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Richard Carrier Steiner

Fundamental Flaws in Mark Steiner’s Challenge to Naturalism in The Applicability of Mathematics as a Philosophical Problem (2003)

Richard Carrier


This is a critical rebuttal to Mark Steiner’s book The Applicability of Mathematics as a Philosophical Problem (1998). Steiner argues that naturalism appears to be false because nature is fundamentally mathematical. Carrier argues otherwise.

Steiner’s thesis in a nutshell is this: physics can only be described in the language of mathematics, the language of mathematics exists only as an invention of a mind (human or otherwise), therefore the universe (which is what physics describes) must be the invention of a mind (presumably God’s). Though Steiner’s approach is too humble to actually argue for God’s existence (he seems aware that his conclusion would be compatible with certain other worldviews like atheistic Solipsism), he does seem convinced by his own arguments and evidence that Naturalism cannot be true, whatever else may be. As he puts it, there is a “correspondence…between the human brain and the physical world as a whole. The world, in other words, looks ‘user friendly.’ This is a challenge to naturalism” (176) because it means “the human species has a special place in the scheme of things” (5).

Confusing the Tools of Description with the Things Described

But Steiner commits several fundamental blunders that topple his entire thesis from the start. First, his argument is formally invalid. Just because X can only be described with a language it does not follow that X cannot exist or be what it is without a language to describe it. In other words, he confuses the tools of description with the thing described. All languages are obviously the invention of a mind, and they are invented specifically for the task of describing what exists. That a language succeeds very well at that task is indeed due to humans being very clever in perfecting language as a tool of description, but that’s all. It is not the result of the universe being ‘suspiciously describable.’

To make Steiner’s argument is like being amazed at what a miracle it is that Earth had Oxygen, just what we needed to breathe—forgetting that we evolved to breathe what was here, not the other way around. Half the life on this planet evolved to breathe what was around before Oxygen came along: Carbon Dioxide. And we would have evolved to breathe that, too, if that remained the only breathable gas around. So there is nothing miraculous about the convergence of Oxygen and Oxygen-breathers. Just so, there is nothing miraculous about the convergence of a describable universe and a tool for describing it. There is certainly no reason to expect, even on Naturalism, that any universe would be somehow undescribable.

Defining Mathematics

Steiner’s error here stems mainly from his failure to properly define “mathematics.” He does not acknowledge what many Naturalists see: that math is just a language, no different from any other (like English or German) except in two respects: its component simplicity and lack of ambiguity. This is, in fact, all that distinguishes mathematics from any other language. Component simplicity is a defining attribute because math is a language we created for describing the simplest elements of complex patterns: primarily, quantities and relations. Every “word” in mathematics refers to a very simple pattern (like a number or a function or a simple relationship), unlike English, for example, where most words refer to very complex patterns (like “tree” or “theorem”). Of course, mathematical sentences and paragraphs can be immensely complex, but this does not change the fact that the words are all simple, and that is what makes mathematics what it is, as opposed to English or German. More importantly, the lack of ambiguity defines mathematics as a language because every mathematical word is deliberately defined by humans in an absolutely precise way, leaving no room for optional interpretations or ranges of vagueness. This was what math was invented for: to escape the perils of imprecision and vagueness, which are fundamental to other languages. Thus, while in English “tree” can refer to a plant or a hierarchical list of computer files, and while applying the word “tree” might run into uncertainty when we are looking at something that straddles the properties of a tree and a bush, and so on, in mathematics no such problems arise—because we humans made sure of it. We created a language without ambiguity, and called it mathematics.

Some people might restrict the definition of mathematics further, thus distinguishing math from logic, by stating that mathematics is a language dealing solely with quantities, relations, and functions applied thereto, whereas everything that is simple and precise but not a quantity, relation, or function falls into the subject of logic instead. But many mathematicians regard logic as a special subset of mathematics, since anything said in a language of logic can also be described in the language of mathematics. However, in practice, logic is used as a bridge for extending the defining advantages of mathematics (simplicity and precision) to normal language and ordinary human thought. In that respect, logic straddles the spheres of math and ordinary language. Either way, though, mathematics is definable, whether broadly as a language characterized by component simplicity and lack of ambiguity, or narrowed further to quantity, relation, and function.

Are Beauty and Convenience Mathematical?

