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3. Geometric algebra.

The term ''geometric algebra'' has been introduced most of all with respect to the II book of Euclid's Elements(HEATH 1956), where the author proves by geometrical techniques many 'algebraic' properties, and by which quadratic equations were there solved. In Science awakening (van der WAERDEN 1963) van der Waerden argues that this book can be ascribed to the Pythagoreans and, in its core reducible to the simple drawing and reasoning about figures, traced back to the Babylonian mathematics.

Before giving some support to such hypothesis, it is useful to remind the geometrical parts of the Elements, which can be credibly assigned to the Pythagorean school: the first book, the 'geometric algebra' achievements, most of all in the II book, many parts of the III and IV books. To Pythagoreans, almost certainly, it had to be ascribed also the oldest part of the theory of proportions, the VII-VIII-IX books, whereas the general theory of proportions, contained in the V book, it is almost surely Eudoxus' work. According to Proclus, Euclid's main contribution was ''putting in irrefutable demonstrable form propositions that had been established by his predecessors.'' ((PROCLUS 1970), 68 9-11) The logic structure of the Elements will be analysed later. Now, we have just to point out the difference between two types of propositions: constructions or problems (how to accomplish the construction of a geometric magnitude) and theorems (proposition concerning geometric entities), as defined with Proclus' words:

 

Again the deductions from the first principles are divided into problems and theorems, the former embracing the generation, division, subtraction or addition of figures, and generally the changes which are brought about in them, the latter exhibiting the essential attributes of each. ((PROCLUS 1970))

The ''problems'' had an explicit 'constructive, creative' character and were not merely 'algorithmic' solutions: this can also be proved considering the 'conclusion phrases' used by Euclid. At the end of a problem he employs the phrase: ''what it was required to do'', and at the end of a theorem the phrase: ''what it was required to prove''. However, at the end of the propositions VII.2 and VII.3, where he gives the Euclidean algorithm for the greatest common divisor, this second conclusion is adopted , even though it is plainly an algorithm. In the Heiberg-Stamatis edition of the Elements (HEIBERG 1969), a note of Heiberg is quoted in which the author argues that Euclid had instead to use the verb eurein, ''found'', for these two propositions were porismata, i.e. propositions dealing ''with something already existing, as a theorem does, but has to find it, and , as a certain operation is therefore necessary it partakes to that extent of the nature of a problem '' ((HEATH 1956).I 13)

''Postulates'' was the name of the elementary constructions , by which to accomplish more complex ones.

It is worthwhile to underline that this do not mean that postulates' role is to found a sort of 'empirical' geometry based on the use of rule and compass. This would be alien to the 'Platonic style' of the 'Elements' and can be falsified by noticing, for example, theorem I.2, where ''to place at a given point a straight line equal to a given straight line'', Euclid describes a very complex procedure, instead of simply using the compass. It is noteworthy the comment of Proclus (227) to this point: he says that to use the compass would mean 'to beg the question', for in such construction we had already to know as to take a line equal to the given straight line. We must remember that often the term 'distance' is employed in Euclid to translate the word 'diastema', which actually means 'interval', without any metric connotation, and that would be alien to the Euclidean geometry. Thus a circle is defined by a well defined 'interval' (and a well defined centre), and not by an abstract 'distance', and in the guessed compass construction the centre of the circle is not part of the interval which lies elsewhere. We can also remark that Proclus considers the centre a 'part' of the 'interval', which maybe is an echo of the platonic thesis of the existence of 'indivisible lines'

In geometric algebra, there are no numbers and all the magnitudes are geometrically represented: numbers as segments, products as rectangles, without distinctions between the geometric entities and their numerical magnitudes. Here, equality means ''equal measure'', and no longer, as in the first part of the I book, ''thorough coincidence''. All the solutions are geometric and there are no symbols for known or unknown quantities and for their relations (see (HEATH 1921) ). In the I book there are some theorems somehow known to Thales and Pythagoras: among them the most important are:

 

I BOOK. Pythagorean theorems

I.5 ''in isosceles triangles the angles at the base are equal to one another, and, if the equal straight lines be produced further, the angles under the base will be equal to one another'' I.15 ''if two straight lines cut one another, they make the vertical angles equal to one another'' I.32 ''in any triangle, if one of the sides be produced, the exterior angle is equal to the two interior and opposite angles, and the three interior angles of the triangle are equal to two right angles'' I.47 ''in right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle'' and the three congruence theorems for triangles (I. 4, 8, 26), of which at least the third had to be known to Thales. Triangles, most of all right-angled, were the earliest core of ancient geometry. Geometric algebra is developed in the I, II and some parts of the VI books.

 

I BOOK. Geometric algebra: Constructions

I.1 ''on a given finite straight line to construct (sunistasqai) an equilateral triangle'' I.2 ''to place at a given point (as an extremity) a straight line equal to a given straight line'' I.3 ''given two unequal straight lines, to cut off from the greater a straight line equal to the less'' I.9 ''to bisect a given rectilinear angle'' I.10 ''to bisect a given finite straight line'' I.11 ''to draw a straight line at right angles to a given straight line from a given point on it'' I.12 ''to a given infinite straight line, from a given point which is not on it, to draw a perpendicular straight line'' I.22 ''out of three straight lines, which are equal to three given straight lines, to construct a triangle: thus it is necessary that two of the straight lines taken together in any manner should be greater than the remaining one'' I.23 ''on a given straight line and at a point on it to construct a rectilinear angle equal to a given rectilinear angle'' I.31 ''through a given point to draw a straight line parallel to a given straight line'' I.42 ''to construct, in a given rectilinear angle, a parallelogram equal to a given triangle'' (fig.4) I.44 ''to a given straight line to apply, in a given rectilinear angle, a parallelogram equal to a given triangle'' (fig.5) I.45 ''to construct, in a given rectilinear angle, a parallelogram equal to a given rectilinear figure'' I.46 ''on a given straight line to describe a square''

Together with the first three postulates, these constructions allow to construct all the basic geometric entities, and, in the same time and with the same methods, to accomplish the first constructions of somehow constrained figures. These kinds of issues are the geometric algebraic approach to the equations solution.

 

I BOOK. Geometric algebra: Theorems

I.36 ''parallelograms which are on equal bases and in the same parallels are equal to one another'' I.37 ''triangles which are on the same base and in the same parallels are equal to one another'' I.38 ''triangles which are on equal bases and in the same parallels are equal to one another'' I.41 ''if a parallelogram have the same base with a triangle and be in the same parallels, the parallelogram is double of the triangle'' I.43 ''in any parallelogram the complements of the parallelograms about the diameter are equal to one another'' (fig.4) These theorems give the basic equalities between areas, which are the geometric algebraic form of the product of numerical magnitudes. Propositions I.42-47 accomplish the area-preserving transformations of figures which are the basic techniques of geometric algebra, connecting ''equality'' and ''identity'' relations: the aim is to transform any rectilinear figure, which always can be decomposed in a set of triangles, in a parallelogram, with a side and angle given. In particular, if the parallelogram is a rectangle, this technique solves the so-called problem of ''application of areas'', which, being the area of a rectangle the product of the lengths of its sides, accomplishes 'geometrically' the division (see (HEATH 1921) I, 347).

The II book shows the full development of the ''geometric algebra'', accomplishing, by geometrical constructions on rectangles, triangles and squares, all the 'algebraic' operations to be used in solving linear and quadratic equations. In the following, we are going to give the theorems of this complete computational theory, giving a semi-algebraic translation of the theorems. The terminology is quite transparent, we have just to recall that the ''gnomon'' is the L-shaped figure obtained by suppressing a rectangle from a greater rectangle.

 

II BOOK. Geometric algebra: theorems and constructions.

II.1 ''If there be two straight lines, and one of them be cut into any number of segments whatever, the rectangle contained by the two straight lines is equal to the rectangles contained by the uncut straight line and each of the segments.''

rect(a,(b+c+d+...))=rect(a,b)+rect(a,c)+rect(a,d)+....

II.2 ''If a straight line be cut at random, the rectangle contained by the whole and both of the segments is equal to the square on the whole''

rect((a+b),a)+rect((a+b),b)=square(a+b)

II.3 ''If a straight line be cut at random, the rectangle contained by the whole and one of the segments is equal to the rectangle contained by the segments and the square on the aforesaid segment''

rect((a+b),a)=square(a)+rect(a,b)

II.4 ''If a straight line be cut at random, the square on the whole is equal to the squares on the segments and twice the rectangle contained by the segments''(fig.3)

square(a+b)=square(a)+square(b)+2rect(a,b)

II.5 ''If a straight line be cut into equal and unequal segments, the rectangle contained by the unequal segments of the whole together with the square on the straight line between the points of section is equal to the square on the half'' (fig.6)

rect(a,b)=gnomon(half(a+b),half(a-b))=square(half(a+b))-square(half(a-b))

a+b=s

II.6 ''If a straight line be bisected and a straight line be added to it in a straight line, the rectangle

contained by the whole with the added straight line and the added straight line together with the square on the half is equal to the square on the straight line made up of the half and the added straight line'' (fig.6)

rect(a,b)=gnomon(half(a+b),half(a-b))=rect((a+b),(a-b))=square(a)-square(b)=square(half(a+b))-square(half(a-b))

where a-b=d=2 beta

a+b=s=2 alfa

II.7 ''If a straight line be cut at random, the square on the whole and that on one of the segments both together are equal to twice the rectangle contained by the whole and the said segment and the square on the remaining segment''

square(a+b)+square(a)=2rect(a,a+b)+square(b)

II.8 ''If a straight line be cut at random, four times the rectangle contained by the whole and one of the segments together with the square on the remaining segment is equal to the square described on the whole and the aforesaid segment as on one straight line''

4rect(a,(a+b))+square(b)=square((a+b)+a)

II.9 ''If a straight line be cut into equal and unequal segments, the squares on the unequal segments of the whole are double of the square on the half and of the square on the straight line between the points of section''

square(a)+square(b)=2(square(half(a+b))+square(half(a-b)))

II.10 ''If a straight line be bisected, and a straight line be added to it in a straight line, the square on the whole with the added straight line and the square on the added straight line both together are double of the square on the half and of the square described on the straight line made up of the half and the added straight line as on one straight line''

square(a+b)+square(b)=2(square(half(a))+square(half(a)+b))

II.11 ''To cut a given straight line so that the rectangle contained by the whole and one of the segments is equal to the square on the remaining segment.'' (fig.7)

II.14 ''To construct a square equal to a given rectilinear figure'' Then, the VI book contains the crucial constructions:

 

VI BOOK. Geometric algebra: constructions.

VI.9 ''From a given straight line to cut off a prescribed part'' VI.10 ''To cut a given uncut straight line similarly to a given cut straight line'' VI.11 ''To two given straight lines to find a third proportional'' VI.12 ''To three given straight lines to find a fourth proportional'' VI.13 ''To two given straight lines to find a mean proportional'' VI.18 ''On a given straight line to describe a rectilinear figure similar and similarly situated to a given rectilinear figure'' VI.25 ''To construct one and the same figure similar to a given rectilinear figure and equal to another given rectilinear figure'' VI.28 ''To a given straight line to apply a parallelogram equal to a given rectilinear figure and deficient by a parallelogrammic figure similar to a given one: thus the given rectilinear figure must not be greater then the parallelogram described on the half of the straight line and similar to the defect'' (fig. 8) VI.29 ''To a given straight line to apply a parallelogram equal to a given rectilinear figure and exceeding by a parallelogrammic figure similar to a given one'' (fig.8) VI.30 ''To cut a given finite straight line in extreme and mean ratio''

The geometric-algebraic solution of quadratic equations can thus be found substantially in propositions II.5, II.6, VI.28 and VI.29. The difference between the II and VI books propositions is, first, in a greater generality of the latter (for example, if the excess or defect is a square, then the diagrams of the VI book are essentially the same as the II book diagrams), and, second, in that the former are theorems instead of constructions and must then be completed to be effective. Probably the II book theory is Pythagorean, whereas the VI book is a later development, maybe Euclidean. In fact, II.5 and II.6 claim that a rectangle with an unknown side having length x and the other having length (respectively) s-x or s+x or x-s is equal (in area) to a gnomon made up from two squares of sides half(s) and (respectively) half(s)-x or x-half(s) or half(s) +x. If the area of the rectangle, G, and the sum or difference of the sides, s, are known, we have a classic quadratic equations in one of the so called ''normal'' forms: x(s-x)=G; x(s+x)=G; x(x-s)=G

