Infinity—Nothing to Trifle With

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I’m preparing a presentation for the Atheist Alliance of America conference in Seattle, August 7–10. I’ll be posting less often for a week. Thanks for your patience.

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.

William Lane Craig scatters tacks on the road

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.

More on infinity

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 such a container 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. It will be concluded in part 2.)

To say that that having a degree in theology
makes you qualified to say that God is real,
is like claiming that having memorized all the Harry Potter books
makes you qualified to say that unicorns exist.
— Unknown

Photo credit: Wikipedia

(This is an update of a post that originally appeared 7/12/12.)

Why the Gospel of Mark Is Likely NOT an Eyewitness Account

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How do we know that Mark wrote the gospel of Mark? How do we know that Mark recorded the observations of an eyewitness?

The short answer is because Papias (< 70 – c. 155) said so. Papias was a bishop and an avid documenter of oral history from the early church. His book Interpretations was written after 120 CE.

Jesus died in 30, Mark was written in 70, and Papias documents Mark as the author in 120 (dates are estimates). That’s at least 50 years bridged only by “because Papias said so.”

Looking through the wrong end of the telescope

But how do we know what Papias said? We don’t have the original of Papias, nor do we have a copy. Instead, we have Church History by Eusebius, which quotes Papias and was written in 320.

And how do we know what Eusebius said? The oldest copies of his book are from the tenth century, though there is a Syriac translationfrom 462.

Count the successive people in the claim “Mark wrote the gospel of Mark, which documents an eyewitness account”: (1) Peter was an eyewitness and (2) Mark was his journalist, and (3) someone told this to (4) Papias, who wrote his book, which was preserved by (5) copyist(s), and (6) Eusebius transcribed parts of that, and (7) more copyist(s) translated Eusebius to give us our oldest manuscript copy. And the oldest piece of evidence that we can put our hands on was written four centuries after Mark was written.

That’s an exceedingly tenuous chain.

The sequence of people could have been longer still; we simply don’t know. Papias was the bishop of Hierapolis, in western Asia Minor. Mark might have been written in Syria, and no one knows how long the chain of hearsay was from that author to Papias. No one knows how many copyists separated Papias from Eusebius or Eusebius from our oldest copies.

Trash talk

It gets worse. Eusebius didn’t think much of Papias as a historian and said that he “seems to have been a man of very small intelligence, to judge from his books” (Church History, book III, chapter 39, paragraph 13). Evaluate Papias for yourself: he said that Judas lived on after a failed attempt at hanging and had a head swollen so large that he couldn’t pass down a street wide enough for a hay wagon. Who knows if this version of the demise of Judas is more reliable than that in Matthew, but it’s special pleading to dismiss Papias when he’s embarrassing but hold on to his explanation of gospel authorship.

Even Eusebius’s Church History is considered unreliable by modern scholars.

The story is similar for the claimed authorship of Matthew. A twist to this story is that Papias said that Matthew wrote his gospel in Hebrew (or perhaps Aramaic), which makes no sense since Matthew used Mark, Q, and the Septuagint Bible, all Greek sources.1

The other gospels besides Mark

What about the other gospels? That evidence comes from other documents with simpler pedigree but later dates.

  • Irenaeus documented the traditional gospel authorship in his Against Heresies (c. 180). Our oldest copy is a Latin translation from the tenth century.
  • Tertullian also lists the four traditional authors in his Against Marcion (c.208), but he doesn’t think much of Luke: “[Heretic] Marcion seems to have singled out Luke for his mutilating process.” Our oldest copy of Tertullian’s book is from the eleventh century.
  • The oldest manuscript labeled “gospel according to Luke” dates from c. 200.
  • The Muratorian fragment, a Latin manuscript from the seventh century, may be a translation of a Greek original from the late second century (or maybe from the fourth). It lists many books of the New Testament, including the gospels of Luke and John.

Evidence arguing that the gospels document eyewitness accounts is paltry. Perhaps only faith will get you there.

If we submit everything to reason,
our religion will have no mysterious and supernatural element.
If we offend the principles of reason,
our religion will be absurd and ridiculous.
— Blaise Pascal

1Randel Helms, Who Wrote the Gospels? (Millennium Press, 1997), 41.

 (This is an update of a post that originally appeared 7/20/12.)

