...Because ultimately it must affirm the conscious mind simply not exist at all (eliminative materialism).
How very Buddhist of you to reason like that. However as Cofty has pointed out:
I have never encountered anybody who denies consciousness. That is different from saying that consciousness is a property of our brains.
I have posed the following question to SlimBoyFat in a previous thread: 'What is the alternative to Science?' I'd like to simplify it a bit more by restating the question in different language more appropriate to this thread: What is the alternative to reductionism?
Slimboyfat didn't answer me but if it was me that had to answer this question I would look for the answer in the opposite of reductionism, in other words Holism.
But Holism (The whole is greater than the sum of the parts) is already included in reductionism in the form of emergent properties. So Holism just forms a part of reductionism.
Footnote: To see emergent properties in action see the software: Game of Life (golly 2.6) and how emergent properties can be generated using simple rules.
But how is Holism useful? There are very good examples of this in scientific history. Historically Chemistry was being probed and prodded independent of Physics. The rules were discovered of how chemicals and compounds interact and today there are libraries full of these chemical heuristics. Yet today we accept that Chemistry is at its most base Physics. This means that chemical reactions can be described in terms of physics alone. Yet this makes explaining even the most basic chemical reactions a complicated affair. So chemists tend to stick to the discovered rules or emergent properties.
So what am I saying? I am saying that Holism or emergent properties are useful in describing new things in simple rule based ways even if we don't understand the underlying levels below this new thing. HOWEVER reductionism fills in the gaps and shows that there exists a continuum in this universe:
Mathematics -> Physics -> Chemistry -> Organic Chemistry -> Systems Chemistry -> Biology -> Ecology and so on (There are many sub categories which have been left out to maintain clarity)
Slimboyfat: Do you want to know more? It goes much deeper than this.