This essay argues that the diversity of the history of mathematics community in the United Kingdom has influenced the development of the subject and is a significant factor behind the different concerns often evident in work on the history of mathematics when compared with that of historians of science. The heterogeneous nature of the community, which includes many who are not specialist historians, and the limited opportunities for academic careers open to practitioners have had a profound effect on (...) the discipline, leading to a focus on elite mathematics and great mathematicians. More recently, reflecting earlier developments in the history of science, an increased interest in the context and culture of the practice of mathematics has become evident. (shrink)

ABSTRACT For some years now, the history of modern mathematics and the history of modern science have developed independently. A step toward a reunification that would benefit both disciplines could come about through a revived appreciation of mathematical practice. Detailed studies of what mathematicians actually do, whether local or broadly based, have often led in recent work to examinations of the social, cultural, and national contexts, and more can be done. Another recent approach toward a historical understanding (...) of the abstractness of modern mathematics has been to see it as a species of modernism, and this thesis will be tested by the raft of works on the history of modern applied mathematics currently under way. (shrink)

This paper surveys central justifications and approaches adopted by educators interested in incorporating history of mathematics into mathematics teaching and learning. This interest itself has historical roots and different historical manifestations; these roots are examined as well in the paper. The paper also asks what it means for history of mathematics to be treated as genuine historical knowledge rather than a tool for teaching other kinds of mathematical knowledge. If, however, history of mathematics is not subordinated (...) to the ideas and methods at the heart of the usual mathematical curriculum – algebraic equations, functions, derivatives, analytic geometry – if it becomes an essential subject for mathematics, then the attempt to find a place for history of mathematics requires refining our understanding of the nature of mathematics education itself. Thus, the paper asks not only how can history of mathematics be incorporated into mathematics education but also how the idea of mathematics education may need to be adjusted to accommodate history. (shrink)

Volume 9 of the Routledge History of Philosophy surveys ten key topics in the philosophy of science, logic and mathematics in the twentieth century. Each of the essays is written by one of the world's leading experts in that field. Among the topics covered are the philosophy of logic, of mathematics and of Gottlob Frege; Ludwig Wittgenstein's Tractatus ; a survey of logical positivism; the philosophy of physics and of science; probability theory, cybernetics and an essay on the mechanist/vitalist (...) debates. The volume also contains a helpful chronology to the major scientific and philosophical events in the twentieth century. It also provides an extensive glossary of technical terms in the notes on major figures in these fields. (shrink)

