Just how certain can we be?

I.  The Setup

On my post Black Swans, I received the following question from St. "naclhv", who is also a physicist... and a Christian... and a blogger who has discussed Bayesian arguments that the Resurrection of Jesus is highly probable!  So that's a fair amount of commonality... and yet there are also some differences, as we shall see!  I had said:

First of all, I should say you should be VERY SUSPICIOUS of any person who starts their argument by making concessions that huge to the other side. Factors of 10^{297} are ridiculous numbers that should never be thrown around in almost any real life situations, and if he concedes something that ridiculous to his opponent, he ought to be guaranteed to lose, plain and simple.  He's like a stage magician who makes a big show of how he's blindfolded and his hands are tied behind his back and so on.  You can be very sure there's a trick somewhere, and that all that patter is there to distract you from the way he actually does the trick.

(The other guy, St. Calum Miller, is also making a fallacy, when he quotes a likelihood factor of 10^{43} for the Resurrection; this number incorrectly assumes that the evidence from each apostle's testimony counts independently.  The odds of a group conspiracy to lie are certainly bigger than 10^{-43}, which is an astronomically tiny number.  No real historical event is ever that certain.  That being said, he's right that the evidence for the Resurrection is extremely strong, as far as historical evidence goes!  It's just that nothing in life is really that certain.)

naclhv responded:

Hey Aron,

Long time lurker here. I love your site and the work you do. I would have stayed lurking longer, but I decided to comment because I happen to be writing my own argument for the resurrection over on my blog (http://www.naclhv.com/2016/03/bayesian-evaluation-for-likelihood-of.html).

Specifically, I'm also getting likelihood ratios around 10^{43} from my own calculations, and I thought they were quite reasonable - very conservative, even. So I thought that I'd run that value by you again, as someone whose opinion I highly value.

[some parallels to physics and history which I will quote in a later section...]

So, I'd love to get your feedback on this way of thinking about probabilities. It forms an important part of my argument for the resurrection, and I'm always looking to refine my ways of thinking.

Thanks in advance for your reply, and thanks again for the work you do here!

You're welcome!

So I read his blog series, which turns out to be quite long, and still continuing.  (This response will also be quite long.)  I find it hard to read long blog series without an outline of where I'm supposed to be in the argument, so I've broken it into some major sections so you can decide for yourself how much you want to read.  Fortunately much of what I want to talk about is in the first four posts:

1. The Main Argument
1 2 3 4

2 Considering possible objections
5
6 7 8 9 10 11 12

3. more examples for calibrating based on testimony
13
14 15 16

4. comparison to other claimed resurrection events
17
18 19 20 21 22
23 + more to come

As requested, I will now provide some friendly fire, against my own side of the argument.  But there's plenty of good stuff in there which I won't be addressing.

II.  Is an individual testimony worth 8 orders of magnitude?

First though, a commendation.  One of the major strengths of this series is that, instead of simply guessing how much evidence a single "seemingly earnest, sincere, personal testimony" is worth, he actually tries to explicitly estimate it using a variety of real-life examples (some of them are thought experiments, while others are taken from his own life, or the news, or gambling situations, and other such situations).

(If you want to decide for yourself how you'd evaluate these decisions, without being tainted by his own suggestions, you should read his first post before proceeding.)

The second post is an interlude in which, for no particularly good reason, he spots the skeptic an enormously tiny prior probability for the Resurrection, namely 22 orders of magnitude: 10^{-22}.  This is, of course, just showmanship—the exact same thing I chewed out Dr. Robert Cavin for at the very top of this post, albeit more modestly—because the goal is to show that the evidence for the Resurrection is powerful enough to overcome even this handicap.  Well I don't think it is, as we shall see below.  If tomorrow I learned a new fact that was 10^{22} more likely to occur if Christianity were false, then if it were true, I'm pretty sure I would deconvert.  I think it's not possible for controversial historical judgements to be that powerful... I intend to explain why below.

In the 3rd post he writes:

Let's use my personal answers, given below, as an example for how to do these calculations. These are my gut answers to the questions, before doing an actual probability calculations. Remember, these are the events that I'm willing to give even odds (50/50 chance) on, based solely on an earnest, personal testimony. It does not mean that I'm willing to believe 100%, and it does not mean that I'd stop looking for more evidence. It only points to how much I'm willing to adjust my beliefs based on someone saying "yes, I know it's unlikely, but it really happened".

For the shared birthday question, I would easily believe that my friend shared a birthday with me. I would also not have any real problem believing that our mothers also shared birthdays. At three people - myself, mother, and father - I would start becoming skeptical, but would probably give my friend the benefit of doubt. Starting with four shared birthdays in the family, I would start leaning more heavily towards skepticism.

On winning the lottery, I would not really doubt that my friend won the lottery. I would start doubting if he says that he won two consecutive lotteries.

On getting a royal flush, I think I could almost believe that my friend got two such hands in a very lucky night at the table. I feel like three would be entering the realm of the fantastical, and I would doubt my friend at around this number.

On pocket aces, I would be willing to believe that my friend had up to four or five pocket aces in a lucky night of Hold'em.

On the multiple births, I would not have any real problems believing that someone was a part of quadruplets. A claim to be in a quintuplet would start to cause a little bit of doubt to me, and a claim of sextuplets would need additional evidence.

On being struck by lightning, I actually had someone around me claim that this had recently happened to her. I had no problem believing it. Even if she had claimed two such accidents I don't think I would have really doubted her. If she had claimed three, I would start to be skeptical.

Now, calculating the numerical probability values for all these things is pretty straightforward:

[He goes on to calculate and gets numbers approximately equal to 10^{-8}]

(In the fourth post, he calculates the testimony of the disciples as being worth a whopping 54 orders of magnitude, but I will hold off on criticizing this number until later.)

There is room to criticize some of the specific examples here.  Maybe I'm just cynical, but I don't think I would believe an acquaintance who claimed to have gotten two royal flushes in the same sitting of poker!

And I also don't think he's right to say that, if someone were to lie on LinkedIn about having a Ph.D. from Harvard, "there is not much concrete negative consequences for lying, while the incentive of getting a job or a business contact can be quite appealing".  There's little point in lying on LinkedIn unless you plan to sustain the lie for your next employer.  But doing that is very high risk, since it's an easily checked fact, and getting caught would result in you getting fired and maybe blacklisted.

But this is quibbling around the edges with the exact numbers.  I think there's a really important point here, namely that sometimes human testimony can really be surprisingly powerful in its effects.

To make my own example, if somebody on a college campus told me, in a nonjocular way, that they'd just seen a building that was on fire, I would think they were probably telling me the truth, even if I was indoors and couldn't check to see if there was smoke.  Even if they looked drunk or disreputable, so long as I had no specific reason to think they were lying, I would certainly entertain the possibility that they were telling the truth.  But, the odds that any given building is on fire at any moment is very small.  If we suppose that a campus has at most one visible building fire (on average) every few years, and that the fire lasts for an hour before being contained, that's a prior odds of at least 1:25,000, brought up to around parity by a not-particularly reliable seeming source.  One could bump the prior odds still lower by adding on some extra details (e.g. somebody jumped out of a window into one of those nets that looks like a trampoline), so long as the extras didn't seem too implausible to be believed.  So I agree that testimony can do a lot!

But I don't think I would interpret this fact in exactly the same way naclhv does.  Suppose it were really true that, in general, "seemingly earnest, sincere, personal testimony" is false only 1 in 10^{8} times.  We can check this by asking how many times in my life have I been lied to?

Now except for pathological liars, people seldom lie about inconsequential facts that they have no emotional stake in; they may lie about trivial matters that make them look bad, but not when you simply ask them the time of day.  Let's instead ask how often people lie about matters of emotional significance.  Things that meet this threshold probably don't come up more than about 10 times a day.  Multiply by about 300 days in a year, and 30 years of life, that's probably about 100,000 situations in my life when somebody has been tempted to lie to me.  If the odds of them lying to me were really 10^{-8}, then that means I might expect to live to be a thousand times my age before somebody would lie to me once. 

Maybe that's is a little unfair because naclhv does specify that the testimony must be "seemingly earnest, sincere, personal testimony", whereas a lot of lies are insincere, easily detectable, or the person backs down immediately when confronted, etc.  But even that sort of really serious lie, surely has happened several times to any of us!  (And there are fewer opportunities for people to make them, too.)  So I think the point stands that the general honesty of human beings ain't 10^{-8}, or anywhere close to it.

