MONDAY 22 MAY:

• Jean Dhombres (EHESS): Do attested historical changes in the concept of a mathematical function reveal a multiplicity of origins?

It is well known that the set-theoretical idea of a function as nowadays used by mathematicians was due to Georg Cantor at the end of the 19th century. Indeed he preferred the German word for “mapping” or “application”. It is also common knowledge that the word “functio”, which entered somewhat by chance the mathematical language thanks to Leibniz, was reserved for curves. An historian like H. Bos asserts that functions played no role in the establishment of the Calculus. This raises the question why Leibniz had not chosen the word parameter, which was then common in mathematical physics from Mydorge and Mersenne for conic sections ? It has now been discovered that Leibniz extensively used the word in various manuscripts as early as 1675 (De functionibus as recently published in Hannover in the Leibniz Edition). Very few historians make an association with Newton’s Variable Quantities. Most historians like Youschkevich who initiated a paper on the subject of functions, quote Euler in 1748 as the one to have declared that “functions are the subject of “Analysis Infinitorum”. But they consider that Cauchy is the only one in 1821 to have defined continuity for functions, thus providing by this specification a new field in Analysis, leading to topology. And also making clear that it is necessary to have a domain of definition for a function : this led to the clever remark of Fourier about periodic functions able to represent an arbitrary (integrable) function on an interval. Pierre Duhem, at the beginning of the XX^th century, ventured to claim that Oresme was the one to have invented the notion of function, by linking it to a graphic representation (different from a curve), and providing an analytic description, something that we may call to-day a functional equation. This is a subject that I studied in some details. The highly complex historiography of the word function seems to imply various origins of the concept. My aim is to disentangle words, concepts, and mathematical practices in various domains (curves, series, analysis, algebra) to come to a particular fact : functions are also linked to computations. I direct the attention to the invention of the expression of “computable function” by Alan Turing, in his famous paper of 1936 where he invented his “machine”. So I’ll attempt to write the history of the concept of function as a vis a tergo to use the wording due to Jean Cavaillès, but accepting diverse origins.

• Emmylou Haffner (Bergische Universität Wuppertal, SPHERE): How Dedekind and Weber's edition of Riemann's collected works reshaped (some of) Riemann's texts.

In 1876, are published Bernhard Riemann's Gesammelte Werke, with a selection of unpublished manuscripts from his Nachlass. This edition is the result of several years of protracted work from the two editors, Richard Dedekind and Heinrich Weber. Indeed, Dedekind and Weber’s editorial work involved the tedious process of deciphering, clearing up and reconstructing Riemann’s manuscripts. I will propose, on the basis of their correspondence and elements of both Riemann's and Dedekind's Nachlässe, to analyse the philological and mathematical practices involved in their editorial work. In particular, I will consider how the path from some of Riemann's original manuscripts to their published version involved considerable efforts from the editors in order to understand, polish, and (sometimes) correct the texts.

• David Waszek (IHPST, Université Paris 1): Representational change in mathematics: a case study from the history of the calculus of operations.

It is conventional wisdom that, in mathematics, a good notation makes a big difference. In the same spirit, historians of mathematics regularly claim that a “representational” or “notational” innovation made a certain progress possible, or at least easier. Can we make sense of such claims? What does it mean to say that a new representation introduces radically new content (a true representational change), or that two representations share the same content, but that one presents it in a more tractable way (a notational change)? I will examine two logical and philosophical ways to clarify these issues, and confront them to a historical example of representational or notational change, borrowed from the history of the calculus of operations in the eighteenth century.

• Francisco Barrios (UNAM): The «Lectures on Algebra» on Newton’s appraisal of Ancient Analysis.

Newton's “Lectures on Algebra” extend formally over a period of ten years: from 1673 to 1683, although they were penned by their author, according to the late scholar D. T. Whiteside, in a few months shortly before 1684. To this day, the only evidence that the content of this manuscript was actually intended as class material —or even developed in the aforementioned period, is Newton's own word. Nevertheless those same years cover the so-called newtonian “transition” from his open commitment to Cartesian formalism to his reappraisal of the ancient tradition of Geometrical Analysis, which is present in his later works, most notably the “Principia”.

