# Infinity—Nothing to Trifle With (2 of 2)

(See Part 1 for the beginning of this discussion in progress …)

We can compare the sizes of two sets of numbers by finding a one-to-one correspondence between them, but in the case of infinitely large sets, strange things can happen.  For example, compare the set of positive integers I = {1, 2, 3, 4, …} with the set of squares S = {1, 4, 9, 16, …}.  Every element n in I has a corresponding n2 in S, and every n2 in S has a corresponding n in I.  Here we find that a subset of the set of integers (a subset which has omitted an infinite number of integers) has the same size as the set of all integers.

Playing with the same paradox, Hilbert’s Hotel imagines a hotel that can hold an infinite number of guests.  Suppose you ask for a room but the hotel is full.  No problem—every guest moves one room higher (room n moves to room n + 1), and room 1 is now free.

But now suppose the hotel is full, and you’ve brought an infinite number of friends.  Again, no problem—every guest moves to the room number twice the old room number (room n moves to room 2n), and the infinitely many odd-numbered rooms become free.

Infinity is best seen as a concept, not a number.  To understand this, we should realize that zero can also be seen as a concept and not a number.  Consider a situation in which I have three liters of water.  I give you a third so that I have two liters and you have one.  I now have twice what you have.  I will always have twice what you have, regardless of the number of liters of water except for zero.  If I start with zero liters, I can’t really give you anything, and if I “gave” you a third of my zero liters, I would no longer have twice as much as you.

Not all infinities are the same.  Let’s move from integers to real numbers (real numbers are all numbers that we’re familiar with: the integers as well as 3.7, 1/7, π, √2, and so on).

The number of numbers between 0 and 1 is obviously the same as that between 1 and 2.  But it gets interesting when we realize that there are the same number of numbers in the range 0–1 as 1–∞.

The proof is quite simple: for every number x in the range 0–1, the value 1/x is in the range 1–∞.  (If x = 0.1, 1/x = 10; if x = 0.25, 1/x = 4; and so on)  And now we go in the other direction: for every number y in the range 1–∞, 1/y is in the range 0–1.  There’s a one-to-one correspondence, so the sets must be of equal sizes.  QED.

(Note that this isn’t a trick or fallacy.  You might have seen the proof that 1 = 2, but that “proof” only works because it contains an error.  Not so in this case.)

The resolution of this paradox is fairly straightforward, but resolving the paradox isn’t the point here.  The point is that this isn’t intuitive.  Use caution when using infinity-based apologetic arguments.

Let’s conclude by revisiting William Lane Craig’s example from last time.

Suppose we meet a man who claims to have been counting from eternity and is now finishing: . . ., –3, –2, –1, 0.  We could ask, why did he not finish counting yesterday or the day before or the year before?  By then an infinite time had already elapsed, so that he should already have finished by then.…  In fact, no matter how far back into the past we go, we can never find the man counting at all, for at any point we reach he will have already finished.

The problem is that he confuses counting infinitely many negative integers with counting all the negative integers.  As we’ve seen, there are the same number of negative integers as just the number of negative squares –12, –22, –32, ….  Our mysterious Counting Man could have counted an infinite number of negative integers but still have infinitely many yet to count.

For a more thorough analysis, read the critique from Prof. Wes Morriston.

And isn’t the apologist who casts infinity-based arguments living in a glass house?  The atheist might raise the infinite regress problem—Who created God, and who created God’s creator, and who created that creator, and so on?  The apologist will sidestep the problem by saying (without evidence) that God has always existed.  Okay, if God can have existed forever, why not the universe?  And if the forever universe succumbs to the problem that we wouldn’t be able to get to now, how does the forever God avoid it?

This post is not meant as proof that all of Craig’s infinity based arguments are invalid or even that any of them are.  I simply want to ask apologists who aren’t mathematicians to appreciate their limits and tread lightly in topics infinite.

Of course, if the apologist’s goal is simply to baffle people and win points by intimidation, then this may be just the approach.

