Mathematics Studies Machines

Abstract
Machines were introduced as calculating devices to simulate operations carried out by human computors following fixed algorithms: this is true for the early mechanical calculators devised by Pascal and Leibniz, for the analytical engine built by Babbage, and the theoretical machines introduced by Turing. The distinguishing feature of the latter is their universality: They are claimed to be able to capture any algorithm whatsoever and, conversely, any procedure they can carry out is evidently algorithmic. The study of such "paper machines" by mathematical means is the topic of our contribution. This is not only in accord with its usual understanding in computer science, but conceptually and historically right, when we recall the purpose for which Turing machines were introduced.
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