<p>I think the answer is this: certain notions – linearity, symmetry, exponentiation (of operators), eigenvalues, are just quite recurrent. They show up again and again because much that happens in the world produces manifestations of them. One example: we discover eigenvectors by thinking of which vectors are nearly unchanged by a simple skewing-stretching-rotating of the plane, and then we notice the same notion applies to much more complicated linear operators (in quantum mechanics, or whatever).</p>
<p>The fact that this happens is an interesting collusion of our adeptness at modeling (something to which Mike alluded) and the recurrence of similar kinds of phenomena in nature. For example, we like to model things linearly, and nature gives us lots of things that we can model reasonably well in a linear way.</p>
<p>Of course, the underlying consistency of nature is still a bit of a mystery. There are lots of functions out there — why does nature seem to like (approximate) linearity and exponentiality so much? But that mystery is much more fundamental, closely related to why nature is “lawlike” at all — i.e., why any principles hold consistently over time and space.</p>
<p>The two-pronged quality of this account also lets you tweak it to your taste, placing most of the emphasis on the part of the explanation you find most plausible. If you think nature really isn’t all that close to the nice math we keep using, you can attribute most of this “miraculousness” to our insistence on modeling the world in ways we understand well. After all, it’s not like physics jumps on you like a giddy toddler and says “I have eigenvalues inside!”. Our decisions about how to model phenomena bear very heavily on what we think the world is like under the hood.</p>