Nominalism Reconsidered

Nominalism is usually formulated as the thesis that only concrete entities exist or that no abstract entities exist. But where, as here, the interest is primarily in philosophy of mathematics, one can bypass the tangled question of how, exactly, the general abstract/concrete distinction is to be understood by taking nominalism simply as the thesis that there are no distinctively mathematical objects: no numbers, sets, functions, groups, and so on. As to the nature of such objects (if there are any), it can be said that it has come to be fairly widely agreed, under the influence of Frege and others, that they are very different both from paradigmatically physical objects (bricks, stones) and from paradigmatically mental ones (minds, ideas). Modern nominalism emerged in the 1930s as a response to the view of Frege and others that numbers, sets, functions, groups, and so on belong to a "third realm".