.
Concerning the Greek numerical signs-system, it apparently seems not really interesting, if contrasted with the previous Babylonian system and the following indo-arab notation.(THOMAS 1993) In Greek mathematics we find two different, both decimal and linked to the writing, numerical systems. The first, ''Herodianic'', can be found in inscriptions and laws since the V century BC. It looks like the roman one, and the basic numbers are represented by the first letter of the denoting word: so P is five, D is ten, and so on. Other numbers are represented by the basic ones by juxtaposition. The same principle rules the representation of money or weight units, accomplished by the initial letter of the denoting word: so T is the 'talanton', and so on. The numerical representation is so scarcely abstract to be sometimes joined with the measure representation to denote a multiple of such measure units by a 'mixed' symbol (for example, a mixture of D and T to denote 'ten talents'). The second system is explicitly referred to the Greek alphabet extended with three old Phoenician letters to reach a total of 27 signs and thus represent the numbers from 1 to 10, the tenths from 20 to 100 and the hundreds from 200 to 900. Other multiples and fractions can be created by apices. For more details, see (van der WAERDEN 1963), (HEATH 1921), (CAJORI 1928).
Historians judged the two Greek numerical systems inferior to the Babylonian one, and probably also to the Chinese, and the second inferior to the first, because it needed more primitive signs and there were more primitive operations to memorise ((HEATH 1921) ). It should be a very odd example of 'involution' in the history of science, terribly odd, if we realise its happening in the heart of the 'Greek miracle'! However we have to remind that, in the abacus computations, the number-signs have just a mnemonic function, to memorise one operand in the simplest operations, or to fill numerical tables, or to 'store' partial results for more complex operations, or to 'print' the final results. Hence, there were no primitive operations at all to remember, because their results were not in a table learned 'by heart', as today, but simply 'shown' by the abacus. The users had to remember nothing but the primitive numerical signs, and the alphabet in the Greek society was the most natural and well known ordered set of signs.
However, the connection between abacus configuration and written number was likely more natural in the Herodianic system, where in addition simple sums were computable by simple sign juxtaposition, without abacus.
Thus, in both the Greek systems, the 'sign' is finally rid of any 'analogic-semantic' connection (finger, hand, etc.), as it was normal in Babylonian and Egyptian systems, and that in the second one it loses also any 'linguistic' connection, taking the alphabet just as the most natural 'linear order'.
In general, beyond the ease or unease of its practical numerical usage, the Greek alphabetic system shows a brand new intertwining with mathematics which will be the starting point for the beginning of more formal terminologies. So, for example: Euclid employes letters to represent geometrical entities, Aristotle employs letters to represent terms in syllogisms, and Diophantus employs letters to represent not only numbers, but also the first algebraic notations (variables, equality).
The earliest Greek philosophers were brand new intellectual figures, and there was been nothing similar before in the ancient world, neither in the eastern nor in the Indo-european tradition: in them previously thoroughly different aspects were joined. They were poets, seers, politicians, businessmen, priests. They travelled through the sea, learning in foreign countries, and then introduced new knowledge in their countries, Italian or Asiatic Greek colonies, founding mysteric sects and gathering disciples. They were also mathematicians, who learnt arithmetic and geometry from Egyptian or Babylonian priests. However, they secularised that knowledge, turning it with Pythagoras in a 'liberal' discipline: opposite to the religious and arithmetic character of Babylonian astrology, the Greek astronomy was laic and geometrical, so that it could not have for them the same value it had for the eastern priesthood, and Vernant underlined the connection between these aspects and the development of the poliV ((VERNANT 1971)). About their deeds and words, it is difficult to distinguish between reality and legend. So, we cannot know exactly what really was Pythagoras' mathematics like. We are quite sure that the greatest discoveries bringing his names were not his own: ''Pythagoras' theorem'' was already known in Babylonian mathematics (see (NEUGEBAUER 1957)) and the ''incommensurability of diagonal and side of the square'' was likely a later result of his school (see (KNORR 1975)). For the former theorem the legend refers that Pythagoras sacrificed an ox to the Gods, and many authors underlined the falsity of this story, reminding the Pythagorean repulse for the sacrifices, most of all of cattle. For the latter another legend refers that the discovery gave rise to a great upset in the sect, and that a member, guilty for the diffusion of the secret, died in a ship-wreck. This story too is credibly false. However, we can recognise in these stories traditional aspects of religious sects: the secret, the sacrifice, the deadly enigma. Nevertheless, the religious and administrative environment, in which the oriental scribes grew the ancient mathematics, is completely over. The secularisation of the Greek culture, while absorbing the oriental culture, looks for a new own foundation, and the vanishing of the ancient mythological framework let philosophy, and hence mathematics too, assume the ontological role of the old myths. The link between reality and truth is no longer stemming from the link between everyday life and narrated stories, but in the connection between social and natural reality and arithmetic. In the Pythagorean teaching ''all is number'' we can only be sure of a crucial ontological role of the integers, in a ''perceptually immediate and figure-like nature'' (Stenzel, quoted in (KLEIN 1934), 62), not separable from the things as 'building' elements of reality. Here, there is something very ancient: music, the constellations in the sky (see (HEATH 1921) ), and the connections between numbers and their geometric patterns in the abacus. Maybe it is antihistorical to ascribe a true 'numerical atomism' to Pythagoras, as Aristotle did. But we can say that the explicit cognitive definition of numbers as 'substance', as something underlying and accounting for the reality, surely shifts mathematics and philosophy in a new theoretical perspective, the same opened for the cosmogony by the ionic philosophers. This way we recognise the beginning of the arithmetic as a distinguished discipline from the traditional logistic, and thus based on substantial spatial properties of the numbers: odd and even are defined with respect to the splitting of the number and from this point of view 1 can be either odd or even, and the Greek mathematics raises those 'geometric' features sketched at the beginning of this section, inherited from Babylonian mathematics and based most of all with the abacus usage, to a theoretical discipline concerning the being of nature and society. As we pointed out in the first report and recalled at the beginning of this, the spatiality is the 'mark' of being, and the sign the new 'mark' of knowledge/truth, almost unknown in the mythological Greek thinking, and crucial in the Greek classical philosophy. It was the new form of the link between reality and truth, that in the mythological framework was grounded on the narrated cosmogony preserved in the memory with Muses' help.
