Boxing Pythagoras

Philosophy from the mind of a fighter

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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8 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

  4. Tyler Scollo on said:

    warebec, what an objective, unbiased, lovely thing to say. Anyways . . .

    I really hope you don’t think that the material presented on Craig’s podcasts or Defenders classes is the be-all, end-all of sophistication exhibited in his books.

    “William Lane Craig does not understand mathematics.”

    Nicely phrased, uncharitable way to get your point across. Doesn’t understand all of mathematics? A part? They said Robert Oppenheimer sucked at mathematics too. So, I guess he didn’t understand mathematics. Why don’t you phrase your theses more charitably? You come off as pompous. Say something like, “It seems to me, from the reasons I’ll share below, that Craig is mistaken regarding the idea that mathematics is a supernatural construct by which a deity composed the cosmos.”

    Then we get an elementary logic lesson that Craig is probably aware of.

    “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.”

    How is it nothing when Craig says, “By contrast, the theist (whether he is a realist of anti-realist) 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. So there is an explanation available to the theist that is not available to the non-theist.”?

    “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 you say. So your reasons for thinking the premise is false are . . .

    “mathematics is a very specific language– one which only describes quantities and the relations between such quantities.”

    and

    “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.”

    You’ve profoundly misunderstood Craig’s argument. Or, as you would put it, ‘you just don’t understand philosophical argumentation.’ Stinks when you put the shoe on the other foot, huh? Here’s where you’re lost. Craig doesn’t deny that mathematics is a description of nature. Craig doesn’t deny that humans can use it to communicate quantity. And just because humans use it to communicate quantities and relations between quantities, how does this prove that mathematics is merely a ‘human’ attempt? Craig’s thesis is in terms of ‘explaining’ why mathematics is such a language, why it functions as the description it is. He then provides ‘arguments’ for why the non-theistic realist and the non-theistic antirealist don’t have ‘explanations’ of such an ‘unreasonable effectiveness’ AS GOOD AS theistic anti-realist explanations. So, you misrepresented him.

    “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.”

    English is a social construction. Mathematics is an objective research program of discovery. So, mathematics IS different from English. Sure, they’re both languages. But that’s like saying that because a whale and I are both mammals, the whale has arms and legs. Peter Higgs didn’t craft mathematical equations that he force-fit onto particle physics just to make his equations applicable to nature; his equations ‘discovered’ a language in nature, already ‘out there’, enabling him to predict, 30 years into the future, the actual existence of a particular particle. Your analogy is not a good one. Sure, English and Mathematics are both languages; that doesn’t mean that English was constructed the same way as Mathematics. The language of mathematics was discovered. The next question is that if nature itself exhibits this objective, mathematical structure, what’s the best explanation for this? Why doesn’t nature exhibit a simpler mathematical structure? Why does nature exhibit mathematical structure at all? What’s the ‘explanation’? Craig hasn’t been given an answer. There may be one, but he hasn’t found an answer, and after going through the explanatory options, he argues that theistic antirealism provides the best explanation. This is normal philosophical methodology. Perhaps he’s made a mistake. No big deal. It happens all the darn time in scientific and philosophy journals. But mistakes aren’t met with pronouncements that a scholar has misunderstood an entire subject-matter. In other words, the conclusion shouldn’t be the petulant idea that a scholar ‘just doesn’t understand mathematics’, especially since the critique is misunderstanding the nature of the argument to begin with.

    “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.”

    Well, you’d know the structure of Craig’s argument better if you didn’t listen to a solitary podcast on the subject. Yes, the option you’re describing – which Craig has addressed in other more scholarly venues – is anti-theistic realism. An actual option in the literature!

    “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.”

    Well, if you’re familiar with Craig at all, you’d know he’s a theistic anti-realist, so he doesn’t even believe in the existence of abstracta at all. You do know that you can still attribute ‘truth’ to mathematical propositions without being committed to the ‘existence’ of such propositions, and what the propositions are about, correct? Craig wrote a 500-page book addressing topics like this. Go read that, instead of critiquing a podcast.

    “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.”