In this talk I will present some insights for a (re)reading of the “Lectures”, which stresses the evolution of their author’s mathematical thought.

• Olivier Rey (IHPST, Univ. Paris 1): How statistics entered physics.

Now that statistics is a branch of mathematics, it is easy to imagine that its use in the field of human affairs is a by-product of modern science's way of looking at the world. Historical study contradicts such an idea: it is in the field of human affairs that quantitative statistics have developed, and it is only afterwards that it became a method for the natural sciences. Most physicists in the 19th century considered statistics all too human to have a place in the scientific study of nature. It took all Maxwell's authority and persuasion to make statistical analysis a new style of scientific thought in physics.

• María de Lourdes Ramírez-Argonza (UAM-I): Two readings from the Morgan’s Canon in the animal behavioural sciences.

A way in which we can find the presence of change in science has been noticed in the study of animal cognition. In first place, as a symbiotic phenomenon, philosophy and science has started the attempt to explain how the behavior comes about in non-human animals. Philosophy, by one hand, started to think about the set out of the proper conditions to consider the non-human animals as subjects of psychological investigation. Science, by the other hand, started with the methodological issues for the study of animals minds. The philosopher Aristotle, for example, has started the systematic study of the animal behavior, then from the scientific approach from different areas, it had been developed specific empirical research programs about the cognitive capacities. However, from the philosophical point of view, there are a few arguments against the animal minds, one of them is the Morgan’s Canon. This argument established that “an action should not be interpreted as the result of the exercise of a higher psychic faculty if it can be interpreted as the result of the exercise of a lower psychic activity on the psychological scale" (Boring, 1950/1983). This assumption can have two implications in theorizing about animal consciousness: 1. Theory of mind is not preferable over the non-mentalist theories as behaviorism, because the first, not the last, attribute too much for not starting from the simplest, so we can`t consider a non-human animal as being conscious, because consciousness is not the simplest form to explain behavior in non-human animals. 2. There is no necessity to imply certain incorrect attributions of mind behavior where there is not, in the sense that our lecture of the behavior of non-human animals, must be read in different way, not implying the most complicated attribute, but the simplest without denying the possibility of a non-human conscious mind. By the way, science has been more inclined to this second way. At least, it has already been created the Cambridge declaration on Consciousness.

• Alfonso Dairon Rodríguez-Ramírez (UAM-I): Charles Darwin on the mental continuity hypothesis.

We will focus on the importance of the concept of continuity among species regarding the explanation of what it means evolution by natural selection. As we will see the idea that there is a cognitive relationship of continuity among several species it is just a corollary followed from the main premises of the evolutionary theory, just as it was formulated by Charles Darwin. I am referring here to the two parts that constitute the evolutionary theory: Descent with modification and natural selection. Both offer a unify vision of life on earth in which the vast diversity of organism, existing or not, hold relationships of affinity among them.

• Gaëlle Pontarotti (IHPST, Univ. Paris 1): From one century to the next: evolutionary theory facing a new science of heredity.

Modern Synthesis evolutionary theory, which constitutes the theoretical background of evolutionary biologists, is based on a strictly genetic view of heredity and heritable variation. However, specialists have recently produced abundant data about non-genetic inheritance, providing information regarding epigenetic, ecological, behavioral and symbiotic channels of transmission. In this context, a few authors have claimed that the unveiled diversity of heritable variation not only weakens our genetic concept of heredity (Mameli, 2005), but also challenges the core tenets of our gene-centric evolutionary theory (Jablonka & Lamb, 2005; Bonduriansky & Day, 2009). In this talk, I argue that a historical perspective is particularly relevant to shed some light on the subversive potential of non-genetic inheritance as far as evolutionary theory is concerned. I show that such perspective highlights that today’s theoretical uncertainty presents striking echoes with the first ‘crisis’ faced by Neo Darwinism at the dawn of the 20^th century, further to the emergence of a new science of heredity soon to be called genetics. I claim that from one century to the next, the main question addressed to biologists is to know whether the nature of heritable variation is compatible with the core tenets of Neodarwinian theory of evolution, among which one finds gradualism and the preponderant role of natural selection in biological evolution. Finally, I assert that a historical analysis allows underlining the specificities of contemporary challenges.

