James Franklin Critiques Applicability of Math Argument

KEVIN HARRIS: We’ve been looking at some video responses to the Reasonable Faith animated videos – the Zangmeister videos. We are going to look at another one today. This is the animated video on the applicability of mathematics^[1] from Dr. James Franklin^[2] – a retired professor of mathematics at the University of New South Wales. He’s another one of those Australian smart guys! Are you familiar with Dr. Franklin?

DR. CRAIG: Yes, I am! In my work on divine aseity I wanted to explore various alternatives to Platonism about mathematical objects. One of the alternatives to Platonism would be to say that mathematical objects are not abstract objects as the Platonist thinks, but rather they are concrete objects. Now, if mathematical objects are concrete objects they could be either physical objects or they could be mental objects – thoughts in the mind either in human minds or in the Divine mind. This sort of conceptualism is probably the most plausible form of mathematical realism which takes these objects to be concrete, mental objects. But Franklin is one of the very few philosophers of mathematics who takes mathematical objects to be concrete, physical objects. He defends this in his book An Aristotelian Realist Philosophy of Mathematics which I read in connection with my study of divine aseity. It's a remarkable view. He thinks that mathematical objects are physical structures and entities in the physical world, and that they are not abstract. They are actually physical. That's his Aristotelian philosophy of mathematics.

KEVIN HARRIS: It is fascinating, and he's going to get into that just a little bit in this interview again from Jordan Hampton and his channel called The Analytic Christian. Let's go to the first clip.

DR. FRANKLIN: Let me explain why there's a serious ambiguity in Craig's argument. What exactly is the premise of his argument? Is it that mathematics is applicable to the world at all, or is it that the mathematical laws of physics are so precise? He does make both claims, but they're very different claims. Here is the first:

“Naturalists cannot provide a reasonable explanation for why mathematics applies to the physical world. It's just a happy coincidence.”

So he said, “Naturalists have no reasonable explanation of why mathematics applies to the physical world. It’s just a happy coincidence.” Here's the second argument,

“This answer still doesn't explain why the physical universe has such a stunningly elegant mathematical structure.”

So he said, “It hasn't explained – it doesn't explain – why the universe has such a stunningly elegant mathematical structure.” Most of the examples at the beginning of physics were about the stunningly elegant mathematical structure. Let's take the first of those – the fact that mathematics applies to the physical world at all. Well, that's not a coincidence, happy or otherwise, because the laws of mathematics are necessary. Mathematics must apply to the physical world, and in fact to any other worlds there might be. There's no escaping them. For example, if God creates circles in flat space, it must be that the distance around them is a little more than three times the distance across. Pi, the ratio of circumference to diameter in a circle, is necessarily 3.14159, etc. And God can't choose otherwise. He can't make it that circles have a ratio of circumference to diameter of say 5.3. It's impossible for God to create the world on a non-mathematical plan because there are no non-mathematical plans. Mathematics must apply to physical and non-physical reality. Therefore, they cannot be an argument to the existence of God from the applicability of mathematics as such.

KEVIN HARRIS: Bill, he says that you're a little ambiguous and then he says that math is necessary.

DR. CRAIG: Yes. I think that Franklin doesn't appreciate that the argument here from Eugene Wigner, from beginning to end, is about the elegant and incredibly accurate mathematics that plays a role in the laws of physics. So whether he wants to characterize this as a design argument (as he does later) or whether it's an argument from the applicability of mathematics, is a matter of indifference to me. It's quite clear from what Wigner is talking about that he's not talking about these elementary concepts of arithmetic and geometry like the circumference of a circle being equal to the diameter times pi. He's talking about the elegant and incredibly accurate sort of mathematics that plays a role in physics. Here's a quotation from Wigner's article showing this. He writes,

. . . whereas it is unquestionably true that the concepts of elementary mathematics and particularly elementary geometry were formulated to describe entities which are directly suggested by the actual world, the same does not seem to be true of the more advanced concepts, in particular the concepts which play such an important role in physics. . . . Most more advanced mathematical concepts, such as complex numbers, algebras, linear operators, . . . and this list could be continued almost indefinitely were so devised that they are apt subjects on which the mathematician can demonstrate his ingenuity and sense of formal beauty.^[3]

So Wigner is not trying to explain the applicability of elementary arithmetic or elementary geometry. This could be a matter of logical necessity, as Franklin says, but he wants to know why is it that the elegant equations of the mathematics of the laws of nature apply to the physical world? If I might just say one thing, even with respect to the elementary mathematical and geometrical concepts, Franklin just assumes that the world could not have been a chaos which exhibited no mathematical order, and that's not at all clear. Albert Einstein thought that the most amazing thing of all was that reality is not a chaos but it does exhibit this sort of mathematical structure. So that's not obvious that reality had to have this mathematical structure. But even if it did have to have some sort of structure corresponding to elementary mathematics and geometry, that doesn't go to explain the applicability of these very difficult and elegant sorts of mathematics that feature in the laws of physics.

