40 research outputs found
Synthetic Philosophy of Mathematics and Natural Sciences Conceptual analyses from a Grothendieckian Perspective
ISBN-13: 978-0692593974. Giuseppe Longo. Synthetic Philosophy of Mathematics and Natural Sciences, Conceptual analyses from a Grothendieckian Perspective, Reflections on âSynthetic Philosophy of Contemporary Mathematicsâ by F. Zalamea, Urbanomic (UK) and Sequence Press (USA), 2012. Invited Paper, in Speculations: Journal of Speculative Realism, Published: 12/12/2015, followed by an answer by F. Zalamea.International audienceZalameaâs book is as original as it is belated. It is indeed surprising, if we give it a momentâs thought, just how greatly behind schedule philosophical reflection on contemporary mathematics lags, especially considering the momentous changes that took place in the second half of the twentieth century. Zalamea compares this situation with that of the philosophy of physics: he mentions DâEspagnatâs work on quantum mechanics, but we could add several others who, in the last few decades, have elaborated an extremely timely philosophy of contemporary physics (see for example Bitbol 2000; Bitbol et al. 2009). As was the case in biology, philosophy â since Kantâs crucial observations in the Critique of Judgment, at least â has often ârun aheadâ of life sciences, exploring and opening up a space for reflections that are not derived from or integrated with its contemporary scientific practice. Some of these reflections are still very much auspicious today. And indeed, some philosophers today are saying something truly new about biology..
Mathematics in the middle: The relationship between measurement and metamorphic matter
This paper revisits philosophical questions regarding the relationship between mathematics and matter. I briefly present four contrary and contemporary perspectives on the speculative force of mathematics, as a provocation for further discussion on the subject of sciento-metrics. I first consider the ideas of the philosopher Quentin Meillassoux, as a way of setting the stage for various kinds of materialist philosophies of mathematics. I then turn to the ideas of two mathematicians - Fernando Zalamea and Giuseppe Longo - and a computer scientist - Gregory Chaitin - and explore how their discussions of contemporary mathematical practice offer important insight (and twist) regarding the relationship between mathematics and matter
Classifying topoi in synthetic guarded domain theory
Several different topoi have played an important role in the development and
applications of synthetic guarded domain theory (SGDT), a new kind of synthetic
domain theory that abstracts the concept of guarded recursion frequently
employed in the semantics of programming languages. In order to unify the
accounts of guarded recursion and coinduction, several authors have enriched
SGDT with multiple "clocks" parameterizing different time-streams, leading to
more complex and difficult to understand topos models. Until now these topoi
have been understood very concretely qua categories of presheaves, and the
logico-geometrical question of what theories these topoi classify has remained
open. We show that several important topos models of SGDT classify very simple
geometric theories, and that the passage to various forms of multi-clock
guarded recursion can be rephrased more compositionally in terms of the lower
bagtopos construction of Vickers and variations thereon due to Johnstone. We
contribute to the consolidation of SGDT by isolating the universal property of
multi-clock guarded recursion as a modular construction that applies to any
topos model of single-clock guarded recursion.Comment: To appear in the proceedings of the 38th International Conference on
Mathematical Foundations of Programming Semantics (MFPS 2022
Strict stability of extension types
We show that the extension types occurring in Riehl--Shulman's work on
synthetic -categories can be interpreted in the intended semantics
in a way so that they are strictly stable under substitution. The splitting
method used here is due to Voevodsky in 2009. It was later generalized by
Lumsdaine--Warren to the method of local universes.Comment: 16 pages. This text is essentially Chapter 6 from author's PhD thesis
arXiv:2202.13132. Updated acknowledgments. Submitted, but comments welcome