Briefly, scientism is the view that science alone gives us knowledge of reality. Of course, that just raises the question of what we mean by “science.” One problem with scientism is that if you define “science” narrowly -- so that it includes physics and chemistry, say, but not philosophy or theology -- then scientism ends up being self-refuting, because it is not itself a scientific claim but a philosophical one. On the other hand, if you define “science” broadly enough so that it avoids being self-refuting, then it becomes vacuous, because it now no longer rules out philosophy, theology, or pretty much anything else adherents of scientism want to be able to dismiss without a hearing as “unscientific.”
A second problem with scientism is that science cannot in principle give us a complete description of the world, both because science takes for granted certain assumptions it cannot justify in a non-circular fashion (such as that perception is reliable, that there is order in the world that is really there and not just projected onto it by the mind, etc.), and because the methods of science of their nature can obscure as much as they reveal. For example, as the philosopher Bertrand Russell -- who was no friend of Scholasticism or of religion -- often emphasized, the methods of physics give us only the abstract mathematical structure of physical reality, but do not and cannot tell us the intrinsic nature of whatever is the underlying reality that has that structure.
A third problem is science cannot in principle provide a complete explanation of the phenomena it describes. Science explains things by tracing them down to ever deeper laws of nature. But what it cannot tell you is what a “law of nature” is in the first place and why it operates. It really is amazing how unreflectively atheists and advocates of scientism appeal to the notion of “laws,” given how deeply philosophically problematic the very notion is. Earlier generations of scientists were aware of the philosophical puzzles raised by the nature of scientific explanation, and some contemporary scientists (such as Paul Davies) are also sensitive to the puzzles raised by the very idea of a “law of nature” (which is actually a holdover from an idiosyncratic theology to which Descartes and Newton were committed, but which Aristotelian and Scholastic philosophers reject just as much as atheists do).
But most contemporary scientists tend not to have the general education that figures of the generation of Einstein, Schrödinger, and Heisenberg did. They don’t know philosophy well, and they also don’t know what they don’t know. This goes double for the more aggressively atheistic ones among them -- people like Lawrence Krauss, Peter Atkins, Richard Dawkins, and Jerry Coyne. Hence they repeatedly commit very crude philosophical mistakes but also refuse to listen or respond when these mistakes are pointed out to them.
Anyway, the main reason scientism has the following it does is probably that people are, quite rightly, impressed with the technological and predictive successes of modern science. The trouble is that this simply gives us no reason whatsoever to believe scientism -- that is to say, it gives us no reason to believe that science alone gives us knowledge. To draw that conclusion you need to assume that if something is real, then it will be susceptible of a precise mathematical description that will make strict prediction and technological application possible. Now that is itself a philosophical or metaphysical assumption, not a scientific one. But it is also an assumption that there is not only no reason to believe, but decisive reason to reject, as I argue in the book.
What the mathematically-oriented methods of modern physics do is to focus on those aspects of nature which can be strictly predicted and controlled and to ignore anything that doesn’t fit that method. As a result, physics tends brilliantly to uncover those aspects of reality that fit that method, and which can therefore be exploited technologically. But it simply does not follow that there are no other aspects of reality. To think otherwise is like the drunk’s fallacy of assuming that his lost car keys must be under the street lamp somewhere, because that is where the light is.
--Dr. Edward Feser
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--Robert Hugh Benson, "None Other Gods".