TUESDAY 23 MAY:

• José Antonio Alonso, Erica Torrens & Ana Barahona (UNAM): The Representations of Human Diversity in the Age of Genomics and the Extended Evolutionary Synthesis.

If we think of concepts that have been decisive for the development of biological taxonomy in general, and of human beings in particular, the term ‘race’ undoubtedly come to mind. Scientific conceptions of race have wandered in space and time since seventeenth century. Naturalists such as Linnaeus and Buffon, philosophers such as Kant and Locke and other scholars such as Hobbes and Bernier have historically contributed to define and restrain human diversity.

In this talk, the authors seek to provide a historiographic account of the concept of race, to comment on its use as a factor of impact on the taxonomy of human beings. In this sense, we are interested in showing how this concept has been represented in the visual characterization of human diversity. Today we know that race belongs to the realm of human culture. However, it is interesting to look at how the world of racial and of human classification have converged into a murky realm of supposed objectivity and of human creativity and imagination. We will review important contributions on the genesis of the concept of race and the racialization of human beings, as well as its recent use in biological and anthropological classifications, in a global context.

To conclude, we will turn our attention on how the new scientific conceptions of race that appeared in late eighteenth century, gave rise to a novel visual culture -focusing on the Mexican scenario-, to show how the discourse of racial hierarchies has been supported by pictorial representation. This in turn, has been important to forge in non-specialists, a deeply entrenched distinction between human diversity and other living entities, positioning humans outside the natural and evolutionary realm.

Also, the authors will put to the fore that the categories used to characterize humans, such as comparative genomic studies, are far from solely technical tools but also incorporate cultural categories that travel from non-scientific to scientific realms and back again.

• Bernardo Yáñez Macías Valadez (Centro Lombardo Toledano): Some Epistemological Issues for Contemporary Biological Anthropology.

The foundation of anthropology as a discipline that concentrates on the human being has its root on the opposite categories of nature-nurture. Since then heated debates have been settled sometimes to support, maintain, or defend, and some others to disqualify, neglect or dismiss such relationship. What cannot be denied is that the discipline has been grounded, among other things, in two perspectives that have been mutually exclusive: social/cultural anthropology and physical/biological anthropology. For more than 150 years these two opposite extremes have mantained a distant relationship. Today there are some proposals that contend a programmatic stance for an integrated anthropology, which consists precisely in the linkage between biological, cultural (material and immaterial), social, linguistic and historic aspects that are the building blocks of our discipline. Here I propose that it is not tenable any more to continue the distinction between the culturalist approach in front of the biological one, and viceversa; but instead, merging both perspectives into a more accurate rapprochement of the human phenomenon. My argument is that contemporary biological anthropology is situated in an analogical circumstance in which ‘The New Physical Anthropology’ was outlined brilliantly in the 1950’s by Sherwood Washburn. What the new advances in the evolutionary perspective have shown until today is the necessity for the amplification of the theoretical toolkit into biological antrhopology; specifically in what has to do with the developmental or ontogenetic dimension. I will take into account some concepts that come from the Evolutionary Extended Synthesis to show that biological anthropology, certainly, requires this conceptual and theoretical extension. I will tackle some relevant aspects of the evolutionary developmental biology (evo-devo) perspective along with fundamental elements of the niche construction theory, as pivotal aspects of the evolutionary extension in as much as it is also useful for the bioanthropological perspective.

• Alba Pérez Ruiz (Centro Lombardo Toledano): The evolutionary perspective of primate social complexity.

From an evolutionary app roach the study of the primatological model on social complex behavior has been relevant to understand the origins of social complexity in humans.

