Metascientific aspects of topoi of spaces

Melville, Charles Edwin Brian (1991). Metascientific aspects of topoi of spaces. MPhil thesis The Open University.

Abstract

This thesis presents a study of the importance of topoi for Science. It is argued that whenever the concept of space enters the practice of Science then formal (mathematical) theories should be interpreted in a topos of spaces. It is claimed that these topoi encode knowledge of space arising directly out of the needs of Science, in that the constitutive questions of the Sciences can be traced back to their leading knowledge interests and these determine the role of mathematics as a methodical device. In the Natural Sciences the constitutive questions involve the study of non-intentional objects, in terms of a causal nexus to be explained geometrically, and this facilitates the introduction of geometric objects as the methodical device for posing questions to Nature. Although the study of intentional subjects in the Human Sciences requires ordinary language, not mathematics, to pose questions to each other, secondary methodological objectifications permit a conception of geometric objects analogous to that of the Natural Sciences. Lawvere*s axioms for the gros and petit topoi illustrate attempts to formalise the idea of topoi of spaces, as a rational reconstruction of categories in which geometric objects satisfying the formal theories of Science can be found. The catalysing function of this knowledge of topoi of spaces can lead to a diagnosis of mathematical difficulties caused by a failure to align mathematical conceptions with these topoi. This is illustrated through Varela's use of self-reference in Biology and Atkin's use of algebraic topology in Social Studies.

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