Although most people know intuitively what the word infinity means, even though all realities we ordinarily encounter are finite, it is more difficult to clearly answer the question: could an infinite reality actually exist, and how would it appear to us? Or could infinity, instead, be merely a concept of the mind, which cannot exist as an external reality?

Some people might answer, immediately, that space is obviously infinite, so that an infinite reality clearly can and does exist. But the concept of space, on closer inspection, does not turn out to be as easy to understand as this assertion would lead us to suppose. To better understand this, we must subject it to a closer examination.

We can begin a closer examination by making a general comparison between the infinite and the finite, as opposites of one another.

If I undertake a simple counting experiment, which involves continually adding one to the previous count total, I can express this as a continuous application of the formula x+1. However high the value of x might become, after any length of time, even if the counting formula is repeated by the fastest supercomputer imaginable, it will always be possible to add another 1 to the total. Thus a count is an intrinsically finite reality, because it can never actually reach infinity. From this we can define the fundamental nature of the finite as that which is countable, or measurable.

All countable realities are thus finite. Mass, distance, volume, energy, time, any kind of numbered collection, and so on, are all finite realities, which can be counted or measured, and which cannot be infinite.

Since mathematics is the science of the countable, it is impossible that the symbol for infinity in mathematics can actually refer to an infinity, since a countable infinity is a contradiction in terms. For a similar reason, the infinitesimal in calculus must always be a finite quantity.

So why are these connected with the notion of infinity? It can be only a matter of convenience, or simplicity. The infinity symbol in mathematics can mean only an arbitrarily large quantity, which can be made so large in the context of any particular problem that making it larger would make no significant difference to the result. Similarly, the infinitesimal should be seen as a value so small that, in the particular context, making it smaller would make no significant difference to the result. In geometry, also, a point cannot be of zero size, or it would not exist. A point, therefore, must refer only to an arbitrarily small volume greater than zero. So, in general, the infinite or infinitesimal in mathematics can refer only to the arbitrarily large or arbitrarily small.

From this it can be seen that we have already defined a property of the infinite, which an infinite reality must display: that is, it cannot be countable or measurable. Nothing countable or measurable is infinite, and we can see the truth of the assertion that 'a countable infinity is a contradiction in terms', which is another way of saying 'impossible'.

The most immediate and convenient candidate for a real infinity is space itself, as already mentioned. But there are problems with this that must be disposed of.

It would seem that, since space has distance and volume, it is countable or measurable, and cannot therefore be infinite. Against this, however, we can say that, if space has a limited volume, it must have a boundary. If, however, it has a boundary, we may ask: what could possibly exist beyond the boundary, if not yet more space? So it would seem that space cannot have a boundary, or limit, and cannot, therefore be finite. Yet we can be certain that a countable or measurable infinity is a contradiction in terms. So it appears that we must conclude that space is finite, and also conclude that space is not finite, but infinite. This situation is impossible and is sufficient to inform us that we so far have no understanding as to what is the real nature of space.

Some people have tried to solve this problem by proposing that space is, at one and the same time, both finite and without a boundary, like the surface of a sphere. The surface area of a sphere is finite and measurable, but has no boundary. Is this the solution? The problem with this is that the sphere itself exists within a space, from which we can view that it is a sphere. We connot contemplate the surface of a sphere existing without being in any external space at all. If we try to get around this by supposing that the space itself, in which it exists, is also some kind of higher dimensional sphere, which is both finite and unbounded, we have to suppose that this, also, must exist within a space. This kind of picture of spheres within higher dimensional spheres cannot be supposed to constitute an infinite regression, since a regression of any kind is countable, and cannot be infinite. So this kind of 'solution' merely shifts the same, original problem so some higher dimensional space, and does nothing to solve it at all.

Perhaps the only hope of finding a solution is to find a way to separate the countable property associated with space from its infinite nature, which must be real, but not countable, if it is to exist at all. We must therefore consider whether the countable or measurable extent associated with space might be a property of space, but not identical with space itself. In this case, extent, or volume, of space, which is measurable, and must be finite, would have to be a property of space in a manner similar, perhaps, to the way in which 'shape', or 'size', are properties of a object, distinct from the object itself. This distinction is visible in that we can imagine an object that can change its shape, or size, while nevertheless remaining the same object. So space could be seen as some kind of 'object', which has no intrinsic size or volume, but which can produce, or support, a finite volume as its property, distinct from itself.

We may ask, how does this enable us to say that space is an infinite reality that supports volume as a finite property of itself? However, before dealing with the possibly infinite nature of space, let us examine how the relationship between space, and volume, as its property, could actually work.

We measure distance in space only by means of some content of space, distinct from space itself. For example, we measure the distance between the earth and the moon, or sun, etc., as separated contents of space. We might therefore propose that measurable space is finite, as defined by its contents. In this way we may think of the entire contents of space as occupying a finite cosmic volume. We may then say that, outside of this volume, no space exists. Against this, however, someone might say, "If I travel in a spaceship out beyond the limit of this volume, will I meet a boundary, or will I not, rather, find more space" I would have to answer "you will find more space", which seems to contradict the assertion that the volume defined by the contents of space is finite, since it clearly has no boundary.