So this raises an apparent conflict with the examples naclhv provides, some of which seem fairly reasonable.  I think the resolution of this paradox requires noticing another important principle, which can be illustrated as follows:

Suppose someone tells you that their license plate number is 4ZIW623.  Discounting the possibilities of a vanity plate, them not owning any vehicles etc. the prior odds of this are 10^{-4} \times 26^{-3} = 5.7 \times 10^{-9}.  But more likely than not, they are telling the truth.  Why?  It is emphatically NOT because the odds of them lying about their license plate number are that low.  Instead, it is for this reason: even if they chose to lie, they would have no particular reason to pick that particular plate.  If they randomly make up a license plate, the odds of getting that specific one are also 5.7 \times 10^{-9}, so those two large factors cancel out.  You're just left with your gut feeling about how likely a lie was (say 1 in 100).  That's why you should be more suspicious if they say their plate was (e.g.) 6DVL666.  The odds of getting that plate by chance are the same (assuming your DMV doesn't throw it out for looking devilish), but the odds of somebody thinking it's funny to lie about having that plate are substantially larger because it's not randomly selected; it's special.

This has a number of implications for evaluating human honesty.

One is that weird things happen all the time, and we tend to talk about them because they are more interesting then all the non-weird things that happen to us.  So if somebody says they got a royal flush in poker, that's the particular weird thing that happened to them.  If it hadn't happened, and instead they'd had an affair with a Soviet spy, they'd talk about that instead.  1-in-a-million things happen to a lot more often than 1-in-a-million people, because every day we do a thousand different things where an interesting thing might happen.

So, supposing it's really true that a typical piece of testimony is worth 8 orders of magnitude, I'm guessing about 6 of those orders of magnitude are due to the license plate effect, while only about 2 of them are due to people being reluctant to lie.  At least 1% of the things you hear are lies, but the 99% that is true is nonrandomly selected from the most interesting things that have happened to a person, so even the stories whose prior odds are 1 in 10^{-8} are still true most of the time.  But you shouldn't believe that even a plausible ordinary fact some schmoe tells you is 99.999999% likely to be true, as you would if you naively slapped 8 orders of magnitude on a 1:1 odds proposition.

This means, that if somebody claims to have gotten two royal flushes in one sitting, that's a lot more improbable than what you'd expect from simply squaring one royal flush.  Because getting one royal flush is just one of a gazillion different noteworthy things that might happen to a person, but getting two in one day is relevantly special, like the numbers matching on a license plate.  A liar can add on an extra royal flush with barely more trouble than it took to lie the first one, but a truth-teller had to be just that lucky.

In other words, if I'm right about the 8 = 6 + 2 split, you can only discount that 6 once.  Any additional improbability of the same sort, is on your own head.

So, a sufficiently implausible story is indeed more likely to be a lie than the truth.  But, the implausibility has to arise from some inherently improbable aspect of the story, which would be more likely to be invented by a liar than it is to really happen.  Merely adding additional details, more information ("and it turned out he was really named Aleksey Smirnov and was dropping off the secrets to a man who drove up in a green car..."), lowers the prior probability, but it doesn't matter to whether you should believe them because of the license plate effect.  (Of course the details do matter if they seem to involve corroborating or suspicious aspects, but the mere presence of lots of detail isn't the crucial thing.)  So this is a magical aspect of testimony, that it can cancel out any amount of low prior probability so long as it's merely due to there being large amount of detail, instead of something intrinsically unlikely happening.

(Of course, with a sufficiently large amount of detail, the odds are good that the person would make at least one mistake of perception recall.  But I am talking about evaluating the odds that the testimony is substantially true, not the odds that it is absolutely inerrant.  Minor mistakes and discrepancies are not to the point here.)

III.  What happens when we stack up multiple testimonies?

This also shows the wisdom of the biblical rule that a person should only be found guilty of a crime on the testimony of at least 2 witnesses.  (Still more or less true in Scots law, although the rule has been adapted to modernity by saying that the witnesses need not be human beings, one of them could be a DNA test or something.)  1 witness can just make up whatever details, but if 2 witnesses agree on the same highly specific thing (the more specific, the better), the probability of all those details being false is infinitesimal unless the witnesses aren't independent.  (For example, if there was a conspiracy to perjure themselves).

Informally, it might seem like this means that 2 witnesses can be more than twice as good as one witness.  That's not really the way the math works though.  What's really happening technically from a Bayesian point of view, is that most of the first witness testimony was used up fighting against the low prior probability of the specific claim (see the "prosecutors fallacy"), leaving the second witness testimony free to provide lots of extra gravy on top!

But what if we keep on stacking on more and more witnesses?  Does each one of them produce an additional new factor of 10^{8}?  No, no, no!  First of all, as I argued in the previous section, I think 10^{8} is already too high for evaluating a single witness.  The odds of getting a liar are at least 1 in 100, for the reasons I said above.  Secondly, conspiracies between multiple people do happen.  (As well as other forms of nonindependence, for example someone being influenced by another person's recollection.)

Suppose that, to the best of our ability to tell, based on the factual details of situation, it looks like the witnesses are all more-or-less independent.  Can we then multiply out all the numbers to get a tiny probability of them lying?  (Say, 10^{-54}, as naclhv claims for the various disciples mentioned in 1 Cor 15.)

Absolutely not.  Because it is always possible you are wrong about the factual details of the situation, and the witnesses are not in fact independent.  How would we go about evaluating the probability of this?  Well, to do proper Bayesian reasoning, you have to think about all the possible scenarios, and assign each one of them a prior probability.  You aren't supposed to assign anything a 0 probability, unless it really is absolutely impossible, nor are you supposed to make it really really tiny without good reason.  So, the probability that the witnesses are not independent should always be assigned some not-gigantically-tiny probability.

Now, consider 2 rival scenarios, one in which N witnesses are e.g. independent and lying, and the other where there is a gigantic conspiracy to lie.  Is it not clear, that, as N gets bigger and bigger, the probability of the second scenario will always exceed the probability of the first?  The plausibility of the independence scenario falls off exponentially with the number of witnesses.  While the plausibility of the conspiracy always remains at a reasonably small (but not too small) tiny value.  Since larger conspiracies are harder to hold together than smaller ones, a big conspiracy is going to be somewhat—perhaps even rather—less likely than a small one, but at least it doesn't fall off at a steep exponential slope, as a function of N.

One can generalize this argument further.  Any time you've successfully argued that some hypothesis which uses independence has a likelihood of 10^{-54}, this pretty much guarantees that any hypothesis which does not assume independence is going to do better.  Unless you think the argument for their independence is itself a 54-orders of magnitude slam dunk, but that just pushes the question back to how one could be so sure of that question.

It's absolutely fine, as a rhetorical technique, to try to show that a viewpoint is implausible by showing that all of the most obvious ways for it to be true would involve the conjunction of several improbable events occurring.  But if one actually multiplies out the numbers, one should not take the final answer too seriously—because the most likely way for you to be wrong, is always going to be that you were in error to multiply out those large numbers in the first place, due to some breakdown of your model (including, but not limited to, failures of independence).

IV.  Why we should not be fantastically certain about almost anything

Here are a couple highly relevant blog posts on the subject, by an expert in reasoning I highly respect, who blogs by the pseudonyms Scott Alexander / Yvain (unfortunately not yet a Christian).  The first is about not taking arguments completely seriously when they lead to hugely confident predictions:

Confidence Levels Inside and Outside an Argument

The second one is about a super-Artificial-Intelligence (AI) taking over the world in the near future.  I don't take this hypothesis anywhere near as seriously as the community of Less Wrong rationalists does, but I have to agree with him that it's way more likely to matter than 10^{-67}.  But you can take this as a general parable about a broader issue:

On Overconfidence

So, when you are evaluating the odds of e.g. the disciples claiming to have seen Jesus risen from the dead, the scenario to worry about is always going to be the one where the disciples are not independent, possibly for some reason that didn't fully make it into the historical record.  So when naclhv says that:

Incidentally, if you thought that I forgot to adjust my calculations for the fact that the testimonies are not independent, this is why - the three named witnesses in my argument ARE largely independent; they come from very different backgrounds and met the risen Christ under different circumstances. Especially in Paul's case, if anything you'd expect his testimony to be anti-correlated with Peter's. For the other witnesses where dependency is expected, I explicitly called it out and severely discounted the Bayes' factor values in the calculation.

for the reasons stated above it's hard to imagine that any three witnesses could ever be "largely independent" for purposes of multiplying many tiny probabilities.  Because the "error" due to them maybe not being independent is always going to swamp the situation where they are.