Non human primate societies have been considered socially complex in many different ways. The context of social complexity in primate groups, that involve competitive and cooperative interactions between individuals, has been linked to the evolution of intelligence at least on functions related to social information processing.

Sociality as a product of evolution is a fundamental topic in the study of primate societies. In a primate group, each individual is part of a net of individualized social relations and each relation has a particular history.

The purpose of this work is to analyze the context of sociality in the primatological model as a way to understand the evolution of social complexity.

• Juan Manuel Rodríguez, Ricardo Noguera & Rosaura Ruiz (UNAM): French influences in eugenics and puericulture in Mexico (1900-1930).

In this work, we want to present the reception and dissemination of eugenics ideas in Mexico from 1900 to 1930, especially in the light of French influences such as puericulture. The period we analyze is the moment of transition between the proposals of childcare and the new eugenic practices that took force after 1920, but which began to spread since around 1912, after the first congress of Eugenics held in London. In those proposals, it can be seen the passage of the analogy of plant cultivation in childcare, to the analogy of animal husbandry in eugenics, under the influence of French ideas, both scientific and social. This last point is understood within the well-known framework of the impact of French science in the development of the incipient Mexican science in the late nineteenth and early twentieth century.

• Marina Imocrante (IHPST, Univ. Paris 1, Vita-Salute San Raffaele Univ., Milan): Dealing with expansion and change in philosophy of mathematics.

In this talk, we discuss the general difficulty for (analytic) philosophers of mathematics in dealing with conceptual change.

Following a suggestion from James R. Brown (1999), we submit that this difficulty is primarily due to the dominant view of definitions of mathematical concepts and theories, famously expressed by Hilbert (1899) and Russell and Whitehead (1910). On this view, the main criteria that a definition must satisfy are those of eliminability and non-creativity.

By considering some simple examples from geometry and graph theory, we will submit that the official view of definitions -although its many advantages- cannot account for the historical development of mathematical concepts, nor for the fact that some mathematical theories have multiple representations.

We conclude with the proposal that a different approach to mathematical definitions, e.g. an approach built upon Meir Buzaglo (2002)’s idea of ‘conceptual expansion’, may be suitable to overcome those difficulties.

References

Hilbert, D. (1899), Foundations of Geometry

Russell, B., Whitehead, A. N. (1910), Principia Mathematica, Vol. 1

Buzaglo, M. (2002), The Logic of Concept Expansion

Brown, J. R. (1999), Philosophy of Mathematics

• Eduardo Noble & Max Fernández (UAM-I): The philosophy and methodology behind three definitions of the classical continuum.

The idea of ​​this talk is to compare various constructions of the continuum (and in more generality, of continua), not so much in their technical aspects but with respect to the justification with which they are introduced. The first two would be Dedekind and Cantor. They both recognize that their definitions are equivalent (say, from the point of view of the analysis that can be done with them), but they both also claim that their own construction is better from another point of view (which is the one we want to examine). The third construction would be that of Frege who has many comments on other authors that greatly illuminate his work in the construction of arithmetic.

• David Rabouin (SPHERE, Univ. Paris 7): Mathematics and Philosophy in Leibniz. What's new?

When presenting, in 1989, an overview of 20^th Century Leibniz’s scolarship, Albert Heinekamp identified a clear (and somewhat radical) transformation. According to him, commentators moved progressively from the search for an ideal Leibnizian system (“à la recherche du vrai système leibnizien”) to its complete denial (“refus du caractère systématique de la philosophie leibnizienne”). This evolution is, however, still in its infancy when it comes to studying Leibniz’s conception of mathematics. In fact, the first reading (“systematic”) is the most widely adhered to amongst scholars interested in studying Leibniz from the perspective of the interrelations between mathematics and philosophy. In this talk, drawing on several case studies, I will present the new picture emerging from the consideration of the various changes and variations occurring in Leibniz’s conception of mathematics.

• Pierre Wagner (IHPST, Univ. Paris 1): Kinds of changes and variations in science.

