Introduction

The discipline of mathematics impinges on cultures in peculiar ways. It is applied to such diverse fields as music and war. In the twentieth century, it has contributed in surprising ways to how we understand the physical world at both atomic and cosmic scales. But, above all, its unique position in education on curricula at all levels, in all jurisdictions and in all languages ensures that it reaches the six billion human inhabitants of the globe more comprehensively than any other subject.

Mathematics is at once familiar and arcane. Like any other discipline, it has a tradition of defending jealously its own territory. In this tradition, physicists and certain statisticians are usually welcome, other scientists and engineers perhaps less so. Philosophers are generally perceived as irrelevant, while journalists, sociologists, adherents of cultural studies and other innumerate disciplines are regarded as having questionable motives. These must quickly be made aware of their incompetence to enter the realm with an attitude other than one of unquestioning awe.

The stand-off between the inhabitants of the mathematical world and those outside is self-perpetuating. A perception of mathematics as being inaccessible to the non-specialist is implicitly promoted. Mathematicians themselves can remain, at least in their professional lives, insulated from the world of lesser mortals knowing confidently that their discipline ‘has a methodology unique among all the sciences. It is the only discipline in which deductive logic is the sole arbiter of truth’ [Bro, p182].

In this short talk, I begin by stating the widely accepted Platonist view of mathematics. As illustrative material of the complexity of mathematics relationship with culture, I present two historical narratives, one set in the Arab world, the other in the Irish. There follows a short section on each of excluded cultures in mathematics, proof and certainty in mathematics, the philosophy of applied mathematics, ‘humanistic’ mathematics, and, finally on how mathematics is taught.

Platonism in mathematics

Mathematics rooted in deductive logic – this is the Platonist’s view of the discipline. God is a mathematician, or, for the agnostics, ‘the universe expresses itself naturally in the language of mathematics … the job of the theorist is to listen to [it] sing and record the tune’ [DH, p68-9]. God – or the universe – is immune from the prejudices of culture and from arbitrary whim.

The formalist, on the other hand, has no interest in objective reality: mathematics comprises axioms, definitions and theorems. In ‘The Mathematical Experience’, published in 1980, Davis and Hersh reckon that the typical mathematician is ‘a secret Platonist with a formalist mask that he puts on when the occasion calls for it’ [DH, p322]. Such an occasion might be when challenged by a philosopher about foundational issues.

Dieudonné suggests that for mathematics to thrive, favourable societal conditions are required, including freedom from religious and political constraints and from exclusively utilitarian preoccupations [D, p2]. Is that all that’s necessary, then? Is Platonist-formalist mathematics impervious to cultural forces in society at large? Let us examine the cultural context of two narratives in the history of mathematics: the foundations of algebra and mathematics in early medieval Ireland.

The narrative of algebra – Arabic or Greek

The word ‘algebra’ is derived from ‘al-jabr wa’l-muqabala’, [Ka, p 228] the title of a work written in Baghdad, the capital of the Abbasid caliphate, between 813 and 833. The author was al-Khwarizmi from whose name the word ‘algorithm’. Its importance was that it emphasised equations rather than geometry or arithmetic, although these equations could represent problems in either of these domains.

All of this happened in the context of institutions of learning such as the House of Wisdom which was the exemplar for wealthy patrons of research which went hand in hand with translation, especially from the Greek. The discipline of translation into Arabic began in the middle of the 7^th century, in Alexandria as the population there transformed from Greek-speaking Christians to Arabic-speaking Muslims [Rashed in N, p 136]. Initially the works translated concerned administration and philosophy, but by 8^th and 9^th centuries the emphasis had moved to scientific and mathematical works, while the location shifted to Baghdad, the centre of power founded by the caliph al-Mansur in 766.

Rashed is very clear on two aspects of this translation: it was on a massive scale, and it was carried out not to learn the material but to continue already existing research in a Greek culture which was in transition to an Arabic one. A work might be translated several times, so that when the work of Diophantus, the 3^rd century Alexandrian mathematician, was translated after the development of algebra, the style, language and interpretation were algebraic. A straight translation of an original text cannot be presumed [Rashed in N, p 142]. This will be important shortly.

In her essay, ‘The creation of the history of algebra in the sixteenth century’, Cifoletti describes humanism as an attempt to build Western knowledge directly on Western sources in opposition to the obvious dependence of European knowledge on medieval sciences, the Arabic branches of which were the most important [C, p123].