However, here the product G is in a ''gnomon'' form, which is not the 'canonical' form for the areas. Simpson (see (HEATH 1956), I 384, 387) accomplished the solving construction for a square of area G. But this construction can not be found in Euclid. The 'real' solving constructions have to be found in the book VI. In the figure 8 left, the theorem says how we can construct, on a given segment a , a parallelogram having the same area of a given figure (lightly coloured rectangle and pentagonal figure), and another parallelogram similar to a given one (heavily coloured parallelograms). This, in the special case in which the given (heavily coloured) parallelogram is a rectangle of sides b and c, and the figure (lightly coloured) has area S, amounts to solve the quadratic equation

ax-(b/c)x^2=S

Observe that the algebraic translation of the theorem is underdetermined, in that (i) we specify the parallelogram as a rectangle, and (ii) we consider only the ''area'', and not also the ''form'', of the generic figure. The theorem of figure 8 right is analogous and solves the problem that, as special case, becomes the quadratic equation:

ax+(b/c)x^2=S

However, the crucial difference between the procedures of II and VI book is more subtle. We are going to show later, dealing with Diophantus, as II book's procedures seem to yield the mnemonic scheme (fig. 6) of the numerical solution of quadratic equations in normal form: i) given s (trace the segment s), compute s/2 (find its midpoint) ii) compute its square (s/2)^2 (trace the square on the half segment) iii) subtract G (delete from the square the gnomon) iv) the square root of this difference (the side of the remaining square) is the number (the segment) to be added or subtracted from s/2 to get the root. We can compare this graphically memorised and proved procedure with the Babylonian example of numerical solution in the following: there is a perfect matching. VI book's procedures are instead thoroughly geometric: perfect for a purely geometric solution, useless for a numerical solution memorisation. They are credibly Euclidean. Anyway these procedures reflect a geometry in a phase in which the memorisation is propositional and not geometric, and geometry becomes an autonomous science based on non-geometric proof techniques. Thus, VI book's procedures are thoroughly geometric solution exactly in that they are no longer the theoretic and mnemonic ground of the general numeric solution, which is instead given by II book's procedures.

Despite of the thoroughly non-syntactic character of the geometric-algebraic procedures, there is no reason to avoid for them the term 'algorithm' in its present meaning. They are easily provable, unambiguously understandable and thoroughly reproducible: the procedure to get the square root of 2 (and it could be generalised to any integer) is perfectly ''reminded'' by the slave in the dialogue Meno. Euclid's employment of segments to denote general 'magnitudes', Descartes' analytical geometry and , last but not least, the modern ''arithmetisation of analysis'' entail that the length of a segment can be considered as a generalisation of an integer, and then that the geometric procedure can be seen, in all the relevant cases, as a procedure defined on integer numbers. We could raise the question: does this geometric procedure violate the Church-Turing thesis ? Obviously not, for the second degree equations can be solved by recursive functions or Turing machines or other 'syntactic' model of computation. Also the general 'geometric' case, that is including the 'form' datum, could be dealt with by analytic geometry techniques. However, we must point out that this algorithm does not share any numerical limitation, e.g. no problem of numerical approximation of irrational solutions. From this point of view no syntactic algorithm is thoroughly equivalent to these and other geometric algebraic procedures. To deal with this problem we could: (i) either rule out the geometrical procedures from our idea of 'effectively computable' procedures, (ii) or reduce the geometrical procedures to syntactical ones, by ''arithmetizing'' the magnitudes, according to the Eudoxus-Dedekind definition of (real) magnitude, and thus implicitly accepting the reduction of the integers to particular ''magnitudes''. To make this point clearer, consider a very simple geometric procedure: <given a segment, to construct the diagonal of the square constructed on this segment>. This is a well defined, always terminating geometric procedure, and not including any 'approximation' aspect. If we want to apply here the Church-Turing thesis, we can: (i) either refuse this geometrical construction as a ''procedure'', (ii) or reduce it to numerical computations by multiplication, sum and square root operations. In general, this procedure do not always terminate, and we can consider the set of results of the infinite terminating approximated procedures as giving the result of the problem, since they yield a set of rational numbers which defines a real number in Dedekind's style. Thus we: (i) either accept that our idea of procedure must be exclusively syntactic, which works only on signs without any connection with the being, (ii) or accept that in this case the connection of the finitistic idea of procedure with the being can only be given by an actually infinite characterisation. Both cases we have that the Greek geometric algebra was a sound and rich mathematical theory, non syntactical in its proof and problem procedures as well, with a connection with the being that cannot be fully reproduced in our modern signs mathematics .

