The fundamental split between realism and nominalism is caused by a fundamental contradiction in conceptualization: that it has an unambiguous front side, but an ambiguous back side.
The front side is all positive statements, like “this is an elk”, or “all mammals have hair”, whereas the back side is all negative statements, like “this is not an elk”, or “no mammals have hair”. The ambiguity in the negative statements resides in that they leave the state of the subject ambiguous, ie, if not an elk, then it can be many different things, and, if no mammal have hair, then mammals can have many different other properties.
Realism is the unambiguous positive approach only accepting positive statements, whereas nominalism is the ambiguous approach accepting both positive and negative statements. (There isn’t any unambiguous negative approach, because this approach is ambiguous and does thereby equal the ambiguous approach, ie, nominalism).
The challenge for us (humans) is to understand the relations in the ambiguous back side, because it is the side of conceptualization that is reality in our conceptualization of reality. (The unambiguous positive side is just a representation in our minds of our different comprehensions of reality.) So, what can we say about this ambiguous back side?
Well, before we can say anything about it, we first have to understand that conceptualization does not have a beginning and an end, but can only rotate around concepts either via synonyms, catalyzing intermediates or orthogonal triangulation. When we understand this, we can understand that the question is how the ambiguous back side in some mysterious way always brings us back to the front side again, although the front side is unambiguous and the back side is ambiguous, ie, that there obviously is a one-to-one correspondence between the back and the front side although the former is unambiguous and the latter is ambiguous.
The key to the the answer to this question is what we call “Russell’s paradox”. This paradox functions as an elevator from the back side to the front side, and the other way around, by being the position for both “both and” and “neither nor”, the difference between which is purely subjective – there simply isn’t any “objective” difference between these two relations, because the former is purely positive and the latter is purely negative, when objectivity is the state between them. This paradox does thus in practice function as an interface between realism and nominalism, providing a path from realism to nominalism, and the other way around, without any contradiction. We can thus answer the question by that Russell’s paradox provides a path to turn ambiguity unambiguous from the back side to the front side, and the other way around in the other direction.
When we understand this fact, we can also understand that the fundamental split between realism and nominalism actually ultimately meet in Russell’s paradox. These two “approaches” are thus not different approaches, but actually two sides of the same coin. The problem for us to understand this fact resides in that we search for a single (unambiguous) truth and do thereby deny ambiguity. If we instead accept ambiguity, then we can understand that unambiguity and ambiguity is connected by Russell’s paradox. The question is thus not whether unambiguity or ambiguity is “true” or “real”, ie, whether realism or nominalism is correct, but rather how they are connected, and the answer is “by Russell’s paradox”.
When we understand this fact, we can formulate the mathematics that mathematicians have formulated today, ie, which derives mathematics from logic. The question is thus not whether realism or nominalism is correct, but rather how we shall join them. The ancient split between realism and nominalism has thus today been closed by modern set theory (or computer mathematics).