Since ancient times all civilizations have had an understanding of mathematics as a rigorous discipline. However different civilizations have shown different inclinations in their mathematical efforts and consequently the content of mathematics practiced in them was different. We would examine the development of mathematics in India, China, and Greece and in the modern West. The emphasis of the study will largely be on the history of concept formation in the realm of mathematics and the underlying philosophical inclinations.
The comparative study of mathematical development in different civilisations shows strikingly distinct paths in their growth or content. Mathematics in India and China showed an algebraic inclination, in contrast to mathematics of Greece which was geometric in its outlook.1 This difference is not just an apparent difference, but is one which pervades even the foundational realm of mathematics and indicates fundamental differences in the paths of development of mathematics. Another significant difference in the methodology of mathematics of these civilizations is that the so called 'method of indirect proof was not accepted in India and China whereas this method was central to Greek efforts.2 Though demonstrative reasoning was in practice in mathematical efforts of all the civilizations the notion of indirect proof, which led to emergence of a peculiar kind of formal logic, was there only in Greek mathematics. Whereas Indians and Greeks paid early and detailed attention to formal logic, the Chinese showed a constant tendency to develop "dialectical logic". Even the formal logical systems developed in India and Greece were entirely different. The Indian logic was structured as an important component of the Indian epistemological analysis and was therefore formal as well as material, deductive as well as inductive. Dasgupta has conjectured that Nyaya logic developed out of medical and linguistic debates. In India, logic seems to have had not much relation with mathematical debates. On the other hand, logic in Greece (as well as in the Modern West) is closely tied with modes of mathematical reasoning. In fact the importance of mathematics to Greek philosophy3 can be accessed from Plato’s attitude towards the knowledge body of his times given in Republic and Timoeus. Knowledge bodies in descending order of importance for him ware Arithmetic, Plane geometry, Solid geometry, Astronomy and Harmonics.
In this study, we shall try to bring out the basic differences in the cognitive structure of mathematics as practiced in the Greek, Chinese and Indian civilizations — differences in their ontological conceptions regarding existence and. nature of mathematical objects and methodological conceptions regarding nature and ways of [establishing mathematical truths. The, differences in cognitive structure of mathematics as practiced in the various civilizations pose many complex problems which call for a closer study of mathematics developed by these civilizations. 'Also the differences in the mathematics of these civilizations have to be identified in a more exhaustive way and can probably be attributed to the differences in the worldviews t prevalent in them, as development of mathematics and its content cannot be seen in isolation" from the general history of the modes of understanding. The modes' of understandings find their overt expression in the philosophical schools of the time and their debates, which influence even the process of concept formation in mathematics. In this paper, we will try to comprehend the plurality of directions of concept formation in mathematics along with their underlying philosophical bases and also try to relate them to some of the philosophical problems of the modern mathematics of the West.
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Greek (mathematics flourished between 6th century BC (Thales: 639-546 BC) and survived till the Ptolemaic dynasty of Egypt. It can doubtlessly be asserted that Greek mathematical tradition drew from achievements of ancient Babylonians and also probably the Egyptian tradition. But the concepts used in Greek mathematics were fundamentally different from those used in Babylonian and Egyptian mathematics and were marked by their philosophical inclination towards anti-empiricism.4
In Greece, the rise of Pythagorean mysticism played a pioneering role in the development of mathematics. ‘Pythagorean experiments with the strings of different lengths led jto the discovery of arithmetic ratio ability of musical notes. This discovery was instrumental in the formulation of Greek theory of proportion. Ratio (logos) 'was understood as the relation between the strings of different lengths represented by two numbers and was equivalent to relation between two lines in terms of their number measure. Greek understanding of the realm of mathematics through peculiar equivalence of geometry and arithmetic’s stems from this principle. For them, measurableness was the root from which countableness stems. Consequently number had a figurative quality. Classification of numbers as plane numbers, similar plane numbers and their further division into square number, rectangular number, triangular number, and pentagonal number finally led to the understanding of realm of numbers as assemblage of classes or species (eidos) of numbers that forms the realm of arithmos. The important point to note here is that number was seen in accord with its measurableness which is essentially a demand posed by the geometric view. Commensurability with geometric unit line was the fundamental criterion for the constitution of the realm of arithmos. Instead of fraction which divides the unit line, concept of ratio was used which transforms fraction into relation of two commensurable complete measures or numbers. Significant trait of Pythagoreanism was its atomistic outlook which remained integral with the rest of Greek mathematics as well. Number was an entity constituted out of an indivisible unit which was understood as in itself monad, as purely abstract and as primordial atom. The fundamental doctrine of Pythagoreanism, the dependence of musical consonances on arithmos5 was not only responsible for formulation of Greek mathematics but also its ontology. The Greeks imagined that the world and its bodies had been built, “out of numbers".
But soon in the realm of arithmos, they came across lengths which were incommensurable when determining the mean proportion of the two side6 of a rectangle which is needed for squaring the rectangle. Even the diagonal of a unit square (= √2) is incommensurable and this discovery was a great scandal for the school of Pythagorean thinkers. As the story goes a pupil of Pythagoras, Hippasus of Metapontum is said to have perished at sea for the rebellious act of disclosing the discovery of incommensurability. Memory of this scandal is retained even in the terminology of modern mathematics. Numbers which are expressible as a ratio of two integers are called rational numbers, whereas numbers like length of a diagonal of a unit square i.e. √2, are not expressible as ratio and hence are called irrational, as Greeks called them, because they were "un-ratio-able". It is an interesting fact that the etymological root of rationalism comes from Latin ratio which is a translation of Greek logos meaning mathematical ratio and which to Greeks, because of its elegance, symbolized reason itself. Corning back to the main point, Greek incommensurability was the result of an attempt to establish equivalence of geometry with arithmetic—it was the result of geometric view of irrationality of numbers. Because of the geometric bias Greeks just did not recognize irrational numbers as numbers and consequently the discipline of numbers was reduced to narrow realm thus robbing away considerable potency of arithmetic’s.
Another significant philosophical movement, which was destined to have lasting effect on Greek mathematics, was the teachings of the Eleatic philosopher Parmenides, a younger contemporary of Pythagoras and his pupil Zeno. They believed that Truth cannot be grasped by means of sense perception but only by reason logo. | They hypothesized on ontological realm of static where there is no motion, no change, no becoming, no perishing, no space and no time. This realm, they believed, was accessible to man by reason- To prove the existence of this realm they devised the rational method of reduction ad absurdum. Zeno's famous paradoxes deny motion, space1 and time by using this method. A salient feature of the method used is the acceptance of singular negation of statement so that statement and its negation give mutually exclusive non-interpenetrating hypothetical existence. True existence of any statement, can be demonstrated if the existence of its negation can be denied and fit can be denied by showing logical inconsistencies, which follow |if the negation is assumed to be true.6 This method opens up a realm, whose, existence is solely demonstrated on 'the ground that its negation is inconsistent. [Realm of existence, which thus opens up, is an ideal realm as it is not arrived at through its empirical accessibility, but through a particular ideation procedure. Here, we see the radical anti-empicsm of Eleatic philosophy. Rejection of empiricism in Greek mathematics after Pythagoras can be attributed to the decisive influence of Eleatic School of philosophy. Though Eleatic philosophers had very little to do with mathematics) directly, still the method they evolved from purely philosophical) arguments played a decisive roie in further evolution of Greek mathematics and is also central to modern mathematics.7
Two centuries later in Euclid's Elements we find complete internalization of Eleatic tradition within mathematics. There was one fundamental tension between Eleatics and Pythagoreans which posed considerable problem for attempting synthesis. That was the preoccupation of Pythagoreans with a conception of space, which is sensuous in orgin. For Pythagoreans number not only had relation with space but also with "material". Eleatic philosophers on the other hand denied being for space. This apparent irreconcilable nature of Pythagorean and Eleatic position was reconciled by Platonic synthesis which climaxed in Euclid. Plato's philosophy (427-347 BC) was influenced by Pythagorean mathematics as well as by Eleatic philosophy. Plato inherited from Eleatics the distrust for sensuous which he readily equated with the realm of changing opinion, the realm of impermanence, the realm of becoming and the realm of things in flux. Plato assigns to 'mathematical' a totally different mode of being than Pythagoreans. Numbers for Plato were separable from the object of senses, so that they appear alongside perceptible things as a separate realm of being. The counting of objects of senses was grounded in the existence of non-sensible idea of number. The difference between counting and the science of numbers was one of the fundamental- doctrines of Platonism. The possibility of calculations was rooted in certain immutable characteristics of the numbers themselves and it is with them that arithmos deals. The immediate consequence of this doctrine, at least within Platonic tradition, was the exclusion of all compu-tational problems from the realm of the pure science of arithmos. Science of arithmos in ideal form was already quite developed by the -time of Plato that is way Plato had good reason to regard arithmetic as the finest science of his time.
It was with the conception of space in Geometry that Plato had problems. Awareness of space is associated with location of things which move, change, are generated and destroyed in it and it was this Plato wanted to avoid. The attempt to develop an extra-sensory conception of space marked the beginning of theoretical geometry in Greece. Plato believed that space is apprehensible not by sense perception but by a kind of "bastard reasoning".8 On the one hand, it was eternal and indestructible, and on the other it was intimately bound up with the phenomenon of the perceptible world. For Plato, the aim of geometry should be to gain knowledge of eternal reality. Arithmetic was based on 'one', its multitude ratios, etc. These were ideal and completely abstract forms and were well suited for Platonic ethos. There was, however, no such ideal starting point for geometry. Hence the idealization of geometry, which was demanded by Platonic connections needed different logical structure than that of Pythagorean arithmetic. It was here that Eleatic method entered into mathematics. Formulation of axiomatic foundation of geometry was a result of Platonic tradition which culminated in Euclid.
In Euclid's Elements forms of geometry (lines points of intersection, angle, figures etc.) were ideal entities whose visible counter-parts served only to represent them. The definitions were intended to eliminate as many sensible features as possible from geometry the axioms and postulates were also aimed at making the foundations of this science purely abstract. Euclid defined a point as that which has no part" and a line as "breathless length". Euclid avoided any mention of motion unlike Proclus who defined line as "uniform" and "undeviating flowing of a point". Here we see that temporality is being avoided. What was sought was the realm of ideas in itself whose simplicity, consistency and beauty was the sole reason for their truth-fullness. This realm of ideas was the true realm of being. Properties of this realm were explored using Eleatic method, which enables demonstration of existence of properties of idea forms by denying consistency of their negation. To develop science of these ideal forms was the self-imposed task of Euclid.
From Plutarch, we know that for Euclid the finest task of his geometry was to be able to explain and investigate Platonic solids. Platonic solids were elemental ideal objects which played j a central role In his cosmology. He thought that distinctiveness of four fundamental elements drew out of the underlying four regular ideal solids: earth-cube (because cube is a most stable solid), fire-pyramid (because fire rises up as symbolized [by a pyramid), air-octahedron and water-icosahedrons. Physical bodies were identified with ideal geometrical forms in accordance with the ' Platonic thesis of identity of idea and being. Platonic matter was held to be a kind of body lacking all qualities except ideal forms which physical bodies imitate. 'The conception of causality was one of a temporal necessity of underlying idea' forms.9
Another significant feature which needs to be pointed out is that the discipline of geometry developed by Euclid avoided difficulties which rose with the discovery of incommensurability. Notions about existence of mathematical objects or the criteria for legitimization of mathematical objects which Pythagoreans upheld were much more stringent than J that of Euclid. It appears there was a major difference between Pythagorean and Euclidean mathematics. Pythagorean atomism and equivalence of geometry and arithmetic inevitably leads to the problem of .incommensurability which stalls any further development. The criterion about "existence" which Euclid adopted from Eleatic tradition transcends this impasse leaving behind the problem of incommensurability. What remains common between Pythagorean and Euclidean mathematics was the acceptance of mathematical objects as monas (indestructible mental objects)
Now we are in a position to summaries the distinct features of Greek mathematics which envelopes its potentiality and its limitations.
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Just where Greek mathematics was weak, Indian and Chinese mathematics were strong, namely in algebra, arithmetic and empirical geometry. Though the mathematical research in India and China was not without differences, still both these civilizations had intrinsic algebraic and empirical inclination. Now we will go into some details of the Chinese and Indian mathematics which indicate their fundamental attitudes.
Unlike Greek civilization which was comparatively short lived, history of Chinese civilization is a story of great continuity. Throughout the history of China refinement of the practical art of calculation played an important role in the process of concept formation in their mathematics. Even till the present day, abacus plays its role in clerical work in China and Japan. Some years back, in a competition between a modern electronic calculator and Soroban (Japanese abacus) the latter proved its efficiency for all the arithmetic operations except multiplication. Such is the efficiency of calculation procedures perfected by Chinese civilization. The principle of place-value10 in numeral notations in its origin is intimately related to calculation procedure of the abacus. Numeral notation of only 9 figures combined with place-value component appears as early as the Shang period in China (1400 SC). This had profound influence throughout the history of China on the process of concept formation and the nature of mathematical enquiry. Chinese never had a separate philological unitary symbol for special fractions (like 2/3, the fraction so important to Mesopotamia). Numbers were constructed out of nine figures only. What followed was early development of decimal system of writing numbers rather than being stuck at in-itself monadic numerical entities and their unfolding into ratios as in Greece. On the other hand Greek preoccupations never allowed appreciation of the principle of place value. The important point we note here is that the Chinese did not view number as an atomic and indivisible entity, and as a consequence the realm of numbers got explored exhaustively.
With their early use of decimal' fraction in expressing roots, Chinese mathematicians never encountered the dilemma of irrational which, perplexed the Greeks. In China extraction of square and cube roots were highly developed by 100 B.C. Chinese rules resemble Aryabhata and Brahmgupta's rules- of extraction given in about 5th-6th century A.D. These very rules influenced a-Kashi (15th century A.D.) in Arabia and Shortly after spread to Europe. Similarly negative numbers were quite acceptable to Chinese and Indian mathematicians. Greeks had no notion of these and their first satisfactory treatment in Europe came only as late as in 1545 A.D. by Jerome Garden. Conception of negative numbers was possible because the notion of arithmetic operation was assigned more potency than the notion of number as an entity.
Solid geometry which developed in China was overtly of an empirical kind whereas plain geometry was essentially algebraic. Problems concerning right-angled triangles were exhaustively solved in a pure algebraic way. In fact, in Needham's view, the invention' of coordinate plane geometry should be attributed to the Chinese civilization."
One noticeable feature of Chinese mathematics was the lack of formulation of the equivalence sign (' = '). All throughout Chinese history algebraic operations were, carried out without the sign of equality. This was one feature which differentiated 1 Chinese algebra and Indian algebra. While without the sign of equality, Chinese solved complex systems of linear algebraic equations, systems of indeterminate linear equations and also higher order, algebraic equations. Chinese in fact developed a peculiar, way of writing equations. The equations were represented by coefficients arranged in a matrix form and the rules of operation developed to solve them. Even Chinese multiplication tables were learned by students in a 9x9 matrix from with 45 elements and the significance of location of number was implicit. We can also find close parallels in the nature of Chinese language and the format or kind of representation that Chinese algebra developed. Chinese mathematics therefore developed at an early date the idea of subtracting columns and rows as in .the simplification [of d determinants. 17th century Japanese scholars transformed this Chinese method into vigorous symbolic form and developed the theory of determinants. Seki Kowa published his theory of determinants in 1683—-a decade prior to Leibnitz's work published in 1693. In fact Needham argues that Leibnitz's philosophy was influenced by Chinese Confucian texts which a Christian jurist brought to him and gives evidence that Leibnitz was familiar with writings translated from Chinese language. For the solution of higher order equations the Chinese developed a format of triangular [representation of J coefficients of different orders. This is the so called Pascal’s triangle of binomial coefficients12 and it was known to the Chinese by 1100 A.D. Another result of. the development of a particular format of representation by the Chinese was their theory of iteration for the solution of algebraic equations.
Chinese called the method of iteration 'too much and the little less this method has also an affinity with the Chinese dialectical thought and the theory of Yin and Yang. Solution to an equation is assumed beforehand and the near exact answer is iterated out. In this method the solution fluctuates between less than actual and more than actual and in this way convergence to actual is realized. Inexactitude of numbers did not put Chinese into uneasy position. This process of solving equations began to be appreciated in Europe only in the 19th century and is more in vogue today after the modern computer has come into being. Methodologically the procedure is inclined towards constructive approach rather than analytic approach. Constructive approach is epistemologically inclined towards empiricism as it is intrinsically related with calculation procedure. In this sense the entire Chinese tradition of algebra and arithmetic was empiricist at its root. This is in line with naturalistic philosophical tradition of China which had little appreciation of ideal entities and their associated ontological realm of statis.
Just as in China, in India also the importance of place value and decimal in numerical notation was realized quite early. In fact the unique feature of early Indian mathematics was the wide "and liberal exploration of the realm of numbers. Word numeral, for _ as big a quantity as 1012 {Parardha) is given in various Samhitas™. On the other hand the biggest word-numeral available in the Greek tradition was 104('Myrioi' — the symbol Mu in Greek numerals). In early India even fractions were regarded as numbers unlike Greeks who did not have such a notion. Sulbasutras denote fractional numbers by the general term 'bhaga'. All mathematical operations were as much valid for fractions as they were for numbers. Unlike Pythagoreans and Euclid, numbers were not regarded as atomic monads existing priori, but as entities whose being is brought forth and dependent on mathematical operations. It is only the outlook which considers mathematical operation as foundational that can bring negative numbers and zero into being. And this did happen in India. Zero and negative numbers were considered at par with other numbers and by the time Brahmagupta (7th century A.D.) explicit rules were formulated for arithmetical operations involving such numbers.
The stress on fundamentally of operations over being of numbers steered Indian mathematics clear of any problem with incommensurability. The Greeks accepted only those entities as numbers that were commensurable with unity, which was for them the principle of being out of which all numbers should arise In India surd numbers known as Karani were accepted as number in very early. Times14 and rules for handling them developed. Though rational - irrational classification did not develop but notion of exact (nitya) and inexact (anitya) numbers was developed.16 Instead of bothering about the nature of being of irrationals, procedures were developed for hhe calculation of these numbers. The Sulbasutras gave the value of √2 correctly to more than five decimal places. Inexactitude of a number for higher calculation was quite acceptable to the Indians.' Awareness of the inexactitude of π which to the Greeks was again a monadic ratio embodying in itself the fundamental truth of the circle, was evident to ancient Indians. Aryabhata gave the value of π= 62832/20000 = 3.1416 and more importantly he was aware of it inexactitude as were other Indian mathematicians. Constant revisions in the value of π occurred from time to time, which eventually lead to the discovery by Indian mathematicians of infinite series converging to the value of Π BY the 15th century. The popularity of Indian methods can be gauged by the fact that Al Kashi in 1427 explained how w can be calculated (correct upto 16 decimal places) by using Indian methods .
In Indian mathematics, Geometry never got idealized in the way it did in Greece. Solid geometry in India as in China was largely empirical. Indian plane geometry was essentially algebraic in character, but in its approach and concept it was quite different from Chinese efforts. What we refer to today as Trigonometry was quite developed in India by the 5th century A.D. Because of their geometric emphasis the Greeks used chords in their astronomical calculations, whereas the Indians developed the notion of sines and versines quite early. Aryabhata was perhaps the first to give a special name to these functions and to draw up a table of sines for each degree. Approximate algebraic expressions were also developed for these functions16. Indian influence flowed to medieval Arab mathematics where Al Battani (858 - 929 A. D.) used the sines regularly with a clear realisation of their superiority over Greek chords and introduced functions akin to the itangent and cotangent functions. Abdul-Wafa (940-998 AD) introduced the recant and cosecant.
In Indian mathematics the realm of numbers was regarded as homogeneous— homogeneous with respect to arithmetical operations. Anuyogadvara Sutra (100 BC) classified numbers into three categories (1) Sankhyata (numerable) (2) Asankhyata (innumerable) arid (3) Ahanta (infinite), and further classifies them into several classes. Implicit within this classification is the idea of homogeneity of numbers as the classification and is based on -the shades of meaning of finite and infinite. Idea that realm of numbers is homogeneous is behind the various rules of operation which are defined in Indian mathematics for positive as well as negative numbers, zero and even for infinity. In contrast the Greek realm of. numbers was colorful and heterogeneous. The Pythagorean classification of numbers into plane numbers, triangular, square numbers, solid numbers etc. was a scheme of classification based on the property of figurative being of numbers.
The idea of realm of numbers as homogeneous is an essential prerequisite for the development of algebra. Idea of provisionally indeterminate number or unknown or unaccomplished quantity can be meaningfully worked out only if its domain of activity is homogeneous. The central imagery behind mature Indian algebra (called Bija Ganita) was that if one sows a seed {bija) in the field of numbers (Ksetra) then one can harvest fruit (phala)—Bija and phafa were technical mathematical terms for the unknown and the solution. Idea of algebraic equations got developed very early in India. Sthananga sutra (300 BC) classified algebraic equations according to the number of unknown in it. Brahmagupta (628 A. D.) classified equations as (1) with one unknown including higher order equations (2) with many unknowns (3) with product of unknowns. Brahmagupta also extended the earlier theory of Aryabhata of Kuttaka, the analysis of indeterminate equations (where the solutionis restricted to integral or rational values only) which was to form a major part of Indian algebra in the later centuries.
Thus the heritage of algebra has been the hallmark of the mathematics of the Chinese and the Indians, and later of the Arabs—though the tradition of algebra was not a homogeneous one as the directionality of the mathematical enquiries of these civilizations was different despite the underlying commonality. In essence the Asian algebraic tradition is fundamentally distinct from the abstract geometric tradition of the Greeks- If in mathematics entities are regarded as having being or nonbeing then the existence criteria would require the method of indirect proof as it evolved in Greek tradition. But if entities are regarded as accomplished and no accomplished then the existence criteria would be constructability. Indeed Indian mathematics may be said to use 'algorithmic logic' in which the central principle is that of constructability.
