Philosophy of science is a fascinating and complex field. The job requires one to philosophize about well-established scientific observations, patterns, theorems, theories, models, laws of nature, and more. A good philosopher of science will also philosophize about the standards of science and the standards of communicating science. And a brave philosopher of science will even attempt to hold scientists to these standards when they stray.

Recently I’ve been reviewing the transcripts of the 2014 debate between Sean Carroll and William Lane Craig. Carroll is one of the brightest minds working in cosmology. In all of his talks and debates he comes across as brilliant, friendly and likable. I watched an online video of the debate a few months after it took place. I was shocked at some of the tactics Carroll used. I was convinced that cosmologists and philosophers of science would take Carroll to task for what I can only describe as his dishonesty.

Aron Wall wrote a series of very good blog posts on the debate and spoke plainly about some aspects of Carroll’s statements, but I think it is time to publish my own analysis. A video of the debate can be seen here. A complete transcript of the debate can be found here.

Carroll was dishonest on two important points.

- Carroll claimed BGV theorem does not imply the universe had a beginning.
- Carroll claimed that quantum eternity theorem (QET) was better than BGV theorem.

Let’s look at each claim in depth.

*Carroll claimed BGV theorem does not imply the universe had a beginning.*

*Carroll claimed BGV theorem does not imply the universe had a beginning.*

“Where Dr. Craig says that the Borde-Guth-Vilenkin theorem implies the universe had a beginning, that is false. That is not what it says.” – Sean Carroll

This claim is perhaps the most shocking in your face denial of science I had ever seen by a well-known scientist. I can only imagine that Alexander Vilenkin was as shocked as I was when he saw this. Regarding BGV theorem, in his book *Many Worlds in One, *Vilenkin writes:

“It is said that an argument is what convinces reasonable men and a proof is what it takes to convince even an unreasonable man. With the proof now in place, cosmologists can no longer hide behind the possibility of a past-eternal universe. There is no escape: they have to face the problem of a cosmic beginning.” (176)

During the debate, Sean implied that BGV theorem is like the Penrose-Hawking singularity theorems in that the description of the classical universe “breaks down.” Not true. In singularity theorems the math “breaks down” in the computer models near t=0. BGV theorem is different. It is based on simple geometry, not complex computer models. Regarding expanding universes, the answer is the answer. The geodesics are incomplete which means there is a boundary, a beginning of space and time

The BGV incompleteness theorem is a mathematical theorem that simply says that any universe like ours that is expanding on average over its history must have had a beginning. It’s a simple theorem, very straightforward and very powerful. BGV theorem is not a singularity theorem, such as the Penrose-Hawking singularity theorem, although it is certainly compatible with these theorems. Both the BGV theorem and the singularity theorems point to the conclusion that all matter, energy, space and time came into existence at the first moment of time. But the BGV incompleteness theorem is even more powerful than the singularity theorems because it implies that the multiverse, if one exists, also must have had an ultimate beginning.

On page 4 of the 2003 BGV theorem paper, the co-authors, Borde-Guth-Vilenkin, make the point that the theorem applies to “cosmology in higher dimensions” meaning the multiverse. If the multiverse exists, then its universe generator is necessarily causing the multiverse to expand and so the multiverse itself must have had a beginning.

Compare Carroll’s comment during the debate with his blog post titled “Let the Universe Be the Universe” dated September 25, 2012. Carroll writes:

The second major point Craig makes is a claim that I ignored something important: namely, the Borde-Guth-Vilenkin singularity theorem. This is Craig’s favorite bit of cosmology, because it can be used to argue that the universe had a beginning (rather than stretching infinitely far backwards in time), and Craig is *really devoted* to the idea that the universe had a beginning.

Isn’t that interesting? If Carroll really believed BGV theorem does not imply a beginning, why would he write that “it can be used to argue that the universe had a beginning”?

We have even more evidence that Carroll understands the BGV theorem implies a beginning in a 2004 paper Carroll wrote with Jennifer Chen called “Spontaneous inflation and the origin of the arrow of time.” There’s a footnote on page 27 that cites the BGV theorem paper. Carroll and Chen write,

“A well-known feature of eternal inflation is that it does not escape the problem of singularities, as it cannot be eternal to the past [93].”

Technically, this statement is not even correct. BGV theorem is an incompleteness theorem, not a singularity theorem. It does not describe a singularity at all. More importanty, Carroll again admits that BGV theorem implies a beginning.