In the article, questions from the field of philosophy of mathematics are studied. The author is driven by the need to achieve a balance between the philosophy of science and the history of science in formation of concepts of the science development. In this regard, the author justifies the reliance on the methodology of implicit knowledge, combined with the epistemology principle of criticism in studying the development of mathematics as the most expedient and effective. The author expresses the necessity (...) of criticizing the methodology of counterexamples typical for mathematical empiricism declared by I. Lakatos. In the article, an explanation of the concept of implicit knowledge, as well as the examples from mathematics are given. The author believes that application of the concept of implicit knowledge in philosophical and mathematical terms allows us to reveal the unique specificity of mathematical science deeper. The author concludes that it is necessary to recognize the fundamental importance of the methodology of implicit knowledge for the epistemology, as well as for the development of philosophy and methodology of science in general. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Method and Mathematics:Peter Ramus's Histories of the SciencesRobert GouldingPeter Ramus (1515–72) was, at first sight, the least likely person to write an influential history of mathematics. For one thing, he was clearly no great mathematician himself. His sympathetic biographer Nicholas Nancel related that Ramus would spend the mornings being coached in mathematics by a team of experts he had assembled, and in the afternoon would lecture on the (...) very same subjects.1 But there can be no doubt that Ramus held mathematics in particular esteem and even played a crucial role in linking philosophical discourse to mathematics and in promoting the use of quadrivial reasoning in the study of the natural world.2The origins for his enthusiasm for mathematics are to be found in his account of the nature of the arts and critique of the curriculum of the universities. From his earliest career, Ramus was a logician and remained one in all his works, whether writing on mathematics or Virgil, and he conceived [End Page 63] of all the arts, especially mathematics, as unchanging structures of necessarily true propositions—not prima facie a position which lends itself to notions of historical change and development.3 His career as a historian of mathematics, I will argue, was directed by problems that arose in his theoretical understanding of mathematics as he began to immerse himself in the sciences of the ancient world.Ramus and the Reform of DialecticRamus set out his fundamental logical positions in his very first printed work, the Dialecticae institutiones (Education in Dialectic) of 1543, a contribution to the ongoing humanist attack on scholastic logic, in which Ramus rehearsed many of the commonplace criticisms of the university dialectic. Humanists complained that logic no longer concerned itself with real human reasoning; instead, it had become a discipline studied for its own sake, using its own incomprehensible jargon, and was of no practical interest. The new humanist dialectics of Lorenzo Valla and Rudolph Agricola attempted to teach the kind of practical reasoning useful for composing a speech or letter, and they borrowed extensively from rhetorical works of Cicero and Quintilian to develop a highly rhetoricized logic. Questions about the formal validity of arguments were of little interest; what mattered was whether the arguments were persuasive.4In his Dialecticae institutiones, Ramus took this humanist reformulation of dialectic a step further. Dialectic—or, in fact, any art or science—consisted of nature, doctrine, and exercise. The natural workings of the mind formed the most significant element. Doctrine was nothing more than a record of natural reasoning; in importance it paled next to nature and practice.5 The logic of the universities bore no resemblance to the true, natural dialectic, as he argued at length in the companion volume to the Institutiones, the innocuously titled but exuberantly offensive Aristotelicae animadversiones [End Page 64] (Observations on Aristotle).6 The question, then, remained: how can we gain access to this "natural reasoning"?Ramus's answer was surprising. He asks us to find a group of men—not scholars, but completely uneducated vineyard workers. Question them about the coming year: the fertility of the soil, the quality and quantity of the crop. "And then," he writes "from their minds as from a mirror an image of nature will be reflected."7 In the reasoned replies of these uneducated men, says Ramus, we discover every part of logic needed for any purpose, whether everyday discourse or composition of poetry: the invention of arguments, the assessment of their truth and their proper and orderly presentation. Other humanists had praised man's natural logical faculties, elevating them over artificial technique, but Ramus was the first to look beyond the walls of the university and the writings of the ancients to find natural dialectic at work.If even the uneducated have some possession of the arts, then the arts taught at the university should do no more than clear away the misleading junk in the mind, and allow its natural clarity to shine through.8 Logic should be easy to learn—not because this is a pedagogic ideal, but from the very nature of the art. And if logic as it is... (shrink)

