Thanks for your response to my question. However, it would probably be worthwhile for me to explain my philosophy of science when it comes to topics like this one.
In general, I think mathematics alone is insufficient to require us to commit to the ontological reality of whatever it is the mathematics represents. We can describe a whole host of concepts mathematically that ultimately fail to actually represent anything real. For example, physicists frequently perform calculations in the "center of mass" frame whereby a relevant parameter of the entire system is represented as though it were carried at a single point. But nowhere does that require that such a thing as the "center of mass" is a real entity. Its merely a mathematical tool. We model the system as though it were a point called the "center of mass" because doing so allows us to solve equations in ways that we otherwise couldn't.
Now, that is a crude example, but it shows the role that mathematics plays in physics. It functions as a language that allows us to describe phenomena, but it is up to us to determine how we ought to understand those descriptions.
In general, I'd say a good theory ought to be taken in a realist sense. But what makes a theory good? Well, not only must a theory make sense of observational data, but it also must make novel predictions. A novel prediction is basically an accidental characteristic of a theory that ends up being validated by observational evidence after the fact. One good example of a novel prediction would be gravitational waves in general relativity. These are unique concepts entailed by Einstein's model as nothing more than a consequence of the math, and it is only because of this that scientists started looking for them. It took 100 years, but just a few years ago the LIGO experiment confirmed measurements of gravitational waves from a binary black hole system (interestingly enough, black holes themselves are a good example of a novel prediction, and you can look up research on the phenomenon of "gravitational lensing" if you're curious). If a theory both explains the original data and makes novel predictions that are later confirmed by further research, then it is a good theory. This doesn't mean it is perfect, but it does give a really good indication that the theory is at least on the right track.
When it comes to the Big Bang Theory, there are a few novel predictions that give us reason to believe it is on the right track. Most importantly would be the Cosmic Microwave Background. This is effectively the leftover radiation from the beginning of the universe. First of all, the mere prediction of this background radiation is novel to the Big Bang Theory, which is why it was so exciting when it was identified a few decades later. But more importantly, we can calculate what temperature that radiation should be given the parameters of the Big Bang Theory (one of which is the age of the universe), and then we can measure the Cosmic Microwave Background. When we do that, we find that our measurements match predictions. There really is no explanation of this other than to say the theory is on the right track.
With that said, there are other considerations that I believe can lead us to doubt the ontological reality of particular parts of a given scientific theory. Philosophically, I think anything that is metaphysically suspect ought not to be considered representative of reality. Theologically, anything entailed by a scientific theory that precludes an essential element of Christian doctrine must also not be interpreted as an accurate description of reality. But, if the theory meets the criteria of a good theory as defined above, we are only justified in rejecting the ontological reality of the suspect part of the theory. We could not, however, use that as a reason to doubt whether or not the theory as a whole is on the right track.
I take your argument about being unable to take an infinite quantity and make it less than an infinite quantity very seriously, and I think it's a good one. In fact, I may borrow it at some time in the future. However, in light of my philosophy of science, I think it works only as a philosophical challenge to the ontological reality of "the singularity" as a physical object, not as a philosophical challenge to the validity of the Big Bang Theory as a whole. What do you say to that?
I think I can see where you’re going with this argument. So let me begin with a few admissions, and then my response.
First, I too think that it is highly interesting that we would identify a theory such as the Big Bang which makes novel predictions that are confirmed by observational evidence on the level of the Cosmic Microwave Background Radiation. The Cosmic Microwave Background Radiation is indeed the strongest argument against my paper, as you have noted. And it is a most remarkable prediction. For this reason, I am not surprised to find a scientist objecting in this way. So for the benefit of my lay readers who are unfamiliar with cosmology, let me briefly explain what the Cosmic Microwave Background Radiation is.
The Cosmic Microwave Background Radiation is a black body radiation which is predicted by the Big Bang Theory. The prediction holds that if the Big Bang occurred in the past, then we should be observing a kind of “relic radiation” left over from the early expansion period. It is believed that this occurred roughly 380,000 years after the singularity during the recombination period in which protons and electrons first appeared in our universe due to photon decoupling.
