Sunday, April 5, 2015

A New Proof

I have just discovered a new proof of the existence of God.

Consider the following sentence:

(S) If this sentence is true, then God exists.

Suppose sentence S is true. Then the first clause is satisfied, so the second clause is true. Thus, God exists.

What the preceding paragraph proves is that if sentence S is true, then God exists. But that is exactly what sentence S asserts. So that means we have proved that sentence S is true! And therefore God really does exist.

As a result of this proof I am longer an atheist. Sorry to disappoint everyone.*

* Note this proof also works for unicorns. Yay, unicorns exist!

Tuesday, October 21, 2014

Craig and BGV

I hate writing more posts about William Lane Craig, because I think he gets way more attention than he deserves already. But.... I can't let this pass.

I just ran across this post from Craig's website. Craig is responding to Carroll and, after quoting him, replies thusly:

Here Carroll claims that to have a singularity in the past does not mean to have a beginning; it means only that SOME [past-directed] geodesics come to an end. He says that others might not. On this interpretation, the BGV Theorem is consistent with some geodesics’ being infinitely extended into the past. But that is precisely what the theorem proves to be impossible. The theorem requires that ALL actual, past-directed geodesics eventually come to an end. In order for the universe to be beginningless, there must be infinite, past-directed geodesics. That’s why Borde, Guth, and Vilenkin take their theorem to prove that any universe which has, on average, been in a state of cosmic expansion throughout its history cannot be past-eternal but must have a beginning.

OK, so, in my last post (yeah, I know, no apologies, I'm only going to post when I feel like it, so there), I quoted the conclusion from the actual BGV paper:

... we see that if Hav > 0 along any null or noncomoving
timelike geodesic, then the geodesic is necessarily
So, it's obvious from the paper that the requirement that a geodesic ends is not applicable to geodesics that are timelike and comoving. In other words, Craig is simply wrong when he writes that BGV requires "ALL actual, past-directed geodesics eventually come to an end."

Either Craig hasn't understood the conclusion of the paper he cites so frequently, or he is deliberately mischaracterizing the paper for his readers so that they will think he has decisively refuted Carroll.

What do you think?

Friday, May 23, 2014


OK, so last time I claimed that our best, experimentally successful model of the early universe is one that is infinitely old and has no initial singularity. If you are savvy about these things (from listening to Craig debates, for instance) you are wondering, "But what about the Borde-Guth-Vilenkin theorem? Doesn't it prove that an inflating universe must have had an initial singularity?" The answer is, "No, it doesn't."

There has been a lot of confusion about this, with clip quotes from one or another of the paper's authors being traded to "prove" that the theorem does, or doesn't, prove the universe had a beginning. So let's look at what the theorem actually says.

"Any theorem is only as good as its assumptions," writes Alexander Vilenkin in a letter to Lawrence Krauss. So let's start with the assumptions of the theorem. Roughly speaking, there are two:
  1. Spacetime is classical.
  2. Spacetime is expanding on average.
(I say "roughly speaking" because there are all sorts of technical issues about spacetime congruences and so forth that one needs to make these assumptions precise, but I think these short versions are sufficient to understand the main philosophical issues involved.)

Now let's jump to the conclusion, which I quote from their paper:
... we see that if Hav > 0 along any null or noncomoving
timelike geodesic, then the geodesic is necessarily
Some translation: a "timelike geodesic" is simply the path that an object will travel on if it is not subject to any forces other than gravity. Similarly, a "null geodesic" is the path that a light ray will travel. "Hav > 0" is the mathematical statement of assumption (2.): the universe is expanding on average. "Past-incomplete" means that if you try to follow one of these paths backwards in time, you can only do so for a finite amount of time.*

OK, so here's my first point: the BGV theorem is not a singularity theorem!

The conclusion says nothing at all about singularities: it only says that certain paths cannot be extended infinitely backward in time. One way that this might happen is if the path encounters a singularity. But that is not the only way it can happen.

The other thing that can happen is that, as we trace the path backward in time, we encounter a region where one (or both) of the assumptions of the theorem is no longer valid.

Start with assumption (2.). The path can enter a region in which spacetime is static, or contracting, or cyclically expanding and contracting. Then assumption (2.) is violated and the theorem's conclusion is avoided. In spacetimes like these, the BGV theorem simply doesn't apply.

