To say that a possible, or potential thing (non-actual thing) is real, is self-contradictory. "Real", by definition refers to the actual thing, as indicating its difference from the possible, or non-actual thing. — Metaphysician Undercover
No; you are ignoring the distinction between "real" and "actual," and instead treating them as synonyms. Realism regarding universals/generals is the view that the real is a broader category than the actual, such that possibilities and regularities (for example) are just as real as actualities. You are simply asserting nominalism - the opposing view that the real and the actual are one and the same. You cannot refute realism by simply insisting on a nominalist definition of "real."
Unless we can establish as a fact, that ideals are what is real, or at least some sort of relationship between ideals and reality, there is no basis for your claim that models must be "of something real", if they are useful. — Metaphysician Undercover
The model itself is ideal, but that which it is intended to represent is real. In particular, a mathematical model - i.e., a diagram - embodies the real relations among the parts of its object; and again, the modeler
selects those parts and their relations as those that are
significant, given the purpose of the model.
What the irrational nature of pi demonstrates is that a circle cannot have both a circumference and a diameter, in any absolute, or "ideal" sense. These two are incompatible, the diameter and the circumference, as the ratio between them cannot be resolved, in any absolute sense. — Metaphysician Undercover
Huh? The ideal sense is the
only sense in which a circle
can have both a circumference and a diameter in the exact ratio of pi. The two lengths are not "incompatible," whatever that means; they are
incommensurable, which simply means - as I said before - that their ratio cannot be precisely
measured as a rational number, not that it is somehow "unreal." The same goes for the ratio between the diagonal and side length of a square. It is not by accident that what we call "real numbers" include not only all rational numbers, but also all irrational numbers.
Even an infinite number of non-dimensional points could not produce a dimensional line, we must assume something in between the non-dimensional points, line segments. — Metaphysician Undercover
I agree with this, since it is the very definition of a true continuum. No multitude of
actual points - not even the infinite total of all real numbers - comprises a line; instead, a line contains
potential points exceeding all multitude. Every part of a continuum is itself a continuum, not an individual; every part of a line is itself a line, not a point. The dispute between realists and nominalists may also be framed as the question of whether there are any
real continua of this sort; realists say yes, nominalists say no.
So the circumference is incompatible with the diameter, one is two dimensional, the other is one dimensional. — Metaphysician Undercover
This, however, is nonsense; if it were true, then the perimeter of a (two-dimensional) square would be "incompatible" with its (one-dimensional) side length, while the (one-dimensional) diagonal length of a square would be "compatible" with its (one-dimensional) side length. On the contrary, the perimeter and side length of a square are
commensurable, but its diagonal and side length are
incommensurable.