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The debate between realists and anti-realists can be traced in the debate between realism, conceptualism and nominalism (Quine as in philosohy of mathematics). Let us recall the debate as promulgated the priests.
Aquinas defends the idea that there are real elements of thinking that is outside the thinker himself. This locution amounts to a realistic scientific exposition, quasi-Aristotelian of sorts.
Duns Scotus remarks that the idea of a thing is more real than the thing itself. It is the concepts that matter and not the thing itself. But that does not necessarily lead to the rejection of real entities for that matter.
Peter Abelard leads the way for the nominalists. I should say that my view lies parallel to this. He ascribes that what he have are mere names. Nothing more... It is to be regarded that the name of a thing, even though thing itself is a name, is what is important to recognize. It could be the case that nominalism amounts to empiricism (as Mach would reagard it).
Well, the main critic is that nominalism views that what we have as real scientific entities are nothing more than names of some objects that we supply in order to perform scientific work. As S. Hawking may say, the idea of black holes is not to say that there are indeed real black holes, but precisely that in asserting that there is as such would make a scientific conjecture provable. We need not say that there are indeed some entity x, for the problem with this is that we cannot ascertain such. What, if any, is the criterion whether an entity is or not.
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