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Naturalism holds that the world is exclusively natural and that the scientific method is the best way to know it, yet it remains vague and overlooks mathematics. Instead of this label, what should be emphasized is the scientific spirit across all disciplines—a spirit that no theory can replace.

Many contemporary philosophers call themselves naturalists. That is, they believe that only the natural world exists and that the best way to understand it is the scientific method. I am also sometimes described as a naturalist. Why do I resist this description? Not because of any religious principles I hold. I am one of the most forthright atheists. But accepting the slogan of naturalism without looking at what lies behind its attractive packaging is an unscientific way of forming a worldview, and that is not something naturalists would want to recommend.
So first, let us see what the natural world is. If we say it is the world of matter, or the world composed of atoms, we are far behind modern physics, because modern physics describes the world in much more abstract terms. In any case, the best current scientific theories will most likely lose their validity with future scientific advances. So perhaps it is better to define the natural world as what the scientific method will ultimately discover. In that case, naturalism becomes the belief in what the scientific method will ultimately discover, and (unsurprisingly) the best way to understand it will be the scientific method. Why shouldn't there be things that are discoverable by methods other than scientific ones, or why shouldn't there be things that are not discoverable at all?
Nevertheless, naturalism is not as exclusionary as it seems. For example, some of its staunch advocates try to posit the soul or God if such a thing is part of the best possible explanation for our experience; because doing so would be an application of the scientific method. Naturalism is not incompatible in its principles with various religions. But in practice, most naturalists doubt that belief in the soul or God can withstand scientific scrutiny.
What is meant by the scientific method? Why assume that science has only one method? For naturalists, although the natural sciences like physics or biology differ in some specific respects, at a sufficiently abstract level, they all use a single general scientific method. This method involves constructing scientific hypotheses and testing predictions through systematic observations and controlled experiments. This method is called the hypothetico-deductive method.
A challenge for naturalism is finding a place for mathematics. The natural sciences are based on mathematics, but can we call mathematics itself a science? If we do, then the description of the scientific method given above will be wrong; because mathematical science will not fit into it, since its results are proven by rational arguments and not the hypothetico-deductive method. Although some naturalists like Quine have argued that the real evidence for mathematics comes from its application in the natural sciences and is therefore indirectly the result of observation and experiment, this view does not align with mathematics as it exists. When mathematicians evaluate a newly proposed theorem, they look at consequences within mathematics, not outside it. On the other hand, if we do not consider pure mathematics a science, then we have excluded mathematical proof in mathematics from the scientific mode, and in doing so, we have discredited naturalism. For naturalism holds the scientific method above all other methods, and mathematics is one of the most spectacular success stories in the history of human knowledge.
What other disciplines count as science? Logic? Linguistics? History? Literary theory? How are we to decide? Here lies the naturalists' dilemma. If what they call science is too inclusive, naturalism loses its appeal. Typically, naturalists criticize some traditional forms of philosophy for not being scientific enough, because they ignore empirical testing. How can they sustain that criticism except by restricting the scientific method to the hypothetico-deductive method? But if they are too exclusive about what they call science, naturalism loses credibility by importing the methods of the natural sciences into domains where they are inappropriate. Unfortunately, instead of clarifying the issue, many naturalists equivocate. When attacking, they treat science as hypothetico-deductive; when under attack, they retreat to a very inclusive understanding of science, which makes naturalism much more diluted. Such maneuvers turn naturalism into a vague matter of faith. I do not call myself a naturalist because I do not want to be complicit in such a vague and indeterminate prejudice. Rejecting an idea for "not fitting with naturalism" is little better than rejecting it for "not fitting with Christianity."
Nevertheless, I sympathize with one of the motivations of naturalism: the ideal of thinking in a scientific spirit. That is a vague phrase, but one can begin to explain it by emphasizing values such as curiosity, honesty, accuracy, exactness, and rigour. What matters is not to pay lip-service to these values – which is easy – but actually to apply them in practice – which is hard. We need not pretend that scientists' motives are pure; they are human too. Science does not depend on indifference to fame, career advancement, money, or comparison with rivals. Rather, the search for truth is best facilitated in social environments, intellectual communities, that minimize the conflict between such base motives and the scientific spirit, and reward work that embodies the scientific virtues. Such traditions exist, and not only in the natural sciences.
The scientific spirit is as relevant to mathematics, history, philosophy, and other disciplines as it is to the natural sciences. When experimentation is the best way to find the answer to a problem, the scientific spirit calls for experimentation. When other methods – mathematical proof, philosophical argument, archive-based research – are more relevant to our problem, the scientific spirit calls for those methods. Although the methods of the natural sciences can usefully be applied more widely than they currently are, the default assumption should be that practitioners of a well-established discipline know what they are doing, and use the existing methods as best they can to find answers to the problems. Exceptions may result from a conservative tradition, or one that does not value the scientific spirit. Nevertheless, intolerance of any methods except those of the natural sciences is a poor basis for identifying those exceptions.
Naturalism attempts to encapsulate the scientific spirit in a philosophical theory. But no theory can substitute for that spirit, since any theory can be used in an unscientific spirit, as a polemical tool to reinforce a prejudice. Naturalism as a prejudice would be just one more enemy of the scientific spirit.
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