Photo credit: Wikimedia

The Inevitability of Gay Marriage

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gay marriage rightsA century ago, America was embroiled in social change. Some of the issues in the headlines during this period were women’s suffrage, the treatment of immigrants, prison and asylum reform, temperance and prohibition, racial inequality, child labor and compulsory elementary school education, women’s education and protection of women from workplace exploitation, equal pay for equal work, communism and utopian societies, unions and the labor movement, and pure food laws.

The social turmoil of the past makes today’s focus on gay marriage and abortion look almost inconsequential by comparison.

Christianity on the right side of social issues

What’s especially interesting is Christianity’s role in some of these movements. Christians will point with justified pride to schools and hospitals build by churches or religious orders. The Social Gospel movement of the early 20th century pushed for corrections of many social ills—poverty and wealth inequality, alcoholism, poor schools, and more. Christians point to Rev. Martin Luther King’s work on civil rights and William Wilberforce’s Christianity-inspired work on ending slavery. (This doesn’t sound much like the church today, commandeered as much of it is by conservative politics, but that’s another story.)

… but maybe not on same-sex marriage

Same-sex marriage seems inevitable, just another step in the march of civil rights. Two years ago, before the tsunami of legal wins for the gay rights side, Jennifer Roback Morse (president and founder of the Ruth Institute for promotion of heterosexual marriage and rejection of same-sex marriage) was asked if she feared being embarrassed by the seeming inevitability of same-sex marriage. She replied:

On the contrary, [same-sex marriage proponents] are the ones who are going to be embarrassed. They are the ones who are going to be looking around, looking for the exits, trying to pretend that it had nothing to do with them, that it wasn’t really their fault.

I am not the slightest bit worried about the judgment of history on me. This march-of-history argument bothers me a lot. … What they’re really saying is, “Stop thinking, stop using your judgment, just shut up and follow the crowd because the crowd is moving towards Nirvana and you need to just follow along.”

Let’s first acknowledge that Morse could be striving to do the right thing simply because it’s right, without concern for popularity or the social consequences. I would never argue that someone ought to abandon a principle because it has become a minority opinion or that it is ridiculed. If Dr. Morse sticks to her position solely because she thinks it’s right, and she’s not doing it because of (say) some political requirement or because her job depends on it, that’s great.

Nevertheless, the infamous 1963 statement from George Wallace comes to mind:

I say segregation now, segregation tomorrow, segregation forever.

That line came back to haunt him. To his credit, he apologized and rejected his former segregationist policies, but history will always see him as having chosen the wrong side of an important issue.

Uh … no, we were on the correct side of that issue all along!

Christianity has similarly scrambled to reposition itself after earlier errors. Christians often claim that modern science is built on a Christian foundation, ignoring the church’s rejection of science that didn’t fit its medieval beliefs (think Galileo and Creationism). They take credit for society’s rejection of slavery, forgetting Southern preachers and their gold mine of Bible verses for ammunition. They reposition civil rights as an issue driven by Christians, ignoring the Ku Klux Klan and its burning cross symbol, biblical justification for laws against mixed-race marriage, and slavery support as the issue that created the Southern Baptist Convention.

Arthur Schopenhauer observed, “All truth passes through three stages: first, it is ridiculed; second, it is violently opposed; third, it is accepted as self-evident.” And then the opposition claims that it was their idea all along!

The same-sex marriage issue in the United States is halfway between Schopenhauer’s steps 2 and 3. Check back in two decades, and you’ll see Christians positioning the gay rights issue as one actually led by the church. They’ll mine history for liberal churches that took the lead (and flak) in ordaining openly gay clerics and speaking out in favor of gay rights.

If someone truly rejects same-sex marriage because their unbiased analysis shows it to be worse for society, great. But it is increasingly becoming clear how history and the public will judge that position.

Truth never damages a cause that is just. 
— Mohandas Gandhi

(This is an update of a post that originally appeared 7/4/12.)

Photo credit: Spec-ta-cles

God is Nonexistent

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christianity atheismDoes God exist? I don’t think so. But can we prove that?

Proving that God doesn’t exist—or, more generally, that no supernatural beings exist—seems impossible. An omniscient being wanting to remain hidden would succeed. That’s a game of hide and seek we could never win.

