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.

God and Evil

Back in the comments section of my post on Giving Thanks, an old college friend and I are discussing the age-old problem of why God permits suffering and other evils.  This is a serious problem; in my view the Argument from Evil is the only really formidable positive argument for Atheism.  (By a positive argument for Atheism, I mean something that provides specific evidence against God's existence, rather than merely making the negative claim that there isn't enough evidence for Theism to believe it.  In order to show that Christianity is plausible, both claims must be addressed.)

The conundrum is famous: If God is the All-Knowing, then he knows what things are evil, if he is the All-Powerful, he should be able to prevent them, and if he is the All-Loving, then he will want to prevent evil.  So why is there evil?

The only way to solve the problem is to postulate the existence of some good thing which cannot exist unless evil either exists, or is at least possible.  (Common "defences" might refer to putative goods such as free will, the opportunity for humans to exercise virtues, the orderliness of the universe, an afterlife of a sort that depends on people having had certain experiences, etc.)  If the good is such that it is logically impossible to get it without (possibly) getting the evil too, then the defence would be successful, since when we say that God can do anything, we don't mean that he can or would create a logical contradiction.  (As C.S. Lewis says in The Problem of Pain, "Nonsense remains nonsense even when we talk it about God.")  I'm not going to attempt a detailed defence here, but I do want to make some general points about the Argument from Evil.

My first point is that God's omniscience actually makes the Argument from Evil weaker, not stronger.  The reason is that we humans are not omniscient.  If we are ignorant, there's no particular reason to assume that we know what is the morally best way to run a world.  Suppose that you wrote down a list of all the things you regard as good (happiness, knowledge, beauty, whatever).  Suppose you figured out a way to weight all of these factors numerically—of course, there's no way we could ever agree on how to do this, and I'm not convinced it even makes sense, but let's run with it—so that you could assert that some possible kind of universe (call it $U$) is optimum: the best possible.

[Note for experts: my kinds of universes $U$ here aren't exactly the same as the "possible worlds" discussed by analytic philosophers.  If the best possible kind of universe contains something like free will or nondeterminism, there will be multiple "possible worlds" $W_1, W_2 \ldots$ consistent with the same overall plan $U$ of the universe, some of which may be morally better or worse compared to the others.]

Now if God knows about even a single kind of goodness that we are ignorant of, or if he weights the various kinds of goodness differently than us in any way, then of course God will view some other kind of universe $U^\prime$ as best.  It seems infinitely unlikely that $U = U^\prime$ just by coincidence, so it seems to be almost certain that the universe will appear to us to contain evils that we can't explain.  One can argue about whether this is a sufficient explanation, but it's definitely something that has to be taken into account.  The idea that a superhuman entity which created the universe will see things exactly the way we do is absurd:

“For my thoughts are not your thoughts,
neither are your ways my ways,”
declares the Lord.
“As the heavens are higher than the earth,
so are my ways higher than your ways
and my thoughts than your thoughts.”
(Isaiah 55:8-9)

The second point I'd like to make, is that the Argument from Evil has emotional force as well as intellectual force.  Atheists tend to get annoyed when Theists suggest that Atheists don't believe in God because they resent him.  I've certainly seen plausible cases of this, but I don't want to speculate that all Atheists are this way, since I don't like making unfounded accusations about individual people's characters.  (Maybe that's why my Politics category only has one post in it so far.)

Nevertheless, leaving the Atheists aside for a moment, I think I can say from an examination of my own heart, and conversations with other people, that it's easy to carry an unconscious grudge against God for various real or imagined grievances in our lives, or the lives of those we care about.  Even if we have no grudge, there can be a deep sense of pain from all the kinds of grief that we don't understand.

So the Argument from Evil carries emotional force as well as intellectual force.  There's no necessary reason why an intellectually satisfying answer should be an emotionally satisfying answer, or vice versa.  One should bear this in mind when evaluating the intellectual arguments, since we may be asking from an argument something that no argument can do.

