This interpretation strikes me as implausible historically. Thomson's integrating machine and Babbage's analytical engine seem to have developed as seperate solutions to the same practical problem long before any actual principles of computation were clearly articulated.
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It's like saying that Ptolemaic system was before the Copernican system and it was semi-effective. That's triviality at best. The Ptolemaic system was already working on equations of motion via Ox calculators and Oresme's formula. They just didn't know the undergirding mechanisms
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By this reasoning not only analog computers but all of mathematics is based on the principles of digital computation. One is left wondering what you could possibly imagine these principles to be.
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Not really. This is just a historical example to use your own term. The issue that wether all math can be seen as computation is not important, what is important are the deep foundational correspondences between mathematical structures and computation.
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The invocation of mathematical foundations is what confuses me so about your position. I don't even know what you think a computer or a computation even is. Whereas to me an analog computation is a recognizably distinct technique from a digital one, full stop.
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Analogue computation is just too sloppy of a term to be taken seriously. My definition of computation is nothing but that of Church and Turing applied to complex behaviors such as asynchronous interactions.
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If you start with Turing's assumptions then naturally you arrive at Turing's conclusions. It is only when it comes to actually building the machine that the difference could possibly matter.
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How do you build a machine without the assumptions such as in the case of the traditional church-turning thesis, Lambda calculus?
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You start with classical mechanics. The world is a set of second order linear differential equations. To solve such equations arrange the world just so and watch what happens.
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Good, but you see classical mechanics in its original form doesn't give you all the goodies you want, until people like Gibbs and Boltzmann reformulate the laws of mechanics in terms of statistical ensembles which can be compartmentalized. Mu-space is proto-computationnal theory.
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Mu-space is the first account of computation-information theoretic of an isolated system. It has discretization, controlled idealization, etc.
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