Questions About Numbers and Math

KEVIN HARRIS: This question is from Tanner in the United States,

Hi, Dr. Craig. I want to thank you for all that you have done for the intellectual case of Christianity and for strengthening the faith of many including myself. I always use your work to answer my questions, and I am very grateful for the abundance of resources there on the Reasonable Faith website. It is on the website that I have come to watch the videos on Leibniz’ contingency argument and the kalam cosmological argument, part 2, philosophical. In the Leibniz video, it is suggested that numbers exist necessarily as “it is impossible for them not to exist.” While in the kalam video, it is suggested that numbers don’t exist at all when the objection of actual infinity and numbers is brought up. It seems that for one argument in support of the existence of God we hold that numbers exist, and in another we hold that they don't exist. Wouldn't the skeptic be able to say that there's a contradiction and therefore be able to cancel one of the arguments? I'm aware that you support the view that numbers don't exist as acknowledging the existence of numbers supports Platonism and suggest the existence of an actual infinity, but how can we justify saying the opposite – that numbers do exist in Leibniz's contingency argument? I'm sure I'm misunderstanding something, but could you explain to me what and perhaps tell me why numbers don't exist when they correspond so well with reality?

DR. CRAIG: I do think that Tanner has a misunderstanding. In the Leibnizian cosmological argument I do not claim that numbers exist. In fact, I think there are no such things as numbers. What I'm simply saying is that, for Platonists, numbers would be examples of something that exists by a necessity of its own nature. I'm trying to help viewers understand what it is to be a metaphysically necessary being. Now, in fact, I think that God is the only metaphysically necessary being, but to try to make this concept understandable I appeal to numbers as an example of what many people (namely, Platonists) think is a necessarily existing being. So the example is purely hypothetical. It's simply to help to get people to understand this distinction between something that exists necessarily and something that exists contingently. If numbers exist, they exist necessarily, not contingently. Now, in fact, I don't think that numbers exist, and in the kalam video one argues that even if they did they could not be the cause of the origin of the universe because numbers do not have any causal power. They are causally effete and therefore cannot explain the origin of the universe. So really in both cases one is talking purely hypothetically in speaking of numbers as necessary objects.

KEVIN HARRIS:

My question is concerning Dr. Craig’s position on anti-realism, numbers, and objective morality. Since morality does exist because it is grounded in God’s nature, could that also be the case for numbers and mathematics. I understand that, for example, it's strange that propositions and properties exist as in the conceptualist view, but mathematics do seem to have some of the characteristics of God such as perfection in the sense of consistency, in being exact, and infinity in a way, especially if they are in the mind of God. So if we separate mathematics or numbers from properties and propositions would you say that probably they exist in the mind or nature of God the same way moral values and duties exist? Secondly, if that's the case, could the conceptualist argument work only with mathematics or numbers? Thank you very much for your time. Francisco.

DR. CRAIG: Let's just address these in reverse order. Certainly if you do think that mathematical entities are somehow grounded in God's nature then you can run a conceptualist argument for God's existence based upon the reality of mathematical objects and the objectivity of mathematical truth. But I'm not convinced that a conceptualist view of mathematics is, in fact, the best. It's important to understand that I don't think that moral values are grounded or exist in the mind of God. Rather, I think that they are grounded in God himself as a concrete object. God is necessarily kind, loving, fair, generous, and so forth. So objective moral values are rooted in this concrete object which is God. Francisco seems to be willing to say that properties and propositions should not be grounded in God as the conceptualists think, but he wants to make room for mathematical entities to be grounded in God. That seems to me to be rather ad hoc. If you can get rid of abstract objects like propositions and properties then I don't see any reason to think that mathematical objects like numbers and sets are any more real than are properties and propositions and therefore should be put in the mind of God. He says that mathematics has the property of infinity. Well, so do propositions and properties. There would be an infinite number of propositions if propositions exist. And the infinity of mathematics is a quantitative concept that I think is inapplicable to God. God is not a collection of an actually infinite number of definite and discrete particulars. When theologians talk of God as infinite they mean it in a qualitative sense not a quantitative sense. God is morally perfect, eternal, omnipotent, omnipresent, and so forth. Moreover, and finally, I think there are certain difficulties the conceptualist faces in identifying mathematical objects with God's thoughts. For example, could sets be thoughts in the mind of God? Alvin Plantinga has suggested that maybe sets are God's collectings – his mental collectings of things into groups. Well, if you think about that, how is it then if sets are in the mind of God as thoughts that I could have access to sets? Suppose I collect mentally the objects on my desk into a unity. That would not be the set of all objects on my desk on this view. Why? Well, because that set exists in the mind of God as his collectings, and so I wouldn't have access to that set even though I talk about the set of all objects on my desk. But how could they not be the same? How could they be diverse? Because in set theory it's membership that determines sets. Sets which have the same members are ipso facto identical. So how could the set of objects on my desk that I mentally collect be different than the set of objects that God collects in his mind? So I think that it's just not very helpful to think of mathematical objects as thoughts in the mind of God.

