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.