Craig’s Eight Arguments for God, Part IV: “God is the best explanation of the applicability of mathematics to the physical world”

This is the fourth installment in a series of blog entries which I am posting in response to a set of theistic arguments published by Christian apologist William Lane Craig. The first three installments can be found here:

Part I: Introduction 

Part II: “God is the best explanation why anything at all exists”

Part III: “God is the best explanation for the origin of the universe”

Craig’s third argument is intended to support his claim that “God is the best explanation of the applicability of mathematics to the physical world.” Craig’s first two arguments have already been shown to be complete failures. Let’s see if Craig’s third argument does any better.

Craig writes:

Philosophers and scientists have puzzled over what physicist Eugene Wigner called “the unreasonable effectiveness of mathematics.”

Thus mathematics presents itself as a prime gap ready for someone like William Lane Craig to come along and insert his god as the “best explanation.” This appears to be a most opportunistic move of expedience on the Christian apologist’s part. After all, if we want to learn about math, would we use the Christian bible as a mathematics text book? I certainly would not.

Craig continues:

How is it that a mathematical theorist like Peter Higgs can sit down at his desk and, by pouring over mathematical equations, predict the existence of a fundamental particle which, thirty years later, after investing millions of dollars and thousands of man-hours, experimentalists are finally able to detect?

I cannot speak on the specifics here, but I don’t have to in order to recognize the familiar cry of “Duh, I donno! Must be God did it!” when I hear it.

Higgs’ achievement is indeed remarkable. But I’m happy to say that it is not entirely unique. What’s important for our purposes here is not to understand the particulars of Higgs’ process in arriving at his conclusions, but how Craig’s treatment of this and related issues is so wrongheaded. In fact, the diametric opposite of what Craig wants to infer from such intellectual feats is a fundamental fact which Craig’s approach only serves to keep hidden from view, and that fact is the power human reason.

Craig’s entire approach is to keep the fundamentals safely out of view in order to sustain the mysteriousness of the relationship between mathematics and epistemology. Indeed, given his devotion to Christianity, I suspect that this relationship is mysterious to Craig himself. If it is a mystery, then he really has no business pontificating about things that he does not understand; on the other hand, if he does understand the relationship between mathematics and epistemology, then he is deliberately concealing this relationship in order to bolster his case for a mystical position. Neither alternative commends Craig’s suitability as a “professional philosopher.”

Other predictions have been made on the basis of reason which have come true. Consider for example Newton’s estimate of the measure of the earth’s equatorial bulge (see for example here) and Halley’s prediction of the regular return of the comet named after him (see here). Arguably even more complex are Ayn Rand’s predictions, which she presented in the form of her novel Atlas Shrugged, for the future state of the American culture and political direction, predictions which she made in the 1950s and which are coming true more and more with each passing day.

What do all of these examples, including Higgs’ boson particle theory, is that they require the ability to make very complex inferences that extend far beyond what is given in perceptual awareness. One of Rand’s major epistemological theses is that it is concepts - given their relation to measurement - which makes this ability possible to man. To begin to understand how concepts make this ability possible, we need to understand the purpose of measurement – something we will not learn by reading Leviticus! Instead, consider what we do learn from Ayn Rand, whom I quote at length (Introduction to Objectivist Epistemology, pp. 8-9):

The purpose of measurement is to expand the range of man’s consciousness, of his knowledge, beyond the perceptual level: beyond the direct power of his senses and the immediate concretes of any given moment. Man can perceive the length of one foot directly; he cannot perceive ten miles. By establishing the relationship of feet to miles, he can grasp and know any distance on earth; by establishing the relationship of miles to light-years, he can know the distances of galaxies.  

The process of measurement is a process of integrating an unlimited scale of knowledge to man’s limited perceptual experience—a process of making the universe knowable by bringing it within the range of man’s consciousness, by establishing its relationship to man. It is not an accident that man’s earliest attempts at measurement (the evidence of which survives to this day) consisted of relating things to himself—as, for instance, taking the length of his foot as a standard of length, or adopting the decimal system, which is supposed to have its origin in man’s ten fingers as units of counting.  

It is here that Protagoras’ old dictum may be given a new meaning, the opposite of the one he intended: “Man is the measure of all things.” Man is the measure, epistemologically—not metaphysically. In regard to human knowledge, man has to be the measure, since he has to bring all things into the realm of the humanly knowable. But, far from leading to subjectivism, the methods which he has to employ require the most rigorous mathematical precision, the most rigorous compliance with objective rules and facts—if the end product is to be knowledge.  

This is true of mathematical principles and of the principles by which man forms his concepts. Man’s mathematical and conceptual abilities develop simultaneously. A child learns to count when he is learning his first words. And in order to proceed beyond the stage of counting his ten fingers, it is the conceptual level of his consciousness that man has to expand.