In 1463, ten years after the fall of Constantinople, Regiomontanus announced the existence of a manuscript of Diophantus containing evidence of algebra, thus feeding into a Greek revivalist movement prominent in Italy [C, pp124/5]. As the algebrists aimed at higher dignity, they strove to raise the status of algebra to a discipline, to transform it from the merely practical into the contemplative [C, p125]. In 16^th century Italy, while freedom of movement between the (practical) abacus schools and the (theoretical) universities was still possible, Cardano specifically credited al-Khwarizmi as the inventor of the art of algebra, identifying the early 13^th century mathematician, Fibonacci as his source [C, pp126/7].

In contrast to this Italian school, the French attitude towards history, including the history of mathematics, was dominated by jurists who were patrons of mathematical publications. The algebraist Jacques Peletier du Mans was not prepared to ascribe the invention of algebra (nor, indeed, most other ‘arts’) to a single author, but saw it rather as the accumulation of knowledge by many. For him there was no need for a universal ancient authority [C, p129-131]. Jean Borrel in his ‘Logistica’ published in Lyon in 1559, denounced the Arabs and (implicitly) the abacus schools as ‘ignorant propagators’, asserting that the essence of the art appeared in Euclid’s Tenth Book. Thus ‘ancient Greece becomes early Europe’ [C, p140].

In The Analytic Art, some thirty years later, François Viète identified Greek analysis with algebra [Ka, p339]. In 1637, Descartes published his ‘Geometry’. With this work European algebra had come of age. Classical geometry was now all but reduced to algebra, an algebra however which had already fashioned its own legitimacy in a Greek lineage.

Høyrup, in his essay ‘The formation of a myth: Greek mathematics – our mathematics’, begins: ‘According to conventional wisdom, European mathematics originated among the Greeks between the epochs of Thales and Euclid, was borrowed and well preserved by the Arabs in the early Middle Ages, and brought back to its authentic homeland by Europeans in the twelfth and thirteenth century. Since then, it has pursued its career triumphantly.’ [Hø, p103] However, he continues, ‘no culture had an exclusive privilege of the Greek inheritance.’ [Hø, p105]

Early medieval Irish culture and mathematics

In the millennium or so between the fall of Rome and that of Constantinople, one can easily be tempted to the conclusion that all civilization and learning ceased in Europe. If it is possible to put classical chauvinism aside for a moment, one might glimpse evidence of learning amongst the Iberian Visigoths and the Insular Irish not to mention, of course, the Franks of Charlemagne’s court.

By the end of the 6^th century, culture in Ireland had gone through a substantial transition: Irish scholars had adopted and mastered Latin, the language of an empire in disarray and of which Ireland had never been a part [Óc] [R]. In Ireland – and northern Britain – a dynamic and civilised culture lasted for about three hundred years, from the 6^th to the 9^th century. However, little attention has been given to the mathematics of early medieval Ireland.

The oldest known Irish writings in Latin concern a matter of mathematical interest, namely the computus or study of the (ecclesiastical) calendar with particular reference to the determination of the date of Easter. Ó Cróinín identifies Mo-Sinu maccu Min, fourth abbot of Bangor as ‘the first of the Irish who learned the computus by heart from a certain learned Greek’. The writings of Columbanus, Mo-Sinu’s student, indicate that he was well acquainted with this material [Óc, p177]. Indeed the computus held a central place in Irish centres of learning in Ireland, Britain and on the European continent. Recent work by McCarthy has recovered for the first time since, it seems, the 7^th century, the technical content of the 84-year Easter cycle used by the Irish [Mc].

Early medieval European mathematics is not treated sympathetically in the literature. Chapter 4 of ET Bell’s The Development of Mathematics, written during the ‘dark ages’ of the late 1930s and early 1940s, is entitled The European Depression. With scathing contempt he writes, ‘The practical outcome of all this [mathematics] was a cumbersome reckoning sufficient for the simple transactions involving money, and for keeping the calendar in order so that the date of Easter might not elude annual recapture. To call any of this computation - or of the debased geometry – mathematics is a gross exaggeration. The significance of mathematics as a deductive system had been forgotten. Science having sunk to the level of superstition, the other half of the Pythagorean vision survived only in the fantastic absurdities of sacred and profane numerology. Number indeed ruled the darkened universe of the European Middle Ages.’ [Bel, pp85-9]

Katz, in 1993, writes of this period, ‘The only schools in early medieval Europe were those connected to the Catholic Church, and it was there that the only significant mathematical problem of the time was considered, the determination of the calendar. In particular, there was a debate in the Church as to whether Easter should be determined using the Roman solar calendar or the Jewish lunar calendar. The two reckonings could be reconciled, but only by those with some mathematical knowledge. Charlemagne, in fact, even before his coronation in 800 as holy Roman Emperor, formally recommended that the mathematics necessary for the Easter computations be part of the curriculum in church schools.’ [Ka, p267]

Although not by any means comparable with the contemporaneous Arabs, the Irish of the early medieval period appear not only to have been abreast of what, admittedly little, mathematics was around in Western Europe, but to have championed computistical studies. Moreover, they demonstrated considerable achievement in decorative geometry. The skill evident in stonework, metalwork and, especially, illuminated manuscripts indicates geometric insight [Ha]. Stevick argues at length that the proportions in the design of the decorated pages of the Echternach Gospels are based on the square root of two and the golden ratio. However, it is still a matter of speculation what formal geometrical knowledge was known in Ireland at the time [S].

Excluded cultures in mathematics

We have seen that mathematical historiography has been strongly influenced by the dominant worldviews in society at large. We should expect this of the historiography of any discipline, however, in mathematics in particular the domination of Platonism has tended to distract attention away from the cultures in which mathematics has evolved.

In this short paper, we have only mentioned two: the Arabic and the Irish, the former of global importance, the latter of regional interest. Recently, the vernacular mathematical heritage associated with particular regions has been attracting increasing interest, not least as an important pedagogic resource.

In their popular book ‘Introducing Mathematics’, Sardar and Ravetz define ethnomathematics as ‘the mathematics of all those people who have been excluded from knowledge and cultural production’ [SR]. More representative of the mainstream, but nonetheless pioneering, is Gerdes’ recent ‘Geometry from Africa – Mathematical and Educational Explorations’ [G].

Proof and certainty in mathematics

Earlier we mentioned the dominant position of Platonism in mathematics. Having glimpsed some specific cases of how cultures in transition influence mathematics, we now turn to some recent trends within the ‘culture’ of mathematics itself.

The role of proof for establishing certainty within mathematics is at least 26 centuries old [DH, p147]. This role however has not gone uncontested. Mancosu analyses the Renaissance debate known as, Quaestio de Certitudine Mathematicarum [Ma, pp10-33]. Central to this debate was how mathematics measured up to Aristotelian science, and on what, if not its logical structure, mathematics could be justified.

In the early 20^th century, the great German mathematician, David Hilbert aspired to formalise classical mathematics. Although his programme gave rise to numerous spectacular by-products, ultimately it failed due to Gödel’s incompleteness theorem of 1938. This very difficult theorem essentially asserts that no axiomatization can capture all the truths of arithmetic and none of its falsehoods [Bro, p73].

More problems with proof were to follow. In 1976, Appel and Haken proved the Four Color Theorem – which had been conjectured in 1852 – but by using a lengthy computer program. A vigorous debate ensued on the validity of the proof on a number of grounds, in particular, that certainty had been abandoned: mathematics appeared more like the inductive natural sciences [Bro, p154]. Some saw no harm in this: ‘as physicists have learned to live with uncertainty, so we [mathematicians] should learn to live with an "uncertain" proof’ [Bro, p156].

Philosophy of Applied Mathematics

If mathematics can no longer boast certainty, what then can it do? For a couple of decades now an emphasis on mathematical modelling has been prevalent. Davis and Hersh, in 1980, lamented this trend away from theory; to them ‘Truth has abdicated and expediency reigns’ [DH, p70]. Such criticism sounds rather old-fashioned today, and especially so by Hersh’s own standards. In 1995, he pinpointed a more serious issue: ‘From Pythagoras to Russell and beyond, philosophy of mathematics rarely paid serious attention to applied mathematics.’ [He, p236]

In his ‘Philosophy of Mathematics’, published last year, Brown devotes an entire chapter to applied mathematics. He identifies three central questions:     1.  How does mathematics ‘hook onto’ the world?
                       2.   Are some mathematical objects merely that or do they have some other status?
                       3.   Is mathematics essential for science? [Bro, p46]

He argues that the nature of applied mathematics is representational rather than descriptive. This view, consistent with the entire history of mathematics, strongly supports the Platonist autonomy of the discipline [Bro, p55].