Steiner nowhere at any time shows any awareness of the above definition of mathematics or indeed anything even remotely like it. Nor does he allow that mathematicians would call “mathematical” anything and everything definable by the nine or so basic axioms of mathematics (which have in common the features I describe above, which is why my definition is more fundamental). The closest thing to a definition he ever comes to is as follows:

Mathematicians today have adopted internal criteria to decide whether to study a structure as mathematical. Two of these are beauty and convenience. The beauty of the theory of a structure is a powerful reason to call it mathematical…[and] mathematicians…introduce concepts into mathematics to make calculations easier or convenient…[so] relying on mathematics in guessing the laws of nature is relying on human standards of beauty and convenience. (7, cf. 63-70)

This means “the concept of mathematics” is “anthropocentric,” which means mathematics could only be successful in physics if the universe, too, were anthropocentric. But there are serious flaws in his method here. Is everything that is beautiful and convenient mathematical? No. Is an ugly, inconvenient theorem not mathematical? No. No matter how ugly or inconvenient Ptolemy’s planetary theory was, it is still universally admitted by all mathematicians to be entirely mathematical. Thus, neither beauty nor convenience are defining attributes of mathematics. Mathematics must be defined by something else, which is obviously what I defined it as above.

Therefore, Steiner’s claim that “the concept of mathematics itself is species-specific” (6) is false. It is true only insofar as language is species-specific, but Language is not identical to Being—it is merely a tool for describing it. Instead, the defining concepts of mathematics—component simplicity, lack of ambiguity, quantity, relation, function—are not at all species-specific or even specific to mind. All these things can exist physically in nature without any mind to create or sustain them. Thus, that we find a language (in this case, a language for describing quantity, relation, and function with component simplicity and lack of ambiguity) useful has nothing to do with the universe being human-centered or “anthropocentric.” It simply means we live in a universe where there are quantities, relations, and functions that are composed of simple and precise components. Naturalism is not only completely compatible with that, but that is generally part of the very meaning of Naturalism: that there is such a universe, and nothing else.

The Heuristic of Beauty and Convenience

But isn’t it at least the case that scientists have found a successful scientific method in focusing on ‘beautiful’ and ‘convenient’ mathematical theories? Not really. Though that has been an effective heuristic for getting at simple and focused problems in comprehensible ways, this is simply the result of human limitations: we have to start small, and solve simple problems first, in the few ways we know how and are best at. But if we were to rely solely on this heuristic, most of the greatest scientific discoveries would never have been made. Far from a “beautiful and convenient” chemistry of four elements, we discovered in the end an incredibly ugly, messy, and inconvenient Periodic Table of over ninety elements and counting (never mind the mind-boggling complexity of the Standard Model of particle physics); far from the “beautiful and convenient” planetary theory of Copernicus, the paths and velocities of the planets are so ugly and inconvenient that we need supercomputers to handle the messy intersection of Newtonian, Keplerian, Einsteinian, Thermodynamic, and Chaotic effects, and even then they are not always entirely accurate in their predictions on astronomical scales of time (like thousands and millions of years).

There are other problems in Steiner’s approach. How do humans come upon their concept of mathematical beauty? Steiner never addresses this. But what if it is a learned reaction to a successful heuristic? If so, that would mean we learned to define as beautiful that which fit the truths of the universe, and not the other way around, making “beauty” just like “oxygen,” something we adapted to, not something arranged for our convenience. Since Steiner does not even discuss the psychological and philosophical literature on aesthetics, much less mathematical aesthetics, he cannot claim that the universe was geared for us any more than that we geared ourselves for it.

Likewise, why do we seek convenient notations and solutions to complex problems? Is it because the universe is convenient for us, or is it simply because we work better with simpler tools rather than complex ones? If a 1000-word description of an apple in English can be rewritten in 100 words, without losing a single iota of information content vis-à-vis the apple, then obviously we will see the advantage in this. And the fact that this could be done would have nothing to do with the apple being suspiciously convenient for short descriptions in English. Rather, it would have everything to do with the natural inefficiency of human language and thought—where it takes effort to analyze our descriptions and locate redundant and unnecessary elements, and discover easier, shorter, better ways of saying the same thing.

Applying the Heuristic

Steiner’s book consists mainly of long and highly technical discussions of the fringes of 20th century scientific discovery, mainly the as-yet-unexplained oddities of Quantum Mechanics, but in some cases formal maneuvers made in other fields like Relativity Theory. Regarding all these examples he declares:

To explain my data away, one must find a natural, or material, property of mathematics as such, and then show how this property accounts for the success of the mathematical discoveries [I] outline [in this book] (8)

I have already done this: “mathematics as such” describes the natural, material properties of quantity, relation, and function. In other words, it describes repeating or repeatable patterns in measurable phenomena, in effect describing structure, behavior, and arrangement, in physical observations. In fact, every example he gives concerns observed patterns in empirical and experimental research and the attempt to describe and thus predict those patterns. When scientists find a different mathematical way of describing a pattern, one that says exactly the same thing but in fewer words, they are not discovering a user-friendly universe, but merely improving their ability to understand what they observe.

That aside, Steiner’s most compelling cases involve situations where one mathematical approach was introduced into the search for an adequate description of certain repeating patterns in empirical observation, simply because there were certain similarities between the new approach and one previously solved. That is, even though the two mathematical descriptions were not identical, they were judged similar, and they were thus tried, and ultimately turned out successful. Steiner says this means human notions of mathematical ‘similarity’ must correspond to real features of the universe, so the universe is suspiciously built for human notions.

There are at least two problems with using his examples in such a way that Steiner never addresses.

First, the heuristic of mathematical similarity does not entail any underlying metaphysics. One can use any method one wants for discovering facts—including picking ideas out of a hat. As long as the result bears out in empirical test, it is acceptable. I do not have to “assume” that there is anything mystically efficacious, anything metaphysically rational, about picking ideas out of a hat. I can still do it, simply because it is easy, and I know I don’t have to trust any results until they bear out in tests anyway. The fact is, even such a totally random method will produce successes. And only the successes would get published and thus heard of. Steiner would then come along, see that all the scientific discoveries in print came from picking ideas randomly from a hat, and wrongly conclude that there is some mystical power inherent in hats to produce knowledge of the universe. He might say the universe had to be Hatrocentric to explain this phenomenon. But he would be wrong. He would have forgotten to consider the hundreds of hatpicked ideas that fell by the wayside. Thus, even if it is the case that scientists have been using a heuristic that was contrary to the metaphysical assumptions of Naturalism, it would neither follow that they were acting irrationally (since they need not assume their heuristic has a metaphysical basis—we do what is easy and engages us, what we know how to do well, because we’re human) nor that the universe was somehow metaphysically linked with that heuristic (since even a totally random method will score hits, and only successes will likely survive in the historical record).

Second, there is still a valid Physicalist reasoning behind the heuristic of mathematical similarity. Since a mathematical description of phenomenon A is a description of a pattern of observed behaviors and effects, when phenomenon B shares a similar pattern of observed behaviors and effects, it is reasonable to expect that its mathematical description will be similar to that of phenomenon A. We do not have to know what the physical basis is for this similarity: we observe a similarity, so we know it’s a fact. We are fully within our rights as Naturalists to assume there is a physical basis to such a mathematically-described similarity until we can identify exactly what that basis is.

Consider it this way: if some underlying physical fact is the cause of phenomenon A, and some other underlying physical fact is the cause of phenomenon B, and phenomena A and B are behaving similarly or have similar external physical features, it is reasonable to hypothesize that the underlying physical facts in both cases also bear certain similar patterns (similar arrangements, similar geometries, similar sequences, etc.), and thus any description of one pattern will have something in common with a description of the other pattern. It is thus still consistent with Physicalism to try out aspects of the mathematical description of phenomenon A on phenomenon B. It might not work out, but the odds of our hitting on something are certainly going to be greater than chance—and as we see it, this is precisely because Physicalism is true. And any heuristic that hits better than chance is reasonable to pursue, especially when we have none better.

