2.4 Irrelevances to the debate
There are two different forms of self-reference total and grounded. Total self-reference is completely self defining, if you follow the causal (or formal) chain backwards, you do not come to a fixed atomic starting place, but find an infinite recursion of definition in terms of itself. Grounded self-reference starts at a specific place (i.e. state or set of axioms), and then the next state is defined in terms of the last state etc. so after a while the current state is almost completely defined in terms of itself and the origin is lost for all practical purposes.
Most examples of self-reference in the natural world would seem to be cases of grounded self-reference: life itself presumably started from some point, which arose from non-living state*1; language is ultimately grounded in our shared experience, either directly as a child learns its first language, or indirectly in the evolution of language in our species*2; even the universe itself seems to have passed through an initial equilibrial stage .
It would seem that total self-reference is difficult to embed in a traditional formalisation*3, if only because in formalising something you need somewhere to start from. It is possible to dissect such a system so that it is representable within a traditional framework (e.g. the technique described in ), but only by effectively grounding it. Thus even if total self-referential formal systems exist they are only usable in modelling and easily communicable if grounded.
Grounded self-reference is formalisable by traditional formal systems, even though in some cases this may be a cumbersome and "unnatural" way to proceed. It is true that some such formalisations are either resistant, or do no have, analytic solutions that allow effective prediction of future (or even description of past) behaviour, but this has always been true of even the most classical of formalisms and thus is not relevant to the absolute reductionist/holist question.
Again whether we use traditional styled or (grounded) self-referential systems for modelling, relates not to abstract but pragmatic considerations.
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