José Ferreirós – Universidad de Sevilla

The emergence of the set-theoretical approach in mathematics  

The history of Cantorian set theory is at present very well known, thanks to the work of Cavaill?s, Meschkowski, Grattan-Guinness, Dauben, Purkert, and others. But the same cannot be said about the history of set-theoretical mathematics. To begin with, the need to differentiate between a theory of sets and a set-theoretical approach to mathematics has frequently not been clear in the mind of historians. This may be due to a certain prejudice – a linear model of developments – according to which theoretical shifts in mathematical research would always be the effect of an application of new theories. The case of set theory – and, I believe, many others – shows that, contrary to that expectation, it is a rather frequent phenomenon that new theories develop in the wake of shifts in mathematical practice. Also important in the relative neglect of some figures (notably Riemann) is the fact that many, and perhaps most, mathematicians and historians do not agree with Cantor’s opinion that “in mathematics, the art of posing questions is of more consequence than that of solving them”.

The talk will explore the gradual rise of a set-theoretical approach to mathematical problems in German authors of the period 1850–1914. This includes Dirichlet’s work on real functions, Riemann’s work on function theory and differential geometry, Dedekind’s work on algebra and algebraic number theory, the work of Cantor and others on real functions and point-sets, attempts to bring to completion the arithmetization of mathematics, and many other later contributions (most notably, those of Hilbert). As can easily be seen from this list, set theory did not just emerge from the field of analysis, as used to be the opinion of historians on the basis of their study of Cantor’s career.

In the talk we shall likewise consider the broad diffusion of set theory as the preferred language for modern mathematics around 1900, and the leading role played by Hilbert in this episode. Also present will be the emergence of logicism, which we shall understand as a result of the rise of set-theoretical mathematics, coupled with an old, “logical” conception of sets; and the problems posed by the set-theoretical paradoxes, showing that this “logical” conception was untenable and calling for axiomatization and formalization.

Ian Mueller  - The University of Chicago

A minimalist Interpretation of Plato's Account of Arithmetic and Geometry in the Republic.

In this paper I describe and discuss what I call the minimalist interpretation put forward by Myles Burnyeat in "Plato on why mathematics is good for the soul," forthcoming in T.J. Smiley (ed.), Mathematics and Necessity in the History of Philosophy (Oxford University Press). I will focus on the passages which are frequently discussed, from the end of book VI (the divided line) and book VII (the mathematical curriculum).

Erhard Scholz – Universität Tübingen

H. Weyl's treatment of the problem of space and the origin of gauge structures   

 In  1918 Hermann Weyl generalized Riemann's concept of differential geometric metric in a manifold. Weyl wanted,  by philosophical and physical reasons,  to build on much stricter principles than Riemann from the "infinitesimal neighborhoods" of the points in the manifold. He took Levi-Civita's interpretation (1917) of the classical Christoffel symbols as defining parallel transport and thus an (affine) connection in the manifold as a starting point for his generalization in a "purely infinitesimal" way. Thus Weyl introduced a "length connection" in some analogy to affine connections. This lead him (in 1918) to the first "gauge" structure for geometry which he immediately tried to use in field theory for a unification of electromagnetism and gravitation and field theory. After Weyl lost faith (in late 1921) in the direct physical meaning of his approach, he undertook a conceptual "analysis" of basic principles for congruence concepts formulated in a purely infinitesimal context. In this seemingly philosophical "analysis of the problem of space" (1921 -- 1923) he explored conceptual structures that turned out useful for his later (1929) reformulation of his gauge ideas in the context of quantum theory (a comprehensive unification of gravitation and the Dirac electron field).  In this talk an introduction to the basic ideas  of Weyl's approach to infinitesimal geometry and in particular the motivation for his generalizations of Riemannian geometry will be given. The link to unified field theory will be hinted at, but not elaborated in detail. 