Up until the 1950s, the history of mathematics was an integral part of the history of science. To George Sarton and his contemporaries, mathematics was the rational skeleton that organized science and held it together, and its history was a fundamental component of the broader history of science. But when historians began focusing on the cultural roots of science rather than its rational structure, the study of mathematics was marginalized and ultimately excluded from the history (...) of science. The alienation between the two fields is detrimental to both, and in recent years there has been a sustained effort to reestablish meaningful communication between the two. This time, however, mathematics is seen not as the static skeleton of science but, instead, as a dynamic and historically evolving field in its own right—just like science itself. The new approach allows for a culturally sensitive study of mathematics, as well as a new and fruitful relationship between the history of science and the history of mathematics. The essays in this Focus section offer a sampling of the new approaches, opening the way to a rapprochement between fields that have gone their separate ways but should by rights be closely interconnected. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Abraham, Planter of Mathematics":Histories of Mathematics and Astrology in Early Modern EuropeNicholas PopperFrancis Bacon's 1605 Advancement of Learning proposed to dedicatee James I a massive reorganization of the institutions, goals, and methods of generating and transmitting knowledge. The numerous defects crippling the contemporary educational regime, Bacon claimed, should be addressed by strengthening emphasis on philosophy and natural knowledge. To that end, university positions were to be created devoted to (...) "Artes and Sciences at large," rather than to the professions. High salaries would render lecturers "able and sufficient," undistracted from their task. Most famously, he argued that teaching of the "operatiue studie of many Scyences" should involve sophisticated technical education. The study of natural philosophy demanded not only books, but globes, astrolabes, and other "instrumentals." Most significantly, yielding reliable and meaningful knowledge from experiential gleanings required a rigorous system of deductive reasoning.The legacy of this colossal proposal has earned Bacon honored status as devisor of the scientific method.1 But Bacon's educational reform extended [End Page 87] beyond the methods of producing and transmitting knowledge. To facilitate more efficient "vse and administration" of the knowledge produced by his system, he also demanded a searching examination of the history of learning. This history would provide a mirror enabling his contemporaries to deploy the fruits of his method, by considering how learning in the past had been used successfully or ill-advisedly. Producing this history was the ambition of the Advancement, for he explained: "no man hath propounded to himselfe the generall state of learning to bee described and represented from age to age, as many have done the works of Nature, & the State civile and Ecclesiastical." He continued:And yet I am not ignorant that in divers particular sciences, as of the Iurisconsults, the Mathematicians, the Rhetoricians, the Philosophers, there are set down some smal memorials of the Schooles, Authors, and Bookes: and so likewise some barren relations touching the Invention of Arts, or usages. But a iust story of learning, containing the Antiquities and Originalls of Knowledges, & their Sects; their Inventions, their Traditions; their diverse Administrations and Managings; their Flourishings, their Oppositions, Decayes, Depressions, Oblivions, Removes, with the causes, and occasions of them, and all other events concerning learning, throughout the ages of the world; I may truly affirme to be wanting.2Bacon thus positioned himself not only as a Father of Modern Science, but also a Father of the History of Science.Following Bacon's suggestion, I will examine the "small memorials" of the history of mathematics—and particularly the mathematical art of astrology—in the sixteenth and seventeenth centuries. My conclusion, however, will not bear out Bacon's claims. Despite his frustration, Bacon was only one of many early modern scholars appraising the role of mathematics within history. And he devoted less energy than others to mapping its origins and tracing its transmissions between communities. In fact, Bacon's 1605 proposal for a history of mathematics was already out-of-date. Discussions [End Page 88] of the history of mathematics had been rife on the continent and in England throughout the previous century.3The evidence that Renaissance scholars inherited was sprawling and inconclusive. Several genealogies for mathematics could be found within classical Greek, Latin, and patristic references. One lineage claimed mathematics began in ancient Assyria, where the priestly caste, the Chaldeans, practiced a form of mathematics that seemed a corrupt admixture of philosophy, medicine, and religion, and relied heavily on observation of the heavens. Other scholars traced the origins to Egypt, claiming that the field developed to survey lands frequently flooded by the Nile. The real problem for ancient, late antique, and medieval scholars had not been the origins of mathematics, but what exactly mathematics were. For some, the term strictly referred to the quadrivium: geometry, arithmetic, astronomy, and music. But for others, mathematics' origins amongst the Egyptians or Chaldeans inextricably linked it to forms of astrological divination, augury, and necromancy that were unsavory to both Latin and Christian traditions. Mathematici were included alongside ghastly lists of Magi, Brahmins, Aruspices, Genethliaci, and other diabolic practitioners of idolatrous magic. These might or might not be distinguished from mathematici such as Pythagoras... (shrink)

This book, a tribute to historian of mathematics Jeremy Gray, offers an overview of the history of mathematics and its inseparable connection to philosophy and other disciplines. Many different approaches to the study of the history of mathematics have been developed. Understanding this diversity is central to learning about these fields, but very few books deal with their richness and concrete suggestions for the "what, why and how" of these domains of inquiry. The editors and authors approach the (...) basic question of what the history of mathematics is by means of concrete examples. For the "how" question, basic methodological issues are addressed, from the different perspectives of mathematicians and historians. Containing essays by leading scholars, this book provides a multitude of perspectives on mathematics, its role in culture and development, and connections with other sciences, making it an important resource for students and academics in the history and philosophy of mathematics. (shrink)

Abraham Robinson’s framework for modern infinitesimals was developed half a century ago. It enables a re-evaluation of the procedures of the pioneers of mathematical analysis. Their procedures have been often viewed through the lens of the success of the Weierstrassian foundations. We propose a view without passing through the lens, by means of proxies for such procedures in the modern theory of infinitesimals. The real accomplishments of calculus and analysis had been based primarily on the elaboration of novel techniques (...) for solving problems rather than a quest for ultimate foundations. It may be hopeless to interpret historical foundations in terms of a punctiform continuum, but arguably it is possible to interpret historical techniques and procedures in terms of modern ones. Our proposed formalisations do not mean that Fermat, Gregory, Leibniz, Euler, and Cauchy were pre-Robinsonians, but rather indicate that Robinson’s framework is more helpful in understanding their procedures than a Weierstrassian framework. (shrink)

