On Infinity and Eternity
As may be evident from my numerous past articles on the subject, I have an avid interest in the philosophy of Time. The nature of time is one of the oldest questions in philosophy, and one which has enormous repercussions on the physical sciences. Since the middle of the 20th Century, the evidence from cosmology has become stronger and stronger for the idea that our universe has a finite starting point, in the past. Many theistic philosophers– especially proponents of the Cosmological family of arguments— have jumped on these reports, claiming vindication for their belief that the universe was therefore created. When I disagree with this claim, I often find that the people with whom I am conversing becoming extremely confused. They ask me if I think the universe is eternal, and I reply that I do. Then, they ask me if I think that cosmologists like Alexander Vilenkin are wrong when they assert that the universe had a finite starting point. I reply that I actually agree with Dr. Vilenkin, and that I believe the universe has a finite past. This is where the confusion abounds: how can something be both finite and eternal?
The concept of Eternity has been defined in numerous different ways by numerous different philosophers, as this article from the Stanford Encyclopedia of Philosophy will attest. For the purposes of this article, I intend to use “eternity” in the sense of “timelessness.” That is to say, something which is eternal is not, itself, subject to time, change, or dynamism of any sort. It is, instead, a completely static entity. I contend that time is a part of the physical cosmos and therefore, the cosmos as a whole must be eternal. Time is a measurement of the universe’s properties; it does not govern the universe.
Since modern cosmologists utilize geometric models to understand the nature of space-time, I’m going to switch tack from pure philosophy to mathematics, now. You see, a part of the problem is that most people have a very limited understanding of geometry. Even the majority of college educated students with experience in calculus have only briefly (if ever) been introduced to any method of performing geometry outside of the Cartesian plane. To illustrate the issue, let’s imagine space-time as two-dimensional, for a moment, having one dimension of space and one dimension of time. Actual cosmological models of space-time are generally four-dimensional (3 space, 1 time), but that becomes incredibly difficult to visualize. Two-dimensional objects are much easier to picture. Our simple space-time, then, might look something like this:
Figure 1: Time versus Space on a Cartesian plane
On a model such as this, time and space are bidirectionally infinite. This is where the confusion arises. A person might reason that if cosmology tells us that the universe began at some time, say 
One way around this might be to simply utilize a Cartesian plane which is finitely bounded. Figure 2 illustrates a variation on our previous graph in which time is unidirectionally infinite. The bold line on the left marks a boundary,
Figure 2: Time versus space, with a past-finite boundary
If the usual Cartesian plane makes for an inadequate or confusing model of space-time, might there be a better model available to us? Let’s look at another type of two-dimensional graph, one which is somewhat less familiar. Now, I’ll fully admit that this is certainly not a perfect model for modern cosmological understandings of space-time, but it does provide a coherent example of a system in which time has some initial point. This is a two-dimensional Polar Graph. The displacement, 
Figure 3: Time versus space on a polar graph
On this model, our pole would represent the time
At first, the polar graph might seem like a completely alien thing, to you. Many people have a very difficult time relating to polar graphs, while Cartesian graphs seem fairly natural and intuitive. However, I’ll bet you’ve actually been exposed to a sort of polar graph, before, without even realizing it. Think about any time you’ve seen a map of the globe. Now, generally, such maps are displayed in a sort of Cartesian, rectangular way; but you know that the Earth is (more or less) spherical, and that a rectangle can’t really do its surface justice.
Looking at the first map, a child might be forgiven for asking, “What is beyond the northern border of the map?” We again see a sort of conceptual wall, in that map, which the human brain wants to believe has another side. However, the second map shows why such a question is nonsensical. If you leave the North Pole, any direction you head will take you to the south. It doesn’t even make sense to ask, “What is north of the North Pole?” Even the statement, “There is nothing north of the North Pole” is not quite right, as it implies that there exists some place north of the North Pole in which nothing resides. The statement, “There was nothing before time,” is nonsensical in exactly the same manner.
This is what I mean when I say that the universe is both past-finite and eternal. The dimension of cosmic measure which we call “time” has a temporal pole, in the past; but the universe, as a whole, is a static entity which is not subject to any transcendental dynamism. Even though there may be a finite, initial point in time, there was never a point at which time did not exist.
Boxing Pythagoras 
Well said, and impressively easy to follow. It’s not an easy subject, so triple points for that.