KEVIN HARRIS: When he says that they're necessary, you pointed out that he means by that they are logically necessary.

DR. CRAIG: Yeah.

KEVIN HARRIS: Yet he holds that they are physical. They would be eternal? Necessary?

DR. CRAIG: This is a real problem for Franklin's own view, and I was surprised in his response that he didn't mention his own view because on his view you would have to say that these mathematical concepts are physically instantiated in the world. They are physical structures. The problem with that is that, again, when you get to the higher mathematical concepts that Wigner is talking about, not all of those are capable of being instantiated in physical reality. When you get to things like complex numbers or imaginary numbers or infinite dimensional Hilbert space which enables the formulation of quantum mechanics then these cannot be physically realized. One of the best books on this problem of the applicability of mathematics is by Mark Steiner who wrote a book called The Applicability of Mathematics as a Philosophical Problem. Steiner focuses particularly on those aspects of mathematics that are useful in science in describing the physical phenomena but which cannot be physically realized. In his own book, at the end of the day Franklin admits that once you get beyond the elementary concepts of arithmetic and geometry you can't regard these things as physical structures. So he adverts to a kind of semi-Platonism. He appeals to Platonism for the advanced concepts of mathematics, and to physical concrete structures for the simple concepts. And since Wigner's argument is from start to finish about the advanced concepts of mathematics, I don't think that Franklin's response is a good one.

KEVIN HARRIS: Let's see what he says next from the interview.

DR. FRANKLIN: Now let's turn to Craig's second argument which starts from the claim that the mathematical structure of the universe is stunningly elegant and thus is coincidental and needs a designer to explain it. Well, it's absolutely right that in physics there is stunningly elegant plans to be seen. So it's reasonable to ask if there's an argument. But if that is the argument then it's a form of design argument. It's not an argument from just the applicability of mathematics as such. It's a design argument so it's in the same family – there's design arguments from the complexity of life or from the fine-tuning of the physical constraint of the universe, physical constants of the universe.

KEVIN HARRIS: Category mistake? It's not applicability?

DR. CRAIG: As I say, it's a matter of indifference to me how you want to classify the argument. I can see why someone would say this is a species of a design argument just as the argument from fine-tuning is an argument for design. The fine-tuning argument is based upon the constants and quantities that appear in these mathematical laws of nature. But Wigner's argument is an argument from the formulation of those very laws of nature. Now, whether you want to call this applicability or design is a matter of indifference to me. It's a good argument for a transcendent designer.

KEVIN HARRIS: When I think of elegant, I think of Audrey Hepburn. But it's used here as a philosophical term. The elegance of mathematical models and formulas.

DR. CRAIG: I originally had said that we're talking here about complex mathematical concepts and formulas because if you've ever looked, for example, at Einstein's gravitational field equations for the general theory of relativity, the mathematics is so hard it's just daunting. It's mind-blowing. But Luke Barnes said, “But they're not complex. Actually, they're simple. One of the striking features of the laws of nature is their simplicity and their mathematical beauty. Use the word ‘elegant’ instead, not ‘complex.’” Well, for me as a layman, when I say “complex” I mean “difficult” or “hard” or “hard to understand.” But in a mathematical sense it would not be right to say that they're complex. You could say that they're remarkably simple, but nevertheless it's a very elegant and advanced mathematics as Wigner emphasized. In his article Wigner begins with very simple examples like Newton's second law of motion and then he advances to quantum mechanics and finally to quantum electrodynamics. It's an ascending hierarchy of independence of the physical world. More and more advanced mathematical ideas that just are not based in the empirical world.

KEVIN HARRIS: Here's the next clip from that interview with Dr. Franklin.

DR. FRANKLIN: Well, design arguments have always been taken seriously since the time of Socrates, and rightly so. However, as design arguments go, the one from the elegance of the mathematical laws of physics is not at the top end. It's rather a dubious one. The reason is that much of the apparent design, complexity, and precision might be the necessary consequence of simpler laws. Well, we don't know. We're getting at the edge of our knowledge in physics and mathematical physics out there and the Big Bang and constants, we quite don't really know where we are. So it would be unwise, to say the least, to rely on the argument from the elegance of mathematical laws to the existence of God.