So a basic understanding of the importance of mathematics, the science measurement, to our knowledge and, more specifically, to the method by which we acquire knowledge of things beyond the immediate range of our perceptual awareness, should now be taking shape. We cannot perceive the equatorial bulge of the earth any more than we can perceive a boson or a fermion. But if we have a method of conceptualizing from the perceptual level which systematically utilizes units of measure which have a perceptual basis, we enable our minds to anchor our knowledge objectively by tying what we learn and know to that which we discover in the world by looking outward (as opposed to looking inward into the realm of the imagination, where we find Harry Potter, Alice in Wonderland, the Wizard of Oz, Craig’s god, etc.).

With respect to Higgs’ discovery of a particle, the point is that the method which he used – using mathematical equations and inferences involving vast, delicate contexts of data – is precisely what we would expect a pioneer in his field to use in order to discover what he discovered, given the nature of man’s conceptual awareness and the role of measurement that makes concepts possible to begin with. Higgs could not shrink himself down to the size of a boson so that he could perceive bosons directly. He had to work within the constraints of the nature of his own consciousness, constraints which dictate the parameters of a methodology suited to that nature. Similarly with Newton in discovering the equatorial bulge resulting from the centrifugal motion of the earth’s rotation: Newton could not inflate himself to the size of the earth (similar to what Craig wanted to do with a ball found by hikers in a forest) so that he could perceive this bulging shape directly; on the contrary, Newton had to work within the constraints of the nature of his own consciousness, just as Higgs did, just as we all do. Unfortunately, these basic facts seem utterly lost on William Lane Craig, who continually comes across as though he expected to accept his claims on his mere say-so, all the while ignoring the need for an epistemological method that is suited precisely to the nature of our consciousness.

Craig writes:

Mathematics is the language of nature.

What exactly does this mean? It almost sounds like something we’d find on a Hallmark card. To understand the relationship between mathematics and the conceptual nature of man’s knowledge, we need more than trite expressions. We need a theory of concepts which elucidates that relationship.

Craig asks, rhetorically:

But how is this to be explained?

Don’t tell me: a supernatural mind zapped mathematics into existence just for the hell of it and created everything else in a manner which conforms to mathematics. Right? Indeed, it seems that all along in the cases he presents for his god, Craig wants us to shrug our shoulders and say “Gee, I donno!” to which he will happily supply “My god did it!”

Here I urge that we proceed with greater caution than Craig’s mysticism can afford, for the kinds of errors which Craig’s apologetic trades on have dangerous consequences. There is only one right answer to the equation 2 + 2 = __; there are billions of wrong answers. If we treat math as some mysterious phenomenon whose relationship to the world and our knowledge must remain enigmatic for the sake of a religious confession, then we cut off from our understanding the true wonder of achievements like Higgs’, Newton’s, Halley’s and Rand’s. Craig says that “mathematics is the language of nature,” but yet he also writes the following:

If mathematical objects like numbers and mathematical theorems are abstract entities causally isolated from the physical universe, then the applicability of mathematics is, in the words of philosopher of mathematics Mary Leng, “a happy coincidence.”

Why should we accept this? Indeed, if “mathematical objects” like numbers and theorems are abstractions, why not suppose that they are abstractions formed on the basis of objective input, which would give them integral relevance to the world around us from their very inception? Craig does not consider this, and yet this is what in fact is the case. Measurement is epistemological, and it begins in perception (such as when we perceive that one object is bigger or smaller than another object), and thus has its roots right here in the world around us and our conscious interaction with it. There is no need to posit some supernatural agency here, nor is there anything to be gained by doing so; in fact, quite the opposite: appealing to the supernatural will simply shut down our minds and halt our understanding in place, stranding us in the very skepticism that elsewhere Craig pretends to rail against.

Numbers are conceptual, and thus to grasp the nature of numbers (and thus formulate a philosophical explanation of them), we need a good theory of concepts. A good theory of concepts will articulate the causal method by which concepts of numbers are formed. Concepts of numbers are formed by a process of abstraction, just as other concepts are formed. And like other concepts, concepts of numbers integrate and condense data into single mental units which man can retain and use in his conscious activity. Echoing Rand’s reference to Protagoras’ dictum that we saw above, Leonard Peikoff writes (Objectivism: The Philosophy of Ayn Rand, pp. 82-83):

Measurement is an anthropocentric process, because man is at its center. His scale of perception—the concretes he can directly grasp—is the base and the standard, to which everything else is related.  