What about assumption (1.)? Classical spacetime is a pretty basic assumption in any sort of cosmology. But it is expected to break down in the quantum gravity regime, where we encounter "spacetime foam" of some sort. This is difficult to discuss, since in the absence of a good theory of quantum gravity, no one has any idea what spacetime foam should look like. But it's possible that the BGV theorem is pointing us to a place where quantum gravity comes into play.

So the BGV theorem does have something very interesting to tell us about the early universe: namely, that the infinitely old, infinitely expanding inflationary period that I discussed in the previous post cannot be the end of the story. Not just because of vague speculations about the Planck epoch, but because of the properties of classical spacetime itself. Here's what the authors said in the paper itself:
Whatever the possibilities for the boundary [where the geodesics come to an end - RNO], it is clear that unless the averaged expansion condition can somehow be avoided for all past-directed geodesics, 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.

But the BGV theorem does not say that spacetime must be singular; still less does it say that there was an initial singularity from which the entire universe arose. This is why I say that the theorem is irrelevant to the Kalam Cosmological Argument: it doesn't say anything about whether the universe had a beginning or not.

* A few more technical points: by "backward in time" I mean in the opposite time direction from the direction in which the universe is expanding. And by "finite amount of time" I mean time as measured by a clock carried along with the moving object (proper time). In the case of null (light) rays, we have to use an "affine parameter" rather than the proper time.

Thursday, May 22, 2014

The Universe Is Infinitely Old (Says Cosmology)

The Secular Student Alliance at my school recently  hosted a debate on the topic "Is there a God?" The theist side was very well prepared and did a great job in the debate, as even the secular students in the audience agreed. One of the arguments they presented was the Kalam Cosmological Argument, which was presented rather the same way that William Lane Craig presents it. The discussion brought up the Borde-Guth-Vilenkin Theorem (BVG for short), which Craig has used as well, as support for the premise "The universe began to exist." I want to talk about why the BVG theorem is irrelevant to the Kalam Cosmological Argument, but first, by way of preliminary, I want to discuss the current state of cosmology.

Cosmology begins with Einstein's equations of General Relativity (GR for short), and asks whether these equations, applied to the universe as a whole, are capable of explaining what we see when we look out into deep space. GR relates the curvature of spacetime to the energy content in the universe, so in order to solve the equations we need to know what the universe is filled with. There are three basic types of energy we need to consider:
  • Matter in the form of galaxies, dust, dark matter, and the like,
  • Radiation, including particles moving so fast that they are relativistic, and
  • Cosmological constant, aka "dark energy."
The three forms of energy behave differently, so different ones are important at different times in the evolution of the universe. For most of the last 13.7 billion years, the expansion has been matter-dominated. But in the far future, the expansion will be dominated by the cosmological constant, and at very early times (the first 50,000 years or so), the expansion was dominated by radiation.

This plot shows the "scale factor" - roughly speaking, the size of some patch of the universe, as a function of time.The universe described by this model fits extremely well with the observations of distant galaxies, supernovas, quasars, etc.

If the early universe is indeed radiation dominated, then the scale factor goes to zero at some finite time in the past: that is, there is a Big Bang - an initial singularity.

However, we now have an alternative account of the earliest moments of the universe. Inflationary cosmology, proposed by Alan Guth in 1980, then in a corrected from by Linde and (independently) by Albrecht and Steinhardt in 1982 supposes that before the radiation-dominated epoch there was another epoch, dominated by a cosmological constant - but a very much larger cosmological constant than the one we measure now. I'm not going to go into the reasons these physicists thought there might have been a very large cosmological constant in the early universe, which then "switched off" (meaning it wasn't really a "constant", obviously): you can read about it at the Wikipedia page if you're interested.

The inflationary model was able to explain several features of the universe that had been puzzling in earlier cosmological models: the flatness problem, the horizon problem, and the monopole problem. In science, though, explanatory power is not enough for a theory to become accepted. In addition, a theory has to make novel predictions that are confirmed by experiment before scientists accept it as (likely to be) true.

(I can't help pointing out how different this is from theistic "explanations," in which God is claimed to be the explanation of things like life, morality, or the universe, but where there is no concern for making testable predictions about these realms.)

We now have several good reasons to think that there was in fact such an inflationary epoch. One of these is the pattern of fluctuations of the cosmic microwave background, that fits extremely well with the predictions of inflation:


Another is the very recent BICEP2 result, that seems to show the effects of quantum gravity on the polarization of the cosmic microwave background, in a way consistent with the predictions of the inflationary model.

So inflationary cosmology replaces the initial singularity with a period of exponential expansion.

