Mathematic — An invention of European scholars?


Mathematics has long been considered an invention of European scholars, as a result of which the contributions of non-European countries have been severely neglected in histories of mathematics. Worse still, many key mathematical developments have been wrongly attributed to scholars of European origin. This has led to so-called Eurocentrism. The neglect of non-European mathematics has become more apparent when studying the discoveries and developments in the Arab sub-continent. Contrary to Eurocentric belief, scholars from Arab and Asia, over a period of some 4500 years, contributed to some of the greatest mathematical achievements in the history of the subject. The peak of Arab mathematics took its peak from 9th century to the 14th century. During this period many key developments took place such as Numerical System, Algorithm, Trigonometry, Combinatory, Geometry and many other. In addition to mighty contributions to all the principal areas of mathematics, Arab scholars were responsible for the creation, and refinement of Algebra, including the number zero, without which higher mathematics would not be possible. The purpose of my project is to highlight the major mathematical contributions of Arab scholars and explain why the Eurocentric ideal is an injustice.


Introduction

The history of science, and specifically mathematics, is a vast topic and one which can never be completely studied as much of the work of ancient times remains undiscovered or has been lost through time. Nevertheless many of the contributions are known and many discoveries have been developed, especially over the last 150 years, which have significantly altered the sequence of the history of mathematics. It is fair to say that by the turn of 21st century many of the key developments were known and there was definite knowledge when these developments took place.

Recent research paints a new picture of the debt that we owe to Arabic/Islamic mathematics. Certainly many of the ideas which were previously thought to have been brilliant new conceptions due to European mathematicians of the sixteenth, seventeenth and eighteenth centuries are now known to have been developed by Arabic/Islamic mathematicians around four centuries earlier. In many respects the mathematics studied now a day is mostly Arab mathematics.

There is a widely held view that, after a brilliant period for mathematics when the Greeks laid the foundations for modern mathematics, there was a period of stagnation before the Europeans took over where the Greeks left off at the beginning of the sixteenth century. The common perception of the period of 1000 years or so between the ancient Greeks and the European Renaissance is that little happened in the world of mathematics except that some Arabic translations of Greek texts were made which preserved the Greek learning so that it was available to the Europeans at the beginning of the sixteenth century.

That such views should be generally held is of no surprise. Many leading historians of mathematics have contributed to the perception by either omitting any mention of Arabic/Islamic mathematics in the historical development of the subject or with statements such as that made by Duhem in [3]:-

… Arabic science only reproduced the teachings received from Greek science.

Before we proceed it is worth trying to define the period that this article covers and give an overall description to cover the mathematicians who contributed. The period we cover is easy to describe: it stretches from the end of the eighth century to about the middle of the fifteenth century.

I became drawn to the topic of mathematics, as there appeared to be a distinct and inequitable neglect of the contributions of the sub-continent. Thus, during the course of this project I aim to discuss that despite slowly changing attitudes there is still an ideology’ which plagues much of the recorded history of the subject. That is, to some extent very little has changed even in our seemingly enlightened historical and cultural position, and, in specific reference to my study area, many of the developments of mathematics remain almost completely ignored, or worse, attributed to scholars of other nationalities, often European.

It is important for me to clarify at this point that the ideology I refer to is that held by (predominantly) European historians of science and mathematics, that mathematics is a European ‘invention’. This ideology leads to an ‘intrinsically’ Eurocentric bias to the history of the subject. As R Rashed comments the ideology can be summarised as such:

…Classical science is European and its origins are directly traceable to Greek philosophy and science. [RR, P 332]

Thus despite the many discoveries that have been made there has been a great reluctance to acknowledge the contributions of non-European scholars. For several hundred years following the European mathematical renaissance of the late 15th and early 16th century there was a commonly held opinion among commentators and historians that mathematics originated in its entirety from Europe and European scholars.

The basic chronology of the history of mathematics was very simple; it had primarily been the invention of the ancient Greeks, whose work had continued up to the middle of the first millenium A.D. Following which there was a period of almost 1000 years where no work of significance was carried out until the European renaissance, which coincided with the ‘reawakening’ of learning and culture in Europe following the so called dark ages.


Figure 2.1: Eurocentric chronology of mathematics history.

