Universal "grammar" for mathematics

I think that ultimately, this is a question without a definitive answer. But it's not a bad one to ask. My feeling is that it contains the germs of a downfall of the 'Universal Grammar' thesis, since unlike language, mathematics can fairly clearly distinguish between the axiomatic and lawful. So while there are many ways in which some mathematical relativity can be conceived, it has no impact on the universal foundations of mathematics. You don't need to look to aliens. The ancient Greeks saw the world through geometry and saw their arithmetic as being reducible to it (I'm grossly simplifying, of course), and we see the world in exactly the opposite way, reducing geometry to arithmetic. However, these two universes are completely equivalent, something that cannot be said for two languages.

However, this first part of the answer takes at face value the very broad reading of the 'Universal Grammar' assumptions. However, I'd say that when read more narrowly (viz when we look closely at the actual principles posited), I'd say that there's no reason to assume that something like this would hold for mathematics. When you look at the grammar of any language, the universal principles posited by the UG theorists only govern a tiny fraction of all the rule-ful patterns present in a language. But with mathematics, you can derive the vast majority of its rules from the underlying axioms. Therefore, I suspect that the Lakoff/Nunez thesis has much more to say about this: http://www.amazon.co.uk/Where-Mathematics-Comes-Embodied-Brings/dp/0465037712.

Note: I should declare an interest here. I'm an ardent anti-Chomskean (http://metaphorhacker.net/2010/08/why-chomsky-doesnt-count-as-a-gifted-linguist/) and I translated one of Lakoff's books, so I certainly come with some theoretical baggage.