July 07, 2013
God and Infinity
Dear Dr Craig
I am a devout Christian and I read your Q&A every week. I have read Q#323 today and I was totally surprised by it because it is basically exactly what I have been thinking about for the last couple of days. Infinity is a concept that has been driving me nuts as I cannot completely fathom it. A few comments I want to make:
1. Is it possible that God could have counted the whole set of natural numbers? The answer according to me is no. Why? Because the set of natural numbers is infinite. You will never stop counting. So even though God is almighty, He cannot do things like these. It simply doesn't make sense. It's like saying God can create a rock that He cannot lift. It's a logical incoherent statement.
2. Regarding the existence of abstract objects: One thing that has bugged me about this is the whole thing about infinity. Does it exist? I have read some of your comments on this, namely that you say that infinity is a mere concept of the mind that doesn't exist in the real word. This makes sense to me. So if there are an infinite amount of natural numbers, how could they all exist? Well I believe numbers exist in two to three ways:
a. Since numbers are mere concepts, they exist in our minds as we think about them.
b. Numbers exist conceptually as real world objects represent them. I.e. the number of planets in the solar system.
c. They exist conceptually as symbols on a piece of paper (or computer screen).
Because only a finite amount of thoughts, objects and written symbols exist, it follows that only a finite amount of real numbers exist in the real world.
3. Abstract objects aren't limited to numbers. It includes things like music, literature and other forms of art. Now there are an infinite amount of songs that can be composed. Do they all exist? Again the same line of reasoning goes. A tune lives in the mind of the one that composed it and the ones who hear it. It is a concept, an idea which didn't exist before someone thought about it. So there are only a finite number of art forms in the real world and always will be even though the number increases all the time.
4. Now I want to say something about the future. God as we know can foresee the future. This is what makes prophecy possible. We know that the world has a beginning, and it will have an end in this age as the Bible predicts, but the age to come will have no end. So the number of future events counts up to infinity. Now is it possible for God to know everything in the future? Again as explained above this is impossible. One cannot know an infinite amount of things. It is simply an illogical thing to ask from someone like God, even though He is almighty, and I am not trying to limit Him in any way. So God knows the future. How much and how far? Well as much as He chooses or as far He deems necessary.
These are my comments. I therefore believe God is almighty, but He cannot do illogical things, or know things that do not exist. Thanks for reading and please let me know if you agree with my views. Also please tell me of articles I can read which might give me more insight into these things.
Finally I want to thank you for the work you're doing. I think you're making a big difference in many people's lives.
These sorts of questions can keep you awake at night, can’t they, Hardus? Let me address each one.
1. Could God have counted all the natural numbers? It’s fairly widely agreed that God could not begin at 0 and successively count all the natural numbers. Here it’s helpful to distinguish between a potential infinite and an actual infinite. A potential infinite is a series which has a beginning and is growing indefinitely; infinity serves merely as an ideal limit of the series which it never reaches. An actual infinite is a collection which includes an infinite number of members, that is, the numbers of members in the collection exceeds any natural number. Counting generates a potential infinite. To say that someone could count to infinity is to say that a potential infinite could be converted into an actual infinite by adding one member at a time. That’s impossible, since for any natural number n, n+1 is always a finite number.
The question is, could someone count all the natural numbers one at a time by never beginning but ending at 0? To my mind that is just as impossible as the first task. If you can’t count to infinity, how could you count down from infinity? If I’m right about this, then the series of past events cannot be infinite, either potentially or actually—potentially because it is not growing in a backward direction toward infinity, nor actual because you can’t get through an actually infinite number of items one at a time. So the series of past events must have had a beginning.
