* As its domain is all observable phenomena, science provides the only valuable explanations of the natural world, and any non-scientific pursuit must be grounded on non-observability and supernaturalism.
First and foremost, this intuition contains a subtle fallacy, which is the idea that science must be the exclusive method for understanding its domain. That is, even though science may rightfully claim the entire universe as its domain of study, it does not necessarily preclude other methods of inquiry or understanding. The idea that science excludes other methods of inquiry would only be correct if science is assumed to provide a complete account of its domain. If that were true, then other methods of inquiry would in fact either be superfluous (or reducible to science) or else be based on the dubious non-natural world. It suffices to prove the intuition false by showing that science does not in fact provide a complete account of its domain and that other approaches can shed understanding on the parts that science does not.
The wikipedia article for scientific method explains: "To be termed scientific, a method of inquiry must be based on gathering observable, empirical and measurable evidence subject to specific principles of reasoning. A scientific method consists of the collection of data through observation and experimentation, and the formulation and testing of hypotheses." As a result, the scientific method can only produce explanations in term of externalist categories. That is, the process of observation and measurement must abstract its subject's existence to that part which effects physical causation on other entities. As a result, if entities do possess an "inner world", then science cannot in fact provide a complete account of its domain. In The World-Knot p.54, David Ray Griffin explains, "If the world contains experiences, points of view, beings that it is like something to be -- and we know that it does -- then every purely externalist description will necessarily leave something out."
One could argue furthermore that the realm of pure abstract reasoning is also beyond the domain of science as defined above. Although the number "2" ingresses upon the world and our experiences in various ways, never do we encounter the pure number in any observable manner. Its existence is purely conceptual; its ontology and status for knowledge a purely philosophical question.
One discipline that combines both of the above areas of non-science (first, concerned with subjective experience and secondly, based in non-empirical reasoning) is the Austrian tradition of economics. This tradition does not "observe" humans acting and model it with equations but rather uses self-evident axioms and verbal logic to deduce what must necessarily be true as humans act. According to this tradition, an economic law like that which says additional units of a good contribute decreasing marginal utility is neither empirically derived nor subject to falsification through experiments. As such, the Austrian tradition is better considered a philosophical discipline than a scientific one, and yet many (including myself) believe it accounts for a better and more truthful understanding of human action than its more scientific rival schools of thought.
Perhaps some would object to the characterization of mathematics and philosophy as non-science. Perhaps they prefer a less rigid definition of science that includes any approach that uses reason. For example, such people might define economic science as any study of economic phenomena and include the Austrian tradition under this umbrella. But then this recasting of what science really is would make any knowledge scientific knowledge. This is a devious form of the No True Scotsman fallacy, where when presented with a counter-example of a claim, one shifts the definition to include the counter-example. Besides, if one is to take all branches of philosophy as actually imparting scientific knowledge, what would one make of theology?
And that question takes us to the real crux of the matter, which has heretofore been avoided. This is that the intuition that science provides the only valuable accounts of the world is not really directed against philosophical disciplines but rather against religious ones. It is argued that rather than being based upon observation and reason, religion is based upon fantasy and faith, largely the product of wishful thinking. Religious explanations are therefore to be always avoided and perhaps even ridiculed.
While I agree that much religious belief may be based in wishful thinking and not have much basis in actual experience, I would also point out that this "scientific" belief is not based on reason but rather fearful thinking. Exponents of the anti-religious worldview view science as a sort of safe-haven of thinking and fear any other way of thinking. According to this line of thought, once you leave science, you leave the reservation. Once you seek such an explanation, you might as well join the Scientologists or some other cult. The irony of such dogmatic thinking is always lost on those who promote it.
In fact, the domain of religion is largely what has been abstracted out of the world by science; the inner life. The questions it seeks to answer have to do with subjective existence, purpose, and decision-making. Religion seeks to account for the meaning of man in the universe in a way that is to guide one's actions.* What is the meaning and purpose of my life? How I ought I to live? How ought I act towards others?
Furthermore, these religious questions are faced by everyone. Everyone acts, so everyone must decide for what they act and how they will act. And just like philosophy, one can consciously develop and explicate religious principles, or one can let them grow organically.
* According to this view, there may be much overlap between religion and philosophy, especially as many great philosophers have developed a conception of God and the moral life of man. However, religion tends to explain not only the nature of man's purposive role in the universe but provide practice for seeking greater understanding and the true inner life.