Example 1: Maxwell’s Anticipation of EM Radiation

I will show how this works on one of Steiner’s most prized examples: Maxwell’s prediction of electromagnetic radiation. This is how Steiner argues the case:

Maxwell noted that the experimentally confirmed laws of Faraday, Coulomb, and Ampère, when put in differential form, contradicted the conservation of electrical charge. By tinkering with Ampère’s law, adding to it the “displacement current,” Maxwell got the laws to be consistent with, indeed to imply, charge conservation. With no other warrant than this (Ampère’s law stood up well experimentally; on the other hand, there was “very little experimental evidence” for the reality of a “displacement current”), Maxwell made the indicated changes. Ignoring the empirical basis….This made electromagnetic radiation a mathematical possibility. [As a result] Hertz exclaimed that the mathematical formulas are “wiser than we are.” (77)

To dismiss Steiner’s entire thesis, Naturalists need only answer one question: Why did Maxwell’s mathematical tactic work in anticipating physical facts? The answer involves the combination of two facts: the reasonable assumptions that follow from the worldview of Naturalism (especially Physicalism), and the nature of mathematics as a human tool for describing observed patterns in the physical world.

The Assumptions of Naturalism: It is reasonable to expect on Naturalism that things don’t pop in and out of existence uncaused…that is, that matter, energy, things like that, are conserved. You don’t get something from nothing. Thus, it is reasonable on Naturalism to expect that ‘charge’, like matter and energy, must be conserved. Since the descriptions of various charge-related phenomena extant in Maxwell’s day did not allow conservation of charge, obviously any Naturalist should have suspected there was something wrong with those descriptions. That this hunch turned out correct was in fact a vindication of Naturalist assumptions, not a challenge to them.

The Function of Math as Description: The basic assumption driving Maxwell’s tactic (that charge must be conserved) followed directly from Naturalism. Then he got to work on the descriptions he suspected were flawed. To that end, the ‘mathematical’ things Maxwell did, from Steiner’s own account, were merely two: to put certain laws describing the behaviors of charge “in differential form,” and then to add a variable to the equations that corrected the conservation error (“displacement current”). The first act is logically necessary on Naturalism: differential equations describe continuities, and Naturalists of the time were well aware that nature works in continuous, not broken, processes (the discovery of Quantum Mechanics changed this, but only after enormous empirical evidence was accumulated), so Maxwell’s first mathematical act was totally explicable on Naturalism. He had to make the descriptions conform to the physical facts before he could do anything else with them.

The second act is a logically sound hypothetical step: if charge isn’t being conserved, then it must be going somewhere. Maxwell rightly picked the simplest imaginable solution first (e.g. that it all went one place, rather than several), which due to human limitations is always the best place to start an investigation, and which statistically is the most likely (simple patterns and behaviors happen far more often than complex ones—since Maxwell’s day, again, the discoveries of Chaos Theory have changed that assumption, but again only after vast amounts of empirical evidence confirmed and thus justified the change in our assumptions).

That Maxwell’s moves anticipated EM radiation was therefore a natural conclusion from entirely Naturalist assumptions. Charge was going somewhere, which we knew because the descriptions of charge behavior that we had, which were empirically well-grounded, left out and thus entailed the disappearance (or spontaneous appearance) of charge, which begged for an explanation. Maxwell hypothesized such an explanation by making some simple and obvious changes to the descriptions that accounted for this discrepancy—changes to the way the pattern of behavior was described that allowed inclusion of another element to that pattern. The changes he made were the simplest ones he could make that didn’t invalidate but instead preserved the predictive success of the existing descriptions, while also bringing them into line with conservation laws. And the changes he made were still, in fact, hypothetical. They could have turned out wrong, and many tinkerings with these equations, by him and others, no doubt preceded this success and failed. But on Naturalism, his final guess was a smart one, and one likely to succeed. So we should not be amazed that it did.

There was even more background to this account that confirmed the Naturalistic assumptions driving Maxwell that Steiner includes but unreasonably dismisses (77-78). Steiner also argues absurd things like “differential equations have many solutions, and there is no reason to believe…that we can produce something just because it solves an equation” (79). Steiner is wrong. Of the solutions to equations that successfully describe physical phenomena Naturalists can rest assured at least one of them must correspond to the truth, to the actual underlying physical facts, and that other results, which are not solutions, will not. Thus, even on Naturalism it is a reasonable heuristic to test only the few solutions available to accurate descriptions, since any other notions would not accord with observation and thus would have to be false.