Michael N. Fried

The Use of Analogy in Book VII of Apollonius' Conica

Apollonius of Perga's Conica, like almost all Greek mathematical works, relies heavily on the use of proportion, of analogia. Analogy as the assertion of a resemblance, however, also plays a role in the Conica. The analogy between the conic sections and the circle is a particularly striking example. But these two senses of analogia, proportion and analogy, are united in Apollonius' principal device in Book VII of the Conica, the 'homologue'. In my talk, I shall suggest that Apollonius' development of the 'homologue' shows that proportion ought to be viewed not only as a vital tool in Greek mathematics but also as a means of making images.

Orna Harari-Eshel - Tel-Aviv University

Syllogistic Logic and Greek Mathematical Reasoning: Reassessment of the Relationship between Aristotle's Posterior Analytics and Euclid's Elements

In spite of the absence of historical evidence, it is widely accepted that Aristotle's Posterior Analytics and Euclid's Elements rest on common theoretical foundations. This view is based mainly on the parallelism between the axiomatic structure of the Elements and Aristotle's account of first principles. However, when the forms of reasoning are considered this parallelism is undermined. Not only is syllogistic reasoning absent from Euclid's Elements, but it also seems inadequate to the task of carrying out a mathematical proof. Faced with this discrepancy, some commentators appeal to Aristotle's lenient attitude towards formalization in accommodating syllogistic reasoning to mathematical reasoning. This paper examines these attempts to expand the range of application of syllogistic reasoning, claiming that such attempts do not fall into line with the presuppositions that underlie Aristotle's theory of syllogism.

The first section of this paper considers Aristotle's reasons for excluding hypothetical deductions from his theory of syllogism. This analysis will show that besides the formal constraints, the theory of syllogism imposes conceptual constraints on the logical relations between the premisses and the conclusion. This analysis will set limits on the possible ways of expanding the scope of syllogistic reasoning. In the second section I will analyze syllogistic reformulations of mathematical proofs, showing that the discrepancy between syllogistic reasoning and mathematical reasoning stems from Aristotle's attempt to accommodate mathematical reasoning with the conceptual constraints, imposed by the theory of syllogism. In conclusion, I will argue that discrepancy between syllogistic reasoning and mathematical practice pose a problem that cannot be settled. Yet, the reasons for this discrepancy are to be detected in Aristotle's normative attitude that attempts to account for mathematical reasoning in terms of his theory of substance.

Jeremy Gray- Open University - Milton Meynes

Anxiety and Abstraction in 19th Century Mathematics

Historians of mathematics like to portray the growth of mathematics in the 19th Century as a success story, but there was also a note, hesitant at first but growing to a crescendo around 1900, of anxiety. I shall argue that the mathematics of the 19th Century is marked by a growing appreciation of error. This mounting disquiet about so many aspects of mathematics after 1850 is seldom discussed. I shall argue here that once the safe havens of traditional mathematical assumptions were found to be inadequate, mathematicians began a journey that was not to end in security, but in exhaustion, and a new prudence about what mathematics is and what it can provide.

Ivo Schneider, Universität der Bundeswehr, Munchen

A question of style: The background of Descartes'claim to the creation of a new mathematics.

Much of the history of mathematics before (and even after) the last 25 years was informed by the conviction based on the personal experience of active mathematicians that the development of mathematics is progressive at least in the sense that the set of known theorems at time t1 is a subset of the set of known theorems at time t2 if t1 < t2. Thomas Kuhn dismissed such a view of a monotonic cumulative growth as not representative for the development of science in his "structure of scientific revolutions". However, several attempts to find mathematical revolutions in the sense of Kuhn or one of his later modifiers were not very successful, at least in my eyes.

I shall here present a case study which shows: A history of mathematics concentrating on answering the question: who solved, when, on the basis of which method, which problem, fails completely to explain the enormous impact of Descartes' "G?om?trie" on his own and on the following generation. The role attributed to mathematics by the social group Descartes belonged to and the role attributed to mathematics in Descartes's epistemology are responsible for the development of a new mathematical style which differs completely from the mathematical style of the professional mathematicians in Descartes' time. The change concerning style did not involve a radical change concerning mathematical content and method, in this sense some kind of mathematical revolution, but a considerable extension of the number of those who could produce new mathematics.

