abstracts

Lakatos's seminal work and Ph.D. dissertation Proofs and Refutations introduced methods of proofs and refutations as a case study. Proofs and Refutations is about the history of the Euler formula V - E + F = 2 for three dimensional polyhedra. Lakatos thought the history of polyhedra presents a very nice example for his philosophy and methodology of mathematics and geometry.

In this study, we focused on the mathematical and topological properties which are incorporated in Lakatos's methodological approach. For each example and counterexample Lakatos utilized, we will briefly give its topological counterpart. In this way, we will be able to present the mathematical background and basis of Lakatos's philosophy of mathematical methodology in the case of the Euler formula, and thereby develop some intuitions about the functions of his notions of positive and negative heuristics.

Gomes Teixeira (1851-1933) was the most prominent Portuguese mathematician after Pedro nunes, who lived in the 15 th century. Gomes Teixeira's contributions to mathematics were internationally recognized given the publication of his mathematical work in the most popular scientific journals. Yet, his work and interests spanned other areas as well. For example, his concern about the diffusion of mathematics led him to found the Jornal de Ciências Matemáticas e Astronómicas [Journal of Mathematical and Astronomical Sciences] (1877) and the Anais Científicos da Academia Politécnica do Porto (Scientific Annals of the Polytechnical Academy of Porto] (1905), two important publications which included comments and articles written by some of the most important mathematicians. Moreover, Gomes Teixeira wrote a number of texts which were quickly used for didactical purposes. In particular, he wrote the Curso de Análise Infinitesimal [Lectures on Analysis], a publication that contributed to a significant change in the teaching of mathematics in Portugal. However, his most internationally recognized work is the Tratado de Curvas Especiais notáveis, tanto Planas como Torsas [Treatise of the notable special curves, both plane and twisted] (1897), which was granted the Spanish Royal Academy of Sciences award, and has been reprinted several times and in various countries. Towards the end of his career, Gomes Teixeira became more and more interested in the history of mathematics in Portugal, giving several talks which were then collected and published: Panegíricos e Conferências [Panegyrics and Lectures] (1925) and Históra das Matemáticas em Portugal (1933).

In this paper I address a dilemma arising from the linguistic behaviour of arithmetical expressions in two basic ways: they occur, either as singular terms in identity statements or as predicates of concepts in adjectival statements. However, those types of syntactical behaviour give rise to opposite accounts of the ontological status of natural number. The substantival use of arithmetical expressions supports the interpretation of natural numbers as abstract particulars while the adjectival use of arithmetical expressions supports the interpretation of natural numbers either as properties of physical collections or as properties of sortal concepts, i.e. as second-order properties.

I take under consideration both interpretations and sketch the special difficulties of each position according to recently discussed aspects of the 'arithmetical platonism' issue. Then I investigate the relationship between the substantival form

The first option is to examine whether a reductionist approach could help at distinguishing the most fundamental of the two accounts. The second option is based on Ramsey's argument that no essential difference between particulars and universals can be asserted on syntactical grounds. Then I move on to show how an equivalence between the substantival and the adjectival account of natural number might be settled and how we can construe them as opposite sides of the same coin.

In this talk I will discuss one or several highlights from the developments in mathematical crystallography after Bieberbach's solution of the 18th Hilbert problem.

A Matter of Opportunities? Johannes Ciermans, a Jesuit mathematician in the early modern Southern Netherlands

Catholicism most certainly does not rule out science. Recently, historians of science redefined the contours of especially the Jesuit encounter with the new science more positively. It is, for the historian of science, not always easy to avoid the manifold pitfalls of misconceptions concerning the nature of the participation of clerics in early modern science. This paper will focus on Joannes Ciermans (1602-1648), a Jesuit mathematician born in the town of 's-Hertogenbosch (the Netherlands) but who trained at Flemish Jesuit colleges. The paper will not be restricted to a mere biographical survey. We only use the biographical data as a means to reflect on the question of how this Jesuit, a cleric in the, at the time, turbulent Spanish Netherlands, became involved in mathematics. In this context, we should definitely bear in mind the multiple identities incorporated in this Jesuit, and also the filters through which this Jesuit received his mathematical input and spat out his mathematical ideas and works. A lot can be said about these things, but my paper will be one about opportunities.

Although in early modern Europe, all secular learning was subservient to theology, being a Jesuit created for Ciermans numerable opportunities to get enrolled in mathematics, opportunities which he never would have had outside the Jesuit Order. Paradoxically, as being a Jesuit gave Ciermans, in the beginning of his career, the required impulses to get involved with mathematics; at the end of his life his Jesuit identity seems to have stood in the way of his further development as a mathematical practitioner.

Helen De Cruz (Centre for Logic and Philosophy of Science, Free University of Brussels)

Contemporary philosophy of mathematics has paid relatively little attention to epistemology. This leaves compelling questions like 'how are humans capable of knowledge of mathematical objects?' and 'does mathematical knowledge differ fundamentally from other kinds of scientific knowledge?' largely unanswered. Even those philosophers of mathematics who focus on mathematical practice have largely ignored the cognitive processes underlying this practice, paying more attention to the socio-cultural level. Experimental studies in neuroscience, developmental psychology and animal cognition have probed the cognitive basis of mathematics. From this nascent cognitive science of mathematics it becomes increasingly clear that mathematical practice depends on the evolved structure of the human brain. Number, for example, depends on an innate number sense that we share with other vertebrates. I shall argue that these experimental findings have profound implications for the philosophy of mathematics. Specifically, I propose that they can provide an answer to Benacerraf's empiricist challenge: how can we acquire knowledge of abstract mathematical objects, if not through experience? This can be addressed through the cognitive science of mathematics, which views mathematical knowledge as an interaction of the human brain and the world. This paper fits in the wider context of scientific naturalism, which views philosophy as continuous with science, rather than as an external arbiter. Cognitive science can serve as a gauge to long-standing ideas in philosophy of mathematics. In this approach, the strong realists' claim that mathematical objects reside outside of space-time does not fit, as there is no cognitive mechanism to describe how we are to acquire knowledge of such objects. This research also has broader implications for the philosophy of science in general, as it views scientific knowledge as the result of ordinary cognitive processes which are available to all neurologically normal human adults.

During my research on the implicit and explicit philosophy of mathematics in the mathematics curriculum of Flanders (Belgium) secondary education (age 12-18), I first discovered that there is small scope for philosophy and history of mathematics.

Second, I discovered a large gap between general and vocational education, where general education is taught the capital M (which stands for mathematics as the Western scientific discipline), and where vocational education is taught the small m (which stands for a set of basic competences). Moreover, the more general the education, the larger the M, and the higher the respect in society. Our Western education system persists social culturally inequality.

This brought me to the idea to elaborate on the role of ethnomathematics in Western curricula, or finding room in the large M for the small m . International comparative research on the results of mathematics educations shows us - in the case of Flanders - nearly the best results all over the world. However, we shall criticise the way in which mathematics in schools makes the selection between the 'elite' and the 'losers' and what should be the role of ethnomathematics in western school curricula to overcome this social stratification.

Aristotle's Argument from Universal Mathematics against the Existence of Platonic Forms

In the historiography of Ancient Greek Mathematics a few lines from Aristotle's Posterior Analytics 1.5 have received a great deal of attention. Aristotle there reports about the recent discovery of a proof belonging to what he calls universal mathematics (the mathematics which proves results for several mathematical fields, such as arithmetic and geometry), namely the alternation of proportionality (in algebraic notation: a : b = c : d if and only if a : c = b : d). In the past, he says, this was proved for each kind of quantity separately, but nowadays there is one universal proof. It is commonly thought that Aristotle must be referring to Eudoxus' theory of proportionality and thus reporting about a crucial development in ancient mathematics.

I argue, however, that this cannot be the case, both for mathematical reasons and for reasons of interpretation of Aristotle's purpose in making these remarks. But because of the very same ground why these remarks tell us little about the history of ancient mathematics, they become very interesting for our understanding of Aristotle's philosophy of mathematics. For at several places he argues for the possibility of universal mathematics, according to which mathematics is true of Platonic Forms. I show that, unlike commonly thought, Aristotle shows successfully that Plato cannot accommodate proofs like the universal proof of the alternation of proportionality.

