Almagest

Dans un article fondateur de la philosophie de la biologie, David Hull (1969) a critique sechement les differentes manieres dont les philosophes s’etaient penches sur des problemes relevant des sciences du vivant tout au long de la premiere moitie du XX siecle. Il a epingle, en particulier, la methode de la ≪ reconstruction formelle d’enonces biologiques dans la notation de la logique mathematique ≫ qui avait ete adoptee par des penseurs lies a la tradition de l’empirisme logique, tels que J. H. Woodger (1929, 1937), afin de clarifier la signification et l’usage de concepts biologiques fondamentaux. Une telle critique s’inscrit dans un recit historiographique plus general, qui attribue les defauts de la philosophie de la biologie d’avant les annees 1970 a l’influence negative de preoccupations ideologiques propres a l’empirisme logique : l’antimetaphysique, le reductionnisme et la volonte d’imposer la physique comme science modele. Cependant, comme le montrent Daniel J. Nicholson et Richard Gawne (2014, 2015), une telle vision de l’histoire de la philosophie de la biologie se fonde sur une lecture selective et sur une interpretation etroite de l’oeuvre de Woodger, ainsi que, plus generalement, de l’empirisme logique en tant que mouvement philosophique heterogene. L’objectif de ce chapitre est ainsi de reevaluer l’apport de l’empirisme logique a l’histoire de la philosophie de la biologie, ainsi que son importance pour la comprehension d’enjeux contemporains, a partir des problemes souleves par l’axiomatisation des fondements des sciences biologiques proposee par Woodger lui-meme.

It is fascinating to consider the remarkable discoveries made in primatology over the centuries. One of the earliest recorded encounters happened twenty-five centuries ago when the Carthaginians stumbled upon what they believed were “gorillas” in today’s Senegal. It turns out that these were actually chimpanzees, but this chance encounter marked the beginning of our understanding of these incredible creatures. Fast forward to modern times, and we have learned even more about primates, including the fact that macaques have cultures that are transmitted between groups and generations through communication, rather than biological inheritance. This discovery, made by a female researcher in Japan, was the result of full-time observation and is now a cornerstone of the theory of cultural evolution. In between these discoveries, we have seen the first descriptions of orangutans, chimpanzees, and gorillas, all of which have contributed to our understanding of primates and our own evolution. It is remarkable to think that these early observations, including Hanno’s description of hairy women who threw stones to avoid capture, laid the groundwork for an entire field of study that continues to fascinate and inform us to this day.

This article is an analysis of Darwin’s attempt to reconcile naturalism and exceptionalism in The Descent of Man. The principle of gradual evolution, which had already been set out in the Origin of Species, was reinterpreted, and extended beyond the morphological characteristics of living things into the range of characteristics considered exceptional for the human species: abstract reasoning, morality, and numeracy. To explain the natural origin of these unique capacities, Darwin will introduce a series of additional hypotheses in which speech, which would have arisen from the ability to produce different sounds at will, serves as a scaffold for the development of the other distinctive human capacities. It is in this way that Darwin will attempt to dispel any doubts about the animal origin of human beings

“The tree of life, understood as an “evolutionary diagram,” is biology’s best-known and most widely used image. It is usually attributed to Charles Darwin, from the only image he presents in The Origin of Species (1859). However, this diagram, usually called “Darwinian”, could have its origin not only in Darwin’s work but also in that of another “Darwinist” author, Alfred R. Wallace. In this paper, we revisit a series of diagrams that Wallace presented before Darwin’s work in 1856, in which, through the relationships between bird species, he sought to present an evolutionary diagram visually. As we will see, the diagrams and their description have limitations that contributed to their lack of understanding. Additionally, we want to discuss the place of the representations proposed by Wallace concerning other visualisations that historically have sought to explain biological evolution.”

The history of North American physical anthropology is a relevant topic for historians and philosophers of science. In this paper I offer a schematic perspective that accounts for a three-level historiographical narrative that includes: i) the “prehistory” of physical anthropology; (ii) the new physical anthropology; and (iii) the construction of a biological anthropology. First, I argue that historiography is a useful methodology for conducting epistemological analyses. Next, I delve into these three historical stages and contrast them in order to establish some guiding principles for each of them. For instance, the “prehistory” of physical anthropology is characterized by a parametric tradition, while the new physical anthropology considers measurement only as the first step of physical anthropological analyses; the second is to contextualize and explain what these measures represent in a biocultural context. Biological anthropology, on the other hand, is an extension of the new physical anthropology that includes theoretical, methodological and conceptual aspects of contemporary evolutionary biology, primatology, molecular anthropology, etc. In addition, it considers the other branches of anthropology: ethnography, archaeology, linguistics as fundamental frameworks towards a comprehensive and integral anthropology. I close with some concluding remarks that highlight the relevance of biological anthropology not only in the context of the Social Sciences and Humanities, but beyond.

