There's a useful summary of the subject of universals in the public domain Encyclopedia.com. In it, we read:
Realist and conceptualist theories of universals are, by long tradition, regarded as opposed because according to realism universals are nonmental, or mind-independent, whereas according to conceptualism they are mental, or mind-dependent. For the realist, universals exist in themselves and would exist even if there were no minds to be aware of them; if the world were exactly what it is now, with the one difference that it contained no minds at all, no consciousness of any kind, the existence of universals would be unaffected. They are public somethings with which we are somehow or other acquainted, and a mindless world would lack not universals but only the awareness of them: They would be available for discovery, even if there were nobody to discover them.
I query the descriptors 'nonmental' and 'mind-independent' here, however. Consider as an example Pythagoras' theorem. It is obviously not a physical object, rather it is a concept or a geometric principle. I agree that it is independent of any particular mind, insofar as it would be true even in the absence of any subject to contemplate it. However, it can only be grasped (or perceived) by a rational intellect. It cannot be perceived by a non-rational mind, such as that of a cow or a horse. So I question the description of such principles as 'mind-independent'. They are real for any rational intellect, so they may be considered independent of your mind or mine, but they exist only as intelligible objects, in other words, they are incorporeal principles.
