## Christianity is True

Last time I wrote a long discussion of what I think is the best evidence for and against Christianity.

When it comes to the historical testimony to the Resurrection, I analyzed the data in terms of a set of minimal facts which I think even a "liberal" biblical critic ought to accept.   By the "liberal" view I mean someone who thinks that the Gospels, Acts, half the letters of Paul, and the general epistles were pretty much all written in the closing decades of the 1st century, by people other than their traditional authors—but that nevertheless the texts are based on identifiable earlier sources which go back to some real, and much earlier, written or oral traditions about Jesus.  (Because some parts of the Gospels are obviously rooted in actual historical events.)

This was a concession, not my actual beliefs: in reality I think this "liberal" view, despite its acceptance by the majority of secular biblical critics, is supported mostly by circular reasoning, and is more than a little reminiscent of the claim that Bacon wrote Shakespeare.  Nevertheless, if a good case can be made for Christianity even under such hostile circumstances, then obviously things would work even better on a more "conservative" view.  For a Bayesian approach based on a somewhat more conservative view of biblical criticism, see this paper by Timothy and Lydia McGrew (they argue for the conservative biblical view, but don't incorporate it into their probability analysis).

I also made no reference to the "inspiration" of the Scriptures by the Holy Spirit; rather I am treating them as ordinary historical documents containing what is claimed to be testimony.  Just to be clear, I do believe that the entire Bible is inspired by the Holy Spirit, and that therefore all of it is an important communication from God to us.  I disagree with fundamentalists in that I don't think the point of the Bible is to communicate scientific facts, or even matters of historical trivia.  For this reason it doesn't bother me that the Gospel accounts contain apparent contradictions: although most of these contradictions would probably be resolvable if we knew the full circumstances, it's not important to me whether if the documents contain mild inaccuracies regarding trivial matters, as all genuine eyewitness testimony does.

So now I have to analyze how I think the probability analysis actually goes.  To summarize the previous post, the main factors weighing against Christianity are:

(A1) prior probability due to specificity of Christian doctrine,
(A2) prior probability due to the weirdness of Christian doctrine,
(A3) prior probability due to Jesus being only one of billions of people.
(B) the Argument from Evil

while some of the factors weighing in favor are:

(C) the Fine-Tuning version of the Argument for Design.
(D) circumstantial facts of Jesus' life prior to his death, making him more likely to be the Messiah
(E) multiple testimonies to the Resurrection,
(F) modern-day miracles.

This is by no means a complete list, but I think it contains most of the highlights.  I've excluded some other arguments for Theism, and various Christian experiences, as less likely to be persuasive to someone coming from a skeptical point of view, even though I rate some of these things as significant.

Each of these factors leads to a shifting of the relative odds of Christianity : Naturalism.  (I'm assuming here that Naturalism is the main rival hypothesis to Christianity for skeptics of a "scientific" sort.)  Since these factors are approximately independent, each of them multiplies the relative odds by some number.  Since we aren't going to agree on precise ratios, I'm going to just estimate how persuasive these are to me, in terms of orders of magnitude (i.e. powers of 10).

I think that (B) and (C) are both good for about 2-4 orders of magnitude and therefore they approximately cancel each other out.  Each of them, despite being grounded in certain observable facts about the universe, is a controversial philosophical argument about metaphysics, and I think it's quite rare for any such argument to establish a conclusion with an error rate less than $10^{-3}$ or $10^{-4}$.  You could say that I'm skeptical about Philosophy, but that's not really fair since shifting the odds by a factor of 1000 should count as a "success" even if this can be easily overwhelmed by more definitive evidence.  For similar reasons I don't think (A2) should be much less than $10^{-4}$.

(A3) and (D) cancel each other out nearly exactly, because if Jesus is in a very small group of people more likely to be the Messiah, then it doesn't matter how many other people there are in the control group.   See the discussion of the prosecutor's fallacy.

(E) contains a factor which cancels out (A3) exactly.  That's because the Christian testimony is to Christianity specifically, and not to any other religion.  Whereas a made up religion is going to be selected from some large set of possible religions.  See the discussion of the Farmer Jones case in the comments of this thread.

The remaining part of (E) has to do with how plausible the testimony is, leaving aside its specificness.  Now the testimony of a single individual is has to be worth at least 2 orders of magnitude, because most of the time people don't lie about things.

Somewhat counterintuitively, the value of testimony towards implausible claims like miracle reports is actually somewhat greater than this, because people are less likely to make up claims that are implausible.  There are many people who claim to have crossed the street, but only a few people claim to be abducted by UFO's.  The number of people who claim to have seen people physically resurrected is quite low; certainly much less than $10^{-2}$ of the population.  To the extent that this effect exists, (E) also partially cancels (A2).  But this cancellation only works up to a certain point, since nothing says that the rate of false claims has to mirror the true a priori probabilities.  For the small fraction of the population who is willing to make absurd claims, there is no limit to how unlikely their assertions may be.  So some residue of (A2) may remain.

What remains to be considered in (E) is the gross improbability of seeing testimony as strong as this about anything false.  The multiplicity of eyewitnesses, the priority of the women, the apparent unanimity in the pre-existing group of Twelve (minus Judas), and the fact that they were willing to die for their testimony all weigh in its favor.  All of these facts are strongly supported by the historical data.  Their initial skepticism and doubt, mentioned explicitly in almost all of the Gospel accounts, weighs heavily in its favor, so the skeptic is probably better off trying to argue (somewhat less implausibly) that this was made up later to prove a point.  The lack of any reasonable ulterior motive also makes the testimony less likely given Naturalism.

St. Paul presents a special case: since we know he was previously hostile, his testimony is more or less independent of the others.  That in conjunction with the unusual circumstances surrounding his conversion is a factor of $10^{6}$ all by itself; the group testimony of the remaining apostles is at least $10^{8}$, and that's rather generous, since on that hypothesis we'd expect to find about 100 similarly strong examples of false dramatic claims by large groups, in the most recent generation of $10^10$ people alone.

Taking everything together I think that the implausibility of a set of testimony looking like (E) given Naturalism is at least $10^{-14}$, and much lower if one takes a more conservative approach towards biblical criticism.  If this is right, then (E) pretty much overwhelms every other factor in play, and (F) is just icing on the cake.  Therefore, with high probability, the Resurrection seems to have really happened.  In other words, (some unspecified version of) Christianity is far more probable than Naturalism is.

Posted in Theological Method | 103 Comments

## Let Us Calculate

In this post I am going to outline what I consider the most relevant evidence for and against Christianity, in the form of a Bayesian probability analysis.  Now, just to be clear in advance, I'm not even going to take seriously the Jesus-myth crank theory that Jesus never existed.  No reputable scholar, no matter how hostile to Christianity, believes this, and one can only believe it by completely discounting almost all of the available evidence, including from nonbiblical sources.  And even the most skeptical or "liberal" biblical critics usually assume that at least some parts of the Gospels go back to real information about Jesus.  So I'm going to assume that Jesus existed, and that we know at least some minimal facts about him and his immediate followers.