They may still be "largely independent" in the sense that postulating a common conspiracy requires making some improbable background assumption.  But, in that case you only pay the price of that background assumption (assuming that is more probable than multiplying out all the numbers on the assumption of independence).

V.  A similar issue with the McGrews

naclhv isn't the only smart person to make this mistake.  In an otherwise very fine article on the evidence for the Resurrection, Sts. Tim and Lydia McGrew claim a Bayes factor of around 10^{44} for the Resurrection, coming largely from the assumption that the testimony of the Twelve Disciples should be independent of each other (together with smaller additional boosts from the women, St. James, & St. Paul).

They then consider the possibility that the disciples were not independent, explaining that:

But when probabilistic independence of testimonial evidence fails, it need not fail in the way sketched above.  Probabilistic relevance can be either positive or negative... [some math follows]

This general statement about probability theory is correct.  But it is not really relevant, once you start claiming that something is really, really implausible.  Suppose that you aren't sure whether the failure of independence is going to be in a positive positive or negative.  In fact it depends on your background assumptions (And in a good Bayesian calculation, you should never really allow yourself to be 100% certain of anything.)

Suppose, just for the sake of argument, we granted to them a 99% chance to Scenario X, where the disciples' testimony would be negatively correlated (or else independent), and only a 1% chance to Scenario Y, where it is positively correlated.  Well, X gets killed by a huge factor of (according to them) > 10^{44}, while the latter gets beaten down by a much smaller factor (since the disciples testimony is now positively correlated).  So Y is always going to win!  (Even if the final result for Y is damped by the 1% factor, that's nothing compared to 10^{-44}!)

They go on to articulate a particular reason to believe that some of the disciples' testimonies might be negatively correlated instead of positively correlated:

If A dies (especially in some unpleasant way) for his testimony to the risen Christ and B hears about it – and there is no serious doubt that the apostles knew when one of their number was put to death – does this make B more likely to stand firm until death in his own testimony? It seems to us that the opposite is true, that knowing of such a death is plausibly and under ordinary circumstances negatively relevant to B’s willingness to remain steadfast. B may well be frightened by the fate of A and drop his claims. In this case, treating A’s and B’s deaths for their testimony – their “martyrdoms” in the original sense of the term “martyr” as “witness” – as probabilistically independent actually understates the case for R.

This correctly identifies a possible mechanism, by which, given certain background assumptions, one disciple's false testimony might make another's (continued) false testimony less likely.

Personally I don't think that this is a more important effect than the sort of obvious social fact that people tend to imitate their friends' behaviors even when those behaviors are self-destructive.  (Consider how gang members react to the death of a gang leader.)

But it doesn't really matter much whether the failure of independence is more likely to be positive or negative.  So long as somebody can articulate any scenario in which the disciple's testimony was positively correlated, that is the scenario to worry about.  (So long as it doesn't also involve implausibilities worth many orders of magnitude, but it's hard to get there without multiplying a bunch of small numbers, and the whole point of these scenarios is that they try to avoid these things...)

Hence, the McGrews analysis provides an overestimate of how likely the Resurrection is.  That doesn't mean there aren't some strong historical arguments in their paper.  But the mathematical statements are hyperbolic and need to be discounted.

VI.  Are alternatives already factored in?

In a later post, naclhv fights against the possibility of alternative analyses here.  After mentioning some specific whacko conspiracy / delusion theories of the usual sort that people bring out to explain the Resurrection—and quite correctly saying that they not are well supported by any of the data that we actually have—he goes on thus:

First, note how weak this argument is, even if we were to grant it everything that it asked for. Remember, the odds for the resurrection are currently at 1e32, so the odds against it are therefore at 1e-32. Now, we'll allow for each independent objection to count as having the full weight of these odds. Never mind that many of these objections contradict one another and therefore reduce the probabilities of the other objections (increasing the probability for 'insanity' decreases the probability for 'conspiracy', because a conspiracy is less likely to succeed with insane people in it). We'll just ignore that. Never mind also that these complex speculations are naturally less likely because of their complexity. We'll also ignore that as well. So, if we can think of a hundred such objections, each of which carries the full weight of the 1e-32 odds for 'no resurrection', the final odds for the resurrection would drop all the way down to... 1e30

Let me first extract a correct and important point from this paragraph.  One doesn't really get out of mileage from simply coming up with large number of fantastically improbable anti-Resurrection scenarios.  For example, the Swoon Theory, the Identical Twin Theory, the Hallucinatory Drugs Theory etc.  For if it is true that each theory contains some individually highly unlikely coincidence (even a 1-in-a-million event) then simply coming up with a hundred or so different theories doesn't get you out of the hole.

But, the skeptic does get some mileage out of suggesting scenarios in which independence of the disciples breaks down, for the reasons explained in the previous section.  naclhv goes on to argue:

But more importantly, this kind of objection is simply, fundamentally wrong: it would not fly in any other investigation into a personal testimony, because it completely ignores the rules about how we evaluate evidence in a Bayesian framework.

Imagine, for instance, that your friend claims to have been struck by lightning. You've taken stock of this claim and have decided to assign it a Bayes' factor of 1e8. But then you say, "well, you may be just a little crazy. And you might have had a nightmare about a thunderstorm last night. Then you might have gone to a hypnotist and who had you recall your dream, which you're now confusing with reality. Or maybe it was the hypnotist who planted the suggestion in your mind first and that caused your nightmare. Really, it might have been any of these things - and isn't it more likely that at least one of these possibilities is true, rather than for you to have been actually struck by lightning?"

Should you or your friend then discount the previously assigned Bayes' factor in light of these new possibilities? Absolutely not. The thing to note here is that the Bayes' factor ALREADY includes all of the ways that this claim may be wrong. It is the numerical estimation of the weight of evidence for a human testimony, and as such already inherently includes the possibility that the evidence may be misleading.

Having established its value, it is simply incorrect to further modify it with no evidence, based on enumerating possibilities that were already included in its evaluation. Your friend's proper reply to your wild speculation would be to say, "what makes you think that I had visited a hypnotist or had a nightmare? Of course, anyone might be wrong about anything in any number of ways - but don't you already know how much you trust me? How does a list of ways that I might be wrong, with no evidence behind any of it, make you trust me less?"

That is quite true and correct for evaluating a single witness, if we have already calibrated the probability of error using everyday examples, as naclhv has attempted to do.

But it does not apply to hypothesis in which independence of multiple eyewitnesses breaks down, because the effects of those scenarios have not already been taken into account.

VII.  On tiny probabilities in physics

You mention that numbers like 10^{43} or 10^{297} are ridiculously large and should not be taken seriously, especially in historical settings. I would, in general, agree with you - but there are exceptions to this rule in some kinds of math, and probabilities is one area where such numbers are not uncommon. Here's how I'm thinking about this:

Let me give some examples from probabilities inherent in everyday objects. The probability of shuffling a deck of cards to a specific order is about 10^{68}. The probability of recreating a game of chess through random play is about 10^{120}.

Even in physics, 10^{43} would be a ridiculously large number if we were talking about something like time (is that in seconds or years? It doesn't matter - it's basically "forever"). But in the branch of physics that deals with probabilities - that is, in statistical mechanics, 10^{43} is nothing.

For example, the standard molar entropy of water vapor is 188.8 J/K/mol. So the number of microstates for a mole of water vapor at standard conditions is e^{(188.8/k_{boltzmann})} - that is, about 10^{(10^{25})}. Lest anyone think that this is so large only because we're talking about one mole of something, even if we take the moleth root of this number we still get about 10^{10} - so, even just five molecules of water vapor will have something like 10^{50} microstates.

The trouble with these examples is that they are all conditional statements of the form:

  • If model M is correct (where independence holds) the probability of event E is tiny.

where the model M is a truly random shuffle, or the statistical mechanics of water, or whatever.  But that does not mean that the probability of an actual shuffle to result in a given configuration is that low.  The cards might be being "shuffled" by a card sharp like Scarne!

