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I'm way out of my depth here, but would there be a place on this page (as well as the Continuous time page) for an inclusion of the philosophical thoughts on discrete time? Starting perhaps with Zeno's pre-Socratic 'Achilles and the Tortoise' and the ensuing history toward calculus and on?
One of the fundamental concepts behind discrete time is an implied (actual or hypothetical) system clock. If one wishes one might imagine the current atomic clock to be the de facto system clock. I will add in an appropriate text for this purpose and make reference to Neil Gershenfeld's book "The Nature of Mathematical Modeling", Cambridge University Press, 1999, p.18:
"... digital systems [...] usually are discretized in time (there is a system clock)"