Doctrine of God (Part 3)

February 25, 2015

The Problem of Platonism

We are studying the attributes of God. The first of God’s infinite attributes that we’ve turned to examine is God’s aseity or self-existence. We saw last time that a robust understanding of this attribute implies not simply that God exists independently of everything else (which would in itself be remarkable), but even more fundamentally that God exists by a necessity of his own nature. So if it is even possible that God exists then it follows that God necessarily exists. Therefore, God’s aseity entails, or implies, two further divine attributes, namely God’s eternality (that God is permanent; he never came into being and will never go out of being), and, second, God’s necessity (that he is a being whose non-existence is impossible; a necessary being.)

In contemporary philosophy, this is usually expressed by saying that God exists in all possible worlds. But that seemed to be a problematic concept to some in the class last week. It shouldn’t be. This is meant to be simply a heuristic device, not a piece of serious ontology. Thinking of possible worlds is simply a way of saying that if something is possible then that means there is a possible world in which that thing exists. If something is necessary then it exists in all possible worlds.

But I’ve been reading a book lately by the philosopher Bob Hale entitled Necessary Beings. Hale is a secular philosopher; very brilliant. This is a study of necessity and necessary beings. The way Hale explicates necessity can perhaps be helpful to those who don’t find the talk of possible worlds helpful. Hale says that something is absolutely necessary if it would be the case no matter what else were the case. No matter what else might be the case, if something would still be the case then that thing is absolutely necessary. So he says you can fill in this schema: “If blank were the case then blank.” You can fill in what you think to be a necessary truth. For example, “If blank were the case then 2+2=4.” If 2+2 would equal 4, no matter what you put in this blank then 2+2=4 is absolutely necessary.

What I am suggesting is that the proposition “God exists” fills this blank. No matter what you put in the other blank, this statement would still be true. If it were the case that the world did not exist then God exists. If it were the case that there were no people then it would be the case that God exists. The proposition that God exists is absolutely necessary in the sense that no matter what else might be the case it would be the case that God exists. This is the notion of God as a necessary being.

This notion of God as a self-existent being and the source of all reality outside of himself faces a very significant challenge from a philosophy called Platonism. Platonism holds that there are objects that are equally uncreated and eternal and necessary. So God is not the sole ultimate reality. In fact, he is just one of an infinite number of uncreated, eternal, necessary beings. The paradigm example of the objects that Platonists are talking about would be mathematical entities or mathematical objects like numbers and sets and functions and so forth. The sort of things that mathematicians talk about.

This raises the very interesting question: do numbers really exist?[1] What do you think? Do you think that numbers really exist? Let’s be sure that we understand the question. We all recognize that numerals exist. For example, this is the numeral two: “2.” But there are many different kinds of numerals. For example, here is the Roman numeral for two: “II.” They both represent the same quantity. So we are not asking: are there numerals? Obviously, there are numerals. We are asking: do numbers themselves exist? I remember coming up from my office when I first began to study this and asking Jan, “What do you think, honey? Do you think the number 2 exists?” We would discuss it over lunch as to whether or not there was such a thing such as the number 2.

Platonists say yes. In addition to these numerals, or these marks on the whiteboard, there is such a thing as the number 2. So if I have two apples on the table, not only are there the two apples, but there is also the number 2. So there is really three things. Well, there are actually an infinite number of things because there is 1, and 1+1, and 2+1, and so forth. But you get the idea. There is not just concrete objects like chairs and apples and people and planets. There are these abstract objects like numbers. These objects are thought by the Platonist to exist just as robustly as concrete objects. Numbers on this view are just like automobiles, only eternal, necessary, and uncreated. But they exist just as robustly as automobiles do.

So the question is: do these sorts of objects really exist? If they do, they would typically be thought to be uncreated, eternal, necessary things and not things that are created by God. So this would compromise God’s role as the sole ultimate reality. It would not be true, as John 1:3 says, that through him all things came into being and that God is the source of all being.