So when Steiner claims:

Maxwell’s reasoning was Pythagorean [i.e. not Naturalistic, because] once he had a mathematical structure which described many different phenomena of electricity and magnetism, the mathematical structure itself, rather than anything underlying it, defined the analogy between the different phenomena. (79)

He is twice wrong. First, the existence of charge defined the analogy between the different phenomena, not the mathematical descriptions of charge’s behavior. Since they all described behaviors involving charge, it was a reasonable Naturalistic assumption that all the behaviors were physically related, and therefore could be described with one description rather than several. And so long as that one description made all the same predictions of the behavior of charge, it would be semantically identical, i.e. describe exactly the same thing, because that is the way humans made mathematics—nothing about the universe makes a three-sentence description of an apple reducible to one sentence. Only the way humans invented language makes that so. The apple remains the same no matter what.

Second, the “mathematical structure” of the equations involved corresponded to the physical structure of the observed behavior of charge. Thus, any manipulation of the description entailed physical differences in the thing described. So if the thing described must conserve charge, and the description of that thing does not conserve charge, it is reasonable to refine the description to resolve the discrepancy. No other description is likely to come close to the physical facts, whereas we know at least one such description must do so. By making our revisions to the description as few and as simple as possible, we would import only one hypothetical solution to the discrepancy (in this case, Maxwell’s “displacement current”). That is fully in accord with Naturalist assumptions, and is logically always the first place to start looking. Thus, everything Maxwell did mathematically has a corresponding physical significance. And he surely knew that. It had that significance even if Maxwell did not yet know what physical facts underlie the physical difference between the two descriptions (the one that described a world without conservation of charge, and the one that described it with conservation of charge). It was a valid, Naturalistic hypothesis all the same, leading to fruitful inquiry.

Example 2: Matrix Mechanics

And all this should alert us in the fringe cases Steiner uses, too, such as his repeated discussions of the efficacy of applying matrix mathematics to quantum phenomena. For the same things follow: (1) The mathematical models are adopted because they successfully describe observed physical phenomena, not because the universe is magically ‘beautiful’ and ‘convenient’; (2) the features and elements of those descriptions that are redundant and unnecessary are not likely to correspond to physical facts but are probably the product of the human inefficiency of our tools of description, and since, also, no physical facts support their retention, it is Naturalistically plausible to revise the descriptions (i.e. the equations) so as to eliminate what has no empirical support or plausible physical basis, and thus to make our descriptions convenient for us; and finally, (3) when the phenomena show patterns of observed behavior that match other patterns of behavior (real or imagined, such as when quantum phenomena show patterns of behavior matching patterns described by matrices) it is Naturalistically reasonable to employ the same description, suitably modified to fit the generic description to the particular cases, because (a) they both make the same predictions, but the old description was flawed by human inefficiency while the new description is more convenient for us, yet either way the universe remains the same; and (b) the same observed pattern in each case is likely due to the same pattern of underlying structure. That is, it is improbable that a totally different internal structure would produce an identical external structure—not impossible, but it is a reasonable heuristic to test first what is physically probable. Just as Maxwell did. This is especially justified when the descriptions we have we know must be wrong, because they exclude something we otherwise know is going on (like the conservation of charge).

And, last but not least, we might even be wrong. Matrix mathematics might be a convenient way for us to predict quantum phenomena and yet be incorrect descriptions after all, just like the erroneous equations Maxwell faced that didn’t conserve charge as they ought. We may be awaiting a new Maxwell to find a better description that is more complete and more successful in inspiring inquiry into the physical facts that underlie the observations, which so far we have not found—possibly because we have distracted ourselves with mathematically convenient but not entirely correct representations, but more likely because we lack the tools to observe the facts we need, like a miraculous ‘microscope’ that could see quantum particles in all their physical structure. It is likely we will never have such means and so will never be in a position to really know what is going on at that level, but human limitations are not limitations on nature. Just because we can never go into a cave does not mean that cave is empty.

Another Worry

Most of Steiner’s examples fall to one or more of the above observations and are thus to be rejected. Some examples used by Steiner are also suspicious. For example, he claims (by citing Peirce as his authority) that there is no physical explanation for the applicability of inverse square laws to physical phenomena, but he rests this on the undefended and unexplained rejection of obvious geometrical explanations (36). Since geometry is certainly the reason for inverse square phenomena, it seems most disturbing that he doesn’t even try to refute this, but assumes the reader will ignorantly accept his assertions to the contrary. This put me on guard: how many other of his examples are scientifically incorrect? Why is he citing a single scientist from forty years ago as his sole authority? One wonders what, say, Hawking would say about the matter today—or, indeed, what your average college textbook says.

