Doctrine of God (Part 3): The Problem of Platonism

(c) The Problem of Platonism

We saw last time that a robust understanding of aseity that God not only exists independently of everything else (which would in itself be remarkable), but that God is the sole ultimate reality. 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? 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 are 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 figure of alternatives discussing this subject. Don’t be overwhelmed by this figure. 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 do not exist. This is just meaningless.

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 is a meaningless statement. Verificationism is a sort of scientism that attempts to dismiss vast tracts of human language 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-20^th century. 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 that mathematical objects are, in fact, concrete objects.

Let’s take the abstract alternative first – that these are abstract objects. 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 pause theologically. It does require you to modify in some significant ways your view of creation. But there are some Christian philosophers today who 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, 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.

You see on the figure next to abstract objects 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.

Consider first the view that mathematical objects are physical objects. One alternative that takes this view would be Formalism, which says that mathematics is basically ink 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 are 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.

Then there is the alternative of taking mathematical entities to mental objects – thoughts in somebody’s mind. This might be either a human mind or God’s mind. The view that mathematical objects are just thoughts in people’s minds is called Psychologism. This view says you have ideas of the number 2 or of 2+2=4 and that is what these mathematical objects are. They are just thoughts in people’s minds. That view also is not very widely adopted today because, again, of the inter-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 an infinite number of mathematical objects and 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.

Many Christian philosophers have chosen instead to adopt the view that numbers are thoughts in God’s mind. This view is called 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. What really exists will be God and his thoughts.