The problem with the first view is that to any mathematician, it seems fairly straightforward to assert that we actually discover something--not just logical relationships between symbols, but actual content. The problem with the second view is that the world of mathematical concepts seems remote; how can we physical beings have access to it?
The alternative is Aristotelian realism, which asserts that mathematical objects inhere in nature. Our minds have access to them initially through observation, then through abstraction and logical reasoning.
This alternative is very attractive for at least two reasons. One reason is that it makes sense of applications far more easily than either Platonism or nominalism. Why should mathematical models be so good at describing real world phenomena? Under the Platonist view, there's not much reason even to wonder about it, since mathematical objects are eternal and inherently separate from the contingent world we live in. Under the nominalist view, the puzzle is why a mere language would be so effective in discovering things about the universe before they are even observed (think about the mathematical development of general relativity). Realism has a simple explanation: we draw mathematical concepts out of the real world, so it's natural that we should use them to explain how it works.
Another reason is that it's satisfying from the point of view of a practicing mathematician. Platonism also has that trait, in that it elevates the objects of mathematical study themselves. But Aristotelian realism allows us to assert that mathematics has real content without divorcing it from common experience. I find this accords well with my own practice of mathematics, both in research and teaching. I always emphasize to my students that common sense should be the starting point for thinking about any mathematical problem. Of course we have to take a long journey out from that starting point, but ultimately each step is grounded in reasoning that any flesh and blood human being can understand.
For me there's a third, more theological reason to appreciate Aristotelian realism. Franklin alludes to theological import himself:
Aristotelian realism stands in a difficult relationship with naturalism, the project of showing that all of the world and human knowledge can be explained in terms of physics, biology and neuroscience. If mathematical properties are realised in the physical world and capable of being perceived, then mathematics can seem no more inexplicable than colour perception, which surely can be explained in naturalist terms. On the other hand, Aristotelians agree with Platonists that the mathematical grasp of necessities is mysterious. What is necessary is true in all possible worlds, but how can perception see into other possible worlds? The scholastics, the Aristotelian Catholic philosophers of the Middle Ages, were so impressed with the mind’s grasp of necessary truths as to conclude that the intellect was immaterial and immortal. If today’s naturalists do not wish to agree with that, there is a challenge for them. ‘Don’t tell me, show me’: build an artificial intelligence system that imitates genuine mathematical insight. There seem to be no promising plans on the drawing board.This paragraph is delightfully provocative. I suspect many proponents of artifical intelligence believe they are not so far off as Franklin believes, but I can neither confirm nor deny such claims. In any case, artificial intelligence is not what interests me most. Instead, I tend to fixate on this question, "What is necessary is true in all possible worlds, but how can perception see into other possible worlds?"
To me the advantage Aristotelian realism has over Platonism is that it lets us see the eternal, even the sacred, in all things. Whereas the Platonist sees objects in the world as mere shadows on the wall, as it were, the Aristotelian sees them as sources of truth in themselves. For this reason I think Aristotelianism can affirm creation in a way that Platonism can't.
It is common for applied mathematicians to point out that their models are only approximations of reality, and that real life, unlike beautiful mathematical theories, is "messy." And I think that both for the nominalist and the Platonist, there is a sense in which one must choose between the beautiful realm of theory and the messy realm of facts. I reject this dualism by taking the radical position that eternal, necessary truths are inherent in real objects. I do not thereby deny the contingency of the universe; of course it could have been different from the way it is. Yet every object reveals necessary truths; paradoxically, we find the infinite and the eternal in the finite and temporary.
To put it in starkly theological terms, I would compare Platonism to gnosticism and nominalism to idolatry. The one would have discovery be a way of escaping the created order; the other would have discovery be entirely about finite, contingent reality. Instead, I think discovery involves an interlocking of the temporal and the eternal. From real world objects we discover eternal, necessary truths; in return, we can use these eternal truths to understand--and also care for--the world we inhabit.
Indeed, is it not the mystery of whether physical laws are truly necessary that drives so much of theoretical physics? One encounters mathematical relations between objects with fundamental constants which can be measured empirically, and it is natural to wonder whether such constants could actually be deduced from some deeper principle. Or whether the laws of physics themselves are actually corollaries of some more fundamental Law. Could the universe have "come into being" through some means other than what we call the "big bang"? Such questions magnify the interlocking of the eternal and the temporal, the necessary and the contingent. God's glory shines in all things, to such an extent that it is difficult to see where his invisible glory ends and the more visible nature of things begins.
As a corollary, I see mathematics not so much as a way of escaping into abstract truths in a higher realm, nor as a mere tool of the sciences, but rather as a humble servant of empirical investigation. We study mathematics not only to understand what the world is like but also how it must be, and in that sense it gives some of the deepest insight of any science. Yet the inspiration for its progress is not so much a desire to ascend toward heaven as to see the heavenly on earth. Whose heart can be so cold as to resist finding the beauty in Euler's formula? Yet if we never saw such things as oscillations in common experience, I'm sure we never would have seen such a beautiful equation.