Here we can see a homology at the foundational level between Indian mathematics, Indian logic and Indian linguistics. The main theoretical concern of Indian linguistics was constructability of accomplished entities of language. Neither did Indian linguistics regard its abstract constructs nor formal universal as real any more than Indian mathematics regarded its formal algebraic universals as real. Algorithmic logic begins from unaccomplished and ends in accomplished entity — so whatever formal apparatus it uses, such formal universal cannot claim being at par with accomplished entities. In fact the methodology of Indian linguistics as brought out even in the most ancient texts such as Rg Pratisakhya explicitly recognizes the importance of Apavada Sutra (exception rules) and Nipatana Sutra (counter-rules) and even goes further to stipulate as a metarule that the exception is stronger than the Samanya Sutra (universal rule) in cases where both are applicable. Thus the fallibility of the universal is accepted in principle as a metalogical truth, this is acknowledging that logical order and real order are not parallel and that the universal is a construct and can never posit itself in the world as infallibly real, arid existing a priori.17 This approach is fundamentally' different from the Platonic theory of ideas which held logic universals to be real and transcendent. It is on Platonic reform of Eleatic philosophy and logic that Euclid built his methodology 'which still gives foundational security to modern formalist and axiomatic logic and mathematics.
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We now come to the transformation which took place in Europe around the 16th and 17th centuries. Underneath this transformation was an assimilation of knowledge within European folds from other cultures as well as birth of new intentionality in the development of knowledge. This intentionally makes European efforts different from the Arab, efforts, though from the 10th century onwards Arabs also assimilated knowledge of. different cultures in a big way and gave algebra a refined status of a separate discipline. While the new science of Europe assimilated knowledge of other medieval sciences its birth is marked by a certain polemical attitude towards these sciences. In place of acknowledging its debt to immediate parental tradition of knowledge, Europe saw itself arising out of natural foundations of distant Greek science. In the process it interpreted Greek mathematics in a way suited to its intentionality. All this had profound impact on concept formation in European mathematics. From 12th century onwards algebraic books of Arabic origin and later Greek books were translated into Latin and other European languages. But only in the 16th century the mathematical research of Europe ascends to different levels with a new awakening. Within a century, Viete (1540-1603), Stevin (1548-1620),[ Descartes (1596 - 1650) and Wallis (1616-1703), gave new foundation to mathematics largely depending on the Greek heritage for its method and on the Asian heritage for its content.
In the preface to Isagoge (subtitled : Introduction to Analytical Art) Viete the father of 'modern algebra' says, "Behold, the art which I present is new, but in truth so old, I so spoiled and defiled by the brabarians, that I considered it necessary, in order to introduce an entirely new form into it, to think out and publish a new vocabulary, having gotten rid of all its pseudo-technical terms (pseudo-categoreniatis) lest it should retain its filth and continue to stink in the old way. And yet underneath the Algebra or Afmucaba/a (of Arabian origin) which they lauded and called the great art, all mathematicians recognized that incomparable gold lay hidden, though they used to find little.18 The very expression 'algebra' belonged for Viete to barbarian pseudo-terminology. He sets forth developing a general analytic art where every number can be expressed by its 'species' (eidos) and the question 'how many?' can be left indeterminate provisionally for analytical purpose. Viete arrives at the conception of a mode of calculation which is carried out completely in terms of the species of numbers and calls it "logistice speciosa". 'Logistice speciosa' he applied to pure algebra, which he understood as most comprehensive possible analytic art, indifferently applicable to numbers and to geometric magnitudes. This new conception of eidos, because of algebraic concern, was ontologically independent (contrary to Greek use of eidos) and was mere symbolic formalism. Therefore the most important tool of mathematical natural science, the 'formala', first became possible. Here we see the metamorphosis of Greek ontological eidos to rationalistic dimensions. From now on ontological science of the Greeks is replaced by a symbolic discipline whose ontological presuppositions are left unclear. , What remain in mathematics are a method, Greek logic and Greek proof. The actualization of this method is sought in the realm of algebra which naturally transcends application part of mathematics, namely arithmetic and geometry in Viete's view. Pure analytic art of algebra is visualized as renewal of the Greek spirit. This idea of renewal played constitutional role for resurrection of the lost spirit in the form of analytic mathematical knowledge.19
Stevin, a contemporary of Viete and Kepler, is also possessed by the idea of a renewal and he casts it into the special image of a unknown 'wise Age' which must be brought back. Barbaric age for him extends from the beginning of the Greeks to the present. Stevin does not consider Greeks to be the only luminaries of the 'wise Age'. Vision of 'wise Age' is constructed out of elements of heritage of India, China, Arabs as well as Greeks. The Arabic digital and positional system in his view is immensely superior to Greek notations and is a heritage of 'wise age'. Algebra, which Europe became acquainted through Arbaic books and whose trace is not to be found in the Chaldaeans, the Hebrews, the Romans and the Greeks again hints at the 'wise Age'. Astronomy of Ptolemy, which has traces of perfect knowledge, is exemplary of near extinction of perfect understanding of the course of heaven whose other elements can be traced to the writings of the Arabic tongue. The heliocentric theory in particular is said to have had an origin of great antiquity. Alchemy again is a science of 'wise age' which he says was unknown to Greeks. Stevin is an interesting thinker in many ways, he wrote and taught mathematics, only in Flemish and held the title of professor of Dutch mathematics what differentiates Stevin from his contemporary thinkers of Europe is the peculiar bent of his cultural sensitivity which is evident from his bias in characterizing wise age'. On the other hand what unites him with his contemporaries is his view that right order must be used in writing and teaching of mathematics. Right order taught by the ‘wise age', is the mode of presentation of the Euclid’s Element, which he calls ‘order natural’. His projection of algebra as well as method of Euclid's Element into 'wise age' unfolded peculiar directions in his mathematics research.
Stevin totally rejected Greek concept of Arithmos as well as its modification in Viete. 'Unit’ to him is a number as much as other numbers rather than a primordial entity or even the beginning of numbers. For him, there are no absurd, irrational, irregular, inexplicable or surd numbers and these terms arise as a result of the schism between actualt mode of understanding of number and the traditional Greek concept of number for him even roots and powers are just plain numbers. This homogenization of numbers into one category was a typically Arabian, Indian and Chinese [notion of number as opposed to Greek Arithmos, a figurative conception of number. Only radical acceptance of Arabian tradition could open up possibility of such view of numbers which escaped grip of Viete who kept Greek notion of species of numbers. He argues that}number is not at all a discontinuous quantity as against the bias arising out of Greek indivisibility of the 'unit'. 'One' can be continuously divided into fractions. Continuous magnitude corresponds to continuous number much like water, which can be added and subtracted continuously. This identification of the realm of] numbers with continuum also led him to see new correspondence between geometry and algebra which was destined to culminate in Descartes Coordinate Geometry and a concept of real line. It is this correspondence (between Asian and Greek1 mathematics) which till today embodies a tension20. This new correspondence was based on a denial of specific nature of Greek corres-pondence at the same time assertion of its spirit in method.
A central feature in the emerging new directions was the notion of natural law. The notion of natural law was a distinct feature of European intentionality and can perhaps be traced back to the culture of Judeo-Christian tradition. The notion of natural law occurs .for instance in Thomas Aquinas (11th century) who thought God was just a prime mover (first cause, etc.,) of the universe, and the universe moves along preordained path and man's duty is to discover the preordained order, which is only accessible through reason, so that he can make this universe his just home. Greek notion of statis is reincarnated to represent static rational order .which has been left by God to be discovered by man. Here, we see there was ample amount of space for Platonism to make a comeback. And it did. Number of historians of science has analyzed resurrection or Platonism in 16th, 17th century Europe21. Stress on the study of space and harmony gradually gave way to deontological Platonism. a rationalist doctrine of methodological monism (Strees on uniqueness of method).
In the 16th century, Fransicus Patritius (1 93) argued that study of space must come before study of matter. Copernicus used mathematical harmony as an argument for his cosmology and wrote 'nothing unfitting [Geometrically] occurs in the course of their [planets] orbit". Kepler spoke of a system of five regular solids, which could be inserted between the sphere of six planets, as the cause of the planets being six in number. Galileo wrote that the ' book of nature is written in geometrical tradition". The phenomenal world imitated the rational order of primary qualities (qualities whose mathematical quantification is possible) and secondary qualities (experimental sensuous world) arise because of the effect of primary qualities on senses. The realm of knowledge now is the realm of primary qualities.
This understanding was in fact; neo-Platonic because primary qualities were not just geometrical magnitudes but also magnitudes of mass and time. Mass and time unlike spatial magnitude are algebraic quantities as they are not infinitely divisible but additive. The difference between infinite divisibility and additivity in my view is the fundamental distinction between space and time." It was this difference which was ignored in Western mathematics and time was geometrized on the plea that space as well as time is continuous quantities. But continuity itself has two meanings, one as continuous divisibility of space, and the other as continuous addivity of time. The fundamental difference between space and time which has affinity towards difference between geometric and algebraic views, was ignored in the analytical unification of these two trends by the West.
The analytic equivalence of algebra and geometry made possible handling of these disciplines at the same plane. However, the earlier tension between geometric and algebraic view has been transformed into, a tension between ideal and empirical mathematics in so far as intentionally of Europe saw geometric Greek tradition as a method and Asian algebraic tradition as an object of mathematics. This transformation was also forced in because the science of physiology of William Harvey, magnetism of Gilbert, and developments in Chemistry showed that logic of experiential science was not absolutely bound to mathematical expression, that there is a realm where mathematical necessity is not demonstrable. This brought about a tendency towards the deontologisation of mathematics and Science. The empirical and experiential streams of science was integrated in mathematical science by transforming the experiential into experimental which is governed by teleology of mathematics. In this process the ontologising aspect of 'Greek method was forgotten and the emphasis was put on method itself and later on epistemology. Mathematics as well was science developed along deontologised rationalism. The aim of mathematics became relation of abstract to abstract and eventually was reduced to the status of language for understanding of reality. A tendency towards formalism, logicism and axiomatic gained ground in Europe and by the end of 19th century and the beginning of 20th century this tendency reached its climax. Analytic art and new symbolism of Viete and Stevin became an end in itself. Mathematics became purely analytical and self-referential discipline. This deontologising of mathematics brought about separation of mathematical realm and physical | realm. It was thought that logic does not commit us to any apriori-assumption about physical reality. It constituted a realm which needs no empirical referent but stands alongside the experiential world and is potentially realizable as the true ground of experiential world.
But in deontologised mathematics what ensures the security of mathematical knowledge? Hilbert (1862-1943) thought that if only completeness and consistency2,2 _of.an axiomatic system can be proved then independent criteria for legitimization of mathematical knowledge with ontology-freeness is achieved. The discipline of Meta-Mathematics was born to demonstrate consistency and completeness of axiomatised Mathematics In ,1931, Kurt Godel disproved what Hilbert wanted to prove, by his [famous undesirability theorem.23 The immediate consequence of Godel's theorem" was that there exists no set of mathematical truths which if assumed, given, all the rest will be derived from them. In other words no formalized system will exhaust mathematical theory. Mathematics as a discipline is logically an open system. This- brought about severe strains on the rationalistic program for mathematics. Mathematics was not a discipline folding onto itself with self-referential completeness that forms a world of its own -either of a symbolic kind [or of a Platonic J realm of ideas. It has the same features as the rest of reality namely, that of open-endedness, of cross-referential growth and of
temporality.
