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Of course not. All modern physicists are inclined to believe that our usual, molar physics, including our space and time concepts, are inapplicable in such circumstances. ([Sil36], p. 2 Weyl’s tiles The above attempts are, of course, the exception and not the rule. Geometry applied in the physical sciences is customarily continuous; the use of the real number system to represent spatial (and temporal) physical magnitudes is characteristic to modern as well as classical physics. To be regarded as a serious alternative, a discrete geometry – assuming it is consistent – must at least reproduce the success of this characteristic applicability.

But since it identifies the class of computable functions with the class of those functions which are computable using a Turing machine, this thesis involves both a precise mathematical notion and an informal and intuitive notion, and hence cannot be proved or disproved. Simple cardinality considerations show, however, that not all functions are Turing-computable (the set of all Turing machines is countable, while the set of all functions from the natural numbers to the natural numbers is not), and the discovery of this fact came as a complete surprise in the 1930s [Dav58].

34 35 If one insists that every dynamical evolution uniquely takes a physical state to one and only one physical state, one has what is sometimes called deterministic causation. A version of this which drops uniqueness but keeps existence is sometimes called probabilistic causation. It is interesting that classical Newtonian mechanics, while considered by many to be the paradigm for determinism, is actually much more hospitable to singularities (collision and non-collision alike) than quantum mechanics, and requires additional structure to rule these out, see [Ear86] and [Ear04].

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Absurdity of the definition of inertial mass in the general theory of relativity

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