To account for the beginning of the axiomatic-deductive method, historians, philosophers and scientists have proposed different hypotheses since the beginning of the method itself. We already recalled the most diffused thesis: the shocking and stressing experience of incommensurability, the ''logical scandal'', gave rise to a real ''crisis of foundations'', and this was the spring for a thorough rebuilding of the whole mathematical method ((van der WAERDEN 1983), (NEUGEBAUER 1957), (BOYER 1968), (WILDER 1968)). Other discrepancies (e.g. rules for the computation of areas) in the old Babylonian mathematics could contribute to the same result, as underlined by van der Waerden(van der WAERDEN 1983). Today this kind of reasoning would be pretty normal: if you find a contradiction or also some little 'bug' in your mathematics, you have first of all to carefully check your axioms and proofs, looking for unjustified steps or incoherent axioms.
This was however everything but obvious in the period between Pythagoras and Plato.The two great streams of formal thinking, mathematics and dialectics, were thoroughly separated till Plato: Pythagoreans never showed any dialectic interest (consider for example the great geometer Theodorus in the 'Theaetetus'), Eleatic philosophers, Sophists and Socrates never contributed to mathematics. In addition, Parmenides, maybe also scholar of Pythagorean teachers, was sharply against Pythagoras' philosophy. The first meeting point between these traditions was Plato and this synthesis was probably the first 'step forward' toward the new Platonic philosophy.
Thus there was no sure ground for the use of Eleatic dialectics in mathematics , there was no explicit 'logic' method beyond the classic 'graphical' procedures, and the 'mathematical' contradiction was most of all a counterevident geometrical construction. There had never been any mathematical 'scandal' at all! Whence the very idea of a 'scandal'? Whence the employment of 'logical' contradiction? Whence the 'logic' rigor?
Other authors stressed Greek peculiarities: they were philosophers and lovers of the beauty (Kline (KLINE 1954)), philosophers and thinkers (Heath (HEATH 1921) ). Social factors, as the slave-based ground of Greek society and economics, or the role of democracy, have been advocated most of all by Marxist historians (Farrington (FARRINGTON 1946), (FARRINGTON 1947), A.N. Kolmogorov in the ''Great Soviet Enciclopedy'', cited in (SZABO 1960), but also Boyer (BOYER 1968)), to explain the love both for the abstract (typical sign of idleness) and for the dialectics (typical democratic tool). In the last years there have been also many contributions about the philosophical background of the 'method question', underlining the Aristotelian, Platonic and Eleatic legacy among its roots. Aristotle's role cannot be denied for his ''syllogism'' and for his idea of ''principles''-based knowledge, Plato's Academia and its idealism were the real cradle for the reform of the mathematics foundations, and to Parmenides the first dialectics and the formal non-contradiction principle can be ascribed. This last heritage in recent years has been suggested by (KNORR 1975) and (SZABO 1960), outlining an interpretation of the new method's birth thoroughly inner to a 'philosophical' debate. We have already quoted Knorr's thesis about the centrality of an autonomous 'coherence' plea, before and also to understand the shocking role of the ''incommensurability'' discovery. Szabo argues about the dialectic origin of the deductive method, even though with probably excessive stress on the 'eleatic' phase of dialectics
...systematic and deductive mathematics was originally a special field of philosophy, more precisely a special field of Eleatic Dialectics. Only with the theoretical foundation of geometry, Greek mathematics became step by step independent from philosophy. ((SZABO 1960)104)
This anti-empirical approach is supported, according to Szabo(SZABO 1960), by two facts: the ''indirect-proof'' procedures, thoroughly alien to any 'empirical' mathematics, and the evolution of the concept of ''evidence'', from a perceptual sense to a theoretical one in the Platonic reform of mathematical reasoning. However, mathematics is not only one of the fields, but the paradigmatic one indeed, of Platonic dialectics, and not only for the special role played in it by the indirect proof. And we have to recall that this kind of procedure was crucial, without any link with mathematics, in the Sophists dialectics too, and was founded on the formal employment of the non-contradiction principle, whose origin in the Eleatic philosophy have been analysed in the first report. Thus, there was also an opposite influence of mathematics on Platonic dialectics, and just then we clearly see a connection between them. The Grundlagenkrisis thesis is rejected also by Lloyd, who advocates a genuine interest in the discovery of ''the 'elements', that is, the fundamental principles which the rest of mathematics presupposes''. ((LLOYD 1987),76), i.e. a philosophic and epistemological interest, at the root of the ''axiomatic-deductive'' method.
Almost all historians have assigned to the ''incommensurability'' discovery the role of the overall explanation for the beginning of the axiomatic-deductive method, and for almost any aspect of the 'Greek miracle' in mathematics as well as in logic and in philosophy. It is sure that this result was at the centre of the process, but its role maybe can be differently regarded, that as a 'causal' one. It is not useless to underline the warning of Knorr against a linear 'causal' connection between the new foundation and the incommensurability/Zenonian crisis:
...to establish that the high standards of rigor characteristic of this evolution were intrinsic to the mathematicians' work ... to counterbalance e prevalent thesis that the impulse toward mathematical rigor was purely a response to the dialecticians' critique of foundations;(1) ...even granted that the existence of incommensurable magnitudes was a counterexample to the naive geometrical theory within which the discovery was made, this in itself does not account for the reorganization of geometry upon revised foundational notions.(3) Many mathematical traditions confronted the computational anomaly posed by such quantities as radical 2... But the Greeks advanced to the assertion that such quantities are irrational. This is a statement of an entirely different theoretical order. Its justification can be based only upon logical deduction, and a theory of such quantities can have but one criterion of validity: logical consistency. ((KNORR 1975),4)
Incommensurability was somehow linked to Zeno's paradoxes, but Aristotle warned against using such paradoxes to prove it. However, the eleatic dialectics was an essential ingredient of the process, as pointed out by Knorr (KNORR 1975) and Szabo (SZABO 1960).
Zeno's paradoxes aimed to fight against pluralistic theses, which claimed the existence of 'many', for the contradictions they showed. We can synthesise his arguments as follows: - anyone of the many is the same of itself and is one. Suppose it has no dimension: a dimensionless thing can not change whatever we join to it, and then does not exist, for something not changing for the addition of anything, is nothing. Now let us suppose, vice versa, it has some dimension and then it can be divided. The division process can continue ad infinitum, without end or determination. Conclusion: if many things exist, they must be both so little to be dimensionless and so big to be not determined. - if there are many, they must be just as many as they are and neither more nor less than they are. Then they should be limited. However, if the things that are, are many, they must be unlimited for there are always other things that can be found in-between them, and again others in-between those, and so on. And so the things that are are unlimited too. - if something is moving, in any instant is in a given place, but what is in a place must be resting. This argument, in the examples of ''Achilles and the tortoise'', can be restated as the problem of the ''infinite divisibility of finite pieces of space or time'', by which a finite magnitude is the sum of infinite non-vanishing magnitudes. In the example of the ''flying arrow'', defining a time interval as union of indivisible instants entails that in each instant the arrow is resting, since otherwise the instant itself should be divisible. Then the motion is impossible.
<Note: Noteworthy the existence of similar themes in chinese culture, in Hui Shih paradoxes ((NEEDHAM 1954-), II,191): -There are times when a flying arrow is neither in motion, nor at rest. -Wheels do not touch the ground. (for the touch point has no size) -If a stick one foot long is cut in half every day, it will still have something left after ten thousand generations.>
About the proof of the ''incommensurability of diagonal and side of the square'', we can follow the reconstruction proposed by Knorr(KNORR 1975), arguing for a 'geometric' version instead of the 'arithmetic' one reported by Aristotle (ARISTOTLE 1952). The fig.9,left is analogous to the one described by Plato in his dialogue Meno(PLATO 1964). Knorr ascribes the earliest proof to the late Pythagorean school, about 430-410 B.C., by using the so-called dot-algebra, i.e. the representation of any length or number by a sequence of 'dots', and by their geometric arrangements (square, rectangle, triangle, gnomon etc.). This way Knorr reconstructs the whole presumed Pythagorean arithmetic and geometry. Among the results, he proves dot-algebraically that ''If in a given Pythagorean triple the largest number is even, then all three terms are even.'' In the figure 9, let 'd' and 'a' be respectively the diagonal and the side of the square, and consider all the possible cases. If 'd' is even, for the above result, also 'a' is even and then we can halve both and repeat the procedure, but it can not be infinite. To avoid the ''infinita regressio'', at the end we have that 'd' must be odd. With dot-algebraic techniques Knorr proves that ''if a number is odd, its square is 1 plus a multiple of 4, and if the number is even, its square is a multiple of 4''. The square of 'd' is however the sum of two squares of 'a', and then the square of 'a' had to be odd and even as well. Hence, 'a' must be both even and odd. In Pythagorean mathematics this is possible only if a=1 dot. Then 'd' had to be an integer greater than 1 and lesser than 2 (dots), and this is an impossible triangle.
A strong support to this reconstruction is that it can be repeated for any non square less then 17, in which some difficulties arise. And this fact accounts for a point in the platonic dialogue Theaetetus, where this young mathematician reports the difficulties encountered for this number by his teacher Theodorus when dealing with the proves of incommensurability for integers greater than 2 (KNORR 1975).
The infinite dichotomy is a classic Zenonian 'topos', and the other employed ideas and techniques were well known inside the V century mathematics: dot algebra and Pythagorean coincidence of number and magnitude, metrical study of geometric figures. Dialectic methods and reduction to absurd arguments were known in philosophical circles and were at least widely discussed as suitable mathematical proof techniques. It is noteworthy another proposed reconstruction of the incommensurability discovery (Christal, cited in (HEATH 1956) III,19) , based on the successive subtractions algorithm, the Euclidean algorithm for the greatest common divisor by successive subtractions.(see fig.9, right). It is probable this algorithm in ancient Greek mathematics played a crucial role, as advocated by Fowler(FOWLER 1987). Here, the side 'a' is subtracted from the diagonal 'd', giving a segment a' = d - a , which in turn is the side of a square with diagonal d' = a - a'. The process can be repeated, giving the side a'' = d' - a' and the diagonal d'' = a' - a'' of a new square, and so on. This process is exactly the Euclidean algorithm to find the greatest common divisor of a and d. Being this process endless, it means that these magnitudes had to be represented by integer numbers without a greatest common divisor, i.e. without any ''common measure''. Also this approach is, despite the algebraic formulas we have employed, thoroughly 'geometric', and it is pretty sure that the earliest Greek geometry was rich enough to allow many of such infinite constructions (not only for the diagonal and side of the square), which could undermine the Pythagorean philosophy.
As already mentioned, Knorr (KNORR 1975) underlines that it is difficult to imagine a 'decisive' influence of incommensurability on the beginning of formalization. The first traces of a foundational problem are in Plato, with his critics to the geometers for their ''lack of rigor''. It was likely linked to the extension of the axiomatization requirements, from the simple 'definitions' introduction (already Pytahgorean) and 'constructions' explicitation, toward the establishment in the Elements of the pair 'axioms/reductio ad absurdum'. This was surely a discussed point, because, according to Proclus, many earlier Elements books rejected such kind of proofs. However, Plato does not connect his critique directly to incommensurability: probably it was known to him at least as problem, but maybe not as a crucial result before his late works. Surely it was one of the most relevant arguments of learning and discussion in his Academia.
To understand the connection of Zenonian and incommensurability questions we can begin recalling the problem we perceived in our analysis of negative judgement paradox. In the first report we sketched the lines of the dynamical link between two fields of concepts: - the reality, the being as real, the being tout court, the Aristotelian ousia, with its spatial metaphors, linked to geometry and the idea of 'resemblance'. It is nevertheless the realm of changing and becoming, on which the knowledge has to introduce some kind of stability, connecting the being as real with the being as true. - the truth, the being as true, is the way of knowledge, it is 'being fixed' and 'fixing the reality'. When the truth was 'mythological', it was genetic and 'ontologically' contradiction-free. Starting from this peculiarity, at the end of the ancient mythological knowledge framework and at the beginning of the new 'syntactic paradigm', the goal of the new philosophy is in founding a new relation between reality and knowledge. The truth becomes the realm of signs, written language and arithmetic, rules and equality. Its non-contradiction turns into a 'formal' one and, lost its religious substance, the knowledge must find a new link with the world.
<Note: The non-contradictory nature of reality is almost a tautology for modern science: so, for example, until Hilbert, the method employed to prove the consistency for mathematical theories was the construction of a ''model'' of the theory. Also in physics, the contraddictions arising from Gedanken-experimenten were felt as weaknesses of the theory. We have to recall, instead, that in the pre-socratic philosophy and in Plato too, reality was the place of the confusion between being and not-being, and the realm of the contradiction. We could say that in modern science the contradiction is a syntactic failure corresponding to a semantic non-existence, while in antiquity it was the reality mark on which the truth, mythological before and philosophical afterwards, had to warrant some steadiness.>
The Pythagorean philosophy tried to find again the ancient connection: the true discourse of knowledge gives the names and the forms to reality. For the Pythagoreans the numerical ''ratio'' was the real logos, i.e. the ''discourse'' which enables to structure and express the reality, by the ''definition'' instead of the ancient ''genesis'', by ''numbers'' instead of ''gods'', by ''mathematics'' instead of ''myth''. In the first report we underlined, remarking the 'geographic' root of the ''definition'', the role played in mathematics by the requirement, which grounds Parmenides' philosophy, of 'fixing' the real by words and signs.
So, Pythagorean philosophy was also an attempt to turn in substantial the adjectival nature of the numbers, which in old civilisations were adjectives (in Greek 1, 2, 3, 4, hundreds, thousands, ten thousands are declined for genders and cases).The establishment of the syntactic paradigm can be recognised in that for Philolaos geometry is ''principle and motherland'' of all sciences, whereas for Architas logistics is superior to all other disciplines, geometry included. For almost two thousand years, the geometric astronomic models required the 'strange' hypothesis of solid crystalline spheres in the sky. Descartes' ''analytic geometry'', maybe more than reduce geometry to algebra, aimed to give reality to the newly created (by Viete and Descartes) algebraic symbolism. The connection between mathematic logic and set theory was maybe founded, less for a logic treatment of mathematics, than to give some meaning to the logic language (steadily required in logic, from the Venn diagrams to the Tarskian extensional semantics). Also 'way''s metaphor from its Greek origin (analysed in Snell ú) to Heidegger's holzweg and unterwegs belongs to the same general 'topological' connection between being and knowledge.
Here, we can understand the role incommensurability did play in the Pythagorical residues of the ancient framework. The ''ratio'' can not be 'expressed' any longer, it becomes not expressible. Out of the Pythagorean philosophy, it would be hardly comprehensible the role played by that discovery. As theoretical result, it had no geometrical 'evidence' so that it could not be 'proved' before the strong establishment of the new axiomatic-deductive method and the full exploitation of the reductio ad absurdum. As a matter of numerical fact, incommensurability was indeed nothing very strange: in the ancient Babylonian mathematics the non-existence of some 'numbers' was common, e.g. in the tables of inverses, 7 was normally skipped for it had no exact inverse, as the incommensurable case was skipped in the Babylonian Pythagorean triples tables. Rarely there was a table with the approximate inverse of 7, as rarely there was an approximation of the square root of 2 (see table 2) (NEUGEBAUER 1945).
The numerical 'fact' of the non-expandibility of the inverse of 7 in a decimal or sexagesimal system was not different from the analogous for the square root of 2. The 'numerical' difference of course is in that the latter holds for any base, whereas the former holds for bases which 7 does not divide. However, this could be hardly appreciated by the Babylonians, but could be surely not appreciated by the Greeks, whose numerical system was not positional at all!
Also the Chinese mathematicians ''seem to have been neither attracted nor perplexed by irrationals, if indeed they appreciated their separate existence.''((NEEDHAM 1954-),III 90). It is hard to realise what really 'was' the 'discovery' of incommensurability. Surely it needed the ''reductio ad absurdum'' applied also to the ''regressio ad infinitum'', and these techniques can be credited to the Pythagoreans (for example these techniques are employed in the VII book of Elements, undoubtedly one of the earliest). But, not all mathematicians before Euclid credited those techniques as sure mathematical tools, and the concept of ''infinite'' was not very well established before Aristotle. Thus, also in the likeliest hypotheses about the first proof, as connected to dot-algebra or to some geometric infinite construction reproducing the 'antiphairesis' technique, that discovery do not seem a 'point', but a 'process' in which the 'theorem' had to be proved together with the establishment of its 'proof methods'. Dot algebra was the core of the Pythagorean philosophy and 'antiphairesis'was connected to a simple measuring approach. Thus, the original sources of the discovery had to disappear together with the strong establishment of the discovery itself: it had to be settled in a brand new environment, grounded on the rigid division between arithmetic and logistic and the end of the Pythagorean dot algebra. Thus, that well known mathematical 'fact' became 'provable' and 'interesting' not for its purely mathematic aspect, but just for the role it played in the new knowledge paradigm coming out from the Pythagorean philosophy crisis: the numbers for the Greeks were becoming the deep architecture of the connection between being and knowledge, and no longer the very substance of the being. The increasing refusal of the ancient numerical duality: 'mysticism' and 'practicality', became in fact deeply embedded in Greek culture, as proved by the steady sharp distinction between logistic and arithmetic . The incommensurability discovery was likely the tip (for it concerned with the most trivial of the reality models, the 'square') of a series of analogous geometrical results, which wrecked the ancient paradigmatic framework using the best weapons of the arising new syntactic paradigm: abstract mathematics and indirect-proof based on dialectics and the non-contradiction principle. And it was credibly the point where these two deeply different streams joined to create the new axiomatic method. So the incommensurability 'issue' has got a 'sense' just in the Pythagorean framework, and it had preserved an important role in the following mathematics as well, for that connection between being and sign, which remains till today maybe the deepest requirement of knowledge.
Zenonian paradoxes reflected the same problem. As pointed out by Cornford:
...we can trace two consequences of Zeno's attack. The first was reflected in the separation of arithmetics and geometry...here also irrational and incommensurable quantities are admitted. ...The second consequence of Zeno's criticism was the distinction between the geometric solid and the sensible body, which the Pythagoreans had confused ((CORNFORD 1939) 60)
In fact, Zeno's paradoxes most of all lived on the opposition between a finite reality and a discrete representation, that seems to allow and require an infinite analysis. The core of the Pythagorean approach was to set a geometric representation as bridge between the being as real of the geometric "sense world" and being as true of the numerical "ontology". The paradoxes crack it in both connections: Pythagorean geometry is neither arithmetic nor real representing. It is so not strange that somebody could think there could be a 'Zenonian' proof of the incommensurability, as guessed in the words of Aristotle:
...suppose for example that a person wishing to prove that the diagonal is incommensurable should attempt to apply the argument of Zeno that motion is impossible, and should reduce the impossibility ...to this; for the false conclusion (of Zeno) is not connected in any way whatever with the original assumption. ((ARISTOTLE 1934) An.Pr. II 17. 65b 16-21)
Any way, also this critique did not deny that mathematic reductio ad absurdum and dialectic paradoxes were become homogeneous procedures . In addition, the connection between incommensurability and Zenonian paradoxes, negated in Aristotle, was likely still in Plato so strong to force him, dealing with the problem of accounting for the ''principle of a line'', to reject the ''points'', regarded as simple 'geometrical fictions', and to resort to the ''indivisible lines'' (as reported by Aristotle, in Metaph., 992 a20): maybe denying reality to some geometrical entities (points, geometric proportions, etc.) was a common way out from Zeno's paradoxes and incommensurability antinomy till Aristotle.
<Note: .The paradoxes were classic toposes , sometimes funny as riddles and sometimes awful as enigmas, of Greek culture: from the 'nobody' ('outis') employed by Ulysses to deceive Polyphemous to Oedipus' enigma. They linked death and joke, knowledge and deception. >
To anticipate a point of the third report, we could confront such ancient connection with the modern relation between logic paradoxes and Goedel incompleteness theorem. In Quinean terminology we could say that Zeno's paradoxes, like Russell's, were ''antinomies'' in way of becoming ''falsidical paradoxes'', by ''a repudiation of part of the conceptual heritage''. Incommensurability instead, as the Goedel theorem, was an ''antinomy'' in the form of a ''veridical paradox''. However this different 'career' was not that simple as it seems today to us. Thus, in Plato the incommensurability was at the end one of the most relevant mathematical results and Zeno's paradoxes a crucial reference for the philosophical debate, whereas in Aristotle the former had become also more relevant, the 'standard' example of 'true' proposition, as entailed in the judgement and not in ontology, whereas the latter was just a piece of sophistry. In his book Physics (VI.1), Aristotle employs the same Zenonian arguments, to reach nevertheless as conclusion not the impossibility of the motion, but the idea instead that lines and times are 'continuous' and infinitely divisible magnitudes, thus paving the road to a brand new notion of infinite.
Infinite in Greek was 'apeiron', in which the prefix a- denies the root peraV, whose meaning is "end, boundary". However, from Anaximander to Plato, apeiron has hardly any 'quantitative' meaning, stressing instead, most of all, the "absence of a clear border" sense. In fact, in Plato, when it has to assume a quantitative meaning, it is almost always associated to plhJoV, which means "multitude". In Aristotle
...the unlimited is really the exact opposite of its usual description, for it is not that beyond which there is nothing, but that of which there is always more beyond. (Phys. III.6 206b34-207a2) ...the existence of something unlimited appears to rest upon the consideration ... that the imagination (nohsei) can always conceive a beyond reaching out of any limit, so that the series of numerals seem to have no limit nor mathematical magnitudes nor the beyond the heavens (exw tou ouranou). (Phys. III.4 203b 17-26)(ARISTOTLE 1929)
This infinite is no longer 'given', 'actual', chaotic and undefined, but it is a 'potential' entity, it is the unlimited possible repetition of a "mental procedure", and this way allows to define two kinds of potentially infinite entities: discrete as number and speech (this latter a continuous entity till Plato), and continuous as time, place and geometrical magnitudes. Both are "quantities", but
A quantity then is a plurality if it can be counted; and a magnitude if can be measured.(Metaph. D3 1020a 10)(ARISTOTLE 1952) ...one thing being continuous with another that those uniting extremes of the two things in virtue of which they touch each other become one and the same thing, and (as the very name indicates) are held together. (Phys. V.3 227a 11-13)(ARISTOTLE 1929) I mean by continuous 'capable of being divided into parts that can in their turn be divided again and so on without limit', and on this definition I say that time is of necessity continuous. (Phys. VI.2 232b 25-27) And if distance is divided, the time is correspondingly divided. And this process may be carried on without limit. ... the continuity of time follows on that of magnitude and also the continuity of magnitude on that of time. (Phys. VI.2 233a)(ARISTOTLE 1929)
The former can give an infinite 'by addition', but there is a limit in the direction of the 'division', for there is a minimum which is the unity, whereas the latter can give an infinite 'by division', for you can not find a magnitude smaller than any given one, but limited 'by addition' for there is nothing as an infinitely great magnitude. This solution avoids Zeno's paradoxes and gives the incommensurability proof the role of the dividing line between an earlier arithmetic-based Pythagorean geometry and a new continuous Euclidean one. Proclus in his Commentary pointed out this connection between incommensurability, infinite and expressibility: "If there were no infinity, all magnitudes would be commensurable, and there would be nothing inexpressible or irrational. ((PROCLUS 1970), 6, 19-21)", underlining as main difference between the Pythagorean and Platonic "Quadrivium" in the finite character of the objects analysed in the former. Geminus, according with Proclus ((PROCLUS 1970), 278), wrote that in geometry "it is an axiom that every continuum is divisible". And, hence it is possible to prove that "not all magnitudes are commensurable with one another", and then that "continuum is also divisible to infinity". In Proclus we find also a sharp expression of the negative character of the "infinite".
The infinite therefore is not the object of knowing imagination, but of imagination that is uncertain about its object, suspends further thinking, and calls infinite all that it abandons, as immeasurable and incomprehensible to thought. Just as sight recognizes darkness by experience of not seeing, so imagination recognizes the infinite by not understanding it. ((PROCLUS 1970), 285)
It is noteworthy that both the mathematical concepts of zero and infinite asked for a full development of a new idea of negative to appear. Considering more closely the parallel 'career' of negation and infinite, we remark that both the ''formal negative'' and the ''potential infinite'' appear in Aristotle, and the same definition of infinite is 'played' on the Aristotelian difference between double negative and affirmative, or on the non-commutativity of negation and copula . In fact, in the above quotation we have the opposition between 'ou meden exo esti'(''nothing is out of that'') and 'aei ti exo esti'(''always something of that is out''). The first reflects the ordinary (Anaximandron) 'actual' meaning of the 'apeiron' as ''having outside nothing ever not of it'' and the second the new (Aristotle) idea of ''having outside always something of it''.
The triumph of the ''actual infinite'' will appear in the modern mathematics in the XIX century (Cantor), together with the full modern formal use of the ''negative'' (Frege). And in the intuitionism the criticism about double negation is strongly linked to the refusal of the actual infinite (Actually only Poincare' and others XIX century mathematicians rejected the actual infinite. Brouwer accepted 'omega', but refused higher infinites and the application to the actual infinite of the third excluded principle). However, also Hilbert underlined the non-realisation of the infinite both in nature and mind (Naturerkennen und Logik, in (HILBERT 1935)), ''since the negation of an existing state (herrschendes Zustandes) is an unsafe abstraction - accomplishable (ausfuhrbar) only by the conscious or unconscious employment of the axiomatic method'', as a sort of 'ideal' entity, as the infinite line in projective geometry or the complex numbers. But its non-contradiction in a formal environment guarantees its mathematical existence. Thus mathematics supplies a signs world coherent and expressive enough to be the ''medium between theory and practice, between thought and observations'', but the realisation of its entities is an experimental business: ''geometry is only a branch of physics''.
The ancient Pythagorean 'proportion' theory, based on the algorithm of successive subtractions to get the greatest common divisor, which can work only for commensurable magnitudes, must be substituted by Eudoxus' 'ratio' theory, which anticipates the Dedekind 'real number' definition. Arithmetics and geometry thus belong to two different realms, so that the numbers do never appear in the Elements: numbers never appear in Euclid in abstract/substantive form, whereas they obviously appear in concrete/adjective form, as in ''a cube has six faces'' . And in Euclid 'diastema' is not really a 'distance', but the 'space between two points'. One figure cannot 'have' the same or double area of another, but it 'is' equal or double of that one. To say that two triangles have the same 'height', Euclid says that their two vertices are on parallel lines. Also the concepts of ''point'' and ''unit'' show the complex evolution of these themes. For the Pythagoreans unit or monad is a foundational concept, defined somehow as ''the boundary between number and parts, because from it, as from a seed and eternal root, ratios increase reciprocally on either side'' or as ''limiting quantity''; then a ''point'' is simply ''a monad having position''. For Plato, as reported by Aristotle, the point is a ''geometric fiction'', the ''beginning of a line''. In Aristotle the unit is ''the indivisible in the quantity, without position'' and the point ''the indivisible in the quantity, having position'', which, by its motion, generates a line. From the presence of a further characterisation for the geometrical concepts (position) Aristotle entails a sort of inferiority of geometry with respect to arithmetics, reflecting his overwhelming role of the formal aspects in the building of science. In the Elements, Euclid defines the point as ''that which has no part'' and the unit ''that by virtue of which each of the things that exist is called one''.(HEATH 1956)
The first major Aristotelian breakthrough, discussed in the first report, was the "syntactic paradigm" establishment and the foundation of the "semantics", as correspondence between a formal and a mental/real world to be defined only for entities of size beyond a given threshold. The second was the development of the concepts of 'continuous' and 'infinite', which had to change radically the ideas of 'number' and 'magnitude'. To give them a closer look, we can recall Klein's words:
The difficulties in the way of an adequate understanding of the greek doctrine of number lie above all...in our own manner of dealing with concepts - in the nature of our own 'intentionality'. ...it remains immensely difficult to leave that medium of ordinary intentionality which corresponds to our mode of thinking ... On the other hand, the ancient mode of thinking and conceiving is not totally strange or closed to us. Rather the relation of our concepts to those of the ancient is oddly 'ruptured' - our approach to an understanding of the world is rooted in the achievements of greek science, but it has broken loose from the presuppositions which determined the greek development. ((KLEIN 1934)117-118)
As Klein pointed out, the concept of 'arithmos' in Greek thinking was that of 'number of things' and had the structure of an adjective. Nevertheless, since the Pythagorean philosophy, it had an increasing role as the connecting link between the being as real and the being as truth, between being and knowledge: this ended the Babylonian usage, simply practical and embedded in an astronomical and religious framework as well, and in which there was no difference between integers and fractions, between exact and approximate computations.
Today we consider the numbers as peculiar mental or real ''entities'', and ''integers'' as particular real numbers, most of all implicitly following the decimal representation of the real numbers. If we leave those ideas and this representation, we can maybe realise that the world of integer numbers and the world of magnitudes (weights, lengths, surfaces, volumes and so on) were quite sharply different. Also the numerical systems preserved traces of this difference: in the earliest Babylonian one, two systems can be recognised, the first ten-based and the second sixty-based.. We can also observe that the first developed most of all for discrete magnitudes and the second for continuous one. Moreover, for example, the Greek abacus had a decimal structure for integers and a duodecimal one for the fractions, currently used to represent continuous magnitudes, which had also often specific numerical signs. English system until today preserves this difference. There are reasons to believe that in antiquity the numbers were common signs used to represent two different kinds of numerical usage: a discrete one, analogical to the finger-counting and then decimal, linked to the increasing signs world, and a continuous one, analogical maybe most of all to the astronomic time representation and then sexagesimal, linked to the being. The earliest Greek definitions revealed this aspect: it was deeply believed that the ''unit'' was not a number, and, according to Iamblichus (quoted in (HEATH 1956), II,279), ''some of the Pythagoreans'' defined the ''unit'' as ''the boundary between number and parts'', i.e. between counting and dividing, and also for Aristotle it is a point ''with position'' and thus preserves its nature of 'seed' of the concept of number as ''number of things'' also in a philosophy which unifies discrete and continuous under the category of ''quantity''. No doubt that the geometry dealt with the being and that, before the Platonic philosophy, we can recognise therein an old practical and constructive structure based on the use of 'tornos', a string with a fixed and a moving point to draw circles, 'stathme', a plumb-line, and 'gnomon', an upright marker of sundial and a carpenter's square for drawing right angles (see (HEATH 1956)I, 371). And, credibly, the 'paradigm' of continuous magnitude as something to be divided was, at the beginning, the time and the geometric entities, most of all 'length'. In fact, the Eudoxian V book of the Elements represents a generic magnitude, an integer too, as a segment.
The state structure organization mirrored this distinction. The ancient division of the Athenian polis was based on the number twelve, connected to theological and astronomical subdivisions. At the end of the VI century in Athens Clistenes reforms such division by grounding it on a decimal base, and at the beginning of V century the pythagoric Iones of Chios and Hippodamus of Miletus spreads this kind of reform throughout the Greek world, together with the beginning of a ruling plan for Athens. The crisis of the pythagoric coincidence between 'signs' and 'being' and the breakdown of the 'flat' semantics is maybe mirrored in the Platonic attempt to restore the duodecimal base for the organization of his polis: he is out of the mythological paradigm, but is also leaving the 'simple' pythagoric version of the syntactic one. Vernant underlined these two ruptures between mathematics and politics: the first, with Parmenides and the ''incommensurability'', destroys the 'ionian' and 'pythagorean' harmony based on the ''equality'', the second, with Architas and Plato, creates a new link, now based on the ''proportionality'' (Vernant). In his ''Politica'', V.1, Aristotle explicitly remarks as the earliest democracies were based on an ''absolute equality'' principle, whereas the following ones were based on a ''proportional equality''. Furtherly Plato's 'Gorgias' (508a) underlines the connection between justice and geometrical equality (isothV gewmetrikh), i.e. 'proportion', under the common headline of the kosmoV, following maybe Architas' ideas. And Plato, in 'Leges' 757, distinguishes between a (worse) simple equality and a (better) proportionality with the personal value. logian ison),
Seidenberg (SEIDENBERG 1962) and van der Waerden (van der WAERDEN 1983) stressed the 'ritual origin' of geometry, underlining for example the connection between the ''duplication of the cube problem'' and the altar constructions. And temples and altars were 'models' of the world: the secularisation process of the ancient functions did not change the connection of geometry with reality. The integers were instead linked to the counting, and hence to a social function of 'signing', so that in the 'palace' administration also continuous magnitudes were represented by numbers. Seidenberg pointed out the 'ritual origin' of counting in rites and myths (SEIDENBERG 1963). There is no need of accepting in toto the Seidenberg-van der Waerden theory of a 'single ritual origin' of mathematics to recognise this ancient distinction between numbers and magnitudes, between arithmetic and geometry, between signs and being, well-documented also for the prehistoric times and sharply traceable in the historic mathematics too. real and the being as truth, between being and knowledge: this ended the Babylonian usage