Doctrine of God (Part 4): Anti-Realism Views

March 04, 2015

Anti-Realist Theories / Practical Application

In our lesson we have been talking about God’s attributes, and in particular his attribute of aseity or self-existence. We saw that the most serious challenge posed to God’s unique incommunicable attribute of aseity is Platonism, which is the view that there exists other uncreated, necessary, eternal objects besides God – things like mathematical objects, numbers, sets, functions, and so forth. Last week we began to review responses to the challenge of Platonism. [Dr. Craig makes reference to the diagram from last week – see previous transcript for that diagram] We have discussed, first of all, the realist alternatives to Platonism, which would take mathematical objects either to be abstract ideas that are created by God or else as concrete objects, namely thoughts in human minds or else, more plausibly, thoughts in the mind of God. These are all realist solutions to the problem posed by Platonism because these solutions agree with the Platonist that, in fact, mathematical objects exist – there really are such things.

But in addition to these realist solutions, you’ll see on the right hand side of the diagram a range of anti-realist solutions to the challenge of Platonism. These are united in denying that there are any such things as mathematical objects. Mathematical objects simply do not exist. There are no such things. These anti-realist solutions immediately remove the challenge posed by the existence of abstract objects to God’s being the sole ultimate reality because on anti-realism there just aren’t any such objects, and therefore God is the only uncreated, self-existent, necessary, eternal being.

Let’s just review briefly some of these anti-realist solutions. For example, Free Logic is a type of logic that has only been developed since about the 1970s. It is a very recent development in the study of logic. According to Free Logic we can use terms to refer to things even though those things do not exist. For example, I can refer to the hole in your shoe. Your shoe exists, but it is not as though in addition to the shoe there is something else, namely, the hole in your shoe. The hole isn’t a thing. It is not an object that exists. What you simply have is a shoe that is shaped in a certain way, but the hole isn’t something different. Or if I say, “There is a lack of compassion in the world.” I am not committing myself to things called “lacks.” There aren’t things out there in the world – objects - “lacks” when I say there is a lack of compassion in the world. Or if I say “Wednesday is the day of the faculty meeting” I am saying something true but I am not committing myself to the reality of Wednesdays. I am not saying that Wednesdays are objects that actually exist. Or if I say, “The pilot’s quick thinking averted the terrible accident on the Hudson river.” The accident never happened but I can still refer to it in a true sentence. So Free Logic is a logic that enables you to talk about and refer to things even though those things don’t exist. What the Free Logician can say is that mathematics sentences, like 2+2=4, are true even though the terms in those sentences don’t actually refer to anything. There is no such thing as 2+2 or 4 anymore than there is such a thing as the hole in your shirt or a lack or a Wednesday. That is the alternative of Free Logic.

Figuralism (the next on the list) is a different form of anti-realism. Figuralism points out that much of our language, a great deal of ordinary language, is figurative in nature.[1] It’s metaphorical in nature. If I say, “It is raining cats and dogs outside,” I’ve said something that is true but it doesn’t mean there are animals falling from the sky. This is a figure of speech for saying that it is raining hard outside. So it would be inept to take that statement literally. It is figuratively true that it is raining cats and dogs outside. Or if somebody is angry, I might say, “She has a bee in her bonnet.” That is true, but not in a literal way. That is a figure of speech. Similarly, the Figuralist will say that mathematical discourse is very plausibly interpreted as a sort of metaphorical or figurative discourse. It isn’t meant to be taken literally as referring to things like numbers. These are what one philosopher calls existential metaphors. They are figurative ways of speaking of things but there really aren’t such things in a literal sense. That would be Figuralism.

Neutralism is yet a third form of anti-realism. Neutralism agrees with Free Logic that we can use terms to refer to things that don’t exist. When we refer to things our statements are just neutral with respect to whether those things exist. So if I say, “The weather in Atlanta today is balmy” I am not committing myself to an object called “the weather” as though “the weather” is something that exists. Or if I say, “The view of the Jezreel Valley from atop Mount Carmel was gorgeous” I am not committing myself to an object “the view” of the Jezreel Valley. It is not as though there is an object that is in the world called “the view of the Jezreel Valley.” Or if I say, “The price of the tickets was ten dollars” I am not committing myself to the reality of objects called “prices.” In many, many different ways we use terms in ordinary language to talk about things without committing ourselves to the reality of those things.

