Indeterminate objects and contemporary platonism in maths

This evening, having dinner with Nenad and Ira at the Kerepesi dorm he asked me the following question [having in mind that I sympathize to Jim Brown's mathematical platonism]: if we take Jim's example [an actually proven theorem in number theory] that

http://www.personal.ceu.hu/students/03/Boris_Grozdanoff/Jim.gif

illustrated by a simple diagram where every [natural] number is substituted by a square like in

http://www.personal.ceu.hu/students/03/Boris_Grozdanoff/Jim1.gif

then it seems that it is not necessary to imagine a determinate number of squares, i.e. a deteminated value of n, since the theorem holds for all n [defined over Z]. Then, if following Jim, we agree that we have a case with a platonic kind of grasp for the truth of the theorem by virtue of the diagram alone it is natural to ask whether we actually grasp an indeterminate abstract object or we indeterminately grasp a [determinate] abstract object? Nenad believes that this question presents us with a problem about the platonist. Intuitively, I do not have an immediate answer about whether there are indeterminate abstract objects in the platonic realm or only determinate ones and also I am not sure that this is specific problem for platonism alone. Nevertheless the issue seems prima facie interesting. One of the options is the traditional answer that there are only determinate abstract objects and sometimes we only manage to get an indeterminate grasp of them. Another option is that all [possible?] sorts of abstract objects are hosted in the realm, that would mean determinate and indeterminate and for a particular grasp there is a paricular object that corresponds. Anyway, questions like grasp of infinite structures and the like does not make the situation easier.