Carnap's Logische Syntax der Sprache (1934) distinguishes a formal mode of speech from a material one. The formal mode is a logical syntax, which universalises a theoretical language through its syntactical conceptualisation. Syntactic predicates are purely formal entities, whose extensions (sets or classes) do not have an independent existence, namely a material foundation. The radical idea resides in Carnap's principle of tolerance, i.e. the conventionality of language forms, such that the universality of an object language is expressed by its conventional, syntactic metalanguage. Carnap did not invent conventionalism out of the blue: he was influenced by Hilbert's Grundlagen der Geometrie (1899), which proposes the first systematic description of an axiomatic geometry. The syntactic formulations of axioms are based on formal relations, which constitute metamathematical definitions about mathematical predicates. Thus, the meanings of geometric concepts depend upon a conventional metamathematics, which has no connection whatsoever with intuitively defined geometric objects. In other words, any mathematical interpretations are acceptable providing that they derive from isomorphic models, which preserve the formal relations of the metamathematical language. The crucial point is that any model may be used as an interpretation, meaning that formal predicates and relations have extensions over a model. Likewise, truth in a model shows that an axiom is true when it is satisfied in a model, or false when it is independent of the other axioms in a model. A typical example of syntactic metalanguage is Carnap's so-called Language II, based on the notion of valuation, such that values are recursively attributed to the variables of a syntactic object language. Valuation implies the substitution of a linguistic entity for an abstract element, taken from the set of all classes of numerals. Yet, Language II avoids Gödel's incompleteness theorems, since these theorems allow that infinite sets are syntactically represented by numerals; in other words, infinite sets arbitrarily fall under syntactic predicates, even though they are undecidable in first-order logic. Therefore, Language II succeeds in conventionally formalising all syntactic properties whatsoever, because the implicit definitions of formal predicates do not assume existence proofs of the predicated properties, which are by contrast, explicitly defined by the material mode of speech.

Generally you may say that the use of history in mathematics education can serve two purposes: (1) to assist in the actual learning of mathematics (mathematical concepts, theories and so forth) and (2) to bring about a dimension of 'meta-mathematics' in mathematics education. By meta-mathematics I am thinking of posing and suggesting answers to questions about what you might call the 'outer issues' of mathematics, like for instance: How does mathematics evolve over time? What forces and mechanisms cause the evolution of mathematics? How does mathematics interact with society and culture? And so forth. So where the second purpose of using history in mathematics education is to teach the students something about these outer issues (perhaps you can even say that it is a matter of general education), the first purpose is concerned with teaching the students something about the 'inner issues' of mathematics.

In my talk I shall first account for some of the arguments usually put forward for using history in mathematics education and propose a categorization of these arguments. Next I shall discuss some of the different ways of involving the history of mathematics in mathematics education and the links between these and the different categories of arguments. Also I shall touch upon the discussion of how history of mathematics is being/or should be done, since this also is linked to the using of history of mathematics in mathematics education. More precisely I am thinking of the internalism/externalism discussion and Rowe's distinguishing between 'cultural historians' and 'mathematical historians'.

At the very end I shall give an outline of my own Ph.D. project, the cases I intend to use in my research and the work which I have to do in the following two and a half years.

The paper will be focused on a little-known article of John von Neumann , first published in 1937: ?Über ein ökonomisches Gleichungssystem und eine Verallgemeinerung des Brouwerschen Fixpunktsatzes". Erg. Eines Math. Coll. Vienna, ed. By K. Menger, 8: 73-82. (A Model of General Economic Equilibrium. Translated into English by C. Morganstern. Rev. Econ. Studies 13: 1-9). This writing contains on the one hand an economic model which outlines a special and unique approach to mathematical modelling in economics of the period and on the other hand a kind of generalization of Brouwer's fixed point theorem . The latter was necessary to prove the existence of equilibria in the economic model. In my presentation firstly I will point out that this article of Neumann played a crucial role both in the history of theoretical and applied mathematics: Kakutani's fixed point theorem, which is also generalization of Brouwer's theorem to the case of correspondences was based on Neumann's generalization. This family of fixed point theorems has proved to be an effective and indispensable instrument of Game Theory , Operation Research and later Mathematical Economics . Secondly I will examine in detail the possible motivations of the author including social factors and interdisciplinary relations. And finally I will argue that among Neumann's papers this is a representative one concerning his special scientific methodology.

It is therefore not strange that the first attempts at developing special methods for these equations were made by astronomers or mathematicians working in this field. In the beginning of the 20th century the Norwegian mathematician Carl Størmer developed a numerical methods of solving these kinds of equations, which he used when he computed trajectories of single electrons in a dipole field. Størmer claims that his method is more efficient than the well-known Runge-Kutta method. He is influenced by a paper of the astronomer G. H. Darwin published in Acta Mathematica in 1897. In addition to his numerical method, Størmer describes a graphical method for solving his problem.

In my talk I will discuss the different numerical methods developed in the beginning of the 20th century for solving systems of second-order differential equations of this kind, and also mention the interaction between the numerical and graphical methods that were used.

The talk is about transferring the interactive model of metaphor to mathematical analogies. I will combine this interactive model with a three-stage scheme to analyse conceptual work in mathematics which was proposed by Pickering and Stephanides. While their underlying question is philosophical - to wit, "How can the workings of the mind lead the mind itself into problems?" - my question is a methodological one for the historian: "How to deal with errors following from the projection of an analogy?" In many cases it seems useful to establish a notion of a "mathematical objects with only a partial meaning".

The talk will discuss the general question in the special case of Kurt Hensel and the genesis of p-adic numbers. In particular, I will exhibit an important example for an object with only a partial meaning in Hensel's theory.

Continuity and Calculus: How assumptions regarding the nature of space and time underpinned William Kingdon Clifford's (1845-1879) development of the bi-quaternion operator.

The 19th century is often referred to as a period of important growth and development in the fields of calculus, algebra, and geometry. Many of those developments were linked to changing conceptions of space. They were also linked to changing conceptions of "continuity", an idea that underlies the technique of measuring movement in space - namely, differential calculus. By looking at the way in which one well-recognised English geometer in the late 1880s - William Kingdon Clifford (1845-1879) - used the concept of "spatial" and "temporal" continuity in his own mathematical and philosophical works, I will argue that the development of non-Euclidean geometrical models, and Clifford's related bi-quaternion operator, were largely the product of the mathematician's materialist belief in a continuous ether medium pervading the universe. I plan to use this case study to also highlight the sociological nature of mathematical thinking as it relates to debate surrounding the distinction between "discrete number" and "continuous quantity" (a distinction originating, for Clifford, in Bernhard Riemann's now-famous 1854 paper, "On the Hypotheses that Lie at the Base of Geometry").

On being industrious: origins of an early nineteenth century Dutch textbook on mechanical engineering

In the early nineteenth century, the Dutch mathematician Gideon Jan Verdam (1802-1866) started to teach mechanical engineering at an industrial school, attached to the Groningen university. His lessons were not intended for university students, but for factory workers who had received little formal education. As he found there was no sufficient Dutch-language textbook available on the subject, he started working on one himself.

In order to write to textbook that was theoretically sound, yet understandable for his students and could also serve as a practical guide on the subject, he had to oscillate between the theoretical and practical approaches of others. He published his work in five volumes between 1828 and 1837. In this talk I will discuss his approach and trace his sources.

Martina Schneider (Saechsische Akademie der Wissenschaften zu Leipzig, Germany)

Shortly after the new theories of quantum mechanics were introduced a few physicists (W. Heisenberg, E. Wigner, H. B. G. Casimir) and mathematicians (H. Weyl, B. L. van der Waerden, J. von Neumann) explored the use of group theory in quantum mechanics. This unfamiliar method was met with a mixed reaction within the physicists' community. The term 'Gruppenpest' (group pestilence) was coined and an alternative group-free approach, developed by J. C. Slater, was more than welcomed. In my talk I will outline this debate. The focus will be around the concept of symmetry, its first appearance, its strategic role in promoting group theory, and its argumentative impact.