Certainly sometimes we do mean to speak in a metaphysically committing way. If I say, “This table is made out of wood laminate,” there I am pretty clearly committing myself to the reality of the table. What will tip us off to whether or not a person is thinking that there is a real object will usually be personal effects – rhetorical statements maybe emphasizing like “it really does exist” or the context. But the Neutralist will agree with the Free Logician that we often use terms to talk about things without thinking there are objects that correspond to those. So he would agree with respect to mathematical objects that, when we say statements like 3 x 3 = 9, those terms are just neutral as to whether or not you are committed to the reality of mathematical objects.

The Neutralist goes farther, however, than the Free Logician because the Free Logician thinks that if you say “there is” something then you are committing yourself to the reality of that thing. The Neutralist would say that even expressions like “there is” are ontologically neutral. I can say, for example, “There are deep differences between Republicans and Democrats” without thinking that I’ve committed myself to objects in the world called “differences” and that some are “deep.” The expression “there is” and “there are” in English are very light in their ontological commitments. It will be, again, personal factors such as context, inflection of your voice, saying “there really is an abstract object” that will tip you off as to whether or not you mean to make an ontological commitment. So the Neutralist goes even further than the Free Logician. The Neutralist will say that really there isn’t anything in language that in virtue of its meaning commits you to saying there really are those objects that either correspond to the terms you use or are the things you say “there is” or “there are.”

Neutralism is a view, I’ll just say personally, to which I am very attracted.[2] It seems to me that this gives a very plausible account of ordinary language. When applied to mathematical discourse, it allows you to affirm the truth of mathematics but to simply say it is neutral in terms of its commitments to objects.

Fictionalism is a quite different form of anti-realism. The Fictionalist, like the Platonist, agrees that if you use terms to refer to something, or if you say “there is” or “there are” something, then you are committed to the reality of the things that you say “there are” or that you refer to. So the Fictionalist accepts those criteria for ontological commitment that the Neutralist and the Free Logician reject. But why is the Fictionalist then not a Platonist? Because Fictionalists think that those statements referring to or saying that “there is” or “there are” certain things are false. They are fictional. They are not true. So the Fictionalist will take the radical line that it is not true that 2+2=4. It is not true that 3 is greater than 1. It is not true that there is a prime number between 2 and 4. If you say that is crazy – those seem to be obvious truths, even necessary truths – the Fictionalist will remind you that on his view to say 2+2=4 is to make a radical metaphysical statement that there is an abstract object named “2+2” and there is an abstract object named “4” and that those two objects are the same object. And that is not at all obvious. So the Fictionalist will say if you accept these criteria for how we make ontological commitments then it is far from obvious that statements of elementary arithmetic are true. That actually turn out to be radical metaphysical assertions that we have no reason to think are true.

The Fictionalist will say these statements are true within the story of mathematics; within standard arithmetic they are true. In this sense they are like statements of fiction. It is not true that Sherlock Holmes lives at 221B Baker Street. There is no such place. There was no such person. But in the Arthur Conan Doyle stories, it is true that Sherlock Holmes lived at 221B Baker Street. So the Fictionalist will say statements of mathematics are true in the story of mathematics in the same way that some of these statements are true in the Conan Doyle stories of Sherlock Holmes. But other statements will be false. It is not true, for example, that Sherlock Holmes met Henri Fauveau in those stories. That would not be true in the story of Sherlock Holmes. Neither is it true in the story of standard mathematics that 2+3=4. That is Fictionalism.