How long is this inflationary period? Well, an exponential function never reaches zero, so the inflationary period is, in principle, infinitely long!

Let's sum that up: According to our best, experimentally verified model of cosmology, the universe is infinitely old and has no initial singularity! 

This is the current state of our understanding of the early universe. Now, I have to admit right away that no physicist thinks the inflationary model is the end of the story. The exponential expansion is so fast that in a very short time the scale factor reaches the Planck realm, where we expect GR to break down and quantum gravity to come into play. So most diagrams of the early universe insert a quantum gravity region before the inflationary epoch. (In this diagram from Andrei Linde it's labeled "Space Time Foam.")

From a lecture by Andrei Linde (link is in the text): a schematic plot of the size of the universe over time.  The extreme left is speculative, the inflationary epoch is the one probed by the recent BICEP2 measurement.

There has been much discussion of what went on before the inflationary epoch: quantum foam, the no-boundary proposal, the cyclic universe, and so on. There is even a version called "eternal inflation," in which some portion of the universe goes on inflating forever, while pocket universes like ours bubble off from time to time. In some of these models, time is finite in the past. In others, it is infinite. But all of them are pure speculation: there is to date no experimental confirmation of any of these scenarios.

Does modern cosmology support the premise that the universe had a beginning? Emphatically, no! Our best model extends infinitely into the past, with no initial singularity. We know better than to take that prediction as the last word: likewise, we know better than to take models that do exhibit an initial singularity as the last word. In short, modern cosmology allows us to draw no conclusion about whether the universe has existed for a finite or infinite amount of time. And anyone who says differently is not being completely honest.

Next time: Why the BVG theorem is irrelevant to the Kalam Cosmological Argument!

Monday, March 17, 2014

Facts, Brute and Otherwise

Prof. Feser has responded at length to some comments I made on one of his posts. As usual, I thank him for his time and attention to my comments.

In those comments, I proposed the example of lightning striking a tree and starting a forest fire. I claimed that the lightning is still an explanation for the fire, even if the lightning itself was a brute fact (i.e. a fact having no explanation).

I realized (eventually) that my example was not the sort of explanation Feser had in mind in his original post. My example was a horizontal causal chain, in which one event causes another, which causes another, and so on, while Feser's original claim was about vertical explanatory chains: one level of explanation is in turn given a more detailed description by a lower-level explanation, which is in turn given a still-lower-level explanation. (The picture I have in mind is, for example, of a broken window that is explained at one level by the rock that hit it, but at a lower level by the fracturing properties of glass and the stresses imposed by the rock, and those properties are in turn explained by the properties of the molecules of which the glass and the rock are made, and so on.) So my example wasn't really relevant to Feser's point.

In his new post, though, Feser clearly does intend his point to apply to horizontal causal chains, so perhaps the forest fire example is relevant after all. Let me add a few more remarks.

For some reason, I'm more sympathetic to the idea that the brutishness of facts propagates vertically. I'm not sure why my intuition differentiates between the horizontal and vertical explanatory chains. The goal of physics is to describe the way the universe is in as simple and efficient a manner as possible. We physicists suppose that everything physical can be explained at bottom by the Standard Model of elementary particles, but we are content to take that theory as a brute fact. (Well, not really "content": we are always striving for a deeper explanation which will explain the structure and parameters of the Standard Model. But if we found such a theory, we would take that as a brute fact.) So in some sense the answer to any physical question is, "That's just the way the universe is." But that doesn't mean such explanations aren't useful.

Any explanation of a fact A will necessarily be in terms of other facts B, C, and D. (Unless A is self-explanatory, whatever that might mean.) B, C, and D, in turn, are either self-explanatory, or brute facts, or they are explained in terms of some further facts E, F, and G. So the whole thing can only bottom out in facts that are either self-explanatory or brute. (It seems to me that this much is true of both vertical and horizontal chains.)

If I read the professor's remarks correctly, he is saying that something can only be a real explanation if it bottoms out in only self-explanatory facts. (And that this is true of both vertical and horizontal explanatory chains.)

My response is that, if this is true, then there are hardly any examples of real explanations. In fact, maybe there has never been a real explanation in the history of humanity. For (nearly?) all actual explanations leave something else unexplained.