Let’s take a look at a PowerPoint of alternatives discussing this subject. Don’t be overwhelmed by this PowerPoint. We will pick it apart piece by piece so that you can appreciate what it says.

Notice we are taking mathematical objects as our point of departure. We could have picked other kinds of abstract objects like propositions, possible worlds, properties, and so forth. But mathematical objects supply the clearest example of what we are talking about – things like numbers. Notice there are three positions with respect to the existence of numbers. There is realism which says that these things exist; there really are such things. On the other hand, there is anti-realism which denies that these things actually exist. Then in the middle is arealism which says this is a meaningless question. There just is no fact of the matter about whether they exist or they don’t exist. This is just meaningless. There are some arealists today.

Taking arealism first. An example of an arealist position would be so-called Conventionalism. This was a philosophy that was popular during the 1930s and 40s. It was based upon the verification principle of meaning. According to that principle, any statement that could not be verified through the five senses was a meaningless statement. It is a sort of scientism that attempts to dismiss vast tracts of human languages as cognitively empty because these statements can’t be empirically verified. Sentences like ethical statements or mathematical statements can’t be empirically verified. These are about abstract objects if they are about something at all. Therefore these sorts of metaphysical questions were regarded as meaningless. It is just a convention that we adopt in order to make science work and get along in society, but there isn’t really any truth or falsity about whether or not the number 2 exists. It is just a convention which is arbitrarily adopted or rejected. That philosophy was prevalent during the mid-20th century.[2] I have to say with the demise of the verification principle this is not as widespread today because that principle of meaning is both too far-reaching (it would dismiss vast reaches of human discourse and language as meaningless), and it also tends to be self-defeating and self-refuting. But there are some arealists who are around today.

Let’s take on the other hand the view of realism. Realism with respect to mathematical objects can be of two types. First, realism could hold that these are abstract objects as a Platonist believes, or there are realists who think that mathematical objects are, in fact, concrete objects.

Let’s take the abstract alternative first – that these are abstract objects. They could be regarded as uncreated. That is the Platonist view. This is the classical Platonist perspective that there are numbers, they are abstract objects, and they are uncreated. That is Platonism. On the contemporary scene, some Christian philosophers have attempted to solve the problem posed to divine aseity by the existence of numbers by adopting a sort of modified Platonism according to which numbers exist all right as abstract objects but these, too, are created by God. He has not only created all of the concrete objects in the world, but God has created all of the numbers. This will force you to modify your view of creation somewhat because in this case these numbers exist eternally and necessarily. So that means that God has been creating from eternity and that there is no possible world in which God alone exists. Creation becomes necessary on this view. That, I think, should give us theological pause. It does require you to modify in some significant ways your view of creation. But there are some Christian philosophers today who would defend Absolute Creationism.

One of the most serious objections to Absolute Creationism is called the bootstrapping objection. That is to say that it involves a vicious circularity. The easiest way to see this is by considering properties. The Platonist thinks that properties are also abstract objects like numbers, and that these exist necessarily and eternally. So consider God on Absolute Creationism having to create properties. Suppose he wants to create the property “being powerful.” He would already have to be powerful in order to create the property of being powerful. So he would already have to have the property in order to create it, which is viciously circular. That is called the bootstrapping objection because it is sort of trying to pull yourself up by your own bootstraps. In order to create the property of being powerful God would already have to have the property of being powerful. You could run a similar paradox with numbers. In order for God to create the number 1, 1 is the number of gods that there would need to be. There would need to be one God in order for God to create the number 1. So, again, you have a kind of vicious circularity or bootstrapping problem. This has caused many contemporary Christian philosophers to have serious reservations about Absolute Creationism. This is not an alternative that has been widely defended today. I think it is largely because of this bootstrapping objection that tends to afflict Absolute Creationism.


Student: My question is very simple. How can the number 2 exist independent of a finite universe? It would be meaningless.

