Particle-free bodies and point-free spaces☆
Section snippets
Introductory remarks
I resisted the temptation to try and appear droll and title this work “Pointless bodies” as I remembered reading the witticism “A celebrated reviewer once described a certain paper (in a phrase which never actually saw publication in Mathematical Reviews) as being concerned with the study of ‘valueless measures on pointless spaces” in Johnstone’s article titled “The point of pointless bodies” (see Johnstone, 1983); I was apprehensive that this paper might suffer a review which actually sees
The need for point free topology to describe a body
The history of natural philosophy bears witness to the fact that the notion of a “point” was a contentious one. A “point” in mathematics is understood to habit an abstract space,11 while in the case of a “particle” it is supposed to lie in “physical space”. We first have to understand what is meant by an abstract mathematical space as well as what is meant by physical
Where the notion of a “point” hinders understanding certain problems of natural philosophy
Let us consider a few important problems in mechanics where the notion of a “point” comes in the way of clearly defining the problem and complicates issues rather than makes them tractable and amenable to meaningful analysis.
Philosophers have been interested in the question of the notion of a boundary, from the metaphysical as well as set theoretic point for a very long time. References to this literature can be found in the papers of Zimmerman, 1996a, Zimmerman, 1996b. Of interest to us here
Meaning of “point” in everyday usage
Most “definitions” of a “point” make a statement that articulates the following sentiment or meaning: “A point is a precise location in space, and it has no dimensions”. Such a “definition” is most unsatisfactory as it does not convey any meaning whatsoever. If a “point” has no dimensions, how are we to ascertain if it is indeed at some location in space? What does it mean to be in some location is space? What is the meaning of “location”? If we are to conceptualize a “point” as a “dot”,
The notion of a “point” in classical Euclidean Geometry
Contrary to popular perception amongst mathematicians, Euclid did not perceive a “point” as being an entity without dimension. In Heath’s authoritative translation of Euclid’s Elements we find the following definition of the notion of “point”45: Σημεῖóν ἐστιν, οῧ μέρος οὐθέν, which according to Heath means
Notion of points in mechanics
The earliest and simplest description of a body within the purview of mechanics is that of a particle, or more precisely a “point mass”, and much of the early studies in natural philosophy were devoted to the investigation of the motion of projectiles and planets within the context of such a description. While the mathematical notion of a point could be considered as an indispensable notion in mathematics, bereft of which it would be no exaggeration to say there would not be much point to
Abstract definitions of a body in continuum mechanics
The remarks of Fransescoe De Oviedo, that can be found in the translation that appears in “Discourse of things above reason” in the Selected Philosophical Papers of Robert Boyle, edited by Stewart, while somewhat harsh seems to have some merit even today, though most mathematicians will tend to view it as being without merit. Oviedo remarks (Stewart (1991)):
“We come to the composition of a continuum, whose hitherto unsurmounted difficulty has sorely taxed the wits of all the learned, and
Some concluding observations
I believe that it is necessary to use the language of point free topology if we are to describe certain problems of natural philosophy and resolve them satisfactorily. While there has been considerable effort in developing “point free” topology, there has been no effort in developing a Calculus where one can talk sensibly about the derivatives of functions that are defined on such topological spaces, the development of ordinary, partial differential and integral equations within such a
Acknowledgements
I thank Josef Malek, Dalibor Prazac, Vit Prusa, Miles Rubin, Brian Seguin, and Benjamin Vejnar for their input based on an earlier draft of the work. I thank the National Science Foundation for its support of my research for the past three decades that has allowed me to think about the issues discussed in this paper. I also thank the Office for Naval Research for support of this work. I also thank the Necas Center at the Charles University in Prague for its support over the years.
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This paper delves into the notion of a “point” as it pertains to its usage in mathematics; it, whatever “it” is has “position but no magnitude”. However, the word is also used in another sense, namely as “essential or indispensable”. This paper is dedicated to Chandrika who is the “point” to my life, in this latter sense of its usage.