Similarly if all the air molecules go to one corner of the room, that would mean there's some natural (or supernatural) effect we didn't take into account.  It would not mean that a 10^{-(10^{25})} event just happened.

In other words, the model M could always be false.

Also, you mentioned that probability values like 98% are actually not at all extreme. I also think that as well. But the five sigma probability of about 10^{-6} is also not all that extreme - it corresponds to something that we're barely certain enough to publish on, at the cutting edge of science.

That's what we do in particle physics, anyway.  But in the soft sciences, they publish at 2 sigma which is why you can't trust anything you read in science news about people.  :-)

However, the 5 sigma rule = 3.5 \times 10^{-6} doesn't actually mean that the odds of being wrong are less than one in a million.  The reason why particle physicists adopted that rule is that, when they used 3 or 4 sigma, they kept getting false alarms!  There seems to have been a recent example of this at the LHC.  This makes it clear that it's an overreaction to guard against biases that weren't taken into account.

One possible source of bias is the Look Elsewhere Effect, where there are a large number of possible theories that you could have checked for, and you just notice the thing that happens to look anomalous.  In Bayesian terms, this is closely related to the fact that theories which predict specific new particles and forces have low prior probabilities.  Finally, there's good old systematic error, the bane of experimentalists everywhere.

So really the 5 sigma rule is just a kludge, which exists precisely because things are never quite as sure as they appear to be, so you need to up the standards a little.

Several independent verification at the 10^{-6} level would easily bring the overall probability to something like 10^{-43}, and any well-established scientific laws would easily break 10^{-100}, by a large margin.

Assuming complete independence, yes.  But systematic error is not independent, nor is failure to properly consider alternative explanations, nor group-think bias, nor grand scientific conspiracies to mislead the public, nor malicious spirits playing jokes on us, etc.

So, even in history, I can easily imagine a statement like "The Roman Empire existed" having an odds of 10^{300} for being true. Basically, my rule of thumb is that probabilities or odds are not "too large" unless their logs are "too large". This makes sense, given the multiplicative nature of probability.

Same as above.

VIII.  Back to Jesus and the Resurrection

So where does this leave probability arguments for the Resurrection?  I made my own attempt to do a probability calculation in these posts:

Let us Calculate
Christianity is True

For the moment let's ignore the philosophical stuff about the argument from evil and fine-tuning, which maybe could also used to be ramped down a bit, and let's discuss the historical stuff.

Well, I still think that all of the basic component arguments here are good.  Well, it's still true that there's good reasons to believe each of the following is true:

a) Jesus was a very special person, apart from the unusual circumstances after his death

b) a few days later his tomb was empty

c) many of his disciples claimed to have seen him alive, including both women (the first eyewitnesses), the full group of 11 remaining apostles, St. James the brother of Jesus, and others.  [Consider "as read" the standard arguments about the testimony of women not being highly regarded among 1st century Jews, and at least some key witnesses being martyred for their faith.]

d) that some highly unusual vision/phenomenon—according to Acts it was noticeable to others and caused him temporary blindness, but even if we consider this to be an exaggeration, it seems likely to have been at least an epileptic fit of some kind—caused St. Paul, an enemy of Christianity, to convert and become a zealous missionary (and eventually get executed himself).

I originally said that (a), taken by itself, roughly cancels out a factor which is basically the Look Elsewhere Effect (discussed in section VII).

I also said that (b,c), taken by themselves, amounts to about 8 orders of magnitude (from many witnesses) and I'm prepared to stand by that given the weirdness of the situation.  Bear in mind that since tens of billions of people have died in historical times, a mere 10^{-8}-level coincidence following somebody's death should still have happened at least a hundred times in history.  For the kinds of skeptical reasons I stated above, it would be hard to get this much above 10^{11} by itself since then we run out of the ability to check how many potential parallels there are.

Finally, (d) taken by itself, is at least a 1-in-a-million event and I stand by that.  I'm pretty sure there are not 40,000 non-Christians alive today who have had similarly dramatic conversion visions leading them to become zealous for a religion they previously disliked.  (It would be circular to count the Christians here, since if we're right God still does dramatic things to convert some people.)  Maybe we need to shave off a factor of 10, because of the existence of multiple possible persecutors in early Christianity whose conversions would have been equally dramatic (e.g. Caiaphas).

Now, under some fairly reasonable background assumptions, if we trust the New Testament texts even a little bit, some of these assumptions seem at least partially independent of the others.  (For example, even very skeptical scholars agree we have at least some information about Jesus' teachings prior to his death.  And that Paul was originally a persecutor of Christians, and therefore not likely to be sympathetic, we have from his undisputed letters, as a testimony against his own current interests.)

But, clearly the right approach for a skeptical attack, the only one that has a hope of success (other than an almost complete skepticism towards the texts which I really don't think is justified), will be to attack the independence of these events.  And there are some ways of doing this that probably do shave off several orders of magnitude.  I just don't feel like they are strong enough to explain all of the data.

For example, it probably IS true that if an unimportant rabbi seemingly rose from the dead due to a coincidence, that people would make up a bunch of stories about him and maybe put some words in his mouth.  But I don't think such an invented composite would end up being plausibly the most insightful and challenging moral thinker the world has ever known.  (And I don't think this is that subjective of a criterion.  The vast majority of people wouldn't pass the "laugh test" for that position.)  Nor would I expect multiple early detailed texts along the lines of the Gospels.

Going in the other direction, if a charismatic religious leader made grandiose claims about his own identity, I quite agree that it makes it more likely for his followers to report grand miracles after his death.  But I wouldn't expect it to involve quite so many coincidences as we find in the New Testament, I wouldn't expect such a large base of eyewitnesses, and I wouldn't expect the whole thing to be so well documented so early.  (Whereas legends that develop over centuries, that can happen to anybody.)

Finally, crankish people converting to a false religion is commonplace, but it's more surprising when one of your biggest persecutors has a vision of Jesus and goes blind until someone comes to baptize him, and it's also a bit surprising when he then goes around doing miracles, all of this described in a text (Acts) which to all appearances looks like a careful historiography, in parts styled very like a personal memoir by a close companion.   (Of course St. Paul's conversion is also mentioned in his own letters, I mean the ones that even anti-Christian theologians think were really written by him.)  You can, of course, say he was a sincere fanatic who (overcome by guilt for his persecution) confabulated multiple miracles, but that still leaves him more or less separate from the others.  To really undercut the independence from (b, c) you have to say he was a plant, or that he was a fraud who made up most of the other disciples' testimonies, but any of these tactics is an uphill battle for various reasons.

So, if you disbelieve the New Testament accounts of the Resurrection you can and should deny the independence of these pieces of evidence.  It's just, you have to pay a price for doing so.  I still think the most parsimonious explanation is that a large group of people deliberately and intentionally conspired to make up the whole thing.  It's more likely than the other naturalistic explanations, it's just not all that likely.

But because naclhv invited me to critique his argument, I'm going to be merciless and observe that he oversteps again when he says this:

Let me reiterate and clarify that, because it's important. There is an utter lack of evidence for disbelieving the resurrection: literally every single record we have from the people who were actually connected to the event to any reasonable degree ALL portray the resurrection as something that actually happened.

If you believe in the resurrection, you have the unanimous support of all the people who were actually close to the event and would know for certain. If you disbelieve the resurrection, literally every piece of evidence - every single testimony of every single person who ever testified about the actual event - is against you.

He has forgotten an important class of witnesses against the Resurrection, namely the guards at the tomb.  St. Matthew's Gospel tells us quite frankly that:

While the women were on their way, some of the guards went into the city and reported to the chief priests everything that had happened.  When the chief priests had met with the elders and devised a plan, they gave the soldiers a large sum of money, telling them, “You are to say, ‘His disciples came during the night and stole him away while we were asleep.’ If this report gets to the governor, we will satisfy him and keep you out of trouble.”   So the soldiers took the money and did as they were instructed. And this story has been widely circulated among the Jews to this very day.

Of course we owe this account to a Christian, but it is hard to imagine anyone would write these words unless either (1) the guards really did report that somebody had stolen the body, or at least (2) some of the Jews claimed that the guards had said this.  Now people do not usually make up, entirely out of whole cloth, arguments against their own position to respond to.  Maybe they unfairly caricature them as strawmen, but usually they are responding to real people.  So it seems historically very probable that there was in fact some kind of anti-Resurrection testimony to this effect.

