A universe without mathematics is beyond the scope of our imagination

Mathematics is a useful way to make sense of the world but it's certainly possible that the universe is not itself a mathematical structure which would mean that mathematics is only an approximation (albeit a very good one when it comes to making sense of and predictions about the world). So I disagree with the author's conclusion that the universe must be mathematical and that we could not make rational sense of it if that wasn't the case. The universe is a big place and even the best theory we have of describing its large scale structure (general relativity) is mathematically incomplete (singularities). Similarly the small scale structure is described by another incomplete theory (quantum mechanics) and progress on reconciling the two has been stuck for some time now.

> Just as there are odors that dogs can smell and we cannot, as well as sounds that dogs can hear and we cannot, so too there are wavelengths of light we cannot see and flavors we cannot taste. Why then, given our brains wired the way they are, does the remark, “Perhaps there are thoughts we cannot think,” surprise you? Evolution, so far, may possibly have blocked us from being able to think in some directions; there could be unthinkable thoughts. If you recall that modern science is only about 400 years old, and that there have been from 3 to 5 generations per century, then there have been at most 20 generations since Newton and Galileo. If you pick 4,000 years for the age of science, generally, then you get an upper bound of 200 generations. Considering the effects of evolution we are looking for via selection of small chance variations, it does not seem to me that evolution can explain more than a small part of the unreasonable effectiveness of mathematics. [1]

Hamming elaborates more on what he means by "the unreasonable effectiveness of mathematics" in Chapter 23 of "The Art of Doing Science and Engineering: Learning to Learn" and the entire chapter is worth reading in full because it's full of other useful comments on mathematics and symbolic systems of reasoning.

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1: The Unreasonable Effectiveness of Mathematics (1980) - Richard Hamming