Hermann Weyl is one of the greatest mathematicians of the last century. Within the last few years also his philosophical writings have gained attention, in particular the influence of phenomenology as mediated by Weyl's Göttingen teacher Edmund Husserl. However, after gaining his first chair in mathematics at the EHT in Zürich, Weyl got, as he himself put it, "deeply involved in Fichte". This involvement in the German Idealist was initialised by his colleague, the philosopher and Fichte biographer and editor Fritz Medicus. As Weyl acknowledges, it was also through Medicus that "the theory of relativity, the problem of the infinite in mathematics, and finally quantum mechanics became the motivations for my attempts to help clarify the methods of scientific understanding and the theoretical picture of reality as a whole."

In the paper I shall provide some historical background on the relationships between Weyl, Husserl and Medicus, to then focus on Weyl's particular readings of phenomenology and Fichte's "constructivism". Whereas around 1918 Weyl defeats logicism by referring to both Husserl and Fichte, he later on sharply distinguishes between the positions of those two philosophers. Around 1925, during the so-called "Grundlagenkrise", Weyl associates Fichte's philosophy with Hilbertian formalism and claims that "if Hilbert prevails over Brouwer, this also means that phenomenology is knocked out as a foundational science." The paper will address the question of whether this is a sensible reading of both Husserl and Fichte and of what their respective philosophical attempts might tell us for current debates in the philosophy but also in the historiography of mathematics.

Is the practice of doing mathematics different from other practices? What are the basic prerequisites for the activity of mathematizing? What does it mean to be a mathematician? Who sets the standards? If understanding is central - what does it mean? Is mathematical understanding different from common sense understanding? A couple of analogies will be presented, which may serve as descriptive tools. It turns out that neither psychological nor sociological nor historical aspects can be neglected. Some major consequences of shifting the focus from the noun "mathematics" to the verb "to mathematize" will be presented.

Getting to know each other: The role of habituation in analysis in the early nineteenth century

During the first decades of the nineteenth century, mathematicians actively pursued a programme to enlarge the domain of mathematical analysis. Following the optimistic applications of analysis to a large number of already "known" functions, the mathematicians were faced with the even bigger challenge of getting to know hitherto unknown functions defined by a variety of abstract methods. Most notorious were Euler's "inexplicable" gamma function and the elliptic functions introduced by Abel and Jacobi. In my talk, I analyse Abel's definition of elliptic functions (1827) and discuss the role of infinite representations in making these new objects "known" to mathematicians. Despite taking up a large part of Abel's first paper of elliptic functions, these infinite representations were of very little importance to Abel's own research agenda and must, therefore, be understood in a larger context involving his contemporaries. In the process, I will briefly discuss the more general role of "habituation" between changing epistemic techniques.

My presentation will discuss aspects of Theon of Alexandria's Commentary upon the Almagest .

Similarities and differences between mathematical and other types of late antique commentary will be noted, along with comments upon contemporary scholarly discussions on the topic of late antique mathematics and its mode of representation.

Different ways in which Ptolemy and Theon of Alexandria introduce and present mathematics, in the Almagest , and in the Commentary upon the Almagest , respectively, will be considered. The different discussions will be shown to reflect the different agendas of the two authors, in respect of their historical and scholarly milieus.

Finally, the philosophical background to Theon's perception of mathematics, as reflected in Book 1 of his commentary, will be set out.

Is there a conceptual difference between Bolyai's and Lobatchewsky's definitions of non-Euclidean parallelism?

The Swedish mathematician Gösta Mittag-Leffler (1846-1927) studied as a "post-doctoral" student in Paris with C. Hermite and in Berlin with K. Weierstrass between the years of 1873 and 1876, when Weierstrass published his influential Zur Theorie der eindeutigen analytischen Funktionen. During this period, Mittag-Leffler elaborated upon this work of Weierstrass' and proved the now-familiar theorem (in present-day notation) associated with his name:

Suppose Ω is an open set in the plane, A Ω, A has no limit point in Ω, and to each α A there are associated a positive integer m(α) and a rational function

Then there exists a meromorphic function f Ω, whose principal part at each α A is and which has no other poles in Ω.

In this paper I will briefly present a background to the Mittag-Leffler Theorem, including the work of Weierstrass regarding entire functions. I will then discuss Mittag-Leffler's extension of these results to the existence of a meromorphic function with arbitrarily assigned principal parts. Finally, I will investigate Hermite's reception of Mittag-Leffler's theorem through their letters of correspondence (of which Hermite's letters have survived) and his addition of the Mittag-Leffler theorem to his lecture material.

Bart Van Kerkhove (Vrije Universiteit Brussel, Belgium)

Questioning the historicity of mathematics - Evolution or Revolution?

Are there epistemic revolutions in mathematics? On the face of it, the answer is as simple as can be. Mathematics being the traditional locus of absolute certainty, what has once been formally proved there, has been proved for evermore, and cannot be relegated to a lesser status by means of a theoretical perspective switch. Hence, if there is one discipline strictly accumulating results, then it must be mathematics. Another way of raising the central question, bringing out more clearly where the historian might enter the arena, is whether mathematical knowledge is eternal or rather time-dependent.

From this particular perspective, we propose to briefly look into what seems to be one of the most obvious and promising candidates for the philosophical status of revolution or breach in the continuity of mathematical knowledge: the conception and subsequent growth plus eventual rigorization of the infinitesimal calculus. Concentrating on the contributions of Leibniz and Cauchy, among others, authors such as Joseph Dauben, Judith Grabiner and Emily Grosholz have actually defended the revolutionary character of the development of this impressive and successful mathematical instrument, either through specific of its episodes or on the whole.

Turning their backs upon each other, the history and philosophy of mathematics have become blind and empty respectively, Imre Lakatos famously noted (thereby paraphrasing Kant). To us, in general, the main importance of the type of discussion introduced above, is that it indeed allows or even forces want to re-examine bonds of interdependence between both disciplines.

Norbert Verdier (Université de Paris 11 (GHDSO, Orsay), France)

Journals of Mathematics in the early ninteenth century in France, in Belgium, in Germany and ... "here", in Scotland!

In 1826 (December 16 th 1826), Joseph Diez Gergonne, who founded "Annales de Mathématiques Pures et Appliquées" in 1810 in Nîmes (France), wrote to W. H. Talbot (See [Gergonne, 1826]). In his letter, he spoke about journals of mathematics: "Correspondance Mathématique et Physique" in Bruxelles and "Journal für die reine und angewandte Mathematik" in Berlin. More exactly: "Since our relations were interrupted, you will doubtless have remarked, Sir, the birth of two collections which imitate my own: the first is the Correspondence published in Brussels by Messers Quetelet and Garnier, and in which the latter has often copied me word for word without naming me. They have there among their collaborators a Mr Dandelin who is of merit. The other collection is that which Mr Crelle publishes in Germany in Berlin."

Gergonne did not understand the situation in England: "It is surprising that, in a country as enlightened as your England, the introduction of writings relative to the sciences should encounter such inconvenience and restrictions". He hoped: "We should presently require a similar [like Quetelet's journal and Crelle's journal] collection published in London and another in some large town in Italy and then we would lack nothing more in order to be well up to date."

In our paper, we will speak about the first journals of mathematics in the early nineteenth century, in France, in Belgium, in Germany and, of course in England. We will study their dynamic of transmission, their public, and the leading role of the first "editors of mathematics": Gergonne, Garner and Quetelet, Crelle, Liouville, Robert Leslie Ellis and Thomson, Fellow of the Royal Society of here in Edinburgh.

Bibilography:

[Gergonne, 1826]. Lettre à Talbot, 16 décembre 1826. Fox Talbot Museum/Lacock Abbey Collection, N° 01512 1 .

Benjamin Wardhaugh (University of Oxford, U.K.)