Consider the proposition Christianity*, defined as something roughly like the following: "There exists one benevolent God who came to Earth in the form of a particular unique human being, died, came back to life again in resurrected form, and then departed again".  Note that: 1) Christianity* differs from Christianity in that it does not indicate which particular human being this is, whereas of course Christianity says that Jesus is that person;  2) We can, if we choose, add additional Christian doctrines to the proposition Christianity*, but I am going to argue that this doesn't matter very much for the end probability result, so long as the additional doctrines are supported by empirical testimony of the kind that I described last post, and we are willing to accept some rate of error in the correctness of our theological interpretations.  3) Although the conditional

is high, because Christianity is as far as I know the only historically credible example of Christianity*, nevertheless the prior probabilities of Christianity and Christianity* are quite different since there's lots of people, and Jesus is only one of them.

Let us now consider the prior probability of Christianity.  The prior probability (A) consists of a product of 3 terms which come from the following kinds of improbabilities in Christianity:

_____A1) Implausibility due to the fact that Christianity* makes a series of several specific claims, whereas prior probability of various hypotheses needs to be distributed amongst all possible kinds of religious claims.  (The size of this factor depends on how many doctrines we include in Christianity*, but I've been arguing in the comments to this thread that the large factor in A1 will be exactly cancelled in cases involving testimony.)
_____A2) Whatever implausibility is inherent in Christianity* due to the fact that it postulates new entities, which behave in weird ways.
_____A3) Implausibility due to the fact that if Christianity* is true, there are billions of people who might have been the special person, and Jesus is only one such person.

Next we have to consider whatever posterior arguments there may be for, or against, generic Theism (with a benevolent deity).  In my opinion there's only one really good philosophical argument against, namely the Argument from Evil:

_____B) One would expect a universe with a benevolent God to have little or no evil in it.  (If we define Christianity as including the existence of evil, this factor would be incorporated into the prior probabilities instead, but that won't make any difference.)

There's several possible arguments for Theism, some good and some bad, but I think the best one is the Argument form Design, which can be made surprisingly precise in the form of the Fine-Tuning Argument.  Eventually I'd like to talk about this in much more detail, but a short summary is here:

_____C) Since, in our current understanding of physics, the overwhelming majority of possible configurations of the constants of nature do not permit life, life is much more probable if those laws were selected by a deity interested in producing life.  The most plausible naturalistic explanations have to propose additional, ad hoc entities, e.g. a gazillion extra universes with different constants, or unknown physical mechanisms with surprising properties.

Next we turn to the arguments for Christianity specifically.  One could consider arguments from religious experiences, but to avoid wallowing in subjectivity I'll focus on miracle claims, particularly the historical testimony to the Resurrection.

Now remember from the Proscutor's Fallacy how even rare events can be expected to occur so long as one draws from a large enough pool of candidates.  In the story I told, the prosecutor needs both the DNA evidence (something strongly correlated with guilt when taken on an individual basis) and also independent circumstantial evidence that the person who matched is among a small group of people more likely to have committed the crime.  We discussed how this circumstantial evidence may be quite weak, in the sense that it only suggests the possibility of guilt without in any way proving it.  Nevertheless, it can make the critical difference when combined with stronger evidence.

This is important because factor (A3) is large, and we need to be able to cancel it out by showing that Jesus is more likely to be special than other people.  This evidence has to be independent of the Resurrection claim, so it's best to look at features of Jesus' earthly life prior to that time.  Exactly how one goes about this will depend on which fraction of the New Testament one accepts, but even from a "liberal" perspective I think it's extremely probable that at least some of the following facts are true:

• Jesus was Jewish (Note here that $P(\mathrm{Judaism} |\mathrm{{Christianity\!\ast} + facts\,about\,B.\!C.\,history})$ is moderate to high since Judaism is one of the few successful monotheistic religions, and if Judaism is true the Messiah should be Jewish—indeed, descended from King David, but that is more difficult to prove.)
• During his lifetime Jesus claimed to be, either the Son of God in a unique sense, or the Messiah, or at the very least he made some highly unusual claims about himself which were understood by some as implying such things.
• Jesus was one of the most provocative and insightful moral teachers who ever lived.  (Of course, if Jesus is the Incarnation of a benevolent God, we'd expect him to be the most insightful moral teacher.  But this is difficult to prove because there are disagreements about morality.  Nevertheless, judging based on the fact that his teaching is widely regarded as great even by saintly people from different religious traditions, it's fair to say he at least makes the top 100 list.)  Note also that even the "liberal" interpretation of New Testament scholarship accepts many of the sayings in the Gospels as really going back to Jesus, and even if some of the insightful sayings were made up by his disciples, those disciples would have had to become unusually morally insightful somehow in order to do that.
• At least some people prior to Jesus' death believed he had performed miracles involving healings, exorcisms, and/or power over Nature.  Otherwise it's difficult to explain the enormous number of realistic-seeming scenes in the Gospels where the crowds are pushing and shoving each other in order to try to get healed, and Jesus has to continually keep escaping from them and hiding in various places to get any rest at all.  Note also that the event where Jesus miraculously feeds 5,000 people is contained in all 4 gospels.

The following data point would also suffice for circumstantial evidence, for those willing to accept more details from the Gospels:

• Some facts about Jesus' life could reasonably be interpreted as fulfilling some of the prophecies found in the Hebrew Scriptures, to a greater degree than expected by chance.  (Note that this counts as circumstantial evidence even if the fulfillment of the prophecies is intentional, because even if this is easy to do, very few people bother to do it.  So it still picks out Jesus as special.  But it doesn't count if the fulfillments of prophecy are fictional reading-back into Jesus' life as a result of the Resurrection experience.)

Let's call this set of facts, and any similar ones, as (D).  Regardless of the precise details, the conjunction of these is sufficient to pick out Jesus as one of a very small number of people who were especially likely, prior to their deaths, to be the special person of Christianity*.  Consequently, the factor (D) cancels out the factor (C).

Since there are multiple circumstantial facts in (D), one could conceivably use them all by themselves to mount a cumulative case that Jesus has the features expected of the Messiah.  However, this would be tricky to do, since the 5 factors listed above aren't very independent of each other.

Next we turn to the testimonial evidence for the Resurrection.  Now, no one claims to have seen the actual instant when Jesus came back to life again, but the New Testament does claim that the tomb was found empty (except for angelic messengers) and that hundreds of people witnessed him physically alive afterwards, on several different occasions before the Ascension.

That's going by the New Testament, but what are the core facts that even someone fairly skeptical of the Gospels should accept?  There has to be some plausible origin story for how Christianity came into being: there must be some core set of claims which led to the later formation of the New Testament.

It would be extremely unparsimonious to postulate the existence of a completely different form of Christianity, of which we amazingly have no records, which transmuted into a completely different thing in just a few decades.  And besides, even the more skeptical biblical critics say that we have genuine letters of St. Paul dating from the late 40's or 50's (recall that the Crucifixion took place around 30), which make reference to doctrines and events taking place many years earlier.  Using just Paul's letters which are agreed to be authentic, together with some not-particularly-controversial chronological details from the Acts of the Apostles, one can date Paul's conversion to just a few years after the Crucifixion, sometime in the 30's.

Furthermore, while Paul might have been able to foist new doctrines off onto Gentile converts to Christianity, it's not very likely he would be able to seize control over the leaders of a previously existing religious group, which he had formerly persecuted, and convince them not only to accept the Resurrection, but also to claim to be witnesses of it.  As Paul says here:

For what I received I passed on to you as of first importance: that Christ died for our sins according to the Scriptures, that he was buried, that he was raised on the third day according to the Scriptures, and that he appeared to Peter, and then to the Twelve.  After that, he appeared to more than five hundred of the brothers at the same time, most of whom are still living, though some have fallen asleep.  Then he appeared to James, then to all the apostles, and last of all he appeared to me also, as to one abnormally born.

For I am the least of the apostles and do not even deserve to be called an apostle, because I persecuted the church of God.  But by the grace of God I am what I am, and his grace to me was not without effect. No, I worked harder than all of them—yet not I, but the grace of God that was with me.  Whether, then, it was I or they, this is what we preach, and this is what you believed.  (1 Cor 15:3-11)

Unless one thinks that Paul made up all of these people, it's clear from this list that the Christian belief in the Resurrection predated Paul's conversion.  So, using Paul's letters and the extremely broad details common to all four gospels (allowing for distortion as time passes), I think one should accept the following Resurrection facts (E) as nearly certain, even from a fairly "liberal" biblical criticism perspective:

• Jesus had an inner circle of twelve disciples, who (except for Judas) later were important leaders in the early church, especially St, Peter, St. James, and St. John.
• Jesus really died on the Cross.  (The theory that he swooned and then recovered is monstrously improbable.  Not only does it require Jesus to be sufficiently "dead-seeming" that the Romans and his friends didn't realize it, he then has to somehow regain enough health to persuade people he's not only alive but triumphant over death.  It doesn't explain Paul's conversion, and it's grossly inconsistent with any of the Gospel narratives.  Since the skeptic is still going to have to explain these things with some independent hypotheses, it's rather a stretch to have a medical improbability on top of all that.)

Furthermore, within a few months or years, the following were all true:

• At least some Christians believed that the tomb was found empty after the Resurrection.
• At least some female disciples believed (or else claimed to believe) that they were the first people to see Jesus, after his Resurrection.  (Since women were not considered reliable witnesses in the 1st century, male disciples would be unlikely to invent this detail out of whole cloth.  It must go back to some original historical fact.)
• Some group of male disciples, including at the very least the Twelve Apostles, believed (or else claimed) that Jesus had appeared to Peter, and to the whole group on more that one occasion, after his death.  (This is only going with the groups mentioned in both the Gospels and Paul.  Paul's list includes St. James, the brother of Jesus, and more than 500 other witnesses, while the Gospels include two additional disciples on the road to Emmeus.)
• Furthermore, either (a) many of these male disciples were originally inclined to be skeptical, or (b) the early Christians liked to invent stories portraying the faith of their most cherished leaders as weak.  Furthermore, they knew that persisting in this testimony was likely to lead to their deaths, which would have been obvious from (i) the fate of Jesus, (ii) the fact that their testimony was inherently uncomplimentary to the Jews and the Romans, and (iii) Paul himself (see below).
• We have Paul's testimony against himself that before his conversion he "persecuted the church of God and tried to destroy it" (Gal. 1:13).  Then, as a result of an experience which he interpreted as a Resurrection Appearance of Jesus speaking to him, he become a Christian and immediately began to preach the new faith.  (If one additionally accepts the accounts in Acts 9, this experience was accompanied by phenomena which affected his travelling companions, and resulted in his temporary blindness until 3 days later, when someone named Ananias came to see him as a result of a vision of his own, after which "something like scales" fell from his eyes.)

This last fact is staggeringly implausible from a naturalistic point of view, even ignoring the additional details from Acts.  This can be seen by imagining yourself in the situation beforehand and then asking how surprised you'd be if it happened.  It's a lot like Hitler converting to Judaism after being struck by lightening, and then later being accepted into the Jewish community and becoming a highly respected rabbi.  That's a 1-in-a-million event right there, folks.  If Christianity isn't right, it's still true that they were fantastically lucky in the case of Paul, at a very critical moment in their history.

If we were to accept even one of the additional corroborating circumstances in Acts, the event becomes even harder to explain naturalistically.  Yet in general the book of Acts (which records 3 different versions of the conversion of Paul), even if it was not written by St. Luke, is filled with so much realistic detail and mundane trivia that it's impossible for me to believe it doesn't incorporate at least some memoirs which go back to the time of Paul himself (especially in the chapters which use the "we" pronoun).

Finally, some of the Apostles were not only willing to die for their testimony, they did die for it.  Christian tradition says that all of the Twelve except for John were martyred, but the later lives of many of the apostles are known only through fanciful legends.  However, at least 4 cases are fairly certain:

• Peter, Paul, the Apostle James (brother of John), and the other James (brother of Christ) were all martyred.  The Apostle James was killed by Herod in the 40's, as described in the book of Acts (hard to imagine why they'd make this up).  That Peter and Paul were martyred in Rome is described in several early post-New Testament Sources, and the martyrdom of the other James in Jerusalem is described by the Jewish historian Josephus.

From Paul's uncontested letters we can also know this:

• Paul believed (or at least claimed) that he and certain other Christians sometimes had the power to do miracles, and that these miracles had been witnessed by the churches he writes to.  (Additionally, the book of Acts records several miracles by Peter and Paul).

Finally, if Christianity is true, one would expect that God would continue to sometimes perform miracles even up to the present day.  And there is indeed evidence for this.  But it would take a whole 'nother post to go into any details, so I'll just reserve a parking space for this by calling it (F).

Posted in Theological Method | 3 Comments

## Christianity and Observations

My first pillar of Science is that it is based on repeatable observations.  In order to see how Christianity measures up, we need to examine whether it is based on observations, and whether it is repeatable.

Observations ultimately boil down to sense-data experienced by individual human beings.  You can't go directly from observations to theories without a certain amount of interpretation, but that's true in Science too.  In order to be regarded as accurate, theories need to be based on empirical evidence, which has to be specific enough to prove that that theory is accurate, rather than just some similar theory in the class of theories.  Not all aspects of the theory need to be directly confirmed to be accepted (otherwise we could never use a theory to make predictions in new circumstances).  So long as the untested aspects of the theory are closely bound to aspects of the theory which are testable, the theory has a whole can be considered empirical.

Now Christian theology is empirical in this sense: that most of the core doctrines are the most reasonable explanations of certain ordinary sense-data reported by actual human beings.  By "ordinary" sense data, I mean things which appear to be interactions with the normal shared waking world, rather than sense-data seen in dreams or hullucinations, experienced in an altered state of consciousness.  (If Christianity is true, God does sometimes communicates using dreams and such, but the most important information was not given primarily in this way.)  The experiences might be "visions" in the sense of revealing truths about things in Heaven, but they are not "visions" in the sense of being subjective and intangible features of a single individual's private experiences.

Here's an example.  One of the most esoteric doctrines in all Christianity—the one that may at first sight seem to have the least to do with the actual physical world—is the doctrine of the Trinity: that God consists of a loving communion between the Father, Son, and Holy Spirit.  But the Gospel of St. Luke tells us that

When all the people were being baptized, Jesus was baptized too. And as he was praying, heaven was opened and the Holy Spirit descended on him in bodily form like a dove. And a voice came from heaven: “You are my Son, whom I love; with you I am well pleased.” (Luke 3:21-22)

There is a specific series of sense-data which communicates something about the relationship between these three entities, two of which are normally invisible.  (The Son would also have been invisible, but for his Incarnation as a human being.)  Combining this with other similar events, such as the Transfiguration and Pentecost, and the fact that Jesus claimed to be divine but also prayed to his Father, the Church came to the conclusion that the Trinity was the best explanation of the observed facts.  If it had been based on philosophical speculation, they would have come up with some more logical way to divine divinity into three aspects (like say, Creator-Preserver-Destroyer).

Note that here I am addressing the question of whether the claimed empirical facts support the theological claim.  Although some might argue that the claimed facts support some other claim (e.g. space aliens), most skeptics would probably think that's pretty silly.  It's a completely different question whether the purported testimony is actually true, whether it really happened or else was made up by somebody.  The accuracy of the accounts is a very important question, but before I go into it, I wanted to make sure to get clear what kind of factual support Christianity claims to have.

Here's a parable to illustrate why this distinction is important.  Water normally boils at 100 $^\circ$C.  Suppose I claim that if I drop an orange into water, it boils at 150 $^\circ$C instead.  I'm pretty sure that's false, but suppose you claim it is true.  And you say that you know it's true because you went into the laboratory and tested it carefully.  Now that doesn't mean I have to believe your claim.  I could accuse you of lying, or making stupid mistakes in the lab.  But if I make fun of you for "believing without any evidence at all" and make derogatory comments about the tooth fairy and Santa Claus, then I haven't understood the nature of your claim.

Admittedly, the degree to which Christian doctrine is founded on sense-data depends on the specific doctrine.  For example, the Second Coming will involve a lot of sense-data when it happens, but because it concerns the future the sense-data hasn't been observed yet.  It is based primarily on his promise that he will do so.  It is related to his Ascension into Heaven, which is itself partially based on sense-data (so far as his leaving Earth is concerned).  The fact that he has the power to rise from the dead and ascend into Heaven is certainly relevant to assessing how likely he is to keep this promise to return to Earth.

The appeal to sense-data becomes particularly clear in the case of the Resurrection.  Here are all the accounts of Resurrection Accounts contained in the four Gospels, Acts, and 1 Corinthians.  (The appendix of the Mark account is a latter addition, which was almost certainly not written by St. Mark.  It might well go back to the first century, but for obvious reasons I won't be putting any weight on it.  I've included it solely for the sake of completeness.)

Note how the accounts refer specifically to the fact that Jesus had a tangible body which could be viewed by multiple people with multiple senses simultaneously, and that he eats fish in order to reassure the disciples that he is not a ghost.  This despite the fact that he was also capable of instantaneously appearing and disappearing.  Note also the role that doubt and skepticism plays in several of these stories, as well as the fact that Jesus could eventually be recognized, but was frequently not recognized at first.

This is most obvious, of course, in the "Doubting Thomas" incident I quoted yesterday.  St. Thomas had an opportunity to believe in the Resurrection on the testimony of others first, but he rebuffs this by saying that he needs to experience the sense-data himself in order to believe it.  He demands that the experiment be repeated.  Jesus appears again a week later, and offers the evidence, but at the same time rebukes him gently by pronouncing a blessing on those who believe without seeing.

At this point the story is likely to raise all sorts of questions to a skeptical mind.  Why should faith play a role at all?  Why not just empiricism?

There's some deep issues here, but let's start with this: Jesus isn't requiring belief without empirical sense-data.  He's asking for belief based on the empirical sense data of other people besides yourself.  That may involve faith in the sense of trusting others, but not in the sense of "belief without evidence".

Leaving aside for the moment questions of why God would choose to arrange things that way, the situation itself is not that unusual.  It is how things always work when you're studying History prior to the 20th century, which is not repeatable.  It's how things usually work in Science itself, since only a very few people have replicated the fundamental experiments in physics personally.  Most of us believe in scientific experiments on the testimony of other people.  As a theorist I certainly haven't done all those experiments myself; in fact, I haven't even interviewed most of the people who did them!

This year, the Higgs boson was discovered at the LHC.  This is Science, so the results have to be "repeatable".  That means, if you have billions of dollars, your own team of hundreds of scientists, and a decade of your life to devote to it, you too can discover the Higgs boson.  But are you going to wait for that to happen before you believe it?  No, you believe it now, on faith!  And what I mean by faith here is not a vaporous sentimentality, but trust based on good evidence that the community of particle physicists is reliable in this particular respect.

(I apologize to anyone who was expecting a Bayesian analysis in this post.  As you can see, I'm still working up to it.)

Posted in Theological Method | 5 Comments

## The Gospel

It's a little bit strange liturgically to have Easter on Christmas Day, but that's how things worked out in this series.  I'm going to quote a passage from the Gospel of St. John, which illustrates the Resurrection claim and connects to many of the issues I'm going to discuss in this series.  (I know that many biblical critics believe that the Fourth Gospel isn't a historically reliable source, but for reasons that will be explained later, I don't agree with them.)  Note that this narrative occurs after the Crucifixion, so according to the Gospels, Jesus has already performed a bunch of miracles in public, and then been killed.

I won't add any more commentary here—that's coming later.  So then, folks, hear the word of the Lord:

Early on the first day of the week, while it was still dark, Mary Magdalene went to the tomb and saw that the stone had been removed from the entrance.  So she came running to Simon Peter and the other disciple, the one Jesus loved, and said, “They have taken the Lord out of the tomb, and we don’t know where they have put him!”