The main reason why scientific change has been much debated among philosophers is that the analysis of the dynamics of science is deeply connected to the issue of scientific rationality. Whence a whole range of central questions (in the philosophy of science) which depend on various possible assessments of scientific change: scientific revolutions, belief revision, test, corroboration, etc. Clearly, scientific changes do not all have the same significance and may depend on various factors: reasons for changes, modes of changes, decisions to change, etc. In this talk, the issue of a taxonomy of changes and variations in science will be raised and principles will be discusses for a possible classification: how to distinguish kinds of changes and variations in science?

WEDNESDAY 24 MAY:

• Marco Panza (IHPST, Univ. Paris 1, Chapman Univ.): What does it mean for a Euclidean theorem to be general?

Euclid’s propositions (problems and theorems) are unquestionably general, but, apparently at least, they are established (namely solved and proved, respectively) by working on a single configuration of objects. How is this possible? This is a classical problem concerning both the understanding and the exegesis of the Elements. It admits two classical answers. The first is that Euclid’s arguments can support general statements, though they concern single configuration of objects, since they are repeatable on any configuration of the relevant sort. The second is that, despite apparences, Euclid’s arguments are not concerned with single configurations of objects, or, at least, not with particular such configurations, but rather with general ones. If I had to chose among these answers, I’d go for the former, since I hardly understand what general (configurations of) objects might be in Euclid’s setting. Still, also the first answer appears inappropriate to me. The reason is that, as well as the second, it grounds on an inappropriate formulation of the question it is supposed to respond. In my talk, I’ll try to explain how this question should be appropriately stated, according to me. To do it, I’ll present an account of generality of Euclid’s propositions (both problems and theorem) according to which this does not depend on a quantification on geometrical items, bur rather on a necessary connections between constructions and objects constructed.

• Eleonora Sammarchi (Univ. Paris 7, SPHERE): Algebraic Vs geometrical proofs in Arabic Mathematics: a case study from al-Zanjani's treatise on Algebra (13th century)

Once the theory for second degree equations is established, algebraists redirect their interest onto new topics. By the end of 10^th century one of them, the mathematician al-Karaji, chooses to investigate the notion of operation and starts to conceive a coherent and exhaustive system of rules for calculating with algebraic entities. His work gave birth to a new tradition of arithmeticians-algebraists, whose aim is to improve algebra by the help of arithmetic and vice versa.

In this talk we will focus on the argumentative tools displayed in this tradition that should justify rules and procedures of the new theory for polynomial algebra. More precisely, we will consider, as a case study, al-Zanjani's Book of Algebra. Extremely influenced by al-Karaji's writings, this 13^th century treatise is a perfect example for our research. Al-Zanjani skips geometrical proofs and seems to prefer other types of argumentative reasoning, that we could qualify as “algebraic”. But can we properly speak of them as demonstrations? And also, despite the fact that, in this arithmetization of algebra, geometry seems to have a marginal place, is this discipline truly on the edges?

After a short overview of the whole treatise, we will uncover evidence to answer to this sort of questions.

• Antoni Malet (CNRS, Univ. Pompeu Fabra): On the social origins of the early modern arithmetization of ratio and proportionality.

Specialization in practical geometry and the proliferation of mathematical instruments in early modern Europe were deeply related to changes in social structure and, in particular, to the sixteenth-century emergence of professions specializing in the measure of specific goods. Measuring professions not only became part and parcel of the fabric of life in early modern Europe but very often were also embodiments of political authority. Some attention has been paid to the increase in mathematical literacy in early modern Europe, but not much attention has been paid to the politics and the sociology of measuring. The paper focusses on one or two case-studies that reveal the ways in which royal and local authorities regulated the practice of measuring, including the measuring instruments and their standardization and the participation of the professional measurers themselves in the coercive apparatus of the state and town. In particular, the standards and measuring instruments were earmarked and kept in symbolically relevant buildings. The paper argues that social conventions, mathematical instruments, and social and political authority shaped conceptual change in mathematics.