The 19th and 20th century also saw the rise of Intuitionist school of mathematics as a trend critical of rationalist mathematics. Another reason for the rise of intuitionism was the discovery of unavoidable paradoxes, in the set theory .of Cantor (1845-1918). The intuitionists do not accept laws of logic as either a priori or eternal. Logic is subject to change. Logic which is valid for finite objects might not be valid for infinite objects like sets containing infinite elements. They do not accept actual infinity as an object of mathematics but only accept potential infinity as was maintained by Hegel, Locke and almost all empiricists. The law of excluded middle was unacceptable to them. This was a fundamental critique of .Rationalist mathematics, as rationalism had accepted Greek logic as the only kind of mathematical logic. For intuitionists, logic was not formal calculus but a methodology, logic of knowledge dealing with transformation of our inferences. Since mathematics is not derived from either logic-or, experience it must originate in a special intuition that presents us the basic concepts and inferences of mathematics as immediately clear and secure. Such basic notion is the notion of natural numbers. As L. Kronecker had said "God gave us the integers, all else are man's handiwork." The criteria now of legitimization of existence of mathematical truth are the principle of constructability. The sole admissible technique of demonstration of existence theorem is effective construction because it permits us to see what it is all about. On the other hand, the method of indirect proof does nothing but points to possibility of existence or of truth, without warranting it. Explicit and effective construction is possible only with finalistic procedures, that is, by means of finite number of signs and operations, as is the case for instance in the computation of the square of a number24. Hence, all propositions involving infinite classes regarded as totalities must be excluded from mathematics. Also theorems that are demonstrated in an indirect way must be eliminated or reconstructed (like most theorems of Cantor's theory of sets) 25. In the rationalist tradition, whose roots go back to Greeks, it was possible to prove the pure existence of objects that cannot be constructed (like most of the modern theorems, using method of indirect proof). On the other hand, intuitionists argued that by according such "ideal" objects the same validity-as real, or finitely cons-tractable objects, mathematics is being robbed away of its potency. Besides, Godel had shown that Hilbert’s rationalist scheme was doomed to failure.
But even after intuitionists, critique and Godel's proof, formalism flourishes in the West. Cantor's set theory and in general, pure-existence proofs are accepted in western mathematical culture, which today is not limited by geographical boundaries of the West but is practiced in India, China as well as in Greece. . Still we can say that this century is characterized by the resurgence of empirical mathematics in the West though it is a rebel tendency and, in miniscule minority. At least the tension which has all along been present in modern Western mathematics (between ideal mathematics and empirical mathematics) has to some extent come to surface today. It is my contention that this tension has its roots in the tension between Asian algebraic tradition and Greek geometric tradition. For the purposes of this study, we may still employ the categories well known in modern philosophical discourse, and summaries these two ideological positions, as they are represented in Modern Western mathematics in the following:
While there has been a serious and ongoing debate on the foundations of mathematics in the West in recent times, it has had little or no impact on mathematical practice or on the development of mathematics, which have been totally conditioned by the formalist methodology derived from the Greek tradition. At the same time, very .little has been achieved by way of comprehending the non-Western mathematical' traditions as only very few of the original source-works have been brought to light or studied (that too mainly in the context of analyzing the European, assimilation 'of the non-European tradition) and even in these limited studies there has been no recognition of the fact that the non-European traditions had their own foundational conceptions and methodologies. This lack of comprehension of what may be called the cultural and philosophical roots of mathematics in different traditions, has totally crippled, our understanding of mathematics and its potentialities and calls for urgent and much more extensive investigations into the foundational conceptions and methodologies of mathematics in different traditions."
Bibliography:
Author: Navjyoti Singh,
Note:
1. " Most of the historians of mathematics agree about this point. Greeks solved difficult algebraic problems in purely geometric way, ex: xS + ab=b2. On the other hand, problems concerning right angled triangle were exhaustively solved by Indians and Chinese in purely algebraic way.
2. Method of indirect proof is also known as method of reduction ad absurdum. Following are the assumptions behind this method :
a) A statement has a singular negation, important corollary of this principle is principle of double negation.
b) If by assuming 'not A' to be true logical inconsistency can be logically demonstrated, then "A-is demonstrated to be true. This method is widely used in modern mathematics.
3. Link between mathematics and philosophy can be ascertained from certain ' crucial Greek categories, namely, marijam-learning matter, ma thesis- study discipline, matbexis- participation and characterization of mathematical truth as mathematic (as thing to be learned).
4. This is the view of most of the historians of Greek mathematics. About Greek geometry 0. NEUGE-BAUER AND ARPAD SZABO and about algebra JACOB-KLEIN have substantiated this view.
5. This doctrine of Pythagoras remained fundamental to the whole of Greek mathematics but was opposed by musicians. For instance, Aristoxenus wrote, "We are attempting to draw conclusions which are in agreement with the experience as opposed the theorists who preceded us. Some of them introduced completely foreign view-point into the subject and dismissed sensory experience as imprecise ; hence they made up intelligible causes and stated that there were certain ratios between numbers and speeds on which the pitch of a note depended. These were all speculations which are completely foreign to the subject and absolutely contrary to appearances", {quoted by SZABO—p. 113). The theorists were the Pythagoreans and their empiricist critique by Aristoxenus had little Impact on Greek mathematics. This tension remained till the time of Plato who sided with Pythagoreans, in denouncing musicians as immature in the debate over the smallest musical interval 'coma'. This debate has been dealt by WILLIAM WHEWELL, Today we know with certainty that some of these musical notes are actually unratioable incommensurable magnitudes and hence the empiricist critique was sound.
6. We may give an example of Eleatic usa of this method to refute Aneximenes. Eleatic assertion was: 'what is, cannot have 'come to being' Proof: Suppose that what is did come to being (assuming negation to! be true,) _then it could only .have come from what is, or from what is not (only mutually exclusive singular negation admitted). There is no third possibility (principle of excluded middle). Now if it had come from what is. rt would already have been existent before it came into being hence to say that it came into being in this way would make no sense. If (on the other hand,' the claim is made that what «came from what is not. this leads immediately to a contradiction {adynat'on) (principle Of consistency). What is can never have been the opposite of itself, what is not, and hence, could not have come into being this way either, ' Hence the assertion is proved (again the principle of excluded middle is used).
7. Vet another contribution of Eleatic philosophy to future Greek Mathematics was the notion of hypothesis. According to Eleatic view, no empirical test can be devised to check whether or not "an assertion" is correct since all physical experience, must in principle be rejected as false-or unreliable. Hence, assertions have to be taken up and .tested in reason. The assertions now have a status of hypothesis. The consequence of' any hypothesis could be examined by their method to see if they lead to contradiction. And if contradiction does not follow the hypothesis stands tested. Likewise, most of the notions in axiomotised mathematics have their root in idealist philosophy of Parmenides and his followers.
8. Jacob Klein, p. 313.
9. This understanding of causality later played important constitutive role for evolution of Kepler's laws of celestial mechanics.
10. The principle of place value is the realization that if divisor is 10 or a power of 10, it is unnecessary to divide as it can be taken care of by shifting the position of numerals with respect to each other. Modern way of writing number uses this scheme whereas the latin numerals do not,
11. Chinese equations ended with implicitly understood ' =: -f- n', n being a constant term. In the early medieval period, it ended having ' = 0', the constant term joining rest of the terms as '-n'.
12. Pascal's triangle is constituted out of coefficients of expansion of (X + Y)n for positive integer n. Arabs as well as Indians also knew of pascal's triangle quite early. In 1252 Nasir al-Din al-Tusi wrote in detail about as pension of (X + Y)n. He explained the method as a current practice, not as a new invention.
13. Datta (1935) p. 9. Later Buddhist text Lalitavistara {100 BC) mentions 'tallaksana' = 1058 and quotes from earlier Jain literature a word numeral 'Sirsaprahe//ka' = order of 102c0.
14. It is important to note that the term- Karani is used for surd root and quantity of root could be rational or irrational. For instance Apastamba Sulbasutra uses chatuskarani = 4 for the number 2.
15. Apastamba Sulbasutra makes this distinction while dealing with squaring of a circle. In later literature terms like 'sthulamana or sthulaganana were used for approximate value or approximate calculation.
16. Sines and" other trigonometrically numbers are not usual numbers. They are expressible only as summation of infinite series (which were to be later discovered by the Indian mathematicians, who kept on calculating approximate values of these functions and improving on them.
17. For a more-detailed discussion of the foundational methodology of Indian linguistics and its similarity with the methodology of Indian mathematics, see Navjyoti Singh (1964)
18. Translation from Latin to English of Viete's Isagoge is included in the appendix of JACOB KLEIN'S book.
19. Incidentally Viete did not choose sides between Ptolemy and Copernicus as he saw between them no fatal tension. On the one hand, he remained committed to Ptolemy's astronomy and on the other he accepted the Copernicus's thesis because it retains the method of Ptolemy's science.
20. See for instance. JOHN MYHILL, "What is real number?" Amer. Math. Monthly, Vol. 79, # 7 P. 748-754, Aug. Sept., 1972. Does every point on real line correspond to a number? If principle of constructivism is used then real I line is full of holes. Most of the real line is non-algebraic numbers called transendental number and there is no method for constructing them.
21. A. Koyre and E. A. Burtt have explicity taken this position.
22. Criterion of consistency: There exists no statement in the axiomatic scheme which can be proved as well as disproved from within. Criterion of completeness : There exists no statement constituted out of objects of the axiomatic scheme which can neither be proved nor disproved from within the system. That is there exists no undividable proposition in the axiomatic scheme.
23. For a 'simple' statement of Godel's Theorem, let us quote the following from 'The Encyclopedia of Philosophy" Ed. Paul Edwards, Mcgraw Hill 1967:
By Godel's theorem the following statement is generally meant : In any formal system adequate for number theory there exists an undividable formula — that is a. formula that is not provable and where negation is not provable. A corollary to the theorem is that the consistency of d formal system adequate for number theory cannot be proved within the system'.
24. Recently another constructionist trend came up with the publication in 1967 of a book by Erret Bishop. Bishop was not much interested in founding mathematics as the intuitionists were but with doing mathematics constructively. Bishop tried to reformulate Cantor's theory by avoiding "idealistic" notions and continually checking the objects and theorems generated for intuitive meaning.
25. One classic instance of pure existence proof is Cantor's proof of infinite number of transcendental (non-algebraic) numbers, which does not give the slightest hint of how even a single transcendental number might be found. It proves existence of such numbers by proving that it would be contradictory for them not to exist. Pure existence proofs now pervade modern mathematics, like, it can be proved that a differential equation has a unique solution without showing a way to reach that solution. Intuitionists and constructivists are critical of such kind of mathematics. 69
The comparative study of mathematical development in different civilisations shows strikingly distinct paths in their growth or content. Mathematics in India and China showed an algebraic inclination, in contrast to mathematics of Greece which was geometric in its outlook.1 This difference is not just an apparent difference, but is one which pervades even the foundational realm of mathematics and indicates fundamental differences in the paths of development of mathematics. Another significant difference in the methodology of mathematics of these civilizations is that the so called 'method of indirect proof was not accepted in India and China whereas this method was central to Greek efforts.2 Though demonstrative reasoning was in practice in mathematical efforts of all the civilizations the notion of indirect proof, which led to emergence of a peculiar kind of formal logic, was there only in Greek mathematics. Whereas Indians and Greeks paid early and detailed attention to formal logic, the Chinese showed a constant tendency to develop "dialectical logic". Even the formal logical systems developed in India and Greece were entirely different. The Indian logic was structured as an important component of the Indian epistemological analysis and was therefore formal as well as material, deductive as well as inductive. Dasgupta has conjectured that Nyaya logic developed out of medical and linguistic debates. In India, logic seems to have had not much relation with mathematical debates. On the other hand, logic in Greece (as well as in the Modern West) is closely tied with modes of mathematical reasoning. In fact the importance of mathematics to Greek philosophy3 can be accessed from Plato’s attitude towards the knowledge body of his times given in Republic and Timoeus. Knowledge bodies in descending order of importance for him ware Arithmetic, Plane geometry, Solid geometry, Astronomy and Harmonics.