It is a separate question whether this anti-Resurrection testimony, as we have it, is at all plausible.  It does nicely undercut the independence of (b) and (c) by postulating that the nefarious disciples conspired to produce both effects, even if their motivations at this stage would be obscure.  But, we can expect that the guards would have been severely punished for sleeping on duty, especially if all of them slept at once.  (This would be true for a Jewish guard, but even more true for a Roman one where the punishment would be execution.  Since Pilate's words were "you have a guard": it is unclear whether he was providing a guard or observing they already had one.)  And, if there was in fact a heavy stone and a seal, it would have been quite challenging to move it without wakening anyone.  And, if the guards were really asleep, how could they possibly know who had stolen the body?

Their testimony may even ultimately favor Christianity, since it's existence helps confirm that there was a guard, which makes the empty tomb a lot more impressive.  But, it is false to say that no one was claiming the Resurrection hadn't happened.  The guard—and apparently the Jewish leaders that allegedly bribed them—were putting forth a different story.  But for some reason, even the skeptics have preferred to tell other tales.

So where does this leave us?  I'm reluctant to slap a number on this now, because earlier I concluded that, if you're really sure something is true, inevitably the best possible skeptical hypothesis is always going to be the thing you didn't think of, something that undermines all of your assumptions.  This means, the more and more sure we get, the harder it is to even calculate just how sure we should be.  But, we should not be too sure.

Leaving aside truly awful skeptical scenarios, like we're all in brains in the Matrix being toyed with, surely we can be pretty darn sure that e.g. Julius Caesar was assassinated.  As I have argued before, the evidence for Christ's Resurrection is almost as strong.  But, very tentatively, it seems reasonable to maybe put a cap on how sure we can be of any particular historical event, maybe 99.99% tops for the final answer, to something we've carefully investigated that seems to require an unlikely "conspiracy" to explain away.  Unless it's something really basic like "The Roman Empire existed", where we should be able to go a bit further.  (Part of me feels a bit dirty assigning some historical conspiracy theories a probability of more than 1 in a million, and maybe that's correct, I'm really not sure where the threshold should be.)

This is just a kludge, until somebody figures out a way of assigning a number to "failures of independence in ways that you haven't even thought of yet".  But, this is good enough for now.  It seems to me one can still be highly confident, on the basis of historical data, that Jesus rose from the dead.  Just not quite as confident as naclhv and the McGrews claim you can be.

(Of course, a complete analysis would have to include all the rest of the evidence from philosophy, experience, etc.  aside from the immediate historical data for the Resurrection.)

IX.  Epilogue

Some people might wonder why I'm spending time criticizing an argument for my own religion, saying that it is too strong.  Most people spend their time arguing against things they don't believe in.

Well, I'm not most people.  I'm hoping to do something a little more unusual, which is trying to follow the truth wherever it leads.  Superficially, it is rhetorically effective to play up the strengths of one's own argument, and the weaknesses of the other side.  Unfortunately, this can lead to a tendency towards dishonesty, ignoring the flaws on one's chosen side.

So I have a different evil plan, which is to evaluate arguments in a fair and unbiased way the way a rational person would.  You see, if I can successfully pretend to be doing that, then people on the other side will say to themselves,

"Here's this reasonable looking person, who doesn't seem biased, crazy, or stupid, and he knows about science, and yet he still thinks it's historically plausible that some dude was God's Son, and came back to life again.  Maybe there's something too it, and I should take another look."

So, there are advantages to pretending to be reasonable.  But I find that the easiest way to pretend to be reasonable, is to actually be reasonable.  And—joking aside—my first priority is to the Truth.  If Christianity is right, Jesus is the Truth, so loving Truth and loving Jesus works out to the same thing in the ultimate analysis.  But, if that weren't the case, I would want to know it, rather than living out my life based on a lie.

Other Christians might say, well what about the certainty which comes through the testimony of the Holy Spirit?  Who cares about probability theory and this historical jibber-jabber?  I kind of doubt whether anyone like that has read this far, but if you have, here's my response:  Obviously I'm not going to tell the Spirit not to bear witness to the truth in people's hearts.  And while much of the time he leaves us to our own devices, sometimes it does seems like he's bearing witness to my heart.  But, although I've had some fairly dramatic spiritual experiences, none of them are so strongly powerful that there's absolutely no chance I could be wrong about their cause.   Which is not unexpected, given that "we live by faith, not by sight" (2 Cor 5:7).

So, they also can't make me perfectly certain as a Bayesian reasoner.  But Bayes' Theorem isn't how people actually think internally.  It's just a somewhat useful model of what a hypothetical Spock-like rational entity would do.

When it comes to emotional certainty, I honestly don't think there's that big of a difference between, a calculation that says you should be 99.5% sure, and one that says you should be 99.999999999999999999% sure.  The heart doesn't really resolve that kind of difference.  Whether or not you trust in Jesus isn't really a matter of having an enormous probability, although you shouldn't do it if you don't think it's true.  It's a matter of making a decision to trust.

Once you've decided to trust, additional percentage points maybe help you sleep at night but I don't think they are all that spiritually valuable one way or another.  Emotional certainty can be spiritually valuable, if it's built up by trusting God in difficult circumstances.  As we all know, it doesn't come automatically from simply being intellectually persuaded.  That's where faith comes in.

To use a classic sermon illustration: what shows you have faith that a plane will arrive at its destination safely?  The answer is if you're willing to get on it.  One person may be trembling in fear, another may be cocksure, but whether or not you get on the plane is a yes/no question, not a continuous probability value judgement.  Maybe the first person gets on and the second doesn't.  So, you can even be a Christian even if you only think it only has a 70% chance of being true, as long as you are willing to get on the plane.  Those who do get on board usually become more sure, while those who don't often become less sure.  Which of these effects is primarily due to bias, I guess depends on who is right!

So, there are credences (i.e. probability assignments), there is the feeling of emotional confidence, and then there is trust, and none of these are exactly the same as each other, even though sometimes they are related.  What we are entitled to is just enough to get by on: "Give us this day our daily bread..."

Posted in Links, Theological Method | 77 Comments

Links of Randomness

♦  My wife has a new blog about quilting.

♦  This Nature article describes the subfield of quantum gravity which I've been working in—I know most of these people.  This is a lot more accurate of a description than your average pop journalism article, so check it out.  (Of course many of the ideas here are speculative and may turn out to be false.)

♦  What if you discovered one day that everyone except you has a magical superpower, and you never knew about it?  A true story, flippantly described yet also deeply moving.

(If you're curious to find out more, here's another decent article on the same subject.  [Warning: includes gratuitous disturbing art involving surreal faces])

♦  This is the best article about lichen I have ever read.  Well, maybe that isn't all that competitive an award, but it's still a pretty good article.  The runner-up lichen article is also pretty good.

♦  You already know that dolphins are really smart animals—but that doesn't mean you won't enjoy reading more about it.

♦  One way to go to college for free.  But not suitable for dolphins...

♦  Once upon a time, people thought that Jews were naturally the best at basketball, because of their short stature and scheming minds!

♦  How we know that the robots didn't take our jobs.

The scary chart (the one that shows how, as a result of poor structuring of government programs, poor people can actually be worse off as a result of getting a job or a pay raise) is from this article.  Honestly, how hard is it to phase out programs gradually with income so as to avoid truly stupid incentives?

♦  This critique of our current primary nomination process, may have changed the way I think about politics.  This article bothers me because, on principle, I dislike pretending to have a democracy when actually the important things are settled in the cliched "smoke-filled rooms" (I've disliked the Democratic superdelegates since I first heard of them) but it seems obvious in this election that that those methods have worked better.

Also one could question whether "democracy" should really mean majority (or plurality!) rule when we are talking about the plurality of a minority (those who vote in a given party).  The main way that the party establishment would like to modify raw democracy is to make the candidates more electable, which means in a way they represent the rest of the nation and make the results more democratic.

♦  How do you warn people thousands of years later about sites where radioactive waste is stored?

♦  Or for a more short term prediction about future developments: some predictions in the year 1900 about the year 2000.  About as accurate as these things ever are, i.e. not terribly but a few of them score some palpable hits.   (Here's a plaintext version if you find the first one hard to read.)

♦  "A Mathematician's Apology" by G. H. Hardy.  Still contains a lot of truth today, although when he lumps Quantum Mechanics and Relativity in with pure mathematics, and says that at least these things can never have any use in war... well, I think we have to count that as another failed prediction.

