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"Rational Scientific Theories from Theism"


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31.7 The recursively nested hierarchy

It is also worthwhile to investigate formally the whole hierarchical structure of degrees, sub-degrees, etc that has been deduced from scientific theism, according to the recursive principles described in Chapter 19.

Some questions might be, for example, is it reasonable to use this structure at only a finite depth, by which I mean $n \ge 0$, when we talk about (sub-)ndegrees? What is the limiting form of this structure as $n \to \infty$? Does it form a continuous set on a line or in a square? Should we form ‘coarse-graining’ approximations for finite creatures if we cannot make restrictions to finite n? Formal arguments may help us see how this structure induces qualitatively new dispositions everywhere, at every depth of analysis. Do we, in the ‘fine-grain’ regime, end up with a continuous spectrum of qualities, or do they forever remain discrete degrees?

Perhaps there is generated some kind of fractal structure, even if it is as simple as the Cantor fractal obtained when lines are divided and extended in their central thirds. I certainly use many self-similarities between (sub-)ndegree triples and (sub-)mdegrees triples for $m \ne n$: these represent possible correspondences of function assumed to occur (in individual ways) at all all levels and between all levels. But what is the full range of self-similarities within the complete structure? What is the range of correspondences, mathematically speaking?

Finally, we would like to know how the full generative structure of (sub-)ndegrees may be represented (in whole, or in parts) by means of physical structures such as a biological body. Are there any formal guidelines for how this may be efficiently accomplished? How is this related to the mapping assumed when I talked in Section 25.6 of how the human functional form is represented and retained in the physical body according to correspondences of function?

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