Some historians made some concessions, by acknowledging the work of Egyptian, Babylonian, Indian and Arabic mathematicians. Modified versions of the Eurocentric model commonly took the form seen below.

Figure 2.2: Modified Eurocentric model.

However, references to the work of these ‘others’ or ‘non-Europeans’ were always brief and hazy, and generally concluded that they were merely reconstructions of Greek works and that nothing of significance or importance was contained in them.

As said this is a vast topic area and although all non-European ‘roots’ of mathematics have suffered neglect and miss-representation by many historians I am not going to attempt to focus on Arab sub-continent of mathematical development. I have chosen to focus on the mathematical developments of the Arab sub-continent, as I consider them not only to be severely neglected in histories of mathematics, but also to have produced some of the most remarkable results of mathematics. Indeed, the research I have conducted has highlighted that many Arab mathematical results, beyond being simply remarkable because of the time in which they were derived, show that several ‘key’ mathematical topics, and subsequent results, indubitably originate from the Arab sub-continent.

The aim of my work is not just to paint an accurate picture of the developments of mathematics by Arab Scholars, but also to attempt to give reasons as to why Europe has chosen to neglect the facts of history.

There are several points that I feel it is important to make before progressing further with my discussion. The first is to make it clear that the chronology of the history of mathematics is not entirely linear. One will often find simplified diagrammatic representations where the work of one group of people (or country) is proceeded by the work of another group and so on. In reality things are far more complicated than this. Particularly in the European dark ages (5th-15th centuries) mathematical developments passed between several countries, being constantly refined and improved. G Joseph states:

… A variety of mathematical activity and exchange between a number of cultural areas went on while Europe was in a deep slumber…

Figure 2.3: Non-European mathematics during the dark ages.

While this is unavoidable it is vital to appreciate the uniqueness and ingenuity of the developments of non-European scholars, even if the results are common-place now.

We also view mathematics from an intrinsically European standpoint due largely to the influence of European scholars over the last 500 years and the colonisation of much of the world by European countries. Perhaps this is why many historians find it hard to accept that many results and developments of mathematics are not European in origin. In short, if we are European, somewhat unavoidably we view history from our indigenous standpoint.

I will now commence with the main body of my work, a discussion of the development of mathematics. I hope to highlight the many remarkable discoveries, and results, and where neglect or incorrect analysis has occurred.


What encouraged the accumulation of knowledge?

The Qu’ran itself encouraged the accumulation of knowledge, and a Golden Age of Islamic science and mathematics increased throughout the medieval period from the 9th to 12th Centuries. The first step to success was the setup of The House of Wisdom in Baghdad around 810, and work started almost instantly on translating the major work of Greek and Indian mathematicians into Arabic.

As George Sarton, (Harvard historian of science) wrote in his book Introduction to the History of Science:

…From the second half of the eighth to the end of the eleventh century, Arabic was the scientific, the progressive language of mankind. When the West was sufficiently mature to feel the need of deeper knowledge, it turned its attention, first of all, not to the Greek sources, but to the Arabic ones…. [17]


Translation Period

While their knowledge of what came before them was incomplete, the mathematicians of the Islamic era were aware of ideas, methods and points of view that originated in India, Persia, and especially Greek. A remarkable translation movement joined with a scholarly tradition of writing commentaries on previous works meant that mathematicians of this era were comfortable with the contents and the methodology of the works of, Euclid, Archimedes, Apollonius, Ptolemy, and Diophantus as well as the basics of Indian decimal arithmetic and trigonometry. They accomplished a great deal with this heritage. What the mathematicians of the Islamic era gave to those who came later was very different in content, style, and approach than what had come before them. [18]

In the twelfth century, Europe became aware of the scientific achievements of the Arabs and embarked upon serious translations of their rich legacy. A special college for translators was founded in Toledo, Spain, and it was there, and in other centres, that some of the great Christian scholars translated most of the Arabic works on mathematics and astronomy. In most European universities Arab work formed the basis of mathematical studies.