Pretense Theory is another anti-realism that takes inspiration from theories of fiction. They work largely off of the brilliant pioneering work of a philosopher at the University of Michigan named Kendall Walton. Walton’s work on fiction holds that fiction is an extension of children’s games of make-believe. Walton notes that children invest enormous amounts of time and energy in games of make-believe. He says it would be very surprising if, when people reach adulthood, they just give this up all of a sudden and no longer make-believe. Walton says, in fact, we don’t give it up. This is what fiction and drama and film and literature and art is all about. These are, in effect, adult games of make-believe. He says what is crucial to fiction is not that the statements are false. A novel about the future like George Orwell’s 1984 could turn out to be true. It could all turn out to be true, but it is still fiction. Or the story of Hamlet might be true on some other planet somewhere in another galaxy in the universe for all we know, but Hamlet is still fiction. It is not the falsity of the story that makes something fictional. Rather, in Walton’s analysis, what makes something fictional is that it is prescribed to be imagined as true. We are to imagine that there is a Danish prince named Hamlet and that he did such-and-such.[3] Or we are to imagine that there was a detective living in London who had a colleague named John Watson who did tremendous exploits. Then within this fictional world of imagination the story can be unfolded. So what is essential to fiction, in Walton’s view, is this act of making believe or imagining something to be true. The statements are prescribed to be imagined as true. They may or may not actually be true. But in either case what is essential to fiction is the prescription to be imagined as true.

Apply this to mathematics. In mathematics, we are, in a sense, prescribed to imagine the axioms to be true. You are prescribed to imagine the elementary arithmetic axioms to be true. Then you can derive all of your theorems. Or you imagine the axioms of set theory to be true. Then the mathematician can derive all of his theorems. So the whole thing is a sort of species of make-believe. Far from being a crazy view of mathematics, this is a view of mathematics that many mathematicians themselves actually entertain. They would say that the mathematical axioms are postulates which you postulate and then you derive your deductions from them. But you are quite free as a mathematician to adopt a different set of postulates, a different set of axioms, and to explore that. So there is a wide variety of set theories that are on offer today. There is not simply a single set theory in mathematics. There is a range of set theories. These have different ontological commitments. Some commit you to sets. Some commit you to a different sort of objects called classes which are different from sets. So Pretense Theory will say that because you are merely pretending or imagining these things to be true, you are not committing yourself to the reality of these objects anymore than you are committing yourself to the reality of Sherlock Holmes in imagining that this fictional world is the case.

Neo-Meinongianism is one of the wildest anti-realisms. This stems from an Austrian philosopher Alexius Meinong who lived at the end of the 19th and into the early 20th century. Meinong was concerned to develop a theory of objects. He called his philosophy “Object Theory” – in German, Gegenstande Theorie. What Meinong maintained is that there are objects that do not exist. He says although it may sound paradoxical, there are things of which it is true that there are no such things. Unicorns. Centaurs. Fairies. The accident that was prevented. Holes. There are things which do not exist, Meinong would say. He develops this whole theory about these objects. On this view the Neo-Meinongian (that is to say, the modern follower of Meinong of which there are several in the world today) would say that mathematical objects are objects that do not exist. That would be one form of anti-realism.

Another one would be Modal Structuralism. This is a view that has been defended by Geoffrey Hellman, who is a philosopher of mathematics. On this view, mathematics is about structures. Without going into great detail, the idea here is that numbers are positions in structures. So the third position in a structure would be 3, the fourth position would be 4, and so forth. There aren’t objects called numbers. They are just positions in a sort of abstract structure. Some Structuralists in mathematics think that structures exist. This would be a kind of abstract object. A Platonist would say there are not numbers but there are structures. But the Modal Structuralist will say there could have been such structures. What he would say is that if there were numbers then 2+2 would equal 4[4]. Or if there were numbers the square root of 9 would be 3. So you can see he reinterprets mathematics in such a way that it doesn’t commit you to the reality of the objects that appear in those statements. These are simply paraphrased in such a way as to say that it is possible that there are such things, and if there were then they would have these properties. So Modal Structuralism is what I referred to last week. It is a kind of Paraphrastic strategy. That is to say you offer paraphrases of mathematical sentences that don’t involve you in commitments to these abstract objects.

