Friday, December 6, 2013

The Infinity Paradoxes and How to Solve Them


The concepts of space and time have been very confusing to philosophers throughout history, because these concepts seem to lead to irresolvable logical paradoxes. If the universe is a logical place, then all paradoxes must be theoretically resolvable. That is, they cannot finally be real paradoxes in the sense of actual contradictions. And yet if we think too deeply about the nature of time and space, we seem to be led inexorably towards actual contradictions. One of these paradoxes has to do with divisibility. Space and time are both matters of dimension or extension. The concept of “space” is about distance, length, height, etc. Space can be measured, and thus is can be divided into parts. The same is true of time. Hence we have centimeters, meters, kilometers, minutes, hours, days, years, and so on. All extended phenomena can be divided (at least theoretically) into parts. Because all material objects occupy space and time, all material objects are extended and thus can be theoretically divided both temporally and spatially. For example, the book sitting in front of me is spatially divisible--it can theoretically be divided into half, into thirds, etc. It is also temporally divisible, in that its existence through time can be divided into various moments--we can distinguish, for example, the book as it was two minutes ago from the way it is now.

The difficulty arises when we start to ask how far the divisibility of material objects, or spatial or temporal lengths or distances, can go. Of course, practically speaking, we can only divide things up so far; but theoretically, there is no stopping point. Every time I divide an extended object or length (let’s say I’m dividing it exactly in half), at the end of the process I will always have two equal parts on each side of my line or point of division. These parts will themselves possess length (half of the original length of the whole), and thus they too can be divided in half. Likewise, these new parts will be able to be divided in half, and apparently so on we could go forever. There can never be a time when we will run out of divisions, because every division must leave some length in the divided parts, which can then be again divided in half. This kind of observation is why many philosophers have spoken of material objects and space and time as infinitely divisible. So then, if the book in front of me is infinitely divisible, how big is the smallest piece that makes up the book? Well, it would be infinitely small; for if it was anything greater than infinitesimal, it would be able to be divided into smaller pieces and thus would not be the smallest piece. If a piece of this book has any dimension--say, length--left in it at all, it will still be divisible into smaller pieces and thus will not be the smallest piece. So my book must be ultimately made up of pieces that are infinitesimal, infinitely small, and which therefore possess no dimension at all. They are precisely zero centimeters (or millimeters, or anything else) long. And, of course, since every division in half produces two equally-sized pieces, and there are an infinite number of divisions, the book must be made up an infinite number of infinitely small pieces. OK, so where’s the problem? Well, if you think about it for a moment, the problem will show itself clearly. For one thing, what exactly is the nature of a piece of matter that possesses no dimension and that therefore takes up no space? Whatever it is, how can we call such a thing matter? A dimensionless object that takes up no space would be the same as no material object at all. For another thing, how many of these infinitely small pieces does it take to make up a book that is, say, about eight inches tall and six inches across? We have an infinite number of them available, so surely that will be enough, right? Well, how long is one of these infinitesimal pieces by itself? As we said, it is dimensionless, and so there is no length at all. How much length would we have if we put two of these pieces together? Well, zero plus zero is still zero; we would still have no length at all. What if we put three of them together, or four, or five, or six thousand, or six million? Obviously, the answer will be the same--there is no length at all. Even if we put an infinite number of such pieces together, we would still have zero length. But my book has length. So my book cannot be made up ultimately of pieces that have no length at all. So we have a situation where it seems both that my book must be made up of an infinite number of infinitesimal pieces (because of the infinite divisibility of extended objects and lengths) and also that it cannot be made up of an infinite number of infinitesimal pieces. That is a problem. How are we going to solve it?

Here’s another problem: How far back does time go? This universe is a temporal universe; time is one of its dimensions. Therefore the universe has a history. How far back does this history extend? Some people believe that the universe is eternal--that is, it never had a beginning; time has been going on forever. And yet this leads us to absurdity. If time has been going on forever, then, as of right now at this moment, an infinite amount of time has already passed in the history of the universe. But there is no way that the universe could have passed through an infinite amount of time, because it is inherently impossible by definition to traverse an infinite. If there are an infinite number of fence posts, how long will it be before I have walked by them all? I could never walk by them all, because it is a contradiction to the very nature of an infinite number of fence posts that I could ever walk by them all. If I could do so, then they would be by definition finite. Any distance I can travel must get me from point A to point B, and therefore must be a finite distance, not an infinite one. If time has been going on forever, then the universe has passed through an infinite number of, say, minutes. But, by definition, it is impossible that an infinite number of minutes has already been passed through. So it would seem that time cannot have been going on forever; it must have started at some moment in the past--say, 14 billion years ago (or whatever).

But now we have another problem. The very concept of a first moment in time is absurd, since every temporal moment implies a preceding moment. Let’s think about the nature of the very first minute. How long did it last? One minute, obviously. Did it come to an end? Of course it did; it came to an end after the minute was up. Did it begin? Of course; it began exactly one minute before it ended. But ending and beginning are events; and all events, by definition, must have a before, during, and after. For the first minute to have begun, there must have been a time before it began. Once, it had not yet begun, and then it began. If there was never a state of affairs before the first minute began, that would be the same as to say that the event of its beginning never took place. For the first minute to have begun is to say that it arose into being, implying that being was empty of it before. For an analogy, imagine the act of opening a door. “Opening a door” is an event that therefore must have a beginning, middle, and end. The act could not be complete unless we start out at a moment in time before I had begun to open the door--that is, when the door was still entirely closed. If we do not start out with the door closed, there is no temporal room for me to begin to open it. Likewise, if there were no time before the beginning of the first minute, there would be no temporal room for the first minute to begin, and yet beginning is essential to the concept of a temporal length such as a minute. Therefore there would have to have been a moment of time before the first minute. And that moment would have had to have been preceded by a preceding moment, and so on ad infinitum. Therefore there would have to have been an infinite number of minutes before the very first minute, which is of course absurd. Therefore, there could not have been a very first minute. Time could not have begun; it must have been going on forever. And now we see a second paradox: We have conclusive logical reasons to think both that time cannot have been going on forever and that it must have been going on forever. (By the way, the same paradox arises when we try to think about how far space extends as well; but for the sake of brevity, I will not go into that now.)

Paradoxes such as these have long been recognized by philosophers. Zeno, the ancient Greek philosopher, famously recounted a number of them, as reported by Aristotle. Immanuel Kant recounted some of them as well in his Critique of Pure Reason and used them to argue that the universe cannot be inherently ordered and logical; we must be imposing order by our own minds on an unordered chaos, the nature of which, since it is non-ordered and chaotic, we can thus know nothing at all about. Theologians trying to talk about the creation of the space-time universe have often run into particularly the latter paradox, although they have often tended to brush it off as a semantic issue. Theologians will often find themselves talking about “before the beginning or creation of time,” and then quickly apologize for the inadequacies of language that force them to speak in such absurd ways. But they have not often enough stopped to think about why they are forced to use such absurd language when talking about the creation of time. I would argue that it is more than an unimportant semantic issue. It points to the same very serious logical problems in understanding the nature of time that we have been talking about.


So how can we solve these paradoxes? There must be some way in which we can do so, or else we will be forced to conclude that the universe is inherently illogical. But, for reasons I don’t have time to go into now, we know that that itself would be an absurd conclusion and can’t be right. Some people would suggest this is simply too difficult a problem to solve for our limited minds, and thus it is not worth pursuing. Well, maybe; but the history of the human race is full of examples of people who have contributed greatly to humanity by continuing to try to do things that other people continually warned them was impossible. So we should prefer to check all possible options before we give up.

I think the answer is this: Extension and divisibility are fundamentally characteristics of a finite, a limited, point of view. If we think about the nature of extension for a moment, we can see that this is so. Whenever we have an extended object or an extended length (or any other dimension) in mind, we find that one of the essential characteristics of that extended object is that it is being viewed from some particular location. It is impossible to separate the concept of an extended distance from the idea of that distance being viewed from some particular, limited, point of view. For example, imagine a line that extends five inches. At one end of that line we have point A, and at the other end we have point B. Point A is in a different location from point B. They are a certain distance apart, which is how we can distinguish them. But notice that these points are in different locations not in some absolute sense but relative to your own viewpoint. That is, your viewpoint, which has you looking at our five-inch line from one possible vantage point, has created a grid in which that line, as well as point A and point B on that line, exists. Point A is in a different location from point B relative to the grid created by your own particular viewpoint. You can always imagine moving your viewpoint to view the line from a different perspective. If you view the line a certain way, point A and point B will appear in the same location. All of this will be true of any extended object or distance that you can see or imagine. The keyboard in front of me is (roughly) about eighteen inches across. The “A” key and the “L” key on the keyboard are in different places, not absolutely, but relative to my viewpoint. Our finite viewpoint provides a necessary ingredient to the very concept of two things being in two different places or being a certain distance from each other, which is the very essence of the concept of extension. I am going to draw a very interesting conclusion from this observation: Extendedness is a characteristic of the viewpoint of finite minds and therefore does not exist outside of the viewpoint of finite minds. Only finite minds, which view things in a limited way from one particular location among many possible locations, and thus can inherently only see a part of reality at a time, have the characteristics necessary to produce extendedness.

This observation, and this observation alone, can solve the paradoxes we discussed earlier. The problem of infinite divisibility arises because it seems that extended objects must be infinitely divisible, and yet it also seems that they can’t be infinitely divisible (since they cannot be made up of an infinite number of infinitesimal pieces, as infinite divisibility would imply). But divisibility is a product of extendedness. Without extendedness, there can be no divisibility. If extendedness can only exist in finite minds, then we can talk about something being potentially infinitely divisible without that something being actually infinitely divided. For example, the book in front of me is potentially infinitely divisible. That is, there is no theoretical point at which I would run into a lack of material to continue to divide. As we noted before, every time I divide, I have divisions that have dimension that can be divided again. And yet, although I will never run into a theoretical barrier to divide further, I never actually see the book in an infinitely divided state. That is, I never perceive in my mind an infinite number of divisions. It is inherently impossible for any mind to perceive an actual infinite number of divisions. Therefore, since extendedness and hence divisibility exist only in the viewpoint of finite minds, as we established a moment ago, since no one ever perceives an infinite number of divisions of my book, those infinite divisions of my book simply do not exist. My book is only ever as divided as some finite mind perceives it to be. Thus, we can say that my book is potentially infinitely divisible and yet is not actually infinitely divided. This allows us to solve the problem of infinite divisibility. The paradox arose because we were imagining that the extended nature of my book existed outside of any finite mind. If this were the case, it would imply that if my book is potentially infinitely divisible (which it must be, for the idea of running into a theoretical point at which there is nothing left to divide is absurd), then it must consist of an actual infinite number of divisions (since the divisions would go on even after they passed beyond the ability of finite minds to perceive them). But if extendedness and divisibility only exist in finite minds, then the potential infinite divisibility of my book would not imply that there is an actual infinite number of divisions. The paradox therefore disappears.

We can also apply the same observation to the other paradox we mentioned--the apparent problem that time cannot have been going on forever and yet seemingly must have been going on forever. The problem here arises because we observe that every moment in time inherently implies a preceding moment in time. This seems to lead to the conclusion that the timeline must extend back infinitely with an infinite number of divisible moments. And yet this can’t be the case, because then the universe would have had to have already traversed an infinite number of moments, which is inherently impossible. But, notice that time, like space, is a dimension that consists of extension (and hence divisibility). Thus, time, like space, only exists in finite minds. We can therefore say that the past is potentially infinite (since we could never find a theoretical first moment that is not preceded by a preceding moment) and yet that the past is actually finite (because any finite mind can only perceive a finite amount of time in the past or anywhere else). This resolves the paradox. The same thing can be applied to space as well. Space is potentially infinite--in the sense that we could never run into a barrier at which space ends--and yet it is actually finite because only finite distances are perceived by finite minds..1  The picture that emerges here is that space and time, consisting of extendedness, are not absolute, but are to be seen as extending out in all directions with potential infinity but actual finitude from a central location which would be some particular finite point of view. However unusual such an idea of time and space is, I think it is the only view that makes sense as we consider the nature of space and time themselves and as we try to solve the paradoxes that philosophers through history have pointed out.2 

This potential/actual distinction exists in all other areas where we have potential infinites in the world as well--another interesting example being the calculation of pi. Pi, famously, is potentially infinite, in that one never can come to the end of calculating it out. It can be calculated out forever. But because actual infinites can't exist--the space-time world being inherently finite--it will ever only be calculated out to a finite degree, no matter how amazing our future computers become. Beyond the point of the most distant calculation yet made, pi goes on with potential infinity. But, as dimension exists only in finite viewpoints or in finite perception, the further decimal places of pi do not exist in actuality but only in potentiality. That is, there is a definite form that will arise, logically connected to what has come before, at any point in the stream of decimals. But the form is only potential and never actualized unless some finite mind actually calculates it out to that degree. Again, this solves the paradox that would exist if we imagined that pi actually exists somewhere calculated out to infinity.

2 For more on the potentially infinite but actually finite nature of time and space, and for an account of how all this relates to classical arguments for the existence of God, see my book Why Christianity is True, particularly the section on “Deeper Philosophical Issues” in chapter three.

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