Harmonics and acoustics: thoughts on the musical work of Brook Taylor (1685-1731)

In the early eighteenth century the mathematical study of acoustics was pursued by a number of individuals, including Brook Taylor (1685-1731), who published the solution of the differential equation for the vibrating string in 1713. However, another way of applying mathematics to musical also existed, one which used only elementary ratio theory and appealed explicitly to ancient Greek models.

The latter tradition ('harmonics') in the mathematical study of music underwent a revival in late-seventeenth-century England, but has often been assumed to have died out after the publication of Newton's Principia (1687) with the mathematical description of the sound wave. In fact this was not the case, and musical ratio theory continued to attract serious attention from mathematicians and, increasingly, musicians, until the end of the eighteenth century. One of its followers only in the century was the same Brook Taylor, as is shown by wealth of unpublished manuscript in Cambridge.

I will explore some of the intriguing questions this situation raises. What was the perceived relationship between these two mathematical studies of music? One might expect them to arise from two separate cultures, say mathematicians and musicians, or academics and amateurs. In fact the two methods could be pursued concurrently even by a single individual, as in the case of Taylor.

I propose to use the case of music as a route into work which I am starting on the emergence in the early eighteenth century of the concept of 'applied mathematics' as opposed to the 'mixed sciences' of the renaissance. These two concepts, though superficially similar, seem to stand for strikingly different set of assumptions about both the nature of the world and the meaning and proper use of mathematics; assumptions which are exemplified in the case of 'harmonics' and acoustics.

Ilana Wartenberg (Tel Aviv University & University of Paris)

Arithmetic on the Mediaeval Mathematical Hebrew Bookshelf

In the first part of the talk, I will give a short survey of the main arithmetical works written in the Hebrew language between the 11th and the 14th centuries, such as Sefer ha-Mispar (The Book of the Number) by Abraham Ibn Ezra (12th century), the treaties that was probably the first to introduce the decimal system into Christian Europe. Within the corpus of the Hebrew arithmetical manuscripts, one finds strong imprints of the Arabic tradition of Hisab (i.e. calculus). This genre is mainly a pedagogical branch of Arabic mathematics, since it is taught in preparatory institutions for people who train as administrators. Hisab books teach practical arithmetic needed for the calculation of taxes, and some geometric calculations as well as algebraic algorithms to find the unknown from the given unknown. In the second part of my presentation, I will focus on the transmission of practical algebra from Arabic into Hebrew, as witnessed in four texts, the first being Sefer ha-Meshiha ve ha-Tishboret (translated into Latin in 1145 as Liber Embadorum) by Abraham bar Hiyya (12th century), and the fourth one, which is probably the first text in Hebrew to include extensive algebraic notions, algorithms and commentaries. The different algebraic elements within these four texts will be juxtaposed.

Steven Wepster (Universiteit Utrech, Netherlands)

Polygons and polynomials, and Van Ceulen's amazing precision

The Dutch mathematician Ludolph van Ceulen (1540-1610) is best remembered for the successful computation of 35 digits of pi, which were eventually engraved on his tombstone. Less well-known, but perhaps more interesting, are his computations for the lengths of sides of regular polygons. This work involved the numerical approximation of roots of polynomial equations of high degree. Although we do not know how he approximated those roots, the scarcity of errors in his results suggests that he possessed a quite efficient method.

Eva Wilhelmus (University of Bonn, Germany)

'Does Professor Jones know that the Jones conjecture is true?' - some empirical results about the conditions for knowledge ascriptions in mathematical practice

In the philosophy of mathematics we deal with the epistemological issue 'when does a person X know that a certain mathematical statement p is true?' or 'what constitutes the knowledge of mathematical theorems?' It seems quite natural to take the availability of a proof of p as an adequate criterion for knowing that p . But what is a proof, or rather: what makes a proof a reliable source for knowledge? At first glance, this has something to do with the notion of formal correctness, and so traditional accounts of mathematical knowledge usually end up with a criterion like:

X knows that p iff X has, at least in principle, available a formalizable proof of p .

But this in turn appears to be somehow at odds with mathematical practice, where formalizable proofs seem to play no major role in fact.

Taking mathematical practice seriously, philosophical approaches in the epistemology of mathematics should also consider the use of 'to know' by those who do mathematics. Methodologically, this means to investigate the empirical questions:

What are the conditions for knowledge ascriptions in actual mathematical practice?

What role does formal proof or formalizability play?

and use the results as input for a philosophical theory of mathematical knowledge. Under this premise, if the above criterion was adequate we should find that mathematicians apply it when ascribing knowledge.

As a part of my PhD thesis in the philosophy of mathematics, I have recently conducted a web-supported survey among research mathematicians to find out about (1) the actual use of 'to know' in mathematical practice and (2) some of their general beliefs about the nature of mathematical knowledge and mathematical truth. In my talk, I will present first results of this empirical project and suggest some possible conclusions from these results concerning the role of formalizability.

Ingo Witzke (Universität zu Köln, Germany)

Leibniz' Calculus from a different perspective

Gottfired Wilhelm Leibniz has enormously contributed to the development of modern analysis by creating the calculus of differentials. Amazingly, until today it seems a difficult task to show that Leibniz's calculus is a consistent theory having an enormous range of applications - the burden of the indefinable infinitely small quantities is heavy and has detained researchers in the "Leibnizian Labyrinth".

Now, there are different ways to approach to Leibniz's concept of the differential and integral calculus; one is to depict Leibniz's calculus with all its inconsistencies; this route has been taken e.g. by H. Bos [ Archive for the History of Exact Sciences 14 (1974/75)].

H. Struve and H. J. Burscheid (University of Cologne) have chosen a genuinely different approach to analyse Leibniz's Calculus, drawing attention to the fact that the calculus provides correct answers to a variety of mathematical problems even though it is founded on the problematic indefinable differentials.

Starting from this point we try to analyse the calculus by reconstructing it in " praxi " and developing a fitting analytical model.

This model, taken from modern analytical geometry, provides us with a solid measure for character, consistency and capability of Leibniz's theory avoiding semantical interpretations of the differentials as infinitesimal small quantities.

The talk will begin with a short introduction on our methods of analysing historical mathematical theories and will progress with an overview of noticeable problems and results regarding Leibniz's Calculus.

The first systematical presentation of Leibniz's Calculus was developed by Johann Bernoulli. Consequently, on the basis of selected examples from Johann Bernoulli's Lectiones de calculo differentialium and the Lectiones mathematicae de methodo integralium aliisque (1691/92) I present our work, resulting in the estimation of Leibniz's approach as a consistent and powerful (empirical) theory, raising the question why it was abandoned.


