Pragmatic Holism - Bruce Edmonds

5 Formal modelling and reductionism

The computability of a well-defined function is a purely formal question. A function is computable if an index exists such that a universal Turing machine (UTM) with that index calculates that function. For example, consider an enumeration of halting computable programs in order of size. Every function represented by programs in this class is computable without us being able to compute (or "write") the programs for these functions, otherwise we could use this enumeration to compute the "halting problem" [19]*1.

This shows that just because if some function is theoretically computable does not mean that we have the means to find the program to compute it. In other words, the characterisation of reducible as computability is too strong for actual use in reducing a problem. To be able to really compute something we have to be able to follow the instructions (or have a machine do it) and write the program to do this (or really compute that). This can form a very long chain (the program that computes the program that computes the program that...) , but eventually it has to be grounded in a program we are able to write ourselves, if the final program is to carry out our intentions. I call this "intensional modelling" (or "intensional computing", depending on the context). This is what we can compute (using computers as a tool), as opposed to what can theoretically be computed by a Turing machine.

If we take a pragmatic view of reductionism only as such intensional modelling, then we come to the surprising conclusion that there may be some computable functions that we can't compute (intentionally), even by using a computer. We may, of course, come across a few of these programs accidentally, for example by genetic programming, but we can never be certain of this and verifying that these programs meet a specification is itself uncomputable in general (although you may be able to analyse a few such particular examples sufficiently).

This raise the possibility that if a form of self-replicating, evolving software life appeared by accident (say as the unintentional result of a genetic programming program) this may be as difficult to model and understand as its more tangible cousins. The code may be so complex and self-referential that it was as difficult to decode as the mechanisms in normal life (as we know it). So even if (a suitable version of) the reductionist thesis were true in the abstract, we might still be forced to use more holistic or uncertain methods to model the phenomena we come across.

Pragmatic Holism - Bruce Edmonds - 22 FEB 96

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