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
6 7 8 9 10 11 12

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

4. comparison to other claimed resurrection events
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.) 1DVL666.  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 a lot 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.  That is, 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 have welcomed his experience—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 scholars 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 would be that a large group of people deliberately and intentionally conspired to make up the whole thing.  It seems 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 in the Gospel 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, these days even the skeptics prefer 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 to 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..."

About Aron Wall

I am a Lecturer in Theoretical Physics at the University of Cambridge. Before that, I read Great Books at St. John's College (Santa Fe), got my physics Ph.D. from U Maryland, and did my postdocs at UC Santa Barbara, the Institute for Advanced Study in Princeton, and Stanford. The views expressed on this blog are my own, and should not be attributed to any of these fine institutions.
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82 Responses to Just how certain can we be?

  1. aleksyl says:

    I agree with the high likelihood of the resurrection, but I have some minor quibbles. Firstly, I think it is somewhat erroneous to assign background knowledge of a miracle because by their very definition, miracles do not happen in a natural environment. So I think it may be difficult to work out a proper prior knowledge probability in a Bayesian framework when non-natural phenomenon are brought under consideration. Nevertheless as a Catholic, I agree that your essay is extremely strong in support of the resurrection.

  2. Andrew wells says:

    Interesting, though I think the weakest chain the argument is the burial of Jesus. Bart Ehrman recently changed his mind about what he thinks happened, he use to grant the empty tomb but has since back tracked and claims that it's unlikely that a cruified victim would be given an honourable burial as this goes contrary to the very purpose of crucifixion, to condemn, humiliate, etc. now if we can't be sure there was an honourable burial, you can't be sure there was an empty tomb.

    Perhaps there was a dishonourable burial (and I'm genuinely sorry to refer to Jesus in this way), but for a dishonourably buried victim, for example thrown in a mass grave , you probably couldn't tell who they were at a much later time due to body decomposition. Obviously, I can't explain the sightings of Jesus on this proposal or the conversion or conviction of the some of the disciples and later followers. But some food for thought.

  3. Andrew wells says:

    *I should said I don't genuinely believe that scenario I laid out, but I think it's one plausible or possible way to attack the argument. By assigning a higher probability to a dishounarble burial and an hounarble one.

  4. JPH says:

    The resurrection is irrelevant.

    God appeared to a nation and gave them 613 commandments. He said they were eternal, everlasting, binding for all generations. There is NOT ONE about worshiping God's son or the Messiah. (Exodus 4:22 says God's son is Israel.) There are horrifying threats for deviating from these commandments in Deuteronomy 28. The thirteenth chapter is devoted to prophets who can perform "signs and wonders" and advocate the worship of gods "whom your forefathers did not know." Their forefathers did not worship Jesus. Deut 13 explicitly grants the possibility of miracles in false traditions and says, "Do not hearken unto that prophet." It says nothing about surviving an execution as an exception or some big standard.

    Why do Christians think the resurrection cancels/changes the Torah? According to what standard? (No, the prophets didn't say so: https://prooftexts.wordpress.com Most of these aren't just wrong, they're cringe-worthy. The prophets received their authority from the 5 Books of Moses. Whatever the prophets were saying, it wasn't to subtract from theses books and approach God via some unheard-of intermediary.)

    Sabbatai Tzvi, too, was considered by many to be the Messiah. He performed signs and wonders. He had his own St. Paul (Nathan of Gaza) who interpreted his conversion to Islam as some humiliating atonement. He still has followers. So what? Miracles don't cancel the Torah. The only reason to think otherwise is because your Bible already has a New Testament attached.

  5. Aron Wall says:

    I agree that miracles do not happen in a natural evironment, but in a Bayesian framework we must always assign everything some sort of prior probability. Some people think supernatural interventions should be assinged an extremely low prior probability. For various reasons I don't think that is true, but it is not part of the topic of this essay.

    However, I think that whenever I mentioned "background knowledge", I was referring to evaluating the probability of alternative naturalistic explanations for the Resurrection data. For this, our background knowledge about the natural environment certainly is relevant.

    Sure, it's unlikely that a randomly selected victim of crucifixion would be given an honorable burial, but Jesus was a person with a large number of admirers. Conditioning on that, it doesn't seem unlikely at all. Turning to the actual historical evidence: All of the Gospel writers say he was given an honorable burial by Joseph of Arimathea, and between them mention 4 specific witnesses to the burial. Also, for the reasons I mentioned in the post, there is some reason for even an enemy of Christianity to accept the existence of the guard. So, if this is the weak point, it's still a pretty well defended one.

  6. Aron Wall says:

    I responded to your comment in a new blog post.

  7. JPH says:

    This is a better treatment of prophecies than the link I gave: http://nojesus4jews.weebly.com/365-prophecies

  8. Elliot Nelson says:

    Hi Aron,
    On the subject of Bayesian reasoning about Christianity, I’m wondering how you would compare the data from science to the historical data. Specifically, let’s suppose that (1) on historical grounds the probability of Christianity is 1-\epsilon, where \epsilon is small, and (2) that the probability of there being new physics that could affect the world on “everyday” biological timescales (as would appear to be necessary in the event of, say, someone’s cancer being suddenly healed in Jesus’ name), is some small number \epsilon_2. Here, I’m thinking that in (1) we are ignoring any scientific data, and likewise in (2) we are ignorning any circumstantial accounts of miracles, just looking at physics. It seems to me that the statement that Christianity is probable requires that \epsilon\ll\epsilon_2, or some similar inequality. That is, there has to be sufficient wiggle room for miracles within the framework of known physics (including our knowledge of where we expect it to break down). Do you agree? I’m wondering how to think about the conditions that unknown physics would have to satisfy, in order to be consistent with miraculous claims (such as a tumor rapidly disappearing following someone’s prayers several hundred miles away, both of which are physical events on macroscopic time and energy scales), which seem inconsistent with the claim that the laws of physics of everyday life are completely understood (http://www.preposterousuniverse.com/blog/2010/09/23/the-laws-underlying-the-physics-of-everyday-life-are-completely-understood/).
    Forgive me if you’ve already addressed this sort of thing — I haven’t managed to look through all the relevant blog posts.

  9. Mactoul says:

    "testimony of the Twelve Disciples should be independent of each other "

    What testimony of the Twelve Disciples? The only testimony we have are of the Gospel writers themselves. The very knowledge that there actually were the Twelve Disciples comes from the testimony of the writer.
    So, at best there are four semi-independent testimonies.
    In reality, what testifies to the Gospel writers is the living Church. Otherwise, we do not know that there were any Gospel writers. Or they were actually writing something about a true happening and not a work of fiction.

  10. Mactoul says:

    Elliot Nelson,

    "there has to be sufficient wiggle room for miracles within the framework of known physics "

    Then why would you call them miracles?
    A miracle is an event that is inexplicable given the laws of nature (pls note, "laws of nature" and not "laws of physics").
    The Virgin Birth is inexplicable, given the laws of human biology, (pls note, I say Virgin Birth and not Virgin Conception).

    I recommend Stanley Jaki's slim volume Miracles and Physics.

  11. Mactoul says:

    Consider Islamic scriptures and traditions. They would seem to be plenty of testimonies. But arguably the whole thing might be an gigantic imposture built on fundamental misunderstandings. It has even been argued that the word "Mohammed" itself may refer to none other than Jesus. So, the Bayesian reasoning assumes a lot of things and suffers from a lack of imagination to think of something spectacularly out-of-the-way.
    PS There have been testimonies to miracles done by various Hindu godmen. Should they be believed
    merely because the testimonies have been written down in some book?

    As for 12 disciples, they are just characters in a story written down by some person. IF they mutually agree, independently or otherwise, how does that relate to the real world outside the book?

  12. Elliot Nelson says:

    I don't think it's necessary to think of a miracle as a violation of the laws of nature, at least not at the deepest level. A miracle may violate the laws of nature as we understand them, but the idea that a miracle is a "suspension" of the laws of nature (or physics) wherein God intervenes to do something different is not very attractive to me at least. I'd rather think of miracles as pointing towards some deeper law(s) or structures which we do not yet understand, and that miracles are no less miraculous when thought of in this way, as opposed to as violations of natural laws.

  13. TY says:


    Great post.

    You got me scratching my head about what is TRUTH and BELIEF.

    In the natural world we can never get to the whole truth and nothing but; so current knowledge is always provisional, which I sense is what make you want to interrogate the data − evaluate arguments in a fair and unbiased way − for rational or “Spockian” certainty. I think you argued persuasively that no Bayesian likelihood factor can nail down historical and theological truths or decide “rational certainty” absolutely.

    After finishing the Post, I kept asking myself: What is it that anchors my religious beliefs? On what basis do I believe the words of the Nicene Creed? Surely my spiritual bedrock cannot be only faith or “emotional certainty” but something that appeals to reason and subject to reason.

    So I don’t think you are dismissing Bayesian analysis or a high probability of the Resurrection (or the existence of God for that matter), but you are cautious of claims of absolute certainty of any particular statistic between 0% and 100%.

    Am I right? Love to get your comments, as usual.

  14. g says:

    Good stuff. As you will remember, I disagree substantially with many of your probability assignments, but it's always nice to see someone thinking about this stuff in a way that actually makes sense.

    TY, not rational certainty; that's essentially never on offer and shouldn't be anyone's goal, and I don't think it's Aron's. I'm a godless heathen myself, but it seems to me that the sort of thing you want as "spiritual bedrock" is a combination of (1) rational evaluation of evidence leading you to the conclusion that the key ideas of your religion are very likely to be right, plus (2) the sort of attitude to those key ideas that makes actual religious devotion possible. (Some people start with #1 and acquire #2; some start with #2 and acquire #1.) Again, "rational certainty" doesn't and shouldn't come into it at all, but for one reason or another many people end up with "emotional certainty" even about things that rationally admit of substantial doubt.

    (Many people also say: "I started out thinking X was probably true, but as I've gone through life looking at the evidence I've become more and more certain of it on rational grounds, and at this point I'd need a colossal amount of evidence to change my mind." Not obviously crazy -- if a thing is true then as you gather more evidence you should become more and more certain about it -- but beware confirmation bias.)

