Philosophers and scientists have puzzled over what physicist Eugene Wigner called “the unreasonable effectiveness of mathematics.”
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?
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.
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.
Mathematics is the language of nature.
Craig asks, rhetorically:
But how is this to be explained?
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.”
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.
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.)
The naturalist has no explanation for the uncanny applicability of mathematics to the physical world.
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.
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.
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.
2. The applicability of mathematics is not just a happy coincidence.
3. Therefore, God exists.
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