Dr. Craig: That is what a Conventionalist says, right?[3] That is adopting arealism. It is meaningless. But I don’t think that that is difficult. Why would there need to be spatio-temporal objects in order for the number 2 to exist? Even if there were no universe, wouldn’t it still be true that 2+1=3? Even more basically, what about this: that 1=1 or 1>0. Surely these sorts of elementary truths of arithmetic are true whether or not anything physical exists.

Student: It would seem to me that what you are doing is coming up with a language to define an infinite universe.

Dr. Craig: That depends on what you mean by “the universe.” For the Absolute Creationist, he does think that these numbers and properties and things are part of creation. So he would say there are sort of two divisions in creation. There is the concrete objects which include things like material objects, souls, angels. These are all part of the concrete world. But then he would say there is another division of creation that we don’t usually talk about, and that would be this abstract realm of numbers and properties and possible worlds and propositions. So if you use the word “universe” very broadly to include everything that is created, yes, this view would say that there is necessarily eternally a created “universe” of abstract objects. If you don’t use it in that broad sense and you restrict the universe to the realm of spatio-temporal objects then, no, the Platonist would say that these abstract entities exist independently of concrete things. They exist a se. They are like God in that respect.

Student: The number 2 only has significance as it corresponds to temporal objects. The concept – the abstraction – of the number 2 or 1 or 1>0 has no meaning apart from a correspondence to physical objects. That is where I see the problem with this. They derive from a number of physical objects.

Dr. Craig: So on your view, in a world in which God didn’t create any physical objects, wouldn’t there still be three members of the Trinity?

Student: Certainly. But that is part of self-defining who God is. We ascribe a value of 3 to the Trinity because we can observe a concept of threeness or twoness or whatever. But it doesn’t have any value if there is no entity.

Dr. Craig: In this case, there is an entity. There is God. There is one God. And there are three members of the Trinity. So you’ve already got arithmetic going right there even in the absence of any physical objects because you’ve got three and you’ve got one and then you’ve got arithmetic operations like 3+1.

Student: But these are abstractions and beings that are tying a value to this in time and space. If you have a being that is not tied to time and space this is meaningless.

Dr. Craig: I’ll just say again one more time, if God is timeless, wouldn’t there still be one God? Wouldn’t there still be three persons in the Trinity? It seems to me that these arithmetic truths don’t have anything to do with temporality or time.

Student: It is true, but we ascribe the value of oneness and threeness. It is for us that are tied to time and space. This is why we have mathematics to use in the real world. It corresponds to things that happen in the real world. It is an expression of science.

Dr. Craig: You do sound sort of like a Conventionalist, I have to say. You are sort of saying there is no objective mind-independent truth about the number of persons in the Trinity. This is just something we use. I am trying to think where on this chart do you fit? Where would he go? [laughter]

Student: I was confused when you were talking about apples. If you had three apples and then you have the number 3 apples. But then you said there was a fourth thing which is this value that you have. What exactly? Then you said you have that value but then you have more values, an infinite number of values. What did you mean by that?[4]

Dr. Craig: I was referring to the numbers. If you have three apples on the table, are there only three things there? Well, the Platonist would say no because he would say there is also the number 3 which is the number of the apples. So there are three apples and there is another thing – the number 3. Then, as I said, once you get the number 3 you get all the other numbers as well. You get an infinitude of numbers just in virtue of there being some objects. The difficulty here is maybe grasping what the Platonist believes. Remember the Platonist thinks that these are real. That these numbers really exist. These are metaphysical realities that are just as real as people and planets and electrons and so forth.

Student: It does seem like numbers are a way of describing reality to me, initially. If you took away one apple then you have one apple, two apples, then the number two. What if you took away all the apples? Does the number 0 still exist there in its place?

Dr. Craig: Well, I think that the Platonist would say so. He would say 0 is the number of apples on the table. That gets into a real interesting question.

Student: I am wondering how you would handle Quine’s objection that mathematics is at least quasi-empirical. He believed that in a world where if you had an apple in the one hand, an apple in the other hand, and put them together, a third apple appears. In that world, 1+1=3. There is at least some empirical element of mathematics, so it is not metaphysically necessary.

