 Notice that as the argument has been stated here, it is not an argument for a platonistic view of properties; it is an argument for the thesis that properties exist, but not for the thesis that properties are abstract objects. Thus, in order to use this argument to motivate platonism, one would have to supplement it with some reason for thinking that the properties in question here could not be ideas in our heads or immanent properties existing in particular physical objects. There are a number of arguments that one might use here, and in  section 4.3,  we will discuss some of these. But there is no need to pursue this here, because there is good reason to think that the One Over Many argument doesn't succeed anyway — i.e., that it doesn't provide a good reason for believing in properties of any sort. In other words, the One Over Many argument fails to refute nominalism about properties.