This might be conflating two concerns, though, which is the reality of numbers and their actual existence.
Compare Plato with Aristotle. Plato held that the all forms exist in some third realm, so numbers would be counted among them. Aristotle, however, held that forms exist in particular instantiations or in minds that abstracted them from reality. (Aquinas could be said to synthesize both views in the sense that forms exist in particulars and in minds, but also have their origin in God, thus making God a sort of third realm, in a way. Neo-Platonists would view the "mind of God" similarly.)
Now, in the Aristotelian view, numbers are quantities abstracted from concrete reality (indeed, quantity is one of the categories), but they are not substantial forms, as you will not see instances of numbers as substances in the world. They're abstractions of accidental forms. Furthermore, a form needn't be instantiated actually, but can exist potentially. This is how he resolves Zeno's paradoxes. You can divide a length an infinite number of times - or in a CS context, you can apply the successor function indefinitely - but only potentially; as a matter of actuality, you have not divided a length an infinite number of times.
So, for Aristotle, you have a finite plurality of things that are potentially infinitely divisible, or a finite series of actions that can be potentially infinitely repeated or whatever.
For a contemporary realist, Aristotelian treatment of math, James Franklin is worth checking out [0].
If something can exist theoretically but not practically, your theory is wrong.
But go ahead and divide a length an infinite number of times then. And actually infinite, not as it tends to infinity.
Take the time to understand the subject matter, because your first sentence doesn't makes sense.
It does.