Translation Period began under the Caliph Harun al-Rashid, the fifth Caliph of the Abbasid dynasty, whose reign began in 786. He encouraged scholarship and the first translations of Greek texts into Arabic, such as Euclid’s Elements by al-Hajjaj, were made during al-Rashid’s reign. The next Caliph, al-Ma’mun, encouraged learning even more strongly than his father al-Rashid, and he set up the House of Wisdom in Baghdad which became the centre for both the work of translating and of research. Al-Kindi (born 801) and the three Banu Musa brothers worked there, as did the famous translator Hunayn ibn Ishaq. [19]

We should emphasise that the translations into Arabic at this time were made by scientists and mathematicians such as those named above, not by language experts ignorant of mathematics, and the need for the translations was stimulated by the most advanced research of the time. It is important to realise that the translating was not done for its own sake, but was done as part of the current research effort. The most important Greek mathematical texts which were translated are listed in [17]:-

…Of Euclid’s works, the Elements, the Data, the Optics, the Phaenomena, and On Divisions were translated. Of Archimedes’ works only two — Sphere and Cylinder and Measurement of the Circle — are known to have been translated, but these were sufficient to stimulate independent researches from the 9th to the 15th century. On the other hand, virtually all of Apollonius’s works were translated, and of Diophantus and Menelaus one book each, the Arithmetica and the Sphaerica, respectively, were translated into Arabic. Finally, the translation of Ptolemy’s Almagest furnished important astronomical material….

The more minor Greek mathematical texts which were translated are also given in [17]:-

Diocles’ treatise on mirrors, Theodosius’s Spherics, Pappus’s work on mechanics, Ptolemy’s Planisphaerium, and Hypsicles’ treatises on regular polyhedra (the so-called Books XIV and XV of Euclid’s Elements) …


Al Khwarizmi and His Successors- Forgotten Figures?

The outstanding Persian mathematician Muhammad Al-Khwarizmi was an early Director of the House of Wisdom in the 9th Century, and one of the greatest of early Muslim mathematicians. Perhaps Al-Khwarizmi’s most important contribution to mathematics was his strong advocacy of the Hindu numerical system (1–9 and 0), which he recognised as having the power and efficiency needed to revolutionise Islamic (and, later, Western) mathematics, and which was soon adopted by the entire Islamic world, and later by Europe as well.

Al-Khwarizmi’s other important contribution was algebra, and he introduced the fundamental algebraic methods of “reduction” and “balancing” and provided an exhaustive account of solving polynomial equations up to the second degree. In this way, he helped create the powerful abstract mathematical language still used across the world today, and allowed a much more general way of analysing problems other than just the specific problems previously considered by the Indian’s and Chinese.

Perhaps one of the most significant advances made by Arabic mathematics began at this time with the work of al-Khwarizmi, namely the beginnings of algebra. It is important to understand just how significant this new idea was. It was a revolutionary move away from the Greek concept of mathematics which was essentially geometry.

Algebra was a unifying theory which allowed rational numbers, irrational numbers, geometrical magnitudes, etc., to all be treated as “algebraic objects”. It gave mathematics a whole new development path so much broader in concept to that which had existed before, and provided a vehicle for future development of the subject. Another important aspect of the introduction of algebraic ideas was that it allowed mathematics to be applied to itself in a way which had not happened before. As Rashed writes in [11] (see also [10]):-

Al-Khwarizmi’s successors undertook a systematic application of arithmetic to algebra, algebra to arithmetic, both to trigonometry, algebra to the Euclidean theory of numbers, algebra to geometry, and geometry to algebra. This was how the creation of polynomial algebra, combinatorial analysis, numerical analysis, the numerical solution of equations, the new elementary theory of numbers, and the geometric construction of equations arose.

Let us follow the development of algebra for a moment and look at al-Khwarizmi’s successors. About forty years after al-Khwarizmi is the work of al-Mahani (born 820), who conceived the idea of reducing geometrical problems such as duplicating the cube to problems in algebra. Abu Kamil (born 850) forms an important link in the development of algebra between al-Khwarizmi and al-Karaji. Despite not using symbols, but writing powers of x in words, he had begun to understand what we would write in symbols as xn.xm = xm+n. Let us remark that symbols did not appear in Arabic mathematics until much later. Ibn al-Banna and al-Qalasadi used symbols in the 15th century and, although we do not know exactly when their use began, we know that symbols were used at least a century before this.