	17th Novembertagung - Edinburgh 2006

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		INVITED SPEAKER Professor David Bloor (University of Edinburgh) OTHER SPEAKERS Can Baskent (Universiteit van Amsterdam, Netherlands) Counterexamples in Proofs and Refutations Lakatos's seminal work and Ph.D. dissertation Proofs and Refutations introduced methods of proofs and refutations as a case study. Proofs and Refutations is about the history of the Euler formula V - E + F = 2 for three dimensional polyhedra. Lakatos thought the history of polyhedra presents a very nice example for his philosophy and methodology of mathematics and geometry. In this study, we focused on the mathematical and topological properties which are incorporated in Lakatos's methodological approach. For each example and counterexample Lakatos utilized, we will briefly give its topological counterpart. In this way, we will be able to present the mathematical background and basis of Lakatos's philosophy of mathematical methodology in the case of the Euler formula, and thereby develop some intuitions about the functions of his notions of positive and negative heuristics. Helena Castanheira Henriques (Universidade do Porto, Portugal) Rosa Ribeiro (Universidade do Porto, Portugal) Rosa Tomás Ferreira (Universidade do Porto, Portugal) Gomes Teixeira: Mathematician, historian, and educator Gomes Teixeira (1851-1933) was the most prominent Portuguese mathematician after Pedro nunes, who lived in the 15 th century. Gomes Teixeira's contributions to mathematics were internationally recognized given the publication of his mathematical work in the most popular scientific journals. Yet, his work and interests spanned other areas as well. For example, his concern about the diffusion of mathematics led him to found the Jornal de Ciências Matemáticas e Astronómicas [Journal of Mathematical and Astronomical Sciences] (1877) and the Anais Científicos da Academia Politécnica do Porto (Scientific Annals of the Polytechnical Academy of Porto] (1905), two important publications which included comments and articles written by some of the most important mathematicians. Moreover, Gomes Teixeira wrote a number of texts which were quickly used for didactical purposes. In particular, he wrote the Curso de Análise Infinitesimal [Lectures on Analysis], a publication that contributed to a significant change in the teaching of mathematics in Portugal. However, his most internationally recognized work is the Tratado de Curvas Especiais notáveis, tanto Planas como Torsas [Treatise of the notable special curves, both plane and twisted] (1897), which was granted the Spanish Royal Academy of Sciences award, and has been reprinted several times and in various countries. Towards the end of his career, Gomes Teixeira became more and more interested in the history of mathematics in Portugal, giving several talks which were then collected and published: Panegíricos e Conferências [Panegyrics and Lectures] (1925) and Históra das Matemáticas em Portugal (1933). Demetra Christopoulou (University of Athens, Greece) Ianus' face of natural number In this paper I address a dilemma arising from the linguistic behaviour of arithmetical expressions in two basic ways: they occur, either as singular terms in identity statements or as predicates of concepts in adjectival statements. However, those types of syntactical behaviour give rise to opposite accounts of the ontological status of natural number. The substantival use of arithmetical expressions supports the interpretation of natural numbers as abstract particulars while the adjectival use of arithmetical expressions supports the interpretation of natural numbers either as properties of physical collections or as properties of sortal concepts, i.e. as second-order properties. I take under consideration both interpretations and sketch the special difficulties of each position according to recently discussed aspects of the 'arithmetical platonism' issue. Then I investigate the relationship between the substantival form and the predicative form and I go on to present two ways to address the dilemma. The first option is to examine whether a reductionist approach could help at distinguishing the most fundamental of the two accounts. The second option is based on Ramsey's argument that no essential difference between particulars and universals can be asserted on syntactical grounds. Then I move on to show how an equivalence between the substantival and the adjectival account of natural number might be settled and how we can construe them as opposite sides of the same coin. Jeanine Daems (Universiteit Leiden, Netherlands) Mathematical crystallography in the 20th century In this talk I will discuss one or several highlights from the developments in mathematical crystallography after Bieberbach's solution of the 18th Hilbert problem. Angelo De Bruycker (Catholic University of Leuven) A Matter of Opportunities? Johannes Ciermans, a Jesuit mathematician in the early modern Southern Netherlands Catholicism most certainly does not rule out science. Recently, historians of science redefined the contours of especially the Jesuit encounter with the new science more positively. It is, for the historian of science, not always easy to avoid the manifold pitfalls of misconceptions concerning the nature of the participation of clerics in early modern science. This paper will focus on Joannes Ciermans (1602-1648), a Jesuit mathematician born in the town of 's-Hertogenbosch (the Netherlands) but who trained at Flemish Jesuit colleges. The paper will not be restricted to a mere biographical survey. We only use the biographical data as a means to reflect on the question of how this Jesuit, a cleric in the, at the time, turbulent Spanish Netherlands, became involved in mathematics. In this context, we should definitely bear in mind the multiple identities incorporated in this Jesuit, and also the filters through which this Jesuit received his mathematical input and spat out his mathematical ideas and works. A lot can be said about these things, but my paper will be one about opportunities. Although in early modern Europe, all secular learning was subservient to theology, being a Jesuit created for Ciermans numerable opportunities to get enrolled in mathematics, opportunities which he never would have had outside the Jesuit Order. Paradoxically, as being a Jesuit gave Ciermans, in the beginning of his career, the required impulses to get involved with mathematics; at the end of his life his Jesuit identity seems to have stood in the way of his further development as a mathematical practitioner. Helen De Cruz (Centre for Logic and Philosophy of Science, Free University of Brussels) How can cognitive science inform philosophy of mathematics? Contemporary philosophy of mathematics has paid relatively little attention to epistemology. This leaves compelling questions like 'how are humans capable of knowledge of mathematical objects?' and 'does mathematical knowledge differ fundamentally from other kinds of scientific knowledge?' largely unanswered. Even those philosophers of mathematics who focus on mathematical practice have largely ignored the cognitive processes underlying this practice, paying more attention to the socio-cultural level. Experimental studies in neuroscience, developmental psychology and animal cognition have probed the cognitive basis of mathematics. From this nascent cognitive science of mathematics it becomes increasingly clear that mathematical practice depends on the evolved structure of the human brain. Number, for example, depends on an innate number sense that we share with other vertebrates. I shall argue that these experimental findings have profound implications for the philosophy of mathematics. Specifically, I propose that they can provide an answer to Benacerraf's empiricist challenge: how can we acquire knowledge of abstract mathematical objects, if not through experience? This can be addressed through the cognitive science of mathematics, which views mathematical knowledge as an interaction of the human brain and the world. This paper fits in the wider context of scientific naturalism, which views philosophy as continuous with science, rather than as an external arbiter. Cognitive science can serve as a gauge to long-standing ideas in philosophy of mathematics. In this approach, the strong realists' claim that mathematical objects reside outside of space-time does not fit, as there is no cognitive mechanism to describe how we are to acquire knowledge of such objects. This research also has broader implications for the philosophy of science in general, as it views scientific knowledge as the result of ordinary cognitive processes which are available to all neurologically normal human adults. Karen François (Free University of Brussels) History and Philosophy in Mathematics Education During my research on the implicit and explicit philosophy of mathematics in the mathematics curriculum of Flanders (Belgium) secondary education (age 12-18), I first discovered that there is small scope for philosophy and history of mathematics. Second, I discovered a large gap between general and vocational education, where general education is taught the capital M (which stands for mathematics as the Western scientific discipline), and where vocational education is taught the small m (which stands for a set of basic competences). Moreover, the more general the education, the larger the M, and the higher the respect in society. Our Western education system persists social culturally inequality. This brought me to the idea to elaborate on the role of ethnomathematics in Western curricula, or finding room in the large M for the small m . International comparative research on the results of mathematics educations shows us - in the case of Flanders - nearly the best results all over the world. However, we shall criticise the way in which mathematics in schools makes the selection between the 'elite' and the 'losers' and what should be the role of ethnomathematics in western school curricula to overcome this social stratification. Pieter Sjoerd Hasper (University of Groningen, Netherlands) Aristotle's Argument from Universal Mathematics against the Existence of Platonic Forms In the historiography of Ancient Greek Mathematics a few lines from Aristotle's Posterior Analytics 1.5 have received a great deal of attention. Aristotle there reports about the recent discovery of a proof belonging to what he calls universal mathematics (the mathematics which proves results for several mathematical fields, such as