So Peter and the other disciple started for the tomb. Both were running, but the other disciple outran Peter and reached the tomb first. He bent over and looked in at the strips of linen lying there but did not go in.  Then Simon Peter, who was behind him, arrived and went into the tomb. He saw the strips of linen lying there, as well as the burial cloth that had been around Jesus’ head. The cloth was folded up by itself, separate from the linen.  Finally the other disciple, who had reached the tomb first, also went inside. He saw and believed.  (They still did not understand from Scripture that Jesus had to rise from the dead.)

Then the disciples went back to their homes, but Mary stood outside the tomb crying. As she wept, she bent over to look into the tomb and saw two angels in white, seated where Jesus’ body had been, one at the head and the other at the foot.

They asked her, “Woman, why are you crying?”

“They have taken my Lord away,” she said, “and I don’t know where they have put him.”At this, she turned around and saw Jesus standing there, but she did not realize that it was Jesus.

Woman,” he said, “why are you crying? Who is it you are looking for?”

Thinking he was the gardener, she said, “Sir, if you have carried him away, tell me where you have put him, and I will get him.”

Jesus said to her, “Mary.”  She turned toward him and cried out in Aramaic, “Rabboni!” (which means Teacher).

Jesus said, “Do not hold on to me, for I have not yet returned to the Father. Go instead to my brothers and tell them, ‘I am returning to my Father and your Father, to my God and your God.’ ”

Mary Magdalene went to the disciples with the news: “I have seen the Lord!” And she told them that he had said these things to her.

On the evening of that first day of the week, when the disciples were together, with the doors locked for fear of the Jews, Jesus came and stood among them and said, “Peace be with you!” After he said this, he showed them his hands and side. The disciples were overjoyed when they saw the Lord.

Again Jesus said, “Peace be with you! As the Father has sent me, I am sending you.”  And with that he breathed on them and said, “Receive the Holy Spirit.  If you forgive anyone his sins, they are forgiven; if you do not forgive them, they are not forgiven.”

Now Thomas (called Didymus), one of the Twelve, was not with the disciples when Jesus came. So the other disciples told him, “We have seen the Lord!”

But he said to them, “Unless I see the nail marks in his hands and put my finger where the nails were, and put my hand into his side, I will not believe it.”

A week later his disciples were in the house again, and Thomas was with them. Though the doors were locked, Jesus came and stood among them and said, “Peace be with you!” Then he said to Thomas, “Put your finger here; see my hands. Reach out your hand and put it into my side. Stop doubting and believe.”

Thomas said to him, “My Lord and my God!”

Then Jesus told him, “Because you have seen me, you have believed; blessed are those who have not seen and yet have believed.”

Jesus did many other miraculous signs in the presence of his disciples, which are not recorded in this book. But these are written that you may believe that Jesus is the Christ, the Son of God, and that by believing you may have life in his name.

(John 20)

Merry Christmas, everyone!

## Can Religion be Based on Evidence?

So I'd like to get kicking soon on the project of actually presenting the positive evidence for Christianity.  In my view the best evidence is the historical testimony of the apostles to Jesus' Resurrection (along with other ancient and modern miracle claims).  However, some people have problems with this because it isn't scientific, and they think that only a "scientific" proof of miracles should qualify as evidence.

The idea that Science is the only very reliable way to gather empirical data is called (usually pejoratively) Scientism.  It is closely related to Naturalism, the belief that the world consists entirely of a certain class of physical things, of a sort which can be scientifically analyzed.  However, the two are not the same, since Scientism is a claim about there being only one good methodology for learning about the world.  A Naturalist is free to believe that there are valid nonscientific methods for learning about the world, as long as they also think those methods don't reveal the existence of any entities they'd consider supernatural.  (There's a bit of a definition problem in defining what exactly natural vs. supernatural can mean, but we more or less know what kinds of things this sort of person doesn't believe in: gods, miracles, spirits or ghosts of any kind, psychic powers, destiny, reincarnation etc.)

Well, Scientism in its strongest form is obviously stupid, since as I pointed out here there exist several other kinds of evidence-based inquiry that involve different methodologies.  Here's another rebuttal by atheistic philosopher Richard Chapell, who points out that Scientism isn't even logically consistent with itself.  So, there may or may not be good reasons to believe in religious claims, but "Science" taken by itself is not one of them.

Well, that was easy.  Maybe too easy.  Because, after all, someone could say this:  Even if there are nonscientific methods of inquiry, hasn't Science at least taught us something about the way the world is?  And hasn't it taught us something about what kinds of evidence are reliable?  Maybe there isn't a sharp contradiction between Science and Religion, but maybe there are things that make it more difficult for a scientifically-minded person to accept religious claims.  I think a lot of people have this idea at the back of their heads, and I'm going to try to address it in my future posts.

For further reflections on the relationship between Science, History, Philosophy, and the various arguments for and against Christianity, see here:

Can Religion be Based on Evidence?

It also explains briefly why I think the Historical Argument for Christianity is quite strong, although I plan to go into that in considerably more detail here.

(Erratum: there are a couple things I'd phrase differently if I were re-writing this essay now.  First of all, my parenthetical statement about "overcredulous Catholics, Pentecostals, and missionaries to Third World nations" was intended as a statement of a skeptical point of view rather than my own view, although there certainly are some overcredulous people in the groups named.  And this book has convinced me that modern day miracles are more frequent than I had previously thought.  Also, the phrase "tortured to death" should really be replaced with "tortured or killed"—in fact the whole sentence is too strongly written, and should make clearer who exactly it refers to.  For now read it referring to "several of the key eyewitnesses", I guess.)

Posted in Scientific Method | 10 Comments

## Bayes' Theorem

Today I'd like to talk about Bayes' Theorem, especially since it's come up in the comments section several times.  It's named after St. Thomas Bayes (rhymes with "phase").  It can be used as a general framework for evaluating the probability of some hypothesis about the world, given some evidence, and your background assumptions about the world.

Let me illustrate it with a specific and very non-original example.  The police find the body of someone who was murdered!  They find DNA evidence on the murder weapon.  So they analyze the DNA and compare it to their list of suspects.  They have a huge computer database containing 100,000 people who have previously had run-ins with the law.  They find a match!  Let's say that the DNA test only gives a false positive one out of every million (1,000,000) times.

So the prosecutor hauls the suspect into court.  He stands up in front of the jury.  "There's only a one in a million chance that the test is wrong!" he thunders, "so he's guilty beyond a reasonable doubt; you must convict."

The problem here—colloquially known as the prosecutor's fallacy—is a misuse of the concept of conditional probability, that is, the probability that something is true given something else.  We write the conditional probability as $P(A\,|\,B)$, the probability that $A$ is true if it turns out that $B$ is true.  It turns out that $P(A\,|\,B)$ is not the same thing in general as $P(B\,|\,A)$.