So, Pythagoras' teaching concerns with the first attempt to articulate the being as real and the signs world, to found the new form of the being as truth . In the reality there are no signs, for signs are repeated individuals and in the reality there are no repeated individuals. In the signs there is no reality, for different sign instances can be the 'same' sign and in the reality there is no sameness or equality between two different things, but only resemblance. In the ancient cultures numbers were adjectives and nothing more than pieces of work-memory for administrative purposes (see Seidenberg(SEIDENBERG 1963)), and had often ritual meanings. Music and astronomy showed however how far numbers could be also the substance of reality. Numerical signs can fix the reality and find there their meanings as 'entities'. Numbers in this brand new perspective can not be confused with measurements and computations, which are real, and hence in the realm of becoming and variability.
<Note: This connection between ''being'' and ''truth'' can be easily shown both for mythologic and modern scientific thought by examples:'Water is a chemical substance whose formula is H2O' and 'Laurel is the plant in which Daphne was transformed by Apollo'. This kind of 'is' is not in the list in the 2. section of the first report, for it is not reducible to a purely syntactic 'be', inner to the ''language'' world. It denotes instead an ontologic link between reality and truth, i.e. between two different levels of the language, between two different 'worlds'. Its nature is also a classic topic in the modern mind/body problem, as we will see in the fourth report. Maybe its first appearance is in the Aristotelean 'on os alethes', more then in his ''substantial being'', for this 'being' belongs more to judgement activity then to the ontology expressed in the substance. We must also remind that of this kind is also a 'not causal' truth as the incommensurability of diagonal and side of the square. >
The core of our analysis of Greek mathematics will be in Euclid's Elements, but we will refer to this book 'rhapsodically', without an ordered analysis, taking the different arguments when needed. As introduction we remind that the Elements comprehend 13 books, each one with a set of specific definitions. At the beginning there are also 5 ''postulates'' (aitemata) (Here and in the following the adopted edition of the Elements is Heath's(HEATH 1956).): Post. 1 ''to draw a straight line from any point to any point'' Post. 2 ''to produce a finite straight line continuously in a straight line'' Post. 3 ''to describe a circle with any centre and distance'' Post. 4 ''That all right angles are equal to one another'' Post. 5. ''That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles''
Proclus, in his Commentary ((PROCLUS 1970),183-184), rejects the last two postulates, because he supposes they can be proved. This confidence derives from the fact that their converses can be proved. Here maybe we can find a late appearance of the ancient coincidence between predicative copula and identity, and hence supposedly between implication and double implication.
Postulates are followed by a set of ''common notions'' ('koinai ennoiai') whose number differs in different manuscripts. The most accepted ones are the following: C.N.1 ''Things which are equal to the same thing are also equal to one another'' C.N.2 ''If equals be added to equals, the wholes are equal'' C.N.3 ''If equals be subtracted from equals, the remainders are equal'' C.N.4 ''Things which coincide with one another are equal to one another'' C.N.5 ''The whole is greater than the parts'' Other ''common notions'', not accepted as original by ancient and modern scholars, concern with subcases of the ''equality'' common notions: c.n.6 if equals be added to unequals, the wholes are unequal, c.n.7 things which are double of the same thing are equal to one another, c.n.8 things which are halves of the same thing are equal to one another, or with the often implicitly used axiom c.n.9 two straight lines do not enclose (or contain) a space. For more details, see Heath's edition of the Elements.(HEATH 1956)
The book of Euclid is the most important of a series of analogous books of Elements, i.e. books based on a, initially roughly and progressively more and more refined, axiomatic-deductive method. So, Euclid's books contain very different parts, belonging to different periods and development degrees of Greek arithmetic and geometry. Maybe the earliest books of 'Elements' appeared at the end of the V century BC and contained first rough axiomatic-deductive approaches to the reconstruction of the ancient mathematics. Some parts of Euclid's Elements are very old: Pythagorean or also Babylonian (books I, II and VII)(HEATH 1956). Other parts can be ascribed to other mathematicians, as Theaetetus or Eudoxus. Knorr (KNORR 1975) recognised two different approaches: the older, 'topologic', ascribed to ionic mathematicians and Hippocrates of Chios, includes I, III and parts of the VI book, the more recent, 'metric', comprehends II, IX,X, XIII and parts of the VI book, due to Theodorus, Theaetetus and Eudoxus. For Neuenschwander (NEUENSCHWANDER 1973) the first four books, but for the Euclidean axiomatization, were already known to the Pythagoreans.