A few months after the debate with Craig, Carroll published a paper called “In What Sense is the Early Universe Fine-Tuned?” On page 13, we read this footnote:

11The Borde-Guth-Vilenkin (BGV) theorem [33] demonstrates that spacetimes with an average expansion rate greater than zero must be geodesically incomplete in the past (which is almost, but not quite, equivalent to saying there are singularities). This has been put forward as evidence that the universe must have had a beginning [34]; there are explicit counterexamples to this claim [35], but such examples are arguably unstable or at least non-generic. However, while the BGV theorem does not assume Einstein’s equation or any other equations of motion, it only makes statements about classical spacetime. It is therefore silent on the question of what happens when gravity is quantized.

Again, it is clear that Carroll understands BGV theorem implies a beginning. He seeks to evade the theorem by bringing up “the question of what happens when gravity is quantized.” Here Carroll is assuming something very significant, that gravity can be quantized. Einstein tried and gave up. Others have been trying for decades with no luck. Some scientists are finally coming to terms with the idea that gravity is simply not an interaction and so will never be quantized.

Carroll enlisted Alan Guth, the G in BGV theorem, for help in this elaborate ruse. Carroll showed a picture of Guth holding a sign that read “[*The universe is] very likely eternal but nobody knows.”* The picture had its desired effect on the debate. It was powerful and shocking, especially for the viewers who were aware that all of Guth’s relevant science papers come to the opposite conclusion – that BGV theorem strongly implies the universe had a beginning.

Consider Guth’s 2007 paper *Eternal inflation and its implications**. *In the abstract Guth writes,

“Although inflation is generically eternal into the future, it is not eternal into the past: it can be proven under reasonable assumptions that the inflating region must be incomplete in past directions…”

Perhaps Guth is saying that inflation is not eternal into the past but the universe itself can still be eternal into the past? No. On page 14, Guth writes,

“If the universe can be eternal into the future, is it possible that it is also eternal into the past? Here I will describe a recent theorem [43] which shows, under plausible assumptions, that the answer to this question is no.”

Guth then describes the 2003 BGV incompleteness theorem. An interesting footnote demands examination. A theorem is considered most powerful when it has the widest possible applicability. The footnote discusses earlier theorems on the topic which were not as powerful as the 2003 version:

There were also earlier theorems about this issue by Borde and Vilenkin (1994, 1996) [44, 45], and Borde [46] (1994), but these theorems relied on the weak energy condition, which for a perfect fluid is equivalent to the condition ρ + p ≥ 0. This condition holds classically for forms of matter that are known or commonly discussed as theoretical proposals. It can, however, be violated by quantum fluctuations [47], and so the applicability of these theorems is questionable.

The added value of the 2003 theorem is that it applies to a much wider set of models. The earlier theorems could be violated by quantum fluctuations but Guth appears to be saying that criticism does not apply to the 2003 theorem.

Guth’s paper then goes on to describe a cosmological model that evades BGV theorem, the Aguirre-Gratton model. Earlier Guth had commented that no model with “reasonable” or “plausible” assumptions could evade BGV theorem. One must conclude that in Guth’s judgment the Aguirre-Gratton model does not have reasonable or plausible assumptions. But this is the model Sean Carroll endorsed in the debate.

When I saw the picture of Guth holding the sign, I thought perhaps he was planning to publish a new paper describing a model with reasonable assumptions that could evade BGV theorem. Six years have passed since the debate. I no longer think a paper is coming or that such a model is possible.

This episode represents a very interesting chapter in the sociology of science. Why would Guth agree to appear in a photograph that publicly undermines an important theorem bearing his name and all of his relevant science papers? That question has never been answered.

So, does BGV theorem imply the universe/multiverse had an ultimate beginning? Yes, of course it does.

*Carroll claimed that quantum eternity theorem (QET) was better than BGV theorem.*

*Carroll claimed that quantum eternity theorem (QET) was better than BGV theorem.*

“If you need to invoke a theorem, because that’s what you like to do rather than building models, I would suggest the quantum eternity theorem. If you have a universe that obeys the conventional rules of quantum mechanics, has a non-zero energy, and the individual laws of physics are themselves not changing with time, that universe is necessarily eternal.” – Sean Carroll

Carroll’s blog “Post-Debate Reflections” cited his paper “*What If Time Really Exists?*” to describe the QET. I approached Carroll’s paper with interest thinking it was going to lay out a widely applicable mathematical theorem constraining all future cosmological models to be past-eternal if they were to be considered viable. That was not what I found.