When we say that the rate of false positives is 1 in a million, we mean that

(note that I'm writing probabilities as numbers between 0 and 1, rather than as percentages between 0 and 100).  However, the probability of guilt given a match is not the same concept:

The reason for this error is easy to see.  The police database contains 100,000 names, which is 10% of a million.   That means that even if all 100,000 people are innocent, the odds are still nearly equal to .1 that some poor sucker on the list is going to have a false positive (it's slightly less than .1 actually, because sometimes there are multiple false positives, but I'm going to ignore this since it's a small correction.)

Suppose that there's a .5 chance that the guilty person is on the list, and a .5 chance that he isn't.  Then prior to doing the DNA test, the probability of a person on the list being guilty is only 1 : 200,000.  The positive DNA test makes that person's guilt a million times more likely, but this only increases the odds to 1,000,000 : 200,000 or 5 : 1.  So the suspect is only guilty with 5/6 probability.  That's not beyond a reasonable doubt.  (And that's before considering the possibility of identical twins and other close relatives...)

Things would have been quite different if the police had any other specific evidence that the suspect is guilty.  For example, suppose that the suspect was seen near the scene of the crime 45 minutes before it was committed.  Or suppose that the suspect was the murder victim's boyfriend.  Suppose that the prior odds of such a person doing the murder rises to 1 : 100.  That's weak circumstantial evidence.  But in conjunction with the DNA test, the ratio becomes 1,000,000 : 100, which corresponds to a .9999 probability of guilt.  Intuitively, we think that the circumstantial evidence is weak because it could easily be compatible with innocence.  But if it has the effect of putting the person into a much smaller pool of potential suspects, then in fact it raises the probability of guilt by many orders of magnitude.  Then the DNA evidence clinches the case.

So you have to be careful when using conditional probabilities.  Fortunately, there's a general rule for how to do it.  It's called Bayes' Theorem, and I've already used it implicitly in the example above.  It's a basic result of probability theory which goes like this:

The way we read this, is that if we want to know the probability of some hypothesis $H$ given some evidence $E$ which we just observed, we start by asking what was the prior probability $P(H)$ of the hypothesis before taking data.  Then we ask what is the likelihood $P(E\,|\,H)$, if the hypothesis $H$ were true, we'd see the evidence $E$ that we did.  We multiply these two numbers together.

Finally, we divide by the probability $P(E)$ of observing that evidence $E$.  This just ensures that the probabilities all add up to 1.  The rule may seem a little simpler if you think in terms of proability ratios for a complete set of mutually exclusive rival hypotheses $(H_1,\,H_2\,H_3...)$ for explaining the same evidence $E$.  The prior probabilities $P(H_1) + P(H_2) + P(H_3)\ldots$ all add up to 1.  $P(E\,|\,H)$ is a number between 0 and 1 which lowers the probability of hypotheses depending on how likely they were to predict $E$.  If $H_n$ says that $E$ is certain, the probability remains the same; if $H_n$ says that $E$ is impossible, it lowers the probability of $H_n$ to 0; otherwise it is somewhere inbetween.  The resulting probabilities add up to less than 1.  $P(E)$ is just the number you have to divide by to make everything add up to 1 again.

If you're comparing two rival hypotheses, $P(E)$ doesn't matter for calculating their relative odds, since it's the same for both of them.  It's easiest to just compare the probability ratios of the rival hypotheses, because then you don't have to figure out what $P(E)$ is.  You can always figure it out at the end by requiring everything to add up to 1.

For example, let's say that you have a coin, and you know it's either fair ($H_1$), or a double-header $H_2$.  Double-headed coins are a lot rarer than regular coins, so maybe you'll start out thinking that the odds are 1000 : 1 that it's fair (i.e. $P(H_2) = 1/1,001$).  You flip it and get heads.  This is twice as likely if it's a double-header, so the odds ratio drops down to 500 : 1 (i.e. $P(H_2) = 1/501$).  A second heads will make it 250 : 1, and a third will make it 125 : 1 (i.e. $P(H_2) = 1/126$).  But then you flip a tails and it becomes 1 : 0.

If that's still too complicated, here's an even easier way to think about Bayes' Theorem.  Suppose we imagine making a list of every way that the universe could possibly be.  (Obviously we could never really do this, but at least in some cases we can list every possibility we actually care about, for some particular purpose.)  Each of us has a prior, which tells us how unlikely each possibility is (essentially, this is a measure of how surprised you'd be if that possibility turned out to be true).  Now we learn the results of some measurement $E$Since a complete description of the universe should include what $E$ is, the likelihood of measuring $E$ has to be either 0 or 1.  Now we simply eliminate all of the possibilities that we've ruled out, and rescale the probabilities of all the other possibilities so that the odds add to 1.  That's equivalent to Bayes' Theorem.

I would have liked to discuss the philosophical aspects of the Bayesian attitude towards probability theory, but this post is already too long without it!  Some other time, maybe.  In the meantime, try this old discussion here.

Posted in Scientific Method | 3 Comments

## All points look the same

I've told you so far that the gravitational field is encoded in a $4 \times 4$ matrix known as the metric.  Here it is, displayed as a nice table:

There's 10 components because the matrix is symmetric when reflected diagonally.  The 4 diagonal components $(g_{00}, g_{11}, g_{22}, g_{33})$ tell you how to measure length-squared along the four coordinate axes.  For example, the length along the $1$-axis is given by

where $\Delta x^1$ is the coordinate difference in the $1$-direction.  The remaining 6 off-diagonal terms keep track of the spatial angle between the coordinate axes.  If you know enough Trigonometry, you can figure out that the angle $\theta$ between e.g. the $1$-axis and the $2$-axis is given by this formula:

However, I've also said that the metric depends on the choice of coordinates, which is arbitrary.  We can use this freedom to choose a set of coordinates where the metric looks particularly simple at any given point.   We can start by choosing our four coordinate axes to be at right-angles to each other.  This gets rid of all those funky off-diagonal components of the metric, which involve two different directions:

If any of the four remaining numbers happen to be 0, we say that the metric is degenerate.  This would correspond to a weird geometry in which you can move in one of the directions for free without it affecting your total distance travelled.  Since we all know that's not the way the real world works, we'll ignore this possibility.

We can also rescale the tick marks along any coordinate axis.  This allows us to multiply each diagonal component of the metric by a positive real number.  So if say $g_{22}$ is positive, we can choose coordinates where it's $+1$, and if it's negative, we can choose coordinates where it's $-1$.  This gives us:

Since it also doesn't matter what order we list the four coordinate directions, all that matters is the total number of $+$'s and $-$'s.  This choice is called the signature of the spacetime.

Now if you remember my very first post on spacetime geometry, $+$ directions in the metric correspond to spatial dimensions, while the funny $-$ sign is what makes for a time dimension.  But the real world has one time dimension, everywhere.  No matter how far you travel, you'll never find a place (so far as we know) where there isn't any time direction, or where there are extra time dimensions.  So that means that the correct signature for spacetime has $(-, +, +, +)$ along the diagonal, which is called Lorentzian (a.k.a. Minkowskian) signature.  (If we had wanted to describe a timeless four-dimensional space, we would instead select the Riemannian (a.k.a. Euclidean) signature $(+, +, +, +)$.)  We conclude that for any point of spacetime, you can always choose a set of coordinates such that the metric takes a special form that we'll call $\eta_{ab}$:

In other words, if you zoom in on any point, you recover Special Relativity.  So after all this fidgeting around, we end up with a somewhat profound conclusion: in General Relativity, every point of spacetime looks the same as every other point.

This is related to what Einstein called the Equivalence Principle, which says that at short enough distances, the effects of acceleration are indistinguishable from being in a gravitational field.  We all know from personal experience that riding in an elevator can make us weigh more or less, and from TV that astronomers in the Space Shuttle are weightless when they're in free fall.  In other words, you can always choose a coordinate system in which there is no gravitational force at any given point.

(Lewis Carroll actually described this principle several decades before Einstein in Sylvie and Bruno, which includes a description of a tea party taking place in a freely-falling house.  Then he describes what happens if the house is being pulled down with a rope faster than gravity would accelerate it, and explains how you could have a normal tea party as long as you have it upside-down.  I like this book better than his more famous classics, but don't read it unless you can withstand LD20 of Victorian sentimentality about fairy children.  Also, Carroll didn't go on to discover a revolutionary theory of gravity based on this principle.)

It might seem now like everything has become too simple.  If the metric looks the same at every single point, then why did we even bother with it?  Where's the information in the gravitational field?  Well, it's true that for any one point, there's a coordinate system where the metric looks just like $\eta_{ab}$.  But there's no coordinate system for which the metric looks like $\eta_{ab}$ everywhere at once.  (Unless there's no gravitational field anywhere, in which case Special Relativity is true).  If you make the metric look simple in one place, it has to look complicated somewhere else.

So in order to describe the gravitational field properly, we have to find a way to compare the metric at different points.  We can do this using something called parallel transport.  I'll give more details later, but basically it tells us how an object moves in a gravitational field when we carry it along a path through spacetime.  When we carry the object around a tiny loop so that it returns to its original position, we might find that it comes back rotated compared to its original orientation.  If so, we say that the spacetime contains curvature.  If the spacetime contains curvature, this is a fact about the gravitational field which is invariant, i.e. objectively true.  You can't eliminate it just by changing your coordinates.

Posted in Physics | 1 Comment

## What is NOT Science?

In my Pillars of Science series, I enumerated six aspects of Science that help explain why it works so well.

It should be clear from my analysis that the characteristics of Science are quite flexible.  All of the criteria are matters of degree, so that they are met more strongly by some fields of study than by others.  Because of this fuzziness, we should expect to find borderline sciences, such as Economics, Anthropology, Psychology, and other social sciences.  It is both futile and unnecessary to try to come up with a criterion to draw an exact line between science and non-science.  In other words, the question of what counts as Science cannot itself be resolved with scientific precision, and is therefore not a scientific question.

This doesn't bother me too much because my parents are linguists.  So when I was growing up, they made sure I was aware that concepts are defined by their centers, not their boundaries.  For example, if I say the word "chair", then what pops into your mind is a thing with four legs at the dinner table.  You might admit under interrogation that a "beanbag chair" is also a chair, but it's hardly the first thing you'll think of.  Concepts can be useful even when they're a bit fuzzy at their boundaries.

Despite their flexibility, the criteria are sufficiently strict that many things don't qualify.  I don't just mean pseudo-sciences such as astrology or homeopathic medicine, but genuine evidence-based fields of knowledge (“sciences” in the archaic sense of the word) which aren't scientific in the modern sense, because they only satisfy some of the criteria.

For example, History and and Courts of Law, despite their empirical character, deal mostly with unique and unrepeatable events.  So they fail the repeatability prong of Pillar I.  Both of these fields are based primarily on testimony of witnesses, although Law Court fact-finding has much stricter rules about admissibility of evidence.  Since much of their subject matter can't be defined with quantitative precision, they don't do terribly well on Pillar IV either.  Academics in History do have a truth-seeking community similar in kind to the Sciences.  But in Law Courts, the role of ethics, community, and authority is completely different.

This does not mean that these fields should be held in contempt; their methods are sometimes capable of establishing specific facts with a very high degree of certainty, “beyond a reasonable doubt” as the saying goes.  They simply lack the particular methodology of science, which has a proven track record of almost routinely proving astonishing facts about the world, to a degree that ends rational opposition.  If you try to increase certainty by imposing a “scientific” approach on a subject that isn't suited for it, you risk generating a pseudo-science which jingles the jargon of science while missing its core value: self-correction through rigorous testing of ideas.

Philosophy is nonscientific for a different reason than the empirical humanities.  While many philosophers strongly value elegance and precision of ideas, typical disputes between philosophers are not very amenable to empirical testing.  That doesn't mean that observation plays no role.  But the way philosophers typically make arguments, they also rely on controversial background assumptions, which can't be definitively settled just by looking at the world.

If, despite the potential for controversy, the argument for the position is sufficiently convincing, this can still establish the philosophical position with great certainty.  In fact, unless the skeptical thesis that no knowledge is reliable could be refuted with near certainty, the result would be that no field of inquiry could produce near certainty.  This potential for certainty does not change the fact that Philosophy operates by a different methodology, which on average does not resolve controversies as easily as the methods of Science or even History do.

For this reason a philosophical thesis based on Science will usually have the degree of certainty associated with Philosophy, not that associated with Science.  A chain of reasoning is only as strong as its weakest link.  So a philosophical argument based on Science should not necessarily trump, e.g. a strong historical argument, simply because Science is normally more reliable than History.

So how do we fit ideas from different fields together?  In a future post, I'll discuss Bayes' Theorem, which is a flexible way to think about all different kinds of evidence-based reasoning, without making specific assumptions about the sorts of evidence we can include.

## Pillars of Science: Summary and Questions

I've now completed my Pillars of Science series.  My goal was to analyze why Science is  such an amazingly effective method for discovering new truths about the world.  Here are the 6 "Pillars" I identified.  Of course, Science is a multifaceted word: it can refer to a method, a set of theories, or a community.  Understanding how Science works really requires thinking about all 3 together.

Intro:

A. How do we test scientific ideas?

B. What kinds of ideas can be tested scientifically?

C. Who can test them effectively?

Having laid this preparatory groundwork, in the next few weeks I'd like to get to a more exciting and controversial topic: I plan to discuss Christianity specifically in the context of each of these 6 Pillars to see how well it holds up.  (But before I get to that, I plan to post a bit about whether there are any other evidence-based ways of looking at the world, besides Science.)