Finally, I believe that Christianity has resources for addressing the Argument from Evil which don't exist in generic-brand Theism, or indeed in any other religion.  It's much too simple to say that the existence of evil contradicts Christianity, when in fact the most basic doctrine of Christianity logically implies the existence of evil.

The basic doctrine is that 1) we human beings are wicked and deserve punishment, and that 2) in order to forgive us, God became an innocent human being and allowed himself to be tortured to death by us, and that 3) this act provides us with spiritual healing now, as well as physical immortality for all eternity.  Now regardless of whether you like this idea, even if you find it implausible or downright incomprehensible, you must admit that it's an idea about how God relates to evil, and uses it for the sake of good.  If there were no such thing as innocent suffering, Christianity wouldn't even be possible.  If Christianity is true, then God has arranged things so that the most important thing that ever happened was a horrible but redemptive evil.  All other evils, we view in the light of the Cross.

Posted in Theology | 27 Comments

A Universe from Nothing?

Today I went to a talk by Lawrence Krauss entitled “A Universe from Nothing”, which had the following abstract:

The question, "Why is there something rather than nothing?" has been asked for millenia by people who speculate on the need for a creator of our Universe.  Today, exciting scientific advances provide new insight into this cosmological mystery: Not only can something arise from nothing, something will always arise from nothing.  Lawrence Krauss will present a mind-bending trip back to the beginning of the beginning and the end of the end, reviewing the remarkable developments in cosmology and particle physics over the past 20 years that have revolutionized our picture of the origin of the universe, and of its future, and which have literally revolutionized our understanding of both nothing, and something.  In the process, it has become clear that not only can our universe naturally arise from nothing, without supernatural shenanigans, but that it probably did.

In the first 45 minutes, he provided an animated and reasonably clear explanation of concordance cosmology, the current version of the Big Bang model, dating from the discovery in 1998 that the expansion of the universe is accelerating (rather than decelerating as one would expect from the attractive gravity of ordinary matter).  This is exciting but now well-established work, which I've heard about a hundred times before, but was probably new to many of the people in the audience.  It was peppered with occasional off-hand snears at Republicans, Theology, and Young Earth Creationism, but for the most part it was a pretty valiant stab at popularizing an important set of 20th century discoveries.

The real reason I was there, of course, was to listen to his claims in the last 15 minutes that modern cosmology somehow points to the nonexistence of a Creator.  His claim was that there is evidence that the universe came from "Nothing" according to physical processes, and this apparently is supposed to undermine the religious view that God created the world supernaturally.  There were so many things wrong with this part of his talk, both a physics and a philosophical perspective, that I'm not entirely sure where to begin.  But let's try anyway.

His Slam on Theology.  Krauss said that Theology wasn't based on empirical evidence, so therefore he didn't believe it.  That was it.  He didn't seem to take any particular theological ideas seriously enough to even try to define them, let alone refute them.  There was no indication that Religion had any other origin besides a bunch of clueless dudes sitting around asking "Why is there Something rather than Nothing?"  (In the case of Christianity, I thought it had more to do with a guy claiming to be God, doing miracles, and dozens of people saying that they saw him alive after he was killed.  But what do I know?)

But let's get back to cosmology, since that was the subject of his talk.  It used to be that Christians believed that the world was created a finite time ago, out of Nothing.  Although some of them, like St. Thomas Aquinas, said that God could have created a universe with an infinitely long past.  Atheists had (and have) a diversity of opinions, but most of them thought that things would make more sense if the universe were around forever, since then maybe you wouldn't have to explain where it came from.  Then Big Bang cosmology came along, and it now seems—provisionally speaking—like the Universe really did have a beginning.  Now some atheists think they can refute the Christian view that God created the Universe from Nothing by arguing that the world did emerge from Nothing.  The role-reversal here is a little strange.

The universe can only come from nothing if you define a certain kind of something as being "Nothing".  Duh, because any explanation by its very nature must explain one thing in terms of some other thing!  This other thing must be taken for granted for purposes of the explanation.  Now, Krauss actually referred to 3 different ideas which he called "Nothing #1, #2, and #3":

Nothing #1: an "empty" spacetime a.k.a. the vacuum.  In ordinary non-speculative quantum field theory (QFT), the "vacuum state" (the configuration of fields with the lowest energy) is actually filled with so-called virtual particles which can affect physics in various ways.  At least, that's what the popularized physics books say; if one actually studies quantum field theory rigorously, people tend to use somewhat different language since the notion of "virtual particle" can be difficult to define.  But let's spot him the terminology since he was talking to a popular audience.

