Metaphysical Reduction

A crude reduction is a reduction according to which entities are reduced in their essence to an essential property or thing that is also manifest in phenomenal world. For example, the debate between whether to reduce the animate to the inanimate, or inanimate to the animate.

A refined reduction is a reduction that is accompanied by conceptual creation. For example, to pose the question that what is the essence of the concept or entity that we call animate or life force. A structural reduction, exemplified by Isbell duality, is a reduction that explicates the machinery according to which entities are defined, and it is an important instance of a refined reduction.

It is crucial that one avoids crude reduction when the initial phase of investigation is over.

Similarly, if one takes \(\mathsf{ZF}\), it flattens, and all mathematical propositions are put in the same level, without an hierarchy - except for those propostions that are about \(\mathsf{ZF}\) itself. Not set-theoretic (and in \(\mathsf{ZF}\)) propositions when reduced to \(\mathsf{ZF}\) ("the phenomenal world") is undergoing a crude reduction.

In \(\mathsf{RCA}_0\), there are enough fine-grainings taking place, so that the proposition \(1+1=2\) and the Hahn-Banach theorem becomes distinguished in terms of hierarchy: \(\mathsf{RCA}_0 \not\vdash \text{Hahn-Banach}\) but \(\mathsf{RCA}_0\vdash (1+1=2)\). We have two refined (relative to to \(\mathsf{ZF}\)) reductions: Hahn-Banach to \(\mathsf{WKL}_0\) and \(1+1=2\) to \(\mathsf{RCA}_0\).
A structured reduction then, e.g., compares the two refined reductions and study what is the shape of the hierarchy itself: then we obtain reverse mathematics.