Those are just some of the anti-realisms that are on offer today. There is a real potpourri of alternatives. There are others that aren’t even on this list. I want to just share these with you to give you an idea of the field of options that is open. Obviously, in this class, we haven’t and aren’t going to discuss any of these in detail, but I simply want to familiarize you with the range of options today lest someone think that the reality of mathematical objects poses an insuperable challenge to divine self-existence and aseity – to the idea that God is the sole ultimate reality. That is not, in fact, true. As you can see, there are a great number of options available to the Christian theist today which would not commit you to the reality of uncreated objects of any sort. Platonism is only one view – a tiny view – in the whole range of views about the reality of these objects. I think that these other views, many of them, are very plausible. In my book that I am going to be publishing on this, I’ll be defending a number of these views as plausible alternatives to Platonism. In order for Platonism to be a defeater so to speak of God’s unique self-existence, he would have to prove that Platonism is true and that all of these alternatives are false. I don’t think anybody believes there is a realistic prospect of doing that.


Student: A lot of these theories remind me a lot of the transcendental argument which I’ve kind of struggled with. The transcendental argument is essentially, as I understand it, an argument for the spiritual existence. It is saying you couldn’t have objects such as Love or Justice and explain that with a naturalistic view. It seems a lot of these theories would be defeaters to the transcendental argument which I find is an attractive argument yet I do not know if I can hold it. I was wondering what your thought was.

Dr. Craig: I’ve never heard what you just described called the transcendental argument. As I’ve heard that expression used, it usually means that in order to affirm rationality and logical thinking, there needs to be some ground of this in God rather than in the evolutionary process because the evolutionary process doesn’t aim at truth. It merely aims at survival. We could survive without having true beliefs. As long as those beliefs are conducive to survival, truth doesn’t need to come into the equation. On naturalism, the argument is that we have a defeater for thinking that our cognitive faculties are reliable. That is the way I hear and understand the transcendental argument. What you seem to be saying is that in order for there to be objects referred to by these abstract terms you need to have God maybe as the ground for those. I think that the Divine Conceptualist might well run an argument like that because he thinks these are thoughts in the mind of God. So I could see someone saying 2+2=4 requires the existence of numbers in order to be true. But what are numbers? Well, they must be thoughts in the mind of God, so this is an argument for God’s existence.[5] I would agree with what you said that anti-realism would undercut that argument. I don’t use that argument. I don’t think it is a good argument because I agree with the Neutralist on this. I don’t see any reason to think that the truth of 2+2=4 commits you to the reality of 2+2 or 4. I usually hear that called the conceptualist argument for God’s existence. I do think that the person who wants to run that argument is going to have to defeat that whole right hand side of the diagram. I think that would be really tough to do.

Student: What you are saying by the diagram is that the left side is referring to mathematical objects being mentally true but not metaphysically actual. Is that what you are saying?

Dr. Craig: The left hand side? No.

Student: The right side is defeating the left side in that sense?

Dr. Craig: The right side are the anti-realist views. These realist views think that there are mathematical objects. There are numbers, and they are either abstract (like the Platonist thinks) or they are concrete (either physical or mental). So the realist views think that there really are numbers.

Student: Oh. Yeah, that’s not right.

Dr. Craig: I don’t think so, but we are leaving that open because there are forms of realism like Absolute Creationism or Divine Conceptualism which are consistent with saying God is the only uncreated self-existent being and these other things are dependent upon God.

Student: And that wouldn’t be the same as saying there are such things as moral perfections like Good or Justice or things like that. How do we then come up with an argument to say that these things are real but they are way more real than numbers can be?

Dr. Craig: What you point out here is, whereas I’ve taken mathematical objects as my point of departure, you could put in other things like that. For example, moral values. Here, I would be a realist. I think that there are moral values. But I would not be a Platonist. I don’t think that there is such a thing as Justice or Rapacity or Greed as some sort of strange abstract object. I would say they are anchored in God who is a concrete object. This would fit very well with the view of God as the concrete paradigm and foundation of objective moral value.