KEVIN HARRIS: OK. I don't know if you've seen this article.^[1] I got it from our friend Tim Stratton – a related topic. This professor, Laurie Rubel of Brooklyn College, says that basic math is culturally problematic. She says math is racist.

DR. CRAIG: Oh, gosh.

KEVIN HARRIS: Two plus two equals four is white supremacist patriarchy.

DR. CRAIG: Oh, Lord, help us.

KEVIN HARRIS: This is just getting insane. It's just a series of tweets. She says, “the idea that math (or data) is culturally neutral or in any way objective is a MYTH. i'm ready to move on with that understanding. who's coming with me?”

DR. CRAIG: Well, not very many people because that would have to be a number of people – right? – that are going along with her; more than one! Wait a minute, Laurie, I thought you didn't believe in mathematics; that two is greater than one?

KEVIN HARRIS: “How many of you are coming with me,” in other words. Many? Wait a minute. [laughter] Well, I don't mean to put her down. There seems to be just kind of a poisoning of the mind.

DR. CRAIG: It’s insanity of political correctness and post-modernism.

KEVIN HARRIS: OK. Let's go to the next question then.

Hello, Dr. Craig. I've noticed that there is a difference between sounds that have meanings (words) and just sounds. For a sound to have meaning there must be something that we causally interact with and want to then communicate. With numbers, to me it is painfully obvious that a number is just a word or a symbol that denotes how many of something concrete exists. Positive cardinal numbers talk about how many finite things there are and positive rationals talk about the infinite part-whole relationship. Negative numbers and imaginary numbers are then simply symbols invented in a game when the rules are changed. Why then do we even entertain the notion of an abstract object as being coherent? Defining something that is not in space and not in time and not causally potent is just halfway then to describing not anything. You have previously mentioned that using numbers as nouns commit us to their existence. Why? For example, “The number of people who died in the plane crash is 115” can simply be rephrased as follows: “The word symbol that represents how many people died in the plane crash is 115.” This commits us to the existence of a word or a symbol, not a non-spatial, non-temporal, and acausal object.

This is from Tariq in Oman.

DR. CRAIG: I agree with Tariq that we shouldn't be realists about numbers. I don't think that numbers do exist. But I don't think they can be dismissed quite as easily as Tariq imagines. I think he's incorrect when he says that for a sound to have meaning there must be something that we causally interact with. That seems plainly wrong. For example, the sound “the equator” clearly has meaning in the English language, but the equator is not something that we causally interact with. It is a geometrical line that encircles the Earth. It's not something that is causally potent. Or we talk about things like unicorns or fairies or leprechauns. These do not causally interact with us, and yet these sounds or words have meaning. So I don't think we can simply say that because numbers don't causally interact with us that the sounds for numbers (like 1, 2, 3, 4, and so forth) are meaningless or do not refer to anything. Rather, the key point is the one that he mentions. How do we ontologically commit ourselves to the reality of certain objects? There is a very widespread view that you commit yourself to the reality of certain objects by using singular terms for those objects in sentences that you regard to be true. So, for example, if I say, “The number of people killed in the crash is 116,” they believe that I've committed myself to the number 116 because I have used that singular term in a sentence that I believe to be true. And here I agree with Tariq. I don't believe in that criterion of ontological commitment. Often we can paraphrase away those commitments by saying something like this: “116 people were killed in the crash.” And there I used the word adjectivally rather than substantively as a noun. So according to this criterion of ontological commitment I'm not committed to the reality of the number 116. But in many cases I think we may not be able to paraphrase away successfully these commitments without meaning loss, but fortunately I just don't think the anti-realist is under any obligation to paraphrase away these singular terms referring to abstract objects. Rather, I think we should simply reject this criterion of ontological commitment as highly implausible and outrageously inflationary. It would commit us to all sorts of fantastic realities if we take it on board. Therefore I think we should be skeptical of this criterion.