In this study, we shall try to bring out the basic differences in the cognitive structure of mathematics as practiced in the Greek, Chinese and Indian civilizations — differences in their ontological conceptions regarding existence and. nature of mathematical objects and methodological conceptions regarding nature and ways of [establishing mathematical truths. The, differences in cognitive structure of mathematics as practiced in the various civilizations pose many complex problems which call for a closer study of mathematics developed by these civilizations. 'Also the differences in the mathematics of these civilizations have to be identified in a more exhaustive way and can probably be attributed to the differences in the worldviews t prevalent in them, as development of mathematics and its content cannot be seen in isolation" from the general history of the modes of understanding. The modes' of understandings find their overt expression in the philosophical schools of the time and their debates, which influence even the process of concept formation in mathematics. In this paper, we will try to comprehend the plurality of directions of concept formation in mathematics along with their underlying philosophical bases and also try to relate them to some of the philosophical problems of the modern mathematics of the West.
Greek (mathematics flourished between 6th century BC (Thales: 639-546 BC) and survived till the Ptolemaic dynasty of Egypt. It can doubtlessly be asserted that Greek mathematical tradition drew from achievements of ancient Babylonians and also probably the Egyptian tradition. But the concepts used in Greek mathematics were fundamentally different from those used in Babylonian and Egyptian mathematics and were marked by their philosophical inclination towards anti-empiricism.4
In Greece, the rise of Pythagorean mysticism played a pioneering role in the development of mathematics. ‘Pythagorean experiments with the strings of different lengths led jto the discovery of arithmetic ratio ability of musical notes. This discovery was instrumental in the formulation of Greek theory of proportion. Ratio (logos) 'was understood as the relation between the strings of different lengths represented by two numbers and was equivalent to relation between two lines in terms of their number measure. Greek understanding of the realm of mathematics through peculiar equivalence of geometry and arithmetic’s stems from this principle. For them, measurableness was the root from which countableness stems. Consequently number had a figurative quality. Classification of numbers as plane numbers, similar plane numbers and their further division into square number, rectangular number, triangular number, and pentagonal number finally led to the understanding of realm of numbers as assemblage of classes or species (eidos) of numbers that forms the realm of arithmos. The important point to note here is that number was seen in accord with its measurableness which is essentially a demand posed by the geometric view. Commensurability with geometric unit line was the fundamental criterion for the constitution of the realm of arithmos. Instead of fraction which divides the unit line, concept of ratio was used which transforms fraction into relation of two commensurable complete measures or numbers. Significant trait of Pythagoreanism was its atomistic outlook which remained integral with the rest of Greek mathematics as well. Number was an entity constituted out of an indivisible unit which was understood as in itself monad, as purely abstract and as primordial atom. The fundamental doctrine of Pythagoreanism, the dependence of musical consonances on arithmos5 was not only responsible for formulation of Greek mathematics but also its ontology. The Greeks imagined that the world and its bodies had been built, “out of numbers".
But soon in the realm of arithmos, they came across lengths which were incommensurable when determining the mean proportion of the two side6 of a rectangle which is needed for squaring the rectangle. Even the diagonal of a unit square (= √2) is incommensurable and this discovery was a great scandal for the school of Pythagorean thinkers. As the story goes a pupil of Pythagoras, Hippasus of Metapontum is said to have perished at sea for the rebellious act of disclosing the discovery of incommensurability. Memory of this scandal is retained even in the terminology of modern mathematics. Numbers which are expressible as a ratio of two integers are called rational numbers, whereas numbers like length of a diagonal of a unit square i.e. √2, are not expressible as ratio and hence are called irrational, as Greeks called them, because they were "un-ratio-able". It is an interesting fact that the etymological root of rationalism comes from Latin ratio which is a translation of Greek logos meaning mathematical ratio and which to Greeks, because of its elegance, symbolized reason itself. Corning back to the main point, Greek incommensurability was the result of an attempt to establish equivalence of geometry with arithmetic—it was the result of geometric view of irrationality of numbers. Because of the geometric bias Greeks just did not recognize irrational numbers as numbers and consequently the discipline of numbers was reduced to narrow realm thus robbing away considerable potency of arithmetic’s.
Another significant philosophical movement, which was destined to have lasting effect on Greek mathematics, was the teachings of the Eleatic philosopher Parmenides, a younger contemporary of Pythagoras and his pupil Zeno. They believed that Truth cannot be grasped by means of sense perception but only by reason logo. | They hypothesized on ontological realm of static where there is no motion, no change, no becoming, no perishing, no space and no time. This realm, they believed, was accessible to man by reason- To prove the existence of this realm they devised the rational method of reduction ad absurdum. Zeno's famous paradoxes deny motion, space1 and time by using this method. A salient feature of the method used is the acceptance of singular negation of statement so that statement and its negation give mutually exclusive non-interpenetrating hypothetical existence. True existence of any statement, can be demonstrated if the existence of its negation can be denied and fit can be denied by showing logical inconsistencies, which follow |if the negation is assumed to be true.6 This method opens up a realm, whose, existence is solely demonstrated on 'the ground that its negation is inconsistent. [Realm of existence, which thus opens up, is an ideal realm as it is not arrived at through its empirical accessibility, but through a particular ideation procedure. Here, we see the radical anti-empicsm of Eleatic philosophy. Rejection of empiricism in Greek mathematics after Pythagoras can be attributed to the decisive influence of Eleatic School of philosophy. Though Eleatic philosophers had very little to do with mathematics) directly, still the method they evolved from purely philosophical) arguments played a decisive roie in further evolution of Greek mathematics and is also central to modern mathematics.7
Two centuries later in Euclid's Elements we find complete internalization of Eleatic tradition within mathematics. There was one fundamental tension between Eleatics and Pythagoreans which posed considerable problem for attempting synthesis. That was the preoccupation of Pythagoreans with a conception of space, which is sensuous in orgin. For Pythagoreans number not only had relation with space but also with "material". Eleatic philosophers on the other hand denied being for space. This apparent irreconcilable nature of Pythagorean and Eleatic position was reconciled by Platonic synthesis which climaxed in Euclid. Plato's philosophy (427-347 BC) was influenced by Pythagorean mathematics as well as by Eleatic philosophy. Plato inherited from Eleatics the distrust for sensuous which he readily equated with the realm of changing opinion, the realm of impermanence, the realm of becoming and the realm of things in flux. Plato assigns to 'mathematical' a totally different mode of being than Pythagoreans. Numbers for Plato were separable from the object of senses, so that they appear alongside perceptible things as a separate realm of being. The counting of objects of senses was grounded in the existence of non-sensible idea of number. The difference between counting and the science of numbers was one of the fundamental- doctrines of Platonism. The possibility of calculations was rooted in certain immutable characteristics of the numbers themselves and it is with them that arithmos deals. The immediate consequence of this doctrine, at least within Platonic tradition, was the exclusion of all compu-tational problems from the realm of the pure science of arithmos. Science of arithmos in ideal form was already quite developed by the -time of Plato that is way Plato had good reason to regard arithmetic as the finest science of his time.
It was with the conception of space in Geometry that Plato had problems. Awareness of space is associated with location of things which move, change, are generated and destroyed in it and it was this Plato wanted to avoid. The attempt to develop an extra-sensory conception of space marked the beginning of theoretical geometry in Greece. Plato believed that space is apprehensible not by sense perception but by a kind of "bastard reasoning".8 On the one hand, it was eternal and indestructible, and on the other it was intimately bound up with the phenomenon of the perceptible world. For Plato, the aim of geometry should be to gain knowledge of eternal reality. Arithmetic was based on 'one', its multitude ratios, etc. These were ideal and completely abstract forms and were well suited for Platonic ethos. There was, however, no such ideal starting point for geometry. Hence the idealization of geometry, which was demanded by Platonic connections needed different logical structure than that of Pythagorean arithmetic. It was here that Eleatic method entered into mathematics. Formulation of axiomatic foundation of geometry was a result of Platonic tradition which culminated in Euclid.
In Euclid's Elements forms of geometry (lines points of intersection, angle, figures etc.) were ideal entities whose visible counter-parts served only to represent them. The definitions were intended to eliminate as many sensible features as possible from geometry the axioms and postulates were also aimed at making the foundations of this science purely abstract. Euclid defined a point as that which has no part" and a line as "breathless length". Euclid avoided any mention of motion unlike Proclus who defined line as "uniform" and "undeviating flowing of a point". Here we see that temporality is being avoided. What was sought was the realm of ideas in itself whose simplicity, consistency and beauty was the sole reason for their truth-fullness. This realm of ideas was the true realm of being. Properties of this realm were explored using Eleatic method, which enables demonstration of existence of properties of idea forms by denying consistency of their negation. To develop science of these ideal forms was the self-imposed task of Euclid.
From Plutarch, we know that for Euclid the finest task of his geometry was to be able to explain and investigate Platonic solids. Platonic solids were elemental ideal objects which played j a central role In his cosmology. He thought that distinctiveness of four fundamental elements drew out of the underlying four regular ideal solids: earth-cube (because cube is a most stable solid), fire-pyramid (because fire rises up as symbolized [by a pyramid), air-octahedron and water-icosahedrons. Physical bodies were identified with ideal geometrical forms in accordance with the ' Platonic thesis of identity of idea and being. Platonic matter was held to be a kind of body lacking all qualities except ideal forms which physical bodies imitate. 'The conception of causality was one of a temporal necessity of underlying idea' forms.9
Another significant feature which needs to be pointed out is that the discipline of geometry developed by Euclid avoided difficulties which rose with the discovery of incommensurability. Notions about existence of mathematical objects or the criteria for legitimization of mathematical objects which Pythagoreans upheld were much more stringent than J that of Euclid. It appears there was a major difference between Pythagorean and Euclidean mathematics. Pythagorean atomism and equivalence of geometry and arithmetic inevitably leads to the problem of .incommensurability which stalls any further development. The criterion about "existence" which Euclid adopted from Eleatic tradition transcends this impasse leaving behind the problem of incommensurability. What remains common between Pythagorean and Euclidean mathematics was the acceptance of mathematical objects as monas (indestructible mental objects)
Now we are in a position to summaries the distinct features of Greek mathematics which envelopes its potentiality and its limitations.
- Atomism: Objects of mathematics were elemental, "complete in itself, "monas"; like point line, number, etc.,
- Criterion for existence of mathematical truth was twofold. For Pythagoreans it was the principle of constructability (genesis) and for Euclid it was the idealist criteria of indirect proof. As Eucljd reinterpreted Pythagorean mathematics in his own terms and the bulk of Greek mathematics is available through his writings, we can regard the methodology of indirect proof as dominant criteria of Greek mathematics. Thus anti-empiricism was its another dominant feature.
- Inclination of Greek mathematics was towards hypothesized ontological realm of stat/'s which was identified with the true realm of being.
Just where Greek mathematics was weak, Indian and Chinese mathematics were strong, namely in algebra, arithmetic and empirical geometry. Though the mathematical research in India and China was not without differences, still both these civilizations had intrinsic algebraic and empirical inclination. Now we will go into some details of the Chinese and Indian mathematics which indicate their fundamental attitudes.
Unlike Greek civilization which was comparatively short lived, history of Chinese civilization is a story of great continuity. Throughout the history of China refinement of the practical art of calculation played an important role in the process of concept formation in their mathematics. Even till the present day, abacus plays its role in clerical work in China and Japan. Some years back, in a competition between a modern electronic calculator and Soroban (Japanese abacus) the latter proved its efficiency for all the arithmetic operations except multiplication. Such is the efficiency of calculation procedures perfected by Chinese civilization. The principle of place-value10 in numeral notations in its origin is intimately related to calculation procedure of the abacus. Numeral notation of only 9 figures combined with place-value component appears as early as the Shang period in China (1400 SC). This had profound influence throughout the history of China on the process of concept formation and the nature of mathematical enquiry. Chinese never had a separate philological unitary symbol for special fractions (like 2/3, the fraction so important to Mesopotamia). Numbers were constructed out of nine figures only. What followed was early development of decimal system of writing numbers rather than being stuck at in-itself monadic numerical entities and their unfolding into ratios as in Greece. On the other hand Greek preoccupations never allowed appreciation of the principle of place value. The important point we note here is that the Chinese did not view number as an atomic and indivisible entity, and as a consequence the realm of numbers got explored exhaustively.
With their early use of decimal' fraction in expressing roots, Chinese mathematicians never encountered the dilemma of irrational which, perplexed the Greeks. In China extraction of square and cube roots were highly developed by 100 B.C. Chinese rules resemble Aryabhata and Brahmgupta's rules- of extraction given in about 5th-6th century A.D. These very rules influenced a-Kashi (15th century A.D.) in Arabia and Shortly after spread to Europe. Similarly negative numbers were quite acceptable to Chinese and Indian mathematicians. Greeks had no notion of these and their first satisfactory treatment in Europe came only as late as in 1545 A.D. by Jerome Garden. Conception of negative numbers was possible because the notion of arithmetic operation was assigned more potency than the notion of number as an entity.
Solid geometry which developed in China was overtly of an empirical kind whereas plain geometry was essentially algebraic. Problems concerning right-angled triangles were exhaustively solved in a pure algebraic way. In fact, in Needham's view, the invention' of coordinate plane geometry should be attributed to the Chinese civilization."
One noticeable feature of Chinese mathematics was the lack of formulation of the equivalence sign (' = '). All throughout Chinese history algebraic operations were, carried out without the sign of equality. This was one feature which differentiated 1 Chinese algebra and Indian algebra. While without the sign of equality, Chinese solved complex systems of linear algebraic equations, systems of indeterminate linear equations and also higher order, algebraic equations. Chinese in fact developed a peculiar, way of writing equations. The equations were represented by coefficients arranged in a matrix form and the rules of operation developed to solve them. Even Chinese multiplication tables were learned by students in a 9x9 matrix from with 45 elements and the significance of location of number was implicit. We can also find close parallels in the nature of Chinese language and the format or kind of representation that Chinese algebra developed. Chinese mathematics therefore developed at an early date the idea of subtracting columns and rows as in .the simplification [of d determinants. 17th century Japanese scholars transformed this Chinese method into vigorous symbolic form and developed the theory of determinants. Seki Kowa published his theory of determinants in 1683—-a decade prior to Leibnitz's work published in 1693. In fact Needham argues that Leibnitz's philosophy was influenced by Chinese Confucian texts which a Christian jurist brought to him and gives evidence that Leibnitz was familiar with writings translated from Chinese language. For the solution of higher order equations the Chinese developed a format of triangular [representation of J coefficients of different orders. This is the so called Pascal’s triangle of binomial coefficients12 and it was known to the Chinese by 1100 A.D. Another result of. the development of a particular format of representation by the Chinese was their theory of iteration for the solution of algebraic equations.
Chinese called the method of iteration 'too much and the little less this method has also an affinity with the Chinese dialectical thought and the theory of Yin and Yang. Solution to an equation is assumed beforehand and the near exact answer is iterated out. In this method the solution fluctuates between less than actual and more than actual and in this way convergence to actual is realized. Inexactitude of numbers did not put Chinese into uneasy position. This process of solving equations began to be appreciated in Europe only in the 19th century and is more in vogue today after the modern computer has come into being. Methodologically the procedure is inclined towards constructive approach rather than analytic approach. Constructive approach is epistemologically inclined towards empiricism as it is intrinsically related with calculation procedure. In this sense the entire Chinese tradition of algebra and arithmetic was empiricist at its root. This is in line with naturalistic philosophical tradition of China which had little appreciation of ideal entities and their associated ontological realm of statis.
Just as in China, in India also the importance of place value and decimal in numerical notation was realized quite early. In fact the unique feature of early Indian mathematics was the wide "and liberal exploration of the realm of numbers. Word numeral, for _ as big a quantity as 1012 {Parardha) is given in various Samhitas™. On the other hand the biggest word-numeral available in the Greek tradition was 104('Myrioi' — the symbol Mu in Greek numerals). In early India even fractions were regarded as numbers unlike Greeks who did not have such a notion. Sulbasutras denote fractional numbers by the general term 'bhaga'. All mathematical operations were as much valid for fractions as they were for numbers. Unlike Pythagoreans and Euclid, numbers were not regarded as atomic monads existing priori, but as entities whose being is brought forth and dependent on mathematical operations. It is only the outlook which considers mathematical operation as foundational that can bring negative numbers and zero into being. And this did happen in India. Zero and negative numbers were considered at par with other numbers and by the time Brahmagupta (7th century A.D.) explicit rules were formulated for arithmetical operations involving such numbers.
The stress on fundamentally of operations over being of numbers steered Indian mathematics clear of any problem with incommensurability. The Greeks accepted only those entities as numbers that were commensurable with unity, which was for them the principle of being out of which all numbers should arise In India surd numbers known as Karani were accepted as number in very early. Times14 and rules for handling them developed. Though rational - irrational classification did not develop but notion of exact (nitya) and inexact (anitya) numbers was developed.16 Instead of bothering about the nature of being of irrationals, procedures were developed for hhe calculation of these numbers. The Sulbasutras gave the value of √2 correctly to more than five decimal places. Inexactitude of a number for higher calculation was quite acceptable to the Indians.' Awareness of the inexactitude of π which to the Greeks was again a monadic ratio embodying in itself the fundamental truth of the circle, was evident to ancient Indians. Aryabhata gave the value of π= 62832/20000 = 3.1416 and more importantly he was aware of it inexactitude as were other Indian mathematicians. Constant revisions in the value of π occurred from time to time, which eventually lead to the discovery by Indian mathematicians of infinite series converging to the value of Π BY the 15th century. The popularity of Indian methods can be gauged by the fact that Al Kashi in 1427 explained how w can be calculated (correct upto 16 decimal places) by using Indian methods .
In Indian mathematics, Geometry never got idealized in the way it did in Greece. Solid geometry in India as in China was largely empirical. Indian plane geometry was essentially algebraic in character, but in its approach and concept it was quite different from Chinese efforts. What we refer to today as Trigonometry was quite developed in India by the 5th century A.D. Because of their geometric emphasis the Greeks used chords in their astronomical calculations, whereas the Indians developed the notion of sines and versines quite early. Aryabhata was perhaps the first to give a special name to these functions and to draw up a table of sines for each degree. Approximate algebraic expressions were also developed for these functions16. Indian influence flowed to medieval Arab mathematics where Al Battani (858 - 929 A. D.) used the sines regularly with a clear realisation of their superiority over Greek chords and introduced functions akin to the itangent and cotangent functions. Abdul-Wafa (940-998 AD) introduced the recant and cosecant.
In Indian mathematics the realm of numbers was regarded as homogeneous— homogeneous with respect to arithmetical operations. Anuyogadvara Sutra (100 BC) classified numbers into three categories (1) Sankhyata (numerable) (2) Asankhyata (innumerable) arid (3) Ahanta (infinite), and further classifies them into several classes. Implicit within this classification is the idea of homogeneity of numbers as the classification and is based on -the shades of meaning of finite and infinite. Idea that realm of numbers is homogeneous is behind the various rules of operation which are defined in Indian mathematics for positive as well as negative numbers, zero and even for infinity. In contrast the Greek realm of. numbers was colorful and heterogeneous. The Pythagorean classification of numbers into plane numbers, triangular, square numbers, solid numbers etc. was a scheme of classification based on the property of figurative being of numbers.
The idea of realm of numbers as homogeneous is an essential prerequisite for the development of algebra. Idea of provisionally indeterminate number or unknown or unaccomplished quantity can be meaningfully worked out only if its domain of activity is homogeneous. The central imagery behind mature Indian algebra (called Bija Ganita) was that if one sows a seed {bija) in the field of numbers (Ksetra) then one can harvest fruit (phala)—Bija and phafa were technical mathematical terms for the unknown and the solution. Idea of algebraic equations got developed very early in India. Sthananga sutra (300 BC) classified algebraic equations according to the number of unknown in it. Brahmagupta (628 A. D.) classified equations as (1) with one unknown including higher order equations (2) with many unknowns (3) with product of unknowns. Brahmagupta also extended the earlier theory of Aryabhata of Kuttaka, the analysis of indeterminate equations (where the solutionis restricted to integral or rational values only) which was to form a major part of Indian algebra in the later centuries.
Thus the heritage of algebra has been the hallmark of the mathematics of the Chinese and the Indians, and later of the Arabs—though the tradition of algebra was not a homogeneous one as the directionality of the mathematical enquiries of these civilizations was different despite the underlying commonality. In essence the Asian algebraic tradition is fundamentally distinct from the abstract geometric tradition of the Greeks- If in mathematics entities are regarded as having being or nonbeing then the existence criteria would require the method of indirect proof as it evolved in Greek tradition. But if entities are regarded as accomplished and no accomplished then the existence criteria would be constructability. Indeed Indian mathematics may be said to use 'algorithmic logic' in which the central principle is that of constructability.
Here we can see a homology at the foundational level between Indian mathematics, Indian logic and Indian linguistics. The main theoretical concern of Indian linguistics was constructability of accomplished entities of language. Neither did Indian linguistics regard its abstract constructs nor formal universal as real any more than Indian mathematics regarded its formal algebraic universals as real. Algorithmic logic begins from unaccomplished and ends in accomplished entity — so whatever formal apparatus it uses, such formal universal cannot claim being at par with accomplished entities. In fact the methodology of Indian linguistics as brought out even in the most ancient texts such as Rg Pratisakhya explicitly recognizes the importance of Apavada Sutra (exception rules) and Nipatana Sutra (counter-rules) and even goes further to stipulate as a metarule that the exception is stronger than the Samanya Sutra (universal rule) in cases where both are applicable. Thus the fallibility of the universal is accepted in principle as a metalogical truth, this is acknowledging that logical order and real order are not parallel and that the universal is a construct and can never posit itself in the world as infallibly real, arid existing a priori.17 This approach is fundamentally' different from the Platonic theory of ideas which held logic universals to be real and transcendent. It is on Platonic reform of Eleatic philosophy and logic that Euclid built his methodology 'which still gives foundational security to modern formalist and axiomatic logic and mathematics.
