Summary: Science must assume that everything can be investigated empirically, but this doesn't force the abandonment of the supernatural, for those who want it.
n philosophy, 'naturalism' is the view that an explanation is justified just so far as it rests on evidence of an empirical kind. It has been very active in the philosophy of mind and moral philosophy, and recently as a tool for the 'conceptual hegemony' of science in opposition to the views of some sociologists and historians of science who would relativise worldviews [Rosenberg 1994]. In the creationist-evolution debate, it tends to mean something else - the view that explanations must not take into account the supernatural or spiritual. These two senses overlap to a degree (because evidence of the supernatural is not empirical, but revelatory).
Notice, though, that the second sense is a view about what exists, while the former is a view about what can be known in science. If there is a spiritual realm which is not open to observation, then science cannot use it in explanation, for science is about explaining things that are observed.
If science cannot be used to explain things in terms of what it cannot see and test, this doesn't rule out other disciplines using non-natural explanations (like theology). It just means that science cannot use it as it undercuts the very notion of science. There are two ways science cannot be non-naturalistic. It cannot make the assumption that phenomena are themselves non-natural - it has to assume that everything observed is amenable to a naturalistic investigation. Call this methodological naturalism.
Science must also avoid non-natural explanations. This is explanatory naturalism. Any explanation that uses a non-natural explanans (thing doing the explaining) fails to be testable. I could propose that some process is the result of an Invisible Pink Unicorn's powers. You can neither falsify nor verify this (in the ordinary senses). The hallmark of science, perhaps the only hallmark, is that explanations are testable. The reason for this lies in what philosophy calls epistemology (from the Greek word for belief, epistemé, but used in the sense of knowledge - hence, 'the study of knowing').
Epistemologies from Plato to Kant were infalliblistic - a belief was not knowledge if there was any chance it was mistaken. Science, on the other hand, is often wrong, and is constantly revised. Nevertheless, what science delivers is by far the most successful form of knowledge gathering humans have ever developed. The epistemology demanded by science is therefore a falliblistic view of knowing. The basis for this lies in testing. A scientific explanation must be open to any competent investigator to test and evaluate. Revelatory experiences are not universally open to all, and intuitions about the universe are wildly different for different people and cultures, so non-naturalistic explanations are ruled out of the domain of science.
A useful way to approach this is to ask what a non-naturalistic explanation would look like. Explanations are equations, of a kind. You explain X by saying it is a Y (and a Z, etc). If a non-natural explanation is to work, it has to put something that is neither empty nor circular on the other side of the equation. What counts as a non-natural explanans? 'Something is non-natural if it isn't natural' is entirely empty until we know how to distinguish between the two.
The usual way to define non-natural is that it is not explicable in terms of natural laws; that is, it breaks the causal chain. If we abandon the methodological assumption of naturalism - that everything is open to empirical investigation - we can say that anything not presently explained by scientific laws is non-natural, but that's not what is meant. We can distinguish between our present ignorance and something that's in-principle not scientifically explicable, surely. We want something that is completely outside the course of physical events [some proponents of the term 'supernatural' use it to mean 'uncaused' - what that actually means is really unclear].
But if we had it, could we incorporate it into a scientific explanation? We could obviously not use empirical observations - they depend on the ordinary course of physical processes. So what else is there? The answer is, nothing. Non-natural explanations are not scientific.
A final form of naturalism is ontological naturalism. This is the opinion that all that exists (Classical Greek: on- , root form of 'to be', from which 'ontology' is derived, hence, 'the study of that which exists') is natural. Many scientists are also physicalists. They argue that if we do not need to postulate the reality of non-physical processes for science, then we can conclude that there are no such things. This argument is too quick. The claim that 'if A then B' explains B may be true, but there may also be a C that explains B. Moreover, many things in the physical world are caused by many things together rather than just a few. So, we might say that a physical event is caused both by God and by the physical causes, without being logically inconsistent.
Your resolution depends on what you are using as basic assumptions. In science, Ockham's Razor ('do not unnecessarily multiply entities in explanation') - also known as parsimony [cf. Sober 1988] - is used to trim as much away as possible in order to achieve the leanest explanation. Extending this outside science is a risky proposition, unless you are willing to make the methodological assumption also work on metaphysics as well as physics. Many are (including myself), but it is not a necessary conclusion from any form of science.
In the philosophical doctrine known as moral naturalism, moral systems are explained in terms of the social or biological properties of humans. This is often a Darwinian approach. The point I want to make is that not only explaining but proposing a moral system in this way commits what GE Moore famously called the "Naturalistic Fallacy". You can give a naturalistic explanation of morals without either justifying or invalidating those moral principles. Explanation and justification are two different activities. So, too, with ontology. You can accept the methodological assumption of naturalism in science without invalidating non-naturalistic ontologies. They just aren't scientific. In my view, ontologies outside science are a matter of personal choice. And as Cicero once said, in matters of taste there is no dispute. In science, there is (legitimate) dispute. Therefore, science is more than a matter of taste.
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