KEVIN HARRIS: Hmm. He doesn’t think it's a good argument.

DR. CRAIG: I think that what he's suggesting is utterly implausible. This is the alternative of physical necessity for explaining the fine-tuning of the universe – that maybe the laws of nature are necessary in their formulation. And that seems to me to be just utterly implausible. Cosmologists deal all the time with the possibility of universes operating according to different laws of nature, and I think it's really plausible to think the laws of nature are in fact contingent rather than necessary. I think this is a rather desperate alternative for the detractor of design to take.

KEVIN HARRIS: In this final clip, Dr. Franklin mentions something that's been in the news a lot lately – artificial intelligence. Here's that segment.

DR. FRANKLIN: Although the applicability of mathematics doesn’t in itself lead to an argument from the existence of God, there is something surprising that does need some explanation when you look at our understanding of necessities of mathematics. There's still something in mathematics, meaning mathematics in human activity – the mental activity – that might give pause to a naturalist view of the universe. So it's not the applicability of mathematics itself, it's the ability of the human mind to understand mathematics. Animals don't. An artificial intelligence doesn't either. If you're conducting a dialogue with Google Translate or ChatGPT, it looks like it's going fine like a human for a while but then it does something ludicrous that reminds you that there's just nothing in there that understands. The text produced by rules is not the same kind of thing as an expression of understanding produced by a human. It's the actual understanding that's different in the human case. So there's not even a start of a plan on how you could make artificial intelligence or make anything mechanical acquire genuine understanding. Well, that's a problem for a strictly materialist atheist theory that says that humans are just material objects and compares human minds to a computer as if human brains or human minds are just, well, the same thing essentially as computers but just implemented in wetware.

KEVIN HARRIS: What do you think about that, Bill? Animals can't comprehend this. Artificial intelligence can't. But human beings can.

DR. CRAIG: Yeah. I love it. I think this is a great independent argument that one might use against naturalism. I was recently speaking at the Harvard Faculty Club, and at the table at which I was seated were Peter Kreeft from Boston College and then a specialist in artificial intelligence. The AI fellow was saying that if you do a so-called Turing test and you could put questions to some entity and that entity can answer your questions then you could infer that there is an intelligence there. And Peter Kreeft reacted so vehemently. He said, “I think that's a terrible argument!” He said, “There’s no one there!” I thought that just so encapsulated it. It's the same thing that Franklin says. There's no one there in these AI mechanisms. I would refer our viewers to my dialogue with Sir Roger Penrose on the Unbelievable podcast with Justin Brierley. Penrose is very constrained to explain what he calls three realms of reality. One is the realm of the mind – the human mind. One is the physical realm – the physical world. And the third realm is the mathematical realm – the realm of abstract mathematical objects. He said, “I cannot understand how to put these three realms of reality together.” Because you obviously can't explain the mathematical realm to the physical realm, and neither can you explain the mental realm to the physical realm, so he just couldn't understand how to put it together. And I said, “Sir Roger, what about theism?” If you have an omniscient mind which transcends the world, it would have in its knowledge all of the mathematical truths that there are and it would be the creator of the physical realm. So the mathematical and the physical realm would find their coalescence in the mind of this transcendent, omniscient being. He said, “I've just never thought of that before. Thank you for giving me something to think about.” He said this after the podcast was over. So I like Franklin's suggestion here very much, and I think it's worth exploring the way in which this points to a mind that transcends the physical realm.

KEVIN HARRIS: Tie it together for us. Where exactly do you think Franklin disagrees with you, or with the video?

DR. CRAIG: I think here we have a multi-faceted argument that is really fascinating. On the one hand, you've got the question of mathematics's applicability. If mathematics is just an a priori discipline that is pursued independent of the physical world and concerns abstract objects which are causally effete then why is it that the physical world is describable by these elegant mathematical structures? Moreover, in these elegant mathematical laws you find appearing certain constants and quantities which are inexplicably fine-tuned to allow for a life-permitting universe. Among those life forms in the universe are self-conscious minds which understand the mathematics that govern the physical universe. So all three of these facets, I think, combine to point to a transcendent, personal creator and designer of the universe who has created the universe on the mathematical plan that he had in mind and has created us in his image as finite minds able to relate to him as the ultimate uncreated mind.^[4]