  15. g says:

    Mactoul, who has proposed that the name Muhammad might actually refer to Jesus, and why? That seems incredibly implausible to me. I'm sure Aron would say that claims of miracles wrought by Hindu godmen should be examined in the same sort of way as claims of Jesus's resurrection; he would presumably expect that they would not come out so well (though, note, he could consistently hold that some such miracles are genuine miracles brought about by God for mysterious reasons or by evil spiritual beings in order to deceive). He certainly wouldn't say that they must be believed simply because they're in a book, and he hasn't claimed that stories of Christian miracles must be believed simply because they're in a book, and if you think he's been saying that then I don't think you've been reading what he wrote.

    (I do think that in some cases Aron trusts testimony found in books substantially more than he should, and indeed if you follow some of the links in his post and look at the comments there then I think you will find me making exactly that argument. But there's a big gap between overestimating the evidential strength of claims made in books and holding that everything found in a book must be believed.)

    Incidentally, your suggestion that Bayesian reasoning gives bad results because of a failure to imagine very out-of-the-way possibilities like outright fabrication in history books is almost exactly one of the key points Aron was making except that he (quite rightly) wasn't portraying that as a problem with Bayesian reasoning but sketching how to deal with it in Bayesian reasoning.

  16. Mactoul says:

    It is a mistake, of scientistic type, to attempt to quantify what is inherently unquantifiable. And that shows-in wildly divergent results for the same calculation i.e. of Resurrection. A Christian finds 10^43 and a atheist finds 10^-43.

    I refer you to "Did Muhammad Exist" by Robert Spencer.

    [made it a link--AW]

  17. Mactoul says:

    Elliot Nelson,
    We must not haste to the deeper levels. Consider the miracle of Cana or the Virgin Birth. Is there no violation of laws of nature? If no, then why would one call them miracles?
    What is precisely the thing that makes an event a miracle?

  18. Pingback: Bayesian evaluation for the likelihood of Christ’s resurrection « Biblical Scholarship

  19. Aron Wall says:

    Nice to hear from you. And, no, I don't at all agree. I think your conception of God as a set of "structures", bound by certain "rules" of a mathematical form, wouldn't make God any different, ontologically speaking, from anything else in Creation. In my own opinion, such a god would at best be an archangel of some kind, and we would need to figure out who created him to get to the fundamental source of reality.

    I also think it is exceedingly unlikely that anyone will ever write down a set of equations which mostly looks like the physics now, but has some sort of extra term or something that every now and then produces something like the Feeding of the Five Thousand. If miracles happen, this means that reality is fundamentally personal, not mathematical, as I argued in the series I just linked to. Just because physics is wonderful doesn't mean everything has to be physics.

    Long time no see! Yeah, you've basically understood what I'm saying, and I pretty much agree with all of your meta-comments too.

    I think I would say that we shouldn't have " `emotional certainty' even about things that rationally admit of substantial doubt"---not that I want anyone to suffer crippling anxiety or anything---but I think it IS probably proper to have a feeling of emotional certainty about something that admits only of infinitesimal doubt. So long as, one recongizes the uncertainty abstractly when having conversations about the state of the evidence rationally considered, and performs actions that are appropriate as well (e.g. looking both ways before you cross the street).

    My argument for this is that human emotions are not of sufficient fidelity to accurately respond to tiny differences in probability judgement, so any emotional worry I might have that (say) a 99.9+% probability proposition is really false is almost certain to be bigger than is justified. Certainly we shouldn't worry at all that the sun won't rise tomorrow, because any feeling of worry that was noticable would for that very reason be disproportionate to the evidence.

    By the way, did you see the Update at the end of the "Christianity is True" post? I wrote that a while back; it doesn't say all that much more than is in this post, but maybe you would still be interested in it as a belated reflection on some of the issues about certainty we were discussing...

    Incidentally, since the time we last conversed, Slate Star Codex has become one of my favorite blogs and I would have read all of the posts by now if he didn't keep on writing them! I know you follow Scott closely so I wanted to let you know I also think he's pretty awesome.

    As for the old books thing, one very small point. If (instead of trying to figure out exactly how strongly we should believe what we believe) we are only interested in the question of what we should believe, then it seems to me that what really matters is not the absolute strength of the evidence, but its relative strength to whatever is on the other side. So if one scales back the degree of certainty of all the arguments simultaneously, it doesn't change what is more probable (this obviously has to include arguments about priors as well as likelihoods). Although I have come to respect certain metaphysical arguments (including some for theism) more than I did a few years ago, I have difficulty putting myself in a brainstate where tricky metaphysics seems more certain than history.

    Arguing that Mohammad didn't really exist is NUTS! You can also find fringe crackpots who believe that Jesus didn't exist, and it looks like this guy uses the same dubious, well-refuted methods on Mohammad, for whom we have literally thousands of multiply-witnessed traditions, not to mention the Quran which (if you actually read it) is pretty homogeneous in style, and was apparently written down within 20 years of Mohammad's death. Yeah, the early Arabs were illiterate so a lot of information was passed down by oral tradition for a couple centuries before being written down, but there is way too much for it to just be some misunderstanding or conspiracy (this this tree diagram shows just one of literally thousands of examples that meet the high standards to be considered authentic by Muslims).

    In fact, I would say there is more evidence for Mohammad's existence than for Jesus' existence (although obviously, since I'm a Christian, I don't think there is as much evidence that Mohammad was what he said he was, as that Jesus was what he said he was).

    George Washington on the other hand? Now he definitely didn't exist. ;-}

  20. g says:

    I also think it is better not to feel "emotional certainty" about things there's substantial scope for rational doubt about, but people often do :-). (And some bits of the Christian tradition would strongly endorse their doing so when the things in question are articles of religion.) I agree that there's little point trying to calibrate our emotions super-accurately when dealing with extremes of likeliness or unlikeliness, but I remark that in many things our strength of feeling/perception varies roughly logarithmically (see e.g. Weber's law as regards perception; and measures of subjective wellbeing tend to increase roughly linearly with log(wealth) or log(income).[1]) There may be something to be said for cultivating a habit of thinking and feeling about probabilities in terms of log odds. Maaaaybe -- some reasons for not doing that will readily occur to one skilled in the art.

    [1] Lest I be misunderstood, I am not suggesting that money either is or should be the main thing that determines our subjective wellbeing. But it's one thing that contributes, and its influence seems to be roughly logarithmic.

    I hadn't seen your addendum to "Christianity is True". It's gratifying to see you moving towards my position on the strength of historical evidence :-). I agree that generally history should outweigh metaphysics, but if you replace "metaphysics" with something like "theory" or "general principles" I'm inclined to think there are plenty of exceptions. (Suppose e.g. the general principles in question are things we regard as laws of physics.)

    Yeah, Scott Alexander is a really smart guy and a very good writer. I don't know how he'd feel about your description of him as "not yet a Christian"...

    Mactoul, thanks for the Robert Spencer reference. I have to say that everything about it screams "crank and/or dishonest" to me; e.g., the outfits praising it on the book's Amazon page are all ones with a long track record of politically-motivated flakiness.

    If Christians get odds of 10^43 for something and atheists get 10^-43, that certainly indicates that something's wrong. But why think that what's wrong is trying to quantify it? Quantifying may be one of the better routes to figuring out more about the source and nature of their disagreements. (Also, the more sensible Christians and atheists alike will tend to agree that they aren't really sure of their position by a factor of 10^43 either way, and that if their calculations end up with such numbers that's at least partly because there are possibilities -- or whole classes of possibilities -- that the calculations ignore.)

  21. TY says:

    Aron and “g”,
    After all of the above, which I bracket as "intellectual skepticism", I circle back to my original question: On what basis should anyone believe in God, the Bible, the Resurrection, the Virgin Birth, the Trinity, and so on down the list?

  22. Mactoul says:

    "Arguing that Mohammad didn't really exist is NUTS! "
    IS the Bayesian probability of Mohammad existing greater than that of Resurrection?

  23. Mactoul says:

    A summary is provided at http://www.newenglishreview.org/custpage.cfm/frm/184451/sec_id/184451.
    It is an interesting view, by no means NUTS, and seemingly supported by numismatic and linguistic evidence.

    I do not understand probability. Esp I do not understand Bayesian analysis. What does it mean to say that a preposition may be believed with probability 5% or 50%?

  24. Mactoul says:

    You estimate the Bayesian probability of Krishna's ascension to be 1/4 * 1/6 = 1/24th of the evidence for Jesus's resurrection. that is 1/24 of 10^43. So, Krishna's ascension is almost as overwhelmingly likely as Jesus' resurrection. You estimate resurrection of a Japanese monk as 1/60 of Jesus' resurrection, Bodhidharma as again 1/60.

    So, by your reasoning, we must believe in the ascension of Krishna, and various resurrections given in various ancient tracts. Doesn't it point to a fundamental error in your method?

  25. Aron Wall says:

    [Several mistakes fixed as per g's comment that follows--AW]

    Presumably you go on believing these things for the exact same reason you did before? Nothing in my post was arguing that you shouldn't believe in them (based on the historical evidence for the New Testament and perhaps also for other reasons) as you know I am also a Christian! I am merely arguing against apologetics of the sort that pulls out giganto probability factors when people try to quantify all this numerically.

    In my opinion yes, the evidence for Mohammad's existence is stronger. If the only way I could refute Islam was by arguing that Mohammad did not exist, I would convert today. As it is, I think I can acknowledge his existence without therefore also acknowledging his prophethood.

    And numismatic evidence? You really think the fact that an early Muslim ruler wanted Zorastrians to also feel good about the currency, implies that Mohammad's existence is sketchy? Or that, it is at all likely that there would be a split in the Islamic community about whether Ali was the natural successor to Mohammad, if neither of those two historical figures had really existed?

    One can't always distinguish between crackpots and non-crackpots based on whether the evidence they present seems superficially plausible if you don't know anything about their subject. These people can be quite dishonest about how they present the evidence. If you read the long 1-star reviews on the Amazon page you'll find some of the reasons why the hypothesis is totally absurd.