Al-Karaji (born 953) is seen by many as the first person to completely free algebra from geometrical operations and to replace them with the arithmetical type of operations which are at the core of algebra today. He was first to define the monomials x, x2, x3, … and 1/x, 1/x2, 1/x3, … and to give rules for products of any two of these. He started a school of algebra which flourished for several hundreds of years. Al-Samawal, nearly 200 years later, was an important member of al-Karaji’s school. Al-Samawal (born 1130) was the first to give the new topic of algebra a precise description when he wrote that it was concerned:-

… with operating on unknowns using all the arithmetical tools, in the same way as the arithmetician operates on the known.

Omar Khayyam (born 1048) gave a complete classification of cubic equations with geometric solutions found by means of intersecting conic sections. Khayyam also wrote that he hoped to give a full description of the algebraic solution of cubic equations in a later work [18]:-

If the opportunity arises and I can succeed, I shall give all these fourteen forms with all their branches and cases, and how to distinguish whatever is possible or impossible so that a paper, containing elements which are greatly useful in this art will be prepared.

Sharaf al-Din al-Tusi (born 1135), although almost exactly the same age as al-Samawal, does not follow the general development that came through al-Karaji’s school of algebra but rather follows Khayyam’s application of algebra to geometry. He wrote a treatise on cubic equations, which [11]:-

… represents an essential contribution to another algebra which aimed to study curves by means of equations, thus inaugurating the beginning of algebraic geometry.

Let us give other examples of the development of Arabic mathematics. Returning to the House of Wisdom in Baghdad in the 9th century, one mathematician who was educated there by the Banu Musa brothers was Thabit ibn Qurra (born 836). He made many contributions to mathematics, but let us consider for the moment consider his contributions to number theory. He discovered a beautiful theorem which allowed pairs of amicable numbers to be found, that is two numbers such that each is the sum of the proper divisors of the other. Al-Baghdadi (born 980) looked at a slight variant of Thabit ibn Qurra’s theorem, while al-Haytham (born 965) seems to have been the first to attempt to classify all even perfect numbers (numbers equal to the sum of their proper divisors) as those of the form 2k-1(2k — 1) where 2k — 1 is prime.

Al-Haytham, is also the first person that we know to state Wilson’s theorem, namely that if p is prime then 1+(p-1)! is divisible by p. It is unclear whether he knew how to prove this result. It is called Wilson’s theorem because of a comment made by Waring in 1770 that John Wilson had noticed the result. There is no evidence that John Wilson knew how to prove it and most certainly Waring did not. Lagrange gave the first proof in 1771 and it should be noticed that it is more than 750 years after al-Haytham before number theory surpasses this achievement of Arabic mathematics.

Continuing the story of amicable numbers, from which we have taken a diversion, it is worth noting that they play a large role in Arabic mathematics. Al-Farisi (born 1260) gave a new proof of Thabit ibn Qurra’s theorem, introducing important new ideas concerning factorisation and combinatorial methods. He also gave the pair of amicable numbers 17296, 18416 which have been attributed to Euler, but we know that these were known earlier than al-Farisi, perhaps even by Thabit ibn Qurra himself. Although outside our time range for Arabic mathematics in this article, it is worth noting that in the 17th century the Arabic mathematician Mohammed Baqir Yazdi gave the pair of amicable number 9,363,584 and 9,437,056 still many years before Euler’s contribution.

Let us turn to the different systems of counting which were in use around the 10th century in Arabic countries. There were three different types of arithmetic used around this period and, by the end of the 10th century, authors such as al-Baghdadi were writing texts comparing the three systems.

Although the Arabic mathematicians are most famed for their work on algebra, number theory and number systems, they also made considerable contributions to geometry, trigonometry and mathematical astronomy. Ibrahim ibn Sinan (born 908), who introduced a method of integration more general than that of Archimedes, and al-Quhi (born 940) were leading figures in a revival and continuation of Greek higher geometry in the Islamic world. These mathematicians, and in particular al-Haytham, studied optics and investigated the optical properties of mirrors made from conic sections. Omar Khayyam combined the use of trigonometry and approximation theory to provide methods of solving algebraic equations by geometrical means.