arithmetic and geometry), namely the alternation of proportionality (in algebraic notation: a : b = c : d if and only if a : c = b : d). In the past, he says, this was proved for each kind of quantity separately, but nowadays there is one universal proof. It is commonly thought that Aristotle must be referring to Eudoxus' theory of proportionality and thus reporting about a crucial development in ancient mathematics. I argue, however, that this cannot be the case, both for mathematical reasons and for reasons of interpretation of Aristotle's purpose in making these remarks. But because of the very same ground why these remarks tell us little about the history of ancient mathematics, they become very interesting for our understanding of Aristotle's philosophy of mathematics. For at several places he argues for the possibility of universal mathematics, according to which mathematics is true of Platonic Forms. I show that, unlike commonly thought, Aristotle shows successfully that Plato cannot accommodate proofs like the universal proof of the alternation of proportionality. Jean-Louis Hudry (University of Edinburgh, U.K.) Logical Syntax and Conventional Metamathematics Carnap's Logische Syntax der Sprache (1934) distinguishes a formal mode of speech from a material one. The formal mode is a logical syntax, which universalises a theoretical language through its syntactical conceptualisation. Syntactic predicates are purely formal entities, whose extensions (sets or classes) do not have an independent existence, namely a material foundation. The radical idea resides in Carnap's principle of tolerance, i.e. the conventionality of language forms, such that the universality of an object language is expressed by its conventional, syntactic metalanguage. Carnap did not invent conventionalism out of the blue: he was influenced by Hilbert's Grundlagen der Geometrie (1899), which proposes the first systematic description of an axiomatic geometry. The syntactic formulations of axioms are based on formal relations, which constitute metamathematical definitions about mathematical predicates. Thus, the meanings of geometric concepts depend upon a conventional metamathematics, which has no connection whatsoever with intuitively defined geometric objects. In other words, any mathematical interpretations are acceptable providing that they derive from isomorphic models, which preserve the formal relations of the metamathematical language. The crucial point is that any model may be used as an interpretation, meaning that formal predicates and relations have extensions over a model. Likewise, truth in a model shows that an axiom is true when it is satisfied in a model, or false when it is independent of the other axioms in a model. A typical example of syntactic metalanguage is Carnap's so-called Language II, based on the notion of valuation, such that values are recursively attributed to the variables of a syntactic object language. Valuation implies the substitution of a linguistic entity for an abstract element, taken from the set of all classes of numerals. Yet, Language II avoids Gödel's incompleteness theorems, since these theorems allow that infinite sets are syntactically represented by numerals; in other words, infinite sets arbitrarily fall under syntactic predicates, even though they are undecidable in first-order logic. Therefore, Language II succeeds in conventionally formalising all syntactic properties whatsoever, because the implicit definitions of formal predicates do not assume existence proofs of the predicated properties, which are by contrast, explicitly defined by the material mode of speech. Uffe Thomas Jankvist (Roskilde University, Denmark) Using History of Mathematics in Mathematics Education Generally you may say that the use of history in mathematics education can serve two purposes: (1) to assist in the actual learning of mathematics (mathematical concepts, theories and so forth) and (2) to bring about a dimension of 'meta-mathematics' in mathematics education. By meta-mathematics I am thinking of posing and suggesting answers to questions about what you might call the 'outer issues' of mathematics, like for instance: How does mathematics evolve over time? What forces and mechanisms cause the evolution of mathematics? How does mathematics interact with society and culture? And so forth. So where the second purpose of using history in mathematics education is to teach the students something about these outer issues (perhaps you can even say that it is a matter of general education), the first purpose is concerned with teaching the students something about the 'inner issues' of mathematics. In my talk I shall first account for some of the arguments usually put forward for using history in mathematics education and propose a categorization of these arguments. Next I shall discuss some of the different ways of involving the history of mathematics in mathematics education and the links between these and the different categories of arguments. Also I shall touch upon the discussion of how history of mathematics is being/or should be done, since this also is linked to the using of history of mathematics in mathematics education. More precisely I am thinking of the internalism/externalism discussion and Rowe's distinguishing between 'cultural historians' and 'mathematical historians'. At the very end I shall give an outline of my own Ph.D. project, the cases I intend to use in my research and the work which I have to do in the following two and a half years. Gergely Khegyi (Corvinus University of Budapest, Hungary) The Fixed Point: An Overview of John von Neumann's methodology The paper will be focused on a little-known article of John von Neumann , first published in 1937: ?Über ein ökonomisches Gleichungssystem und eine Verallgemeinerung des Brouwerschen Fixpunktsatzes". Erg. Eines Math. Coll. Vienna, ed. By K. Menger, 8: 73-82. (A Model of General Economic Equilibrium. Translated into English by C. Morganstern. Rev. Econ. Studies 13: 1-9). This writing contains on the one hand an economic model which outlines a special and unique approach to mathematical modelling in economics of the period and on the other hand a kind of generalization of Brouwer's fixed point theorem . The latter was necessary to prove the existence of equilibria in the economic model. In my presentation firstly I will point out that this article of Neumann played a crucial role both in the history of theoretical and applied mathematics: Kakutani's fixed point theorem, which is also generalization of Brouwer's theorem to the case of correspondences was based on Neumann's generalization. This family of fixed point theorems has proved to be an effective and indispensable instrument of Game Theory , Operation Research and later Mathematical Economics . Secondly I will examine in detail the possible motivations of the author including social factors and interdisciplinary relations. And finally I will argue that among Neumann's papers this is a representative one concerning his special scientific methodology. Kristine Lohne (Adger University College, Kristiansand, Norway) Numerical methods developed in the beginning of the 20th century In celestial mechanics most of the differential equations are of the form y '' = f ( x , y ). It is therefore not strange that the first attempts at developing special methods for these equations were made by astronomers or mathematicians working in this field. In the beginning of the 20th century the Norwegian mathematician Carl Størmer developed a numerical methods of solving these kinds of equations, which he used when he computed trajectories of single electrons in a dipole field. Størmer claims that his method is more efficient than the well-known Runge-Kutta method. He is influenced by a paper of the astronomer G. H. Darwin published in Acta Mathematica in 1897. In addition to his numerical method, Størmer describes a graphical method for solving his problem. In my talk I will discuss the different numerical methods developed in the beginning of the 20th century for solving systems of second-order differential equations of this kind, and also mention the interaction between the numerical and graphical methods that were used. Birgit Petri (TU Darmstadt, Germany) The interaction theory of metaphor for the New in mathematics The talk is about transferring the interactive model of metaphor to mathematical analogies. I will combine this interactive model with a three-stage scheme to analyse conceptual work in mathematics which was proposed by Pickering and Stephanides. While their underlying question is philosophical - to wit, "How can the workings of the mind lead the mind itself into problems?" - my question is a methodological one for the historian: "How to deal with errors following from the projection of an analogy?" In many cases it seems useful to establish a notion of a "mathematical objects with only a partial meaning". The talk will discuss the general question in the special case of Kurt Hensel and the genesis of p-adic numbers. In particular, I will exhibit an important example for an object with only a partial meaning in Hensel's theory. Josipa Petrunic (University of Edinburgh, U.K.) Continuity and Calculus: How assumptions regarding the nature of space and time underpinned William Kingdon Clifford's (1845-1879) development of the bi-quaternion operator. The 19th century is often referred to as a period of important growth and development in the fields of calculus, algebra, and geometry. Many of those developments were linked to changing conceptions of space. They were also linked to changing conceptions of "continuity", an idea that underlies the technique of measuring movement in space - namely, differential calculus. By looking at the way in which one well-recognised English geometer in the late 1880s - William Kingdon Clifford (1845-1879) - used the concept of "spatial" and "temporal" continuity in his own mathematical and philosophical works, I will argue that the development of non-Euclidean geometrical models, and Clifford's related bi-quaternion operator, were largely the product of the mathematician's materialist belief in a continuous ether medium pervading the universe. I plan to use this case study to also highlight the sociological nature of mathematical thinking as it relates to debate surrounding the distinction between "discrete number" and "continuous quantity" (a distinction originating, for Clifford, in Bernhard Riemann's now-famous 1854 paper, "On the Hypotheses that Lie at the Base of Geometry"). Wijnand Rekers (Vrije Universiteit