    But if your goal was to remind me that there are historical propositions (such as the existence of Mohammad) to which I assign greater than 99.99% certainty, then you've succeeded. :-)

    If you want an introduction to Bayesian epistemology, there are plenty of explanations on the Internet. I got into it initially from reading this discussion,

    The "not yet" was intended as a droll expression of optimism, somewhat like St. Paul's statement to King Agrippa in Acts 26:29. I didn't mean to imply that I think his conversion is imminent or that there cannot exist intellectually honest people who (for whatever reason) assess the evidence very differently than I do. (Although, since I believe I'm right, I also believe the odds of an eventual conversion by an exceptionally intellectually honest person are greater than normal.) If by some remote chance Scott is reading this and feels weird about it, sorry ;-|

    but if you replace "metaphysics" with something like "theory" or "general principles" I'm inclined to think there are plenty of exceptions. (Suppose e.g. the general principles in question are things we regard as laws of physics.)

    I think there is an important qualitative distinction between, the kinds of evidence we have from physics that e.g. something like the Standard Model + General Relativity is a really great description of the physical world nearly all of the time, and grand totalizing principles like Naturalism which claim that nothing outside the "natural" world exists and there are no exceptions to the laws of physics ever. In Bayesian terms, the precision and predictive power of science only gets you high likelihood for "no exceptions" if you think that the exceptions would have been likely to already been noticed based on the scientific experiments that you've actually done.

    Given this, I think the philosophical arguments specifically for Naturalism are fairly weak. (Of course, somebody could still accept it as a default hypothesis until convinced of some specific supernatural hypothesis, citing Occam's razor.)

    I guess I would assign a pretty low prior to claims that violate the laws of physics, when there is no conceptual apparatus in place (or relevant parallels) to explain why this is a situation where that might be likely to occur. For example, if a person is not claiming to be a prophet or anything, they are just trying to make some farfetched claim which (say, unbeknownst to them) can be ruled out as impossible on physics grounds, then that is a pretty strong reason to disbelieve them.

  26. g says:

    I was going to make a snarky comment about the idea of Scott's putative conversion (or anything else about him) being immanent ... but then I looked it up and discovered that one of the meanings of that word is "entirely internal to the mind and having no external effect". I suppose we'd never know if he had that sort of conversion. But I think you probably meant "imminent" :-).

    I'm not quite sure why you're offering me pointers to introductions to Bayesian epistemology. I'm wondering whether perhaps you rearranged your comment a bit and something originally intended for Mactoul found its way into something addressed to me? (If it really was meant for me, then I would be interested to know what I said to suggest that I'm in need of such help.)

    I agree about the distinction between "present-day science plus naturalism almost always gives almost exactly the right answer" and "naturalism is The Perfect Truth" and with your formulation in terms of whether one should expect to have seen whatever exceptions there are. But yes, Occam, and I suggest that in a Bayesian framework the way to think about this isn't in terms of "default hypotheses" but in terms of priors, and the point is that "naturalism" is a much much simpler hypothesis than any version of "naturalism with very occasional deviations" concrete enough to get predictions out of, and that it's reasonable to assign it much higher prior probability. I agree about "conceptual apparatus", which I would cash out in terms of whether you actually have a version of not-quite-naturalism concrete enough to get predictions from.

    (We approach these things in such similar ways. But, alas!, you are not yet an atheist :-).)

  27. Aron Wall says:

    Yikes! Just like you guessed, the comment about Bayesian epistemology was definitely intended for Mactoul, and I'm sorry it somehow inadvertantly ended up in the response to you---who certainly do not need a Bayes 101 refresher!!! I've corrected the comment below (including the typo, amusing as your interpretation).

    I agree that Occam's razor is really an issue of priors. I guess my prior must just be flatter than yours. The other issue is that I'm influenced by some philosophical arguments for Theism---I think that Naturalism leaves some important questions unanswered which Theism helps to explain. So even before examining the historical evidence I'm more sympathetic to religion.

  28. TY says:

    Aron, that was a satisfactory reply with which I fully agree. I was a bit concerned that atheists and people who are not Christians, or who make it a profession to attack Christianity, might use your post in a self-serving way. Your reply of September 1, 2016 at 8:23 AM plugs that opening. Thanks.

  29. Mactoul says:

    "naturalism" is a much much simpler hypothesis than any version of "naturalism with very occasional deviations"

    It needs to be emphasized the the laws of physics apply only to inanimate bodies. As Aquinas has put it, the stones move by necessity, the sheep move by instinctive judgment, and man by free judgment.

    Animals have qualia that physics does not (and can not) account for. The inanimate things more over have aspects that physicists handwave away by talking of emergent properties. It has not been found possible to reduce even chemistry to physics. For instance, the quantum chemistry solvers assume the solution in the form of molecules and try to obtain the shape of molecular orbitals. But that, the matter would take a molecular form is assumed and not derived.

    That many theists talk of miracles in an erroneous way is a big obstacle towards a proper discussion of naturalism. It is simply incorrect to define miracles as infrequent events that are directly caused by God or have a deeper meaning. Meaning of an event is subsequent to what the event is. And who directly or indirectly caused the event is also comes after understanding what the event is.

    A miracle is an event that is inexplicable, irrespective of the frequency. A resurrection is inexplicable, given the facts of human biology. The Virgin birth is inexplicable, turning of water into wine is inexplicable. They are not infrequent events. they are events that could not have happened, in the course of nature.

    [Inserted some \i (close italics html tags---AW]

  30. g says:

    Mactoul, there is a great deal of evidence that the laws of physics apply just fine to animate bodies as well as to inanimate ones. Science has progressed a lot since Aristotle! What is true is that sheep and men are very complicated systems admitting higher-level descriptions that are often more useful for understanding and prediction than analysing them in terms of fundamental particles would be. As you observe, that holds even for many systems much simpler than sheep. But there's no valid inference from "we don't have the computational resources to analyse X from first principles in terms of physics" to "X does not in fact obey the same laws of physics that things easier to analyse do".

    I don't think any definition that's widely used can rightly be said to be incorrect, though you could argue that it's unhelpful or something. In the present case, I'm not convinced that even that is the case. You say that such things as a resurrection or a virgin birth are miracles because they "could not have happened, in the course of nature" -- but I suggest that that last phrase is in fact saying something about frequency. Imagine that someone somehow discovers that everything that ever happens involves some direct intervention by God -- would you want to say in that case that literally everything is a miracle? I wouldn't; it seems to me that there would still be an important distinction between events that fall into the usual pattern and ones that are, so to speak, dramatically divinely caused.

    (I also suggest that "inexplicable" isn't the best term to use in defining miracles. Suppose it turns out that there is nothing mind-like underlying the operation of the universe -- no gods, spirits, etc. -- but that there's a little element of randomness in everything that happens. Then in a way everything would be "inexplicable" but you surely wouldn't want to say that everything in such a universe is a miracle. Or, contrariwise, suppose it turns out that a god is in charge of everything, and that all his actions (including designing the laws of nature, and including any ways in which he brings about deviations from those laws) are chosen for the optimal achievement of some goal -- human flourishing, the god's glory, maximal number of insects, whatever. Then nothing would be inexplicable; everything would be perfectly explained by the god's pursuit of its goal. But it would still be helpful to distinguish between cases where the usual low-level regularities hold, and cases where they are smashed to pieces because they get in the way of that goal.)

    Anyway. Could you explain how different definitions for "miracle" make an important difference to how we understand naturalism? It seems to me that naturalism isn't and shouldn't be defined in terms of any particular person's notion of miracles; if you adjust your preferred definition of "miracle" it might make a difference to whether "miracles" are compatible with naturalism or something, but it won't change anything about what "naturalism" means, nor about what claims it makes about the world.

  31. TY says:

    Mactoul writes:
    “A miracle is an event that is inexplicable, irrespective of the frequency. A resurrection is inexplicable, given the facts of human biology. The Virgin birth is inexplicable, turning of water into wine is inexplicable. They are not infrequent events. they are events that could not have happened, in the course of nature.”

    Let’s not conflate “the miracle (call it X)” and “the X happening (call it Y).

    I interpret Mactoul to mean that X is inexplicable by purely natural laws of physics, chemistry, etc. that we have today. Maybe in the future, some chemist will explain how you get straight from H2O to C2H6O. So in this specific sense, he is right! But If Mactoul is saying Y is inexplicable, then it depends on which side of the fence you sit (and pray). As a Christian, I don’t have a problem and I don’t feel compelled to produce any fantastical Bayesian factor approaching 100%: there is nothing to explain God’s intervention, which is my explanation.

    If you are an atheist and hence a naturalist, then those X’s don’t happen, period. Explanation is thus irrelevant.

  32. g says:

    If you are an atheist and hence a naturalist, then those X's don't happen, period. Explanation is thus irrelevant.

    TY, explanation of the miracle may be irrelevant for a naturalist, but they still need an explanation of the actually-observed facts, and actually I suggest that that's what everyone, believer or un-, needs. So, e.g., John's gospel says that Jesus turned water into wine and says a bit about how the partygoers responded. No one is obliged to explain the fact that Jesus turned water into wine because it's debatable whether there's actually any such fact; but everyone needs (in some rather weak sense of "needs") to explain the fact that that story is there.

    I haven't heard anyone claim that the existence of such a story is much of a challenge for naturalists, but an almost exactly parallel claim is very commonly made for the NT accounts of the resurrection. And it's perfectly reasonable for a Christian to challenge someone who isn't a Christian to explain how the particular stories that are in the NT come to be in the NT.