    Huh? There’s nothing about ‘imbue’ that has anything to do with agents at all. What definition of ‘imbue’ gives you that idea? And serves to assume his intended conclusion? Well, since he’s talking about a NON-THEISTIC platonist, how in the world would ‘imbue’ help his conclusion, if it’s loaded the way you say it is. Craig critiques this position as explanatorily deficient on entirely different grounds.

    “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.””

    Begs all kinds of questions against the Platonist. But either way, it doesn’t matter at all, because Craig doesn’t go for this option anyway. He’s an anti-realist!

    “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.”

    What the heck are you saying? Please listen to what you’re saying, lol. Per the non-theistic NOMINALIST, Craig is asking how, if such nominalism is assumed to be the case, the physical world could be structured in terms of fictions in our minds. And you’re saying this is backward: that a non-theistic NOMINALIST should be asking why such fictions are structured in terms of the physical world! If they’re structured IN TERMS OF THE PHYSICAL WORLD, then, by definition, they’re not fictions; you’re tacitly assenting to non-theistic realism! The nominalist, BY DEFINITION, must account for how mere fictions STRUCTURE the world, NOT, as the realist needs to do, account for how the mathematical “non-fiction” is structured IN TERMS of the world. It’s, by definition, ‘non-fiction’ if put this way, because the structure of the world that determines the “fictional”-structure wouldn’t be a fiction!

    “Once again, these fictions are applicable because they are intentionally constructed to be applicable.”

    Great point! And because mathematics as a language is discovered, and the applicability of mathematics to nature is INTENTIONALLY CONSTRUCTED, Craig would be more than happy to concede this as a key premise in his argument that the best explanation for such applicability is theistic non-realism. Agents have intentionality. Mathematics is NOT like English.

    “Firstly, the ability to give an explanation for something does not, in any way, imply that the explanation is correct.”

    Craig would know this! All over his publications, you’ll find various criteria for explanatory acceptance! The same criteria you could find if you pick up any book on inference to the best explanation or abduction or probabilistic inference.

    “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.” ”

    Wretched misrepresentation.

    “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.”

    Dude, even in science, explanations aren’t confined to ‘why’ questions, and aren’t disqualified on that score. There’s ‘how’ and ‘what’ questions, as well! I mean, just off the top of my head, establishing the rest mass of a particle without believing in an explanation for why the rest mass is this value rather than that.

    “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.”

    Of course, such a strange question can only be met with ‘it depends!’ I’d imagine it would depend on whether it’s the least non-arbitrary stopping point for an explanation, and whether we’re talking about ‘how’, ‘why’, or ‘what’ questions.

    “Mathematics does not underlie the physical cosmos. Mathematics is our attempt to understand that which does underlie the cosmos.”

    That’s your assertion based on a disanalogy between Mathematics and English. Right. When the math links up with the structure, the argument is asking what the best explanation of this is! You’re not addressing the argument. Why is the structure such that it agrees with mathematics. Think of the Higgs example.

    “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.”

    Begged question! Altering mathematics when it doesn’t apply is completely compatible with its applicability being due to nature’s mathematical structure. If you bump your head against the wall when the lights go out, and alter your course, does that automatically mean the wall doesn’t have an impermeable structure which explains the alternation of my direction of walking? Or that walking is just a social construction to explain walk-behavior that conveniently avoids these theory-laden things we’ve labeled ‘walls’? You’re too quick to dismiss this.

    And you’re so dogmatic in the tone of your blog. I’m almost positive that if Craig were to read this, he wouldn’t just sit there and think, “Wow. I suck. I don’t understand mathematics. I think I’ll be a fisherman.” I bet a million bucks that, if he had nothing better to do, he would completely rout you. At the very least, this initial blog would need some substantial editing.

    • I really hope you don’t think that the material presented on Craig’s podcasts or Defenders classes is the be-all, end-all of sophistication exhibited in his books.

      I don’t. However, none of the works by Dr. Craig which I have read show much greater sophistication on the subject of mathematics than do his podcasts or Defenders series, and many of them make similar mistakes. If you would like to suggest a particular book which you feel does a better job of laying out Dr. Craig’s position, I’d be more than happy to address it directly.

      Nicely phrased, uncharitable way to get your point across. Doesn’t understand all of mathematics? A part?

      The statement was intentionally blunt, for rhetorical purposes. The rest of the article makes it quite clear that I am particularly making the case that Dr. Craig does not understand the nature, history, and development of mathematics.

      Craig’s thesis is in terms of ‘explaining’ why mathematics is such a language, why it functions as the description it is. He then provides ‘arguments’ for why the non-theistic realist and the non-theistic antirealist don’t have ‘explanations’ of such an ‘unreasonable effectiveness’ AS GOOD AS theistic anti-realist explanations. So, you misrepresented him.

      No, he doesn’t provide arguments for why the non-theistic realist and the non-theistic antirealist cannot have explanations of such “unreasonable effectiveness” which are as good as theistic anti-realist explanations. He simply baldly asserts that this is the case. At no point in the podcast does he actually engage with any philosophy on the subject.

      English is a social construction. Mathematics is an objective research program of discovery.

      There is an element of discovery to mathematics, to be sure. That doesn’t make it any less of a social construction.

      Mathematics begins with definitions and axioms which are set up by human beings in order to effect some pre-determined goal. Now, those definitions and axioms quite often lead to unforeseen implications– discoveries– which are incredibly interesting and useful. However, this doesn’t make mathematics any less a human invention. Change those definitions and axioms, and you create a whole new set of unforeseen implications to discover. The choice of definitions and axioms by which one operates is human. It is not determined by some objective necessity of nature.

      Well, you’d know the structure of Craig’s argument better if you didn’t listen to a solitary podcast on the subject. Yes, the option you’re describing – which Craig has addressed in other more scholarly venues – is anti-theistic realism. An actual option in the literature!

      I’m not sure how this is meant to be a response to the portion of text which you are quoting. I agree that this is talking about the realist position– though it is not necessarily anti-theistic. I agree that it is an option in the literature. I never said otherwise. What is your point?

      Well, if you’re familiar with Craig at all, you’d know he’s a theistic anti-realist, so he doesn’t even believe in the existence of abstracta at all.

      I do know that, and have explicitly noted it elsewhere in my work. What is your point?

      Craig wrote a 500-page book addressing topics like this. Go read that, instead of critiquing a podcast.

      If you will note the date at which I published this article– or even if you had listened to the podcast to which I respond, or read its transcript– you would know that I wrote this well before Dr. Craig’s book on abstracts had been published. That book is now in my reading queue, but I have not yet gotten to it.

      Huh? There’s nothing about ‘imbue’ that has anything to do with agents at all. What definition of ‘imbue’ gives you that idea?

      Well, there’s these:
      https://www.merriam-webster.com/dictionary/imbue

      The word “imbue” is a transitive verb which implies both a subject and an object– “X imbues Y.” When using the passive voice, the object and subject are traded, “X is imbued by Y.” One can omit the “by Y,” in that passive voice, but the implication of its existence remains. Nothing can be imbued without something to do the imbuing.

      Begs all kinds of questions against the Platonist.

      No, it doesn’t. Even for the Platonist, there cannot be any structure of non-spatio-temporal, abstract entities. Plato was fairly adamant in his description of the universals as being wholly removed from the physical world. He surely would have scoffed at the claim that these abstracts could be acted upon in any way, including being structured.

      And you’re saying this is backward: that a non-theistic NOMINALIST should be asking why such fictions are structured in terms of the physical world! If they’re structured IN TERMS OF THE PHYSICAL WORLD, then, by definition, they’re not fictions

      A fiction, in the sense which Dr. Craig utilizes the word, is not necessarily a falsehood. To call something a “fiction” is merely an admission that it is a concept invented by a mind. It is entirely possible that such a fiction can describe the physical world. As it so happens, this is precisely the case on the nominalist view of mathematics.

      And because mathematics as a language is discovered, and the applicability of mathematics to nature is INTENTIONALLY CONSTRUCTED, Craig would be more than happy to concede this as a key premise in his argument that the best explanation for such applicability is theistic non-realism.

      Mathematics, as a language, is invented. The implications of the invented rules of that language are discovered.

      Craig would know this! All over his publications, you’ll find various criteria for explanatory acceptance! The same criteria you could find if you pick up any book on inference to the best explanation or abduction or probabilistic inference.

      Craig SHOULD know this. He should ALSO know that “I am not convinced of A, B, or C” does not therefore imply that D is the best explanation– which you would also learn from those books which you recommend. Even if it is the case that one does not think A, B, or C are good explanations, that does not necessarily imply that D must therefore be a good explanation.

      Wretched misrepresentation.

      Really? And what explanation does Dr. Craig offer for why his proposed deity so structured the universe?

      Dude, even in science, explanations aren’t confined to ‘why’ questions, and aren’t disqualified on that score. There’s ‘how’ and ‘what’ questions, as well!

      Dr. Craig’s argument is based around a “why” question: Why is mathematics applicable to the universe? “How” and “what” questions are irrelevant to the syllogism which lies at the heart of his argument.

      When the math links up with the structure, the argument is asking what the best explanation of this is!

      And the answer is that humanity has explicitly invented rules for mathematics which describe that structure. Despite your protestations to the contrary, mathematics is every bit as much an invented language as is English.

      Begged question! Altering mathematics when it doesn’t apply is completely compatible with its applicability being due to nature’s mathematical structure.

      It’s not a begged question, at all. The previous mathematical language was not abandoned because it was inconsistent or illogical or malformed. It is perfectly possible to continue to utilize and explore that previous mathematical language. It does not cease to be mathematics simply because it does not describe the physical world. There are quite a number of mathematical systems which do not describe the physical world. Heck, even Dr. Craig will argue that there are perfectly consistent and interesting mathematical languages which are not applicable to the physical world.

      There is no single “mathematics” which is objectively true to the exclusion of all others. There are mathematical languages which more accurately describe the physical world, and those which less accurately describe the physical world, and they were all invented by humans every bit as much as English was.

      And you’re so dogmatic in the tone of your blog. I’m almost positive that if Craig were to read this, he wouldn’t just sit there and think, “Wow. I suck. I don’t understand mathematics. I think I’ll be a fisherman.”

      I don’t either. Especially since understanding mathematics isn’t really necessary for a Theologian and Philosopher of Religion. He can continue to completely misunderstand mathematics for the rest of his life, and yet still remain successful in his occupation.

      I bet a million bucks that, if he had nothing better to do, he would completely rout you.

      I would welcome his attempts to do so.

  5. Tyler Scollo on said:

    “If you would like to suggest a particular book which you feel does a better job of laying out Dr. Craig’s position, I’d be more than happy to address it directly.”

    His work on God and Abstract Objects (chapter 7), his essay “Mary Leng’s Mathematics and Reality”, his essay “God and the ‘Unreasonable Effectiveness of Mathematics’”#406 Is the Applicability of Mathematics Necessary?”, #305 God and the Applicability of Mathematics, #277 The Applicability of Mathematics, and his book God Over All.

    “The statement was intentionally blunt, for rhetorical purposes. The rest of the article makes it quite clear that I am particularly making the case that Dr. Craig does not understand the nature, history, and development of mathematics.”

    Well, it comes off as pompous, just so you know, unless you want your audience to be reduced to head-nodders. If you want to expand your audience to interact with possible Christian scholarship, I’d change this part. It doesn’t make for good PR.

    “No, he doesn’t provide arguments for why the non-theistic realist and the non-theistic antirealist cannot have explanations of such “unreasonable effectiveness” which are as good as theistic anti-realist explanations. He simply baldly asserts that this is the case. At no point in the podcast does he actually engage with any philosophy on the subject.”

    Then what in the world is Craig talking about when he says this, “Neither the realist nor the anti-realist who is a non-theist has a good explanation for the uncanny applicability of mathematics to the physical world. This is self-confessed. They both admit that they have no account to give. That is why I say it is just a happy coincidence.”? SELF-CONFESSED. Or, what about, “It is just this happy coincidence to use Mary Leng’s words.” TO USE HER WORDS. He’s talking about self-confessions of such people admitting there’s no explanation. THIS ISN’T BALD ASSERTION. If you’re ever disappointed that something wasn’t touched on, be charitable and attribute it to something like the podcast-format or time-constraints, not to misunderstanding or idiocy.

    “There is an element of discovery to mathematics, to be sure. That doesn’t make it any less of a social construction.”

    Fine. How isn’t this non-theistic anti-realism? Answer Craig’s questions: “if you are a non-theistic anti-realist and 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?” Craig says: “Neither the realist nor the anti-realist who is a non-theist has a good explanation for the uncanny applicability of mathematics to the physical world. This is self-confessed.” Now maybe Craig hasn’t read the solutions of almighty Boxing Pythagoras. One can only deal with what one has read, and the non-theistic anti-realists he’s read (probably a tad more informed than you on the subject, I don’t know your background), CONFESS that they don’t have an explanation.

    “Mathematics begins with definitions and axioms which are set up by human beings in order to effect some pre-determined goal. Now, those definitions and axioms quite often lead to unforeseen implications– discoveries– which are incredibly interesting and useful. However, this doesn’t make mathematics any less a human invention. Change those definitions and axioms, and you create a whole new set of unforeseen implications to discover. The choice of definitions and axioms by which one operates is human. It is not determined by some objective necessity of nature.”

    Fine. Make it a human invention. Like I said above, you’re a non-theistic anti-realist! How does this have any impact on Craig’s argument? He takes this position into account. Here’s a quote from Craig’s essay [God and the ‘Unreasonable Effectiveness of Mathematics’]: “Now consider anti-realism of a non-theistic sort. Leng says that on anti-realism relations which are said to obtain among mathematical objects just mirror the relations obtaining among things in the world, so that there is no happy coincidence. Philosopher of physics Tim Maudlin muses, “The deep question of why a given mathematical object should be an effective tool for representing physical structure admits of at least one clear answer: because the physical world literally has the mathematical structure; the physical world is, in a certain sense, a mathematical object.” [6] Well and good, but what remains wanting on naturalistic anti-realism is an explanation why the physical world exhibits so complex and stunning a mathematical structure in the first place. Perhaps the universe had to have some mathematical structure–though couldn’t the world have been a structureless chaos?–still, that structure might have been describable by elementary arithmetic. For example, one thing and another thing make two things. But modern physics shows the physical world to be breathtakingly mathematically complex. When Albert Einstein was struggling to craft his General Theory of Relativity, for example, he had first to go to a mathematician to be tutored in tensor calculus before he could advance further to formulate an adequate theory of gravitation. Balaguer admits that he has no explanation why, on anti-realism, mathematics is applicable to the physical world or why it is indispensable in empirical science. He just observes that neither can the realist answer such “why” questions.”

    So, on this view, THERE’S NO HAPPY COINCIDENCE. Craig: “Well and good, but what remains wanting on naturalistic anti-realism is an explanation why the physical world exhibits so complex and stunning a mathematical structure in the first place. Perhaps the universe had to have some mathematical structure–though couldn’t the world have been a structureless chaos?–still, that structure might have been describable by elementary arithmetic. For example, one thing and another thing make two things. But modern physics shows the physical world to be breathtakingly mathematically complex.” So, ACCORDING TO WHO CRAIG HAS READ, they have NO EXPLANATION because THERE’S NOTHING TO EXPLAIN. At this point, Craig does what any metaphysician would do: intuition pumps for seeing why there is something here to be explained. There’s nothing here that’s just woefully irrational.

    “If you will note the date at which I published this article– or even if you had listened to the podcast to which I respond, or read its transcript– you would know that I wrote this well before Dr. Craig’s book on abstracts had been published. That book is now in my reading queue, but I have not yet gotten to it.”

    Other things we’re available on the website.

    “Well, there’s these:
    https://www.merriam-webster.com/dictionary/imbue

    The word “imbue” is a transitive verb which implies both a subject and an object– “X imbues Y.” When using the passive voice, the object and subject are traded, “X is imbued by Y.” One can omit the “by Y,” in that passive voice, but the implication of its existence remains. Nothing can be imbued without something to do the imbuing.”

    Well, the first usage implies intentionality. Go for another one, then! Maybe it’s a metaphor. The principle of charity! It’s in the context of talking about theistic or non-theistic REALISM; since the latter is an option, contextually, Craig is using the word literarily to intimate at the realism, not doing any kind surreptitious slight-of-hand to sneak intentionality through the backdoor!

    “No, it doesn’t. Even for the Platonist, there cannot be any structure of non-spatio-temporal, abstract entities. Plato was fairly adamant in his description of the universals as being wholly removed from the physical world. He surely would have scoffed at the claim that these abstracts could be acted upon in any way, including being structured.”

    You’re missing the point. Craig addresses Plato in his essay! https://www.reasonablefaith.org/writings/popular-writings/existence-nature-of-god/god-and-the-unreasonable-effectiveness-of-mathematics/

    “A fiction, in the sense which Dr. Craig utilizes the word, is not necessarily a falsehood. To call something a “fiction” is merely an admission that it is a concept invented by a mind. It is entirely possible that such a fiction can describe the physical world. As it so happens, this is precisely the case on the nominalist view of mathematics.”

    Do you know what fictionalism is? Fictionalism denies that abstract object talk is true. You’re saying something literally false. That means that mathematical propositions are literally false. They are to be IMAGINED as true. They quantifiers of fictional discourse are ontologically non-committing. It’s make-believe. THAT IS NOT NECESSARILY THE SAME AS NOMINALISM, as Craig is a nominalist, but not a fictionalist. If you’re a NON-THEISTIC fictionalist (a kind of anti-realist), then see above about why this is denying the ‘happy coincidence’ thesis.

    “Craig SHOULD know this. He should ALSO know that “I am not convinced of A, B, or C” does not therefore imply that D is the best explanation– which you would also learn from those books which you recommend. Even if it is the case that one does not think A, B, or C are good explanations, that does not necessarily imply that D must therefore be a good explanation.”

    He DOES know this. That’s NOT how he’s arguing at all. Your MISREPRESENTATION is an obvious argument from ignorance gaffe. He provided REASONS for why A doesn’t explain the happy coincidence, and B admits there’s no happy coincidence (whereupon he provides a brief intuition-pump for why there is: it’s a short podcast!), and then argues that THERE’S GOOD REASON TO THINK C is the best explanation. Craig says: ” . . . the theist (whether he is a realist of anti-realist) 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. So there is an explanation available to the theist that is not available to the non-theist.”

    Address Craig’s argument here: https://www.reasonablefaith.org/writings/popular-writings/existence-nature-of-god/god-and-the-unreasonable-effectiveness-of-mathematics/

    “Dr. Craig’s argument is based around a “why” question: Why is mathematics applicable to the universe? “How” and “what” questions are irrelevant to the syllogism which lies at the heart of his argument.”

    Point taken. Your original point was: ““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.”

    You don’t need to know ‘why’ God did it this way rather than that to have a good reason ‘that’ He did it ‘in this way’. Why do you need to know this first? This is theology or divine psychology.

    “It’s not a begged question, at all. The previous mathematical language was not abandoned because it was inconsistent or illogical or malformed. It is perfectly possible to continue to utilize and explore that previous mathematical language. It does not cease to be mathematics simply because it does not describe the physical world. There are quite a number of mathematical systems which do not describe the physical world. Heck, even Dr. Craig will argue that there are perfectly consistent and interesting mathematical languages which are not applicable to the physical world.”

    Not my point. Craig’s point is that precisely that subset of mathematics that DOES apply to the world is that which needs explaining. All math doesn’t apply to the world; but all the structure of the world is mathematical.

    “I don’t either. Especially since understanding mathematics isn’t really necessary for a Theologian and Philosopher of Religion. He can continue to completely misunderstand mathematics for the rest of his life, and yet still remain successful in his occupation.”

    He can, nice. But he isn’t.

    “I would welcome his attempts to do so.”

    He’d be more likely to, or people up his alley would be more likely to if they didn’t have to be met by rhetorically disparaging blog titles like “X doesn’t understand an entire subject matter.”

    • His work on God and Abstract Objects (chapter 7), his essay “Mary Leng’s Mathematics and Reality”, his essay “God and the ‘Unreasonable Effectiveness of Mathematics’”#406 Is the Applicability of Mathematics Necessary?”, #305 God and the Applicability of Mathematics, #277 The Applicability of Mathematics, and his book God Over All.

      Thank you. I will find and review these.

      If you want to expand your audience to interact with possible Christian scholarship, I’d change this part. It doesn’t make for good PR.

      I’ve had many wonderful, interesting, and irenic conversations with Christian apologists, theologians, and scholars. My rhetorical tone hasn’t seemed to do me any harm, on that front, and many with whom I have conversed have explicitly stated that they rather enjoy talking with me.

      If you find my tone offensive, I can only say that it is not my intention to be insulting. Dr. Craig would be the first person to tell you that he is not a mathematician, nor is he a philosopher of mathematics, nor does he hold more than a layman’s grasp of most fields of mathematics. To say that he doesn’t understand mathematics is by no means an overstatement of fact. Nor should it be taken as an insult; the unfortunate fact of the matter is that MOST people do not understand mathematics.

      They both admit that they have no account to give. That is why I say it is just a happy coincidence.”? SELF-CONFESSED. Or, what about, “It is just this happy coincidence to use Mary Leng’s words.” TO USE HER WORDS. He’s talking about self-confessions of such people admitting there’s no explanation. THIS ISN’T BALD ASSERTION.

      Cherry-picking a single quote from a single philosopher can hardly be considered a fair assessment of the field. Nor is it fair to generalize from this one philosopher’s quote that the totalized “they” representing all non-theist views of the subject self-confess that there is no explanation.

      There have been multitudinous responses to this question in the literature of the philosophy of mathematics proposing numerous different explanations. Despite Dr. Craig’s bald assertion to the contrary, non-theist realists and non-realists, alike, have offered explanations for the applicability of mathematics to the universe.

      Fine. How isn’t this non-theistic anti-realism?

      Because it’s completely agnostic regarding deity. It is entirely possible to hold that mathematics is a social construction while also positing that it was a divinely inspired one, for example.

      Craig is using the word literarily to intimate at the realism, not doing any kind surreptitious slight-of-hand to sneak intentionality through the backdoor!

      Regardless of whether Dr. Craig consciously intended to utilize a transitive verb in order to imply an undisclosed actor or not, the fact remains that the word DOES carry such an implication and was therefore a poor choice in the context of the conversation.

      You’re missing the point. Craig addresses Plato in his essay!

      I’m actually willing to concede this point. On re-reading Dr. Craig’s claim from the podcast in light of the essay you posted, it seems I misunderstood his meaning. He wasn’t talking about the abstracts, themselves, being structured in any manner, but rather posited a physical structure which confirmed to those abstracts.

      Do you know what fictionalism is? Fictionalism denies that abstract object talk is true. You’re saying something literally false.

      I do know what fictionalism is. Particularly in the mathematical sense. Fictionalism denies that the concepts involved are directly associated with some actualized entity in the world. It does not deny that those fictions are deliberately modeled after the physical world. Quite the contrary, that is the main tenet of fictionalism.

      However, this is fairly irrelevant. The fact that Dr. Craig was discussing “fictions,” does not imply that he is necessarily discussing “fictionalism.” He’s simply saying that on the non-realist perspective, mathematical objects are invented concepts rather than extant entities, as on the realist view.

      and then argues that THERE’S GOOD REASON TO THINK C is the best explanation. Craig says: ” . . . the theist (whether he is a realist of anti-realist) 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. So there is an explanation available to the theist that is not available to the non-theist.”

      So, even according to your own quotation, Dr. Craig’s stated reason for claiming that the theist explanation is the best is the simple fact that a theist can offer an explanation, and that he is unaware of any other good explanations.

      Which of those books on inference to the best explanation would allow such a blatant Argument from Ignorance as a criterion?

      All math doesn’t apply to the world; but all the structure of the world is mathematical

      Not necessarily. You’ve already noted one philosophical position which vehemently denies that all the structure of the world is mathematical (mathematical fictionalism). There are other views, both realist and anti-realist, which can take either side of that question.

      Again, I do not see how the applicability of mathematics to the universe is supposed to be entirely inexplicable on non-theist views. And on the nominalist or fictionalist views, in particular, it is exceedingly simple to explain: mathematics describes the real world because it has been deliberately developed for that purpose.

      He can, nice. But he isn’t.

      I dunno. He seems to be fairly successful in his occupation, to my mind.

      He’d be more likely to, or people up his alley would be more likely to if they didn’t have to be met by rhetorically disparaging blog titles like “X doesn’t understand an entire subject matter.”

      I doubt it. Dr. Craig has engaged directly with people who have said far worse things about him than I have. I think it’s more likely that he would avoid interaction with me because I neither hold a PhD nor am I widely followed. Which is perfectly fine! The man can’t be expected to personally address everyone on the Internet who disagrees with him.

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