Amsterdam) On being industrious: origins of an early nineteenth century Dutch textbook on mechanical engineering In the early nineteenth century, the Dutch mathematician Gideon Jan Verdam (1802-1866) started to teach mechanical engineering at an industrial school, attached to the Groningen university. His lessons were not intended for university students, but for factory workers who had received little formal education. As he found there was no sufficient Dutch-language textbook available on the subject, he started working on one himself. In order to write to textbook that was theoretically sound, yet understandable for his students and could also serve as a practical guide on the subject, he had to oscillate between the theoretical and practical approaches of others. He published his work in five volumes between 1828 and 1837. In this talk I will discuss his approach and trace his sources. Martina Schneider (Saechsische Akademie der Wissenschaften zu Leipzig, Germany) The role of symmetry in quantum mechanics around 1930 Shortly after the new theories of quantum mechanics were introduced a few physicists (W. Heisenberg, E. Wigner, H. B. G. Casimir) and mathematicians (H. Weyl, B. L. van der Waerden, J. von Neumann) explored the use of group theory in quantum mechanics. This unfamiliar method was met with a mixed reaction within the physicists' community. The term 'Gruppenpest' (group pestilence) was coined and an alternative group-free approach, developed by J. C. Slater, was more than welcomed. In my talk I will outline this debate. The focus will be around the concept of symmetry, its first appearance, its strategic role in promoting group theory, and its argumentative impact. Norman Sieroka (EHT Zürich, Switzerland) Between Husserl and Fichte: Hermann Weyl's Philosophy of Mathematics Hermann Weyl is one of the greatest mathematicians of the last century. Within the last few years also his philosophical writings have gained attention, in particular the influence of phenomenology as mediated by Weyl's Göttingen teacher Edmund Husserl. However, after gaining his first chair in mathematics at the EHT in Zürich, Weyl got, as he himself put it, "deeply involved in Fichte". This involvement in the German Idealist was initialised by his colleague, the philosopher and Fichte biographer and editor Fritz Medicus. As Weyl acknowledges, it was also through Medicus that "the theory of relativity, the problem of the infinite in mathematics, and finally quantum mechanics became the motivations for my attempts to help clarify the methods of scientific understanding and the theoretical picture of reality as a whole." In the paper I shall provide some historical background on the relationships between Weyl, Husserl and Medicus, to then focus on Weyl's particular readings of phenomenology and Fichte's "constructivism". Whereas around 1918 Weyl defeats logicism by referring to both Husserl and Fichte, he later on sharply distinguishes between the positions of those two philosophers. Around 1925, during the so-called "Grundlagenkrise", Weyl associates Fichte's philosophy with Hilbertian formalism and claims that "if Hilbert prevails over Brouwer, this also means that phenomenology is knocked out as a foundational science." The paper will address the question of whether this is a sensible reading of both Husserl and Fichte and of what their respective philosophical attempts might tell us for current debates in the philosophy but also in the historiography of mathematics. Emil Simeonov (Fachhochscule Technikum Wien, Austria) Foundations of Mathematical Practice Is the practice of doing mathematics different from other practices? What are the basic prerequisites for the activity of mathematizing? What does it mean to be a mathematician? Who sets the standards? If understanding is central - what does it mean? Is mathematical understanding different from common sense understanding? A couple of analogies will be presented, which may serve as descriptive tools. It turns out that neither psychological nor sociological nor historical aspects can be neglected. Some major consequences of shifting the focus from the noun "mathematics" to the verb "to mathematize" will be presented. Henrik Kragh Sørensen (Aarhus University, Denmark) Getting to know each other: The role of habituation in analysis in the early nineteenth century During the first decades of the nineteenth century, mathematicians actively pursued a programme to enlarge the domain of mathematical analysis. Following the optimistic applications of analysis to a large number of already "known" functions, the mathematicians were faced with the even bigger challenge of getting to know hitherto unknown functions defined by a variety of abstract methods. Most notorious were Euler's "inexplicable" gamma function and the elliptic functions introduced by Abel and Jacobi. In my talk, I analyse Abel's definition of elliptic functions (1827) and discuss the role of infinite representations in making these new objects "known" to mathematicians. Despite taking up a large part of Abel's first paper of elliptic functions, these infinite representations were of very little importance to Abel's own research agenda and must, therefore, be understood in a larger context involving his contemporaries. In the process, I will briefly discuss the more general role of "habituation" between changing epistemic techniques. Denise Sumpter (Imperial College, London, U.K.) Mathematics in Theon of Alexandria's Commentary on the Almagest , Book 1 My presentation will discuss aspects of Theon of Alexandria's Commentary upon the Almagest . Similarities and differences between mathematical and other types of late antique commentary will be noted, along with comments upon contemporary scholarly discussions on the topic of late antique mathematics and its mode of representation. Different ways in which Ptolemy and Theon of Alexandria introduce and present mathematics, in the Almagest , and in the Commentary upon the Almagest , respectively, will be considered. The different discussions will be shown to reflect the different agendas of the two authors, in respect of their historical and scholarly milieus. Finally, the philosophical background to Theon's perception of mathematics, as reflected in Book 1 of his commentary, will be set out. János Tanács (Budapest University of Technology and Economics, Hungary) Is there a conceptual difference between Bolyai's and Lobatchewsky's definitions of non-Euclidean parallelism? The talk is going to examine the differences between the conceptual-terminological systems of János Bolyai's and Lobatchewsky's non-Euclidean geometry. Last year I came forward with the new and astounding thesis that the traditional double-lined symbol of parallelism ('\|\|') introduced into the work in the 22 nd paragraph of Bolyai's Appendix replaces gramatically the "parallela" term and is used in the terms of equidistant and not in the terms of non-intersecting or first non-intersecting. The consequence of it is that Bolyai and Lobatchewsky have different conceptual-terminological systems concerning the central term "parallel" of their non-Euclidean geometry. This forecasts that the divergence has to be observable when the two systems meet each other in Bolyai's manuscript written in Hungarian, Comments made on Lobatchewsky's Geometrische Untersuchungen . The careful examination of this primary source can help us to answer the question whether there is a difference or not. In the case of finding differentiation we will see how Bolyai managed to handle the problem of translation or interpretation of Lobatchewsky's system in his own conceptual systems. Also it would be a phenomenon of becoming a translator in the sense as Thomas S. Kuhn described in the Afterwords of The Structure of Scientific Revolutions . Laura Turner (Simon Fraser University, Canada) The development and reception of the Mittag-Leffler Theorem The Swedish mathematician Gösta Mittag-Leffler (1846-1927) studied as a "post-doctoral" student in Paris with C. Hermite and in Berlin with K. Weierstrass between the years of 1873 and 1876, when Weierstrass published his influential Zur Theorie der eindeutigen analytischen Funktionen. During this period, Mittag-Leffler elaborated upon this work of Weierstrass' and proved the now-familiar theorem (in present-day notation) associated with his name: Suppose Ω is an open set in the plane, A Ω, A has no limit point in Ω, and to each α A there are associated a positive integer m(α) and a rational function Then there exists a meromorphic function f Ω, whose principal part at each α A is and which has no other poles in Ω. In this paper I will briefly present a background to the Mittag-Leffler Theorem, including the work of Weierstrass regarding entire functions. I will then discuss Mittag-Leffler's extension of these results to the existence of a meromorphic function with arbitrarily assigned principal parts. Finally, I will investigate Hermite's reception of Mittag-Leffler's theorem through their letters of correspondence (of which Hermite's letters have survived) and his addition of the Mittag-Leffler theorem to his lecture material. Bart Van Kerkhove (Vrije Universiteit Brussel, Belgium) Questioning the historicity of mathematics - Evolution or Revolution? Are there epistemic revolutions in mathematics? On the face of it, the answer is as simple as can be. Mathematics being the traditional locus of absolute certainty, what has once been formally proved there, has been proved for evermore, and cannot be relegated to a lesser status by means of a theoretical perspective switch. Hence, if there is one discipline strictly accumulating results, then it must be mathematics. Another way of raising the central question, bringing out more clearly where the historian might enter the arena, is whether mathematical knowledge is eternal or rather time-dependent. From this particular perspective, we propose to briefly look into what seems to be one of the most obvious and promising candidates for the philosophical status of revolution or breach in the continuity of mathematical knowledge: the conception and subsequent growth plus eventual rigorization of the infinitesimal calculus. Concentrating on the contributions of Leibniz and Cauchy, among others, authors such as Joseph Dauben, Judith Grabiner and Emily Grosholz have actually defended the revolutionary character of the development of this impressive and successful mathematical instrument, either through specific of its episodes or on the whole. Turning their backs upon each other, the history and