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The central philosphical argument is that the similarity of structure of the physical world and that of many branches of mathematics shows that God made them both, and therefore that mathematics is Platonically pre-existing and hence discovered, not invented, by man.
The fallacy, I think, is that most of mathematics, and especially most of the mathematics Nickel cites, was occasioned and designed for the explicit purpose of providing models for nature. In short, I cannot be surprised that the models resemble the things modeled, any more than that a map should resemble the terrain it describes. .
As an evangelical Christian mathematician and computer scientist, I was hoping for a book that set forth the "discovered, not invented" argument with logical clarity, for I have been puzzled by that view. Disappointingly, this book isn't it. Fundamentally it presses the argument first by assertion, and then by testimonials of agreement by various mathematicians. Nickels also extensively uses quotations from the many works of theologian and philosopher of science Stanley Jaki. He does not, so far as I see, address the "mathematics as modeling tool" argument.
Nickels does a fair and clear job of expressing what he calls "the majority view", that mathematics is invented, and he cites various scientists holding that view.