But this is not quite an accurate description of what is happening. If the person travels, in a spaceship,

beyond the volume defined by the contents of the space, he, himself, is a content of space and, by thus travelling, he is therefore redefining the volume of space that the contents are occupying. In other words, in this case, we must imagine that space, as an object, always provides the finite extent that the contents of space require. We cannot speak of what exists beyond this finite extent without automatically supposing that some content of space is sent into it in order to determine that it is there. But, if any content is sent into it, space itself automatically provides the spatial extent required. So we are not justified in supposing that space is infinite in extent. Indeed, it cannot be, because nothing measurable can be infinite, as mentioned at the start.

I must now return to the question as to how to justify the conclusion that space, as an underlying object, distinct from volume as its property, is infinite in nature. There is one consideration that points in that direction. The underlying space-object, though it must support some finite volume, can support any arbitrarily large volume, as required. This means that it must be something like an infinite source of finite volume. If not, then it must be a finite source of volume, which would mean that there is some absolute, finite limit to the volume that can exist. But, since the volume of space can be arbitrarily large, this cannot be so, and therefore space, as an object that supports any arbitrarily large finite volume as its property, must be infinite in nature. It, itself, therefore, cannot be measured, and can be observed only indirectly via the finite volume that is its property.

The measurement of a finite distance in space is essentially a comparison between one distance and another, between the distance measured as the fundamental unit of distance and whatever distance is said to be thus 'measured'. If all distances were to be multiplied by an arbitrary number, n, without the definition of the fundamental unit of distance being altered, then all the results of the measurements of distances would remain the same. That is a demonstration that distance, in an absolute sense, is arbitrary within the background space, and therefore a measurement of distance is not a measurement of the underlying space itself.

Another way to view the infinite nature of space as object is to examine it in terms of smallness rather than largeness. If we want to measure the distance between two objects in space, we must assign a finite unit for the measure of the distance. The definition of such a unit (or 'quantum') is arbitrary, in that, no matter how small such a basic unit might be, it is always possible to make it even smaller, but it can never be actually made to be zero, since one cannot measure a distance with a basic unit of measurement of zero. So the finite unit for the measure of distance can become forever smaller, or more infinitesimal, but never actually become zero. Since the underlying space can support an arbitrarily small unit of the measure of distance, it has a quality of infinity in terms of smallness as well as that which exists in terms of largeness. We refer to this aspect of its infinite nature as 'continuity', which cannot be actually measured.

We may note that the necessity of assigning a basic unit for the measurement of distance means that the measurement of distance is quantized. This is not an actual, material quantization, in the sense that the chosen basic unit marked on a ruler does not acutally quantize the ruler itself in a material sense. It is known, however, from quantum theory, that matter itself is actually quantized, and it can be argued that this quantization is necessary for matter or energy to exist at all in a finite manner. That is, energy and matter are themselves spatial, in that they always occupy finite spatial volume in some way. If energy was not quantized, it would have to be unbounded, and would therefore coincide with space itself and be indistinguishable from it. We can say, therefore, that energy or matter, unlike space itself, are limited, from the perspective of largeness, but not necessarily from the perspective of smallness.

We cannot suppose that the indivisible quantum of matter or enerty is indivisible in an absolute sense, but only in the context of a particular physics. We cannot suppose that there could not be a physics in which the indivisible quantum of energy would be even smaller, and thus, potentially, arbitrarily small. This leads us to understand that energy, or matter, which are spatial, also have an underlying continuity, like space itself, which suggests the existence of an infinite object that underlies finite energy as its property, in just the same way in which an infinite object underlies the volume of space itself as its property. We might try to say that the spatial quality of energy does not imply a separate underlying object for energy, even though energy is seen as distinct from its spatial quality. I would say, however, that this is not really satisfactory, since the energy, though distinct from space, is spatial and, in effect, appears as if it were a different kind of space. This therefore implies either a different kind of infinite object than that which underlies ordinary space, or that the one infinite object can support more than one property, i. e., that of finite spatial extent in the ordinary sense, and that of energy, as a finite spatial extent of a different kind. In this case, the infinite object that underlies space would be also, and at the same time, an infinite energy-object.

If we were to attempt to say that there are two separate infinite objects, one for space and another for energy, the two infinities would have to be separately located in some way, in order to be distinguishable. But this would mean that they would have to appear as finite at their respective locations, so that they would not overlap and be indistinguishable from one another, which contradicts their infinite natures, and their ability to manifest their properties throughout one and the same space. So we are really forced to the only satisfactory conclusion that there cannot be more than one infinite object, and all properties that may be discovered must be finite properties supported by one and the same infinity.

Time is another measurable reality which we regard as being able to be arbitrarily long. We are able to describe it in terms of a spatial dimension, with the past in one direction and the future in the opposite direction, with the total length being arbitrary and unbounded, as is the case with ordinary space. This leads to the consideration that the one possible infinite object also supports finite time as its property, and it can thus be referred to as infinite in the sense of 'eternal'.

In this way we have identified an infinite reality that represents infinity in terms of volume, energy, and time, with the actual manifestations of these being its finite properties.

I would emphasize, finally, that the characterization of the underlying infinity as an 'object' is illuminating, but also of limited value. The term object merely refers to the fact that the relationship between an object and its size, or volume, corresponds to the relationship between the underlying infinity and the total cosmic volume, which is its finite property. A real object, of course, is not an infinity.


© Alen, October 2015
January 2016
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