I’ve often disagreed with Steve Landsburg, sometimes on this blog and sometimes in Slate. So it seems worth mentioning that I’m totally on board with his take on the reality of numbers and other mathematical objects. (Scroll down — and down, and down, and down — to item 9 for the part I’m talking about.)
To me, by far the most satisfying solution is a full-fledged Platonic acknowledgement that numbers are indeed just “out there” and that they are directly accessible to our intuitions. I embrace this view for (at least) three reasons: A. After a lifetime of thinking about numbers, it feels right to me. B. Pretty much every one else who spends his/her life thinking about numbers has come to the same conclusion. C. It seems enormouosly more plausible to me that numbes are “just out there” than that physical objects are “just out there”, partly because there is in fact a unique system of (standard) natural numbers, whereas the properties of the physical universe appear to be far more contingent and therefore unnecessary.
Right on! The view that mountains, clouds, and frogs are not real things can’t really be refuted, but it’s universally judged to be a boring view that’s not worth holding, right? So in order to decide to deem numbers “out there” we don’t have to defend the claim that they’re real, but only that they are at least as real as mountains, clouds, and frogs. This last, weaker claim seems to me obviously correct.
You don’t have to outrun the bear!
Quomodocumque 
I would find this a more compelling argument if we didn’t have lots of mathematical examples where it turned out that there was *no* right answer, and that you could do mathematics, say, with or without the axiom of choice, or geometry with or without the parallel postulate, and still have sensible models. Note that he has to qualify the uniqueness of the “standard” natural numbers— “standard” is a weasel word if I ever heard one. Does the Platonic realm provide us with a limited, unique set of models? Or if it provides models of *all* consistent systems, what good is it?
I am willing to believe that numbers are as real as “clouds” and “the United States of America” and “the game of chess” though there seems to need to invoke any Platonic underpinning to believe in the objective existence of such ill-defined entities.
Those advocating a Platonist answer for the unreasonable usefulness of mathematics (or the unreasonable effectiveness of mathematical intuition) must also deal with its unreasonable usefulness at producing the wrong answers as well as the right ones!
1) I’m not sure ‘Platonism’ means what you think it means. Platonism is not just realism; it is a specific form of realism that specifies an explicit way (i.e. the existence of Platonic forms in a Platonic universe) in which ideas are real.
2) You have an important point in here, which is that philosophy of mathematics is not something that can be divorced from the rest of philosophy (or at least metaphysics/epistemology). I admit to being a formalist mathematically, but this is because I am also formalist about everything else, perhaps with the exception of direct sensory experience. (In philosophy in general, this position is usually called nominalism.) I don’t think numbers are real, but I don’t think electrons are real either. To me, they’re both fictions, though very convenient perhaps to the point of being essential fictions. (The advantage of calling them essential fictions is that the phrasing suggests the questions ‘essential to whom and for what’, which is now a question that can be studied historically and anthropologically/sociologically.)
I feel uneasy with the statement “the properties of the physical universe appear to be far more contingent and therefore unnecessary” because I’m not sure what he’s saying. The sentence needs to be completed somehow: unnecessary for [fill in the blank].
I’ve always thought the concepts of number (at least to some degree) and set are deeply wired into our brains and related to the evolution of perception. When a crow or raccoon finds an untended nest of eggs, they undoubtedly see a eggs related by the enclosing form of the nest (a set defined by an equivalence relation) and undoubtedly have some internal measure of size which could be as primitive as one, a few, and many. (I believe that there is one known isolated human tribe that has a similar primitive number system.) If you’re a mother wolf trying to keep track of your little ones, it certainly helps if you have some fast and reliable sense that all of them are present, such as all four of them are here (though of course they don’t use those words). This is still a long way from a formal number system, but in humans it would be further elaborated with the beginning of human bartering and the (possibly) superior human capacity to name (things, colors, numbers, etc.). If you’re a cougar and looking at a line of elk and think you spot one which may be weaker than others, it’s very useful to be able to immediately relocate that elk if you have to temporarily take your eyes off the line, and the only way to do that is with some visual system construct of order and low integer number.
That doesn’t yet address what is real and what is “out there” but I’m getting too tired to continue with this tonight. Quick thought though: I think a number (named or not) constructible within some formal system in my brain and then posited as an object by my brain is as real to my brain as a tiger perceived through my eyes, but the number/object can not directly terminate my perception of itself like the tiger can terminate my perception of itself.
I thought that I would continue this tomorrow, but I’m not sure that I want to go there at the moment. The last paragraph seems to be pointing the question of the existence of numbers to issues of consciousness in general, and I’ve often found that to be a can of worms, though an interesting can of worms, and I have some other things I can’t avoid worrying about right now.