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In the end, however, it is physical reality, not mathematics, that determines what is true in fact. “[T]he mistake is to confuse an abstract attribute with a physical one of the same name. Since it is possible to prove theorems about the mathematical attribute, which have the state of absolutely necessary truths, one is the misled into assuming that one possesses a priori knowledge about what the laws of physics must say about the physical attribute.” Deutsch, The Beginning of Infinity, p.183. Thus, mathematics may not always reflect the existing physical relationships.

In addition, Deutsch argues that the conclusions of Hilbert, Turing and Gödel about the limits of computing or proving mathematical statements²² are the results of the existing laws of physics, reflected in the operations of our brains and our computers. “Different physical laws would make different things infinite, different things computable, different truths—both mathematical and scientific—knowable. …[I]f the laws of physics were in fact different from what we currently think they are, then so might be the set of mathematical truths that we would then be able to prove, and so might the operations that would be available to prove them with. The laws of physics as we know them happen to afford a privilege status to such operations as “not”, “and” and “or”, acting on individual bits of information… .” Id., p.186.

Deutsch asserts “a computation of a proof is a physical process in which objects such as computers or brains physically model or instantiate abstract entities like numbers or equations, and mimic their properties. …It works because we use such theories only in situations where we have good explanations saying that the relevant physical variables in those objects do indeed instantiate those abstract properties.” Id., p.188. Thus, mathematics will not necessarily provide good predictions, let alone good explanations, of the physical world. In addition, there may be significant aspects of our reality that are not susceptible to mathematical modeling, that are “non-computable.” “So there is something special—infinitely special, it seems—about the laws of physics as we actually find them [to date], something exceptionally computation-friendly, prediction-friendly and explanation-friendly.” Id., p.189.