Stewart in his preface to the second edition of Courant & Robbins’ classic ‘What is Mathematics?’ essentially agrees with this: ‘Mathematics links the abstract world of mental concepts to the real world of physical things without being located completely in either.’ [CRS]

Humanistic mathematics

So, ‘What is Mathematics, Really?’ This, in fact, is the title of Hersh’s 1995 manifesto in which he makes a spirited case for humanist – otherwise known as humanistic – mathematics. In the preface, he opens his case: ‘Repudiating Platonism and formalism, while recognizing the reasons that make them (alternately) seem plausible, I show that from the viewpoint of philosophy mathematics must be understood as a human activity, a social phenomenon, part of human culture, historically evolved, and intelligible only in a social context. … In attacking Platonism and formalism and neo-Fregeanism, I’m defending our right to do mathematics as we do. To be frank, this book is written out of love for mathematics and gratitude to its creators.’ [He]

Curiously, Hersh’s attitude reminds us of Berkeley’s empiricism: experience forms the basis of all human knowledge. Berkeley was critical of mathematics, which he regarded as a science of empty abstractions. [Ber, p3]

In keeping with Hersh’s philosophy, Thurston sees mathematicians not as people who ‘make progress in mathematics’ but who ‘advance human understanding of mathematics’ [T, p162].

How mathematics is taught

In ‘Essays on Humanistic Mathematics’, there are five essays on teaching and learning, including one provocatively titled ‘Towards a Pedagogy of Confusion’ [W]. Hersh, too, emphasises pedagogy: ‘If mathematical objects were an other-worldly nonhuman reality (Platonism), or symbols and formulas whose meaning is irrelevant (formalism), it would be a mystery how we could teach or learn it. Its teachability is the heart of the humanist conception of mathematics.’ [He, p238]

Courant in his classic collaboration of 1941, ‘What is Mathematics?’ clearly identifies why mathematics should be taught: ‘The teaching of mathematics has sometimes degenerated into empty drill in problem solving, which may develop formal ability but does not lead to real understanding or to greater intellectual independence.’ [CRS]

More recently, Hughes-Hallett and her collaborators emphasised the importance of protecting the teaching of calculus from its power: ‘Calculus has been so successful because of its extraordinary power to reduce complicated problems to simple rules and procedures. Therein lies the danger in teaching calculus: it is possible to teach the subject as nothing but rules and procedures – thereby losing sight of both the mathematics and of its practical value.’ This, so-called Harvard Consortium set out its stall to restore that insight. One of their instruments is the Way of Archimedes: ‘that formal definitions and procedures evolve from the investigation of practical problems’ [HG, p vii].

The Harvard Consortium is the cutting edge of the calculus reform movement, as it is known. However, this movement has, perhaps predictably, suffered criticism of those who want to restore emphasis on skills and rigour [Bro, p181].

Another vital strand in mathematics teaching today is the use of history. About a century, the great French mathematician, Poincaré, identified ‘the task of the educator [as one of making] the child’s spirit pass again where its forefathers have gone, moving rapidly through certain stages but suppressing none of them. In this regard, the history of science must be our guide.’ Bressoud, in his recent ‘A Radical Approach to Real Analysis’ summarises Poincaré’s injunction succinctly: ‘let history inform pedagogy’! [Bre, p vii]

Abstraction, Application and Appreciation of narrative

I hope that by this tour of two mathematical narratives, spanning well over 2000 years, together with some views on current thinking on philosophy, practice and teaching of mathematics, all necessarily selective, the overall impression is that mathematics cannot be described as ‘the only discipline in which deductive logic is the sole arbiter of truth’. There is a strong cultural context both within mathematics and between mathematics and other human cultures – hence the title of this talk.

Ptolemy the First asked Euclid if there was not a shorter road to geometry, to which the geometer replied: ‘There is no royal road to geometry’. For mathematics as a whole, such a question defies response. Let me dare to suggest three elements are essential at once individually and as they relate to one another: Abstraction, Application and Appreciation of narrative.

Bibliography

page updated: 20 March 2001