Dans cet article je présente une analyse critique de l’approche habituelle de l’identité mathématique qui a son origine dans les travaux de Frege et Russell, en faisant un contraste avec les approches alternatives de Platon et Geach. Je pose ensuite ce problème dans un cadre de la théorie des catégories et montre que la notion d’identité ne peut pas être « internalisée » par les moyens catégoriques standards. Enfin, je présente deux approches de l’identité mathématique plus spécifiques: une avec la (...) fibration catégorielle et l’autre avec des catégories supérieures faibles. (shrink)

Dans cet article je présente une analyse critique de l’approche habituelle de l’identité mathématique qui a son origine dans les travaux de Frege et Russell, en faisant un contraste avec les approches alternatives de Platon et Geach. Je pose ensuite ce problème dans un cadre de la théorie des catégories et montre que la notion d’identité ne peut pas être « internalisée » par les moyens catégoriques standards. Enfin, je présente deux approches de l’identité mathématique plus spécifiques: une avec la (...) fibration catégorielle et l’autre avec des catégories supérieures faibles. (shrink)

The popular view according to which category theory provides a support for mathematical structuralism is erroneous. Category-theoretic foundations of mathematics require a different philosophy of mathematics. While structural mathematics studies ‘invariant form’ (Awodey) categorical mathematics studies covariant and contravariant transformations which, generally, have no invariants. In this paper I develop a non-structuralist interpretation of categorical mathematics.

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)

This article asks whether a country that suffers from serious environmental problems caused by another country could have a just cause for a defensive war? Danish philosopher Kasper Lippert-Rasmussen has argued that under certain conditions extreme poverty may give a just cause for a country to defensive war, if that poverty is caused by other countries. This raises the question whether the victims of environmental damages could also have a similar right to self-defense. Although the article concerns justice of war, (...) we will concentrate only on the issue of what can be just causes of war, instead of evaluating the entire justification of war. This is to say that we will limit our discussion to the question concerning just cause and leave aside more general questions concerning justness and moral permissibility of war. Our aim is to list the questions that must be made and settled if defensive war in the case of serious environmental problems is said to have (or not to have) a just cause. We will argue that there are three questions that are most important in this context. They are the question concerning liability, the question of collective responsibility, and the question whether environmental harms may create a “sufficient reason” for raising a war. (shrink)

In 1900 David Hilbert announced his famous list of then-opened mathematical problems; the problem number 6 in this list is axiomatization of physical theories. Since then a lot of systematic efforts have been invested into solving this problem. However the results of these efforts turned to be less successful than the early enthusiasts of axiomatic method expected. The existing axiomatizations of physical and biological theories provide a valuable logical analysis of these theories but they do not constitute anything like their (...) standard presentation,which can be used for transmission, evaluation, and justification of physical and biological knowledge. This state of theart in the axiomatization of physics is strong evidence that the standard notion of axiomatic theory stemming from Hilbert and Tarski is not appropriate for thet ask. However in the recent years in mathematics there emerged a new axiomatic approach best represented by the Homotopy Type theory (HoTT). We argue that the constructive axiomatic architecture used in HoTT has better chances to be successfully applied in physics as well as in computer science and engineering. (shrink)

Category theory doesn't support Mathematical Structuralism but suggests a new philosophical view on mathematics, which differs both from Structuralism and from traditional Substantialism about mathematical objects. While Structuralism implies thinking of mathematical objects up to isomorphism the new categorical view implies thinking up to general morphism.

In addition to the “universal glue,” which is the local mechanical causation, the standard explanatory scheme of classical science presumes two “universal vessels,” which are global space and time. I call this outdated metaphysical setting “black-and-white” because it allows for only two principal scales. A prospective metaphysics able to bind existing sciences together needs to be “colored,” that is, allow for scale relativity and diversification by domain.