This is an argument that could only be had in religious and philosophy departs.
It is of no substance, the reality is everything is physic's and so everything can be investigated using the scientific method.
What knowledge can't be explored through reduction is knowledge not worth having, in the sense that it will not help us materially.

Sorry this is just so confused it is hard to know where to start. The "supremacy" of science is a study in epistemology - the study of the question "what is knowledge".

There is no definitive answer to the question - nor is there a definitive answer to the question "What is this thing called Science?" Alan Chalmers excellent book by that title traverses the field well.

Science does, however, have the following "principles":
1. The theory needs to be "consistent" with all the known observed instances
2. The theory needs to have explanatory value - it is not just an equation that relates input A to output B but describes a mechanism by which it occurs.
3. (Desirable but not necessary)the theory is capable of making predictions the accuracy of which can be observed.

Science makes no claim to completeness - it doesn't have an answer to everything. Science acknowledges that it is a growing study - that there will be events that conflict with theory. How these are handled is a large part of the debate in philosophy of science (Popperians regard it as the moment a theory is abandoned, Kuhn identifies a paradigm that simply ignores anomalies till it is overthrown in a revolution, Feyerabend claims scientists simultaneously use a range of incommensurable theories for specific events, Bayesians regard the data points as building to a probability of accuracy).

Perhaps the word "Scientism" can be used for those who overclaim for science - but real science does not claim perfection nor completeness. It does claim to a systematic method.

PS Art is not "knowledge" it is however "communication" - very different.

Most brilliant scientist are not dumb enough to ignore the fantasy of order from chaos. They also don't have to imagine in their mind the idiotic notion that one species becomes another species. These are simple facts that expose to a 2 year old the foolish fairytales that pseudo science has as its dogmas.

Remember everyone in"runners" world it's the angles that keep the planes in the sky.

Another thought-provoking piece, thanks Peter.

David Deutsch has taken on the problem of reductionism and of causality with his Constructor Theory approach. I won't bore the reader with an explanation here, if anyone is interested they can look his work up. It's in early stages, but is intended to be a completely general approach to both the epistemological and ontological bases of Popperian philosophy with the intent of defining a new language of physics.

In a nutshell, in that model the fallacy you point out in Hume's reasoning disappears :

Everything which is directly observable is the result of interactions between things that may not be directly observable.

If there is a conceivable way that something could be possible [the product of an interaction], then it must be assumed to be real unless it is already known to be impossible through experiment.

In fact, this is now becoming a very common view among scientists, especially physicists and mathematicians. Feynmann certainly held strongly to a very similar view.

You might be interested in looking up Sir Roger Penrose's musings on the subject as well.

Craig,
Thanks, I will look into your suggestions. I was going to include something on indeterminacy but found myself out of my depth. It appears that unstable nuclei decay at random and that they do out of the usual effective/final cause structure of scholasticism. It seems to me that not knowing about causality in this instance is no excuse for separating cause and effect.
Peter