When the traditional distinction between a mathematical concept and a mathematical intuition is tested against examples taken from the real history of mathematics one can observe the following interesting phenomena. First, there are multiple examples where concepts and intuitions do not well fit together; some of these examples can be described as “poorly conceptualised intuitions” while some others can be described as “poorly intuited concepts”. Second, the historical development of mathematics involves two kinds of corresponding processes: poorly conceptualised intuitions are (...) further conceptualised while poorly intuited concepts are further intuited. In this paper I study this latter process in mathematics during the twentieth century and, more specifically, show the role of set theory and category theory in this process. I use this material for defending the following claims: (1) mathematical intuitions are subject to historical development just like mathematical concepts; (2) mathematical intuitions continue to play their traditional role in today's mathematics and will plausibly do so in the foreseeable future. This second claim implies that the popular view, according to which modern mathematical concepts, unlike their more traditional predecessors, cannot be directly intuited, is not justified. (shrink)

In this paper we discuss in some depth the main theorems pertaining to Carnot’s theory of transversals, their initial reception by Servois, and the applications that Brianchon made of them to the theory of conic sections. The contributions of these authors brought the long-forgotten theorems of Desargues and Pascal fully to light, renewed the interest in synthetic geometry in France, and prepared the ground from which projective geometry later developed.

I present a systematic interpretation of the foundational purpose of constructions in ancient Greek geometry. I argue that Greek geometers were committed to an operationalist foundational program, according to which all of mathematics—including its entire ontology and epistemology—is based entirely on concrete physical constructions. On this reading, key foundational aspects of Greek geometry are analogous to core tenets of 20th-century operationalist/positivist/constructivist/intuitionist philosophy of science and mathematics. Operationalism provides coherent answers to a range of traditional philosophical problems regarding classical mathematics, such (...) as the epistemic warrant and generality of diagrammatic reasoning, superposition, and the relation between constructivism and proof by contradiction. Alleged logical flaws in Euclid can be interpreted as sound operationalist reasoning. Operationalism also provides a compelling philosophical motivation for the otherwise inexplicable Greek obsession with cube duplication, angle trisection, and circle quadrature. Operationalism makes coherent sense of numerous specific choices made in this tradition, and suggests new interpretations of several solutions to these problems. In particular, I argue that: Archytas’s cube duplication was originally a single-motion machine; Diocles’s cissoid was originally traced by a linkage device; Greek conic section theory was thoroughly constructive, based on the conic compass; in a few cases, string-based constructions of conic sections were used instead; pointwise constructions of curves were rejected in foundational contexts by Greek mathematicians, with good reason. Operationalism enables us to view the classical geometrical tradition as a more unified and philosophically aware enterprise than has hitherto been recognised. (shrink)

RésuméCet article est consacré à l’histoire de la théorie locale des courbes “à double courbure”. Initiée par Clairaut en 1731, cette théorie se développe en parallèle à la théorie des surfaces et trouve son achèvement avec les formules de Serret et Frenet et leur interprétation par Darboux, en 1887. Au delà de l’analyse des contributions de nombreux mathématiciens, parmi lesquels Monge bien sûr mais aussi Fourier, Lagrange et Cauchy, notre étude donne un regard particulier sur l’évolution conjointe de l’Analyse et (...) de la Géométrie, dans une longue période riche de nombreuses remises en cause théoriques. (shrink)

In this paper, we try to understand what considerations and possible sources of inspiration Desargues used to formulate his concepts of involution and transversal, and to state the related theorems that are at the basis of his Brouillon project. To this end, we trace some clues which are found scattered throughout his works, we connect them together in the light of his experience and knowledge in the field of perspective, and we investigate what were his motivations within Mersenne’s academy. As (...) a result of our research, we can safely say that were his great geometrical insight and his projective vision of space which, guided by some classical theorems, led him to these completely new concepts in the panorama of the geometry of that time that were destined to remain misunderstood for about two centuries. (shrink)

We introduce new first-order languages for the elementary n-dimensional geometry and elementary n-dimensional affine geometry , based on extending equation image and equation image, respectively, with new function symbols. Here, β stands for the betweenness relation and ≡ for the congruence relation. We show that the associated theories admit effective quantifier elimination.

The history of the continuous inclusion of mathematics in liberal education in the West, from ancient times through the modern period, is sketched in the first two sections of this chapter. Next, the heart of this essay (Sects. 3, 4, 5, 6, and 7) delineates the central role mathematics has played throughout the history of Western civilization: not just a tool for science and technology, mathematics continually illuminates, interacts with, and sometimes challenges fields like art, music, literature, and philosophy – (...) subjects now universally considered to be liberal arts. Section 8 adds an international perspective to the contemporary liberal arts story by describing some instructive mathematical achievements from many cultures and societies. Finally, Sect. 9 addresses how contemporary mathematics teaching can use the history of mathematics viewed as a liberal art to enhance the appreciation and understanding of mathematics for all students. (shrink)

El artículo analiza los fundamentos epistemológicos y metodológicos del esquema cruzado en el campo de las ciencias sociales y en particular del análisis cualitativo, considerando que el origen de tal esquema se encuentra en las coordenadas cartesianas. Tales coordenadas tienen un conjunto de elementos que a la vez están en la base de la geometría analítica y a partir de ella se resaltan tres aspectos que han sido “exportados” mediante el uso de este esquema, hacia las ciencias sociales: la lógica (...) de opuestos, la “espacialización” o “espacialidad” y la fecundidad conceptual. A partir de esta triada de elementos se analiza el aporte de la lingüística estructural y del estructuralismo antropológico en torno al esquema en cuestión y en relación al objetivo de formalización en ciencias sociales. El texto ilustra algunos usos del esquema analizado realizados por distintos autores contemporáneos y termina por clasificar a los mismos de acuerdo a la integración, en tales análisis, de elementos epistémico-metodológicos implícitos en las coordenadas cartesianas.This article analyses the epistemological and methodological bases of crossed diagram in the field of social sciences and especially in qualitative research, considering that the origin of this diagram is located in the Cartesian coordinate system. Such coordinates provide elements that are at the bases of analytic geometry and from it, we highlight three aspects that have been ‘exported’ in the use of this diagram by the social sciences: the logic of the opposites; ‘spacialisation’; and conceptual fecundity. From these three aspects, we analyse the contribution of structural linguistics and anthropological structuralism to this diagram, particularly regarding the goal of formalization in the social sciences. The article shows some of the uses given to this diagram by different contemporary authors, to end classifying those authors according to their analytical integration of epistemic and methodological aspects implicit in the Cartesian coordinate system. (shrink)