Mathematical realism defends that mathematical objects have independent existence outside minds. In this view, mathematics is more about discovering an unknown world than just creating formal coherent maps. This position seems to me an indisputable matter of principle, an aesthetic position that cannot be decided, a modern version of the ancient theory of Plato's forms.
Math is the paradigm of objective knowledge but I feel suspicious when that is used to imply their external existence. Beliefs can be objective or subjective but how can that tell us anything about external reality? An objective belief is a belief that does not depend on the agent's state of mind, but a belief nonetheless It does not directly follow that without minds objective beliefs would still exist (or could exist even before minds). What are the arguments to justify this step?
Another way to argue is to state the uncanny usefulness of mathematics. Some math models are surprisingly useful for science. However, there are potentially an infinity of different mathematical models. In our finite world we will always use an infinitesimal fraction of the mathematical formal structure. So, usefulness does not seem a strong argument in the defense of math realism: most of Math would be useless to explain a finite Universe (not enough world to use all those theorems).
Even restricting the realm of 'real' Math like Kronecker did when he said God made the natural numbers; all else is the work of man does not make things easier. In some alien world, the subset of Math they'll use may be very different from our own. Eg, in a plasma world -- where everything would be in flux and no solids would exist -- natural numbers (0,1,2...) might not make the least practical sense for its inhabitants and they would be as known to the average plasma-mind as manifold theory is to our carbon-minds. So, even those most basic of math concepts, like naturals, might not be as natural as we think they are. And stating that Human «natural Math» is the one that is «real» seems just provincialism.
Original posting & discussion: https://plus.google.com/u/0/107157321007785956289/posts/4abeMMDY2Qx