Prior to the use of “Arab” numerals, as we know them today, the West relied upon the somewhat clumsy system of Roman numerals. Whereas in the decimal system, the number 1948 can be written in four figures, eleven figures were needed using the Roman system: MDCCCXLVIII. It is obvious that even for the solution of the simplest arithmetical problem, Roman numerals called for an enormous expenditure of time and labour. The Arab numerals, on the other hand, rendered even complicated mathematical tasks relatively simple.

The scientific advances of the West would have been impossible had scientists continued to depend upon the Roman numerals and been deprived of the simplicity and flexibility of the decimal system and its main glory, the zero. Though the Arab numerals were originally a Hindu invention, it was the Arabs who turned them into a workable system; the earliest Arab zero on record dates from the year 873, whereas the earliest Hindu zero is dated 876. For the subsequent four hundred years, Europe laughed at a method that depended upon the use of zero, “a meaningless nothing.”
 Had the Arabs given us nothing but the decimal system, their contribution to progress would have been considerable. In actual fact, they gave us infinitely more. While religion is often thought to be an impediment to scientific progress, we can see, in a study of Arab mathematics, how religious beliefs actually inspired scientific discovery.
 The mathematics that the Arabs had inherited from the Greeks made such a division extremely complicated, if not impossible. It was the search for a more accurate, more comprehensive, and more flexible method that led Khawarazmi to the invention of algebra. According to Professor Sarton, Khawarazmi

…influenced mathematical thought to a greater extent than any other medieval writer…

Both algebra, in the true sense of the term, and the term itself (al-jabr) we owe to him. Apart from mathematics, Khawarazmi also did pioneer work in the fields of astronomy, geography and the theory of music.


Numeral System

Prior to the use of “Arab” numerals, as we know them today, the West relied upon the somewhat clumsy system of Roman numerals. Whereas in the decimal system, the number 1948 can be written in four figures, eleven figures were needed using the Roman system: MDCCCXLVIII. It is obvious that even for the solution of the simplest arithmetical problem, Roman numerals called for an enormous expenditure of time and labour. The Arab numerals, on the other hand, rendered even complicated mathematical tasks relatively simple.

The scientific advances of the West would have been impossible had scientists continued to depend upon the Roman numerals and been deprived of the simplicity and flexibility of the decimal system and its main glory, the zero. Though the Arab numerals were originally a Hindu invention, it was the Arabs who turned them into a workable system; the earliest Arab zero on record dates from the year 873, whereas the earliest Hindu zero is dated 876. For the subsequent four hundred years, Europe laughed at a method that depended upon the use of zero, “a meaningless nothing.” Had the Arabs given us nothing but the decimal system, their contribution to progress would have been considerable. In actual fact, they gave us infinitely more.


The Decimal System and the Concept of Number

For Euclid — the preeminent mathematician of Greek Alexandria — “number” meant a rational number. In his work, irrational numbers were called magnitudes and were treated quite differently from numbers. In fact, Euclid’s very influential book Elements contains few numbers and hardly any calculations. Starting with Khwarizmi of Khwarizm (c. 780–850 c.e.), the principles of the positional decimal system that had originally come from India were organized and widely disseminated. Hence, with the use of 10 symbols it was possible to carry out all arithmetic operations. Over the following centuries, the methods for these arithmetical operations were improved and included working with decimal fractions and with large numbers. In fact, in the process, the Euclidean concept of number was gradually enlarged to include irrational numbers and their representation as decimal fractions. The mathematician Kashani (c. 1380–1429), also known as al-Kashi, worked comfortably with irrational numbers and, for example, was able to produce an approximation that was correct to 16 decimal places. The Arabic texts on the decimal number system were translated to Latin and were the basis for what are now called the Hindu-Arabic numerals.


Algebra

While it is possible to recognize algebraic problems in ancient mathematics, algebra as a discipline distinct from geometry and concerned with solving of equations was developed during the Islamic period. The first book devoted to the subject was Khwarizmi’s Al-kitab al-muhtasar fi hisab al-jabr wa-l-muqabala (Compendium on Calculation by Completion and Reduction).

