Higher Theory and the Three Problems of Physics

According to the Butterfield--Isham proposal, to understand quantum gravity we must revise the way we view the universe of mathematics. However, this paper demonstrates that the current elaborations of this programme neglect quantum interactions. The paper then introduces the Faddeev--Mickelsson anomaly which obstructs the renormalization of Yang--Mills theory, suggesting that to theorise on many-particle systems requires a many-topos view of mathematics itself: higher theory. As our main contribution, the topos theoretic framework is used to conceptualise the fact that there are principally three different quantisation problems, the differences of which have been ignored not just by topos physicists but by most philosophers of science. We further argue that if higher theory proves out to be necessary for understanding quantum gravity, its implications to philosophy will be foundational: higher theory challenges the propositional concept of truth and thus the very meaning of theorising in science.Comment: 23 pages, 1 table

Kategoriak analogia matematikorako

Analogy in Mathematics has to be taken from a triple point of view:Conceptual analogy, analogical reasoning, and analogy-based approaches to generalisation in some relevant mathematical fields. My basic point in this short notice is that Category Theory can play an important role in defining that idea of analogy, and therefore in providing new ideas and methods for the construction of a more unified conception of Mathematics

Kategoriak analogia matematikorako

Analogy in Mathematics has to be taken from a triple point of view:Conceptual analogy, analogical reasoning, and analogy-based approaches to generalisation in some relevant mathematical fields. My basic point in this short notice is that Category Theory can play an important role in defining that idea of analogy, and therefore in providing new ideas and methods for the construction of a more unified conception of Mathematics

On Constructive Axiomatic Method

In this last version of the paper one may find a critical overview of some recent philosophical literature on Axiomatic Method and Genetic Method.Comment: 25 pages, no figure

Some proposals for the set-theoretic foundations of category theory

The problem of finding proper set-theoretic foundations forcategory theory has challenged mathematician since the very beginning. In this paper we give an analysis of some of the standard approaches that havebeen proposed in the past 70 years. By means of the central notions of class and universe we suggest a possible conceptual recasting of these proposals. We focus on the intended semantics for the (problematic) notion of large category in each proposed foundation. Following Feferman (2006) we give a comparison and evaluation of their expressive power

Some proposals for the set-theoretic foundations of category theory

The problem of finding proper set-theoretic foundations for category theory has challenged mathematician since the very beginning. In this paper we give an analysis of some of the standard approaches that have been proposed in the past 70 years. By means of the central notions of class and universe we suggest a possible conceptual recasting of these proposals. We focus on the intended semantics for the (problematic) notion of large category in each proposed foundation. Following Feferman (2006) we give a comparison and evaluation of their expressive power