We now come to the transformation which took place in Europe around the 16th and 17th centuries. Underneath this transformation was an assimilation of knowledge within European folds from other cultures as well as birth of new intentionality in the development of knowledge. This intentionally makes European efforts different from the Arab, efforts, though from the 10th century onwards Arabs also assimilated knowledge of. different cultures in a big way and gave algebra a refined status of a separate discipline. While the new science of Europe assimilated knowledge of other medieval sciences its birth is marked by a certain polemical attitude towards these sciences. In place of acknowledging its debt to immediate parental tradition of knowledge, Europe saw itself arising out of natural foundations of distant Greek science. In the process it interpreted Greek mathematics in a way suited to its intentionality. All this had profound impact on concept formation in European mathematics. From 12th century onwards algebraic books of Arabic origin and later Greek books were translated into Latin and other European languages. But only in the 16th century the mathematical research of Europe ascends to different levels with a new awakening. Within a century, Viete (1540-1603), Stevin (1548-1620),[ Descartes (1596 - 1650) and Wallis (1616-1703), gave new foundation to mathematics largely depending on the Greek heritage for its method and on the Asian heritage for its content.
In the preface to Isagoge (subtitled : Introduction to Analytical Art) Viete the father of 'modern algebra' says, "Behold, the art which I present is new, but in truth so old, I so spoiled and defiled by the brabarians, that I considered it necessary, in order to introduce an entirely new form into it, to think out and publish a new vocabulary, having gotten rid of all its pseudo-technical terms (pseudo-categoreniatis) lest it should retain its filth and continue to stink in the old way. And yet underneath the Algebra or Afmucaba/a (of Arabian origin) which they lauded and called the great art, all mathematicians recognized that incomparable gold lay hidden, though they used to find little.18 The very expression 'algebra' belonged for Viete to barbarian pseudo-terminology. He sets forth developing a general analytic art where every number can be expressed by its 'species' (eidos) and the question 'how many?' can be left indeterminate provisionally for analytical purpose. Viete arrives at the conception of a mode of calculation which is carried out completely in terms of the species of numbers and calls it "logistice speciosa". 'Logistice speciosa' he applied to pure algebra, which he understood as most comprehensive possible analytic art, indifferently applicable to numbers and to geometric magnitudes. This new conception of eidos, because of algebraic concern, was ontologically independent (contrary to Greek use of eidos) and was mere symbolic formalism. Therefore the most important tool of mathematical natural science, the 'formala', first became possible. Here we see the metamorphosis of Greek ontological eidos to rationalistic dimensions. From now on ontological science of the Greeks is replaced by a symbolic discipline whose ontological presuppositions are left unclear. , What remain in mathematics are a method, Greek logic and Greek proof. The actualization of this method is sought in the realm of algebra which naturally transcends application part of mathematics, namely arithmetic and geometry in Viete's view. Pure analytic art of algebra is visualized as renewal of the Greek spirit. This idea of renewal played constitutional role for resurrection of the lost spirit in the form of analytic mathematical knowledge.19
Stevin, a contemporary of Viete and Kepler, is also possessed by the idea of a renewal and he casts it into the special image of a unknown 'wise Age' which must be brought back. Barbaric age for him extends from the beginning of the Greeks to the present. Stevin does not consider Greeks to be the only luminaries of the 'wise Age'. Vision of 'wise Age' is constructed out of elements of heritage of India, China, Arabs as well as Greeks. The Arabic digital and positional system in his view is immensely superior to Greek notations and is a heritage of 'wise age'. Algebra, which Europe became acquainted through Arbaic books and whose trace is not to be found in the Chaldaeans, the Hebrews, the Romans and the Greeks again hints at the 'wise Age'. Astronomy of Ptolemy, which has traces of perfect knowledge, is exemplary of near extinction of perfect understanding of the course of heaven whose other elements can be traced to the writings of the Arabic tongue. The heliocentric theory in particular is said to have had an origin of great antiquity. Alchemy again is a science of 'wise age' which he says was unknown to Greeks. Stevin is an interesting thinker in many ways, he wrote and taught mathematics, only in Flemish and held the title of professor of Dutch mathematics what differentiates Stevin from his contemporary thinkers of Europe is the peculiar bent of his cultural sensitivity which is evident from his bias in characterizing wise age'. On the other hand what unites him with his contemporaries is his view that right order must be used in writing and teaching of mathematics. Right order taught by the ‘wise age', is the mode of presentation of the Euclid’s Element, which he calls ‘order natural’. His projection of algebra as well as method of Euclid's Element into 'wise age' unfolded peculiar directions in his mathematics research.
Stevin totally rejected Greek concept of Arithmos as well as its modification in Viete. 'Unit’ to him is a number as much as other numbers rather than a primordial entity or even the beginning of numbers. For him, there are no absurd, irrational, irregular, inexplicable or surd numbers and these terms arise as a result of the schism between actualt mode of understanding of number and the traditional Greek concept of number for him even roots and powers are just plain numbers. This homogenization of numbers into one category was a typically Arabian, Indian and Chinese [notion of number as opposed to Greek Arithmos, a figurative conception of number. Only radical acceptance of Arabian tradition could open up possibility of such view of numbers which escaped grip of Viete who kept Greek notion of species of numbers. He argues that}number is not at all a discontinuous quantity as against the bias arising out of Greek indivisibility of the 'unit'. 'One' can be continuously divided into fractions. Continuous magnitude corresponds to continuous number much like water, which can be added and subtracted continuously. This identification of the realm of] numbers with continuum also led him to see new correspondence between geometry and algebra which was destined to culminate in Descartes Coordinate Geometry and a concept of real line. It is this correspondence (between Asian and Greek1 mathematics) which till today embodies a tension20. This new correspondence was based on a denial of specific nature of Greek corres-pondence at the same time assertion of its spirit in method.
A central feature in the emerging new directions was the notion of natural law. The notion of natural law was a distinct feature of European intentionality and can perhaps be traced back to the culture of Judeo-Christian tradition. The notion of natural law occurs .for instance in Thomas Aquinas (11th century) who thought God was just a prime mover (first cause, etc.,) of the universe, and the universe moves along preordained path and man's duty is to discover the preordained order, which is only accessible through reason, so that he can make this universe his just home. Greek notion of statis is reincarnated to represent static rational order .which has been left by God to be discovered by man. Here, we see there was ample amount of space for Platonism to make a comeback. And it did. Number of historians of science has analyzed resurrection or Platonism in 16th, 17th century Europe21. Stress on the study of space and harmony gradually gave way to deontological Platonism. a rationalist doctrine of methodological monism (Strees on uniqueness of method).
In the 16th century, Fransicus Patritius (1 93) argued that study of space must come before study of matter. Copernicus used mathematical harmony as an argument for his cosmology and wrote 'nothing unfitting [Geometrically] occurs in the course of their [planets] orbit". Kepler spoke of a system of five regular solids, which could be inserted between the sphere of six planets, as the cause of the planets being six in number. Galileo wrote that the ' book of nature is written in geometrical tradition". The phenomenal world imitated the rational order of primary qualities (qualities whose mathematical quantification is possible) and secondary qualities (experimental sensuous world) arise because of the effect of primary qualities on senses. The realm of knowledge now is the realm of primary qualities.
This understanding was in fact; neo-Platonic because primary qualities were not just geometrical magnitudes but also magnitudes of mass and time. Mass and time unlike spatial magnitude are algebraic quantities as they are not infinitely divisible but additive. The difference between infinite divisibility and additivity in my view is the fundamental distinction between space and time." It was this difference which was ignored in Western mathematics and time was geometrized on the plea that space as well as time is continuous quantities. But continuity itself has two meanings, one as continuous divisibility of space, and the other as continuous addivity of time. The fundamental difference between space and time which has affinity towards difference between geometric and algebraic views, was ignored in the analytical unification of these two trends by the West.
The analytic equivalence of algebra and geometry made possible handling of these disciplines at the same plane. However, the earlier tension between geometric and algebraic view has been transformed into, a tension between ideal and empirical mathematics in so far as intentionally of Europe saw geometric Greek tradition as a method and Asian algebraic tradition as an object of mathematics. This transformation was also forced in because the science of physiology of William Harvey, magnetism of Gilbert, and developments in Chemistry showed that logic of experiential science was not absolutely bound to mathematical expression, that there is a realm where mathematical necessity is not demonstrable. This brought about a tendency towards the deontologisation of mathematics and Science. The empirical and experiential streams of science was integrated in mathematical science by transforming the experiential into experimental which is governed by teleology of mathematics. In this process the ontologising aspect of 'Greek method was forgotten and the emphasis was put on method itself and later on epistemology. Mathematics as well was science developed along deontologised rationalism. The aim of mathematics became relation of abstract to abstract and eventually was reduced to the status of language for understanding of reality. A tendency towards formalism, logicism and axiomatic gained ground in Europe and by the end of 19th century and the beginning of 20th century this tendency reached its climax. Analytic art and new symbolism of Viete and Stevin became an end in itself. Mathematics became purely analytical and self-referential discipline. This deontologising of mathematics brought about separation of mathematical realm and physical | realm. It was thought that logic does not commit us to any apriori-assumption about physical reality. It constituted a realm which needs no empirical referent but stands alongside the experiential world and is potentially realizable as the true ground of experiential world.
But in deontologised mathematics what ensures the security of mathematical knowledge? Hilbert (1862-1943) thought that if only completeness and consistency2,2 _of.an axiomatic system can be proved then independent criteria for legitimization of mathematical knowledge with ontology-freeness is achieved. The discipline of Meta-Mathematics was born to demonstrate consistency and completeness of axiomatised Mathematics In ,1931, Kurt Godel disproved what Hilbert wanted to prove, by his [famous undesirability theorem.23 The immediate consequence of Godel's theorem" was that there exists no set of mathematical truths which if assumed, given, all the rest will be derived from them. In other words no formalized system will exhaust mathematical theory. Mathematics as a discipline is logically an open system. This- brought about severe strains on the rationalistic program for mathematics. Mathematics was not a discipline folding onto itself with self-referential completeness that forms a world of its own -either of a symbolic kind [or of a Platonic J realm of ideas. It has the same features as the rest of reality namely, that of open-endedness, of cross-referential growth and of
temporality.