KEVIN HARRIS: This came in from the Philippines.

Dr. Craig argues for the existence of God through the unreasonable applicability of mathematics to the physical world. He often cites examples of scientists precisely discovering a certain thing through the use of complex mathematical calculations. However, it's still hard for me to fully grasp his explanation as to why mathematics is unreasonable to be applied to the physical world or why the absence of mathematics is possible in the physical world. Wouldn't it be better explained by considering mathematical laws as mere necessity to the universe, and therefore it's actually impossible for mathematical laws not to apply to the universe? Take this as an example. If you take one thing and another thing together, there will be two things, and it is just impossible for it to be three or four (which is similar to 1 + 1 = 2). I don't get why 1 + 1 equals 2 is unreasonable to apply to the real world because it seems to me that its applicability is perfectly reasonable and logically necessary. Can I get clarifications of this argument because it's really hard for me to grasp. Irwin, Philippines.

DR. CRAIG: Let me first clarify the use of the word “unreasonable.” Eugene Wigner in putting forward this argument spoke of the unreasonable effectiveness of mathematics in the physical sciences.^[2] The reason that it appeared to Wigner to be unreasonable was because Wigner was a naturalist. He wasn't a theist. As a result he had no explanation for the applicability of mathematics to the physical phenomena. So it was as a naturalist that Wigner said that the applicability or effectiveness of mathematics is unreasonable. Now, as a theist, however, I think we do have a good explanation of the applicability of mathematics. Wigner in his article said that the effectiveness of mathematics is a miracle for which we have no good explanation. But he never considered whether God might not in fact be a good explanation for the applicability of mathematics. If God has designed the world to operate according to the mathematical laws that he had in mind then it is not unreasonable for mathematics to be applicable to the physical phenomena. So rather than the word “unreasonable,” I think it would be better to use the word “surprising.” It is surprising that mathematics would be so effective in describing the physical phenomena. Why is that? Well, it is because, first of all, mathematics is pursued for aesthetic reasons or a priori reasons. It is not pursued for its scientific utility but pursued quite apart from its scientific usefulness. Yet over and over again scientists have found that these mathematical concepts are applicable. Secondly, mathematical objects (even if they exist) are abstract objects and therefore causally effete. They have no causal powers and so have no ability whatsoever to influence the physical world. Therefore, given the a priori nature of mathematics and the causal inefficacy of mathematical objects, it is surprising that the laws of nature would be characterized by these elegant mathematical formulations. Now, Irwin is quite right that you could maintain that the truths of elementary arithmetic like 1 + 1 equals 2 or elementary truths of geometry are logically necessary. But when philosophers and physicists puzzle about the applicability of mathematics, they're not talking about elementary arithmetic or geometry. Wigner gives example after example of difficult elegant mathematics like complex numbers or infinite dimensional Hilbert spaces or the use of matrices in Heisenberg's quantum mechanics. These are not logically necessary. The laws of nature are not logically necessary, otherwise we could pursue physics a priori and mathematically rather than experimentally. So the universe could have been characterized by different laws of nature, or perhaps only by these sort of elementary arithmetic truths that Irwin mentions. But it is puzzling as to why the universe exhibits this complex mathematical describability that is not something at all that you would have expected. So that is the basic puzzle that has troubled not only Wigner but many physicists and philosophers since he first spoke.

DR. CRAIG: Hello! This is William Lane Craig. I'm really excited about our spring campaign for strategic partners of Reasonable Faith. We are offering as a free premium two books, Assessing the New Testament Evidence for the Historicity of the Resurrection of Jesus and The Historical Argument for the Resurrection of Jesus, both of which were previously available only in editions that cost literally several hundred dollars apiece. And now, because these books are being reprinted, we're able to offer them free to you for anyone offering a sustaining campaign gift of $75 monthly on an ongoing basis. If you're unable to give that much, we have many other fine premiums at lower levels like $50 a month, $30 a month, but this top award is so extraordinary, so unprecedented given the previously unaffordable and exorbitant price of these books that I really do hope you'll take advantage of it. This is my scholarly work done at the University of Munich on the historicity of Jesus’ resurrection, and I think it will be a real benefit to you if you can get a hold of it. So I hope you'll participate in this year’s spring campaign.^[3]