As a result of this tendency, I was left uncomfortable trusting many of his claims. For example, he often asserts that certain scientists (like Einstein) used no physical reasoning in their choice to apply a particular mathematical solution to a problem, and he bases this on the fact that the relevant published papers state no such reasoning. But they don’t have to—indeed, the principle of parsimony generally requires that one not include unproven assumptions in a scientific paper. What a brief and empirically rigorous paper says does not tell us what was in the mind of the actual scientist during his process of hypothesis abduction—and I find it highly suspicious that a man like Einstein would disregard physical reasoning in advancing an idea, a man who was so committed to physical reasoning that he refused for the longest time to accept many of the claims of Quantum Theory of his day because they lacked such reasoning. The only way to check Steiner’s claims, then, is to redo all his research—to discover what Einstein actually was thinking at the time, from all his notes and papers and memoires—and any writer who puts you in such a suspicious state of mind is hard to trust generally.

Covert God of the Gaps Argumentation

In the end, when we wipe away every argument in Steiner’s book that is based on the false assumptions outlined earlier, his book stands with only one formally valid but still incorrect argument left: scientists have not been able as yet to provide a physical explanation of certain observations in areas like Quantum Mechanics and Particle Physics, therefore there is probably no such explanation, ergo Naturalism (or at least Physicalism) is false. He pushes for the falsity of Naturalism generally by arguing that the only explanation left is anthropocentric: the universe simply behaves according to complex mathematical rules that have no physical basis and therefore it must have a super-mental basis anticipating the human species. He does not even seem aware of Platonic Naturalists like Paul Davies who see no incompatibility between non-physical entities and Naturalism. Davies would add a middle solution, I imagine: that though the complex mathematical behaviors of the universe may not be based in anything physical, they are not based in anything uniquely intelligent or mental either, but are simply brute facts of the nature of the universe, which we humans have evolved an adept skill at spying out and describing. As a Physicalist, I disagree with this idea, but if one wants to take on Naturalism, one has to be able to refute all forms of Naturalism, not just Physicalism.

But Physicalism is not actually in danger from Steiner’s only valid argument. Why? Because it is met with another valid argument that carries greater weight: scientists have consistently found physical explanations for every phenomenon they have been able to thoroughly examine, constantly and without exception, for millions of physical facts and attributes of our universe, and have not found such explanations only where they have not been able to thoroughly examine the facts (and as it happens, though still hypothetical, Superstring theory now offers a complete physical basis for the success of matrix mechanics, something Steiner seems to think is impossible). Therefore, that scientists have yet to explain such facts has much more probably to do with their inability to “get to” those facts than with those facts somehow being fundamentally different than all the millions and billions of other facts scientists have gotten to in the past three thousand years.

In other words, the trend of history is entirely against Steiner, and offers no support whatever for his conclusion. There is not a single instance on record of any fact that has been thoroughly examined by scientists that turned out to have no identifiable physical origin. That is why almost all of Steiner’s examples are on the fringes of science, not the settled facts of science: he can only find his “failures” where scientists have been unable to make the needed observations to resolve the matter. And as a matter of fact, scientists all remain committed to finding physical causes of the very observations Steiner uses as his examples. No physicist has thrown up his hands and said “Hey! We’re wasting our time! There is no physical basis to these effects, it’s just a mathematical fact of the universe!” Indeed, all physicists would find such an approach, entailed by Steiner’s argument, to be quite absurd, even antiscientific. You can say “I can’t get into that cave, so there must be nothing in it” if you want to, but you would be betraying the very principles of science if you did. And going against all the evidence and precedent of history as well: scientists have always found things in the caves they’ve gotten into. Why lose confidence in their tactics now?


The bottom line: any universe composed of conserved and discrete objects arranged into patterns in a multidimensional space will always be describable by mathematics. We invented mathematics just for that purpose: to describe such things. But are patterns of conserved and discrete objects in a multidimensional space at all anthropocentric? Do they anticipate or assume in any way the eventual arrival of Homo sapiens? No. And that is the ruinous end of Steiner’s thesis. The mathematization of nature does not require or even imply anything anthropocentric about the universe and thus offers no challenge at all to Naturalism.

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