The paper uses the example of historical paper research in Vienna around 1900 in order to analyze the dynamics of scientific cooperation between the natural sciences and the humanities. It focuses on the Vienna-based plant physiologist Julius Wiesner (1838–1916), who from 1884 to 1911 studied medieval paper manuscripts under the microscope in productive cooperation with paleographers, archaeologists and orientalists (Josef Karabacek, Marc Aurel Stein, Rudolf Hoernle). The paper examines why these cooperations succeeded and how they developed over time. Here (...) we distinguish between two forms of cooperation: while Wiesner initially worked only reactively, in a “closed cooperation”, he later entered into “open cooperations”, in which both parties defined research questions and methods. This form of cooperation proved particularly successful, but at the same time was especially demanding because, in addition to contributing one’s own skills, it required considerable “interlocking expertise” (Andersen 2016). This was favored because the historical auxiliary sciences were in a phase of upheaval. Wiesner contributed his knowledge of technical microscopy and developed a genuine interest in historical questions, while the humanists were prepared to open themselves to scientific processes and ultimately acknowledged Wiesner as a historian of paper in his own right. (shrink)

If anachronism is an historian's unforgivable sin, as Lucien Fèbvre told us long ago, it is nonetheless unavoidable, if only in the sense that we are constrained by our own point of view, anchored...

Wittgenstein’s philosophy of mathematics is often devalued due to its peculiar features, especially its radical departure from any of standard positions in foundations of mathematics, such as logicism, intuitionism, and formalism. We first contrast Wittgenstein’s finitism with Hilbert’s finitism, arguing that Wittgenstein’s is perspicuous or surveyable finitism whereas Hilbert’s is transcendental finitism. We then further elucidate Wittgenstein’s philosophy by explicating his natural history view of logic and mathematics, which is tightly linked with the so-called rule-following problem and Kripkenstein’s paradox, (...) yielding vital implications to the nature of mathematical understanding, and to the nature of the certainty and objectivity of mathematical truth. Since the incompleteness theorems, foundations of mathematics have mostly lost their philosophical driving force, and anti-foundationalism has become prevalent and pervasive. Yet new foundations have nevertheless emerged and come into the scene, namely categorical foundations. We articulate the foundational significance of category theory by explicating three forms of foundations, i.e., global foundations, local foundations, and conceptual foundations. And we explore the possibility of categorical unified science qua pluralistic unified science, arguing for the categorical unity of science on both mathematical and philosophical grounds. We then turn to an issue in conceptual foundations of mathematics, namely the nature of the concept of space. We elucidate Wittgenstein’s intensional conception of space in relation to Brouwer’s theory of space continua and to the modern conception of space as point-free structure in category theory and algebraic geometry. We finally give a bird’s-eye view of mathematical philosophy from a Wittgensteinian perspective, and further sheds new light on Wittgenstein’s constructive structuralism and his view of incompleteness and contradictions in mathematics. (shrink)

Wittgenstein’s philosophy of mathematics is often devalued due to its peculiar features, especially its radical departure from any of standard positions in foundations of mathematics, such as logicism, intuitionism, and formalism. We first contrast Wittgenstein’s finitism with Hilbert’s finitism, arguing that Wittgenstein’s is perspicuous or surveyable finitism whereas Hilbert’s is transcendental finitism. We then further elucidate Wittgenstein’s philosophy by explicating his natural history view of logic and mathematics, which is tightly linked with the so-called rule-following problem and Kripkenstein’s paradox, (...) yielding vital implications to the nature of mathematical understanding, and to the nature of the certainty and objectivity of mathematical truth. Since the incompleteness theorems, foundations of mathematics have mostly lost their philosophical driving force, and anti-foundationalism has become prevalent and pervasive. Yet new foundations have nevertheless emerged and come into the scene, namely categorical foundations. We articulate the foundational significance of category theory by explicating three forms of foundations, i.e., global foundations, local foundations, and conceptual foundations. And we explore the possibility of categorical unified science qua pluralistic unified science, arguing for the categorical unity of science on both mathematical and philosophical grounds. We then turn to an issue in conceptual foundations of mathematics, namely the nature of the concept of space. We elucidate Wittgenstein’s intensional conception of space in relation to Brouwer’s theory of space continua and to the modern conception of space as point-free structure in category theory and algebraic geometry. We finally give a bird’s-eye view of mathematical philosophy from a Wittgensteinian perspective, and further sheds new light on Wittgenstein’s constructive structuralism and his view of incompleteness and contradictions in mathematics. (shrink)

