William Lane Craig and his sidekick, Cameron Bertuzzi, recently responded to a video by Rationality Rules’ rebutting Craig’s five-minute cartoon presentation of his argument for god from mathematics that was based on a paper by Eugene Wigner. Being mired in the reductionist and trivializing nature of analytic philosophy, Rationality Rules did not mount a very impressive challenge and we won’t deal here with his arguments, but instead will focus on just a few of Craig’s responses. Craig’s argument rests heavily on a paper delivered in 1959 by Eugene Wigner titled **The Unreasonable Effectiveness of Mathematics in the Natural Sciences **to a gathering at NYU. Craig makes claims about the Wigner paper that betray a combination of misunderstanding and distortion born of religious fervor. From this, Craig attempts an argument dependent on empty metaphysical speculation and the inevitable theist argument from ignorance: god of the gaps. The entire Bertuzzi and Craig video lasts over an hour and consists of no more than the above theme and several variations on the same distortions and plea from ignorance, so we will focus on just a few statements. The entire video, however, is linked below.

Before getting to the issues Craig offers this prefatory comment:

(All videos are cued to the start time, nut not end time. The relevant segments should end after Craig finishes his statement)

Craig usually praises those who present the weakest opposition, and I proudly contrast my last video on Craig’s 14 ridiculous responses to any irenic approach. Grifters and charlatans need to be exposed and becoming one of those that Craig welcomes is nowhere on my agenda, although I would love the opportunity to debate him. He also went on to say that the 71,000 views to Rationality Rules video warranted a response. Again, my approach here will be different. Unlike Craig’s cartoonish presentation (Literally. His five-minute video is actually a cartoon), I will go into a substantive discussion of Wigner, mathematics, epistemology and physics. I doubt there are 71,000 people on YouTube who could even understand the Wigner paper and I would be happy to get 71 views from those who do. What follows requires serious attention to fully grasp, and in the next several days I will have a text version available on my blog: toolateforthegods.com

Let’s start with an overview of the Wigner paper. Wigner was a physicist and mathematician of enormous importance in the development of quantum mechanics. He won the Nobel Prize in Physics in 1963 for his contributions to the theory of the atomic nucleus and the elementary particles, particularly through the discovery and application of fundamental symmetry principles. It is important to note that as a mathematician he was a formalist, which means he understood mathematics to be inventions logically derived from reason and not ontologically connected to the physical world. He was also an atheist.

His paper was delivered at an event to celebrate the 70^{th} birthday of Richard Courant, a mathematician who emphasized the creative and esthetic nature of mathematics rather than any practical application. From that perspective, the seeming ability of a purely rational a priori sort of “game” to describe the laws of nature is simply inexplicable – a wonder and mystery, and Wigner’s paper was to address that mystery. In presenting Wigner as that night’s guest speaker, Kurt Friedrichs said:

“One may think that one of the roles mathematics plays in other sciences is that of providing law and order, rational organization and logical consistency, but that would not correspond to Courant’s ideas. In fact, within mathematics proper Courant has always fought against overemphasis of the rational, logical, legalistic aspects of this science and emphasized the inventive and constructive, esthetic and even playful on the one hand and on the other hand those pertaining to reality. How mathematics can retain these qualities when it invades other sciences is an interesting and somewhat puzzling question. Here we hope our gift will help.”

And indeed, Wigner went on to explore this dual nature of mathematics: an essentially esthetic pursuit enjoyed by pure mathematicians through the simple elegance and harmony of proofs on one side, and practical instrumentality employed in everyday life and physics on the other. The central question then becomes why such a subjective esthetic enterprise should have any practical application at all in the physical world. Throughout the paper Wigner refers to this applicability as a “miracle”, or sometimes as a mystery, but as an atheist he did not mean this in the sense Craig twists it into, but rather as a dramatic effect for what seems inexplicable. He resolutely refrains from offering any supernatural element to this mystery. He does, however, suggest an explanation which he subtly develops through his paper. A solution that Craig fails to grasp.

Wigner starts with the beginning of mathematics as the invention of arithmetic, algebra and geometry. This had a purely practical motivation and was based on what was observable in the surrounding environment. Numbers of goods for sale and the geometric aspects of construction were the founding drivers of these inventions. Wigner identifies these practical concerns as the first tier of mathematical thought, ground in actual events encountered in the world. As he wrote in the paper:

“[they] were formulated to describe entities which are directly suggested to us by the actual world”

This is the first ontological connection between pure mathematics and the physical world, which ultimately demystifies the approximation of math as description of reality. Its importance will come to light later.

The next tier is abstraction from these practices, which gives us concepts that can be generalized. The third and highest tier is then shown to be invariability principles, which allows these concepts to be extended as laws of nature. These posit that within a limited scope of space, time and chosen events, the observed abstractions hold as a law of nature. It is critical to understand, however, that he limits the applicability to the physical. For example, what appears invariable within Newtonian physics does not hold for Relativity, and what appears invariable for relativity does not hold in quantum physics. We will return to this fundamental point in a moment, but first we need to also look at the esthetic nature mathematics which precedes any practical function.

As mentioned earlier, Wigner was a formalist who viewed mathematics as the playing out of pure reason in the faculties of the mind. While initially practical in purpose, these founding elements of mathematics were only possible as the result of turning this purely esthetic rational faculty toward the practical through the invention of tools formed specifically by physical questions. This is the initial ontological contact and relation between mathematics and the physical world – the application of the pure concept of numbers to physical objects suggested by encounters in the world and the logical manipulation of these numbers. The pure abstraction of mathematics is thus causally shaped by external events to answer questions of physical reality. But keep in mind these events are exceptions to the real practice of mathematics as essentially esthetic and unconcerned with practicality, as seen for example in the mystical mathematics of Pythagoras.

Starting with Newton, however, the increasing demands of physics conjoined science with mathematics and fostered the invention of new mathematical tools. The invention of calculus by Newton and Leibniz, for example, was intended to quantify the laws of motion without any regard for its esthetic qualities. From that moment on, physics appropriated mathematics as its official language. This led to the consensus among physicists that the universe itself was a mathematical construct that naturally cohered with the rational mind. To this day many physicists naively retain this belief, which Wigner will go on to subtly amend.

Now back to where we left the issue of the limitations of invariability principles. Despite what we will see as Craig’s claims to the contrary, the shifting of the time, space and event framework resulted in a need for new mathematics. Not as Craig will claim as merely a new physical law, but a mathematics with new axioms contradictory to earlier invented mathematical systems. This brings us to the seminal paper by Henri Poincaré on non-Euclidean Geometry which is crucial for Wigner’s paper. Poincare demonstrated that it is possible to devise internally consistent rational systems with contradictory basic assumptions. To do so he compared Euclidean, Lobachevskian and Riemannian geometries, where “The number of parallel lines that can be drawn through a given point to a given line is one in Euclid’s geometry, none in Riemann’s, and an infinite number in the geometry of Lobachevsky.” To this, Poincare added a fourth geometry even stranger than the other two non-Euclidean geometries, yet equally internally consistent. The significance of this points to Wigner’s limitations on the law of invariability, where each is a rational construct which can differ from other rational constructs formed with different assumptions and chosen events. This is exactly what Wigner refers to when has asks

“How do we know that, if we made a theory which focuses its attention on phenomena we disregard and disregards some of the phenomena now commanding our attention, that we could not build another theory which has little in common with the present one but which, nevertheless, explains just as many phenomena as the present theory?” It has to be admitted that we have no definite evidence that there is no such theory.”

Wigner refers to this as “nightmare of the theorist”: the existence of numerous examples of theories, with elegant mathematical formulation, and “alarmingly accurate” description of a group of phenomena, which are nonetheless considered to be false.

This takes us to the final crucially important element of Wigner’s paper: what he calls the Empirical Law of Epistemology. With this he eliminates the mystery inherent in the metaphysical question of why the universe operates under the same laws as our innate reason by transforming it to the epistemological question of how our faculties of understanding create our representations, concepts and explanatory systems of the universe. The answer: our innate reason is the source of pure mathematics. 5000 years ago, this was put to practical use, and in that process the ontological connection was shaped by observed events which we then abstracted to generalizations. This formed a connection between mathematics and the physical world that worked in two directions. Our innate reason provided the framework for how we intuit the world and the events we encountered formed the contents of those shapes. Through further abstraction, which is enabled through the esthetic nature of reason, we generalized from our observed events and found we could form systematic understandings of the world that held within certain limits. The coherence between the universe and our subjective reason isn’t a “miracle”, but rather a constructed understanding conditioned by external events ans formed through our faculties of understanding. This enabled our creation of modern physics. In time this engendered the illusion of a mathematical universe.

As we will see, Craig confuses this. He first claims that the immutable laws of nature and their mathematical precision, along with our innate ability to understand it, point to a divine creator. Without the claim of a mathematical universe, his argument disintegrates. Yet, he later discusses the nature of mathematical understanding of physics as epistemological representation, which in fact was Wigner’s point but contradicts the belief in a mathematical world.

Let’s now turn to the opening of Craig’s comments:

It’s clear we can reasonably claim that Craig has also failed to digest Wigner’s paper. Craig diverts attention from his own indefensible argument to a claim that the critic’s real problem is with Wigner’s argument, which Craig himself fails to understand, and the claim that Wigner has refuted all of Rationality Rules’ objections. Craig fails, however, to tell us what the objections and refutations are. This is a typical Craig move. He rests assured that his followers are never going to read Wigner, and even fewer could ever understand the paper. This frees him to make unsupported claims exploiting the trust of the less educated.

Craig then makes an early introduction of his god of the gaps argument:

He takes what Wigner initially terms inexplicable and in literally the exact same breath concludes that it then must be god.

The segment starts with a cut from Craig’s cartoon which explicitly claims that “the physical universe operates mathematically”. This is important because without that claim there is no miracle to assert, although the claim is actually at odds with Wigner’s paper. Maybe it’s Craig whose argument is with Wigner:

Here Craig moves to obscure the issue by focusing on logicism, which in fact is a failed enterprise but in no way detracts from the derivation of mathematics from logic. This was not only Wigner’s view, but was famously demonstrated in Kant’s Critique of Pure Reason as the working of pure reason within the innate sensibilities of space and time. Again, Craig counts on the ignorance of his followers in his facile and false dismissal of a much larger issue. In doing so Craig evades the stickier issue for him that logical truths are entirely dependent on premises verifiable through the senses, or else they themselves are as empty as pure mathematics and as such merely constructions. Pure reason, including mathematics, can never make existential claims, which is a defeating fact for Craig’s entire metaphysical enterprise. His assertion that a god exists as an explanation is no more valid than the assertion that “an infinite set exists.” His criticism of logicism equally applies to his own metaphysics, and his continued god of the gaps arguments are errors of this kind.

The next segment brings about all at once Craig’s failure to understand Wigner, his own confused mishmash, and eventual undermining of his own argument.

He begins by dismissing the element of invention in mathematics as a post-modern view held by a small minority. Despite the obvious fallacy of argumentum ad populum, it dishonestly characterizes the role of invention. As noted several times, Wigner also saw mathematics as invention separate from external ontology, as did Courant and most formalists. As Wigner wrote in the paper: mathematics is

“a science of skillful operations with concepts and rules invented just for this purpose”

Craig then attempts to make some indiscernible point by claiming that Wigner considers mathematics a purely esthetic pursuit, ignoring that Wigner presented both the esthetic and practical natures of mathematics and grounded the beginning of mathematics in the practical – the observed events.

Perhaps he hoped to demonstrate that the ability of such a pure esthetic mode of thought to describe reality is truly miraculous. However, what he claims as support is false.

He then makes the preposterous claim that new discoveries in physics did not cause changes in mathematics, but merely a change in physical laws. The major problem Einstein faced in developing the theory of relativity is that the assumptions of Euclidean geometry failed to describe the newly discovered reality. As a result, Einstein embarked on the struggle to devise a new geometry which allowed for parallel lines to meet in the curvature of space until his friend Grossman pointed out to him that Riemann had already done so. Soon after, quantum mechanics, which had no use for Riemannian geometry, reintroduced calculus to account for relative motions, but had to introduce the infinite dimensionality of Hilbert space, again a reinvented geometry. It is also interesting that the further removed from the initial observable reality which was the beginning of mathematics, the more abstract and unreproducible in reality the mathematics becomes, leading to unimaginable results such as many dimensions and superposition – all artifacts of mathematical invention. Again, the limitations of the principles of invariability come to the fore, with Relativity describing a world essentially incompatible with quantum physics, yet both mathematically internally consistent – a sure indication that reality is far different from one calculable mathematical universe. Again, this demystifies the original question in that there is no magical correspondence between mathematics and the universe, but rather various rational systems of understanding that are approximately true within limits of space, time and chosen events. Later, Craig tries to obscure his problem by again claiming the mathematic axioms never change by misrepresenting the differences between Euclidean, Lobachevskian and Riemannian geometries as merely differing in the shapes they apply to, omitting Poincare’s demonstration of conflicting axioms that make the systems contradictory.

Craig then further demonstrates his failure to understand Wigner when he claims that Rationality Rules has confused natural law with mathematics when Wigner explicitly demonstrated that in modern starting with Newton physics natural laws are in fact mathematical expressions.

Now listen to this statement and keep in mind his previous claim that the universe is mathematically knowable because of a divine creator. Afterwards, we’ll listen to a segment where he contradicts this and undermines his own argument:

And now:

Craig is right for once. Wigner was addressing the epistemological question of why our representations approximate reality under limited conditions. That is why Wigner terms his law the Empirical Law of Epistemology. And as Craig concedes, this does not assume a mathematical universe, only our mathematical representations of it. Without this mystical connection between reason, mathematics, and a mathematical world, which Craig earlier posited as proof of a divine creator, his argument from god collapses. Clearly, Craig has lost the plot here.

The critical point is that with the Empirical Law of Epistemology Wigner has shifted from the mystery of metaphysics to epistemology – the connection of innate sensibility to the suggested world events as founding of mathematics, the abstractions from the observations, and within the limits of the invariability principles, the determination of Natural laws, which are conditioned by time, space and chosen events and are provisional.

After having created a muddle, Craig seems to abandon Wigner entirely and attempts to return to long-discarded metaphysics in order to posit god as the answer.

Once again, Craig’s last gasp is the dying metaphysical breath of god of the gaps.