The universe is spatiotemporally finite

Let us construct a model of a physical universe. I am going to assume a quasi-classical picture of reality. I think, even though the technical detail might not entirely work with relativity or quantum theory, the philosophical intent would still be salvageable. My intentions here are philosophical, rather than physical.

Points exist in spacetime. These points have various properties, such as mass or electrical charge. Now, we can convert this model into one in which points have no property other than location, by treating the properties such as mass or electrical charge as additional dimensions. Vacuous points, not occupied by any particle, would lie upon the origin of the property dimensions. We also need to introduce some geometrical restrictions, since a single point can only have one possible mass or one possible electrical charge. We can divide the dimensions of this model into two subsets:

— positional dimensions: x, y, z, t

— property dimensions: m, q, etc.

For each possible value of x, y, z, t, there is only one possible value of m, q. This is a geometrical constraint.

This new space (I might christen it property-space) I have defined is somewhat similar to the concept of phase space. However, in phase space, each point of the space represents a different possible state of the system as a whole. Whereas, in this space, each point of the space represents a possible state of a single individual particle, so a possible state of the system is not a single point (as in phase space), but rather a collection of points. Thus, in phase space the number of dimensions is a multiple of the number of particles, whereas in this space it is constant irrespective of the number of particles.

Now, our spacetime is finite dimensional (at least, up to the limits of observation), and specifically 4-dimensional. Let us assume that every particle only has a finite number of properties. Then, our property space is also finite dimensional.

We are familiar with the notion of spatial volume. This is possible because the dimensions of space are commensurable, and indeed naturally so. Can we extend this notion to one of spatiotemporal volume? Well, relativity provides a natural relationship between space and time. But, even supposing relativity was false, and there was no natural relationship, for the purpose at hand we could equally well choose an arbitrary, conventional, relationship. So, for any region of spacetime, we can consider its spatiotemporal volume.

Now, let us consider property space. We can also define a notion of property space volume. There may be some natural relationship between the spatiotemporal dimensions and the non-spatiotemporal dimensions of property space. Planck units, for instance, may provide such natural units. But again, if there are no natural units, then arbitrary ones will do just as well. Whatever the choice of units, the resulting units of volume will differ only by a constant factor.

Now, consider a finite spatiotemporal volume. Is the corresponding property space volume finite? Equivalent statements: Is there a greatest possible mass? Is there a greatest possible electrical charge (of either sign)? etc. If yes, then for every finite spatiotemporal volume, the property space volume is also finite.

Now, property space may be continuous or discrete. If it is continuous, then even a finite property space volume will contain an infinite number of points. But if it is discrete, then the number of points in a region of property space will be infinite only if the volume of the property space is infinite.

So, we can derive the following:

— If property dimensions are bounded, and property space is discrete, then a finite spatiotemporal volume can only be in a finite number of possible states.

Let us consider “Earth-like planets”. Clearly, there is an upper-bound to planetary size after which a planet should no longer be considered Earth-like, set by the laws of physics. Even if, through some miraculous violation of physical laws, there existed a terrestrial planet the size of the Milky Way, supporting Earth-like lifeforms, one would still say such an entity is too large for the term Earth-like. Thus, there is an upper bound on the spatial volume of Earth-like planets.

Is there a limit on the temporal volume of Earth-like planets? Science implies so — the Earth has only existed for a finite period of time, and after a finite period of time will likely cease to exist; and, long before its existence ceases, it will have become inhospitable to life, and thus no longer able to be called Earth-like. But, let us suppose that the Earth is in fact infinitely old; even if that were true, we are unaware of that fact, and thus we should say that a later finite temporal part Earth-like.

The point is, that Earth-like planets are important because they constitute the life-worlds of humanoid beings, at least up to current levels of technology. Thus, they are the entities whose possible or actual existence we as humanoid beings should be most interested in. “Earth-likeness” is not predominantly a scientific concept defined purely in terms of physical characteristics; rather, it is defined in terms of humanoid characteristics. Now, given known physical laws, the humanoid definition implies a physical definition; but, potentially, there may be other (actual or possible) universes with different physical laws, in which there exist planets which are not Earth-like by a scientific definition inspired by the physical laws of this universe, and yet we might still recognize as Earth-like in a humanoid sense.

So, let us say that the spatiotemporal volume of Earth-like planets is bounded. Are the associated property dimensions bounded?

— Physics suggests there may well be limits, e.g. a maximum possible temperature, maybe Planck temperature.

— But, even if there is, for instance, no maximum possible temperature, it would seem that, for instance, there is a maximum temperature T, achievable on an Earth-like planet, without threatening that planet’s status of Earth-like, in the generic case. By the “generic case”, I mean to exclude, for example, temperatures achieved in contexts such as nuclear weapons, fusion reactors, particle accelerators, etc.

— Let us suppose there is an upper limit T, to the temperature achievable in the general case on an Earth-like planet. But, suppose, there is no upper limit to the temperature achievable on an Earth-like planet in particle accelerators, etc. Imagine that we took such a planet, on which the maximum temperature in particle accelerators exceeds T, and created a duplicate, in which, by some supernatural means, it was ensured that every experience ever had by any observer on that planet was exactly the same as in the original planet, even though the temperature inside particle accelerators was limited to exceed T. So, for instance, even though the temperature inside the particle accelerator was limited to T, our hypothetical supernatural force would intervene to cause all experimental readings of the temperature inside the particle accelerator to be greater than T, precisely the same reading as in the original planet, even though the actual temperature was in fact limited to T. Thus, even though, the two planets would physically differ, all observations ever made by any humanoid being would be identical, between the two planets. Let us use this as a criterion of identity for Earth-like planets; therefore, we can define a maximum property space volume for Earth-like planets, such that, if any Earth-like planet has a greater property space volume, there is any identical Earth-like planet with no greater property space volume.