This brings us to Ayn Rand’s momentous discovery: the connection between measurement and conceptualization. The two processes, she observes, have the same essential purpose and follow the same essential method. In both cases, man identifies relationships among concretes. In both cases, he takes perceived concretes as the base, to which he relates everything else, including innumerable existents outside his ability to perceive. In both cases, the result is to bring the whole universe within the range of human knowledge. And now a further, crucial observation: in both cases, man relates concretes by the same method—by quantitative means. Both concept-formation and measurement involve the mind’s discovery of a mathematical relationship among concretes.  

Ayn Rand’s seminal observation is that the similar concretes integrated by a concept differ from one another only quantitatively, only in the measurements of their characteristics. When we form a concept, therefore, our mental process consists in retaining the characteristics, but omitting their measurements.

Every time we use a concept, then, we are – whether we realize it or not – using a measure-related process. The concept ‘man’, for example, includes all men – who currently exist, who have lived in the past, and who will live in the future, regardless of when and where they live (time and place are omitted measurements), how tall or short they are (height is an omitted measurement), how skinny or fat they are (weight is an omitted measurement), whether they were bald or sporting a mullet (hirsuteness is an omitted measurement), etc. This is why, when we have a specific man in mind, we must use qualifiers – e.g., this man, the man standing at the bus stop, etc. – to distinguish one member of the class ‘man’ from the rest.

But notice the great economy that concepts afford the human mind: with a single mental unit, a concept, we can denote a potentially infinite quantity of members. As Peikoff points out, this is without a doubt a uniquely anthropomorphic process. An omniscient being would have no need or use for such a process, for it would be able to hold in one moment of its awareness all members of a class without the need to condense them into a single unit (for more on this, see my blog entry Would an Omniscient Mind Have Knowledge in Conceptual Form?). This could only imply that the Christian god or anything like it would have no use for mathematics as such, since the nature of its consciousness would not require measure-related procedures.

But that’s not the case with man; man is not omniscient. His consciousness has a specific nature, and it can perform only so many functions at any given moment, and his method of knowing must be suited to the limited nature of his consciousness. This is how measurement finds its epistemological value, as Rand explains (Introduction to Objectivist Epistemology, p. 63):

Since consciousness is a specific faculty, it has a specific nature or identity and, therefore, its range is limited: it cannot perceive everything at once; since awareness, on all its levels, requires an active process, it cannot do everything at once. Whether the units with which one deals are percepts or concepts, the range of what man can hold in the focus of his conscious awareness at any given moment, is limited. The essence, therefore, of man’s incomparable cognitive power is the ability to reduce a vast amount of information to a minimal number of units—which is the task performed by his conceptual faculty. And the principle of unit-economy is one of that faculty’s essential guiding principles….  

A “number” is a mental symbol that integrates units into a single larger unit (or subdivides a unit into fractions) with reference to the basic number of “one,” which is the basic mental symbol of “unit.” Thus “5” stands for │││││. (Metaphysically, the referents of ”5” are any five existents of a specified kind; epistemologically, they are represented by a single symbol.)

Thus, I would argue, in response to Craig’s assault on the human mind, that an understanding of the nature of concepts is crucial to grasping the actual nature of the achievements of Newton, Halley, Higgs, and so many more. Man’s mind is not a passive lump of clay which some supernatural dictator manipulates according to its own whimsical pleasure, choosing arbitrarily to enlarge some while choosing arbitrarily to cut down others. And where does the Christian bible provide any understanding on the nature of concepts and how the human mind forms them? There is no such thing as a distinctively Christian theory of concepts. In fact, a good understanding of the objective theory of concepts (cf. Ayn Rand’s Introduction to Objectivist Epistemology for starters) simply demystifies what Craig would prefer to remain mysterious and therefore susceptible to apologetic hijacking.

Craig asserts:

The naturalist has no explanation for the uncanny applicability of mathematics to the physical world.

Notice that Craig offers no argument here whatsoever. In fact, Craig puts himself in the dubious position of having to argue for a negative.

Craig writes:

By contrast, the theist has a ready explanation: When God created the physical universe He designed it in terms of the mathematical structure which He had in mind.

What Craig offers here is essentially nothing more than an appeal to magic to explain something he himself does not understand. Notice how all of this is couched in vague descriptions which never penetrate into the deeper issues of measurement and cognition. Craig’s appeal to his god does not explain how the human mind quantifies measurements or applies quantification in abstract uses. Instead, Craig opportunistically seeks to ride a wave of ignorance, never really explaining anything at all while pretending to have solved some great mystery. But his “explanation” does nothing to further our understanding of the world, of mathematics’ relationship to the world, of our knowledge, or of the proper method by which we can and should acquire knowledge.