For instance:
  • I can explain why that pot of water is boiling by noting that it has been on a hot burner for 15 minutes. 
  • I can explain why the window broke by noting the rock that hit it.
  • I can explain why I slipped and fell by noting the ice on the sidewalk.
People do not normally require a deeper explanation in order to consider these real explanations. I do not need to understand the molecular structure of water and its relation to the boiling point in order to consider the hot stove to be the explanation for the boiling pot. I don't need to know about the breaking stress of glass to consider that the rock explains the broken window. I don't need to understand how ice lowers the coefficient of friction to think the ice explains why I slipped.

What I'm saying is, any actual example of an explanation always leaves some loose ends. The regularities themselves are enough for us to claim we have an explanation: heat boils water, rock breaks window, ice makes sidewalks slippery.

Now, what Feser seems to be saying is that, though we might not know what the explanation is for the explaining facts B, C, and D, we must at least believe that there is an explanation for those facts. Otherwise we don't really have an explanation.

To this I can only respond as Keith Parsons did: I don't see why I should think this. If all actual examples of explanations leave something else unexplained, why should I deny that these are true explanations? It makes more sense to me to provide an account of explanation that reflects how we actually use explanations than to provide an account which declares by fiat that no real-world examples of explanation are true explanations.

Feser challenged me to provide an alternative account of explanation. I have done so before in previous discussions, and have not to my recollection had a response, but I am happy to repeat it here.

Consider the D-N model of scientific explanation. According to this model, an explanation of an event A consists of two things:
  1. A list of natural laws L1, L2, L3....
  2. A list of conditions C1, C2, C3.... that guarantee the laws apply in the case A.
So we can provide a D-N explanation of the forest fire as follows:
  1. L1: Lightning causes fires.
  2. C1: There was a lightning strike.
Under the D-N model, the lightning strike is an explanation of the forest fire, even if we have no explanation of the lightning itself (i.e, it was a brute fact).

Let's return to the boiling pot. I can, in principle, carry my explanatory chain vertically downward, explaining the molecular properties of water in terms of the quantum mechanical properties of the atoms,  and the properties of the atoms in terms of the Standard Model. There I bottom out in brute facts, from my physicist's point of view.

So here's my counter-challenge for Professor Feser: give a real explanation - in his own sense - of why the water is boiling: an explanation that bottoms out only in self-explaining facts or necessary truths.

Finally let me note that scientific explanations of the kind I've been talking about have a stunning record of success. Engines, TVs, computers, cell phones - all of modern technology stems from our ability to explain things in terms of unifying regularities. In contrast, Aristotelian explanation has been around for more than 2000 years: what practical successes can it claim?

Friday, March 14, 2014

Tuesday, March 4, 2014

Bicycles and Universes

I have often imagined debating William Lane Craig myself, and thought out the ways I would respond to his arguments. I have often, while listening to Craig's debates, wondered why his opponent didn't call him on some claim that was simply untrue. Were they just being polite, or did they not realize the falsity of the claim?

I think I may be cured of these fantasies. Sean Carroll did brilliantly in the debate - far better than I could ever have done. He didn't hesitate to say outright, "That's just false!" And his deep expertise in cosmology was the perfect counterpoint to Craig's quote-mining of partially-understood physics papers.

I have only a couple of comments on style and content. I thought Sean did a good job of pointing out where Craig failed to respond to the argument. (This is an area where Craig usually excels.) But instead of merely pointing it out, he ought to have taken the opportunity to summarize his argument again, for those who might not have understood it completely the first time.

Craig, as usual, excelled in his logical organization and presentation of his argument. His concluding summary nicely recalled his original point: not that he was out to prove God's existence, but that modern cosmology lends support to one of his premises.

Here Carroll really missed an opportunity. He ought to have said, briefly and succinctly, that modern cosmology lends no support at all the premise that the universe had a beginning. What we can say for sure is that the universe was a very different place 13.7 billion years ago. But any statement about what happened before that is very speculative and unfounded in established science. There are models in which time has a beginning, and there are models in which it doesn't: none of these models are established science, and so nothing can be deduced from them about a beginning.

One final missed opportunity: when Craig asked, quite reasonably, "If universes can just pop into existence, why not bicycles? What's the difference?" (from memory, not an exact quote) Sean could have responded that there is an obvious and crucial difference: bicycles are things that exist within time, while universes are not. On the contrary, time exists within a universe. For all Craig's bluster about simultaneous causation in the Q&A session, causality has to do with what brings about a change. And for there to be change, there must be time. Since a universe is not something that happens in time, the causality issue doesn't apply.

I think Sean probably had something like this in mind in his argument about the a cosmological model as a self-contained description needing no outside cause, but it would have been nice to respond to Craig's question with a specific difference that clearly matters.