A realist, a conceptualist, and a nominalist walk into a bar. The bartender says, "We serve your kind and your kind, but not your kind."
That's a subtle philosophy joke from an acquaintance of mine.
The reason I'm not a nominalist is simple: I believe that abstract objects, such as numbers, sets, propositions, laws of logic, moral obligations, and so forth, are indispensable to rational inquiry. It seems absurd to me that something can be indispensable, while simultaneously being a "useful fiction" or a "social convention," as nominalists would have it. One can summarize the argument easily:
1. Whatever is indispensable exists. (Premise)
2. Laws of logic are indispensable. (Premise)
3. Therefore, laws of logic exist. (From 1 and 2)
I chose the laws of logic because logic is an area of philosophy I do quite well in, at least academically-speaking. If you think I'm illogical or non-logical in my personal interactions, that's quite a different story. However, I digress.
The argument appears to me, at any rate, to be intuitively obvious. While intuition does not constitute proof, it should not be dismissed outright as a rational basis for accepting the argument. Intuition, I wager, is a rational means to base one's beliefs on barring some defeater. However, I've also argued in my book, Faith and Philosophy, that nothing that possesses an attribute can be nonexistent. Nonexistent things do not have any instantiated attributes, or attributes found in the real world. Since indispensability is an attribute of the laws of logic, it follows that the laws of logic in fact exist.
A popular counter to this argument (at least on some internet forums) is that unicorns possess attributes, but do not exist, at least as far as we know. I think the difference is that the indispensability of the laws of logic is instantiated in actuality, whereas the unicorn's attributes (roughly, a magical horse with a horn) are not. It really is impossible to reason apart from the laws of logic, which is why I say their indispensability is instantiated.