1,384 research outputs found
Mathematical Models of Abstract Systems: Knowing abstract geometric forms
Scientists use models to know the world. It i susually assumed that mathematicians doing pure mathematics do not. Mathematicians doing pure mathematics prove theorems about mathematical entities like sets, numbers, geometric figures, spaces, etc., they compute various functions and solve equations. In this paper, I want to exhibit models build by mathematicians to study the fundamental components of spaces and, more generally, of mathematical forms. I focus on one area of mathematics where models occupy a central role, namely homotopy theory. I argue that mathematicians introduce genuine models and I offer a rough classification of these models
Mathematical Abstraction, Conceptual Variation and Identity
One of the key features of modern mathematics is the adoption of the abstract method. Our goal in this paper is to propose an explication of that method that is rooted in the history of the subject
Canonical Maps
Categorical foundations and set-theoretical foundations are sometimes presented as alternative foundational schemes. So far, the literature has mostly focused on the weaknesses of the categorical foundations. We want here to concentrate on what we take to be one of its strengths: the explicit identification of so-called canonical maps and their role in mathematics. Canonical maps play a central role in contemporary mathematics and although some are easily defined by set-theoretical tools, they all appear systematically in a categorical framework. The key element here is the systematic nature of these maps in a categorical framework and I suggest that, from that point of view, one can see an architectonic of mathematics emerging clearly. Moreover, they force us to reconsider the nature of mathematical knowledge itself. Thus, to understand certain fundamental aspects of mathematics, category theory is necessary (at least, in the present state of mathematics)
Vérité partielle et réalisme scientifique: une approche bungéenne
Le rĂ©alisme scientifique occupe une place centrale dans le systĂšme philosophique de Mario Bunge. Au cĆur de cette thĂšse, on trouve lâaffirmation selon laquelle nous pouvons connaĂźtre le monde partiellement. Il sâensuit que les thĂ©ories scientifiques ne sont pas totalement vraies ou totalement fausses, mais plutĂŽt partiellement vraies et partiellement fausses. Ces Ă©noncĂ©s sur la connaissance scientifique, Ă premiĂšre vue plausible pour quiconque est familier avec la pratique scientifique, demandent nĂ©anmoins Ă ĂȘtre clarifiĂ©s, prĂ©cisĂ©s et, ultimement, Ă ĂȘtre inclus dans un cadre thĂ©orique plus large et rigoureux. Depuis ses toutes premiĂšres publications sur ces questions et jusquâĂ rĂ©cemment, Mario Bunge nâa cessĂ© dâinterpeller les philosophes afin quâils dĂ©veloppent une thĂ©orie, au sens propre du terme, de la vĂ©ritĂ© partielle afin de clarifier les enjeux Ă©pistĂ©mologiques liĂ©s au rĂ©alisme scientifique. Bunge a lui-mĂȘme proposĂ© plusieurs parties de cette thĂ©orie au fil des annĂ©es, mais aucune de ces propositions ne lâa satisfait pleinement et la construction de cette thĂ©orie demeure un problĂšme entier. Dans ce texte, nous passerons rapidement en revue certaines des approches proposĂ©es par Bunge dans ses publications et nous esquisserons certaines pistes qui devraient servir Ă tout le moins de desiderata pour la construction dâune thĂ©orie de la vĂ©ritĂ© partielle
Stairway to Heaven : The abstract method and levels of abstraction in mathematics
In this paper, following the claims made by various mathematicians,
I try to construct a theory of levels of abstraction. I rst try to clarify
the basic components of the abstract method as it developed in the rst
quarter of the 20th century. I then submit an explication of the notion of
levels of abstraction. In the nal section, I brie
y explore some of main
philosophical consequences of the theory
Menger and Nöbeling on pointless topology
This paper looks at how the idea of pointless topology itself evolved during its pre-localic phase by analyzing the definitions of the concept of topological space of Menger and Nöbeling. Menger put forward a topology of lumps in order to generalize the definition of the real line. As to Nöbeling, he developed an abstract theory of posets so that a topological space becomes a particular case of topological poset. The analysis emphasizes two points. First, Menger's geometrical perspective was superseded by an algebraic one, a lattice-theoretical one to be precise. Second, Menger's bottomâup approach was replaced by a topâdown one
The Structuralist Mathematical Style: Bourbaki as a case study
In this paper, we look at Bourbakiâs work as a case study for the notion of mathematical style. We argue that indeed Bourbaki exemplifies a mathematical style, namely the structuralist style
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