In response to the bit about logical positivism: Logical positivism would, more than anything I think, label this question as meaningless. However, my understanding is that logical positivism has been mostly discredited, largely thanks to Quine. His "Two Dogmas of Empiricism" attempts to break down the analytic/synthetic distinction that the verification principle so heavily relies upon. Furthermore, it seems impossible to formulate the verification principle in such a way that it gets rid of metaphysical questions such as "What is math?" and hang on to the terminology required to do science.
Carnap gave up on it, convinced by Quine. Wittgenstein, the often cited father of the movement, completely rejected it in his later years.
However, your point with regards to pragmatism I think is important, though not insurmountable. These questions are important for determining what we consider legitimate mathematics and thus how we should proceed in the study of mathematics. Consider the foundational questions in the first half of the 20th century: Frege's, Russel's, Zermelo's, and Hilbert's programs all had wildly different consequences in terms how one should proceed in mathematics (though they all tried to preserve the main body of it). The most concrete example of this is the bizarre machinery that is required to do calculus: have you ever tried to convince anyone that there are have to be different levels of infinity in order to do that which physics takes as central? When Cantor (and Newton for that matter) first came up with their ideas, they were laughed at! What they showed was not considered legitimate mathematics at first, though it turned out to be crucial for modern science.
Furthermore, these questions help decide what mathematics' place in the world: what exactly are we trying to accomplish with it? What is mathematics studying and what predictive powers should we expect from it?
I don't know if this will convince anyone these questions are worth asking, but hopefully they will make people curious. The deeper one goes, the stranger things become. For instance, the Banach-Tarski theorem says that one can break up a ball into a whole bunch of points, and then reassemble them into two balls identical to the first! And this is a theorem, a direct result from ZFC! Or (and this screwed with my head for a long time) Skolem showed that any first order set theory satisfiable by a nondenumerable model is also satisfiable by a coutable one (note: I made that all jargony so that people unfamilar with set theory won't know what I am talking about; dangerous idea if you don't know/believe Cantor's stuff and very easy to misinterpret if you don't know set theory to an okay extent. Even then, takes a while...)
Second, my answer to the question:
Mathematical structure is discovered, but such abstractions do not exist as universals/forms/etc. Mathematics is the study of abstract structure. This structure is inherent in/a result of the causal properties of objects in the universe. There are no forms/universals, just systems of objects and causal relations, and isomorphisms between these systems.
(I'm a trope nominalist when it comes to properties: http://plato.stanford.edu/entries/tropes/. Further, I believe that the resemblance classes central to the theory are defined by causal similarity. I think this is possible because causal properties ARE qualitative properties. Sorry, just wanted to answer any phil majors up front:).)
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In response to the bit about logical positivism: Logical positivism would, more than anything I think, label this question as meaningless. However, my understanding is that logical positivism has been mostly discredited, largely thanks to Quine. His "Two Dogmas of Empiricism" attempts to break down the analytic/synthetic distinction that the verification principle so heavily relies upon. Furthermore, it seems impossible to formulate the verification principle in such a way that it gets rid of metaphysical questions such as "What is math?" and hang on to the terminology required to do science.
:).)
Carnap gave up on it, convinced by Quine. Wittgenstein, the often cited father of the movement, completely rejected it in his later years.
However, your point with regards to pragmatism I think is important, though not insurmountable. These questions are important for determining what we consider legitimate mathematics and thus how we should proceed in the study of mathematics. Consider the foundational questions in the first half of the 20th century: Frege's, Russel's, Zermelo's, and Hilbert's programs all had wildly different consequences in terms how one should proceed in mathematics (though they all tried to preserve the main body of it). The most concrete example of this is the bizarre machinery that is required to do calculus: have you ever tried to convince anyone that there are have to be different levels of infinity in order to do that which physics takes as central? When Cantor (and Newton for that matter) first came up with their ideas, they were laughed at! What they showed was not considered legitimate mathematics at first, though it turned out to be crucial for modern science.
Furthermore, these questions help decide what mathematics' place in the world: what exactly are we trying to accomplish with it? What is mathematics studying and what predictive powers should we expect from it?
I don't know if this will convince anyone these questions are worth asking, but hopefully they will make people curious. The deeper one goes, the stranger things become. For instance, the Banach-Tarski theorem says that one can break up a ball into a whole bunch of points, and then reassemble them into two balls identical to the first! And this is a theorem, a direct result from ZFC! Or (and this screwed with my head for a long time) Skolem showed that any first order set theory satisfiable by a nondenumerable model is also satisfiable by a coutable one (note: I made that all jargony so that people unfamilar with set theory won't know what I am talking about; dangerous idea if you don't know/believe Cantor's stuff and very easy to misinterpret if you don't know set theory to an okay extent. Even then, takes a while...)
Second, my answer to the question: Mathematical structure is discovered, but such abstractions do not exist as universals/forms/etc. Mathematics is the study of abstract structure. This structure is inherent in/a result of the causal properties of objects in the universe. There are no forms/universals, just systems of objects and causal relations, and isomorphisms between these systems.
(I'm a trope nominalist when it comes to properties: http://plato.stanford.edu/entries/tropes/. Further, I believe that the resemblance classes central to the theory are defined by causal similarity. I think this is possible because causal properties ARE qualitative properties. Sorry, just wanted to answer any phil majors up front