The notion that a supernatural mind created the universe in conformity with mathematical principles is not new with Craig. This error has been perpetuated by otherwise highly intelligent thinkers throughout the centuries. But the root error involved here can be traced in part to a faulty understanding of causality. David Harriman, in his lecture The Philosophic Corruption of Physics, Part 2, explains (7:14 – 8:14):

In the early 17th century people were surprised when [Johannes] Kepler discovered that the planetary orbits could be described by mathematical laws. Kepler’s explanation was that God loved mathematics and therefore designed the solar system accordingly. In other words, the fact that nature could be described mathematically was regarded as a miracle, not as a consequence of natural causality. But in fact causality is the metaphysical basis for the mathematical description of nature. It’s not a miracle that mathematical laws can be described mathematically. A mathematical law is simply an exact, quantitative statement of our inductive generalization. Causality says that entities must act in accordance with their natures, not just most of the time, but all the time. And not just in a qualitative way, but down to every last detail, which we can describe quantitatively.

Like Kepler before him, Craig buys into the idea that a supernatural consciousness created and ordered the universe according to mathematical principles, thus ignoring the fact that mathematics has its basis right here in the universe, not in some otherworldly mind that is accessible to us only by means of the imagination. In fact, Craig has an entirely backwards understanding of causality: having accepted the primacy of consciousness as his fundamental metaphysical assumption, Craig believes that the things which exist throughout the universe must obey some prescriptive force that originates independently of the universe and on which the very existence of the universe depends. In essence, Craig wants to begin with a magic consciousness whose afflatus shapes and directs all that exists. Without this, goes the assumption, everything that exists would simply collapse into chaotic randomness, thus the order we observe in the universe is “evidence” of a supernatural mentality behind it all. No, we don’t actually have evidence of such a thing, but we’re supposed to imagine it’s there all the same.

The problem here is that such thinking has causality completely reversed. Existence exists, and to exist is to be something specific. Identity is not the product of conscious activity, but rather a fact that is absolutely concurrent with existence. It is because identity obtains throughout existence without exception that we have what we commonly call “orderliness” or uniformity throughout nature. For nature to be uniform essentially means that it is uniform with itself, that is to say: identity is concurrent with existence. Actions similarly have identity: walking is distinct from swimming, reading is distinct from sleeping, etc. Thus the basis of causality – i.e., the law of identity applied to action – is existence as such, not some conscious activity which comes along and commands things to obey divinely ordained edicts. All evidence shows that existence exists and that to exist is to be something specific, and no evidence shows that any such conscious activity could possibly have such power over objects.

Given this failure to grasp the nature of causality along with his ignorance of the nature of concepts and their relationship to measurement, Craig’s case is doomed from its very roots.

Craig gives the following summary for his third argument:

1. If God did not exist, the applicability of mathematics would be just a happy coincidence.

Craig is all too happy to repeat the words of some thinkers who are, apparently, as baffled about mathematics as he hopes his audience is, expressly to perpetuate a state of ignorance which he can “explain” by asserting the existence of a god. Such machinations are passed off as the prestigious acumen of a “professional philosopher,” which is simply a disgrace. Without any understanding of the nature of concepts and the role measurement plays in cognition, Craig is able to execute his predatory apologetic tactics in order to give the world yet another rendition of the god-of-the-gaps.

2. The applicability of mathematics is not just a happy coincidence.

This is true, but not for the reasons Craig wants to believe. The source of our knowledge of mathematics is the universe itself, not some supernatural realm which resides only in our imagination, and our cognitive interaction with what we discover in it. Since measurement plays an inherent role in our conceptual methodology, mathematics is wholly in line with an entirely this-worldly epistemology, one which one would never be able to glean from the myths and tales of the bible.

3. Therefore, God exists.

So here we have yet another “Duh, I donno! Must be God did it!” moment dressed up in syllogistic form to make it appear that Craig has a solid argument for his god when in fact he is simply trying to cram his god into another deep dark gap in his knowledge.

Again, Craig seeks to explain the known by appealing to the unknown, and the objective theory of concepts shows why his maneuvers are philosophically useless from the very outset. Mathematics is a conceptual enterprise, and Craig attempts to “explain” it with his storybook worldview. Notice that Craig's position moves us no closer to understanding how Higgs or Newton or any other thinker accomplishes great things, but understanding how the mind forms concepts from the basis of perception and how measurement plays an integral role in knowledge, does. Craig's "explanation" therefore is certainly not "the best explanation"; indeed, it's hard to see how one could consider it to be a serious explanation in the first place. Fail.

But this was only Craig’s third argument. He still has five more or us to examine. So far Craig has scored a zero out of three. Next I will examine his fourth argument and see if he can salvage any opportunity to score his first point.

by Dawson Bethrick