The Leibnizian Lineage of Deleuze's Theory of the Spatium

This paper examines the Leibnizian influence in Deleuze's theory of the spatium. Leibniz's critique of Cartesian extension and Newtonian space leads him to a conception of space in terms of internal determination and internal difference. Space is thus understood as a structure of individual relations internal to substances. Making some Nietzschean corrections to Leibniz, Deleuze understands the spatium in terms of individuating differences instead of individual relations. Leibnizian space is thus transformed into a genetic space producing both extension (quantity) and quality.

Kant’s Asynchronicity Concerning Newtonian Space and Gravity in his Pre-Critical Writings

Kant’s ‘Newtonianism’ has been rightly highlighted by figures like Friedman. The follow-up debates led to a more adequate view on Kant’s natural philosophy and in particular his relation towards Newton. But the discussion that evolved did not point to the asynchronicity that takes place in Kant’s struggle with the central Newtonian concepts. Newtonian space and gravity, in revised form, are of central concern to Kant’s critical philosophy. But Kant adapted and re-evaluated these two concepts in an asynchronous way. While Kant tries to integrate a notion of gravity into his theory of matter in his very first published writing, he has at this stage no adequate notion of space. At this time, as in regard to space, he can neither be called a Newtonian nor a proper Leibnizian and misconceives the necessity of an independent space for the foundations of physics. This perspective changes under the influence of Euler at the end of the fifties of the eighteenth century and finally leads to his writing of 1768 and the adoption of transcendental idealism in 1770. In the following, I depict this asynchronicity by taking central pre-critical writings into account while discussing Kant’s concept of space and gravity. This sharpens the picture of Kant’s work and the different stages his philosophy of nature went through. Further, it helps to understand the influence of Euler on Kant’s development in natural philosophy and his critical philosophy in general, as Kant under the influence of Euler formed deeper-going reflections on Newton’s theory of space and these mark turning points of his development.

Space

In some of its uses, the word ‘space’ designates an empty or potentially empty expanse among things, for example, when a driver finds a space in a crowded parking lot, or when a typesetter increases the space between words on a page. In other uses, ‘space’ is meant to stand for a boundless extension which supposedly contains everything, or every thing of a certain sort. The former sense is well-grounded in ordinary experience and can be traced back to the etymology of the word (from the Latin word spatium, meaning ‘race-track’, or generally ‘distance’, ‘interval’, ‘terrain’). The latter sense originated in scholarly circles – possibly as late as the fourteenth century – by a bold extrapolation of the former; it does not refer to anything that can be exhibited in sense-perception; and yet, through the influence of Newtonian science on Euro-American common sense, it has become so entrenched in ordinary usage that it is normally viewed as the primary meaning of ‘space’, from which all others are derived. According to Cornford, the ‘invention of space’ as a boundless, all-encompassing container happened in the fifth century bc. However, it is more likely to have occurred in the late Middle Ages. At any rate, the idea was rampant in Cambridge in the 1660s, when Newton made it a fundamental ingredient in his framework for the description of the phenomena of motion. In a posthumous paper, Newton stressed that space evades the traditional classification of entities into substances and attributes, and has ‘its own manner of existence’. Until the publication of this paper in 1962, philosophers took Newtonian space for a substance, and most of them thought this to be utterly absurd. In view of the role of all-encompassing space in Newtonian physics, Kant opted for regarding it as a precondition of human knowledge, contributed once and for all by the human mind. Newton had written that the points of space owe their individual identity to the relational system in which they are set. Nineteenth-century mathematicians vastly extended this concept of space by conceiving many such relational systems. They thus made it possible for relativity theory to substitute four-dimensional spacetime for Newtonian space and time, and for current string theory to countenance a ten-dimensional physical space. These developments confirm the productivity, but not the fixity, of the knowing mind.

Spacetime

Spacetime is the four-dimensional manifold proposed by current physics as the arena for Nature’s show. Although Newtonian physics can very well be reformulated in a spacetime setting, the idea of spacetime was not developed until the twentieth century, in connection with Einstein’s theories of special and general relativity. Due to the success of special relativity in microphysics and of general relativity in astronomy and cosmology, every advanced physical theory is now a spacetime theory. Spacetime is undoubtedly an artificial concept, which our hominid ancestors did not possess, but the same is true of Newtonian space and time.