In this title, “al-jabr” — the origin of the word “algebra” — means “restoration” or “completion” and refers to moving a negative quantity to the other side of an equation where it becomes positive. Al-muqabala means “comparison” or “reduction” and refers to the possibility of subtracting like terms from two sides of an equation. While all algebra problems were stated and solved using words and sentences — symbolic algebra did not arise until much later in the fifteenth century in Italy — an algebra of polynomials was developed by Abu Kamil (c. 850–930), Karaji (c. 953–1029), and Samu’il Maghribi (c. 1130­–1180+, also known as al-Samaw’al). Powers, even negative powers, of unknowns were considered and many algebraic equations were classified and solved. Khwarizmi gave a full account of second-degree equations, and Khayyam (1048–1131) gave a geometric solution to equations of degree three using conic sections. Here, we give a problem — translated to modern notation — solved by Abu Kamil. Some three hundred years later, this exact same problem appeared in Chapter 15 of the 1202 text Liber Abaci by Leonardo Fibonacci. Abu Kamil gave a solution to the following system of three equations and three unknowns:

x + y + z = 10
x2 + y2 = z2
xz = y2

Abu Kamil first started with the choice of x = 1 and solved the latter two equations for y and z. Since, for the latter two equations, any scalar multiple of the solutions continues to be a solution, he then scaled the solutions so that the first equation was also satisfied. He simplified the answer to get:


Geometry

Geometrical methods and problems were ubiquitous in the Islamic era. While algebraic problems were solved using the newly developed algebraic algorithms (the word “algorithm” itself is derived from algorismi, the Latin version of the name of the mathematician al-Khwarizmi), the justification for the algebraic methods was usually given using geometrical arguments and often relying on a distinctively Euclidean style. Guided by problems in astronomy and geography (for example, finding, from any place on Earth, the direction of Mecca for the purpose of the Islamic daily prayers), spherical geometry was developed.

But new work in plane geometry was also carried out. Khayyam and Nasir al-din Tusi (1201–1274), for example, studied the fifth postulate of Euclid and came close to ideas that much later on led to the development of non-Euclidean geometries in Europe. However, as is the case with much of the mathematics of this era, applications play an important role in the choice of questions and problems.

For example, Abu’l Wafa Buzjani (940–997) reports on meetings that included mathematicians and artisans. A problem of interest to tile makers is how to create a single square tile from three tiles. A traditional mathematician, Abu’l Wafa explains, translates this problem into a ruler and compass construction and gives a method for constructing a square of side.

While logically correct, this construction is of little use to the tile maker, who is confronted with three actual tiles and wants to cut and rearrange them to create a new tile. Abu’l Wafa also gives the customary practical method that is actually used by tile makers to solve this problem, and proves that their method, while practical, is not precise, and the final object is not exactly a square. While stressing the importance of being both practical and precise, and the virtues of Euclidean proofs, he presents his own practical and correct methods for solving this and related problems.


Trigonometry

The origins of trigonometry begin with the Greek study of chords as well as the Indian development of what is now called the “sine function.” Claudius Ptolemy’s table of chords and Indian tables of sine values were powerful tools in astronomy. However, a systematic study and use of all the trigonometric functions motivated by applications to astronomy, spherical geometry, and geography begins in the Islamic era. Abu’l Wafa had a proof of the addition theorem for sines and used all six trigonometric functions; Abu Rayhan Biruni (973­–1048) used trigonometry to measure the circumference of Earth; and Nasir al-din Tusi gave a systematic treatment in his Treatise on the Quadrilateral that helped establish trigonometry as a distinct discipline.


Combinatorics

One of the earlier known descriptions and uses of the table of binomial coefficients (also known as the Pascal triangle) is that of Karaji. While his work on the subject is not extant, his clear description of the triangle survives in the writings of Samu’il Maghribi. Binomial coefficients were used extensively, among other applications, for extracting roots. Kashani, for example, used binomial coefficients to give an algorithm for extracting fifth roots. He demonstrated it by finding the fifth root of 44,240,899,506,197. Other combinatorial questions were treated as well. Ibn al-Haytham (c. 965–1039, also known as Alhazen) gave a construction of magic squares of odd order, and Ibn Mun’im (died c. 1228) devotes a whole chapter of his book Fiqh al-Hisab to combinatorial counting problems.


