Logic (or logical thought processes) is an ideological tool. While it helps us navigate the universe, it's imperative that we don't make the mistake of mystifying logic to the point that it becomes inherent within nature. There is nothing to suggest that the logic used by human beings is somehow pre-existent in the world around us.
To take a crack at mathematics, one need only read Zeno's paradoxes; the paradoxes of motion, for instance. We have units of measurement that help us to determine how great a distance exists between objects. So you start at point "A," and need to get to point "B." The distance between them is fifty feet. We obviously know that a person can cross between these points, and arrive at one from the other. However, Zeno claims that this is an illusion (at least, it's an illusion based on our representative system of quantification). Zeno claims that, before one can reach the opposite point, one must first reach the halfway point; and before one can reach the halfway point, one must first reach the one quarter mark; but before one can reach the one quarter mark, one must first reach the one eighth mark, and so on ad infinitum.
What this should show us is that there is a fundamental disjunct between the way we represent reality, and the way it actually works.