Looking for parallels

To see what we can say about God, let’s look for parallels in how we handle other beings not acknowledged by science—Bigfoot, the Loch Ness monster, space aliens, leprechauns, fairies, or Merlin the shape-shifting wizard. Any evidence in favor of these beings is sketchy, far too little to conclude that they exist. Do we reserve judgment? Do we say that the absence of evidence is no evidence of absence? Of course not. There’s plenty of evidence (or lack of evidence) to make a strong provisional case. As a result, we typically say that these things don’t exist.

While we’re at it, note the error in the adage “absence of evidence is no evidence of absence.” Of course it’s evidence! Absence of evidence is no proof of absence, but it can certainly be strong evidence. If you’ve spent five minutes poking through that drawer looking for your keys and still can’t find them, that’s pretty strong evidence of their absence.

Examples in the animal kingdom

Note also the difference in the claim that Bigfoot doesn’t exist versus the claim that God doesn’t exist. Science has been surprised by new animals in the past. The gorilla, coelacanth, okapi, and giant squid were all surprises, and Bigfoot could be another. After all, Bigfoot is just another animal and we know of lots of animals. But the very category of the Christian claim is a problem because science recognizes zero supernatural beings.

As definitively as science says that Bigfoot doesn’t exist, how much more definitively can science say that God doesn’t exist when the category itself is hypothetical? Perhaps more conclusively, what about the claim that a god exists who desperately wants to be known to his creation, as is the case for the Christian god?

Let’s be careful to remember the limitations on the claim, “God doesn’t exist.” Science is always provisional. Any claim could be wrong—from matter being made of atoms to disease being caused by germs. Austin Cline observed that a scientific statement “X doesn’t exist” is shorthand for the more precise statement,

This alleged entity has no place in any scientific equations, plays no role in any scientific explanations, cannot be used to predict any events, does not describe any thing or force that has yet been detected, and there are no models of the universe in which its presence is either required, productive, or useful.

The Christian may well respond to science’s caution, “Well, if you’re not certain, I am!” But, of course, confidence isn’t the same as accuracy. That bravado falls flat without dramatic evidence to back it up.

Does God exist or not?

Now, back to the original question, Does God exist? Does this look like a world with a god in it? If God existed, shouldn’t that be obvious? What we see instead is a world in which believers are forced to give excuses for why God isn’t obvious.

Or, let’s imagine the opposite—a world without God. This would be a world where praying for something doesn’t increase its likelihood; where faith is necessary to mask the fact that God’s existence is not apparent; where no loving deity walks beside you in adversity; where natural disasters kill people indiscriminately; where far too many children live short and painful lives because of malnutrition, abuse, injury, or birth defects; and where there is only wishful thinking behind the ideas of heaven and hell.

Look around, because that’s the world you’re living in.

But this isn’t an anarchist’s paradise; it’s a world where people live and love and grow, and where every day ordinary people do heroic and noble things for the benefit of strangers. Where warm spring days and rosy sunsets aren’t made by God but explained by Science, and where earthquakes happen for no good reason and people strive to leave the world a better place than it was when they entered it. God isn’t necessary to explain any of this. Said another way, there is no functional difference between a world with a hidden god and one with no god.

Listen closely to Christian apologists and you’ll see that they admit the problem. The typical apologetic approach is to:

  1. make deist arguments (for example, the existence of morality or design demands a deity to create it), and
  2. argue that this deity is the Christian god rather than the god of some other religion.

Are these deist arguments convincing? If so, the apologist should be a deist, not a Christian. And why is the first step necessary? It’s because the Christian god is functionally nonexistent, and this step admits as much.

The God hypothesis isn’t necessary. God has no measurable impact on the universe, and science needn’t sit on the sidelines. There is enough evidence to render a provisional judgment (and scientific conclusions are always provisional).

Prayers are answered with the same likelihood whether you pray to Zeus, the Christian god, or a jug of milk. Religion is what you invent when you don’t have Science.

Can we say that anything doesn’t exist? With certainty, probably not. But with the confidence that we can say that anything doesn’t exist—leprechauns, fairies, or Merlin the wizard—we can say that God doesn’t.

The universe we observe has precisely the properties we should expect 
if there is, at bottom, 
no design, no purpose, no evil and no good, 
nothing but blind, pitiless indifference.
– Richard Dawkins

Photo credit: Philosophy Monkey

 

The God Debate, 2 of 3 (Fiction)

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From Part 1 of this excerpt from my book Cross Examined:

Jim set the cups on their saucers and swirled the tea in the pot. “It was a revelation—all the convoluted and flimsy rationalization that had been necessary before just vanished. My God hypothesis was a poor explanation of reality, and when I no longer insisted that it was correct and simply followed reality where it led me, things made vastly more sense.”

“Moving from love to nothing is a harsh change,” Paul said. “You must at least agree that the atheist position repels many people.”

“Some do feel that way, but that has nothing to do with whether or not God exists. I have no use for the happy explanation but quite a bit for the correct explanation. You can’t be arguing that it’s disagreeable to imagine that there is no God, so therefore he exists—surely we have higher standards of evidence than that.

“Many Christians admit that the problem of evil is the most difficult problem for Christianity—why God lets bad things happen to good people,” Jim said. “The Greek philosopher Epicurus had an excellent way of putting it. He said that if God is willing to prevent evil but not able, then he is not omnipotent. If he is able but not willing, then he is malevolent. If he is both able and willing, then why is there evil? If he is neither able nor willing, then why call him God?”

“Well, difficulties build character,” Paul said. “My own life started out pretty bad, but it made me who I am.”

“That’s just what an atheist would say—we don’t seek out misfortune, but good can come of it. It’s a simple and workable understanding of the world, and it doesn’t need a supernatural element. But the Christian’s challenge is to make sense of evil permitted by a supernatural being who could stop it in an instant if he wanted to—a tall order.”

“Here’s how Reverend Hargrove explained it to me,” Paul said. “Free will is mandatory in a good world. God could have created us like machines without choice so that we would always do the right thing, but then we wouldn’t be human.” He remembered a metaphor from one of Samuel’s sermons. “What’s the point in walking a maze if there’s a sign pointing the correct direction at every junction?”

Jim dropped the Bible onto the carpet and put his bare feet up on the table where it had been. They were practical feet, ugly with calluses, and they looked like they were rarely confined in shoes. His feet made a sharp contrast with his natty gray trousers. “If God cares so much about free will, I wonder why he allows the free will of the victim to be trampled by the thief or murderer,” Jim said. “And tell me this: does free will exist in heaven?”

“I would think that it must.”

“And does it cause the same problems in heaven that it does here on earth?”

Paul swirled the tea in his cup and watched the dark bits of leaves make patterns in the liquid. “I suppose the spirits in heaven are enlightened. They’d have no desire to do bad things. Otherwise, heaven would have the problems we have here.”

“Then God could give us that enlightenment now.”

“God can’t just give us wisdom. Then we’d have no opportunity to learn. My point about the maze was that a lesson learned is more powerful than a lesson given. Wisdom is more valuable if we earn it.”

“Why is that?” Jim asked. “God could enlighten us with the same lessons as profoundly as if we’d learned them through experience, and no trials would be necessary. If free will is mandatory, why was the enlightenment needed to properly use it reserved for the spirits in heaven? Why would God shortchange us like that? It’s like an automobile without an instruction manual. The God hypothesis is unnecessary and it complicates the explanation. And why does God not show himself? Why not make clear his purpose? Why the mystery, why the test, when he knows the outcome already? Why not just tell us?”

“He did tell us. He told us through Jesus’s ministry.”

“That’s a story, not concrete evidence. What we see, including the legend of Jesus and the emphasis on faith instead of reason, is exactly the kind of thing primitive people would give us. The natural explanation is far more plausible than the existence of a divine maze maker.”

Paul leaned back in his chair. The defensiveness of the past was almost gone. Instead of focusing on his own crumbling argument, Paul reflected on the construction of Jim’s. He tried for another analogy. “I think of life as a school. We learn and then we graduate into heaven.”

“But some fail and go to hell. It’s a poor school that fails such a huge fraction of its students. Isn’t God a skilled enough teacher that everyone could pass?”

Paul fingered the seams on the arm of the chair as another avenue came to mind. “I’ve always found comfort in everyday miracles—not parting the Red Sea, but a child rescued from a fire or a miraculous recovery from illness. Don’t events like that make you stop and think?”

“Good news pleases me as well, but let’s be clear about what causes it. I read in the paper a few weeks ago about a woman whose home was destroyed in April’s earthquake. She lost everything, but charities and the government have given her food, clothes, and a temporary place to live. Do you know what her reaction was? She said, ‘Thank you, Jesus.’ ”

“And you say that it wasn’t Jesus who helped.”

“Exactly. Give thanks to those who provided the help—the people who have opened their homes to the homeless or fed the hungry. It doesn’t happen often enough, but it happens. People from as far away as Idaho and Washington are still sending bread by train to feed people in San Francisco. The prayer ‘give us this day our daily bread’ is being answered, but by people, not God. We don’t need to invent supernatural beings to explain what we see.”

Paul had seen enough callousness to last a lifetime—and yet, he had seen plenty of generosity, too. “Some would say that Jesus caused those people to do good, that He worked through them.”

“Right, and Santa Claus works through parents. ‘God will provide’ really means that people will provide. It’s amazing how far Christians will go to rationalize a positive role for Jesus. The Creator of the universe apparently stands by to let disaster consume a city, and then his apologists want to credit him with the people cleaning up afterwards. No—I’d rather give people the credit they earned.”

“Many would say that disasters are part of God’s plan—a short-term loss for some greater gain in the future.” Paul surprised himself as he shored up the Samuel side of the argument. He didn’t think it the stronger position, but he wanted to hear Jim’s response.

Jim said, “People don’t say, ‘This disaster must be for the greater good’ and sit back to watch dispassionately. They help where they can. We don’t say, ‘Smallpox is supposed to be deadly and to change that would interfere with God’s plan’—we create vaccines. We don’t say, ‘Injuries are supposed to hurt’ or ‘Bones just break sometimes’—we create laudanum and splints. People talk about how there must be a greater good behind God’s plan so they can salvage the claim that God is good, and yet they don’t hesitate before using modern medicine to help the sick and injured or using charity to help people displaced by a natural disaster. They don’t hesitate a moment before interfering in ‘God’s plan.’ ”

Paul sipped from his cup as he considered Jim’s argument. He was beginning to enjoy this tea—harsh but with a sweet aftertaste. “I heard a story about a woman tending her garden.” Paul wasn’t much for telling jokes, but this one took on a new meaning. “The pastor walks by and says, ‘Isn’t it marvelous what God can do in a garden?’ She wipes the sweat from her forehead and says, ‘You should have seen it when He had it all to Himself.’ ”

Jim stood and let out a whoop. “There’s hope for you yet!” He picked up the tea tray. “Let’s continue in the kitchen.”

To be continued.

God is as Believable as Unicorns

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God UnbelievableA chapter in Carl Sagan’s The Demon-Haunted World is titled “The Dragon in My Garage.” In the spirit of Sagan’s story, here is an imagined exchange between you and me about my unicorn.

Me: I have a unicorn in my garage!

You: Wow—let’s see!

Me: You don’t want to just take my word for it?

You: Of course not—I want to see.

(I open the garage door.)

Me: Okay, here you go.

You: Uh … this garage is empty.

Me: No … uh, the unicorn is invisible. They can do that, you know.

You: Okay … can you make him make a sound?

Me: No. He’s silent, too.

You: Can we see food vanish as he eats it?

Me: Of course not—he’s magic. He doesn’t need food.

(You wander through the garage with your hands out in front.)

Me: What are you doing?

You: Trying to feel for it.

Me: Uh … no—he’s really small and he scampers away.

You: Can you hear him running? Like the sound of hooves on concrete?

Me: No—I told you he’s silent.

You: Well, how about spreading flour on the floor so we can see the footprints.

Me: Nope. He can float. And I’m sure he would, because he doesn’t like to be detected.

You: Can we can catch him with a net and weigh him? Can we put a sheet over him so I can see him moving underneath? Could we spray paint and see it on his body?

Me: No—he’s … he’s noncorporeal. Yeah, that’s it. Noncorporeal.

Of course, by now you’ve lost interest in this “unicorn.” Still, you haven’t falsified my claim, and I win!

But no one would accept this conclusion. By slithering away from every possible test, this supernatural claim has no evidence to support it. Any unicorn that has this little impact in the world is the same as no unicorn at all. We can’t prove it’s nonexistent, but it’s functionally nonexistent.

“You haven’t been able to falsify my claim” is true but irrelevant. This is backwards reasoning. The proper conclusion is: There is no evidence to support this claim, so there’s no reason to accept this claim.

Isn’t this how Christians evaluate the miracle claims of other religions? Let’s handle those of Christianity the same way.

Jesus is Santa Claus for adults
(seen on a bumper sticker)

(This is an update of a post that originally appeared 6/17/12.)

Photo credit: Wikimedia