Boxing Pythagoras

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Mathematics: Natural or Supernatural?

Yet again, Dr. William Lane Craig’s weekly Reasonable Faith podcast discusses a topic with which I am keenly interested. Unfortunately (and unlike last week), I once again find myself in a state of disagreement with the famous apologist. Dr. Craig’s discussion, this week, is entitled “God and Math,” and centers around a claim that mathematics is “unreasonably” effective. WLC builds his argument off of an article published in 1960 by a physicist named Eugene Wigner, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.” Having mentioned the Wigner quote, Dr. Craig attempts to show that the mathematical foundation which underlies all of the natural sciences is not, itself, natural. He intimates that mathematics is a supernatural construct by which a deity composed the cosmos.

William Lane Craig does not understand mathematics.

In order to unpack my criticism of Dr. Craig, a bit, let’s explore his argument. Our famed apologist takes Wigner’s opinion on the matter and uses it to construct the following syllogism:

  1. If God did not exist, the applicability of mathematics would just be a happy coincidence.
  2. The applicability of mathematics is not a happy coincidence.
  3. Therefore, God exists.

Logically speaking, the syllogism is well constructed. The Minor Premise (#2) is a direct reference to the Major Premise (#1), and the Conclusion (#3) follows validly from their combination. Of course, a well constructed syllogism does not necessarily make a good argument. For example, imagine I claimed the following:

  1. All apples are red.
  2. The fruit which I ate was not red.
  3. Therefore, the fruit which I ate was not an apple.

Again, the Minor Premise relates directly to the Major Premise, and the two combine to yield a valid Conclusion. However, the Major Premise is, itself, false. Not all apples are red. Since one of our premises is false, our whole syllogism is flawed– I could very well have eaten a green fruit or a yellow fruit which was an apple, for example. In order for a syllogism to be useful, its premises must be true. Unfortunately for Dr. Craig, his Major Premise is a complete non sequitur. It does not follow that, if God did not exist, the applicability of mathematics would just be a happy coincidence. Firstly, even if God does exist, the applicability of mathematics might simply be a happy coincidence– Dr. Craig has given nothing to show otherwise. However, more importantly, WLC implies a false dichotomy with the premise: either mathematics was utilized by God, or else it is a happy coincidence.  These two cases, however, are not logical opposites. There are numerous ways in which mathematics can be highly applicable to the natural world while simultaneously being neither an instrument of the divine nor a joyous accident. The reality behind mathematics is such that we can see it is not simply a happy coincidence (I’m happy to agree with Dr. Craig’s Minor Premise, here), yet neither does it necessitate utilization by a deity. William Lane Craig’s Major Premise, here, is false.

So, then, what is mathematics, and why does it so keenly apply to the cosmos? You may have heard the saying, “Mathematics is the language of science.” This is not simply a metaphor. Mathematics is a language, an invented method for communicating information between two people. However, mathematics is a very specific language– one which only describes quantities and the relations between such quantities. For example, arithmetic can help us to communicate the number of apples on a particular tree, but it cannot tell us anything about those apples except for their quantity. It cannot tell us what color the apples are, or how juicy, or whether they are ripe. In fact, it cannot even tell us that they are apples. The only thing it can relate to us is their quantity.

Mathematics, therefore, is a description of nature. It is not the foundation which underlies nature, but rather a very human attempt to describe and communicate that foundation. So, the Major Premise which Dr. Craig has formulated is no less ridiculous than if he had claimed, “If God did not exist, the applicability of English would be just a happy coincidence.” The applicability of English is due to the fact that human beings have defined and refined that language with the deliberate purpose of making it applicable to the world around them. Mathematics is absolutely no different. Over the past several thousand years, humanity has deliberately sculpted mathematics into a system which applies to the world around them.

Marveling at the applicability of 1+2=3 in describing the universe is just as silly as marveling at the applicability of “red” in so doing.

One might try to argue that the language which we use to describe mathematics is beside the point, and that the underlying abstract concepts which are being described are what WLC’s actually referencing by his claim. That is, one might claim that WLC is not referring the the symbol “1” or the word “one,” but rather the underlying “oneness” that these things describe. However, the key to showing the unintelligibility of this is to point out that Dr. Craig is specifically referring to the “applicability” of mathematics. Abstract concepts are not applied to anything. These abstracts are referents– things to which we refer, by symbols and language. Abstract concepts cannot be acted upon– that is precisely what we mean when we claim that they are “abstract.” The word comes from the Latin abstrahere, which means “drawn away from” or “removed from.” That is, these concepts are separate from those things which we can directly access. Regardless of whether you are a platonist or a nominalist or anything in between, in regard to abstract concepts, it remains completely incoherent to discuss such abstracts as being “applied” to anything.

Dr. Craig claims that, for a non-theist platonist, “it is just inexplicable why the physical world would be imbued with a mathematical structure of these causally irrelevant, non-spatio-temporal, abstract entities.”  There are some problems with this. Firstly, the word “imbued” is a particularly loaded term, as it specifically connotes that some agent acted upon the physical world in order to give it a mathematical structure. The word serves to assume WLC’s intended conclusion before it has been adequately demonstrated. Secondly, the idea that there could be a “mathematical structure” of abstract entities is completely lacking in cogency, since abstracts cannot be acted upon and therefore cannot be structured in any particular way. There cannot be a structure, of any sort, of “non-spatio-temporal, abstract entities.”

Dr. Craig then asks of non-theist nominalists, “if… you think that these are just fictions in our minds then how is it that the physical world is structured in terms of these fictions?” Of course, this question is precisely backwards. In this case, it is not the physical world which is structured in terms of these fictions, but the fictions which are structured in terms of the physical world. We come up with the concepts of “oneness,” “twoness,” “redness,” and “blueness” as a response to things which we find in the physical world. These fictions are constructed in an attempt to understand the physical world. For example, when I say, “My apple is red,” the world is not structured upon my concept of “redness.” Rather, I have informed my concept of “redness” based upon observation of similarities in the properties of disparate entities which exist in the physical world. Once again, these fictions are applicable because they are intentionally constructed to be applicable.

After claiming that the non-theist has no good explanation for the applicability of mathematics, WLC goes on to say that, “the theist (whether he is a [platonist or nominalist]) can give an account or explanation of the applicability of mathematics to the physical world by saying that God structured the physical world on the pattern of the mathematical structure that he had in mind.” There are, of course, two major problems with this assertion. Firstly, the ability to give an explanation for something does not, in any way, imply that the explanation is correct. For example, I could explain how the US Constitution was written by saying that Martians implanted mind-control chips in the brains of our founding fathers and remotely controlled the muscles in Thomas Jefferson’s hand. Simply being able to offer an explanation for something is largely irrelevant, especially if you are trying to disparage opponents for not offering any “good” explanations. Which brings us to the second issue: the explanation that WLC here offers is a particularly bad one. It amounts to a tautology– he’s basically saying, “It is because it is.” Even if God did structure the physical world on the pattern of the mathematical structure he had in mind, this explanation does nothing to tell us why God had that structure in mind, nor why he structured the physical world upon it. WLC simply kicks the can down the road, a bit, with this entirely speculative explanation. If a naturalist had offered the reason “that’s just the way it is,” Dr. Craig certainly would not have thought it a good explanation. So why does he consider “That’s just the way God did it” to be a good explanation?

William Lane Craig does not understand mathematics. He does not understand the origin, development, and nature of the field. Mathematics is not some divine, supernatural force which directs the natural world. Mathematics is just one of humanity’s attempts to describe the manner in which the natural world functions. Mathematics does not underlie the physical cosmos. Mathematics is our attempt to understand that which does underlie the cosmos. There is nothing unreasonable about the applicability of mathematics to the natural sciences, because we actively alter mathematics every time we find that it doesn’t apply to the natural world. Irrational numbers. The concept of Zero. Infinitesimal calculus. Non-Euclidean geometry. We should not be surprised that something which has been deliberately designed and refined by humanity to better describe the natural world actually does a decent job at describing the natural world.

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4 thoughts on “Mathematics: Natural or Supernatural?

  1. This is excellent. It’s always neat to see someone tear down a WLC argument.
    I first encountered the “mathematics proves god” argument in a book titled Gravity: True for You, but Not for Me, and I think it is what put the final nail in the coffin of my hope for finding anything convincing in Christian apologetics.

    • Thanks for your input!

      While I was a Christian, I often made similar claims about Math and its applicability to the universe. I thought of Math as the Language of God, and it was one of the strongest points which lay under my own faith. Between personal experience, and the Language of God apologetic, I was convinced in the existence of God.

      In the years since I stopped being Christian, Math History became an incredibly interesting subject for me, and one which I have pursued vigorously. The more I have read, the more I realized how very human Mathematics really is. I appreciate Math much more now that I don’t view it with supernatural awe.

  2. Excellent piece.

  3. Pingback: On Bad Apologetics | The Complete History of the Universe: A Personal Journey

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