Hans-Joachim Waschkies, Universität Kiel

Operative foundations and deductive proofs in Greek arithmetic

As texts written by Plato and Aristotle dealing with philosophy of science show mathematicians of those days aimed at founding arithmetics in an axiomatic - deductive way. However, a quick glance at Euclid's Elements and Diophantus' Arithmetica reveals that this goal was never achieved. The very first axioms the old Greeks had in mind when trying to found arithmetics where definitions supposedly describing the essence of unity and number. But as the Euclidian definition of number and its ancient variants as 'multitude composed of units' shows they did not make explicit the concept of number but of figure. Another point to be considered is that Euclid's first arithmetical proposition stating a criterion by which to decide wether two numbers are prime to one annother using the well known algorithm producing the greatest common divisor of them is proved without recourse to an explcitely stated axiom. These are pieces of circumstantial evidence that old Greek mathematicians actually founded their science in first sentences induced by their competence in calculation. Figures are representatives of numbers and therefore numbers can be regarded as equivalent classes of figures. On the other hand Greek, Roman and Arabic numerals as well as ps?phos - configurations representing numbers are figures constructed according to fixed rules. This is the idea behind an operative foundation of arithematics presented by Paul Lorenzen in 1950. It starts with rules for the construction of admissible figures. These give rise to statements about equivalence classes represented by subsets of those figures (and afterwards interpreted as numbers). The thruth of this statements is not founded in logic conlusions form a set of axioms. They are true because they are statements about classes of correctly constructed figures. These statements on their part then can be used premiss of deductive proves. The aim of the paper is to show that old Greek arithmetic tacidly was done in this way.

Chr. Marinus Taisbak

Completing Euclid's Data

Since 1981 (twenty years by any counting) I have endeavoured to complete, if not Euclid's Data, then my own understanding of that booklet. After spending one decennium in reading and thinking of the subject, another in also talking about it, I still do not know of any completion, rather I am aware of a certain impatience in our learned republic with my lingering in such trifles. So let me present to you one hurdle that I need help to get
over: The fourth definition in the Data and the propositions "proved" by that, e.g. Dt 28 (presented in the following page).

Some questions to be answered:

What does it mean to "always hold the same place" (Def 4)?

What is meant by the Greek verb "metapiptein" (Dt 28 and alibi)?

Would Dt 28 fit into the Elements Book I after proposition 30?
Why not?

If the word "uniqueness" keeps popping up in your mind when you read about the property "Given in Position", what is wrong, then, with translating the term by "Unique"?

Suppose that we agree on some answers to these questions, are we entitled to "repair" or "supply" a text that has been used by generations of mathematicians without any scruples? Whom do we think we are?

Such was the state of my mind by April 1, anno Domini MMI. May may change it.

Ruth Glasner

From Physics VI to VII: A turning point in the application of mathematics to physics

Aristotle's first task as a natural scientist, I suggest, was the extension of the concept of continuity, which was a mathematical concept and defined as infinite divisibility, to physical entities, notably time and motion. This task was accomplished in Physics VI, which, I argue, is the earliest book of the Physics. Aristotle's approach to the study of natural science at this stage is geometrical. In book VI Aristotle studies the motion of one body. In this book VII he studies a more complex situation, that of the interaction between two bodies. At this stage he realizes that the geometrical approach of VI is no longer sufficient, and develops a new concept of continuity which significantly changes his approach to the study of natural science. The new approach leads to the definition of continuity of Physics V. The paper studies Aristotle's early geometrical approach, the gradual abandoning of this approach and the development of the first physical concepts.

Alain Bernard

Analysis and sophistics in Pappus's 'Collectio Mathematica’

The mathematical Collection written by Pappus of Alexandria in the 4th cent. AD has usually been studied as a useful commentary upon Greek mathematics of older times (that of 5th, 4th or 3rd cent BC). It is only recently that historians like Serafina Cuomo have shed a new light on Pappus's work by setting it in its own cultural context: that of mathematics in Late Antiquity. Thus, it has been shown that while Pappus shared with many of his contemporaries a strong respect for the past, his way of doing mathematics is original and deserves to be studied for its own sake. Even the use Pappus makes of the past appears to be a present-oriented reconstruction.

