Argument for holism

This is an argument for holism and against reductionism, in which I argue that the whole is more than the sum of it's parts and that you cannot reduce the behaviour of a physical thing to the properties of the elementary particles it is made of.
I use as a parallel the properties of mathematical systems such as algebra where there is no finite set of axioms which can be used to deduce the truth or falsity of all theorems in that system, as proved by Gödel, and the halting problem in computers.

Firstly, what is the property of these systems which makes it impossible to simply take basic properties of numbers or turing machines and just work it out? I believe it is the extension of numbers to infinity and in computers the possibly never ending machines. This may sound fairly obvious but it is important, since if for example the number set was finite you could write a computer program to determine the truth or falsity of any theorem, and then run it until it gave the answer, and there would be no undecidability. I believe the same thing hold for physics, due to the continuous nature of space and time. Thus space and time may be subdivided ad infinitum and there may be no computer program that models a physical system (or no digital computer program at least), since a computer program proceeds in discrete steps.
I believe that many if not just about all of the systems in nature are impossible to model exactly. If as in maths there is no way to reduce the system to component parts then there is either a solution specific to that system or maybe no model exists, short of an exact replica. (I think I remember someone saying that this is optimum information content).

What does this mean for science? Physics may be no longer reducible to maths, nor chemistry to physics or biology to chemistry. Empirical study is the only answer, and where systems may be modelled the systems may produce behaviours which are not reproducible even in theory on a computer or other modelling device. Computers, with their digital rather than analogue processes and completely predictable from component behaviours, may well never become conscious. I believe consciousness to be a large scale electrical system in the brain, with this holistic property.

Of course this rather dodges the issue of determinism, since the future may still be determined under these holistic ideas, but this can be considered as a separate issue. It also leaves the need for an equivalent of Gödel or Turing's theorems in physics, to put it on a secure footing. However I have faith in the basic idea as a justification for holism.

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