Related posts:

Related articles:

• “Aleph number,” Wikipedia.
• Wes Morriston, “Must the Past Have a Beginning?” Philo, 1999.
• William Lane Craig, “The Existence of God and the Beginning of the Universe,” Truth Journal.

# Infinity—Nothing to Trifle With

The topic of infinity comes up occasionally in apologetics arguments, but this is a lot more involved than most people think.  After exploring the subject, apologists may want to be more cautious.

Philosopher and apologist William Lane Craig walks where most laymen fear to tread.  Like an experienced actor, he has no difficulty imagining himself in all sorts of stretch roles—as a physicist, as a biologist, or as a mathematician.

Since God couldn’t have created the universe if it has been here forever, Craig argues that an infinitely old universe is impossible.  He imagines such a universe and argues that it would take an infinite amount of time to get to now.  This gulf of infinitely many moments of time would be impossible to cross, so the idea must be impossible.

But why not arrive at time t = now?  We must be somewhere on the timeline, and now is as good a place as any.  The imaginary infinite timeline isn’t divided into “Points in time we can get to” and “Points we can’t.”  And if going from a beginning in time infinitely far in the past and arriving at now is a problem, then imagine a beginningless timeline.  Physicist Vic Stenger, for one, makes the distinction between a universe that began infinitely far in the past and a universe without a beginning

Hoare’s Dictum is relevant here.  Infinity-based arguments are successful because they’re complicated and confusing, not because they’re accurate.

One of Craig’s conundrums is this:

Suppose we meet a man who claims to have been counting from eternity and is now finishing: . . ., –3, –2, –1, 0.  We could ask, why did he not finish counting yesterday or the day before or the year before?  By then an infinite time had already elapsed, so that he should already have finished by then.…  In fact, no matter how far back into the past we go, we can never find the man counting at all, for at any point we reach he will have already finished.

Before we study this ill-advised descent into mathematics, let’s first explore the concept of infinity.

Everyone knows that the number of integers {1, 2, 3, …} is infinite.  It’s easy to see that if one proposed that the set of integers was finite, with a largest integer n, the number n + 1 would be even larger.  This understanding of infinity is an old observation, and Aristotle and other ancients noted it.

But there’s more to the topic than that.  I remember being startled in an introductory calculus class at a shape sometimes called Gabriel’s Horn (take the two-dimensional curve 1/x from 1 to ∞ and rotate it around the x-axis to make an infinitely long wine glass).  This shape has finite volume but infinite surface area.  In other words, you could fill it with paint, but you could never paint it.

A two-dimensional equivalent is the familiar Koch snowflake.  (Start with an equilateral triangle.  For every side, erase the middle third and replace it with an outward-facing V with sides the same length as the erased segment.  Repeat forever.)  At every iteration (see the first few in the drawing above), each line segment becomes 1/3 bigger.  Repeat forever, and the perimeter becomes infinitely long.  Surprisingly, the area doesn’t become infinite because the entire growing shape could be bounded by a fixed circle.  In the 2D equivalent of the Gabriel’s Horn paradox, you could fill in a Koch snowflake with a pencil, but all the pencils in the world couldn’t trace its outline.

Far older than these are any of Zeno’s paradoxes.  In one of these, fleet-footed Achilles gives a tortoise a 100-meter head start in a foot race.  Achilles is ten times faster, but by the time he reaches the 100-meter mark, the tortoise has gone 10 meters.  This isn’t a problem, and he crosses that next 10 meters.  But wait a minute—the tortoise has moved again.  Every time Achilles crosses the next distance segment, the tortoise has moved ahead.  He must cross an infinite series of distances.  Will he ever pass the tortoise?

The distance is the infinite sum 100 + 10 + 1 + 1/10 + ….  This sum is a little more than 111 meters, which means that Achilles will pass the tortoise and win the race.

Some infinite sums are finite (1 + 1/2 + 1/4 + 1/8 + … = 2).

And some are infinite (1 + 1/2 + 1/3 + 1/4 + … = ∞).

(And this post is getting a bit long.  Read Part 2.)

