My take here starts by saying it is a false dichotomy to think the situation would be EITHER empiricism OR platonism (or constructivism or intuitonalism, or however else we want to phrase this traditional divide between "looking out" and "looking in"). Instead - logically - it must always be BOTH. As the complementary extremes of "what can self-consistently be".
This is what happens because I chose asymmetric dichotomisation as the foundation of my logic. This is of course the unfamilar bit, even though it starts from ancient greek metaphysics (Anaximander, Aristotle), was messed about a bit by the likes of Hegel, and reappears in modern times with Peirce.
Now asymmetric dichotomisation says that any (vague) state of possibility or potential can only be (crisply) divided if that act of separation goes in two exactly "opposite" directions. And by opposite, this is not symmetric as in left/right or other kinds of symmetry breakings which have just a single scale. It must be an asymmetric breaking that is across scale and so results in completely unlike outcomes (as opposed to merely mirror reflections of the same thing).
If you are with me so far, then the classic examples of asymmetric dichotomies in metaphysics are local-global, substance-form, discrete-continuous, stasis-flux, chance-necessity, matter-mind, vague-crisp, subjective-objective, atom-void, space-time, location-momentum (and the list goes on, but these are among the "strong ones").
You can see that each is both the very opposite of the other, and yet also logically mutual or complementary. That is because each is defined actively as the exclusion of the other. Pure substance would be a stuff that has absolutely no form, and form is that which has absolutely no substance. (Even Plato had to have the BOTH of the forms and the chora).
So this is an emergentist and interactions-based logic or causality (a logic being a generalised model of causality in my book). You cannot have one side arise into being, into existence (or persistence) without also forming the other. As one arises (in thought or reality) by becoming everything that the other is not.
As I say, Anaximander was the first to articulate a vagueness => dichotomy => hierarchy approach to modelling causality, the logic of reality. Aristotle then polished it up (as in the law of the exclude middle). Today, you can see mathematical sketches of the idea in the symmetry breaking models of condensed matter physics, in hierarchy theory, and even in some basic stabs at maths notation.
Check out Louis Kauffman's musings on this...
http://www.math.uic.edu/~kauffman/Peirce.pdf
The laws of form are another stab...
http://en.wikipedia.org/wiki/Laws_of_Form
A gateway to Peirce's writings (which are only a precursor to what I'm talking about)...
http://www.cspeirce.com/
And others currently treading some of the same ground (though I would have many criticisms of Kelso's actual approach)...
http://www.thecomplementarynature.com/
Anyway, I hope you can appreciate that this is like swapping in, swapping out, a complete computational architecture. There is standard logic based on atomism, mechanicalism, locality, and other good stuff which is like your classic sturdy von Neumann serial processing engine. It works, no question. Then over here in left field, there is an attempt to build an architecture of thought, a way of modelling, that is founded on very different basic computational principles. It is like the attempt to get neural networks off the ground. Some kind of global, holistic, hierarchical version of logic. And while it looks promising, it is still a long way from commercialisation.
But anyway, let's take these still developing ideas and apply them to the question you asked.
Again, for me on the grounds of logic (all reality always works this way) I would come with the expectation that the story is going to be not either/or but instead both, and interactionist. So yes, strong dichotomies always emerge, and then the whole point is that they emerge because their existence is self-consistent in the wider view. They are mutually causal, or synergistic as asymmetric extremes.
Therefore it does seem that the creation of mathematics has this basic divide. There is either the pure development of ideas, or the discovery of ideas from observation. And my logic would force me to expect a mutually emergent story. The firming up of ideas inside a person's head allows them to make more detailed observations of the world, which in turn allow for more development of ideas inside their head. And these two parts of the action are driving each other ever further apart in scale. As the observations get ever smaller, ever finer, ever more particular, so the ideas get ever more general, ever more global and universal, ever more lacking in picky detail.
Now to take the specific example of non-euclidean geometry. The tale of the discovery follows this dichotomous logic. At first, forms got separated from substances in a way that divided the flat 3D world of immediate experience. Then as mathematicians realized that just three dimensions is a rather particular choice, and likewise just flat space was a rather particular choice, they could make a leap of generalisation to allow infinite dimensionality and any curvature. Their ideas became less particular, and so more general.
At the same time, this step in one direction brought with it a matching step in the ability to make ever finer "observations". It became possible to model some world with some particular curvature or number of dimensions. Maths could start exploring imaginary worlds of any crisply chosen design (and science could then use this new technology to test our actual world against the new variety of predicted designs).
So dichotomisation is the logic by which humans stepped back to see more. And then I would go further - from epistemology to ontology. Dichotomisation also is how the world probably actually emerges.
Taking non-euclidean geometry, we can see for example that "flat space" is precisely the average, the sum over histories, of curved space. If you have a dichotomous spectrum from purely locally hyberbolic space (disconnecting sea of points) to purely global hyperspheric space (curvature which makes a continuous or perfectly closed space) then flatness is the average, the equilibrium outcome, of these extremes "in interaction".
Of course this is still a hypothesis as I'm not sure how to go about constructing a mathematical proof of the idea. But I am just sketching the kind of answer I would expect to be the case if dichotomous logic is a valid logic.
There is another argument about why there would be just three spatial dimensions. But I can save that for some other time as it is even more left-field if Peircean semiotics is unfamiliar terrain.
To sum up, all my arguments stem from applying a different computational architecture. And it is not an arbitrary choice as - dichotomously - there would have to be exactly two deep models of logic/causality. Standard logic is one pole, and now I am working with people in developing the other pole. I see this as great news for good old fashioned atomistic logic as it cements its authority in place. It can be "right" because there is also the asymmetric view now making it "right" - that is, together they exclude the middle, all other possible approaches to logic.
So dichotomies rule. And the division over whether maths is derived from intuition or perception is a classic example of how both in interaction, creating a virtuous spiral of development, is the answer.
Then the logic of our minds is also the logic of reality itself. Dichotomies or symmetry breakings are also how things happen "out there" - how systems develop into being, complex hierarchies arising out of vaguer potentials because they are the self-consistent way a vagueness can be stabily, self-persistently, divided.
I am sure this is still indigestible. But just focus on some dichotomy and see for yourself if you can break it down differently.
Local-global is the most fundamental dichotomy I believe - pure scale. Though (dichotomously) it is then paired with an equally fundamental dichotomy vague-crisp. One talks about what exists, the other how what exists has developed.
But substance-form is the Athenian set-piece debate. Or you could back up a bit to consider the weaker dichotomies of stasis-flux or chance-necessity or atom-void.
! I almost agree with what you're saying here, except for one minor point.
That is why maths moved on to category theory in its search for its fundamental ground. Structure-preserving change, patterns or symmetries that can persist.