• Sébastien Maronne (Univ. Toulouse): Variations in Cartesian Geometry: 1637-1730.

In my talk, I will study the variations in Cartesian Geometry from the publication of La Géométrie in 1637 to the late commentary of Rabuel in 1730, Commentaires sur la Géométrie de M. Descartes. For that, I will consider the corpus composed of the three volumes of Correspondence edited by Clerselier from 1657 to 1667, and of the commentaries given by Schooten in 1659-1661, and by Rabuel in 1730. I will show that Cartesian Geometry varies, sometimes in contrast with Descartes’ original conceptions.

• Vincent Jullien (Univ. Nantes): The surprising paths of mathematization in the Dynamica of Leibniz.

It is said that mathematization of the sciences of nature began in the 17th century with the Galilean program. The works of Descartes and Leibniz and of course, of Newton, feed the reflection on these beginnings.

The problem of the origin of mathematization exists if we think that at one time a physics was born not only mathematized but also a mathematical physics. So we must seek this moment when the alchemists of ontology have succeeded in transmutation: physical objects and/or mathematical objects have become physico-mathematical objects. This new essence has given birth to a new science, with various names.

I wonder if it is not in vain we look for this date of birth, insofar as I wonder whether this emergence of physico-mathematical objects (and the science that goes with them) is not as chimerical as the transmutation of lead into gold.

My point will be to consider a very remarkable contribution of Leibniz to this case, as it appears in his great treatise, the Dynamica de potentia and see, more precisely, how this attempt is conducted. We’ll see Leibniz constructing these amphibious objects to develop a physical science that would not exist without mathematics.

This work and the attempt it represents to associate the two natures has made me more convinced that physico-mathematical objects cannot really exist.

• Cecilia Bognon (IHPST, Univ. Paris 1): From Stoffwechsel to the logic of metabolism: the rise of a chemical biology.

In this paper I investigate the historical development of the « epistemic space » (Müller-Wille, Rheinberger, 2012) of the concept of metabolism throughout the 19th century, starting with the german notion of « Stoffwechsel » (the metamorphosis of matter in living tissues) and contrasting it with Claude Bernard’s theory of « indirect nutrition ». I will focus on the conceptual shifts that have enabled an understanding of living organisms in terms of self-sustaining machines (that persist over time as the same entity), able to convert the substance of others into the self. I will show that the emergence of this philosophical problem was at least partially determined by the development of a chemical understanding of vital processes and laws of organization. Finally I will draw some conclusions on the role played by this conception of metabolism on the emergence of the modern notion of organisms as individuals.

Françoise Longy (IHPST, Univ. Paris 1): "Biological species": an investigative category as well as a multitask concept.

First, I will make plane why I think that today's notion of "biological species" is like the 19^th century's category of "chemical element" an investigative category, and specify what that means and implies. Second, I will draw some lessons from the history of chemistry. And, then I will explain how these lessons can help clarify both the philosophical and scientific debates about the nature of biological species by specifying how methodological issues relate to ontological ones. One of my conclusions will be that we are not faced with the alternative of being either a realist or a pluralist about biological species.

• Jean Gayon (IHPST, Univ. Paris 1): Are population genetics models reversible?

Repetitiveness and reversibility have long been considered as characteristic features of scientific knowledge. In theoretical population genetics, repetitiveness is illustrated by a number of genetic equilibria realized under specific conditions. Since these equilibria are maintained despite a continual flux of changes in the course of generations (reshuffling of genes, reproduction…), it can legitimately be said that population genetics reveals important properties of invariance through transformation. Time-reversibility is a more controversial subject. Here, the parallel with classical mechanics is much weaker. Time-reversibility is unquestionable in some stochastic models, but at the cost of a special, probabilistic concept of reversibility. But it does not seem to be a property of the most basic deterministic models describing the dynamics of evolutionary change at the level of populations and genes. Furthermore, various meanings of ‘reversibility’ are distinguished. In particular, time-reversibility should not be confused with retrodictability.