A further problem with theists trying to swim in cosmology is they cherry pick Vilenkin and simply ignore that actual conclusions of the BGV, namely that all models break down. This, of course, is because the BGV is based only on Special Relativity and makes no provision whatsoever for QM. They can’t seem to understand that the BGV Theorem is not saying “the universe had a beginning,” but that inflationary models cannot go infinitely into the past, and require physics other than inflationary models to describe the boundary condition. As the paper itself states:
inflation alone is not sufficient to provide a complete description of the Universe, and some new physics is necessary in order to determine the correct conditions at the boundary. This is the chief result of our paper
I completely agree. I’ve expounded on that very same subject in two previous posts– most recently in my article On the Kalam Cosmological Argument.
BP this article actually helps me understand what you were trying to tell me over on my blog. Thanks. My first question is probably rather obvious- why should we prefer the polar graph over the second Cartesian graph listed above?
Models of space-time geometry are generally non-Euclidean, in nature. As such, it is preferable to use tools which have been explicitly designed to represent non-Euclidean geometry, and a polar graph is one such tool.
Again, it’s similar to attempting to map the globe onto a Cartesian plane. You end up creating some fairly problematic distortions. Look at that first map which I posted, in the article. If I were to mark the military station at Nord, Greenland, it would appear to be extremely far from Siberia, on this map, when it is actually closer to Siberia than to Ontario.
We can use either the Cartesian or the polar graph to make a coherent picture of the mathematics, and both can certainly be useful tools. The polar graph simply makes it easier to visualize and comprehend the reason why “north of the North Pole” and “before time” are nonsensical questions.
I knew I read something like this before but was not sure where until I began looking in my cosmological journal papers folder. In similar yet different manner Rüdiger Vaas in his 2003/2004 paper “Time before Time : Classifications of universes in contemporary cosmology, and how to avoid the antinomy of the beginning and eternity of the world”(http://arxiv.org/pdf/physics/0408111.pdf)
I am looking forward to responding your well thought and argued article.
Thank you so much for linking me to the Vaas paper! I found it intensely interesting!
On your polar graph, where the lines “come together” and t=0, does that point represent the big bang?
It represents the initial point in time, whether that means the Big Bang or some hypothetical pre-Big-Bang moment, given a past-finite universe.
Does it represent the initial point in space as well? i.e. the initial point in spacetime?
It represents a boundary to space-time, but it would not necessarily be the initial point of space-time. Spatial dimensions could coherently provide an infinite number of points within that temporal boundary. That is to say, it represents an initial state of space in the temporal dimension.
Very enjoyable reading. Thanks for sharing.
“there was never a point at which time did not exist.”
Out of interest, have you read any of the academic writing / philosophy that argues (or points out, depending on your point of view) that this very phrasing is confused/misleading?
And its converse, of course. I don’t mean to limit it just to the above sentence; I mean the phrasing in general so as to include ‘time does exist’ etc.?
Not so much as I’d like, as yet. I’ve read G.J. Whitrow’s wonderful The Natural Philosophy of Time, and I am working through Julian Barbour’s The End of Time, but I am always looking to read more on the subject.
As opposed to works specifically on the subject, I suppose I had in mind an alternative, holistic perspective; i.e. approaching the subject mater from the perspective of the Ordinary Language philosophers, Ludwig Wittgenstein’s, or others’? From a perspective that’s not mathematical or scientifically orientated?
I haven’t yet had much opportunity to read on that. Would you have any suggestions? Which of Wittgenstein’s texts should I look to?
It was only a suggestion as I thought it might be of interest given your general interests. Not that any of it necessarily provides a contradictory view to anything you’ve expressed; it’s just that it would approach the subject matter from a completely different angle/perspective, and this is where the interest might lie.
In respect to L.W., on the whole, his “Philosophical Investigations” would be of general interest but I’m not sure he specifically dedicates much to ‘time’– but it is one of many concepts / words he does address. I’m sure I remember specific paragraphs in his drafts of PI (“The Blue and Brown Books”), in case that would be of more interest. Arguably more digestible, too.
I’m sorry if this seems rather vague– what I mean is that he has a fundamentally different approach to these subject matters (as did, say, the Ordinary Language philosophers) and so if you grasp his way of thinking on the whole, you can turn this to discussions of ‘time’, ‘ethics’, ‘mathematical language’, etc. and it assumes a completely different perspective.
L.W. I think might be of particular interest to you given that he was a mathematical/logical prodigy who approached every avenue of life from this perspective before abandoning it all completely in later life in respect to certain subject matters, e.g. religion, ethics, aesthetics, etc.).