You see, in this blog I am taking seriously the "What about Science?" objection to Christianity.  Many people think that the basic principles of Science are somehow refute or undercut religious views.  These are supposedly based on something called "faith" which is diametrically opposed to "evidence".  While everyone knows that some scientists are religious, many people think this is only possible because of "compartmentalized thinking" in which the two different approaches to life are somehow sealed off in different compartments so that the "evidence" compartment isn't allowed to explode the "faith" compartment.

Now those of us who practice the spiritual discipline of Undivided Looking obviously approve of UN-compartmentalized thinking, in which we think of reality as a whole, without making special exemptions for parts of life we don't want to subject to critical scrutiny.  Somewhat paradoxically, this does not require us to disapprove of compartmentalized thinking.  In certain respects Science itself is based on compartmentalized thinking (see Pillar III).

And we couldn't stop doing it even if we tried, because our brains are wired for compartmentalized thinking.  (Especially the male brain, which is more likely to delegate tasks to particular regions of the brain, whereas the female brain is more likely to think using connections between different parts of the brain.  See e.g. this study.)  But what we can and should do sometimes, is make a conscious effort to look at things together, rather than separately.

Since I'm going to be referring back to these six Pillars of Science, I'd like to ask for some reader feedback.  Do you think my discussion of these Pillars could be improved?  I'd like to solicit criticisms on any of the following issues, or anything else you can think of:

• Is there any practice which is important to Science which I have not included in the Pillars?  Or which I should have emphasized more?
• Is there anything which I've said is important for Science, which actually isn't?  Are there branches of Science which do without any of these things?
• My perspective is that of a physicist who works on fundamental issues.  But there's lots of other scientific fields: Biology, Geology, Chemistry, etc.  Do you think someone from these fields might have prioritized different aspects of scientific practice than I did?
Posted in Scientific Method | 38 Comments

## Coordinates don't matter

In my last post about spacetime, I explained how the geometry of spacetime is determined at each spacetime point by a set of 10 numbers.  These 10 numbers are packaged together into a $4 \times 4$ matrix called the metric, which is written as $g_{ab}$.  The subscripts $a$ and $b$ stand in for any of the 4 coordinate directions (in a 4-dimensional spacetime).  Since the metric is symmetric, i.e. $g_{ab} = g_{ba}$, there are 10 possible numbers in this matrix.  The value of these 10 numbers depends on your position and time,which makes them a field, specifically the gravitational field.

However, there is an important caveat in all this.  The coordinates which you use to describe a given spacetime are totally arbitrary.  For example, a flat 2-dimensional Euclidean plane can be described using Cartesian coordinates $-\infty < x < +\infty$ and $-\infty < y < +\infty$.  In this coordinate system, the distance-squared is given by the Pythagorean formula

which can be written in terms of the metric as

On the other hand, for applications involving rotations, it's often useful to use polar coordinates: $0 \le r < +\infty$ (the distance from the origin) and $0 \le \theta < 2\pi$ (the angle around the origin, measured in radians).  They're related to the original coordinate system by

In polar coordinates, the distance-squared is given by

where the extra $r^2$ factor comes in because circles that are a greater distance from the origin have a larger circumference, so there's more space as you move outwards.  This can be written in terms of the metric like this:

(Note: I've given these coordinate systems their traditional coordinate names to make them look more familiar, but this is actually just a redundancy to make it easier for humans to think about it.  I could have written the two coordinates as $(x^0, x^1)$—the superscript being a coordinate index, not an exponent—and then you could tell whether it was Cartesian or polar coordinates just by inspecting the formula for the metric.)

Now the point is, these two coordinate systems describe the same geometry in a different coordinate system.  If we were playing pool (or billiards) on a planar surface, and you wanted to describe how billiard balls bounce off of each other, you could equally well describe it using either coordinate system.  The physics would be the same.

Of course, the language you use to describe the system differs.  Suppose that I analyze a collision using Cartesian coordinates, while you use polar coordinates.  And suppose we had to communicate to each other what happened.  If you say to me, "The cue ball had a velocity in the $x^1$ direction", then I'll get confused because $x^1$ means something different to me than it does to you.  These kind of statements vary under a change of coordinate system, they are "relative" to your coordinate-perspective.  So if you want to communicate with me, you have to find a way to describe what's going on which does not refer to coordinates in any way.  For example, you could say "The cue ball hit the 3 ball, which knocked the 8 ball into a pocket."  Since the two balls and the pocket are unique physical objects, we can all agree on whether or not this happened, no matter what coordinate system we use.  These kind of statements are invariant under a change of coordinate system.  The goal of coordinate-invariant physics is to describe everything in this sort of way.

Here's another way in which coordinate systems can let you down: when you use polar coordinates, there are places where the coordinates go kind of funny.  For example, when you're going around the origin clockwise in the direction of increasing $\theta$, and you arrive at $\theta = 2\pi$, you immediately teleport back to $\theta = 0$ since you've come full circle.  Even stranger, space seems to come to an end at $r = 0$ (the origin) since there's no such thing as negative $r$.  And if you're sitting right at $r = 0$, the different values of $\theta$ all refer to the same point as each other.  However, in reality we know that nothing weird is happening to the geometry at any of these points, since nothing strange happens in Cartesian coordinates.  (A similar issue comes up in black hole physics.  The original set of coordinates found by Schwarzschild blow up at the event horizon, but actually nothing unusual happens there in classical general relativity.)

The upshot of all this for general relativity is the following: I told you above that you can describe general relativity using the metric $g_{ab}$, which involves 10 numbers at each point.  But this description actually has some redundancy in it, since there's infinitely many possible coordinates systems you could use (one for each way of labelling the points uniquely with four numbers), and the metric looks different in each one—it isn't an invariant object.

When a theory has redundancy like this, we say there is a gauge symmetry.  A regular symmetry says that two different states (i.e. configurations) of a system behave in the exact same way as each other.  A gauge symmetry is stronger than a regular symmetry: it says that the two configurations are actually the same physical state of affairs.  In general relativity, the choice of coordinates is a gauge-symmetry.  It is a mere human convention which doesn't correspond to any actual physical thing in Nature.

Of course, even if you aren't doing general relativity, you can still use whatever coordinate system you like!  Most games of billiards can be understood in the approximation where space is flat (unless you like to spice up your games with black holes and gravity waves, like the cool kids do!)  In flat space time, all coordinates are equal, but some are more equal than others.  Although nothing stops you from calculating in horrible coordinates, the laws of physics look especially simple in ordinary Minkowski coordinates, where the symmetries of spacetime look especially simple.  Since Newton's First Law of motion holds in these coordinates, we call it an inertial frame.  (Here I'm ignoring the downward pull of gravity, since in billards we're only interested in horizontal motions.)

However, if you're doing general relativity, then there's a property of spacetime which forces you to describe physics in a coordinate-invariant way; at least if you want the equations of the theory to look elegant and lovely instead of like horrendous cludge.  This property is called curvature—but we're out of time for today.