♦  Interesting article in the NY Times about a two sets of identical twins (2 x 2 = 4) where one twin from each set was swapped at birth, and what happened after they found out.

♦  An interesting series by St. Jason of Triablogue on some of the less well-known evidence in favor of the traditional authorship of the Four Gospels.

♦  Does wishing to believe in religion put one in a better or worse position, for learning whether it is true?  An interesting fictional dialogue [google books] on the subject by a Catholic author.  I read this on the strength of a quotation excepted at Siris.

♦  An actual dialogue about religion at First Things, with a Catholic and Muslim, both authors, about religion and their friendship with each other.

♦  Nobody Expects the Spanish Inquisition … to Be Explained Fairly, a review of a book addressing anti-Catholic history.

♦  Speaking of agencies that use the methodology of inquisition (the accused must prove themselves to be innocent)... please don't call Child Protective Services on parents for trivial issues unless you hate both them and their offspring.  Followup posts here, here, and [added later] hereSpoiler: happy ending.

Posted in Links | 13 Comments

Is Divine Simplicity compatible with other Doctrines?

Introductory Warning: please note that when theologians call God "simple" that's a technical term.  It doesn't mean he is easy to understand.  It means that he is not composed of any parts of any kind.  (Depending on who you talk to, this could also mean stronger statements which are believed to follow from that, e.g. that all of God's attributes are really different ways of talking about the exact some thing, and that they only appear different to us from our limited earthly perspective.)

A certain St. Matthew J. Thériault, whom I met at Ratio Christi, sent me the following questions about whether divine simplicity is compatible with other Christian doctrines such as God's Omniscience, the Trinity, and the Incarnation:

Attached [click on this for a word file---AW] is the abstract to a presentation I intend to deliver at Ratio regarding the doctrine of Divine Simplicity. Regarding the Trinitarian objection there's no relation to physics, and you already personally addressed the Incarnation objection when you last visited Ratio (though I'd be interested if you've given any more thought to the matter of the simultaneity of the ascended Christ and the Church on earth). However, I imagine you'd be able to offer immense insight regarding my objection to Omniscience, which is partially informed by a short article linked within. Thank you in advance for any feedback you might be able to offer.

My reply email spiraled out of control, and having gotten permission to share it on my blog, I will now do so:

Dear Matthew,

Although as you say the simplicity of God is easier supported by philosophical than by scriptural arguments, nevertheless there is a little bit of Scriptural support for the doctrine of simplicity.  Traditionally, the doctrine was held to be taught in the Shema: "Hear O Israel, YHWH your God is one YHWH" (Deut 6:4).  Jews interpreted this to mean not just that there are no other gods, but also that this God has some kind of absolute unity of being.  Traditional Christianity, rather than deny this interpretation to make room for the Trinity, has tended to affirm it and then to assert that the Trinity does not contradict it, because the sense in which God is three is different from the sense in which God is one (the persons of the Trinity are not parts, or additional deities).

One could also gather some indirect support for divine simplicity from the tendency of Scripture to sometimes refer to God as "I am [attribute]" or "God is [attribute]".  Also, obviously a lot of the philosophical arguments depend on God's self-existence which is taught more explicitly by "I am who I am" in Ex 3, John 8, among other passages.

Nevertheless, I think one needs to proceed cautiously.  While it would be heretical and unreasonable to say that God is actually divided into parts, it seems to me that some theologians have subscribed to extensions of the doctrine of simplicity which go beyond what can necessarily be deduced from it.  As you know the very strongest formulations of divine simplicity can lead to a number of philosophical paradoxes which are difficult to resolve.

We must always remember that (whether we are talking about Natural Theology or Scripture), "for now we see through a glass, darkly; but then face to face: now I know in part; but then shall I know even as also I am known", and also that "no man shall see my face and live".  Mortal human beings are not capable of seeing the divine nature directly, but only deducing its existence through either revelation or more remote lines.  God may be known to exist from philosophy but he is also the Invisible and Incomprehensible Glory, the numinous Sanctus, that haunts all our experiences but can never quite be contained in any of them.

Therefore we should not be surprised if different attributes of God seem to be in some degree of tension with each other, in fact it would be more surprising if we could fully understand how all of the various attributes can be consistently combined.  Christianity has never shied away from paradox, as can be seen from the doctrines of the Trinity and the Incarnation.  A paradox is not a logical contradiction per se, it is only a sign that our own understanding is limited, yet we can make progress by approaching the same thing from two or more viewpoints which appear to give conflicting information (just like our two eyes, by seeing slightly differing data, can produce a 3d stereographic perspective).

As St. Lewis said about a different theological quandary: "Heaven will solve our problems, but not, I think, by showing us subtle reconciliations between all our apparently contradictory notions. The notions will all be knocked from under our feet. We shall see that there never really was any problem."

What I have just said about paradox is really my primary answer to all of your questions.  It would be grossly misleading if I gave the impression that I could 'resolve' paradoxes in the divine nature in the sense of providing a clear logical schema in which the nature of divinity could be fully grasped with the mind; any such scheme would necessarily be misleading and even impious.  But nevertheless, as a secondary matter, I think I can say some specific things to resolve some of the specific difficulties you list:

A. Omniscience.  This is only an issue if you assume that God's knowledge is, like ours, representational, that is, that it proceeds by means of making something like an image or duplicate of the object known, in some other physical system (in our case, the brain).  That viewpoint seems excessively anthropomorphic, and I have already argued against it here:

Fundamental Reality VII: Does God Need a Brain?

To put it into bite sized arguments, God's knowledge cannot be representational because:

1. The Redundancy Objection: If God is omniscient then his thoughts about the universe would necessarily be an exact copy of the universe not differing in any details.  But that is silly because if two things are identical in every respect, they may as well be identified.  Furthermore it would imply that God's knowledge is limited to the form of the thing (the structural attributes which are the same between the image and the reality) rather than the essence of the thing (which would not be shared between the universe and the image in its mind.  But that would be a limitation on God's intellect.  So instead we must assert that there is no division between God's knowledge of a thing, and the thing itself.

2. Infinite Regress Objection: Even human knowledge is not purely representational, because that would threaten an infinite regress.  For example, suppose we look at a lamp and form a mental image of the lamp somewhere in our mind/brain.  We are then aware of the lamp outside ourselves by means of the lamp inside.  But how is it then that we know the lamp inside?  By means of a second image of the lamp inside of us?  That would threaten an infinite regress.  Instead we must somehow have the power to directly perceive, without any intermediary representations, some things that reside in our own brains.  But God would have the power to directly perceive anything, without any limitations.

3. And of course because it contradicts Simplicity, as you point out.  But that just means we have the wrong model of how God's knowledge should work.

I think the article you link to ["Information Storage and the Omniscience of God", by Hollis R. Johnson & David H. Bailey] is completely off-base when it proposes that God's knowledge must be understood as if God were a giant computer.  They should have realized that it was this ridiculous idea of their own which they were refuting, and not anything in the Bible or theology as traditionally understood.  They say that:

"Some defenders of the traditional doctrine of God’s omniscience may respond to this argument by simply declaring that God is omnipotent and thus omniscient, in the sense of residing and operating completely outside the confines of the Universe and the natural laws that govern our Universe. In short, they may assert omnipotence and omniscience by fiat: God can store knowledge, even an infinite amount of knowledge, without any plausible physical storage mechanism or medium. This is because God’s ways are not ours, and our finite mortal minds cannot possibly hope to comprehend the means employed by this supreme Being. Against such reasoning there is no counter argument."

The reason there is no counterargument against this position is that it is obviously correct.  The idea that God is outside the confines of the physical universe and therefore does not store his knowledge on some physical medium such as a film reel, is not some arbitrary stipulation made to avoid falsifying Theism.  It is part of the definition of Classical Theism that God is outside the universe, for goodness sakes.  To explain God's knowledge by means of ordinary causal mechanisms, far from being a "scientifically tenable theology" would simply be the denial of classical theology, which holds that God as creator is not subject to the limitations and natural laws which govern creation.  If we found the giant film reel it would refute, not confirm, Classical Theism.  One may as well say that any theism compatible with modern biology would need an evolutionary explanation of why God the Father has a beard!

