Mathematical aristotelianism

Aristotle's philosophy was a reaction to that of his predecessor and teacher Plato. One particular detail that separates the two philosophers is the question of the forms, or universals. While Plato claimed that the forms are eternal, existing independently of their physical actualisations, Aristotle thought that the forms were inherent in the objects, without any possibility of existing independently of the objects. Consequently, mathematical Platonism sees physical objects as instantiations of general forms, while mathematical Aristotelianism views mathematical forms as attributes of particular concrete objects. The question is, do particular objects (such as physical bodies) require the platonic forms for their existence, or do, as Aristotle claimed, the forms depend on particular objects for their existence?
At least when it comes to mathematics, the Aristotelian view runs into trouble because there are mathematical objects and results which cannot possibly have a physical basis. Let's list some examples:

• Infinite sets. Physics tells us that there are only a finite number of (observable) physical entities. (There may exist infinitely many distinct universes, but if you believe this, then you don't need to be convinced of mathematical Platonism.) Aristotle tried to solve this problem by rejecting the existence of 'actual' infinity, and replacing it by 'potential' infinity. It's not entirely clear whether this distinction makes any sense when analysed closer, but in any case modern mathematics could definitely not work without something like actual infinity. Moreover, it is a fact that modern mathematics has tremendous applications to all aspects of science. Hence mathematics is something that uses concepts without physical ground, and at the same time it is something whose consequences deeply affects physical reality.
• The Banach-Tarski paradox. If this result had a physical grounding, it would most likely have to be its standard interpretation involving two physical objects and decompositions of these. However, the mathematical result then implies something which is physically impossible, namely that one solid ball can be decomposed and the pieces rearranged so that to form two balls, each of the same volume as the first. The Banach-Tarski theorem can therefore not possibly have any physical grounding. Thus we have here another example of a mathematical form which cannot possibly be realised physically.

There are more examples of this kind, but it will not change anything to include these. To these examples the Aristotelian can reply that 'real' mathematics is grounded in physical objects, such as a number of pebbles on the ground, and then mathematicians dream up all sorts of abstractions on top of that, which are are just mental constructions with no connection to the real world. Well, the problem with this view is that these 'dreamt-up' abstractions have applications to the physical world which are just as intimate as the relation between three pebbles and the number three. To take a simple example, consider differential and integral calculus discovered by Newton and Leibniz. Operating with infinite sums of infinitely small quantities led to modern calculus and the continuum, techniques essentially incorporating infinite objects, and without which most of physics and its applications in technology, would not have seen the light of day.