ABSTRACT In a series of papers read to the Cambridge Philosophical Society through the 1820s, the Cambridge mathematician George Peacock laid the foundation for a natural history of arithmetic that would tell a story of human progress from counting to modern arithmetic. The trajectory of that history, Peacock argued, established algebraic analysis as a form of universal reasoning that used empirically warranted operations of mind to think with symbols on paper. The science of counting would suggest arithmetic, arithmetic (...) would suggest arithmetical algebra, and, finally, arithmetical algebra would suggest symbolic algebra. This philosophy of suggestion provided the foundation for Peacock's “principle of equivalent forms,” which justified the practice of nineteenth-century English symbolic algebra. Peacock's philosophy of suggestion owed a considerable debt to the early Cambridge Philosophical Society culture of natural history. The aim of this essay is to show how that culture of natural history was constitutively significant to the practice of nineteenth-century English algebra. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Histories of Science in Early Modern Europe:IntroductionRobert GouldingIn 1713, Pierre Rémond de Montmort wrote to the mathematician Nicolas Bernoulli:It would be desirable if someone wanted to take the trouble to instruct how and in what order the discoveries in mathematics have come about.... The histories of painting, of music, of medicine have been written. A good history of mathematics, especially of geometry, would be a much more interesting (...) and useful work.... Such a work, if done well, could be regarded to some extent as a history of the human mind, since it is in this science, more than in anything else, that man makes known that gift of intelligence that God has given him to rise above all other creatures.1Such a history of mathematics was attempted by Jean-Etienne Montucla in his Histoire des mathématiques (first printed in 1758, and reissued in a [End Page 33] greatly expanded form in 1799).2 Montucla's great work is generally acknowledged as the first genuine history of mathematics. According to some modern historians, previous attempts at such a history had amounted to little more than collections of anecdotes, biographies or exhaustive bibliographies: "jumbles of names, dates and titles," as one writer in the Dictionary of Scientific Biography characterized them.3 Montucla, on the other hand, was thoroughly animated by the Enlightenment project expressed in Rémond's letter, and his Histoire was a philosophical history of the "development of the human mind," as he himself described it.4 It was precisely Montucla's vision of what mathematics meant and his conviction that mathematics itself must change in order to reflect the historical elevation of the human intellect, that allowed him to transform the scattered dates and anecdotes of his predecessors into a genuine history. All subsequent histories of mathematics—until the most recent social histories5 —have been in a sense "footnotes to Montucla."The papers here argue that there were indeed histories of mathematics before Montucla which are worthy of scholarly attention, and that the dismissal of Renaissance histories of science as nothing more than a "cloud of fine adjectives and metaphors" is unfair (although it contains a germ of truth). Anthony Grafton has argued in an article on the historical writings of Cardano, Rheticus, and Kepler, that the purpose of such texts was not so much to trace what actually happened, as to justify the study of a subject often derided—by both humanist and scholastic writers—as obscure, useless and undignified.6 The following four papers go further in arguing that writings in this genre (or, at least, some writings in this genre) were not limited to this characteristically humanist objective. Authors did not just use their histories to persuade others. They themselves relied on historical narratives in order to think about their discipline, define its parts, distinguish among its acceptable and unacceptable forms and prescribe its content [End Page 34] and method of teaching. By placing their discipline into a historical context shared by other, more mainstream humanistic arts, moreover, they could draw upon the large, narrative structures which Renaissance humanists had adapted to understand human intellectual and cultural development, origins, progress and decline.James Byrne's article highlights a disparity of vision between those humanists who treated the origin of the arts as just another subject for humanistic display; and those (like Regiomontanus) who were invested in the arts and wrote their histories from an insider's perspective. Alongside Regiomontanus, one could include several other authors treated in this collection, such as Thomas Vaughan, Peter Ramus, and Henry Savile. In each case, history provided a means for working through problems of the legitimacy and nature of their disciplines. While their writings are not "histories of the human mind," their interests went beyond mere rehearsal of anecdotes. Ramus's histories, for instance, written 200 years before Montucla's Histoire, had the kind of unifying vision which de Montmort demanded, and shared Montucla's concern to find a meaning in the development of mathematics. Ramus, and the supporters and detractors he spawned, were convinced that constructing the history of this science was crucial to understanding how and why human beings have... (shrink)