[Not in my original email: I looked it up, and the authors are actually Mormons.  That explains a lot.  Mormons are polytheists who believe that God the Father is merely an exalted human being, one of many deities who worked his way up in some sort of cosmic pyramid scheme, and that he has a physical body.  So according to them God isn't omniscient and probably does have a literal beard, and a brain with finite information storage capacity.  This is, of course, completely different from the classical Jewish-Christian-Islamic concept of God as the absolutely powerful and wise being who is the source of all existence.   Such a being, if he happened to exist, would not be a God at all in the traditional Classical Theist sense.  Why should I worship a being simply because he happens to be a finite amount more powerful, wiser, or moral than I am?  There are already human beings who are better than I am, that doesn't mean they are worthy of my worship!]

B. Trinity.  Here I think I need to quibble with some of your language.

While it is true that the Second and Third Persons in some way originate from the First, I do not think it is orthodox to say that only the Father is the First Cause, apart from the other two persons.  That would seem to gloss over the crucial distinction between "making" and "begetting".  The Father is not a separate metaphysical entity from the Son and the Spirit, indeed the Father has no separate existence apart from his act of begetting the Son and breathing forth the Spirit, since these acts were by metaphysical necessity; it could not have been otherwise.  (This does not mean it was involuntary, for God is spirit and his acts are therefore by will and love, not physical compulsion.)  The persons are so united that you can't have any of them without having the other two as well.  For this reason, I would say that the Triune God taken as a whole is the First Cause, rather than the Father alone.

John 5:26 says that "For as the Father has life in himself, so he has granted the Son also to have life in himself", that is even though it is a gift, the nature of the gift is that the Son has life intrinsically, according to his identical divine nature, rather than derivatively and externally through grace.  Otherwise Christ could not say, in the divine sense, "I am", but rather should have said "I was made to be".

As you recognize the persons of the Trinity are not parts (since they are indivisible), but they are real distinctions in the divine nature.  It therefore seems inevitable that the analysis of the "relations" which define the persons (things like Paternal, Begotten, and Proceeding) must necessarily differ from the divine "attributes" such as power, wisdom, or love which are common to all persons because they belong to the single divine nature.  To that extent I agree with you.

But if this admission seems to contradict some specific analytic formulation of "Property Simplicity", why not simply acknowledge an accidental misstep, coming from an over-strong formulation of Simplicity, and retreat to a slightly weaker version of the doctrine?  For example, one might tentatively say that if God is One, then any of his properties must either be identical to himself OR ELSE to one of the persons, and then say that the latter possibility does not contradict simplicity because each person of the Trinity contains the other two within by "perichoresis".

You say that "Begotten" is "unidentifiable with and alien to" the First Person, but this seems to be stating it too strongly, since even though the Father is not Begotten he does have the reciprocal property of "Begetting", which is the exact same thing viewed from the other side.  The one implies the other.  Apparently God can have real distinctions within himself, but only involving relational terms, of the kind we are discussing.

Thus, since the doctrine of the Trinity is clearer in Scripture than the doctrine of Simplicity, we should adjust the latter to make room for the former, but without of course abandoning the doctrine of Simplicity altogether!  Implicit in this is the idea I sketched in my introduction, that God is mysterious, and that the philosophical "proofs" of his attributes, while perhaps compelling, do not amount to strict logical implications.  And therefore that there is room for "adjustments" in our very provisional understandings when we run into trouble!

C. Incarnation.  As you note, I was asked about this during the Ratio Christi meeting, but for clarity I'll repeat myself a little.  According to the Chalcedonian understanding, the Incarnation involves the union of a complete divine nature with a complete human nature into a single person, Christ.  Properties like Simplicity would apply only to the divine nature, and therefore it would not contradict Simplicity to note that Christ's physical body had parts and could change etc.

After the Resurrection and Ascension, Christ continues to have a human nature, but now his body and soul are glorified, possessing additional abilities and attributes.  This glorified Resurrection body transcends our current earthly state (although we too will be glorified when our bodies are raised from the dead.)  I would love to know more about this but our data from the Gospels and Acts is limited.  What we do know is that Christ's body was capable of being recognized (though not always immediately) and touched, that he could speak and eat, and that he was capable of teleporting instantaneously.  After which he ascended into "heaven" (i.e. somewhere else outside of spacetime as we know it, in which God's will is more fully done as in the Lord's Prayer, as in the angelic world), which he conceptualized as a return to the Father from whom he came, triggering the pouring out of the Holy Spirit on the Church.

Obviously this cannot be understood in an excessively anthropomorphic way.  While Christ has a body by virtue of the Incarnation, God the Father does not, and therefore "sitting at the right hand of the Father" cannot be taken literally, to mean the proximity of two physical bodies at a common time.  Instead it is an Aramaic way of saying that Christ is placed in a position of full welcome and authority, that a formerly crucified and rejected man is now being given the governance of all Creation, with rebel angels and authorities now fully subject to him.  (Try searching the Psalms for "right hand".)  Christ's body continues to have an objective and real existence---and our earthly imaginations cannot conceptualize this except by imagining him residing in something like a "place"---but the nature of that "place" is not one that we can understand, until we ourselves follow him there (John 14:2-3).  There is no reason to think that the visible universe as we know it is anything other than a small portion of God's creation:

"There may be Natures piled upon Natures, each supernatural to the one beneath it, before we come to the abyss of pure spirit; and to be in that abyss, at the right hand of the Father, may not mean being absent from any of these Natures – may mean a yet more dynamic presence on all levels."--St. Lewis, "Miracles" (essay in God in the Dock).

How "time" works in this "place" is not something which I think we are in a good position to know.  I agree that Einstein's theories suggest that time is a feature of our own material universe, so that a completely disconnected universe would probably have a different timestream, if any.  However, if there are interactions between two such universes, then there would presumably still be causal relations between them, and hence (I suppose), some partial notion of prior/posterior events.

This agnosticism about the details might seem a little depressing, but I am afraid it may be the best we can do right now, fun as it may be to speculate on the details of the "control room" from which Christ currently reigns!

I am not sure why you think that there needs to be any "simultaneity" between Christ's body in heaven and the "physical universe in between his Ascension and Return", any more than there needs to be simultaneity between the eternal God and us, in order for God to answer our prayers.  Christ is present in the Church in a number of ways; as the Head who gives the Body life, through the presence of the Spirit, sacramentally in the Eucharist, and so on; but none of these ways seem to involve or require any one-to-one map between individual moments of Christ's existence in heaven and our individual existence on earth.  Even Christ's everlasting intercession for us comes not through continual labor, "offer[ing] sacrifices day after day", but rather by presenting his wounds to the Father once for all, as an eternal atonement for the sins of the whole world: past, present, and future.  We may be sure that he knows and cares for all of our needs, which suffices for practical spirituality, without getting into the mechanics of exactly how his glorified human nature shares in the universal knowledge of his divine nature.  Psalm 131.

All right; this has become quite a treatise.  Hope it helps!

Blessings,
Aron

PS Do you mind if I post this exchange on my blog?  This could involve as much or little of your identifying information and words as you wanted.

Posted in Theology | 33 Comments

In the Red Light District

I'm in the middle of a six week trip in Europe; currently I'm attending the Amsterdam String Workshop.

I'm reminded of something that happened to me a year-and-a-half ago December when I visited the String Theory group in Amsterdam.  I didn't realize until I starting doing touring on Sunday that my hotel was close to the main "red light" district, where the alleyways are full of semi-naked women in booths selling their bodies to the tourists.  The main red light district is right in the middle of the oldest part of town, well worth seeing for the architecture, if you can ignore the vice peddling (which is easier during the daytime).

I was absolutely shocked in the red light district—but not by the prostitutes or the drug use, which I had expected.  (Although these things are bad and degrading, don't do them.)  There is a beautiful old Dutch Reformed church there, dating from the 1300's, which I wanted to see.  I went in to see the church, but whoever was in charge had allowed an artist to set up a crass avant garde multi-media work of art in the interior, with disturbing images of unwholesome faces projected on the blank walls speaking nonsense phrases, and even representations of bright neon casino scratch pads, glowing on the floor!  I felt it was an extremely disrespectful, if not diabolical, use of a space dedicated to our Father in heaven, and in which faithful Christians were buried.

