Indeed, other mathematicians were interpreting “continuity” in radically different ways at the same time that Clifford was pushing his empiricist agenda. However, a mathematical philosophy of the type that he was advocating was not easily accepted by the whole of the mathematical community either. For example, continuity implies that “energy” (then a newly emergent concept in physics) can only pass from one piece of matter to another it cannot be transferred through a void.ĤClifford’s contentions with regards to continuity would not have been out of place among many of his contemporaries in the 1870s, given that many of his colleagues sympathized with the empiricist undertones in his foundational arguments. They must accept that material interactions alone can result in changes of motion. Clifford argued that scientists and mathematicians who accurately accept the implications of analytical techniques must accept certain limitations to their scientific theorizing. Such action amounts to a “discontinuity” in space, meaning that physical theories often devolve into metaphysical speculation. Newtonian gravity is a case in point-the belief that two bodies simultaneously express an attraction towards one another implies that the “force” of attraction operates across immense tracks of space instantaneously and thereby causes bodily motion. ![]() Forces, by their very nature, are a-physical they exist independently of the material bodies they act upon. However, Clifford contended that if scientists correctly adopted the assumption that continuity is true of the structure of the universe (as Clifford himself believed it to be), then they must avoid the notion of “force” as a causal explanation of phenomena. Following in the footsteps of a quintessentially British-empiricist tradition, Clifford accepted that there is no way to know certainly whether the universe really is continuous he acknowledged that many physical phenomena might in fact be the product of discontinuities in space, rendering analytical techniques useless and inappropriate. The accuracy of the descriptions gained from the use of such tools (for example, descriptions depicting the rate of acceleration of an object falling towards Earth) is fundamentally dependent upon the truth of the assumption that continuity exists in the ontological structure of the universe. Specifically with regards to the physical aspect of continuity, Clifford invoked the image of a continuous medium that pervades the entire universe ( i.e., an ether).ģThus, for Clifford, the techniques of calculus are conventional tools that describe phenomena in space. Continuous space implies that there are no gaps or moments of non-existence in the fabric of the universe continuous time implies that there are no gaps in the fabric of forward-moving time. Its mathematical definition is an abstraction of the assumed existence of continuity in space and time. The aim of this paper is to show how one mathematician, namely William Kingdon Clifford (1845-1879), conceived of mathematical “continuity”, how he used it, and how he subtly redefined it as part of his grander philosophical project-to prove that scientific theories based on action-at-a-distance principles ( i.e., instantaneous action across excessively large or infinitesimally small expanses of space) constitute poor means of explaining physical phenomena.ĢAs a mid-19 th century empiricist-influenced as much by the ideas of Charles Darwin and Herbert Spencer as he was by the ideas of Bernhard Riemann, Sir William Rowan Hamilton and Hermann Grassmann-Clifford’s approach to continuity can be summarized as follows: continuity is a conventional mathematical tool based on empirical evidence. “Continuity” is one of the fundamental concepts in calculus, and its history is by no means simple or straightforward. ![]() 1 Likewise, histories of “continuity” can introduce historians to the broadest themes in eighteenth- and nineteenth-century mathematics. ![]() S Hodge write that, in presenting a volume composed of various case studies in the history of ether concepts, their book introduces readers “to the broadest themes in the scientific thought of the eighteenth and nineteenth centuries”. 1 In their account of ether histories, Cantor and Hodge identify five categories of “ether' concepts (.)ġIn the edited collection Conceptions of Ether: Studies in the history of ether theories, G.
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