As a metaphysical position, naturalism, like most metaphysics, is untenable and perhaps even incomprehensible. What does it mean for anything, e.g. the natural world, to exist -- I mean really exist? Position two glasses on a table. Ask a mereological nihilist (someone who believes does not believe in objects with parts) how many objects exist on the table, and she'll answer, "Two: the first glass and the second." Ask a mereological essentialist (someone who believes in such objects) and he'll answer, "Three: the first glass, the second glass and the mereological sum of the two glasses." But does the mereological sum really exist? One philosopher says "yes", the other "no." Which, if either, is right? I think each philosophers speaks the truth because each interprets the proposition "The sum exists" from within vis own worldview, in terms of vis own ontology. To her, 'exists' means 'belongs to the set of objects without parts,' while he reads 'exists' as 'belongs to the set of objects including those with parts'. Does the sum exist in the sense that it belongs to the set of objects without parts? Trivially, no. But does it exist in his sense? Trivially, yes. I'm tempted to believe that the dispute is merely terminological. But then, whose definition is better? I don't think one can choose between worldviews without appealing to some primitive concept of existence, which I don't seem to possess.
The naturalistic claim that "there is only the natural world" means nothing more than that only the natural world belongs to some arbitrary set such as 'the natural world', 'the world discovered by science' (aka, 'the natural world') or something equally trivial. A similar reduction to tautology can be made to most metaphysical theses. Only when one clarifies what she means by existence given the context of her inquiry can she ask questions capable of prompting substantive answers. The predicate 'to exist' becomes a strongly context-dependent shorthand, and ontology is revealed to study only the most trivial claims about object identity.
Let's accept naturalism for the moment: I'll pretend that thinking "there is only the natural world" is somehow the right way of thinking. What types of objects exist? I think the naturalistic notion of existence is an ambiguous notion at best. The scientific method does not define a clear ontology. According to naturalism, an object exists if it belongs to a model of the universe that has survived scientific scrutiny; that is, a falsifiable hypothesis left standing by experiment. The author notes that the 'best model' of the universe is constantly changing as science progresses, and this is surely a concern. I think there is a similar, larger problem with defining an allegedly objective concept such as existence on a subjective preference of models: the scientific method does not prefer a particular model, rather it approves of some and invalidates others. It is certainly not the case that there is only one scientific model now -- more than a few string theories are effectively "complete" in that they explain everything we observe -- but even if experiment eventually eliminates all but one of them, there will still be a range of metaphysical systems that account for experience. Scientists have historically employed vague standards of parsimony and elegance (not to mention cultural domination) to select a single model from the mix. Unless naturalism clarifies how these tools complement the scientific method in determining a preferred model and hence an objective ontology, it remains ambiguous. Put in other terms, science cannot test existence. Metaphysical questions like "Do electrons exist?" can only be answered with the tautology P->P, however convoluted our arbitrary interpretation of P.
I feel similarly about naturalism as a methodological stance. A methodology is a means of acquiring truth, and the truth of a proposition is relative to the interpretation of the symbols that constitute it. Is it true that P->Q? The answer to this question depends on what is meant by P, on what is meant by Q and even on what is meant by ->. Let's assume we have a scientifically perfect theory of the universe. Are the dynamical equations deduced from this theory true? The answer to this question depends on what is meant by the mathematical equality relation in the logical sentence that is each dynamical equation. Given a scientific interpretation (or methodology), equality represents a statistical correlation not falsified by experiment and so, presumably, the proposition is true. Another interpretation -- for example, the negation of 'scientific equality' -- does not necessarily yield this result. To claim, as methodological naturalism does, that the scientific method determines truth is to claim the obvious: that propositions made true by the scientific interpretation are true by exactly this interpretation.
The author asks, "[are there] things discoverable by non-scientific means, or not discoverable at all?" In terms of metaphysics, the answer depends on what ontology is presupposed and whether such things can be found on that list of things that exist. For example, a Christian ontology includes such things. In terms of methodology, the answer is a definite "yes." No doubt will some non-scientific logical interpretation, established by some non-scientific methodology, 'discover' some truth. But do these things really -- I mean, really -- exist? Considered in the abstract, this question is a bad one. I don't know: does P->Q?
As a methodology, naturalism is less susceptible to my second attack on naturalistic metaphysics. Science does extremely well at generating unambiguous truths. It is only when multiple theories, each compatible with experiment, are abstracted from direct observation that scientific truth becomes relative. And I think the more abstract scientific claims can be read as metaphors for basic causal relations, as shorthand for lengthy lists of events and the relations between them. Most scientists seem to have forgotten that each of these patterns originated from the language of a lower level model and that legitimate interpretations of their physical meanings are limited. If we restrict or reduce our notion of truth to claims about observations (rather than objects like electrons), methodological naturalism is unambiguous, though still arbitrary.
Perhaps the most devastating argument against naturalism -- and, more generally, logical positivism -- turns the methodological thesis against itself: if the only truths are those revealed by science, how is naturalism, the thesis that such truths are the only truths, true? It simple isn't.
What do ontological and conceptual relativism mean for science? I see little conflict. Instead of asking which way of thinking is right, ask which is more practical. Scientific claims not only draw causal relations between past events but also predict the future. The scientific method generates these truths. A naturalistic metaphysics aids the development of scientific theory. A statement like "Theory P is true by the standards of science" is no less practical than an unjustified appeal to a primitive concept of truth and is certainly more philosophically coherent.
Nonscientific metaphysics and methodologies can be equally practical when applied to their appropriate areas of thought. For example, the majority of historical claims are fundamentally distinct from scientific claims, so a naturalistic methodology can never apply to history. Yet historians still make claims to truth. Are they mistaken in doing so? No, but their truth is distinct from scientific truth, as it is generated by a different methodology, and many historical objects exist only in nonscientific metaphysics. Per the above, no preference between these standards can be justified; that is, beyond what is practical to assume. Whether historical, literary critical or mathematical methodologies achieve what their respective partitions of academia set out to achieve is the only question worth asking.