The paper did not, in fact, attempt to prove a new theorem at all. And the term “quantum eternity theorem” does not even appear in the paper. In fact, the term does not appear anywhere in the scientific literature until after the 2014 debate.

Carroll’s paper began as an appeal to scientific anti-realists to consider the possibility that time is real. Carroll is not arguing that time is absolute or relative or anything in particular. He is simply arguing only for the reality of time. As a scientific realist, it would be hard for me to disagree with Carroll’s perspective here. The paper goes on to describe the fact that QM’s Schrodinger equations can move backward and forward in time. That is to say, once you know the wavefunction at a specific point in time, then you can calculate the wavefunction at any point along an infinite timeline from infinity past to infinity future.

Carroll writes:

John Wheeler, following Niels Bohr, liked to admonish physicists to be radically conservative – to start with a small, reliable set of well-established ideas (conservative), but to push them to their absolute limits (radical) in an effort to understand their consequences. It is in Wheeler’s spirit that I want to ask what the consequences would be if we take time seriously. What if time exists, and is eternal, and the state of the universe evolves with time obeying something like Schrodinger’s equation?

Here I must throw a penalty flag. Time can be real and not eternal. Carroll is committing circular reasoning. First, Carroll’s paper presupposes time is eternal and then after the debate Carroll claims his paper demonstrates a “theorem” that time is eternal. This is false and dishonest. There is nothing to stop time from coming into existence when the universe comes into existence and nothing to stop time completely if/when the universe stops changing. Time is simply a measure of change. At some point in the future we know the universe is going to run out of usable fuel, the stars will go out and the universe will no longer be life-supporting as it reaches maximum entropy. If nothing meaningful is changing, then I would argue that time stops. What good are Schrodinger’s equations at that point?

Needless to say, Carroll’s “quantum eternity theorem” doesn’t require any cosmological model to be past eternal. It only demonstrates that if an eternal universe existed, then Schrodinger’s equation would be able to calculate the wavefunction anywhere along an infinite timeline. But everyone working in QM knew that already.

Aron Wall wrote an interesting piece on Carroll’s use of QET. He begins with a quick overview of quantum mechanics and then makes this statement,

“It’s a little bombastic for Carroll to even refer to this as a ‘theorem,’ since it’s just an elementary restatement of one of the most basic principles of QM.”

In his Post-Debate Reflections, Carroll basically admits this “theorem” is very weak and easy to evade:

The time parameter in Schrödinger’s equation, telling you how the universe evolves, goes from minus infinity to infinity. Now this might not be the definitive answer to the real world because you could always violate the assumptions of the theorem but because it takes quantum mechanics seriously it’s a much more likely starting point for analyzing the history of the universe. But again, I will keep reiterating that what matters are the models, not the abstract principles.

I understand that Carroll likes models, but his attack on theorems is unwarranted. We use theorems to constrain and judge the models. For example, the Steinhardt-Turok eternal cyclic universe model is no longer highly regarded precisely because it violates BGV theorem. Carroll knows this and his attack on theorems is really an attack on science (*Steinhardt-Turok eternal cyclic universe model*).

Carroll also knows that QET is not really a theorem at all and so cannot honestly be described as better than BGV theorem. Any cosmological model can violate Carroll’s concept of the QET and no cosmologist would care.

*Conclusion*

*Conclusion*

Uninformed viewers of the 2014 Carroll-Craig debate may think that Carroll won the debate. After all, Carroll is a cosmologist, he’s brilliant, confident and likable. He attacked and undermined BGV theorem, the science upon which Craig often bases his arguments. Carroll even enlisted the help of Alan Guth to undermine his own theorem. Then Carroll sprung the quantum eternity theorem on Craig, who was caught off-guard by the term since it had never appeared in the scientific literature.

Informed viewers of the debate came away with a different view. Carroll’s denial that BGV theorem implies the universe/multiverse had an ultimate beginning was shocking and dishonest. Also, informed viewers saw it as rather underhanded for Carroll to claim “quantum eternity theorem” was a recognized theorem that implies the universe is eternal into the past.

On the basis of the science, Craig was truthful with the audience and Carroll was not.

Truth will win out as they say.

Carroll’s behavior can only be seen as harmful to science.