Student: Would it be possible for the anti-realist ideas to backfire? Someone, for instance, asks if you believe that God is an uncreated object. On anti-realism, how can you say there is a God?

Dr. Craig: How do you think I would answer that question?

Student: Because God is uncreated or because God . . . I don’t know! That’s why I am asking.

Dr. Craig: I would run cosmological, teleological, moral, ontological arguments for God’s existence.

Student: Which you don’t have for numbers?

Dr. Craig: We’ve got arguments. We’ve got reasons to think that God is real. We are not just postulating God’s existence on the basis of language – that there are sentences like “God is good” or something like that, and therefore the term “God” must refer to an object, therefore God must exist. I agree that that kind of argument, I don’t think, works – to use this kind of linguistic argument. But I think you’ve got to have a cause of the origin of the universe, for example. That is not a linguistic argument.

Student: What keeps the Divine Conceptualist from saying Modal Structuralism is just one point God accepts us and then creates this reality?

Dr. Craig: That is what the Conceptualist would say. These views are, to a certain extent, incompatible with each other. The anti-realist says there are no mathematical objects. The Divine Conceptualist says yes there are, and they are thoughts in the mind of God. He would say God has the thought of these structures, and these structures are ideas in God’s mind. That would be a realist view of Structuralism. You can be a Structuralist and be a realist or you can be an anti-realist.[6] Either option is available to you. I’ve not tried to decide in our class here which of these options is correct. That would be far to ambitious a project. But just to let you know there are lots of different options on the table today. It would be very difficult to rule them out in such a way that Platonism is the only alternative left.

Student: I am struggling from all that you have. For the realist there are concrete objects or abstract objects. I understand how a physical object is concrete. How is a mental concrete object different from an abstract object?

Dr. Craig: I have to apologize somewhat for skimming the surface here because it leaves these unanswered questions. The question that you ask is a profound one. You are saying how do you differentiate between an abstract object and a concrete object. The most widespread and I think plausible answer to that question is that concrete objects are objects that have causal powers to do things. Whereas abstract objects are causally impotent. They are causally effete. The number 7 has no causal effect upon anything. But obviously a mind or thoughts in someone’s mind would have causal effects, both on God’s part and on our part. Thoughts would be concrete events in some person’s mind whereas mathematical objects as usually construed would be causally impotent and therefore abstract, not concrete. So concrete here doesn’t mean material. It means causally efficacious or capable of having causal influence.


What practical application does all of this have to our lives? Let me mention just two.

First of all, because God is the sole ultimate reality, God ought to be our ultimate concern in life. The theologian Paul Tillich actually defined God as the object of ultimate concern. Whatever is your object of ultimate concern is god for you. Since God is the sole ultimate reality, he is and ought to be our proper ultimate concern. To substitute anything else for God would be idolatry. If I were to ask for a show of hands in the class today, how many idolaters do we have in the class today there would probably be very few. Some have realized I think already the implications of what I am saying. If there is anything else in life that is more of concern to you than God, you are guilty of idolatry. If your ultimate concern is not knowing and serving God better then you are worshiping a lesser god. You are falling into idolatry. God’s aseity and ultimate reality is a powerful reminder to us of where our ultimate concern ought to be.

Second, God’s self-existence ought to exclude our selfishness. Another word for self-existence is independence. God is independent of everything else that exists. This is what man and Satan want, isn’t it? Independence. They want to go their own way; to challenge God’s self-existence by opposing to it their own independence. We want to oppose our selfhood to God’s “I am.” Selfishness, I think, can seem very natural until we reflect upon the being of God. But when we understand who God is and his self-existence then I think we can see how foolish it is, how insane it is, to oppose our selfhood to God’s self-existent being and to not treat him as our ultimate concern and to submit ourselves to him. Living for God, denying self in favor of God’s self-existence, I think makes good sense once we understand God’s self-existent nature.[7]



[1] 5:01

[2] 10:00

[3] 15:01

[4] 20:02

[5] 25:02

[6] 30:00

[7] Total Running Time: 35:50 (Copyright © 2015 William Lane Craig)