The historiography of mathematical functions already includes many references of often respectable size, and the present work is neither a new course, nor a kind of summary relating to some original texts not yet analyzed on this subject. It is intended to be a reflection on the practice of the history of mathematics. On the one hand, this is because this work is based on various presentations made previously which are generally unpublished, in the now venerable framework created by the seniority and a number of Franco-Mexican meetings. On the other hand, it is also because this work raises the question of the objective, and also of the issue of objectivity, that exists in telling the history of a theorem, a concept or a mathematical object. And I should not forget the way of using both the sources and the technical knowledge of prospective readers. It is in fact important to signal how deeply the writing of a text in the history of mathematics depends on such readers. While three words seem, wrongly, to frame very specific things that I wish to discuss : history of the fundamental theorem of algebra, concept of variable quantity, or the function object. In the three cases, the difficulty is to deal with things which have acquired their name only long after the nature of the theorem, the properties of the concept or of the object have circulated in various forms. The dangers, that of anachronism in particular, are to freeze a situation, such as a photograph with a dummy background shot, or to neglect or overestimate the importance of the contemporary situation. Our question here is to concretely analyze these dangers in the case of functions, not to destroy previous accounts, but to bring out prejudices, and to allow us to grasp the differences that there are in qualifying a concept or an object.

In 1923, Wiener proved that the sample paths of Brownian Motion are almost surely nowhere differentiable and then, have infinite variation. This led to an interest in the study of continuous time stochastic processes. Initially, there were two main approaches: one based on Kolmogorov’s work on Markov processes, and another based on Levy’s approach to Brownian Motion. In the 1940’s, Kiyosi Ito began to investigate continuous time stochastic processes with independent increments. He found the way to define a Lebesgue-Stieltjes type integral, with respect to the Brownian Motion, and developed the now called Ito’s formula, thereby unifying both approaches and giving rise to the Theory of Stochastic Differential Equations. In this paper, we present the two different points of views, an integral with respect to the Brownian Motion developed previously by Wiener, and the construction of Ito’s Integral.

I examine the famous debate between Poincare and Russell around impredicativity. After highlighting the main points in the emergence of the concept, I review some elements of Poincare's philosophical position against impredicative definitions. I aim to isolate among his arguments those that target a broader public because they are less dependent on his philosophy.

In 1876 Bernhard Riemann's Gesammelte Werke, with a selection of unpublished manuscripts from his Nachlass are, are published. 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. In this paper, using the editors' correspondence and elements of both Riemann's and Dedekind's Nachlasse, I analyse the philological and mathematical practices involved in their editorial work. In particular, I 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.

In the 17th century, Antoine Arnauld published his work Nouveaux Elements de Geometrie. Through this, he put into practice some methodical principles introduced in his previous work, Logique de Port Royal. This resulted in the reconstruction of elementary plane geometry with respect to its traditional form in Euclid’s Elements. In this article we will emphasize the coherence of Arnauld’s reconstruction project, and we will present one of its most notorious and novel parts, which is a “geometry of straight lines” independent of the notion of angle and of the properties of plane figures.

I defend the thesis that physico-mathematics does not exist. Of course, physics closely associated with mathematics does exist, but the nature of physical activities and propositions remains distinct from mathematical activities and propositions, even within the context of this close association. I’ll use a great text, Leibniz’s Dynamica de Potentia, to show how, in the midst of the rise of this supposed “new science” of physico-mathematics, this distinction cannot be ignored.

In A Half-Century of Mathematics written in 1951, Hermann Weyl aims to account for the mathematics of the first half of the twentieth century. Regarding Lebesgue's integral and measure theory he stares that: "I must mention the, in all probability final, form given to the idea of integration by Lebesgue at the beginning of our century." However, there exist over a hundred integrals to date so Weyl's comment regarding the final form of the integral seems rather stringent. Our goal here is to compare two integrals in particular, Denjoy’s and Daniell’s, and to analyse the conceptual change that took place after the introduction of Lebesgue's integral that made them possible

The concept of proof in mathematics has evolved through centuries from the intuitive notions of justification and validation. Many aspects around proofs and their acceptance have been part of philosophical and mathematical discussions since long time ago, but the insight of what counts or not as a valid mathematical proof has been changed since the advent of computers. Nowadays computer-assisted proofs and methodologies play a relevant role in many scientific areas and in the case of today mathematics some of those proofs can be put on a par with traditional paperand- pencil proofs. This raises the question whether they are just two incarnations of the same idea of proof or some kind of conceptual change is required to elucidate whether a computer-assisted proof can be considered as a valid demonstration in the mathematical practice. In this paper we give some initial remarks and arguments to explore the emergence of possible conceptual changes related to a specific kind of computer-assisted proofs, namely those developed by a human agent with the help of a modern interactive theorem prover or proof-assistant.

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 nineteenth 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.

This paper revisits a famous episode in the history of mathematics: the 1695 discovery by Leibniz, in correspondence with Johann Bernoulli, of an “analogy” between the powers of a sum and the differentials of a product. It is often suggested that this discovery essentially depended on specific notational choices previously made by Leibniz: his use of a separate ‘d’ symbol, which, unlike Newton's notation, separated the differentiation operation from the variables it operates on; and his introduction of an “exponential” notation for differentials (e.g., d3x for dddx, d⁻1x for ∫x, etc.). Can we really make sense of the idea that the notations themselves—independently of the conceptual motivations Leibniz may have had to introduce them in the first place—played an essential role in this discovery? I argue that the notation did indeed play a part in this discovery and subsequent ones, but a rather subtle one, which cannot be understood (as is often suggested in the literature) in terms of an increase in “expressive power”; I also argue that Newton's notation could have served just as well.