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As far as we know, the natural world is all there is. By weakening naturalism to a belief held in the absence of evidence to the contrary, weak naturalism becomes an even more formidable opponent to supernaturalism. Naturalism entails the closed system of the known universe, whereas supernaturalism refers to a possible realm outside of the universe.
Weak naturalism is analogous to "weak" atheism. A weak atheist simply disbelieves in God given the lack of evidence. Weak naturalism disavows the supernatural for the same reason.
Without relying on the accepted definitions of naturalism, or of the rich philosophical history, weak naturalism is justifiable on its own terms. In only asserting the existence of the natural world, and by leaving itself open to disproof by evidence of a supernatural realm, the burden of proof is transferred to the "supernaturalist."
Proposing weak naturalism does not require positive evidence showing why it's probable that nothing transcends nature. Rather, it appeals to the lack of evidence for anything supernatural, period.
Further, we might wonder what "positive evidence" could be provided for things that don't actually exist. Things like ghosts, Bigfoot, or Atlantis, for example. Surely, the lack of any evidence for the existence of such things amounts to at least a prima facie case for their nonexistence. If we have to wager a guess, our default position should be that they don't exist, rather than that they do exist.
Thus, the supernaturalist must provide evidence of the existence of a supernatural world. Where is Exhibit A? Unfortunately, the claims of the supernaturalist fall at the first hurdle. The distinction between "nature" and "supernature" is fraught with difficulties because it relies on a comparison between known entities. We don't and cannot know anything about any proposed transcendent or outside realm. Thus, the word "supernatural" relies on nature itself for its meaning. It has no specific agreed upon definition, and is an umbrella term housing an immense pantheon of disparate religious claims, including places such as Hell and Valhalla, and entities such as souls, angels, and gods.
Indeed, we should observe the special pleading routinely used to justify beliefs held sans evidence. Pope Benedict claimed that science cannot answer "questions that go beyond its methodical canon and cannot be answered within it." The supernatural is routinely defended in this way—by proposing methods of enquiry beyond the scope of empirical science. If the supernatural is defined as being beyond the scope of the natural world, it is neither governed by natural laws nor measurable by science, so the supernaturalist posits other, presumably supernatural, means of evidence. In order to smuggle in unevidenced beliefs about the existence of supernatural realms and entities, an extraordinary leap is often made from a distinction between religious beliefs held about values/morals, and empirical facts.
But this is an illegitimate move that could be used to justify any proposition whatsoever. If this was really an acceptable move, then all that we would need to claim about a proposition is that it is beyond the purview of science, and "Voila!" we can demand equal epistemic justification for positing the existence of fairies, unicorns, and the Loch Ness Monster. We could also continue to superimpose new unseen layers upon the universe indefinitely.
Furthermore, it's not entirely clear what constitutes this "evidence." In fact, the move invokes a circular argument by seeking to justify the existence of the supernatural by inventing its own realm of supernatural evidence. If such a realm is beyond the limits of empirical investigation, then by definition it is unknowable. Thus, we cannot assign it any probability of existing. In addition, our chances of predicting the nature of what is unknowable would seem to be infinitely poor. There is an infinite number of possible supernatural realities that might be posited. And since in practice belief in the supernatural always pertains to belief in a specific version of supernaturalism, the choice is not simply a choice between two views. For example, it is not a choice between belief in the natural world alone and belief in the supernatural in addition to belief in natural things. It is a choice between belief in the natural world alone and belief in the Christian version of the supernatural—or the Hindu version, or the ancient Greek version, and so on—on top of belief in the natural world. It is indisputable that the myriad possible versions of the supernatural reduce the likelihood of any particular version of it existing.
We also see an appeal to "immaterial" realities in the appreciation of poetry, literature, music and such, as if to suggest that things that cannot be seen or are not explainable by science point to the possibility of a supernatural cloak upon reality. But we may briskly walk from the science lab over to the humanities building and see books of poetry. We can play musical instruments, read music, and even use an MRI machine to view our neurons firing as we listen. Our invisible thoughts are no more evidence for supernaturalism than the invisible operation of gravity.
Alternatively, supernaturalists often seek to frame the debate in philosophical, rather than scientific, terms. In this way, the inconvenient hard evidence of science can be replaced with less restrictive philosophical argument, providing a softer approach combining reason and argument with a lesser threshold for evidence. One can point out that philosophical naturalism and theism are both metaphysical points of view. But contextualizing the conversation as philosophical does not remove the demand for evidence in deciding between them; it does not realign or artificially equate their probabilities. All it does is seek to reframe the conversation as philosophical in order to draw on our rich religious history of philosophical arguments for a supernatural realm. But philosophy does not sit in its own self-enclosed bubble, distinct from empiricism, with no relation to the real world.
Imagine that we have found an incontrovertible answer to how the universe came to be. Would we be surprised if this was discovered by a team of cosmologists? Or of theoretical physicists? Probably not.
But what if it the announcement mentioned a team of philosophers? The reason why such an announcement would be greeted with derision is because we know that without evidence, testing, and verification, a claim made purely on philosophical grounds is lacking. A discovery of this type would only find general acceptance through scientific confirmation. Moreover, we know that it's scientists who are working on these questions, not philosophers. They are questions of fact, not argument.
Thus, it's not scientism to expect knowledge-claims to be verifiable or testable. The scientific method has become the accepted method for ascertaining which empirical claims are true or not for a reason.
Using philosophical argument, theists sometimes invoke Thomas Aquinas' "Five Ways," or the cosmological argument, to claim that the universe must have had a cause, as though this argument was evidence. But, alas, the cosmological argument is not accepted as establishing empirical knowledge. In fact, the majority of contemporary philosophers remain unpersuaded by it. Its proponents illegitimately claim that the existence of the world is "evidence" that something preceded it, or sits outside of it, because of the operation of causality—particularly the notion that "everything has a cause." However, this is a semantic trick and a fallacy of division. What is true for a part of a system is not necessarily true for the sum of its parts.
Weak naturalism limits itself to what we know. If there are realms that we cannot know, then there is no use in speculating upon them. The question of what constitutes the relevant evidence becomes the key issue. If "evidence" is not measurable and knowable, then it is not really evidence. And in the absence of useful evidence, it's a false dichotomy to suggest that weak naturalism is on a par with supernaturalism. Our default position must be that, as far as we know, nature is all there is.