<Nota:To rule out these geometric procedures, we could raise the question of the non-euclidean possible character of the geometry, and then of some not well-defined features of the procedure. However, we must remind that the euclidean or not-euclidean nature of the geometry is entailed by the geometry/physics relationship we choose. Then, the peculiarities of the geometrical procedures are due, once more, only to their link with the 'being'.>

We can ask whether the substance of these results was already known in Babylonian mathematics. The answer is substantially positive: not only the clay tablets show the steps of analogous computations, but also some minor technicalities resemble the Greek ones. In fact, we can recognise the presence of expressions recalling classic Babylonian techniques, as in the phrase ''cut into equal and unequal segments'', where we can recognise the ''sum and difference'' Babylonian method to be applied when it is given the sum 's' or the difference 'd' of two segments 'a' and 'b'. In the first case we have to set

a=half(s)+t, b=half(s)-t

and in the second:

a=t+half(d), b=t-half(d)

One of the main problems of the history of Greek mathematics is the rationale of this 'geometric algebra', as set for example by B.L. van der Waerden :

 

Why did the Greek transform these (Babylonian) algebraic methods into the geometrical form we find in Euclid's Elements?((van der WAERDEN 1983), 88)

Van der Waerden, as Neugebauer (NEUGEBAUER 1957) before him, started from the correct idea that the Greek methods to solve quadratic equations had a Babylonian origin, and in Science Awakening (van der WAERDEN 1963) considered ''geometric algebra'' as a simple translation of Babylonian algebra in geometric language. However, Babylonian texts show only solutions which seem to reflect the steps of the ordinary algebraic calculus. Neither geometry seems to play any special role. From this analysis of Babylonian texts it is difficult to realise the rationale of the geometric embedding of the Greek approach as connected with the Babylonian tradition, and the history of mathematics, since Descartes, has generally followed the opposite direction: to translate geometry in arithmetic/algebra. It is then hardly comprehensible the ''geometric algebra'' just as an ''algebra in geometric clothes'', for we can not find its rationale, and, more important, we can not understand the structure of the old Babylonian algebra, 'bare' without geometric cloths.

The traditional thesis reinforced the habit of 'translating' Greek and Babylonian geometric algebra in its 'correct' modern algebraic form. Against it, Klein (KLEIN 1934) and, recently, Unguru (UNGURU 1975) recoiled arguing for the unity of form and content in mathematics. To this critique van der Waerden (van der WAERDEN 1976) answered criticizing his overvaluation of the symbolism . These reports advocate the thesis the role of symbols in mathematics and in western civilisation as well is so crucial that it is simply impossible to overevaluate it.

Van der Waerden furtherly , to answer this problem, argued that

 

Greeks combined two traditions, which both originated in the Neolithic Age: one tradition of teaching mathematics by means of problems with numerical solutions, and one of geometrical constructions and proofs.((van der WAERDEN 1983), 89)

and Seidenberg claimed

 

There are two great traditions, easily discernible, in the history of mathematics: the geometric or constructive and the algebraic or computational. ((SEIDENBERG 1978), 301)

From this assumption it follows that the change from the numerical to the geometric tradition had to be caused by a 'trauma': the Pythagorean discovery of the ''incommensurability of diagonal and side of the square''. Such discovery has often been the 'deus ex machina' to account for any mathematical aspect of the 'Greek miracle': any change has been explained by the historians as a consequence of this discovery, from the ''geometric algebra'' to the ''deductive method'', to the ''logic'' and to the special role of the integers. Tannery judged that discovery a ''a veritable logical scandal''. But it is scarcely credible a 'logical' scandal as substantial cause of the birth of the 'logic' method! I think that that discovery, which presumably happened at the end of the V century, too late to cause all those changes, is worth a more detailed analysis, where we are going to point out the role it likely played. Now we want to stress that there is no need of it to explain the problem of ''geometric algebra''.

Our idea is that the geometrical framework was the proof theory already for Babylonian scribes, but it was not written, for the apprenticeship style of their learning. From this point of view Euclid's Elements are also a formalised and written version of the ancient Babylonian teaching. The Euclidean work was most of all the mark of a culture in which the intellectual reproduction shifted from the oral to the written medium, as remarked in the words of Plato's Phaedrus quoted in the first report (PLATO 1964) about the role of writing in changing the role of the memory in knowledge and learning. However, at the beginning of the axiomatization problem, the mathematical substance was quite the same. Recently Serres (SERRES 1993) pointed out the origin of geometry at the meeting of a semitic (intuitive, graphic, measuring-based) and an indoeuropean (formalist, linguistic, reasoning-based) tradition. In fact, if I am not mistaken, it is impossible to see any way of proving, showing and remembering, without any kind of formal algebra, the ancient equation-solving procedures, but by geometric tools.

To support this thesis, first of all we have to underline the persistence of a geometric 'flavour' in the Greek idea of proof (see also (GRAY 1979), (CAMBIANO 1967), (UNGURU 1975)): (a) the role of the constructions in Greek geometry is evident in the very beginning of Euclid's Elements: the first three postulates and the first three theorems are constructions. This fact is much more relevant if we remind that Euclid and the whole earlier tradition of Elements geometry books grew in a Platonic, and then 'idealistic', environment. In addition, every theorem in Euclid has got its own figure, even in the books which deal with general magnitudes (V) or integer numbers (VII, VIII, IX). Sometimes there are theorems whose relative figure is a simple segment without any other feature. (b) the usage of words like 'gramma', 'egraphe', 'graphein', all stemming from the root 'grapho', originally meaning ''to scratch, scrape, graze'', to mean ''demonstration, proof, theorem''. Heath in ((HEATH 1921) I, 203, footnote 2) underlines but underevaluates this terminology. Also words like 'idea' or 'theorem' stem from roots linked to the idea of 'seeing', underlining the substantially visual 'perspicuity' of the ancient proof theory. (c) in spite of their repeated statements about the purely exemplifying role of the figures in a proof for the mistakes their drawing can cause, still in Plato and Aristotle there are traces of similar suggestions. For example, the same Aristotle (ARISTOTLE 1952) accepts that only by the real construction there is knowledge:

 

Propositions ('diagrammata') too in mathematics are discovered by an activity (i.e. by actual working); for it is by a process of dividing-up that we discover them. If the division had already been performed, the propositions would have been manifest; as it is they are present only potentially ... Hence it is manifest that relations subsisting potentially are discovered by being brought to actuality: the reason is that the exercise of thought is a (bringing to) actuality. (Metaphysics 1051a 21-31)

Heath observes: ''I feel no doubt that 'ta diagrammata' are geometrical propositions including the proof of the same and not merely 'diagrams' or even 'constructions''', but it adds ''...'dividing-up' is evidently meant in a non-technical and even a literal sense, and there is no reference to the method of mathematical analysis. The dividing-up is affected by inserting additional lines, etc.'' ((HEATH 1949),216). In another point (An Post.77a) Aristotle underlines that ''the geometer bases no conclusion on the particular line which he has drawn being that which he has described, but (he refers to) what is illustrated by the figures''. In Metaphis. 998a the proof appears as functional to a figure, and in Metaphys. 1014a they appear as different, although both grounded on 'elements'.

And in Theaetetus 147d Plato (PLATO 1964) writes:

 

Theodorus was proving for us by diagrams something about the powers

Knorr remarks.

 

So close was this association of theorem and diagram that the two terms might be used as synonymous (72) In short diagrams here are constructions on the one hand, 'geometrical theorems' on the other (74)(KNORR 1975)

The proof techniques were then 'essentially' geometric and the ''effective constructions'' played a crucial, well-defined role, not purely didactic or elementary, or something to be given only in addition to the real proof, but accomplishing instead the 'whole' proof. It is difficult in our algebraic cultural environment to realise what this sort of 'geometric proof' theory and practice was like. Nevertheless, we could imagine also today a sort of geometric-analogical computer (maybe by some sort of laser technology), with some constructive primitives, by which to accomplish both more complex constructions to solve problems and 'visual' demonstrations. It is evident that the open problem would be the risk of 'displaying' properties deriving from the particular drawing and not from the general hypotheses. However, the risk of 'wrong' proofs has not been thoroughly vanished also in our syntactic proofs, as the history of mathematics shows. With respect to the old Babylonian ancestors, Greek geometric algebra had got three main differences: - the written form, originally mirroring the original oral learning, - the sentence form to be memorised together or instead of the old geometric figure, and, most of all, - the axiomatic-deductive framework to substitute visual perspicuity.