Continuing along the same line of thought, I shall present my research upon what I call the sophistic context of Pappus's work. Namely, some passages of the Collection show that Pappus has a lot in common with the sophists of his time who were very influential in the Greek world under Roman control. In particular, an accurate understanding of the sophistic techniques can shed a new light on the beginning of book three of Pappus's Collection, as I shall argue. The comparison does not merely illustrate the importance of the second sophistry to Pappus's style. It could also lead toward another understanding of the technique of analysis than the usual one which refers to the 7th book of Pappus's Collection.

Moritz Epple

The importance of local traditions: The making of modern knot theory in Vienna and Princeton during the 1920's

A decisive event in the formation of the modern topological theory of knots was the invention of the first effectively calculable invariants distinguishing between different types of knots and links. These invariants were invented twice, at about the same time (1925-26) but in two quite different intellectual milieus. The mathematicians primarily involved were J. W. Alexander, working in Princeton, and K. Reidemeister, strongly influenced by Vienna's mathematical and philosophical traditions. While a focus on the mathematical results of Alexander and Reidemeister might lead historians to underline the international and universal character of modern mathematics, a closer analysis of their research activities reveals how strongly local traditions and knowledge ressources influenced both Alexander's and Reidemeister's mathematical productions. One of the points of such an analysis is that the local nature of mathematical research is not only a matter of "local context vs. universal content". Even the precise formulations of Reidemeister's and Alexander's results, and the techniques used to prove these results, bear traces of different intellectual environments. As a historiographical tool for understanding such differences, I will propose to use a notion of 'epistemic configurations of mathematical research'.

Vassilis Karasmanis

Proclus (in Eucl., 213) gives a description of a geometrical method called 'apagoge' (reduction) and after that adds that "the first to effect reduction (apagoge) of difficult geometrical propositions was Hippocrates of Chios". Also Aristotle, in his Prior Analytics, speaks about a kind of syllogism called apagoge and gives a mathematical example that reminds the squarings of Hippocrates of Chios. Now, relying on these testimonies, but mainly analysing Hippocrates' fragments on the squaring of lunes (reported by Eudemus), I try to find more about this method. My conclusion is that 'apagoge' is an early form of the method of analysis and synthesis and consists roughly in reducing one problem (or theorem) to another. Reductions can be continued until we arrive at something already known or something that is possible to be proved directly.

David Rowe

Mathematical Productivity as an Expression of Oral and Written Culture: The Case of G?Öttingen

Historians of mathematics have traditionally been concerned with an exclusive set of "end products" of mathematical activity: famous theorems and great works as found in those texts that have "stood the test of time." True, a great deal of recent activity has been devoted to non-Western mathematics, so that our horizons have been broadened and our awareness of cultural differences and their importance deepened. But these do not bear directly on the historiography of traditional European and North American mathematics during the modern era. In this respect our discipline may benefit by adopting new approaches that aim to illuminate the process of making mathematics rather than the end results that find their way into certain famous or less well-known texts. Accounts of how mathematics gets produced, why it is made and for whom, are of course far more elusive than the published works of professionals. Yet, despite this scarcity, historians with a trained eye can still find plentiful clues of the mathematicians' workaday world. If one is willing to speculate beyond the "hard evidence" by exercising a bit of historical imagination, I'm convinced that it is possible to reconstruct a compelling picture of various communities and local communal enterprises.
After setting the scene with some remarks about the G?ttingen mathematical community in the heyday of Klein and Hilbert, I will sketch the significance of a single event that took place there in the summer of 1915 when Einstein delivered six lectures on an early form of his general theory of relativity. My purpose is not to recount one of the more famous events in Einstein's career. On the contrary, I wish to illustrate how a totally new picture of what transpired emerges by merely focusing on the precise circumstances involved. Doing so, means focusing on the event and its aftermath as phenomena within an intensely oral culture. I will suggest that with this approach one can go beyond the recent work of Corry, Renn, and Stachel in order to give a holistic account in which the profound importance of Einstein's visit to G?ttingen is finally brought to light.
If time permits, I'll close with some brief remarks that point to Richard Courant's role in stabilizing the largely oral knowledge he encountered as a student and later professor in G?ttingen. The written word gained ascendancy with Springer's yellow series, which Courant edited, as well as the classic text of Courant and Hilbert, Mathematische Methoden der Physik (1924). Courant's personal relationship with the publisher Ferdinand Springer proved to be indispensable in bringing about this transformation from the spoken to the written word.