Numerical Mathematics

The prominence of applied problems, the development of Hindu-Arabic numerals and calculation schemes, and the development of algebra and trigonometry led to a blossoming of numerical mathematics. One prime example is Kashani’s Miftah al-Hisab or Calculators’ Key. In addition to his approximation of:

and his extraction of fifth roots, he also gave an iterative method for finding the root of a third-degree polynomial in order to approximate the sine of one degree to as close as an approximation as one wishes.

Recent research paints a new picture of the debt that we owe to Arabic/Islamic mathematics. Certainly many of the ideas which were previously thought to have been brilliant new conceptions due to European mathematicians of the sixteenth, seventeenth and eighteenth centuries are now known to have been developed by Arabic/Islamic mathematicians around four centuries earlier. In many respects the mathematics studied today is far closer in style to that of the Arabic/Islamic contribution than to that of the Greeks.

There is a widely held view that, after a brilliant period for mathematics when the Greeks laid the foundations for modern mathematics, there was a period of stagnation before the Europeans took over where the Greeks left off at the beginning of the sixteenth century. The common perception of the period of 1000 years or so between the ancient Greeks and the European Renaissance is that little happened in the world of mathematics except that some Arabic translations of Greek texts were made which preserved the Greek learning so that it was available to the Europeans at the beginning of the sixteenth century.


Conclusion

I wish to conclude initially by simply saying that the work of Arab mathematicians has been severely neglected by western historians, although the situation is improving somewhat. What I primarily wished to tackle was to answer two questions, firstly, why have Arab works been neglected, that is, what appears to have been the motivations and aims of scholars who have contributed to the Eurocentric view of mathematical history. This leads to the secondary question, why should this neglect be considered a great injustice.

I have attempted to answer this by providing a detailed investigation (and analysis) of many of the key contributions of the Indian subcontinent, and where possible, demonstrate how they pre-date European works (whether ancient Greek or later renaissance). I have further developed this ‘answer’ by providing significant evidence that a number of Arab works conversely influenced later European works. I have also included a discussion of the Arab numeral system which is undoubtedly the greatest Arab contribution to the development of mathematics, and its wider applications in science, economics (and so on).

Discussing my first ‘question’ is less easy, as within the history of mathematics we find a variety of ‘stances’. If the most extreme Eurocentric model is ‘followed’ then all mathematics is considered European, and even less extreme stances do not give full credit to non-European contributions.

Indeed even in the very latest mathematics histories Arab ‘sections’ are still generally fairly brief. Why this attitude exists seems to be a cultural issue as much as anything. I feel it is important not to be controversial or sweeping, but it is likely European scholars are resistant due to the way in which the inclusion of non-European, including Arab, contributions shakes up views that have been held for hundreds of years, and challenges the very foundations of the Eurocentric ideology. Perhaps what I am trying to say is that prior to discoveries made in technically fairly recent times, and in some cases actually recent times it was generally believed that all science had been developed in Europe. It is almost more in the realms of psychology and culture that we argue about the effect the discoveries of non-European science may have had on the ‘psyche’ of European scholars.

However I believe this concept of ‘late discoveries’ is a relatively weak excuse, as there is substantial evidence that many European scholars were aware of some Arab works that had been translated into Latin. All that aside, there was significant resistance to scientific learning in its totality in Europe until at least the 14th/15th century and as a result, even though Spain is in Europe, there was little progression of Arabic mathematics throughout the rest of Europe during the Arab period.

However, following this period it seems likely Latin translations of Arabic works will have had an influence. It is possible that the scholars using them did not know the origin of these works. There has also been occasional evidence of European scholars taking results from Arabic works and presenting them as their own. Actions of this nature highlight the unscrupulous character of some European scholars.

Above all, and regardless of the arguments, the simple fact is that many of the key results of mathematics, some of which are at the very ‘core’ of modern day mathematics, are of Arab origin. The results were almost all independently ‘rediscovered’ by European scholars during and after the ‘renaissance’ and while remarkable, history is something that should be complete and to neglect facts is both ignorant and arrogant.

