My Eschaton buddy JR isn't sure if numbers themselves, rather than the sets they describe, exist. I do think numbers exist; if I can think about them, manipulate them and use them, they have to, at some level at least. And they're wicked useful. Problem is, they're willfully misunderstood. Over and over again, throughout our culture and society, the attempt is made to simplify complex entities by placing them in correspondence with numbers on the real number line. Makes things easier to compare, define, act on, but does violence to reality. It can be shown that the law of trichotomy--any real number must be greater than, less than, or equal to any other--fails even with the relatively simple expansion of complex numbers. There are things which cannot be quantitated. Working with them as if they can be isn't likely to succeed.
Examples abound: speeding over reckless driving. A breathalyzer number over intoxication. IQ over intelligence. The US News college rankings. Testing as a measure of teacher and school competence. Economic models as sufficient descriptions of human society, privileged over and trivializing social, moral, ethical considerations--what Tony Judt calls 'economism'. Quantitative scientific observations and theorizing in theory able to explain all phenomana, and, if they can't, it bespeaks the limitations of our current scientific knowledge rather than of science itself.
Not just intellectual laziness, this, but a significant, sometimes even dangerous error. Arising, in part, I think, from an uncritical celebration of the Enlightenment, which many intelligent folk with whom I mostly agree find seductive in the wake of the Busherdammerung, in the face of right wing know-nothing Lysenkoist or fundy madness. We need to reject ignorance, especially willful ignorance, the dogma and intolerance which fuels it and results from it. But we, too, need to understand, appreciate and act mindfully of the limitations of 'reason' alone in describing our lives in this vale of wrath and tears. Not easy to do it right, that...
Anthony, another buddy who comments on Eschaton and blogs on Echidne,rants about this, too, from more of a humanities perspective, and is well worth reading, in general as well as on this specific topic, if this stuff interests you: