Reality is a Concept - Or, What's Wrong with Immanuel Kant

Kant's Theory

Immanuel Kant is one of the most important of the modern philosophers.  In my opinion, his importance lies more than anything else in his epistemology, whereby he attempted to take empiricism to its logical conclusions.  (For those who don't do a lot of philosophy, epistemology simply means "our ideas about how we can know things."  Empiricism is the epistemological view that says that all knowledge comes through the senses in some way or another.)  His formulation of how we know, what we can know, and what we can't know, has fundamentally shaped modern thought on these subjects ever since.  He provided a strong philosophical foundation for the empiricist Agnosticism that much of our culture tends to take for granted today.

Kant's basic theory goes like this:  Information about objective reality only comes through the senses.  But sense data is a big, unorganized mess.  Our minds give order to that data and so create the appearance of the ordered cosmos that we see.  Our minds impose categories--like space, time, number/quantity, and other logical categories--on the chaos of sense data, and the world we know is created out of that.  Sense data is like someone throwing at us a bunch of wires, bits of metal, glass, and other things, all piled in a heap.  But then someone comes along and puts the whole thing together and builds a car--that is like our mind, with its categories, organizing sense data into the world we know.  The end result of this is that we really know nothing at all about the objective world outside of our minds.  All the characteristics we can gather from what we experience (quantity, color, taste, sound, shape, spatial extension, temporal extension, etc.) have been imposed on that experience by our minds.  So what we are really experiencing is the creation of our own minds, not the true, objective world.  The latter we can never know, for we cannot get beyond or outside of our own mental categories.

Kant was very interested to distance his view from a view called idealism.  He was thinking of the theory of idealism put forward by another Enlightenment philosopher, George Berkeley.  Berkeley had proposed that, actually, everything that exists is either mind or the perceptions of mind.  In his view, there is no such thing as "external matter" conceived of as some reality that exists outside of the experience of minds.  He based this conclusion on many of the same arguments Kant would later use.  He pointed out that all our concepts--color, texture, sound, taste, smell, shape, distance, extension, and therefore space and time--only make sense when understood as perceptions of minds.  The color red, for example, is meaningless apart from the idea of someone seeing red.  Taste is meaningless without the idea of someone tasting something.  Even something like shape, which seems objective, implies a perceiving mind, for shape depends on extension (this part of object A is at some distance from that part of object A), which implies a perceiver in relation to whose viewpoint things can be in different places.  Without some viewpoint connected to someone viewing, there would be no meaning in talking about point A being in a different place from point B, for the viewpoint of the viewer provides the grid on which different places can exist.  There is no characteristic of matter whatsoever that is intelligible without, for its context, the idea of a perceiver perceiving it.

Berkeley concluded from all of this that the idea of "matter" as some entity existing apart from the experience of minds is meaningless gibberish, and so is non-existent.  Berkeley's critics accused him of denying the existence of the external world.  Berkeley replied that he was not denying the existence of the external world, but merely describing more accurately what the external world is--not an entity existing apart from the experience of perceivers, but an entity existing by means of such perception.  "To be is to be perceived."  But Kant was not convinced.  He didn't like the idea that there is no world external to the experience of minds.  He wanted to maintain the idea of such a world.  But he also agreed with Berkeley that all our perceptions and experiences are of things created by our own minds.  Therefore, he concluded, a mentally external world exists, but we can know nothing at all about it.  Here is Kant describing all of this in his own words, from his classic work, Prolegomena to Any Future Metaphysics (First Part, Section 13, Remark 2, from the public domain translation by Paul Carus, provided by Wikisource under the Creative Commons Attribution-ShareAlike License, found here):

Whatever is given us as object, must be given us in intuition. All our intuition however takes place by means of the senses only; the understanding intuits nothing, but only reflects. And as we have just shown that the senses never and in no manner enable us to know things in themselves, but only their appearances, which are mere representations of the sensibility, we conclude that "all bodies, together with the space in which they are, must be considered nothing but mere representations in us, and exist nowhere but in our thoughts." Now, is not this manifest idealism? 

Idealism consists in the assertion, that there are none but thinking beings, all other things, which we think are perceived in intuition, being nothing but representations in the thinking beings, to which no object external to them corresponds in fact. Whereas I say, that things as objects of our senses existing outside us are given, but we know nothing of what they may be in themselves, knowing only their appearances, i. e., the representations which they cause in us by affecting our senses. Consequently I grant by all means that there are bodies without us, that is, things which, though quite unknown to us as to what they are in themselves, we yet know by the representations which their influence on our sensibility procures us, and which we call bodies, a term signifying merely the appearance of the thing which is unknown to us, but not therefore less actual. Can this be termed idealism? It is the very contrary. 

Long before Locke's time, but assuredly since him, it has been generally assumed and granted without detriment to the actual existence of external things, that many of their predicates may be said to belong not to the things in themselves, but to their appearances, and to have no proper existence outside our representation. Heat, color, and taste, for instance, are of this kind. Now, if I go farther, and for weighty reasons rank as mere appearances the remaining qualities of bodies also, which are called primary, such as extension, place, and in general space, with all that which belongs to it (impenetrability or materiality, space, etc.)---no one in the least can adduce the reason of its being inadmissible. As little as the man who admits colors not to be properties of the object in itself, but only as modifications of the sense of sight, should on that account be called an idealist, so little can my system be named idealistic, merely because I find that more, nay, all the properties which constitute the intuition of a body belong merely to its appearance. The existence of the thing that appears is thereby not destroyed, as in genuine idealism, but it is only shown, that we cannot possibly know it by the senses as it is in itself. 

I should be glad to know what my assertions must be in order to avoid all idealism. Undoubtedly, I should say, that the representation of space is not only perfectly conformable to the relation which our sensibility has to objects---that I have said--- but that it is quite similar to the object,---an assertion in which I can find as little meaning as if I said that the sensation of red has a similarity to the property of vermilion, which excites this sensation in me.

So here is where Kant has become very influential.  Most people today, at least those immersed in Western culture and Western thought, take it for granted that all our knowledge comes through the senses.  They would claim that we cannot know anything in any other way.  But what about logic?  Can we gain knowledge through an application of logic to our concepts?  For example, consider the idea that the past is infinite--that is, that there was no beginning to the universe, but that it has always been going on, that no matter how far back you went in a time machine, you would never run into a beginning of time because there is no such beginning.  I submit that the concept of an "infinite past" is logically absurd.  It is absurd because it contains a contradiction--the concept of "infinite" contradicts the concept of "past".  The "past" is that part of the timeline that we have already got through.  But an "infinite", in terms of number or quantity, is by definition something that can never be got through.  If you try to count to infinity, for example, you will never arrive.  So an infinite past would be a length of time that could never be gotten through, it could never be completed, and yet the "past", by definition, is a length of time that has already been gotten through, it is already completed.  Yes, we are adding to the past as we move into the future, but that part of time which is already the "past" is already through.  So an "infinite past" would be a length of time that both cannot be gotten through and also has already been gotten through--a manifest contradiction.  But contradictions cannot exist, for, by definition, being excludes non-being, and all beings exclude their opposites.  Whatever is, by definition it is what it is and isn't what it isn't.  So the conclusion of logic is that an infinite past cannot exist, and therefore the past must be finite--that is, it must be limited.

Now the Kantian empiricist responds in this way:  "OK, I'll grant that the concept of an 'infinite past' is contradictory and therefore illogical.  But that doesn't tell us at all about the real world.  It only tells us about our concepts.  We're just playing around with ideas inside our own heads.  Such logical games can tell us nothing about objective reality, which exists outside our heads and our concepts.  So such logical analyses provide us with no knowledge.  Knowledge can only come through the senses, not through logic."

My First Objection to Kant

Now here's where I want to provide a fundamental critique of the Kantian view.  I want to make two objections, the second more important than the first.

My first objection is that, if the Kantian is right, then he is indeed right that our logical analyses provide no actual knowledge of the objective world.  But it also follows that our senses provide no knowledge of the objective world.  Kant himself admitted as much, and my experience suggests that the more perceptive among the modern empiricist Agnostics will also admit as much when pressed.  But I think they often fail to live up to the full implications of this admission.  So I think this conclusion needs to be pressed.  If this Kantian empiricist view is right, then nothing at all, neither logic nor our senses nor anything else, gives us any knowledge of reality.  Logic gives us no knowledge of reality because it only provides an analysis of the concepts in our heads and doesn't touch the real, external world.  And our senses provide us with no knowledge of reality because they never actually give us any access to the external world.  Kant appears to say at first that they do grant such access, but then he takes it all back by affirming that it is the categories of our minds ordering the sense data that really is the source of everything we actually know and experience.  Space, time, extension, quantity, shape, taste, color--in short, everything at all in our experience and knowledge--is a creation of our minds.  It is not what the external world is really like.  We know nothing at all about the external world.  You can be sure that Kant did not reach this conclusion because he wanted to.  He would have liked to have shown us how we can have actual knowledge of reality.  But he rightly recognized that his way of thinking about how we gain knowledge made that impossible.  Empirical sense data, without adding anything else, even logic, to it, can tell us absolutely nothing about reality.  I look across the room and see an apple on the table.  How do I know that there really is an apple on the table?  Perhaps it is an illusion.  How do I know that if there is an apple on the table, it isn't also true at the same time that there is NOT an apple on the table?  Only logic could make that inference.  But if I cannot even deduce that from my experience, if I cannot even, by my experience, exclude the opposite, then my experience tells me nothing, for there is nothing there that can be translated into any concept that could possibly have any meaning to me.  If all I've got, literally, is empirical sense data to go on ultimately, then all I can know, literally, is nothing at all.  And if all the characteristics that make up our idea of what a physical object is--color, shape, size, etc.--only exist as experiences of our minds, as Kant says, and cannot be attributed to objects in the external world, then we can really have no idea at all of what a "physical object" is or is like.  The very phrase, "physical object", becomes simply a meaningless collection of sounds.

I think it is worth bringing this out because I think it is missed by a lot of people who like to think like empiricist Agnostics and deny that logic can tell us about reality.  Even those more acute thinkers who will grant this in principle seem to forget it in the practice of their lives.  They keep going on as if they know something about the world, while denying in theory that they do.  They seek to gratify their curiosity, they argue with people, they hold opinions, they act on their opinions and seem to feel they are doing something meaningful in some way, they do science--all of which must be ultimately meaningless if their epistemology is correct.

My Second Objection to Kant

But this is not, I think, the most important objection to the Kantian theory.  The most important objection is that it is plainly wrong.  Its error consists in a simple forgetfulness with regard to the definition of words.  The Kantians--and George Berkeley before them--are quite right in pointing out that all our concepts--color, shape, texture, distance, extension, divisibility, quantity, etc.--are, well, concepts.  That is, they are ideas that exist in the minds of thinking and perceiving beings.  But then the Kantians make their fallacious move--they draw a distinction between our concepts about reality and reality itself, asserting that since all we know is the former, we can never know the latter.  Do you see the fallacy in drawing this distinction?  Think about it for a moment before reading on, and see if you can figure it out before I tell you what it is.

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OK, did you figure it out?  The fallacy, of course, is that reality itself is also a concept, just like all the other concepts.  It is just as much an idea in our minds as any other concept.  So the distinction between our concepts and reality itself is a false distinction.  There is no reality outside our concepts, by definition, since reality is itself one of those concepts.  The whole Kantian skeptical conclusion rests on the assumption that because we can never get beyond our concepts, we can never get to that which is outside those concepts, namely reality itself.  So our concepts block us off from reality, leading to the conclusion that we can know nothing at all.  But if reality itself is a concept, then just the opposite is true.  Experience of (and analysis of) our concepts does give us true knowledge of reality, because these concepts are nothing else than reality itself.  It turns out that George Berkeley had already given us the answer to Kant's skepticism decades before Kant ever wrote, when he pointed out that it is a fallacy to worry about idealism leaving out the external world.  Idealism is only leaving out the external world if we assume an idea about what the "external world" is contrary to what the idealist view thinks it is--that is, if we assume that the "external world" is something that exists outside of our concepts, outside of the experiences of minds.  But since reality itself is a concept, it is contradictory gibberish to talk about a reality that exists outside of our concepts.  So even though we can know nothing beyond our concepts, we have lost nothing of reality, for concepts are reality.  They are simply one and the same thing.  The "reality" that Kant thought lies forever beyond our knowledge turns out to be itself unreal, nothing but a phantom born of confusion in how we are using our words.

It is easy to be confused by such phantoms of language, because language can be used in ways that create illusions or impressions that don't correspond to reality.  I think of comic literature like Alice in Wonderland, where much of the humor is carried on by playing around with the illusions of language.  Or we can think of the example with the cats that is often used in philosophy classes:  "I can prove to you that a cat has nine tails.  No cat has eight tails.  One cat has one more tail than no cat.  Therefore, since 8+1=9, one cat has nine tails."  Of course, the problem here is a trick in the use of language.  Some classic arguments against the existence of God are based in such fallacious use of language.  For example, some Atheists argue that there cannot be an all-powerful being, because such a being could not make a rock so big that he couldn't lift it.  "If he can lift the rock, then it's not a rock so big that he can't lift it, so he can't create such a rock.  But if he can't lift it, then he can't lift it, so there is something he can't do.  Either way, there has to be something he can't do, so he cannot be all-powerful."  The problem with this argument is simply that the arguer is assuming that the word can't always implies a lack of power.  But this is not the case.  Sometimes something is impossible to someone because of a lack of power, but other times the impossibility lies not in a lack of power but in the absurdity of the concept.  The reason why God can't make a rock so big he can't lift it is not because he lacks power, but, on the contrary, it is because he has all power.  Being all-powerful, he can't make a rock more powerful than himself, which would be a contradiction.  Similarly, God cannot make a square circle, not because he lacks power, but because the concept of a square circle is meaningless gibberish.  But because the word "can't" often implies a lack of power, we neglect to consider whether the present argument is such a case, we just assume it is, and so we fall into the fallacy.

It seems like we must be out of touch with external reality if all we can know are our own concepts.  But this seeming is just an illusion.  It comes from having a false impression of what a concept is and what the external world is.  We vaguely imagine someone's head, and concepts floating around in that head, and then we imagine a world outside of that head.  But the idealists don't deny the existence of objects outside of people's heads; they deny the existence of objects outside of people's perceptions and concepts.  The very concept of outside is itself a concept.  So, by definition, there is nothing outside concepts.  Therefore, a logical analysis of concepts can give us true knowledge of actual reality.  We are not limited to the empiricist way of knowing things.  If logic excludes an "infinite past" because the concept involves a contradiction, then we know that an "infinite past" does not and cannot exist.  We know something true (and very important) about the real world.

So recognizing Kant's fallacy here can help us get beyond the false empiricist epistemology that has cut off from modern culture and much of modern philosophy a very important source of knowledge about the real world.  By adopting this false epistemology, we have blinded ourselves and doomed ourselves to an unresolvable skepticism.  But it is all unnecessary, for our epistemology is based in nothing other than a simple fallacy.  Once we realize this, we can get back to exploring the real world using all of the resources at our disposal.

For more, see here, here, and here.  It is evident that recognizing the inextricability of reality from the concepts and experiences of minds points to a view of reality in which mind is fundamental, as opposed to a view where mind is somehow an eventual by-product produced by non-mind.  To see how all of this relates to arguments for the existence of God, see chapter three of my book, Why Christianity is True, including the section on "Deeper Philosophical Issues".

The Infinity Paradoxes and How to Solve Them

THE PARADOXES OF SPACE AND TIME

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.

SOLVING THE PARADOXES

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 

1  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.