Krauss claimed that if you start with an empty space which has no virtual particles in it, virtual particles will appear, and this is "something" coming from "nothing".  This is bosh, since strictly speaking, there's no such thing in QFT as a state with no virtual particles.  (If there were, it would be infinitely different from the vacuum state, and would therefore have an infinitely large energy.  That's not nothing at all!)  If anything can colloquially be called "Nothing" in QFT, it is the vacuum state.  But this state already has all those virtual particles in it.  And as time passes, this vacuum evolves to....wait for it....itself!  That's right, if you agree to call the vacuum state Nothing, then Nothing comes out of it.  (He seemed to think this story might change once you take gravity into account, due to negative energies, but I didn't really understand this suggestion so I won't comment on it.)

The QFT vacuum isn't nothing.  Of course, from a strict philosophical point of view, the vacuum state of QFT is not Nothing since it's filled with all those virtual particles, and even aside from that, there's the space and time geometry, which is not Nothing.  To fix this he started taking up a different kind of nothing:

Nothing #2: the absence of any space or time.  This actually connects to an interesting quantum gravity idea known as the "Hartle-Hawking state" or the "no-boundary boundary condition".   (Jim Hartle is on my floor at UCSB, by the way.)  The suggestion is that the laws of physics not only tell you how the universe at one time evolves to a later time, they also tell you what the initial state of the universe is.

In some sense, one can think of this state as emerging out of Nothing #2.  However, the sense in which this is true is subtle.  There's another sense in which the Hartle-Hawking state does not emerge from Nothing; rather it has existed for an infinite amount of time— the popular physics articles never mention this, for some reason!  This is an interesting and important idea, but I think it deserves to be in it's own post, after I've explained QFT better.  The important thing to know is the following:

The crucial physics here is totally speculative!  It was entirely based on speculative ideas about quantum gravity which anyone working in the field would admit are not proven.  This is because we currently have no experimentally testable theory of quantum gravity!  (Nor do we even know how to formulate a consistent theory of quantum gravity mathematically, except perhaps in some special situations that probably don't apply to the beginning of our universe)

I mentioned this in the Q&A afterwards.  My comment seemed to aggravate him a little, since he thought he'd been sufficiently clear about this.  But I discovered that at least one member of the audience was still unclear on which parts were speculative, and which weren't, at the end of the lecture.  In my experience, one has to be crystal clear about this sort of thing when speaking to a popular audience, or they tend to walk away thinking that "Science" has proven things when it hasn't.

Atheists such as Krauss scorn theology as being completely non-empirical.  They claim it is not based on evidence of any sort.  I find it extremely ironic when this sort of atheist thinks that speculative quantum gravity ideas are just the right thing to further bolster their atheism.  Suppose you think that Science is better than Religion because it is based on evidence, and suppose you also want to refute Religion by using Science.  Here's a little hint: consistency would suggest using a branch of Science that actually has some experimental data!

The universe has zero energy.  Krauss thinks that the universe coming out of Nothing has been made more plausible by cosmology.  To understand his terminology, you have to know that roughly speaking) a closed universe means that space at one time is finite in volume, and shaped kind of like a sphere, so that if you travel around the universe far enough you come back to where you started.  On the other hand, in a flat universe, space at one moment of time is shaped like ordinary Euclidean geometry, and is infinitely large.  Current observations indicate that the universe is flat.  As far as I could tell, Krauss' argument can be translated into these terms:

1. The total energy of a closed universe is zero.  (It's tricky to define energy in general relativity, but according to one commonly used definition, this is true.)
2. Conservation of energy suggests that if the universe came from Nothing, it should have zero energy.
3. If there was a period of extremely rapid expansion at the beginning of the universe (as evidence suggests there was—this is called inflation), then whether or not the universe started out closed, it should look flat today.
4. But the universe does look flat,
5. Therefore Science suggests that the universe was created out of Nothing,
6. Therefore there is no need for God.