Dr. Craig: I don’t think that is his view myself. Quine, who was a naturalist philosopher, felt forced to adopt Platonism about sets at least. So this naturalist believed that there are these abstract entities because they are referred to in our scientific theories. In our scientific theories we have reference to things like numbers and functions and so forth. So they must actually exist. He was a Conventionalist about necessity. Maybe that is what you are thinking of.

Student: Maybe. I’ve read that he believed mathematics to be at least somewhat empirical. There were at least some empirical elements that grounded . . .

Dr. Craig: The only thing I can think of that would connect in that way would be that he didn’t think that necessity and possibility were objective. These were just conventions, as I described a moment ago.

Student: Do you have a response to the Formalist who says mathematics isn’t metaphysically necessary; it is grounded in physical reality.

Dr. Craig: That is later on in the chart as you will see. Let’s put that aside.

Student: Backing up one step, is it really critical that God is the only uncreated and necessary being? You mentioned John 1:3. And it does seem like it is a problem there when it says, “All things came into being through him.” But then it qualifies and says, “Apart from him nothing came into being that has come into being.”

Dr. Craig: Right. The second clause of John 1:3 is weaker than the first clause. The first clause is a universal statement: “All things came into being through him.” You are quite right. If you punctuate the text the way you read it then it would be followed by a weaker clause that isn’t the same as the first clause. There it simply says, “Not one thing that has come into being came into being without him.” One of the interesting things is that the punctuation of that verse is very uncertain. Many scholars think that the punctuation should be “All things came into being through him and without him not one thing came into being.” Then the next verse starts, “What has come into being through him is life and the life . . .” So the question of the punctuation of this verse actually even comes into play. If you look in your Greek New Testament they will have a footnote about the uncertainty of how to punctuate the verse which is amazing because the punctuation isn’t in the original Greek.[5] Why are they having a footnote about how English translators punctuate this sentence when it has absolutely nothing to do with the Greek text? I’ve asked some Greek scholars about that, and they said this is really extraordinary that they would have this sort of comment. I think you can show many other passages that we did review last week where it says, “For from him, and to him, and through him, are all things.” So John 1:3 is just a piece, I think, of a broader textual testimony to God’s being the unique uncreated being.

But you are certainly correct in saying some Christian philosophers are just willing to bite the bullet and say, yeah, there are things that are uncreated by God and co-eternal with him, and necessary, and independent of him. They just don’t see any problem with it. I have difficulty understanding how such a view could be reconciled with the Jewish concept of God which seems to me to be clearly that God is the source of everything outside of himself. God is not to be praised and worshiped because he has created this little small part of reality, namely the concrete realm, but he is to be praised and worshiped because he is the creator of everything that exists other than himself. That certainly is something that is part of the debate. Some philosophers would respond by just biting the bullet and saying, yeah, there are things independent of God and co-necessary, and co-eternal. But I am not willing to go that route myself.


You see next to abstract objects there is a kind of realism that says that these things exist as concrete objects. These could be two types of concrete objects. They could either be physical objects or they could be mental objects. That is to say, thoughts in somebody’s mind.

Physical objects. One view or alternative that takes this view would be Formalism which says that mathematics is basically scratch marks on paper. There is no significance beyond that. Mathematical entities just are these marks on paper which are manipulated by mathematicians in accordance with certain rules, and that is all there is to it. There is not many people that find that point of view persuasive today because it certainly seems that the number 2 isn’t to be identified with the mark on your piece of paper or the mark on my piece of paper. When we say 2+2=4 we are talking about a general truth, not some specific mark that has been made on a piece of paper. It is difficult to see how this view would be consistent with the necessity of mathematical truth.