The above uses the example of temperature; but, we could reason the same for any property dimension.

So, an Earth-like planet has a finite property space volume. Is its property space volume discrete or continuous? Quantum theory would suggest that it is discrete. But, consider a hypothetical universe in which quantum theory is false. Even then, if we do not have quantum theory to discretize reality, we can appeal to our criterion of identity instead. For, it seems, that in any humanoid being, experiences are ultimately discrete rather than continuous. For all senses, there would be some finite amount, such that if the sensory input differed by less than this amount, the conscious experience would be indistinguishable. Any being, whose sensory system did not obey this principle, would not be a humanoid being. Now, let us consider, not direct sensory perceptions, but measurements taken through scientific instruments. In a quantum universe, these are inevitably discretized. But, what about non-quantum universes, in which measurement is possible to arbitrary accuracy? Let us suppose we have a planet in a continuous universe; let us transpose it to a discrete universe, using the same supernatural force as earlier to ensure the scientific measurements are identical. Thus, the property space volume of Earth-like planets is discrete, such that, if there is any Earth-like planet in a continuous property space, there is an identical Earth-like planet in a discrete property space.

Thus, we can conclude that:

— All Earth-like planets exist in a discrete, finite-dimensional, property space

— There is a finite upper bound on property space volume, such that every Earth-like planet has a property space volume less than or equal to that bound.

Therefore, there are only a finite number of possible Earth-like planets.

Hence, these are three possibilties with respect to the universe as a whole (let us also assume that spacetime is Archimedean):

1. The universe is spatiotemporally finite

2. The universe is spatiotemporally infinite, but Earth-like planets only exist within a finite, contiguous, sub-region of spacetime [we need the Archimedean property to prove that the sub-region is finite and contiguous]

3. The universe is spatiotemporally infinite, and an infinite number of Earth-like planets exist, but there are only a finite number of distinct Earth-like planets, but the universe contains an infinite number of identical copies of at least one of those planets

So, above is a sketched argument that there are only a finite number of Earth-like planets. I believe, one could sketch a similar argument, that there are only a finite number of humanoid observers. These two arguments are essentially related, in that one concerns the type of entities in whom we are interested (entities fundamentally like ourselves), and the other concerns the domain of their existence.

This should not be taken to mean that a space-faring civilization could not be humanoid. Clearly, it could be. Then, it would seem, that “Earth-like planet” is an insufficiently small domain. And yet, I think the point remains, that to be a humanoid, a being must be both finite, and also, finite beyond some upper limit. For example, a species with twice the average intelligence of homo sapiens would still be humanoid; however, a species with a billion times the average intelligence of homo sapiens would post-humanoid or non-humanoid. And, if there is a finite bound on the size of humanoid entities, then there is also a finite bound on the size of their domain, even if that domain could be interplanetary or interstellar. Indeed, what matters here is not the domain of the civilization as a whole, but the domain of the individuals. A single human being, in the space of one lifetime, could in principle visit every country on earth; every city on earth; every town on earth; etc. Thus, the actual domain of our civilization is more or less identical to the potential domain of each individual. But, conceivably, for a vast interstellar civilization, it might be far too vast for any single individual to visit all of it in one lifetime, even in principle, even if individuals in that civilization had lifetimes vastly exceeding our own.

Suppose the civilization was actually infinite. An infinite civilization could only contain a finite number of distinct humanoid individuals, so it must either contain an infinite number of identical copies of at least one humanoid individual, or a finite number of copies of humanoid individuals and an infinite number of non-humanoid individuals. Supposing the civilization contained only humanoid individuals, I think one might even be able to show that it must consist of an infinite number of disconnected finite segments, all absolutely identical, and thus be finite. For, how could two subregions that were absolutely identical (up to the identity of humanoid beings) sensibly interact? I suppose this issue is a bit like closed time-like curves.

Now, back to the three cases:

1. The universe is spatiotemporally finite

2. The universe is spatiotemporally infinite, but Earth-like planets only exist within a finite, contiguous, sub-region of spacetime [we need the Archimedean property to prove that the sub-region is finite and contiguous]

3. The universe is spatiotemporally infinite, and an infinite number of Earth-like planets exist, but there are only a finite number of distinct Earth-like planets, but the universe contains an infinite number of identical copies of at least one of those planets

Cases (1) and (2) differ only in inobservables, with respect to humanoid beings. So, we might say that, up to the identity of humanoid beings, they are the very same universe. And likewise, case (3) is, identical to (1) / (2), up to the identity of humanoid beings as well. Furthermore, why should we assume that there are an infinite number of absolutely identical copies, rather than a single copy, possibly with some complex topology? I mean, an infinite spacetime with an infinite number of repeating subregions is indistinguishable from a finite but unbounded spacetime, such as the surface of a hypersphere.

So, it seems, that in any case, the universe is spatiotemporally finite.