There were a small number of middle aged couples roaming around looking a bit perplexed.  I was outraged.  I said to myself "How DARE they do this to my Father's house!" and I couldn't stay there any longer because I could not contain my rage.  (I said something about it to the poor lady handing out tickets at the entrance.  I tried to make it clear to her that my anger was not directed at her, but I had to say it to somebody.)

As I was wandering around in a daze, I noticed that there was another church in the district, a Roman Catholic church, which was free for anyone to enter.  (The first church had had a 10 euro entrance fee, which is also wrong—what if one of the prostitutes felt a sudden urge to go into a church and pray?—but one quickly becomes desensitized to fees for entering famous churches in Europe).  It was full of tourists but pious ones, and I felt such relief to know that, despite the theological differences, there was some place in the area dedicated to God which was still held sacred, and where the people had natural feelings.  I sat down in an empty pew and wept.

Posted in Ethics, History | 9 Comments

Open and Closed

A reader asks:

This seems to be as good a place as any to ask a question about closed universes.

See, in a lot of popular science books, they teach you that an "open" universe is one where space is infinite, saddle-shaped, and keeps expanding forever; a "flat" universe is infinite, plane-shaped, and the rate of expansion eventually peters out to zero; and a "closed" universe is finite, sphere-shaped, and eventually contracts in a big crunch. They then talk about the cosmological constant and "dark energy," which make our universe expand at an accelerating rate, something that doesn't fit the taxonomy of possibilities for the universe's topology, and which they do not relate back to that taxonomy in any way.

Can a universe with lots of dark energy be a closed universe? Will a closed universe with dark energy keep on expanding and accelerating, or will it eventually collapse in a big crunch like a "normal" closed universe? Is the three-type Taxonomy only relevant given certain energy conditions? (Strong/weak/null)

Oh, and I almost forgot:

are there any good reasons to think that the universe is closed in the first place, other than Kalam-esqe arguments against actual infinities?

David,
It sounds like these books were just adding the new material about the cosmological constant to the old discussions without doing the hard work of going back and revising it so that it makes sense.

The Bad Old Days

In the old days (pre circa 1998) people didn't know about the acceleration of the universe, and they thought that the universe just consisted of ordinary radiation and matter (where for these purposes, dark matter is a form of matter).  In the old days, the model of closed, flat, and open works exactly as you say: a closed universe (spherical geometry) will recollapse, and open one (hyperbolic geometry) will trend to a constant rate of expansion (in terms of distance / time) and a flat one is right on the edge and will expand forever at a slower and slower rate (but still getting arbitrarily large).

Given the rate of expansion, it takes a certain amount of energy density to get a flat universe.  Too much, and you get a sphere, too little and you get hyperbolic space.  (The expansion or contraction of the universe makes it hyperbolic in the absence of matter.)  These are the 3 kinds of geometries which are homogeneous (the same everywhere) and isotropic (the same in every direction).  On average, the observable universe seems to be homogenous and isotropic, so it's got to be one of these three (a.k.a. an "FRW cosmology").

However, this was confusing for several reasons.  One is that the cosmological data kept suggesting that there wasn't enough energy in matter to get anywhere close to a flat universe, yet other data seemed more consistent with a flat universe.  A flat universe is also a natural consequence of inflation since it stretches out the pre-existing geometry to exponentially large distance scales.  Also, the universe seemed like it wasn't quite old enough to explain all the structures in it.

Concordance Cosmology

Now we know that there is an additional form of energy which is confusingly called "dark energy" (but I dislike this name, because it makes people think it has something to do with "dark matter".)  Most likely it is just a cosmological constant, a constant energy density in all of space.

Now it turns out that for purposes of determining the spatial geometry, a positive cosmological constant counts positively (so it helps to close the universe).  But when you calculate its effect on the expansion of the universe, it counts negatively, as repulsive gravity.

This may seem like odd behavior because energy and mass are equivalent, and we all know that mass causes gravitational attraction, not repulsion.  But in turns out that in General Relativity, both energy density (associated with time) and pressure (associated with space) lead to attractive gravity.  Negative pressure is called tension, and tension therefore causes antigravity.

In ordinary matter travelling at low speeds, the amount of pressure/tension is typically very small compared to the energy density.  Radiation which travels near the speed of light has a lot of pressure, but that only makes gravity stronger.

On the other hand, a positive cosmological constant has tension equal to its energy density.  Something has tension if, when you stretch it out, it's energy increases.  But the energy of the cosmological constant is proportional to the volume, so when the volume increases the energy increases proportionally.  Hence the tension in each spatial direction is equal to the energy.  Since there are 3 dimensions of space and only 1 of time, the antigravity due to the tension is 3 times larger than the gravity due to the energy density.  Hence the antigravity wins!  So paradoxically, the gravitational effects of this tension just make the universe want to grow faster!  Unlike the usual effects of tension, which cause things to shrink in on themselves.

On the other hand, if the cosmological constant were negative (it isn't, but suppose) its effects would be reversed: it would make the spatial geometry more hyperbolic, but would decelerate the expansion.

So, once you include a cosmological constant, the rules change (as you guessed).  You can still have the same 3 types of spatial geometry (the words "open", "flat", and "closed" describe the spatial geometry, not the dynamics).  But with a positive cosmological constant, even a universe with closed topology can sometimes expand forever, if it gets big enough for the cosmological constant to take over.  (Matter thins out, while the CC doesn't, so when the universe is small the matter is more important, and when it gets larger the CC is more important.)  On the other hand, with a negative cosmological constant, even an open cosmology will always eventually recollapse when it gets big enough.

(The various energy conditions you mention place limits on the allowed energy density and/or tension/pressure, so not surprisingly these have certain implications for what a cosmology can do.  Note that a positive CC violates the strong energy condition—which allows for a bounce, at least in the case of a closed universe.  While a negative CC violates the weak energy condition, which requires that any FRW cosmology which is neither expanding nor contracting at some time, must be closed.  (OK, technically it also allows space to be flat, but only if the matter energy is exactly 0, which is unrealistic.))

Our universe seems to have a positive cosmological constant, which fixes all of the problems I mentioned above.  The cosmological constant seems to give us exactly the extra energy density we need to get a flat universe.  Yet it also causes the universe to be currently accelerating in its expansion (lengthening the projected time back to the Big Bang); this acceleration has been confirmed by surveys of supernovae in the past.  So everything seems to hang together consistently.

As far as we can tell from current observation, the universe is exactly flat (with experimental error of about 1-2% over scales comparable to the observable universe)  However, a flat geometry is right on the knife's edge between the spherical and hyperbolic cases, so actually this is perfectly compatible with the universe having a tiny positive or negative curvature, as long as the radius of curvature is big enough.

So really it could still be any of the three cases, or else something more irregular.  As I said, inflation blows up the size of the universe, so regardless of the initial geometry, the observable universe will look flat after enough inflation.  Outside the observable universe, for all we know, it could be some other shape, perhaps it isn't even symmetrical.

There is really no particularly good physics reason, apart from aesthetics and philosophical bias to think that the universe should be closed or open.  I personally don't think much of the "Kalam" argument that actual infinities are impossible, but I do find it distasteful that in an infinite homogeneous universe everything (including all possible histories of the Earth) would happen infinitely many times in different places.

Also on the speculative hypothesis that the universe originated from some kind of quantum fluctuation, or no-boundary condition, I think one expects it to be closed.  But this kind of thing is extremely speculative.

If I had to place a bet with a metaphysical bookie, my money would be on closed (but enormously large so that we could never tell).  But this is my own personal guess, not a conclusion of Science!

(Incidentally, even if the topology of space is flat or hyperbolic, it would still be possible for the universe to be finite in size and therefore closed, so long as it has nontrivial topology.  For example, space could be a really big "torus" where if you go far enough in one direction, you come back around on the other side, like in some video games.  Locally, such a universe couldn't be distinguished from the infinite case, but globally it would be different.  Astronomers have done measurements looking for nontrivial topology in the sky.  They haven't seen anything, but of course they wouldn't if it happened on a scale much bigger than the observable universe!)

On the other hand, if the universe really does have a positive cosmological constant than (regardless of its spatial geometry) the final outcome seems secure.  If we extrapolate the current laws of physics to the far future (assuming no changes or interventions), we get an exponentially growing universe.  The matter thins out and becomes unimportant, and you end up with a very tiny final temperature (corresponding to the analogue of Hawking temperature but for cosmological horizons instead of black hole event horizons).

Posted in Physics | 13 Comments