    It's not usually posed as "OK, then, if Christianity is wrong then how do you explain the existence of these stories in our scriptures?". It usually goes more like this: "Here are some facts about the events surrounding the resurrection that everyone agrees on. How do you explain those if Christianity is wrong?", where it just happens that the only evidence for those facts comes directly or indirectly from the Christian scriptures. E.g., William Lane Craig is fond of making an argument along these lines where the alleged facts are (1) Jesus was crucified and his body buried in a tomb by Joseph of Arimathea; (2) on the following Sunday morning the tomb was found empty by a group of female followers of Jesus; (3) after this, various people and groups had experiences of encountering Jesus post mortem; (4) Jesus's disciples believed he was risen from the dead despite a predisposition to think otherwise. See e.g. here. Every one of these propositions derives all its credibility from its assertion in the New Testament, and although Craig tries to avoid putting it this way the challenge he's really posing is "explain why our scriptures are the way they are".

  33. kashyap vasavada says:

    Excuse me for a completely off-topic question, but this is of current interest to me. I read in the paper that sainthood was bestowed upon Mother Theresa. One of the requirement is performance of two miracles. I understand as a devout Christian, you believe that resurrection (a miracle) was so important that God made exception to the laws of nature to make it possible. I do not have any problem in people believing in their faith. But my problem would be that if you require exceptions (miracles) for every sainthood, there would be too many exceptions to the laws of nature. What is your opinion on this? If you have talked about such things before, please just give a reference. Thanks.

  34. TY says:

    Thanks for the comments.

    It is not a sufficient argument that (just) because Jesus’ resurrection and the empty tomb, for example, was reported by Christian sympathisers, then reports of the facts don’t count as evidence (or good evidence). Independence of the eyewitnesses to the facts must count for something.

    In the article cites, William Craig writes: “The appearance traditions in the gospels provide multiple, independent attestation of these appearances. This is one of the most important marks of historicity. The appearance to Peter is independently attested by Luke, and the appearance to the Twelve by Luke and John. We also have independent witness to Galilean appearances in Mark, Matthew, and John, as well as to the women in Matthew and John.” And he doesn’t mean that because the “events are mentioned in more than one gospel they thereby enjoy multiple, independent attestations.” (see A Reasonable Response, Answers to tough Questions on God, Christianity and the Bible, Moody Publishers, Chicago, pages 297 – 300).

    But I would agree with you that if one can show that the eyewitnesses were not independent but acted collusively to lie or misrepresent, or had reason or motive for so doing, then the reporting based on their accounts ought not be accepted as evidence. I think a sufficient condition for veracity of the facts is the corroboration by at least two independent witnesses.

  35. Aron Wall says:

    I've responded to your question here.

    I would agree if you had said that the primary or best historical evidence we have for these propositions comes from the New Testament. (Leaving aside arguments for Christianity based on "religous experience".) But at least some of it is also in extrabiblical sources, for example, I think if the New Testament itself were completely excluded, we would still have enough evidence from other sources to conclude that (a) Jesus was crucified and that (b) early Christians believed he was resurrected from the dead. That doesn't mean the residue would be enough to conclude that the early Christians were right to believe this, but I think there is some evidence for it outside the New Testament. (Mostly, of course, embedded in the form of other texts.)

    As for how much evidence might be located in texts in general, one point to make is that the New Testament contains quite a lot of information in it. Of course most of the bits of information aren't evidentially loaded, but let's rather stingily suppose that there is only 1 evidentially relevant "bit" of evidence on average for every 5 chapters of the New Testament (there are 260 chapters total), and that even these bits don't all point in the same direction. (By a "bit of evidence" I mean something in the text which is twice as likely to have been written if Christianity were true, than if it were false, or vice versa. Here Christianity is true/false is an umbrella generalization, that depends on the results of various proxy fights over authorship of the texts and the status of individual miracles, and maybe ethical tests as well. Presumably because of dependencies, the rate of bits / chapter goes down as one adds more and more chapters to a holy book; I am assuming that these dependencies have already been factored out.)

    Suppose for simplicity that these bits of evidence arise in a symmetrical way regardless of which hypothesis we adopt. So that, on the hypothesis that Christianity is [true/false] then we might expect that 1/3 of these bits are misleading (seeming to indicate that Christianity is [false/true] even though it isn't) and the other 2/3 point in the correct direction (seeming to indicate it is [true/false], which it is). Then, after cancellation this would get you 17.3 bits of information total, which gets you a Bayes factor of about 165,000 [for/against] Christianity. My point here is that even in the absence of "smoking guns", the accumulation (in a long text) of a large number of relatively small chunks of evidence can still lead to a sizable amount of total evidence. So it is not necessarily irrational for a person to have fairly strong intuitions about whether Christianity is historically plausible or not, after reading the New Testament, even if they may have difficulty putting into words why they believe this (because it involves the accumulation of many small indicia of reliability). To my mind, the main source of systematic error in this process is questions about one's own ability to judge these indicia without bias.

    (Of course, I am not conceding the absence of "smoking guns" providing well over 1 bit of evidence, just making a point about the evidence potentially available from reading texts. The model above is grotesquely oversimplified and ought to be taken with an enormous grain of salt.)

  36. naclhv says:

    Hello! I've begun my reply to Aron's post here.


    It'll go on for the next several posts, so if you're thinking "but what about...", then it may be something that's being covered in an upcoming post.

    Oh, and since Mactoul mentioned me by name:

    You estimate the Bayesian probability of Krishna's ascension to be 1/4 * 1/6 = 1/24th of the evidence for Jesus's resurrection. that is 1/24 of 10^43. So, Krishna's ascension is almost as overwhelmingly likely as Jesus' resurrection. You estimate resurrection of a Japanese monk as 1/60 of Jesus' resurrection, Bodhidharma as again 1/60.

    So, by your reasoning, we must believe in the ascension of Krishna, and various resurrections given in various ancient tracts. Doesn't it point to a fundamental error in your method?

    Mactoul, you seem to be confusing evidence and Bayes' factors. Recall Bayes' rule: posterior odds is prior odds times Bayes' factor. Each piece of evidence has a Bayes' factor, which means that for two pieces of evidence, the Bayes' factor would be multiplied twice.

    That is to say, when you ADD another piece of evidence, you MULTIPLY the Bayes' factors. Or, when you DOUBLE the amount of evidence, you SQUARE the Bayes' factors.

    So, for Krishna's ascension, there is 1/24th of the evidence for Jesus's resurrection.Then the Bayes' factor for that evidence would not be 1/24th of 10^54. Rather, it would be the 24th root of 10^54, or 10^2.25, which is about 178.

    Well, instead of being too high, that actually seems too low. But remember that I said that 10^54 was an underestimate (under the condition that we're ignoring crackpot theories for the time being, to satisfy Aron's objection). For a slightly different condition, I said that "If I were to carry out a more full and reasonable calculation, using all the different lines of evidence that a modern Christian has at his or her disposal, I do not doubt that the odds would turn out to be far in excess of 1e100" So, let's approximate the actual Bayes' factor for Christ's resurrection at about 1e100 (roughly doubling the evidence), and then Krishna's ascension would also increase correspondingly to about 10^4.5, or about 30000.

    This seems to be about the right number for Krishna's ascension. This means that there is still about 10^9 / 10^ 4.5 = 10^4.5 = 30000 cases like Krishna's ascension to be found in world history. This is still a little to large - I think maybe 3000 or 300 would have been better - but in a rough calculation that involved taking the 24th root of numbers like 10^100, getting within 2 orders of magnitude is actually quite good.

    So, no, my method does not say that we must believe in Krishna's ascension. It in fact correctly says that there would be numerous cases of a "resurrection" with similar levels of evidence as Krishna's ascension, which is what we in fact find. So this actually gives me additional confidence in my method.

  37. g says:


    I agree that the mere fact that the NT documents were written by Christians doesn't stop them being evidence.

    I don't think we are entitled to treat reports in the NT of independent eyewitnesses as actual independent observations, for the same sort of reasons Aron discusses in the post we're commenting on. If a text says falsely that one person saw the risen Jesus, it can almost as easily say falsely that a hundred did, which means that a text describing a hundred independent eyewitnesses provides nothing like 100x as much evidence as a text describing just one. (It is a bit more, all else being equal, because a false claim that 100 people saw something is a little easier to refute, which makes it slightly more likely that a text with such a false claim would have been discarded early in its lifetime. But not nearly so much more likely as some Christian apologists appear to think.)


    I agree that if we didn't have the NT we would still have evidence, e.g., from earlier Christians' words and actions, that Jesus was crucified and that some people thought he rose from the dead. But I suggest that all -- maybe that's just a little too strong, so let's say almost all -- of that evidence reaches us via the NT somewhere along the line. E.g., if I'd never heard if the New Testament and all copies of it vanished tomorrow from the earth, I could probably still go into a few churches and figure out from crucifixes and stained glass and so forth that they believed Jesus was crucified, and the fact that they believe it is evidence in its favour. But why did the people who built those churches and made their artworks believe that? Well, to a large extent because they found it in the New Testament.

    If you go right back to the earliest days of Christianity, there are some quite early people for whom we have some evidence that they believed some of those things, and it's at least plausible that they did so partly because of an oral tradition that doesn't go via the written scriptural texts. But I think it's fair to say that only a very small fraction of the evidence we have for the crucifixion and early belief-in-resurrection of Jesus doesn't come via the NT.

    I think there's an enormous amount of non-independence in the NT evidence for propositions about Christianity, and don't see any reason to think that the right way of looking at it is "so many bits per chapter" at all -- that suggests a basically-linear relationship, whereas it seems clear to me that if the NT were twice as long it would not provide anything like twice as much evidence.

    Indeed, it's fairly clear that there's a roughly constant upper bound on how much information the NT can provide for anything; it's not outright impossible that the whole thing was a deliberate fabrication, and the probability of that surely has only a very weak dependence on the length and number of the documents as such. (For the avoidance of doubt, I am not claiming that this is at all likely. But e.g. I wouldn't reckon it less likely than 10^-7 and this wouldn't change if the NT were, say, 10x longer.)

  38. Mactoul says:

    there would be numerous cases of a "resurrection" with similar levels of evidence as Krishna's ascension, which is what we in fact find.

    Are these 300 or 3000 cases of resurrection/ascension miraculous or not?
    If Christ's resurrection was the chief miracle He performed and the bedrock of the Christian belief, then what does it say about 300 historical resurrections, esp when they are unrelated to Christianity?