To realise the cognitive effectiveness of this pre-syntactic mathematics, both to prove a theorem and to 'remember' a procedure, we can consider fig.6 left: (i) the 'proof' of the theorem is simply the acknowledgement of the 'equality' between the 'vertical stripped' and the 'horizontal stripped' rectangles; the whole 'algebra' is in the decomposition and union of figures. (ii) the resolution of the 'normal form' quadratic equation is straightforward as well. Given the sum s of two numbers and their product G, the difference between the square of side s/2 and the rectangle of area G is the little 'white' square, whose side is the difference between the greatest of the two numbers and s/2. So, this value can be found subtracting G from the square of s/2. Adding and subtracting it from s/2, we get the two numbers. This describes exactly the same 'procedure' we give in our school by the quadratic equations solution formula.

The 'flavour' of the evolution from Babylonian to Diophantine techniques can be grasped, inside the same simple equation solving, facing a real Diophantus problem in which are given of two numbers the sum 20 and the sum of their squares 208 (HEATH 1910), with an analogous Babylonian text. In Diophantus, beyond the alphabetic notation for the numbers, 'sigma' is the 'unknown' number, 'Delta y' is its 'square', 'dynamis', 'iota sigma' means '=', 'mi zero' characterises an 'integer constant', and ' arrow' means '-'. (On the right side we translate the procedure in algebraic and English terms).

The analogous Babylonian problem BM 345658 VsII 10, where the sum is 23, the diagonal (the square root of the sum of squares) 17 is:(NEUGEBAUER 1935),(GOETSCH 1968)

<length and breath summed is 23 and 17 the diagonal. the magnitudes are unknown. 23 times 23 is 8,49. 17 times 17 is 4,49. 4,49 from 8,49 you subtract and it remains 4,0. 4,0 times 2 is 8,0. 8,0 from 8,49 you subtract and it remains 49. What must I take to get 49? 7 times 7 is 49. 7 from 23 you subtract it remains 16. 16 times 0;30 you take it is 8. 8 is the breath. To 7 you add 8 it is 15. 15 is the length.>

or another similar problem, in normal form, AO 6484 Rs 10-14, where the sum is 2;0,0,33,20, and the product is 1:

<divisor and dividend is 2;0,0,33,20. with 0,30 multiply gives 1;0,0,16,40. 1;0,0,16,40 with 1;0,0,16,40 multiply gives 1;0,0,33,20,4,37,46,40. I subtract 1 It remains 0;0,0,33,20,4,37,46,40. What must we multiply with 0;0,0,44,43,20 with 0;0,0,44,43,20 multiply gives 0;0,0,33,20,4,37,46,40. 0;0,0,44,43,20 to 1;0,0,16,40 sum gives 1;0,45 divisor. 0;0,0,44,43,20 from 1;0,0,16,40 subtract gives 0;59,15,33,20 dividend.>

Despite the simplicity of the problem, substantially the 'normal form', there is in the Diophantine version something, which is new with respect to the Babylonian clay tablet versions, beyond the 'syncopated' form of his notation and the different numeric system. Both of them, the Babylonian mathematician and the Greek one, could not have any formula to remember. But the Babylonian looks as the 'computer' was following a figure like fig.7 right or 6 left, and the 'time' of the solution was the 'time' of the construction, while Diophantus seems instead to be managing different pieces, each one corresponding to a 'figure-text' theorem of geometric algebra, and these pieces are in form of ''equalities'', to make up something we could call a 'formal' procedure, by employing the formal properties of ''equality'' as 'glue' to put the whole together. The 'time' of the solution is functional, i.e. to get a result we have recursively to compute before the required data. In the mind of the 'computer' there is no figure at all to follow, but a set of equality relations, 'elements', to use, and a set of formal principles, 'equality axioms', to accomplish the solution. The presence of the variable and the 'substantially' written form allow to follow a no longer linear temporal sequence during the execution. The appearance of the concept of equality in the formal framework will be analysed in the following, but now, to strengthen this interpretation, we can introduce a comparison with Chinese algebra:

 

...it brought into play an abundance of abstract monosyllabic technical ideograms indicating generalised quantities and operations. If these were not yet symbols in the mathematical sense, then they were more than merely words in the ordinary sense. And then in the course of the work the counting-board with its numbers was laid out in such a way that certain positions were occupied by specific kinds of quantities (unknown, powers, etc.). But since the types of equation always retained their connection with concrete problems, no general theory of equations developed. However the tendency to think in terms of patterns finally evolved from the counting-board a positional notation so complete (as far as it went) that it rendered unnecessary most of our fundamental symbols. Unfortunately, though the achievement was magnificent, it led to a position from which no further advance was possible.(III,112). In Chinese mathematics ...the algebraic operations were carried on entirely without the use of the = sign, and the terms were arranged in tabulated columns.(III,113-114). It is a matter for reflection how far Chinese algebra was inhibited...by its failure to produce a sign which would permit the setting up of equations in modern form. Nor did Chinese algebra have any sign for exponents and powers; the value of a quantity, whether x2 or x3, depending entirely on its place in the 'matrix' tables which the Chinese used.((NEEDHAM 1954-),III,115)

We can see that, credibly in absence of the alphabetic revolution, in China the modern algebraic form did not appear, substituted by a 'smooth' transition from the 'abacus' computation to a 'disposition-algorithmic' form as the one we learn in our simplest elementary 'one below the other' numerical operations (sum, subtraction, product, division, root, matricial calculus). Here we find another strong support to the crucial and autonomous role of signs: in absence of the alphabetic writing technology, in China the earlier naive geometry evolved following only the spatial arrangement of the abacus (and of Chinese writing). Thus they discovered and gave a central role (not only for arithmetic operations, but also for the matrix-like solution of equations systems) to those spatial algorithms, that in Europe arrived (only for arithmetic operations) with the Arab and Italian medieval 'algorisths', and are still today taught in our elementary school.

 

The Chinese went a step further in arithmetizing these concepts of geometric algebra on the counting board, where positions played a vital role. In taking this important step, mathematical thinking was switched from the verbalised geometric form to a rod numerical notational form.((LAM-LAY-YONG 1986))

Now, we can guess that the whole relation between geometry and logic reasoning was more complex that the relationship between a 'discipline' and a 'method'. It was instead a problem thoroughly inner to the method evolution. Our hypothesis is that in the Babylonian and early Greek scenario, geometry, beyond a disciplinary role, was the core of a real proof theory and practice, a sort of metageometry, whose ingredients were: - visual evidence as 'proof', and figures remembrance as 'memory' tool - the figure as the 'space' of the possible moves, and its construction as the 'time' of the proof, - 'epharmozein' method, i.e. ''superposition'' as condition for the ''equality'', - dissection of a figure and ''equality'' between the whole and the parts, to prove geometric equality relations, - orally-described construction of the figure, starting from evident constructions and terms.

It is worth noticing the absence of the any explicit negative aspect. For, also in reductio-ad-absurdum proofs, in the earliest Greek geometry the negation was probably shown as a counter-intuitive positive construction: typically, contradiction was the non-coincidence of the positive constructions of two different entities which had instead to coincide. In a following section we will analyse the presence of this ancient framework in the birth of the axiomatic-deductive method and its connection with the Euclidean geometry, but we have to underline since now the major breakthroughs to be accomplished toward the modern mathematic formalization: - the 'space' of proofs and problems has to lose its geometric and realistic nature to become a logic and symbolic space, and hence the 'time' must be subject no longer to construction constraints, but only to logic and functional requests, - the 'figure' must be substituted by the 'signs', which must show a functionality becoming more and more autonomous, in that they must have a syntax, that in turn contains also the rules for the evolution of the solving procedure, both for theorems and for problems: the modern form to build is the idea of ''proof'' as an ''algorithm'' whose data and results are formulas.


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