To summarise, the main reasons for the neglect of Arab mathematics seem to be religious, cultural and psychological. Primarily it is because of an ideological choice. R Rashed mentions a concept of modernism vs. tradition. In terms of consequences of the Eurocentric stance, it has undoubtedly resulted in a cultural divide and ‘angered’ non-Europeans scholars. In order to maximise our knowledge of mathematics we must recognise many more nations as being able to provide valuable input, this statement is also relevant to past works. Eurocentrism has led to an historical ‘imbalance’, which basically means scholars are not presenting an accurate version of the history of the subject, which I view as unacceptable.

At the very least it must be hoped that the history of Arab mathematics will, in time become as highly regarded, as I believe it should. As D Almeida, J John and A Zadorozhnyy comment:

…Awareness is not widespread. [DA/JJ/AZ1, P 78]

R Rashed meanwhile explains the current problem:

…The same representation is found time and again: classical science, both in modernity and historicity appears in the final count as work of European humanity alone…

He continues:

…It is true that the existence of some scientific activity in other cultures is occasionally acknowledged. Nevertheless, it remains outside history or is only integrated in so far as it contributed to science, which is essentially European. [RR, P 333]

In short, the doctrine of the western essence of classical science does not take objective history into account.
 Finally, beyond simply alerting people to the remarkable developments of Arab mathematicians between around 9th and 11th century, and challenging the Eurocentric ideology of the history of the subject, it is thought further analysis and research could also have important consequences for future developments of the subject.

Clearly there is massive scope for further study in the area of the history of Arab and other non-European mathematics, and it is still a topic on which relatively few works have been written, although slowly significantly more attention is being paid to the contributions of non-European countries.
 In specific reference to my own project, I would have liked to have been able to go into more depth in my discussion of Indian algebra, and given many more worked examples, as I consider algebra to be both remarkable and severely neglected.

As a final note, many question the worth of historical study, beyond personal interest, but I hope I have shown in the course of my work some of the value and importance of historical study. I will conclude with a quote from the scholar G Miller, who commented:

…The history of mathematics is the only one of the sciences to possess a considerable body of perfect and inspiring results which were proved 2000 years ago by the same thought processes as are used today. This history is therefore useful for directing attention to the permanent value of scientific achievements and the great intellectual heritage, which these achievements present, to the world. [AA’D, P 11]


Bibliography:

Books:

1. P Duhem, Le système du monde (Paris, 1965).

2. E S Kennedy et al., Studies in the Islamic Exact Sciences(1983).

3. J L Berggren, Mathematics in medieval Islam, Encyclopaedia Britannica.

4. A A al’Daffa, The Muslim contribution to mathematics(London, 1978).

5. R Rashed, Entre arithmétique et algèbre: Recherches sur l’histoire des mathématiques arabes (Paris, 1984).

6. R Rashed, The development of Arabic mathematics : between arithmetic and algebra (London, 1994).

7. Joseph, G. G. (2000). The Crest of the Peacock, non-European roots of Mathematics. Princeton and Oxford: Princeton University Press.

8. Sarton, George. Introduction to the History of Science, 3 vols. Baltimore: Carngie Institution of Washington, 1927–1948.

9. R Rashed, L’extraction de la racine n-ième et l’invention des fractions décimales (XIe — XIIe siècles), Arch. History Exact Sci. 18 (3) (1977/78), 191–243.

10. Encyclopaedia of Mathematics and Society, Editors: Sarah J. Greenwald and Jill E. Thomley, 3 vols, October 2011 .

11. Al-Daffa, A. A. (1977). The Muslim Contribution to Mathematics. USA: Humanities Press.

Journal article:

12. Almeida, D. F., John, J. K. and Zadorozhnyy, A. (2001). Keralese Mathematics: Its Possible Transmission to Europe and the Consequential Educational Implications. Journal of Natural Geometry 20, 77–104.

Internet:

13. Mathematical biographies
 (Written and compiled by Professor Edmund F Robertson and Dr John J O’Connor, University of St Andrews. I used 38 articles from this website.) http://www-history.mcs.st-and.ac.uk/history/Indexes/500_999.html

Videos:

14. UstadKadirMisiroglu, “Science and Islam, Jim Al-Khalili — BBC Documentary” Online Video Clip, Youtube, 27 Mar 2013, Web, 11 December 2013-http://www.youtube.com/watch?v=qL41gX0fJng.

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