philosophy of mathematics have become blind and empty respectively, Imre Lakatos famously noted (thereby paraphrasing Kant). To us, in general, the main importance of the type of discussion introduced above, is that it indeed allows or even forces want to re-examine bonds of interdependence between both disciplines. Norbert Verdier (Université de Paris 11 (GHDSO, Orsay), France) Journals of Mathematics in the early ninteenth century in France, in Belgium, in Germany and ... "here", in Scotland! In 1826 (December 16 th 1826), Joseph Diez Gergonne, who founded "Annales de Mathématiques Pures et Appliquées" in 1810 in Nîmes (France), wrote to W. H. Talbot (See [Gergonne, 1826]). In his letter, he spoke about journals of mathematics: "Correspondance Mathématique et Physique" in Bruxelles and "Journal für die reine und angewandte Mathematik" in Berlin. More exactly: "Since our relations were interrupted, you will doubtless have remarked, Sir, the birth of two collections which imitate my own: the first is the Correspondence published in Brussels by Messers Quetelet and Garnier, and in which the latter has often copied me word for word without naming me. They have there among their collaborators a Mr Dandelin who is of merit. The other collection is that which Mr Crelle publishes in Germany in Berlin." Gergonne did not understand the situation in England: "It is surprising that, in a country as enlightened as your England, the introduction of writings relative to the sciences should encounter such inconvenience and restrictions". He hoped: "We should presently require a similar [like Quetelet's journal and Crelle's journal] collection published in London and another in some large town in Italy and then we would lack nothing more in order to be well up to date." In our paper, we will speak about the first journals of mathematics in the early nineteenth century, in France, in Belgium, in Germany and, of course in England. We will study their dynamic of transmission, their public, and the leading role of the first "editors of mathematics": Gergonne, Garner and Quetelet, Crelle, Liouville, Robert Leslie Ellis and Thomson, Fellow of the Royal Society of here in Edinburgh. Bibilography: [Gergonne, 1826]. Lettre à Talbot, 16 décembre 1826. Fox Talbot Museum/Lacock Abbey Collection, N° 01512 1 . Benjamin Wardhaugh (University of Oxford, U.K.) Harmonics and acoustics: thoughts on the musical work of Brook Taylor (1685-1731) In the early eighteenth century the mathematical study of acoustics was pursued by a number of individuals, including Brook Taylor (1685-1731), who published the solution of the differential equation for the vibrating string in 1713. However, another way of applying mathematics to musical also existed, one which used only elementary ratio theory and appealed explicitly to ancient Greek models. The latter tradition ('harmonics') in the mathematical study of music underwent a revival in late-seventeenth-century England, but has often been assumed to have died out after the publication of Newton's Principia (1687) with the mathematical description of the sound wave. In fact this was not the case, and musical ratio theory continued to attract serious attention from mathematicians and, increasingly, musicians, until the end of the eighteenth century. One of its followers only in the century was the same Brook Taylor, as is shown by wealth of unpublished manuscript in Cambridge. I will explore some of the intriguing questions this situation raises. What was the perceived relationship between these two mathematical studies of music? One might expect them to arise from two separate cultures, say mathematicians and musicians, or academics and amateurs. In fact the two methods could be pursued concurrently even by a single individual, as in the case of Taylor. I propose to use the case of music as a route into work which I am starting on the emergence in the early eighteenth century of the concept of 'applied mathematics' as opposed to the 'mixed sciences' of the renaissance. These two concepts, though superficially similar, seem to stand for strikingly different set of assumptions about both the nature of the world and the meaning and proper use of mathematics; assumptions which are exemplified in the case of 'harmonics' and acoustics. Ilana Wartenberg (Tel Aviv University & University of Paris) Arithmetic on the Mediaeval Mathematical Hebrew Bookshelf In the first part of the talk, I will give a short survey of the main arithmetical works written in the Hebrew language between the 11th and the 14th centuries, such as Sefer ha-Mispar (The Book of the Number) by Abraham Ibn Ezra (12th century), the treaties that was probably the first to introduce the decimal system into Christian Europe. Within the corpus of the Hebrew arithmetical manuscripts, one finds strong imprints of the Arabic tradition of Hisab (i.e. calculus). This genre is mainly a pedagogical branch of Arabic mathematics, since it is taught in preparatory institutions for people who train as administrators. Hisab books teach practical arithmetic needed for the calculation of taxes, and some geometric calculations as well as algebraic algorithms to find the unknown from the given unknown. In the second part of my presentation, I will focus on the transmission of practical algebra from Arabic into Hebrew, as witnessed in four texts, the first being Sefer ha-Meshiha ve ha-Tishboret (translated into Latin in 1145 as Liber Embadorum) by Abraham bar Hiyya (12th century), and the fourth one, which is probably the first text in Hebrew to include extensive algebraic notions, algorithms and commentaries. The different algebraic elements within these four texts will be juxtaposed. Steven Wepster (Universiteit Utrech, Netherlands) Polygons and polynomials, and Van Ceulen's amazing precision The Dutch mathematician Ludolph van Ceulen (1540-1610) is best remembered for the successful computation of 35 digits of pi, which were eventually engraved on his tombstone. Less well-known, but perhaps more interesting, are his computations for the lengths of sides of regular polygons. This work involved the numerical approximation of roots of polynomial equations of high degree. Although we do not know how he approximated those roots, the scarcity of errors in his results suggests that he possessed a quite efficient method. Eva Wilhelmus (University of Bonn, Germany) 'Does Professor Jones know that the Jones conjecture is true?' - some empirical results about the conditions for knowledge ascriptions in mathematical practice In the philosophy of mathematics we deal with the epistemological issue 'when does a person X know that a certain mathematical statement p is true?' or 'what constitutes the knowledge of mathematical theorems?' It seems quite natural to take the availability of a proof of p as an adequate criterion for knowing that p . But what is a proof, or rather: what makes a proof a reliable source for knowledge? At first glance, this has something to do with the notion of formal correctness, and so traditional accounts of mathematical knowledge usually end up with a criterion like: X knows that p iff X has, at least in principle, available a formalizable proof of p . But this in turn appears to be somehow at odds with mathematical practice, where formalizable proofs seem to play no major role in fact. Taking mathematical practice seriously, philosophical approaches in the epistemology of mathematics should also consider the use of 'to know' by those who do mathematics. Methodologically, this means to investigate the empirical questions: What are the conditions for knowledge ascriptions in actual mathematical practice? What role does formal proof or formalizability play? and use the results as input for a philosophical theory of mathematical knowledge. Under this premise, if the above criterion was adequate we should find that mathematicians apply it when ascribing knowledge. As a part of my PhD thesis in the philosophy of mathematics, I have recently conducted a web-supported survey among research mathematicians to find out about (1) the actual use of 'to know' in mathematical practice and (2) some of their general beliefs about the nature of mathematical knowledge and mathematical truth. In my talk, I will present first results of this empirical project and suggest some possible conclusions from these results concerning the role of formalizability. Ingo Witzke (Universität zu Köln, Germany) Leibniz' Calculus from a different perspective Gottfired Wilhelm Leibniz has enormously contributed to the development of modern analysis by creating the calculus of differentials. Amazingly, until today it seems a difficult task to show that Leibniz's calculus is a consistent theory having an enormous range of applications - the burden of the indefinable infinitely small quantities is heavy and has detained researchers in the "Leibnizian Labyrinth". Now, there are different ways to approach to Leibniz's concept of the differential and integral calculus; one is to depict Leibniz's calculus with all its inconsistencies; this route has been taken e.g. by H. Bos [ Archive for the History of Exact Sciences 14 (1974/75)]. H. Struve and H. J. Burscheid (University of Cologne) have chosen a genuinely different approach to analyse Leibniz's Calculus, drawing attention to the fact that the calculus provides correct answers to a variety of mathematical problems even though it is founded on the problematic indefinable differentials. Starting from this point we try to analyse the calculus by reconstructing it in " praxi " and developing a fitting analytical model. This model, taken from modern analytical geometry, provides us with a solid measure for character, consistency and capability of Leibniz's theory avoiding semantical interpretations of the differentials as infinitesimal small quantities. The talk will begin with a short introduction on our methods of analysing historical mathematical theories and will progress with an overview of noticeable problems and results regarding Leibniz's Calculus. The first systematical presentation of Leibniz's Calculus was developed by Johann Bernoulli. Consequently, on the basis of selected examples from Johann Bernoulli's Lectiones de calculo differentialium and the Lectiones mathematicae de methodo integralium aliisque (1691/92) I present our work, resulting in the estimation of Leibniz's approach as a consistent and powerful (empirical) theory, raising the question why it was abandoned.	abstracts
	History and Philosophy of Mathematics		abstracts
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