There is the alternative of taking them as mental objects – thoughts in somebody’s mind. This might either be a human mind or God’s mind. The view that mathematical objects are just thoughts in people’s minds is called Psychologism. This would say you have ideas of the number 2 or of 2+2=4 and that is what these mathematical objects are. They are just ideas in people’s minds. That view, again, is not very widely adopted today because, again, of the inner-subjectivity of mathematics. If Kevin has the idea of 2+2 and 2+2 is an idea in Kevin’s mind, then what is Stephanie thinking of when she thinks 2+2? The idea or thought that is in Kevin’s mind isn’t in her mind. Different people have different thoughts. So how could these mathematical objects just be your thoughts? Moreover, there are infinite numbers of mathematical objects and infinite mathematical truths. There aren’t enough people to have all those thoughts. So you can’t ground them in human minds. Moreover, human beings aren’t necessary. They only have existed for a period of time on this planet. Are we to think then that these mathematical objects haven’t always existed or that it hasn’t always been true that 2+2=4? These are the sorts of problems that attend Psychologism that has made it unpopular today.[6]

More Christian philosophers have chosen to adopt Divine Conceptualism. This is historically the mainstream Christian position from Origen and St. Augustine, through Thomas Aquinas, through William Ockham, on into the Late Middle Ages. The standard Christian view has been that what Plato thought were these abstract entities are really thoughts in the mind of God. So the church fathers moved the realm of Platonic ideas into the mind of God and made them God’s thoughts. This is immune to the sort of objections that Psychologism falls prey to because in this case, for example, the number 2 is uniquely that object that God is thinking when he thinks 2. That is the number 2. Because God is eternal and necessary, he can be the ground of necessary mathematical truths. Because he is infinite and omniscient he can ground an infinite number of mathematical truths and have an infinite number of mathematical objects as objects of his thought.

So Divine Conceptualism is an alternative that finds quite a few defenders on the contemporary scene. In this way one would avoid having entities outside God as it were – entities apart from God which would be numbers and other mathematical objects. They don’t really exist. What really exists will be God and his thoughts.


Student: Can we simply see numbers as adjectives? So adjectives can come in two ways. Man sees nouns and describes them with adjectives. But God actually has this conceptualism – the adjective in him – then he creates nouns. So it is a different perspective.

Dr. Craig: All right. I am simplifying. I am skating over the surface here. This question that you just asked is very penetrating. So it requires me to say a little bit more. You are quite right. The adjectival use of numbers isn’t committing to objects. So if I say, “There are 3 members of the Trinity” that doesn’t commit you to the number 3. In order to commit you to numbers you have to use the word as a noun, as you said. “3 is the number of the members of the Trinity.” Or if I say “3 men enter the tavern” there is no commitment there to the number 3. It is just an adjective. But if I say “3 is the number of men who entered the tavern” then I have committed myself to the reality of the number 3 because I used it as a substantive; as a noun that refers to some object out there. The difficulty is that those two sentences seem to be synonymous, right? The number of men who entered the tavern is 3 – 3 men entered the tavern. Does that sentence convey the metaphysical commitments that the Platonist thinks, or that the anti-realist thinks? This gets into a huge debate over whether or not we can reduce all of our commitments by expressing them adjectivally. I would say that there is a pretty general consensus that that cannot be done. Do you see anti-realism on the chart? One of the strategies for anti-realism – it is not on this one – one of the alternatives that is not listed would be what we could call Paraphrastic strategies where you would paraphrase away your commitment to numbers by using adjectives instead of nouns.[7] I think, as I said, it is fairly widely acknowledged that a Paraphrastic strategy is going to face huge obstacles. So many Platonists would say this isn’t going to avoid the problem. But there are others who have offered these kinds of strategies. I can think of a couple. For example, there is a philosopher at Berkeley called Charles Chihara. There is another philosopher named Jeffery Hellman. Both of them have offered what are essentially ways of paraphrasing arithmetic and numbers so that you get rid of them and you aren’t committed to them. So, yes, your question is very good and would represent one of the anti-realist alternatives.


With that we need to draw it to a close.

[Closing prayer][8]

[1] 5:00

[2] 10:03

[3] 15:00

[4] 20:02

[5] 25:05

[6] 30:13

[7] 35:03

[8] Total Running Time: 36:28 (Copyright © 2015 William Lane Craig)