The cosmic microwave background radiation (or, CMB for short) is observably isotropic (or, uniform) in all directions and exhibits very tiny, minute perturbations (or anisotropies) which resemble those precisely predicted by the Friedmann-Lemaitre model. For this reason, many scientists hold that the CMB is our best direct evidence for the Big Bang. It is, admittedly, a strong argument in support of the Big Bang Theory. But not strong enough. Let me explain...
The argument you have used here necessitates that the present expansion phase of our universe began from something called a “non-singular” expansion. The reason why this is true is simple: Infinite quantities are definitional to singularities. As American philosopher of physics, John Earman, writes: “A maximal spacetime is singular if and only if it contains an inextendible path of finite generalized affine length.”  It is this “inextendible pathology” which amounts to the singularity possessing the various infinite quantities I have innumerated.
Thus in order for your argument to be right, the CMB must be the relic radiation of a non-singular expansion—not possessing such important qualities as infinite temperature, pressure, etc. This makes sense because the early universe is either singular or non-singular, either infinite or finite in size. And, as you yourself have admitted, the universe could not have expanded from the state so described. Therefore, our original predictions must be wrong.
To be fair, your claim is certainly plausible. We might reason for example that due to the evidence of the CMB, it is proper for us to conclude that the current expansion phase must have had a non-singular beginning; that it surely must have lacked the important qualities I have noted, such as infinite spacetime curvature, temperature, pressure and so on. The sheer existence of the CMB with its minute anisotropies is surely enough evidence to conclude that the movement we see in the heavens above us was generated by a non-singular expansion.
Such a supposition however is precluded on the basis of what the CMB itself predicts, namely—the presence of closed trapped surfaces of inextendible length which virtually guarantee a true singularity. As Hawking and Ellis write in their now classic volume, The Large Scale Structure of Spacetime: “...past-directed closed trapped surfaces exist if the microwave background radiation in the universe has been partially thermalized by scattering” which it has.  This would therefore mean that the evidence of the CMB is a strike against the hypothesis of a non-singular expansion which you posed as an alternative.
Of course such models have been proposed which argue that the current expansion we see comes about as a result of a non-singular “bounce” from a previous expanding phase. But as Borde, Guth, and Vilenkin show, no “bounce” universe can ultimately avoid the need for a singularity in the Friedmann-Lemaitre sense. Therefore, it would appear that we have two choices: (1) argue that the current expansion phase was indeed non-singular, despite the previous points, or (2) admit that the current description of the singularity is correct, and that therefore the CMB, however uncanny in its predictions, is simply not a relic of the Big Bang. So which direction does the evidence point?
Well, concerning the first alternative, there is virtually no discussion amongst scientists on mathematically redescribing the classical singularity so as to render its quantities finite. For, as Hawking and Ellis note in their formative volume: “The theorems on the existence of singularities could possibly be refined somewhat, but in our view they are already adequate.”  In fact, this is why such great scientists as Sean Carroll hold that the singularity proves that General Relativity is wrong. Since it predicts a singularity, as the professor describes, then, despite our best empirical evidence, we must hold that Einstein was mistaken.
But such an alternative should be rejected for good reason since it would have us throwing out what we have already experimentally detected (such as the gravitational ripples you mentioned) solely on the grounds that it makes the classical singularity unavoidable. And this is simply confused. We should rather conclude that our observations are right, and that we live in a universe governed by General Relativity, rather than claiming that General Relativity must be wrong since it leads us into mathematical irresolvables which preclude a naturalistic cosmic creation.
But that only seems to make the second alternative all the more appealing, that the CMB, however uncanny in its predictions, is simply not a relic of the Big Bang. Instead, it becomes inexplicable, like dark matter. For now, those are the issues, as I see them.
So I thank you Lucas once again for your stirring engagement. Such discussions, I believe, are vitally important to our understanding of both God and his universe.
Ben Fischer <><
. Earman, J., 1995, Bangs, Crunches, Whimpers, and Shrieks: Singularities and Acausalities in Relativistic Spacetimes, (New York: Oxford University Press), 36.
. S. W. Hawking and G. F. R. Ellis, 1973. The Large Scale Structure of Spacetime (Cambridge University Press, Cambridge, 1973), 348.
 Ibid. 363.