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Leibniz’s Theory of Space

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Abstract

In this paper I offer a fresh interpretation of Leibniz’s theory of space, in which I explain the connection of his relational theory to both his mathematical theory of analysis situs and his theory of substance. I argue that the elements of his mature theory are not bare bodies (as on a standard relationalist view) nor bare points (as on an absolutist view), but situations. Regarded as an accident of an individual body, a situation is the complex of its angles and distances to other co-existing bodies, founded in the representation or state of the substance or substances contained in the body. The complex of all such mutually compatible situations of co-existing bodies constitutes an order of situations, or instantaneous space. Because these relations of situation change from one instant to another, space is an accidental whole that is continuously changing and becoming something different, and therefore a phenomenon. As Leibniz explains to Clarke, it can be represented mathematically by supposing some set of existents hypothetically (and counterfactually) to remain in a fixed mutual relation of situation, and gauging all subsequent situations in terms of transformations with respect to this initial set. Space conceived in terms of such allowable transformations is the subject of Analysis Situs. Finally, insofar as space is conceived in abstraction from any bodies that might individuate the situations, it encompasses all possible relations of situation. This abstract space, the order of all possible situations, is an abstract entity, and therefore ideal.

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  1. Department of Philosophy, McMaster University, Hamilton, ON, L8S 4K1, Canada

    Richard T. W. Arthur

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Correspondence to Richard T. W. Arthur.

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This essay draws on earlier published work, including the introductory essay in my (Leibniz 2001), as well as several papers based on joint work with Graham Solomon, including my (Arthur 1987, 1988). Graham’s tragic early death prevented any further collaboration between us. I publish this in his memory. I am deeply indebted to Ed Slowik for his detailed comments and suggestions on an earlier draft, and also to Steve Savitt for his comments on the penultimate one. All translations of Leibniz’s Latin or French are my own, except where noted, and many of them from Leibniz (2001). I have used the notation ‘ôr’ for the ‘or’ that translates seu or sive, the ‘or of equivalence’, to distinguish it from disjunction or alternation. It has the sense of “that is to say”.

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Arthur, R.T.W. Leibniz’s Theory of Space. Found Sci 18, 499–528 (2013). https://doi.org/10.1007/s10699-011-9281-4

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