Karine Chemla

 This talk aims at giving an overview of results obtained in the last decades about the practice of mathematical proof in ancient China. By doing so, it will highlight some of the distinctive features of the practice of mathematics in this tradition. First, the proofs produced address the correctness of algorithms, which were the basis of mathematical activity in ancient China. Secondly, one type of proof is referred to by a technical name : the proof that brings into play a visual auxiliary, and we shall indicate the specificities of these auxiliaries and their uses. Thirdly, proofs as practised in ancient China seem to aim at bringing to light some fundamental algorithms underlying the variety of different procedures. We shall show that problems play a key part in this aspect of the practice of proof. Lastly, we shall evoke how proofs relate to the activity for which they were developed : the exegesis of canons.

Ian Mueller

 A minimalist Interpretation of Plato's Account of Arithmetic and Geometry in the Republic.

 In this paper I describe and discuss what I call the minimalist interpretation put forward by Myles Burnyeat in "Plato on why mathematics is good for the soul," forthcoming in T.J. Smiley (ed.), Mathematics and Necessity in the History of Philosophy (Oxford University Press). I will focus on the passages which are frequently discussed, from the end of book VI (the divided line) and book VII (the mathematical curriculum).

Annette Imhausen, Dibner Institute MIT

Egyptian Mathematical Texts in their Contexts

Our knowledge of mathematical techniques in Ancient Egypt is exceedingly limited. Papyri are fragile artifacts, so we need not be surprised that only a few have survived, documenting the mathematical training and skills of the scribes who oversaw a variety of economic, technological, and administrative activities within Egyptian society.
Yet the historiography of ancient Egyptian mathematics has suffered from another deficiency, namely the lack of communication between the two scholarly communities interested in it: historians of mathematics and Egyptologists. Each of the groups has mostly ignored the results of the other which has often resulted in an anachronistic presentation of the supposed algebraical content of the Egyptian problem texts. Having been educated in History of Mathematics and Egyptology, my thesis aimed to rework the entire mathematical problem texts, focusing on three main aspects:
preparing a new edition of the sources, including one formerly unpublished mathematical fragment, linking the settings of the problems to their appropriate backgrounds, and analysing the problems according to their internal algorithmic structure. As the title indicates, my talk concentrates on the second aspect, the embedding of the problems in their concrete context. Using several examples, I will argue that the consideration of this background is necessary to estimate the significance of the problem. A modern reader can only appreciate the significance of a mathematical exercise dealing with calculations for conversions of grain into quantities of bread and beer if he takes two factors into account:
The heart of the Egyptian administrative workforce was designed to implement a ration system.
The Egyptian ration system was based on quantities of grain, and of bread and beer of a specific quality which is expressed by its grain-content. Moreover, the consideration of the background and the socio-economic functions is in some cases the only way to understand the meaning of certain calculations found in the problems. Not only is the mathematical terminology derived from everyday vocabulary, but there are also quite a few instances where expressions used in administrative or handicraft practices have made their way into the mathematical texts. Furthermore there are a number of constants that also derive from the underlying processes, and these are used often without explicit explication within the problems.
In summary, paying attention to the close connection between mathematics and daily life found in Ancient Egypt provides modern readers with further tools to analyses individual problems that make it possible to understand the scope of Egyptian mathematics.