A central figure in Victorian science, William Whewell held professorships in Mineralogy and Moral Philosophy at Trinity College, Cambridge, before becoming Master of the college in 1841. His mathematical textbooks, such as A Treatise on Dynamics, were instrumental in bringing French analytical methods into British science. This three-volume history, first published in 1837, is one of Whewell's most famous works. Taking the 'acute, but fruitless, essays of Greek philosophy' as a starting point, it provides a history of (...) the physical sciences that culminates with the mechanics, astronomy, and chemistry of 'modern times'. Volume 1 studies Greek physics and metaphysics, attributing their failure to a method that derived its principles from the common use of language. It surveys the state of the physical sciences in the middle ages, and deals with the rise of 'formal' astronomy - based on observation rather than calculation - as exemplified by Copernicus. (shrink)

One influential interpretation of Newton's formulation of his calculus has regarded his work as an organized, cohesive presentation, shaped primarily by technical issues and implicitly motivated by a knowledge of the form which a "finished" calculus should take. Offered as an alternative to this view is a less systematic and more realistic picture, in which both philosophical and technical considerations played a part in influencing the structure and interpretation of the calculus throughout Newton's mathematical career. This analysis sees the (...) development of Newton's calculus not principally as a calculated movement toward "rigorous" justification (in the sense of Cauchy, Weierstrass, and their 19th century contemporaries) and refinement of techniques, but as an evolution in the light of his involvement in debates over philosophical and scientific issues central to the rise of modern philosophy and science in the 17th and early 18th centuries. (shrink)

ArgumentIn the decade between 1983 and 1993, Herbert Mehrtens in West Berlin and I in East Berlin communicated closely about our parallel work on mathematics under the National Socialist (NS) regime. For a short period (1988–89), we worked on a joint book on this topic. We agreed that the book should be based primarily on empirical historical material, using a theoretical approach largely guided by Mehrtens’ work on social systems in mathematics (Mehrtens1981, 1987a). However, parallel work for his habilitation thesis (...) on modernity in mathematics increasingly captivated Mehrtens’ interests. This, together with ongoing resistance among older mathematicians and the insecurities of our individual careers during times of considerable political change, particularly in East Germany, led to the failure of our joint project.This paper aims at reconstructing—mostly by drawing on correspondence—parts of the dialogue between Mehrtens and me at the time. Our dialogue was influenced by our different upbringing and socialization in the two parts of Germany, and by different, though complementary, views about the combination of empirical and theoretical approaches to the historiography of mathematics.In the epilogue, I describe how my later work in the decades from the 1990s on the history of mathematics profited much from Mehrtens’ early theoretical and empirical work but was less influenced by his book on modernity. That later work of mine was, however, mostly done at a time when Mehrtens had largely left the field and considered himself increasingly as a “cultural scientist” (Kulturwissenschaftler). (shrink)

This book is part of a major project undertaken by the Centre for Studies in Civilizations , being one of a total of ninety-six planned volumes. The author is a statistician and computer scientist by training, who has concentrated on historical matters for the last ten years or so. The book has very ambitious aims, proposing an alternative philosophy of mathematics and a deviant history of the calculus. Throughout, there is an emphasis on the need to combine history (...) and philosophy of mathematics, especially in order to evaluate properly the history of mathematics in India, in particular the history of the calculus.The pivotal goals of the book are to oppose the Eurocentric account of the history of science in general and mathematics in particular; to avoid the usual philosophical idea of the centrality of proof for mathematical knowledge, in favour of the traditional Indian notion of pramāṇa [validation] encompassing empirical elements and emphasizing calculation; to analyze the thousand-year-long development of infinite series in India, starting in the fifth century; and to show ‘how and why the calculus was imported into Europe’ from about 1500. The result is a picture in which inputs from the Indian subcontinent and epistemology are the driving forces of the history of mathematics, as people in the European subcontinent struggle to adopt new calculation techniques from the East in spite of Western philosophico-religious biases . Thus, a ‘first math war’ involved the algorismus de numero indorum, adopted for practical reasons, which forced Europeans to modify their epistemology of number and quantities. A ‘second math war’ revolved around infinite series and the calculus, since the background of Western epistemology created ‘spurious difficulties’ about infinities and infinitesimals, partially resolved with theories of real numbers. And a ‘third math war’ is …. (shrink)