The 19th and 20th century also saw the rise of Intuitionist school of mathematics as a trend critical of rationalist mathematics. Another reason for the rise of intuitionism was the discovery of unavoidable paradoxes, in the set theory .of Cantor (1845-1918). The intuitionists do not accept laws of logic as either a priori or eternal. Logic is subject to change. Logic which is valid for finite objects might not be valid for infinite objects like sets containing infinite elements. They do not accept actual infinity as an object of mathematics but only accept potential infinity as was maintained by Hegel, Locke and almost all empiricists. The law of excluded middle was unacceptable to them. This was a fundamental critique of .Rationalist mathematics, as rationalism had accepted Greek logic as the only kind of mathematical logic. For intuitionists, logic was not formal calculus but a methodology, logic of knowledge dealing with transformation of our inferences. Since mathematics is not derived from either logic-or, experience it must originate in a special intuition that presents us the basic concepts and inferences of mathematics as immediately clear and secure. Such basic notion is the notion of natural numbers. As L. Kronecker had said "God gave us the integers, all else are man's handiwork." The criteria now of legitimization of existence of mathematical truth are the principle of constructability. The sole admissible technique of demonstration of existence theorem is effective construction because it permits us to see what it is all about. On the other hand, the method of indirect proof does nothing but points to possibility of existence or of truth, without warranting it. Explicit and effective construction is possible only with finalistic procedures, that is, by means of finite number of signs and operations, as is the case for instance in the computation of the square of a number24. Hence, all propositions involving infinite classes regarded as totalities must be excluded from mathematics. Also theorems that are demonstrated in an indirect way must be eliminated or reconstructed (like most theorems of Cantor's theory of sets) 25. In the rationalist tradition, whose roots go back to Greeks, it was possible to prove the pure existence of objects that cannot be constructed (like most of the modern theorems, using method of indirect proof). On the other hand, intuitionists argued that by according such "ideal" objects the same validity-as real, or finitely cons-tractable objects, mathematics is being robbed away of its potency. Besides, Godel had shown that Hilbert’s rationalist scheme was doomed to failure.
But even after intuitionists, critique and Godel's proof, formalism flourishes in the West. Cantor's set theory and in general, pure-existence proofs are accepted in western mathematical culture, which today is not limited by geographical boundaries of the West but is practiced in India, China as well as in Greece. . Still we can say that this century is characterized by the resurgence of empirical mathematics in the West though it is a rebel tendency and, in miniscule minority. At least the tension which has all along been present in modern Western mathematics (between ideal mathematics and empirical mathematics) has to some extent come to surface today. It is my contention that this tension has its roots in the tension between Asian algebraic tradition and Greek geometric tradition. For the purposes of this study, we may still employ the categories well known in modern philosophical discourse, and summaries these two ideological positions, as they are represented in Modern Western mathematics in the following:
| Earlier manifestation of ideological tension Ontological inclination | Algebraic approach (Indian,Chinese) | Geometric approach (Greek) |
| Epistemology Phenomenal | Dynamis stress on becoming | Statis stress on being |
| inclination Existence criteria | Reasonableness (Empiricism) | Rationalism (Idealism) |
| Mathematical objects | Temporality Constructibility Finitistic, blur, fuzy. | Spatiality Indirect proof |
| Dominant Character of continuity | Continuous additivity | In-itself monads, need not be finitistic. Continuous divisibility |
While there has been a serious and ongoing debate on the foundations of mathematics in the West in recent times, it has had little or no impact on mathematical practice or on the development of mathematics, which have been totally conditioned by the formalist methodology derived from the Greek tradition. At the same time, very .little has been achieved by way of comprehending the non-Western mathematical' traditions as only very few of the original source-works have been brought to light or studied (that too mainly in the context of analyzing the European, assimilation 'of the non-European tradition) and even in these limited studies there has been no recognition of the fact that the non-European traditions had their own foundational conceptions and methodologies. This lack of comprehension of what may be called the cultural and philosophical roots of mathematics in different traditions, has totally crippled, our understanding of mathematics and its potentialities and calls for urgent and much more extensive investigations into the foundational conceptions and methodologies of mathematics in different traditions."
Bibliography:
- Africa, Thomas W ; Science and f State in Greece and Rome.
- Bag, A. K: Mathematics in Ancient and Medivial India-Chaukhambha Orientalia, Varanasi, 1979.
- Bishop, Erret: Foundations of Constructivist Analysis, 1967.
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- Dutta B. and Singh A. N. : History of Hindu Mathematics 2 Vols. Motilal Banarsidas, 1935.
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- Lokatos, Imre : Mathematics, Sceince and Epistemology.
- Mikami, Y The Development of Mathematics in China and Japan. Feubner, Leipzig, 1913.
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Author: Navjyoti Singh,
Note:
1. " Most of the historians of mathematics agree about this point. Greeks solved difficult algebraic problems in purely geometric way, ex: xS + ab=b2. On the other hand, problems concerning right angled triangle were exhaustively solved by Indians and Chinese in purely algebraic way.
2. Method of indirect proof is also known as method of reduction ad absurdum. Following are the assumptions behind this method :
a) A statement has a singular negation, important corollary of this principle is principle of double negation.
b) If by assuming 'not A' to be true logical inconsistency can be logically demonstrated, then "A-is demonstrated to be true. This method is widely used in modern mathematics.
3. Link between mathematics and philosophy can be ascertained from certain ' crucial Greek categories, namely, marijam-learning matter, ma thesis- study discipline, matbexis- participation and characterization of mathematical truth as mathematic (as thing to be learned).
4. This is the view of most of the historians of Greek mathematics. About Greek geometry 0. NEUGE-BAUER AND ARPAD SZABO and about algebra JACOB-KLEIN have substantiated this view.
5. This doctrine of Pythagoras remained fundamental to the whole of Greek mathematics but was opposed by musicians. For instance, Aristoxenus wrote, "We are attempting to draw conclusions which are in agreement with the experience as opposed the theorists who preceded us. Some of them introduced completely foreign view-point into the subject and dismissed sensory experience as imprecise ; hence they made up intelligible causes and stated that there were certain ratios between numbers and speeds on which the pitch of a note depended. These were all speculations which are completely foreign to the subject and absolutely contrary to appearances", {quoted by SZABO—p. 113). The theorists were the Pythagoreans and their empiricist critique by Aristoxenus had little Impact on Greek mathematics. This tension remained till the time of Plato who sided with Pythagoreans, in denouncing musicians as immature in the debate over the smallest musical interval 'coma'. This debate has been dealt by WILLIAM WHEWELL, Today we know with certainty that some of these musical notes are actually unratioable incommensurable magnitudes and hence the empiricist critique was sound.
6. We may give an example of Eleatic usa of this method to refute Aneximenes. Eleatic assertion was: 'what is, cannot have 'come to being' Proof: Suppose that what is did come to being (assuming negation to! be true,) _then it could only .have come from what is, or from what is not (only mutually exclusive singular negation admitted). There is no third possibility (principle of excluded middle). Now if it had come from what is. rt would already have been existent before it came into being hence to say that it came into being in this way would make no sense. If (on the other hand,' the claim is made that what «came from what is not. this leads immediately to a contradiction {adynat'on) (principle Of consistency). What is can never have been the opposite of itself, what is not, and hence, could not have come into being this way either, ' Hence the assertion is proved (again the principle of excluded middle is used).
7. Vet another contribution of Eleatic philosophy to future Greek Mathematics was the notion of hypothesis. According to Eleatic view, no empirical test can be devised to check whether or not "an assertion" is correct since all physical experience, must in principle be rejected as false-or unreliable. Hence, assertions have to be taken up and .tested in reason. The assertions now have a status of hypothesis. The consequence of' any hypothesis could be examined by their method to see if they lead to contradiction. And if contradiction does not follow the hypothesis stands tested. Likewise, most of the notions in axiomotised mathematics have their root in idealist philosophy of Parmenides and his followers.
8. Jacob Klein, p. 313.
9. This understanding of causality later played important constitutive role for evolution of Kepler's laws of celestial mechanics.
10. The principle of place value is the realization that if divisor is 10 or a power of 10, it is unnecessary to divide as it can be taken care of by shifting the position of numerals with respect to each other. Modern way of writing number uses this scheme whereas the latin numerals do not,
11. Chinese equations ended with implicitly understood ' =: -f- n', n being a constant term. In the early medieval period, it ended having ' = 0', the constant term joining rest of the terms as '-n'.
12. Pascal's triangle is constituted out of coefficients of expansion of (X + Y)n for positive integer n. Arabs as well as Indians also knew of pascal's triangle quite early. In 1252 Nasir al-Din al-Tusi wrote in detail about as pension of (X + Y)n. He explained the method as a current practice, not as a new invention.
13. Datta (1935) p. 9. Later Buddhist text Lalitavistara {100 BC) mentions 'tallaksana' = 1058 and quotes from earlier Jain literature a word numeral 'Sirsaprahe//ka' = order of 102c0.
14. It is important to note that the term- Karani is used for surd root and quantity of root could be rational or irrational. For instance Apastamba Sulbasutra uses chatuskarani = 4 for the number 2.
15. Apastamba Sulbasutra makes this distinction while dealing with squaring of a circle. In later literature terms like 'sthulamana or sthulaganana were used for approximate value or approximate calculation.
16. Sines and" other trigonometrically numbers are not usual numbers. They are expressible only as summation of infinite series (which were to be later discovered by the Indian mathematicians, who kept on calculating approximate values of these functions and improving on them.
17. For a more-detailed discussion of the foundational methodology of Indian linguistics and its similarity with the methodology of Indian mathematics, see Navjyoti Singh (1964)
18. Translation from Latin to English of Viete's Isagoge is included in the appendix of JACOB KLEIN'S book.
19. Incidentally Viete did not choose sides between Ptolemy and Copernicus as he saw between them no fatal tension. On the one hand, he remained committed to Ptolemy's astronomy and on the other he accepted the Copernicus's thesis because it retains the method of Ptolemy's science.
20. See for instance. JOHN MYHILL, "What is real number?" Amer. Math. Monthly, Vol. 79, # 7 P. 748-754, Aug. Sept., 1972. Does every point on real line correspond to a number? If principle of constructivism is used then real I line is full of holes. Most of the real line is non-algebraic numbers called transendental number and there is no method for constructing them.
21. A. Koyre and E. A. Burtt have explicity taken this position.
22. Criterion of consistency: There exists no statement in the axiomatic scheme which can be proved as well as disproved from within. Criterion of completeness : There exists no statement constituted out of objects of the axiomatic scheme which can neither be proved nor disproved from within the system. That is there exists no undividable proposition in the axiomatic scheme.
23. For a 'simple' statement of Godel's Theorem, let us quote the following from 'The Encyclopedia of Philosophy" Ed. Paul Edwards, Mcgraw Hill 1967:
By Godel's theorem the following statement is generally meant : In any formal system adequate for number theory there exists an undividable formula — that is a. formula that is not provable and where negation is not provable. A corollary to the theorem is that the consistency of d formal system adequate for number theory cannot be proved within the system'.
24. Recently another constructionist trend came up with the publication in 1967 of a book by Erret Bishop. Bishop was not much interested in founding mathematics as the intuitionists were but with doing mathematics constructively. Bishop tried to reformulate Cantor's theory by avoiding "idealistic" notions and continually checking the objects and theorems generated for intuitive meaning.
25. One classic instance of pure existence proof is Cantor's proof of infinite number of transcendental (non-algebraic) numbers, which does not give the slightest hint of how even a single transcendental number might be found. It proves existence of such numbers by proving that it would be contradictory for them not to exist. Pure existence proofs now pervade modern mathematics, like, it can be proved that a differential equation has a unique solution without showing a way to reach that solution. Intuitionists and constructivists are critical of such kind of mathematics. 69