Perhaps I'm missing some crucial steps in his argument.  But there seem to be several enormous leaps of logic in there.

The Hartle-Hawking state isn't Nothing either.  Strictly speaking, even the Hartle-Hawking idea doesn't strictly get the universe out of Nothing, since it says that the initial state of the universe depends on the laws of physics.  Now the laws of physics aren't nothing.  So if, for example, you are wondering if there is any role left for the Creator, then one might say he picked the laws of nature.

Now, there's all sorts of difficult philosophical issues involved in what's called the Cosmological Argument for the existence of God.  But it's hard to get into them with someone like Krauss who is so dismissive of Philosophy.  The trouble with people like that is that it isn't possible to just find things out using Science instead of Philosophy.  That's because you have to do Philosophy to know what is or is not implied by Science.  People who dismiss Philosophy still end up doing it; they just do it badly, without a critical examination of their premises.

Nothing #3: the string theory multiverse.  Krauss acknowledges that the laws of phyiscs themselves might seem to call  for an explanation.  Especially since the various constants of Nature seem to be "fine-tuned" to allow the existence of life (I'll go into this in much more depth later).  On the face of it, this seems to be at least some mild evidence for the existence of God, but Krauss would never admit such a thing.

He suggests that we can explain this fine-tuning if string theory turns out to be true.  That's because string theory has an enormous number of different possible configurations, that look like universes with different laws of physics.  Some people have suggested that if there's a gazillion different universes (known as the "multiverse"), each with its own laws of physics, that it's not surprising that one of those universes should support life.  Krauss admitted that there was some dispute as to whether this idea counts as "Science", what with it being totally speculative and arguably untestable.  But what I want to know is, why the \$@#& would we ever refer to an infinite number of universes, governed by the principles of string theory, as a Nothing?

I should say that this review is based entirely on Krauss' talk.  I have not read his book, but I have read this negative review by philosopher St. Feser.

Posted in Reviews | 2 Comments

Pillar of Science VI: Community Examination

Scientific Results are Examined Collaboratively.

Scientists do not work alone, but in a particular kind of community.  The last stage of a research project is publishing and explaining the results.  Assuming these results get noticed, this begins the process of further review, critique, confirmation and rebuttal by other scientists.  No one person is smart enough to see things from all angles.  We need help from others to look in a clearer, less fragmented way.  Perhaps one could call this undivided looking?

Science is not just a set of facts, or an abstract procedure for testing ideas.  It is an ethical, truth-seeking community.  The love of truth is embodied in the alliance of particular, fallible humans, united by a common geeky interest in finding something out.  Together we create a public deposit of information which can be used to find new things out.

Because the community as a whole is truth-seeking, in the long run it reduces the need to trust the competency and ethics of the original researchers.  If someone fakes an experiment (or else just makes an innocent mistake), other people will be unable to replicate the result, and eventually the truth will come out.

Healthy scientific collaboration encourages reasonable dissent.   Otherwise group-think can insulate the community from effective criticism of accepted ideas.  Some people say that scientists should proportion their beliefs to the evidence.  However, there's also some value in diversity of opinion, because it permits subgroups to work on unpopular hypotheses.  I suppose things work best when the scientific community taken as a whole proportions its research work to the evidence.

One might argue that collaboration is not strictly necessary to Science.  Imagine a solo scientist doing careful experiments in secret, and drawing the correct conclusions from them.  (Even in this case the scientist would be drawing on public ideas which had gone before, "standing on the shoulders of giants", as the saying goes.)  But in practice, the benefits of discussion are so great that it's hard to imagine a successful modern scientist working completely alone.  Hence the symbiosis of Science with the Academy.

Individuals who think they can revolutionize Science all by themselves are almost always crackpots, the sort of crazy person I described one pillar ago.  If you want to see clearly, you have to expose yourself to the light.

Posted in Scientific Method | 1 Comment