  39. Mactoul says:

    It is not a question of computational resources. Physics simply does not and can not account for qualia, in principle. Animals have qualia ergo animals are not fully describable by physics.

    "No one is obliged to explain the fact that Jesus turned water into wine because it's debatable whether there's actually any such fact; but everyone needs (in some rather weak sense of "needs") to explain the fact that that story is there."

    That is precisely why it is misguided for theists to start with NT miracles. One should start with modern miracles, esp Lourdes. They are well-attested. Eg from official Lourdes site:

    Danila Castelli : 69th cure of Lourdes recognized as miraculous by a Bishop

    Danila Castelli, born on 16 january 1946, wife and family mother, has lived a more or less normal life until the age of 34 when she started having spontaneous and severe blood pressure hypertensive crisis. In 1982 some Rx and ultrasound tests detect a right para-uterine mass and a fibromatous uterus. Danila is operated for hysterectomy and annexectomy. In november 1982 she undergoes partial pancreatectomy. A scintigraphy the following year proves the existence of «pheochromocytoma » (a tumor that secretes high amounts of catecholamines) in the rectal, bladder and vaginal region. More surgical interventions follow in the attempt to stop the triggers to the crisis until 1988 but with no bettering at all. In may 1989, during a pilgrimage to Lourdes, Danila gets out of the Baths where she had been immersed and she feels an extraordinary feeling of wellbeing. Shortly after she reported to the Lourdes Office of Medical Observations (Bureau des Constatations Médicales de Lourdes) her instantaneous alleged cure. After five meetings (1989, 1992, 1994, 1997 and 2010) the Bureau certified the cure with an unanimous vote : « Mrs Castelli was cured, in a complete and lasting way, from the date of her pilgrimage to Lourdes -- 21 years ago -- of the syndrome she had suffered and with no relation with the treatments and the surgeries she received ». Danila Castelli has since gone back to an absolute normal life. The CMIL (Lourdes International Medical Committee) in it's annual meeting of 19 november 2011 in Paris has certified that the cure « remains unexplained according to current scientific knowledge ». On June 20th 2013 Monsignor Giovanni Giudici, Bishop of Pavia, the diocesis where Danila Castelli lives, has declared the « prodigious-miraculous » character and the value of « sign » of this cure. It is the 69th cure of Lourdes recognized as miraculous by a Bishop.

  40. Aron Wall says:

    In your series, the analysis of (the extremely scanty evidence for) resurrections in other religions was interesting, but it seems to presuppose that the amount of evidence stacks independently, which I argued against. (Also, it should be mentioned that you are being quite generous in what you consider evidence for these other miracles.)

    Also, why are we considering only Resurrection claims? Wouldn't a better reference class be something more like, oh "dramatic miracles attributed to a religious founder"?

    And, one thing you specifically need to guard against is the possibility that the Apostles were for some reason especially unreliable as a group, diverse though they were. (Obviously, as a Christian I don't think this is true. But we are discussing small probabilities here. If I found out that St. Peter was lying, and then I found out that St. Paul was lying, then my conditional odds that e.g. St. James was also lying would certainly go up.)

    The strategy of starting with modern day miracles is an interesting one, but there's no reason why every approach has to be the same. There is not generally only a single possible order of presentation when leading people to the truth.

    Ideally, to establish the truth of claim one would like to maximize both (1) the number of written reports and (2) the number of claimed eyewitnesses. (The two categories sometimes overlap when a written report is claimed to be authored by one of the eyewitnesses.)

    You are right that large numbers for (2) is only a little bit effective against outright fraud; for that a large number of (1)'s is more useful. But (2) is useful for eliminating other skeptical hypotheses such as honest mistakes (honest on the part of the author, I mean).

    I wasn't proposing a "basically linear relationship", as I wrote in this caveat:

    Presumably because of dependencies, the rate of bits / chapter goes down as one adds more and more chapters to a holy book; I am assuming that these dependencies have already been factored out.)

    Rather, I have no idea how to calculate how the probability would go down, but at the same time I was using a stingy number to begin with, so I hoped it would balance out. This argument is still under construction; I'm not sure how I want the precise details to go...

    But I don't see why things would necessarily asymptote to some fixed upper bound (unless it is a fairly high one). As a criminal keeps committing crimes, the odds of them getting caught goes up. It's not at all linear because some criminals are cleverer than others, but it is not as though the possibility of accidentally incriminating yourself ever goes away.

    Nor am I quite clear on what you mean by "the whole thing was a deliberate fabrication". Do you mean a conspiracy by multiple authors (in which case the odds of it being successful should decrease as some function of the size of the conspiracy) or do you mean something more radical like all of the NT in fact being written by a single clever fraudster? (Because I think the odds of something like the latter are VERY low since some books of the NT are written in radically different styles. It would take a linguistic supergenius, among several other things.)

    [made a few edits later--AW]

  41. g says:


    If animals have qualia and having qualia guarantees behaviour inconsistent with the laws of physics, then we should be able to identify actual violations of those laws in (I guess) the brains of animals. Do you believe any such violations have been found? Do you believe that they would be found if looked for?

    (My own opinion, with which of course you need not agree, is that it's not in any way clear that having qualia implies any sort of failure to obey the laws of physics, nor is it clear that all animals have qualia.)

    The question of whether specific miracles claimed e.g. at Lourdes actually have (1) good evidence that the claimed phenomena occurred and (2) good evidence that the claimed phenomena require something beyond the usual course of nature is, I think, far removed from the methodological discussion I thought we were having, and I suggest we not get into it here.


    Yes, I agree that a document that claims lots of independent eyewitnesses is less likely to be wrong because of an honest mistake.

    My apologies for missing how explicitly you'd disclaimed any sort of linear relationship between length and amount of evidence; I suggest that if you don't want to claim such a relationship then it's best to avoid describing the amount of evidence in terms of "bits per chapter" :-).

    Some authors write in a wide variety of styles but yes, I think the super-paranoid conspiracy theory explanations of the origins of Christianity would probably require multiple writers to have been in on it. That would indeed yield some increase in evidence as a function of (not the length but) the stylistic diversity of the documents, but I suggest the increase would likely be pretty slow.

    Of course since no religion's scriptures are infinitely long the distinction between "really slowly increasing but eventually going to infinity" and "bounded above" is not really the one that matters here. I think we are at least qualitatively agreed that the amount of evidence offered by a big pile of closely related documents may grow very slowly as a function of the quantity of documents, and I suggest that deciding how much actual evidence for the resurrection there actually is in the NT is (1) really rather difficult, (2) something that requires careful detailed analysis not well suited to the medium of blog comments, (3) something both of us have actually made some attempt at in the past -- and therefore probably not worth trying to relitigate right now :-).

  42. Mactoul says:

    Animals behavior is directed towards their own good. This also can not be captured within a physics framework. Physical systems do not act to further their good.

  43. g says:


    There is nothing in the laws of physics to stop a physical system acting to further its own good.

    (Doing so in the fancy ways that animals -- especially clever ones like us -- do is really complicated, and therefore may require a really complicated physical system and be infeasibly difficult for us to analyse to see exactly how they do it. But I know of no reason why a physical system should be unable to act in its own interests.)

  44. Mactoul says:

    A physical system, a collection of atoms, does not have "interests" and can not have them either.

  45. Aron Wall says:

    Agreed that this is not the right time to relitigate our previous debate. But thanks for the comments and discussion.

    It would be helpful if you actually argued for your views instead of merely asserting them.

    For what it's worth, I don't see a necessary contradiction between:
    "Animals have qualia that physics does not (and can not) account for" (Mactoul),
    "it's not in any way clear that having qualia implies any sort of failure to obey the laws of physics" (g).

    for example an epiphenomenalist would agree with both of those statements, and so might a property dualist like myself.

    A similar thing might be said about the "good" of an animal. I am not at all sure that the concept of "good" is reducible to physics language. But that does not impeach the ability of Darwinian evolution to explain why there exist complex systems that (in ways that are sometimes very hard to analyze) act in order to obtain certain physical states of affairs conducive to their continued existence and reproduction (and which are, in fact, good for the animal in question).

    Also, maybe not all animals have qualia, but I would be very surprised if the "higher animals" like cats & dogs & chimpanzees did not. Basically this is for the same reason I believe other people are conscious (argument by analogy to myself, plus trusting my sense of empathy as providing basically reliable knowledge about the world), though with a lesser degree of confidence (because they are more dissimilar to me).

  46. g says:

    Mactoul, when you say 'A physical system, a collection of atoms, does not have "interests" and can not have them either.' you are begging the question. Suppose I say: I am a physical system and I have interests. (Which, if correct, is a refutation of your claim.) What reason do you have for denying this, other than your assertion that physical systems don't have interests?

  47. Cedric Brown says:

    Excellent essay. I appreciate the point about independence. It's more likely that one person could persuade others to lie about the resurrection than that several people would choose to lie about it spontaneously. However, a conspiracy itself seems fantastically unlikely. It would be utterly remarkable if even one person chose to pretend that Jesus had risen from the dead. And I can't imagine that even one other person could be persuaded to join the pretence.

    I don't think sceptics have ever got to grips with the difficulty of explaining the belief in the resurrection.

  48. Aron Wall says:

    Welcome to my blog, Cedric. I'm glad you agree.

  49. Mactoul says:

    A physical system has a hamiltonian. This is what physicists talk of. But they never talk of good of a physical system. QED.

  50. Mactoul says:

    "the ability of Darwinian evolution to explain why there exist complex systems that (in ways that are sometimes very hard to analyze) act in order to obtain certain physical states of affairs conducive to their continued existence and reproduction (and which are, in fact, good for the animal in question)."

    Complexity is just an all-purpose cop-out. I have worked in chaos theory and this term "complexity" was invariably used to pretend to an understanding one does not have. In this case of living beings, the physicists tend to have it backwards. Complexity is not the cause of life but life is the cause of complexity one sees in living beings.

    Also, Darwinian evolution is very far from being reduced to physics. Even so, Darwinian evolution has simply not explained why there exist complex systems that are alive. This is not even the purpose of Darwinian evolution which is speciation.

    Argument would proceed from definition of physics. Physics starts by considering inanimate bodies. The animate bodies may be considered in aspects they share with inanimate bodies such as mass.
    But clearly, animate bodies are more than inanimate bodies. But that "more" is simply physics can not deal with, by definition. Thus, animate bodies can not be fully described by physics or laws of physics.

  51. g says:

    Mactoul, that is an extraordinary argument. Physicists doing physics don't generally talk about physical systems' ability to indicate correspondences between French and English words ... therefore a French/English dictionary is not a physical system? Physicists doing physics don't generally talk about physical systems' role in political history ... therefore an original copy of the US Constitution is not a physical system? Physicists doing physics don't generally talk about whether a physical system embodies a chess position where white can mate in five moves ... therefore a chess set is not a physical system?

    What physicists do or don't talk about tells you something about what properties of a system are amenable to investigation by the methods of physics. It doesn't tell you anything about what properties the system actually has.

    (What properties are amenable to investigation by the methods of physics changes over time, of course, as science and technology make progress.)

  52. Mactoul says:

    You are using the term "physical system" rather loosely and equivocally. Is dictionary a physical system?
    The pages of the dictionary and the ink marks therein are physical but the meaning they convey isn't.
    A person is certainly not a physical system for he possesses an immaterial intellect. That can be shown by arguments such as argument from reason. A physical system can be abstracted from a person however, that system has mass and other physical properties.

    What is not amenable by the methods of physics is by definition not a matter of physics and hence it can not be called as "physical system". Otherwise, it is a metaphysical claim that everything must be ultimately amenable to physics, even though at present we are unable to say how. This claim is generally called scientism.

  53. g says:

    Mactoul, I am certainly not intending to equivocate. A dictionary is a physical system in the sense that its behaviour can, so far as we know, be fully explained in terms of the laws of physics. Even if e.g. it turns out that "meaning" is utterly irreducible to physics, it is still the case that any object with the same physical properties as the dictionary (say, a particle-for-particle duplicate) bears the exact same meaning as the original; whatever meaning-evoking qualities the dictionary has are not dependent on non-physical properties of the dictionary, even if they are dependent on non-physical properties of the people reading it.

    (I do not think the "argument from reason" shows anything much; in so far as it proves anything it does so only by smuggling in presuppositions that are at least as open to doubt as the claims it purports to prove.)

    I think it is completely wrong to say that if a system has properties not amenable to study by the methods of physics then it is not a physical system. It would be more reasonable to claim that in that case we cannot be sure that it is a physical system, that those properties aren't at least partly non-physical. But I think even that is wrong. Let me illustrate why with a hypothetical example.

    Suppose we come up with an entirely artificial set of physical laws; perhaps something like those of Conway's "Life". Let's suppose those laws are fully deterministic; given the complete state of any system obeying the laws and sufficient computational resources, we could work out its entire future. And suppose that, again like "Life", these laws (simple though they may be to state) turn out to support extremely complicated behaviour. Then it might turn out that given an incomplete description of some system obeying these laws, and given large but limited computational resources, we could be entirely unable to predict some important aspects of the future behaviour of the system. I am pretty sure that there are in fact theorems along these lines that one can prove concerning Conway's "Life". For instance, I think something like this is probably provable: "Suppose some configuration in Life is divided into 4x4 regions and you are told only the number of live cells within each region. And suppose you are allowed to watch its evolution -- but only in this reduced-resolution manner -- for as long as you like. Then for any N it is possible to construct two configurations that you cannot distinguish by watching them for N units of time, such that after that time one of the configurations grows for ever and the other destroys itself completely and ends up with no live cells."

    Now, I propose that observing such a system at the limited resolution permitted to us and trying to predict its behaviour is very much like physics. And I suggest that if a fact of the kind I just described is true, then there are questions about these systems that are not amenable to resolution by means of "physics". But there is surely no useful sense in which these are not physical systems. They perfectly obey our physical laws; these laws are even perfectly deterministic; the only obstruction to our ability to predict everything they do is our limited observational power.

    I am not making the metaphysical claim that everything must be ultimately amenable to physics. You, however, are making the metaphysical claim that various classes of thing are definitely not amenable to physics even in principle, and I think you are making it with far greater confidence than the evidence actually warrants.

  54. Aron Wall says:

    In order to prove your theorem I think you just need to put either a "Beehive" or a "Toad" in one of your 4x4 regions (both fit and have exactly 6 live cells at any given time), and then set up a glider to run into it, in such a way that the evolution of the system depends in the desired way on which one it hits. (If one glider is too trivial, you could try two.) Then you can prove your theorem by simply letting the glider begin a distance away that is proportional to N---although this may be more trivial of an example than you were looking for!

  55. Mactoul says:

    You again make the not-well-hidden assumption that living things are nothing but extremely complex physical systems and nothing but a lack of computational power hinders us making perfect predictions of their behavior.

    It is an assumption, unjustified empirically and unjustified on metaphysical grounds. You need to keep the definition of physics before you. Physics has always been defined for inanimate bodies. As applied to animate bodies, it treats them as inanimate, for instance an animal falling due to gravity.

    PS A dictionary is, strictly speaking, NOT a physical system. It has no dynamics. It is, however, a physical object.

  56. naclhv says:


    "Are these 300 or 3000 cases of resurrection/ascension miraculous or not?"

    They are highly unlikely to be. Now, do you understand how Bayesian reasoning allows me to conclude, in a mathematically consistent way, that these other miracles claims are unlikely to be true, while the resurrection is overwhelmingly likely to be true?

    Aron Wall:

    "In your series, the analysis of (the extremely scanty evidence for) resurrections in other religions was interesting, but it seems to presuppose that the amount of evidence stacks independently, which I argued against. (Also, it should be mentioned that you are being quite generous in what you consider evidence for these other miracles.)"

    Well, we're jumping the gun here a bit - these things will be discussed in the upcoming posts in my series. But the summary goes something like this:

    You can either assume independence or not in order to interpret these other reported miracles. They both lead to productive conclusions.

    If you assume independence, then these other resurrection reports serve to validate my Bayes' factor of 1o^54. (again, with the caveat that we're not using this for the full, aggregated probability for the resurrection) This is still immensely useful, because it essentially forces the skeptic into a crackpot theory.

    If you don't assume independence, you can still use these other reports to state that the Bayes' factor for the testimonies for Christ's resurrection is FAR, FAR in excess of 1e9. Basically, the idea is that since these other reports are actually empirical data from history, they automatically include things like conspiracy theories and alien interference at their correct probability values. So we don't have to worry about actually performing the full dependence probability calculation, because reality already did that, and gave us the result, as empirical data. That is to say, if you think that the evidence of Christ's resurrection was faked by a conspiracy, why couldn't a conspiracy produce just as much evidence for any of the other resurrection claims? Since these things did not in fact produce the same level of evidence, we can safely say that Christ's resurrection reports are not likely to be caused by these crackpot theories. Isn't empirical data wonderful?

    Of course, you're right that I'm being very generous with what I consider evidence for these other miracles. That only serves as further hedge for my conclusion outlined above. Assigning more realistic values for these evidence of other resurrection stories would only further strengthen my conclusion.

    "Also, why are we considering only Resurrection claims? Wouldn't a better reference class be something more like, oh "dramatic miracles attributed to a religious founder"?"

    That would certainly be a useful and interesting exercise - one that I hope I might get to one day. But I chose against doing this for a couple of reasons.

    1. It makes everything more murky and complicated. "Did Jesus rise from the dead?" is a simple question, one that witnesses have a very low chance to be wrong about. We have very good ideas about what it means for a human to be alive or dead. But something like "Is Buddhism true?" is a complicated question, with many parts, which is hard to get at even with historical, personal testimonies. Religions like Hinduism can't even really be said to have a founder.

    2. Focusing on other religions and their claims about their founder doesn't do much to answer an atheistic skeptic. They would simply say that while the evidence for Christianity is better than that of other religions, they all fall short anyways so it doesn't matter. But by focusing on a historical, physical event instead, my approach forces everyone to face the reality of that event.

    I do really hope to get to the "dramatic miracles attributed to a religious founder" question. I don't doubt that the conclusions there will be similar to what I found by looking at other resurrection events.

    "And, one thing you specifically need to guard against is the possibility that the Apostles were for some reason especially unreliable as a group, diverse though they were."

    I see two separate objections crouched in this sentence. One is the question of dependence - maybe the the disciples were lying as a group? This was briefly covered above, and will be covered more in my upcoming posts in the series.

    The other objection is of the "have you considered..." type, as in "have you considered that the disciples might not be trustworthy for whatever reason?". Fortunately for us, this is precisely the kind of objection that utterly withers in the face of empirical evidence. This is like your friend claiming to have been struck by lightening, and then you saying "are you sure you weren't hypnotized to imagine that?" When we have empirical evidence, merely mentioning the possibility of the evidence being wrong does NOTHING. The Bayes' factor of that empirical evidence already takes that fully into account.

  57. naclhv says:

    One more thing - the comparison with other resurrection stories also validates my value of 1e8 for the disciple's testimony, because of the 1e9 other reportable deaths in history, not one of them (apart from Christ's) came anywhere close to having any historical figure testify that the death was reversed via resurrection.

  58. TY says:

    Thanks for this post for stimulating a “helluva” lot of good discussion, which brings me to the exchange between Mactoul and g.

    Mactoul & g
    At the risk of simplification, the debate might be stated thus: Be it resolved that physics can explain everything -- both physical and non-physical phenomena.”

    Mactoul is obviously on the nay side. It seems to me that g is on the yea side but with conditions.

    A naturalist would naturally (no pun intended) take the position that even things we associate with the mind and consciousness are fully reducible to a physical system, and thus explainable by the (actual?) laws of physics. But regular readers of this Blog aren’t sure about that (see, for example, Aron’s post, Fundamental Reality VIII: The Hard Problem of Consciousness).

    g’s stated: “Let's suppose those laws are fully deterministic; given the complete state of any system obeying the laws and sufficient computational resources, we could work out its entire future.” But realistically, are the actual laws of physics deterministic? (See, for example, Aron’s blog post: Quantum Mechanics I: Interference; and Quantum Mechanics III: Wavefunctions).

    Are we asking too much of physics and, for that matter, the physical sciences? The same question might be asked of theology, to be fair.

    In the end, that both are needed might be a more reasonable thesis.


  59. g says:


    No, I am very decidedly and deliberately not making the assumption you accuse me of making (and of trying to hide, and of being incompetent at hiding it). I do, as it happens, think it likely that that assumption is correct. But I have been going out of my way not to make that assumption in this discussion. If you think I have been unsuccessful, how about showing what I've said that relies on it?

    It is simply untrue that "physics has always been defined for inanimate bodies" (assuming, as seems clearly the case, that you mean "... and not for animate ones"). Maybe that was true for, say, Aristotle, but it is flatly incorrect as applied to present-day physics. It might turn out, as you apparently believe, that the laws of physics apply only to inanimate bodies; but it is not at all true that physics by definition doesn't apply to them, and I would guess that a very large fraction of actual practising physicists regard their field of study as applying to animate as well as inanimate bodies.

    And it is simply untrue that a dictionary is "not a physical system" or "has no dynamics". Its dynamics may not be very interesting (though they could be -- the pages can move somewhat separately from one another) but so what?

    Aron: You may very well be right. (It's 3am local time and I am too lazy to think it through right now.) I don't mind at all if the facts of Life (heh) to which I'm appealing turn out to be trivial; all the better, in fact.

    naclhv: You say "of the 1e9 other reportable deaths in history, not one of them (apart from Christ's) came anywhere close to having any historical figure testify that the death was reversed via resurrection" but surely this is just wrong; for instance, there are other claimed resurrections in the Christian tradition and some of them are reported by some of the same people who reported the resurrection of Jesus. What am I missing?

    TV: what I'm mostly arguing here is that Mactoul is entirely unjustified in claiming to know that certain phenomena are definitely beyond physical explanation. (I happen to be of the opinion that probably everything is "physical", but whether that means everything has a physical explanation in any useful sense is not so clear.)

    I wasn't at all claiming that the actual laws of physics are deterministic. Rather, I was saying: consider the most flatly physicalist hypothesis one can imagine: everything happens according to fixed deterministic rules that can be modelled exactly using, say, a sufficiently capable computer; well, even that hypothesis is compatible with what Mactoul was offering as conclusively refuting physicalism, namely with some properties of some systems being resistant to investigation in physical terms. Therefore, we can let Mactoul be as right as he pleases about physics being unable to explain some things that living beings do; we can take the strawiest strawman version of physicalism; but his argument still wouldn't be sound.

    I would guess that Mactoul agrees with you that "both are needed", and presumably that's a reasonable thesis if, and only if, there is in fact a god of the sort studied by the theology in question :-).

  60. Aron Wall says:

    g writes to naclhv:

    naclhv: You say "of the 1e9 other reportable deaths in history, not one of them (apart from Christ's) came anywhere close to having any historical figure testify that the death was reversed via resurrection" but surely this is just wrong; for instance, there are other claimed resurrections in the Christian tradition and some of them are reported by some of the same people who reported the resurrection of Jesus. What am I missing?

    Yeah, I was thinking the same thing! Of course additional miracles supporting Christianity ought to provide further evidential support, but they slightly muddy the waters when one wants to calculate a background rate. Presumably naclhv really wanted to make this claim about non-Judeo-Christian resurrection claims.

    (Of course there is also a distinction in Christian theology between Resurrection in the sense of a temporary reversal of death, and Resurrection in the sense of a permanent transformation to a new state of being, but I don't think this is immediately relevant to naclhv's claims, and arguably Christianity has multiple Resurrection claims of both types.)


    I see two separate objections crouched in this sentence. One is the question of dependence - maybe the the disciples were lying as a group? This was briefly covered above, and will be covered more in my upcoming posts in the series.

    The question of dependence was indeed my point. I am not saying that I currently have any good reason to believe the apostles were unreliable---which has a lot to do with why I am a Christian---but if I hypothetically imagine finding out that some of them were, it would certainly make it seem more likely that the rest were. This issue is mitigated, but not entirely eliminated, by the diversity and former skepticism of certain of the witnesses.

    My point is not, of course, that the evidence is not strong. It is just that the evidence does not stack exponentially in the way you would get with perfect independence.

  61. TY says:

    I totally agree: “ I would guess that Mactoul agrees with you that "both are needed", and presumably that's a reasonable thesis if, and only if, there is in fact a god of the sort studied by the theology in question.”

    It all comes down to reasonable faith and there ARE reasonable grounds to believe in God the Creator of the universe. I think the God hypothesis trumps all others, and this is essentially what this post (motivated in this instance by naclhv) is all about.

  62. Cedric Brown says:

    Something that needs to be taken into account with the resurrection is the emotional commitment involved. It is easier to lie or be careless with the truth when no emotional commitment is required. For example, if you see a glimpse of something in the woods it wouldn't be too hard to convince yourself that it was Bigfoot. There isn't much at stake for the individual who may or may not have seen Bigfoot.

    But imagine that you catch a glimpse of a face in the crowd that looks like your dead best friend. You can't casually say to yourself, "Ah, it looks like Bob has come back from the dead." Believing that Bob has come back from the dead would require an enormous emotional commitment. And in a case like that you just can't accept weak evidence.

    So there is a huge difference between the resurrection and other claims of the paranormal. A strong belief in the resurrection implies exceptionally strong evidence. We know this because of what is at stake.

  63. g says:

    Cedric, I think that cuts both ways: it's also easier to fool yourself into thinking something if it's a thing you really want to be true. I think a surprisingly large fraction of recently bereaved people report having what seem to them to be encounters of some sort with the deceased.

    (Not usually anything so dramatic as meeting bodily with the (ex-?)deceased and talking to them at length, though. If the stories of that sort in the NT are not true, they are probably the result either of outright fabrication or (more likely) of a sort of Chinese Whispers process where at each stage the story gets a little more impressive, rather than being things that the people concerned genuinely thought had happened just as described in the NT.)

  64. Mactoul says:

    No. It does not "come down to reasonable faith". I was making a point about what physics is.
    It is entirely unexceptional to say that physics treats animate bodies as if they were inanimate. Thus, physics does not claim competence upon the living bodies as such.

  65. Cedric Brown says:

    Actually, g, that supports my argument. It is quite common for bereaved people to "see" the deceased. But when that happens the bereaved person will think that he is hallucinating or that he has seen a ghost. So the experience may be interpreted as evidence for the supernatural but it isn't interpreted as evidence for a resurrection. If you see Bob's ghost, Bob is still dead. He hasn't been resurrected.

    So a strong desire to believe that Bob isn't dead combined with some sort of evidence - a vision of Bob - doesn't lead to a belief that Bob is really alive.

    The resurrection of Jesus seems to be in different category.

  66. TY says:

    Go back to AD 30 (or approximately) in Jerusalem, nine o’clock in the morning, not the hour people usually have wine with breakfast. A massive polyglot crowd is gathered in Jerusalem for the day of Pentecost. In this crowd are a fisherman named Peter and followers of Jesus, a familiar face to many in the same gathering. The fisherman gets up and addresses an audience, proclaiming with a straight face the resurrected Jesus. Christianity is launched. Peter and fellow disciples are willing to face horrible persecution and death in the years to come. Call this insanity.

  67. g says:

    This discussion threatens to turn into yet another rehash of "Christianity: right or wrong?" which is rather on the large side for the comments thread on a blog entry about something slightly different. I shall therefore be (I hope) brief enough that no one gets the impression I'm trying to give complete answers to anything :-).

    Mactoul, it is simply untrue that physics treats animate bodies as if they are inanimate. I will give two examples. (1) I know someone whose PhD (in physics) was concerned with how to get good MRI images of bodies that are constantly moving a bit due to breathing, heartbeat, etc. These are issues that arise only because the bodies in question are animate. (2) There are some lovely books by Stephen Vogel about (to quote one of the titles) "life in moving fluids", relating fluid dynamics to the needs and behaviours of living things. Again, most of the stuff in these books arises precisely because the bodies concerned are animate. What is true, as I have already pointed out, is the weaker statement that physics is generally not an effective tool for working out the complicated details of what animals will do; this statement, however, offers no appreciable support for the idea that living things must operate in ways not even in principle covered by the laws of physics.

    Cedric, if you believe yourself to have seen a ghost then you believe that in some fashion the deceased lives on. It is true that ghosts and resurrected bodies are different things, but I was not claiming that the two are identical.

    TY, the very early history of Christianity post-Resurrection (e.g., the Pentecost story) is subject to the same sort of historical skepticism as the Resurrection itself; someone who considers the NT resurrection story untrue is under no compulsion to believe that the rest of the narrative surrounding it is accurate.

  68. TY says:

    g, I would have to go with naclhv on the overwhelming probability claims (subject to the comments by Aron) and on the Christian life experience -- call that personal -- regarding the Resurrection. I see no other persuasive counter-arguments. But I do respect your position, and more so, if it was outcome of weighing all the evidence, and not vice versa.

  69. Mactoul says:

    Physics in dealing with various physical and biochemical processes that take place within animate bodies necessarily ignores two salient features of the animals.
    1) They possess a unity that is not a feature of physical systems. For instance, eye exists as a part of the whole organism and can not be understood without it.
    2) Animal eyes see as our do. Animal ears hear as our do. In short, animals are conscious, have experiences of qualia. Physics is utterly incompetent to describe and understand these phenomena.

  70. g says:

    TY, I don't think anyone can ever claim to have weighed all the evidence, but I have done the best I know how to do.

    Mactoul, interdependence of subsystems is a very common feature of inanimate systems as well as of animate bodies, and there is nothing in physics that requires it to assume e.g. that you eyes don't interact with your skull; quite the reverse. You probably have some more specific notion of unity in mind, but you haven't said what it is. Physics doesn't at present have a good analysis of qualia, it's true, but the track record of the argument "we don't have a good physical analysis of X, therefore it is the result of something non-physical that will remain for ever beyond scientific investigation" is not an impressive one.

  71. Mactoul says:

    That physics is, in principle, incompetent to account for qualia is well-known philosophical point to which no satisfactory rejoinder exists.

  72. Aron Wall says:

    Instead of saying the arguments are "well-known", it might be more helpful to link to some specific article...


    Physics doesn't at present have a good analysis of qualia, it's true, but the track record of the argument "we don't have a good physical analysis of X, therefore it is the result of something non-physical that will remain for ever beyond scientific investigation" is not an impressive one.

    The track record of people claiming that long-standing philosophical puzzles can be definitively resolved by Science, is also not very impressive! :-) Of course scientific knowledge increases with time (essentially monotonically) and it seems like that would lead to some examples of people being wrong about the limits of science, even if there really are some things it can never explain.

    In any case there seem to be qualitative distinctions between consciousness and say, previous claims about vitalism. I think this Chalmers essay does a pretty good job of explaining why these claims are essentially different. (Normally I'd say that attacking vitalism is beating a dead horse, but St. Mactoul here seems to be a real live vitalist so I guess I can't really say that.)

    The question is, what would a supposed physical "explanation" of consciousness even look like? Doesn't it seem that however much physical details you specify about what is taking place between neurons in the brain, at some point you have to take a philosophical leap when you say "and such a structure would in fact be a conscious experience"? In a very loose sense I can imagine a correct theory of quantum gravity (some consistent mathematical structure which reduces to GR and QFT in various limits and gives probabilities for various final states if you collide gravitons with black holes etc.) but I have difficulty even loosely imagining what a physical "explanation" of consciousness could look like, unless it is simply an eliminationism, the denial that there is anything to be explained.

    I mean, take the Life cellular automaton again. Suppose hypothetically that it contained structures which moved around and interacted with the world in ways kind of like animals do. (In reality such structures are almost certainly unstable to a stray glider, in other words Life can't actually evolve life, but ignore this point.) Do you think it is meaningful to ask questions like, what is the minimum number of "live" cells needed to construct a being capable of having a phenomenal experience? Do you think there is any way of reducing the concept of conscious experience to some big conjunction and disjunction of statements about individual cells being "live" or "dead"? Or would you think that the very concept "consciousness" is too slippery imprecise and should be elminated entirely---in which case, how would you reconcile that with your own experience of being conscious right now?

    (As long as we are on the subject of the Game of Life, maybe it is worth mentioning that it had a creator (John Conway) who fine-tuned its laws to produce interesting behavior...)

  73. g says:

    Mactoul, there are certainly philosophers who say confidently that physics could not possibly ever explain qualia. There are also philosophers (not obviously incompetent ones, either) who say the reverse. I am getting the very strong impression that you have absolutely no interest in justifying any of the controversial claims you make with such confidence. I can see how that might be an effective rhetorical strategy (though I suspect less so here than in many other venues) but I am growing rather tired of it.

    Aron, it seems to me that science has actually done rather a good job of putting to rest what used to be philosophical controversies. Of course, in cases where it's widely agreed that science has resolved them what happens next is that everyone stops considering them part of philosophy.

    I think the most important differences between consciousness and life, as candidates for Things Science Cannot Explain, are these: 1. We've got much further along the road to explaining life, so vitalism is less credible than mysterianism about consciousness. 2. Consciousness-mysterians have in effect adopted a strategy that guarantees that their questions cannot be answered. There simply isn't any evidence one could possibly present, any argument one could possibly make, that would count as showing that consciousness is a physical phenomenon. #2 is the key difference that Chalmers points out, though of course he does so in terms more sympathetic than mine to consciousness-mysterianism. And it's also what you draw attention to, again with a spin different from mine :-).

    But, really, doesn't making that argument trigger at least a feeling of unease? What you're saying comes down to this: nonphysicalism about consciousness is unfalsifiable even in principle: no possible evidence could ever suffice. Usually unfalsifiability is a serious problem for a theory. Personally, I'm only comfortable holding an uncheckable-even-in-principle belief with much confidence if (1) I think I can actually prove it from first principles (note: observing that it's unfalsifiable doesn't count!) as with pure mathematics or statements that are true by definition, or (2) I can't avoid holding it because it's an unavoidable load-bearing element of my cognitive apparatus, as with those first principles themselves. In Bayesian terms, an uncheckable belief can't accumulate evidence, so it has to come from your prior, and I prefer my priors without too much unnecessary stuff built into them :-). And nonphysicalism about consciousness seems to me very much not the sort of thing covered by either #1 or #2. -- Of course, your attitude to unfalsifiability need not be the same as mine.

    As for consciousness and Life: I don't know whether the question "is this entity conscious?" is one that always has a definite answer. Not only because of marginal cases like someone half-asleep or drugged or suffering neurological damage, but also because our notion of consciousness has been developed in the context of life on earth and I don't see any reason to think it has a single canonical extension to situations very very different from that. But I can certainly imagine situations in which I would say of a particular Life configuration "at any rate it has something that in important ways closely resembles consciousness" and, e.g., in that case I would be very reluctant to wipe that Life configuration from the computer's memory.

    I would expect the task of finding a precise specification in terms of Life cells of what it means for some configuration to be consciousness to resemble that of finding a precise specification in terms of quarks and leptons of what it means for some configuration of those to constitute a basketball game. You'd need to do it by building up higher- and higher-level notions and any criterion that gave "right" answers would need either those higher-level notions or something else capable of doing the same work. But (modulo my concerns that "consciousness" is not well enough defined for there to be a single right way to extend it to Life configurations, and that in any case it has fuzzy edges) I would expect it to be possible in principle, though far too difficult for our present state of understanding, to give such a criterion.

    I shall not rise to the bait in your final parenthesis. You can probably work out the shape of my reply on your own :-).

  74. Mactoul says:

    Chalmers begins by asking
    "But how and why do physical processes give rise to the experiences?. ...This is the central mystery of the consciousness".
    It is a mystery only to those that assume that it is the physical processes generate consciousness, in the first place. But I see no reason to assume any such thing.

  75. Mactoul says:

    "nonphysicalism about consciousness is unfalsifiable even in principle."
    Not necessarily. We wait for someone to construct a conscious creature.
    Let me tell you my vitalist view. Animals are material but have non-computable aspects. That is, even though there is nothing immaterial about animals, their motion is not predictable, even in principle and this non-predictability has nothing to do with quantum mechanics, chaos or statistical mechanics.
    Your "physicalism" confuses two different things. a) Are animals material (b) Can physics deal with the material body that an animal is.
    That non-computable aspects of material bodies exist and may play a role in biology is an old idea, e.g
    Penrose "The Emperor's New Mind: Concerning Computers, Minds and The Laws of Physics "

  76. Mactoul says:

    Those philosophers that espouse the view that physics can explain consciousness talk of the Hard Problem of consciousness. They haven't made any progress in explaining consciousness and merely assert that it can be done.

    Can physics even explain chemistry? A discussion excerpted from branemrys blog.

    "The notion of structure has different meanings at different levels of description; while quantum mechanical structurs have been used to make excellent predictions of chemical properties, this isn't sufficient to prove that complete reduction is possible in principle, because (again) the higher-level properties get smuggled into the alleged reducing theory . Thus, for instance, quantum chemistry is to a great measure based on the Born-Oppenheimer approximation, which itself is grounded in the classical chemical view, "the picture of a semi-rigid framework of atoms connected by bonds that rotate and translate in ordinary space as time elapses". Likewise, you cannot get the concept of electronic configurations from quantum mechanics; you only get them from spectral observations. And so forth. As van Brakel points out, "numerical methods are governed by what experimental data have to be predicted; or, if new predictions are made, the choice of parameters is governed by extrapolations made from calculations performed on other molecules or general experience derived from experimental work on chemical compounds and reaction mechanisms". Quantum chemistry assumes the existence of a molecule, and adapts quantum mechanics to that; it does not pull the molecule out of quantum mechanics, and, in particular, (as is commonly noted in philosophy of chemistry) there is no clear way you can get chemical structure (molecular shape) out of quantum structure. The view that chemistry is simply reducible to physics is based on the assumption that science is heading toward microreductionism; it is not based on any actual success in the reduction.

  77. g says:

    Mactoul, if someone claimed to have constructed a conscious creature how would you be convinced it was actually conscious?

    I promise that I am not confused about the distinction between "are animals material?" and "can physics deal with the material bodies of animals?".

    Penrose thinks brains may be non-computable, but he doesn't think they're non-physical; he holds that consciousness arises from the laws of physics. So your view is importantly different from his.

  78. Pingback: Love Makes Christianity More Rational | Veritas Christo et Ecclesiae

  79. NWR says:

    Hi, Aron,

    I was raised Roman Catholic, had looked into all of this here and there before, but just accepted that a God was there. But my dad passed away a few months back, so now I've been reading, researching, stressing, and obsessing at times about Jesus and the afterlife. Comments like Sean Carroll's claims about no spirit particles and "impossible" really struck me as dubious(especially since he also said, at another point, that it's possible and plausible, if unlikely), especially when compounded with the ever-increasing preponderance of data and studies on nde's, obe's, etc., but I am no expert. Also, if you do a quick Google search on Jesus, the opposition would have you believe that he is a nonsense myth. Only I got enough knowledge there long ago to know that the events (in all likelihood) took place. But your posts have really helped in putting me at ease regarding all of this. I just wanted to reach out and say thank you. I'm in school for finance, not theoretical physics or theology. Also, I want to say how impressed I am with the comment sections of your posts. Largely worthwhile discourse without endless snarkiness like on so many other sites I have went to when trying tp read about all of this. Thank you again, and I'll be a regular lurker going forward

  80. James H says:

    I am not sure if this already familiar to readers of this blog but there is an interesting Bayesian analysis of the Resurrection-an MA dissertation- here:

  81. Pingback: Bayesian evaluation for the likelihood of Christ's resurrection (Part 50) - NaClhv

  82. Pingback: Bayesian evaluation for the likelihood of Christ's resurrection (Part 25) - NaClhv

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