2. Does an infinite number of numbers exist? My answer to that question is no, not because I think that the number of numbers is finite but because I think that there are no such things as numbers! Numbers are just useful fictions, like the Equator or the center of mass of the solar system. Do numbers exist in any of the three ways you suggest? Not really. Consider your alternatives:
a. Certainly I have the idea of the number 2, for example. But that thought is not the number 2 itself. The nineteenth century mathematician Gottlob Frege called the view that numbers are ideas in our minds “psychologism” and subjected it mercilessly to criticism. That 2 and my idea of 2 are not identical is evident in that they have different properties; for example, my idea of 2 comes and goes, but 2 itself, if it exists, doesn’t depend upon my thinking about it!
b. Certainly, things that exist in the world can be numbered; for example, Mars has two moons. But this adjectival use of number terms doesn’t require that numbers themselves exist. As Frege showed, we can express that Mars has two moons without using any number terms at all by saying that there is some entity x which is a moon of Mars and some entity y which is a moon of Mars and x is not identical to y, and for any other object z, if z is a moon of mars, then either x is identical to z or y is identical to z. (In logical notation: (∃x) (∃y) (Mx & My & x≠y ((∀z) (Mz ⊃ z=x ∨ z=y)).) Pretty slick, eh?
The hotly disputed question is whether the use of number words as referring terms commits us ontologically to the existence of numbers. For example, the statement “Two is the number of Mars’ moons” is thought by many philosophers to commit its user to the reality of the number 2. This strikes me as perverse. Metaphysics is surely a lot more difficult than that! These sorts of ontological commitments happen, as Wittgenstein said, when “language goes on holiday.” You can’t read metaphysical commitments off language—at least so I think.
c. Obviously, we make marks on paper in doing mathematics. But here we need to distinguish between numbers and numerals. There are many different ways to represent the number two: 2, II, úú , and so on. But these are numerals, not numbers. There are many numerals but only one number 2, if such a thing exists. So numbers do not exist on paper or computer screens.
So what you should say, I think, is that there is at most only a finite number of numerals in the world, not that there is a finite number of numbers in the world. There are no numbers anywhere.
3. What about other kinds of abstract objects? You are absolutely right that philosophers who believe in the existence of abstract objects think that novels, plays, musical compositions, fictional characters, and so forth, exist as abstract objects. What is disputed is whether these are created by their writers and composers or whether these people just happened to stumble upon these pre-existing objects. Many people feel quite uncomfortable in saying, for example, that Leo Tolstoy did not create Anna Karenina but just found it. This view seems to seriously depreciate the creative genius of authors and composers. So many want to say that people created these abstract objects. Still, it’s hard to see why, once you grant that such abstract objects exist, these collections of words or notes did not pre-exist their discovery by these folks. I think it’s better to just deny that such abstract entities exist and maintain that our ability to talk truthfully about them (e.g., “Sherlock Holmes is the most famous detective in English fiction”) doesn’t entail their existence.
4. Does God have complete foreknowledge of the future? Yes, why not? Your statement that “the number of future events counts up to infinity” is ambiguous. We’ve already agreed that it’s impossible to “count up” to infinity. So the series of future events “counts up to infinity” only in the sense of a potential infinite: infinity is the limit to which the series of events strives but never reaches. There will never be an actually infinite number of events. From any point in time that you pick the number of events future from that point is always finite and always increasing. If you pick the present event as your point of reference, the number of future events is 0! That’s because temporal becoming is a real and objective feature of the world.
You say that it’s impossible for God to know everything in the future. That doesn’t follow from anything we’ve said. To get an objectionable, actually infinite number of things out of this, you have to think that God’s knowledge is broken up into propositional bits that actually exist. But such a view of God’s knowledge is not obligatory for the theist (and traditionally has been denied by theists). Suppose God’s knowledge of reality, including the future, is non-propositional in nature, and we finite cognizers represent what God knows non-propositionally by breaking it up into propositional bits. (For an analogy think of your unbroken visual field, which someone could represent by breaking it up into pixels.) Then there is no actual infinity of ideas, thoughts, propositions, or what have you.
So do not limit God by denying His complete foreknowledge of the future. There is no good reason to adopt such a view and it impugns God’s greatness.