Photo credit: Wikipedia

Related posts:

Related articles:

• “Is God Actually Infinite?” Reasonable Faith blog.
• Peter Lynds, “On a Finite Universe with no Beginning or End,” Cornell University Library, 2007.
• Mark Vuletic, “Does Big Bang Cosmology Prove the Universe Had a Beginning?” Secular Web, 2000.
• Wes Morriston, “Must the Past Have a Beginning?” Philo, 1999.
• William Lane Craig, “The Existence of God and the Beginning of the Universe,” Truth Journal.

# Women at the Tomb? Weak Evidence for the Resurrection.

Let’s consider an incident from that first supposed Easter.  All four gospels say that women were the first to discover the empty tomb.  (Of course, who was actually at the tomb varies by the gospels, as do many other important details about the resurrection, which makes the gospels unreliable as history.  But let’s ignore that for now.)

Many apologists point to the women as an important fact arguing that the gospels are reliable.  Greg Koukl says:

Women, disrespected in the ancient world, are the first to witness the risen Christ.  Why include these unflattering details if the Gospels are works of fiction?

I don’t know who argues that the gospels are fiction.  I don’t think they’re history, but I certainly don’t think that they were deliberately invented.  But let’s set that aside as well.

William Lane Craig says:

The discovery of the tomb by women is highly probable.  Given the low status of women in Jewish society and their lack of qualification to serve as legal witnesses, the most plausible explanation … why women and not the male disciples were made discoverers of the empty tomb is that the women were in fact the ones who made this discovery.

That is, having women make this momentous discovery is embarrassing.

This is an application of the Criterion of Embarrassment, which argues that you’re likelier to delete something embarrassing than add it to your story.  And if a story element is embarrassing, that points to its being historical fact.

But what’s embarrassing?  Things that look embarrassing to us may not have been so to the author.  For example, all four gospels show Peter denying Jesus three times.  That’s pretty embarrassing … or is it?

Paul’s relaxed approach in converting Gentiles conflicted with the more traditional approach of Peter and James (the conflict is shown in Galatians 2:11–21, where Peter is called “Cephas”).  Maybe supporters of Paul built their case by circulating a story in which Peter looks foolish, and this story became part of the canon.

So our question becomes: is it embarrassing to have women discover the empty tomb?  These apologists certainly think so, and historical records agree on women’s unreliability.  Josephus, a first-century historian, stated, “Let not the testimony of women be admitted because of the levity and boldness of their sex,” and the Mishnah (a Jewish legal text written in 220 CE) concurs.

However, this flimsy argument is much more popular than it deserves to be.  I’ll respond in several ways.

Give the original authors credit for being good storytellers.  As plot twists go, having women make the discovery instead of men isn’t particularly shocking.  But if you find it a powerful argument for the truth of the story, then you can imagine why that element might have become attached to the story.

The gospel story wasn’t made up.  The point that women were unreliable witnesses is relevant only in rebutting the charge that the story was deliberately invented, a claim I don’t make.  I’ve never heard this hypothesis except by apologists.  Instead, what best fits the facts for me is that the story documented in the gospels (in incompatible versions, but that’s another story) is the result of forty or more years of oral history.  Each gospel is a snapshot of the tradition of a different church community in widely different places (perhaps Alexandria, Damascus, or Rome?) and over decades of time.

Believers might demand, “Well, how do you explain the empty tomb?”  But of course, that assumes the accuracy of the gospel story to that point.  It’s like saying, “How do you explain Jack’s cutting down the beanstalk any other way than that there really was a giant climbing down after him?”

Who cares about women’s “unreliability”?  Women discover the empty tomb, they tell men, the men verify the story, and then the men spread the word.  If you don’t like women as witnesses, you’ve got the men.

That women were less reliable as witnesses in court doesn’t matter because there is no court in the story!  The women were trustworthy where it mattered—in conveying the story to people who knew and trusted them.

Tending to the dead was women’s work in this culture.  Instead of women discovering that Jesus had risen, imagine this incident:

On the first day of the week, at early dawn, Simon Peter and James entered the kitchen to prepare bread for the community.  In the darkness of the kitchen, a voice called out to them saying, “Why do you tend to minor matters when there is the LORD’s work to be done?”  And they took hold of His feet and worshiped Him.

What’s wrong with this story?  It’s that preparing bread is women’s work in this culture.  It makes no sense to have men come across Jesus in the kitchen.  And the same is true for men dealing with the dead.  According to the Women in the Bible web site:

It was the women’s task to prepare a dead body for burial. … Tombs were visited and watched for three days by family members. On the third day after death, the body was examined. … On these occasions, the body would be treated by the women of the family with oils and perfumes. The women’s visit to the tombs of Jesus and Lazarus are connected with this ritual.

The Bible also gives clues to women’s role in mourning in Jer. 9:17–20 and 2 Sam. 14:2.

Mark focuses on reversals, and the other gospels followed Mark’s lead.  Richard Carrier gives a detailed discussion of this topic and argues that a the-last-shall-be-first philosophy led Mark to add this touch.

Given Mark’s narrative agenda, regardless of the actual facts, the tomb has to be empty, in order to confound the expectations of the reader, just as a foreign Simon must carry the cross instead of Peter, a Gentile must acknowledge Christ’s divinity instead of the Jews, a Sanhedrist must bury the body, and women must be the first to hear the Good News.

He continues with a fascinating hypothesis about the relevance of the names of the three women that Mark places at the empty tomb.  You can read the argument and decide for yourself if it’s well founded.

Seeing the gospel story as no more supernatural than any other myth from the past best explains the facts.

Photo credit

# It’s Funny Until Someone Gets Hurt, then it’s Hilarious

I’ve been amazed at the popularity of Creationism/Intelligent Design among Christian pundits.

Old-earth Creationism accepts the consensus within the field of cosmology about the Big Bang and the formation of the solar system 4.6 billion years ago but rejects evolution.  Young-earth Creationism also rejects evolution and argues that the earth is less than 10,000 years old.  This view is predominant among evangelical pastors.

Dr. Karl Giberson recently pointed out an interesting downside of this mindless rejection of science.  He begins by citing a Barna survey that lists six reasons why most evangelical Christians disconnect from the church, at least temporarily, after age 15.  The most interesting reason: “Churches come across as antagonistic to science.”

• 23% say they had “been turned off by the creation-versus-evolution debate”
• 25% say “Christianity is anti-science”
• 29% say “churches are out of step with the scientific world we live in”
• 35% say “Christians are too confident they know all the answers”

As an example of this rejection of science, Giberson points to the technique recommended to schoolchildren by Creation Museum founder Ken Ham.  Ham encourages students to ask, “Were you there?” when the biology teacher says that life on earth appeared roughly 4 billion years ago or the physics teacher says that the Big Bang gave us the universe in its present form 13.7 billion years ago.

Ham proudly blogged about nine-year-old Emma B., who wrote to tell Ham how she attacked a curator’s statement that a moon rock was 3.75 billion years old with “Were you there?”

Biologist PZ Myers nicely deflated Ham’s anti-science question with a gentle reply to Emma B.  Myers recommends using instead “How do you know that?” which is a question from which you can actually learn something.

Contrast that with Ham’s “Were you there?” which is designed simply to shut down discussion and to which you already know the answer.

“Were you there?” is a subset of the more general question, “Did you experience this with your own senses?”  To Science, this question lost significance hundreds of years ago.  The days when Isaac Newton used taste as a tool to understand new chemicals are long gone.  Modern science relies heavily on instruments to reliably provide information about nature—from simple ones like compasses, voltmeters, and pH meters to complex ones like the Pioneer spacecraft, Hubble space telescope, and Large Hadron Collider.

Personal observation is often necessary (finding new animal species, for example), but this is no longer a requirement for obtaining credible scientific evidence.

From the standpoint of mainstream Christianity, Ham’s position as a young-earth Creationist and Bible literalist is a bit extreme, but higher profile figures like William Lane Craig also give themselves the option to pick and choose their science.  Craig uses science a lot—at least, when it suits his purposes.  The Big Bang suggests a beginning for the universe, so he takes that.  Evolution suggests that life on earth didn’t need God, so he rejects that bit.

He imagines that he’s Hanes Inspector Number 12: “It’s not science until I say it’s science.”  It may be fun to pretend that, but what could possibly make you think that’s justifiable?

That reminds me of a joke:

Scientists figure out how to duplicate abiogenesis (the process by which molecules became something that could evolve).  They are so excited that they email God to say they want to show him.  So God clears some time on his calendar and has them in.

“Sounds like you’ve been busy,” God said.  “Show me what you’ve got.”

“Okay—first you take some dirt,” said one of the scientists.

“Hold on,” God said.  “Get your own dirt.”

And to William Lane Craig’s pontificating about science, I say, “Hold on—get your own science.”

You either play by the rules of science and accept the scientific consensus whether it’s compatible with your preconceptions or not, or you sit at the children’s table.  If you want to hang out with the adults, you can’t invent reasons to rationalize why this science is valid and that is not.

Evangelicals may want to rethink this picking and choosing of science.  Giberson ends his article:

The dismissive and even hostile approach to science taken by evangelical leaders like Ken Ham accounts for the Barna finding above.  In the name of protecting Christianity from a secularism perceived as corrosive to the faith, the creationists are unwittingly driving the best and brightest evangelicals out of the church….  What remains after their exodus is an even more intellectually impoverished parallel culture, with even fewer resources to think about complex issues.

Perhaps I should be more welcoming to Christian anti-science in the future.

Photo credit: commandoscorch

Related posts:

• Karl Giberson, “Creationists Drive Young People Out of the Church,” Huffington Post, 11/19/11.
• “Six Reasons Young Christians Leave Church,” Barna, 9/28/11.
• PZ Myers, “Dear Emma B,” Pharyngula blog, 10/3/11.
• Ted Olsen, “Go Figure,” Christianity Today, 11/14/11.

# What Does the Bible Say About Abortion? Not Much.

The Old Testament patriarchs would scratch their heads at the problem conservative Christians have invented and seized upon.  “That’s not what ‘Thou shalt not murder’ means!” they’d say.  “It means that you shouldn’t take a stick and beat someone over the head until he’s dead!  We kill people around here at the drop of a hat—both our own people when they transgress the Law and people of other tribes when we get into border squabbles.  And God has no hesitation in killing people.  To simply make someone not pregnant is vastly different.  People try lots of folk remedies to bring about that very thing, and our only complaint is that they’re not effective.”

All this hand-wringing about the safety of a single cell, less than one trillionth the size of an infant, would baffle them.  God is happy to slaughter (or order slaughtered) lots ’n lots of humans—men, women, and children.

If the Big Man doesn’t care, why should we?  That’s a rhetorical question—of course we should care.  It’s just that we shouldn’t imagine an argument against abortion based on what the Bible says.

About Babylon, it says, “Happy is the one who seizes your infants and dashes them against the rocks” (Ps. 137:9).  And: “Their children also shall be dashed to pieces before their eyes; their houses shall be spoiled, and their wives ravished” (Is. 13:15–16).  Whether God uses genocide against the other guys, poisonous snakes against his own people, or an old-fashioned global flood against everyone, God has a broad palette of options when it comes to death, and he makes no special provision for children, infants, or fetuses.

The Bible even describes a potion to deliberately induce a miscarriage, used by the priest when a woman is suspected of adultery.

God himself has a hand in abortions.  Roughly half of all pregnancies end in spontaneous abortion, a far greater rate than that of clinical abortions.  If God exists, he’s the biggest abortionist of all.

Why imagine that the Bible is against abortion?  Maybe it’s that whole “thou shalt not murder” thing.

But you do know that “thou shalt not murder” isn’t in the Ten Commandments, right?  Let’s review the story.  Moses comes down from Mt. Sinai with the Ten Commandments (Exodus 20) and then smashes them when he sees the golden calf.  He goes back up for another set (Ex. 34), but God must’ve been stoned when he dictated them the second time because it’s quite a different set of rules.  But these rules aren’t just an addendum of some sort; these are the replacement Ten Commandments.  Exodus 34:28 makes this clear: “[Moses] wrote on the tablets the words of the covenant—the Ten Commandments.”  In other words, if you’d been able to peek inside the Ark of the Covenant to see this Ten Commandments 2.0, nowhere would it have said, “Thou shalt not murder.”

But let’s ignore that and assume that the scripture say not to murder.  What is “murder”?  Is capital punishment murder?  It’s illegal in Europe, and many people think it’s murder in the U.S., and yet it’s legal in 34 U.S. states.  What about killing in wartime?  Or killing in self-defense?  Or killing accidentally?  Or killing animals?  Or euthanasia?  Murder is undefined, so “Thou shalt not murder” is meaningless.

You’d think that this vaguely supported legal opinion that God is against abortion would give Christians pause, but I guess the hearts of pro-life Christian soldiers are resolute.  They’re quick to argue that God’s actions are beyond our understanding when it suits them—when confronted with the Problem of Evil or the justice of hell, for example—but at other times they acknowledge no vagueness and know for certain what God wants.  In particular, they know that God is against abortion!

Why is abortion that big a deal from the Christian standpoint when abortions send souls to heaven without the risk of doing the wrong thing in adulthood?  That murdered babies go straight to heaven was one way William Lane Craig tried to wriggle out of the moral consequences of God ordering the Canaanite genocide (“Christianity Can Rot Your Brain”).

Using Craig’s logic, abortion clinics may save more souls than churches!

Photo credit: Wikimedia

Related posts:

# What’s Wrong with the Pro-Life Position?

One commenter to this blog made the excellent point that the label “pro-life” for the anti-abortion movement is a bit odd.  In this contentious debate, I wanted to label those in each group as they prefer, but who’s not pro-life?

In the Christian view, life on earth is “the cramped and narrow foyer leading to the great hall of God’s eternity” (William Lane Craig).  What a dismal view of life—something simply to be endured as we wait for the real Life to begin.  By contrast, the atheist, certain of only the one life we all know exists, is the one who lives life to the fullest.  It can be argued that the atheist is the one who’s truly pro-life.

But let’s leave the conventional labels alone and consider the pro-life position.  If there were no downsides of carrying a fetus to term, if carrying the fetus to term were nothing more than a minor inconvenience for the mother, the abortion question wouldn’t be an interesting issue.  But of course there are downsides—big ones.  To bring a child into the world, poorly cared for in the womb, unwanted and unloved by its mother, abandoned by its father, neglected or abused, or growing up in squalor or in an abysmal home—for me, that potential harm eclipses the harm of denying a cell the chance to grow into a person.  Demanding that the state step in and declare that it knows the consequences better than the mother seems an odd position to take for typically conservative Christians.

The pro-life advocate has a quick answer: carry the child to term and give it up for adoption.  But this does nothing to address the problem of the woman unable to or uninterested in caring for herself and the baby properly during the pregnancy.  Or of the baby with identified birth defects.  Unhealthy babies are far more likely to live out their childhood in foster care.

“Just put it up for adoption” is hopeless naïve when only two percent of all births to unmarried women ended in an adoption.  For teen mothers, the rate is even less.  Let’s not pretend that if the mother’s life and home situation aren’t conducive to raising a baby until adulthood that she’ll always put the baby up for adoption.

Even if a teen mother chose to have her baby adopted, the consequences of the pregnancy are dramatic.  She’ll miss school, she’ll be ostracized, and she’ll go through an emotional meat grinder when it comes time to give up her baby.  And since the statistics say she won’t, that she will almost surely keep the baby, she’ll have no chance to get back on track for the life she had planned.

I have a mental image of an anti-abortion activist looking with satisfaction on the girl he just talked out of having an abortion, with no understanding of the shackles he may have placed on her life or the hellish environment to which he has may have consigned that child-to-be.  Infuriating.

The alternative to abortion rights is compulsory pregnancy.  My claims are simple: that (1) some lives are truly abysmal and (2) creating such a life (for the mother or the child) is a bad thing.  I doubt that my argument has convinced any pro-lifers to budge in their position, but I do demand that they acknowledge the terrible burden that making abortion illegal would place on a million women each year.

Photo credit: Wikimedia

Related posts: