When we discuss reality, we generalize groups of things into what we call “classes”. Examples of classes are “humans”, “chairs” and “garbage cans”. Such classes consist on one hand of an idea about the class, an abstract type, which defines which things that belongs to the class, called the intension of the class, and on the other hand of all things in reality that fits this idea, called the extension of the class. The intension is thus infinite, whereas the extension is finite. It means that “classes” bridges the insurmountable aisle between abstraction and reality, which is a contradiction.
This contradiction gives rise to the fundamental split between us humans concerning which of these aspects of classes that are “real”, ie, exist – practicians (like me) consider it self-evident that the extension is real, whereas theoreticians have a tendency to instead consider the intension to be real. The former approach is traditionally called nominalism, whereas the latter is called (class-)realism. (The prefix “class-” is often left out, making the approach easy to conflate with “pragmatism”. But, “realism” in this sense has nothing to do with accepting the world as it is, but rather with believing that classes are real in their abstract sense, ie, as types.)
This split, in turn, is responsible for most of our fights concerning opinions. Within each approach, different opinions are merely a matter of varieties in a theme, but between the approaches, different opinions is a matter of different themes. Proponents of the different approaches simply think that proponents of the other approach is totally stupid. One simply can’t understand the other in any part.
The problem for (class-)realism is that IF the intension aspect of classes is real, THEN there ought to be a single class of all classes, which Bertrand Russell demonstrated in 1901 is a paradoxical contradiction. It can thus be true only if (paradoxical) contradiction is real, which (ie, “contradiction”) on the opposite is the definition (ie, intension) of the class “false statements”. Russell’s demonstration did thus actually falsify (class-)realism by demonstrating that it is paradoxically contradictory.
The problem for nominalism is that IF the extension aspect of classes is real, THEN there are infinitely many classes and thus infinitely many different ways to describe reality correctly. This problem is traditionally handled by empirical testing of different ways to describe reality (ie, traditional science), which, however, splits questions into rational (ie, scientific) and irrational (ie, unscientific) questions. The problem with this split is that it makes questions concerning process rational and questions concerning pattern irrational.
Our problem (as humans) is thus that our discussion on reality splits us into nominalists and (class-)realists, whereof (class-)realists are wrong per definition, and nominalists can only answer questions concerning process. It leaves questions such as “what is this thing really?” and “how are A:s related to B:s?” unanswerable – the fundamental problem thus being that classes are not real by their intension, but by their extension.
This fact means that mathematics is correct, although it may not appear so for cladists and particle physicists. The problem is not to find “the true tree of life” or “Higgs particle”, but that there are no such thngs.