Parallels to Christians’ Selective Use of Science

Advanced Test Reactor at Idaho National Laboratory (the blue glow is Cherenkov radiation)The film Pandora’s Promise (2013, 86 minutes, $4) explores nuclear power as it interviews prominent environmentalists who switched from being against it to being in favor. I’d like to highlight some of the features of the transition these environmentalists went through. There are surprising parallels with the transition people make when leaving Christianity, and there are parallels between a dogmatic anti-nuclear attitude and a dogmatic religious attitude.

The charges against nuclear power

Dr. Hellen Caldicott (a medical doctor) is used as the representative of anti-nuclear environmentalism. She has been called “the world’s foremost anti-nuclear campaigner.” She has received many prizes, 21 honorary doctorates, and a nomination for the Nobel Peace Prize by Linus Pauling and has been called by the Smithsonian Institution “one of the most influential women of the 20th Century.”

Caldicott uses nuclear accidents to make her case and claims that 985,000 people died as a result of the 1986 nuclear accident at Chernobyl. She says that the 2011 Fukushima power plant accident will be even worse. Seven million will die in the next two decades, and tens of millions more will suffer from “debilitating radiation-induced chronic illnesses.”

And the rebuttals

The World Health Organization disagrees. About Fukushima, it concluded in 2013, “The increases in the incidence of human disease attributable to the additional radiation exposure from the Fukushima Daiichi NPP accident are likely to remain below detectable levels.” No deaths due to radiation have been attributed to the accident.

Caldicott’s source for the nearly one million deaths due to Chernobyl has been widely discredited. A consortium of United Nations organizations and others said about the Chernobyl disaster:

According to [the United Nations Scientific Committee on the Effects of Atomic Radiation] (2000), [Acute Radiation Syndrome] was diagnosed in 134 emergency workers. … Among these workers, 28 persons died in 1986 due to ARS. … Nineteen more have died in 1987–2004 of various causes; however their deaths are not necessarily—and in some cases are certainly not—directly attributable to radiation exposure.

There were no radiation deaths in the general population, though there have been close to 7000 cases of thyroid cancer among children. These would have been “almost entirely” prevented had the Soviet Union followed simple measures afterwards.

The report estimates an increase in cancer mortality due to radiation exposure of “a few per cent” in the 100,000 fatal cancers that would be expected in this population.

In other words, Caldicott is about as wrong as it is possible to be. This is not to dismiss the problem—the Chernobyl and Fukushima nuclear accidents were indeed disasters—but it doesn’t help to see them incorrectly. The Fukushima earthquake and tsunami caused 16,000 deaths, while the power plant accident caused none.

Not seeing the problem correctly causes its own problems. The World Health Organization concluded twenty years after Chernobyl that “its psychological impacts did more health damage than radiation exposure did,” and childhood obesity in the Fukushima area is now the worst in Japan because children are not allowed to play outside, in most cases without any valid reason.

Environmentalists—aren’t they the ones who should be following the science?

One critic compared environmentalists with climate change deniers.

Failing to provide sources, refuting data with anecdote, cherry-picking studies, scorning the scientific consensus, invoking a cover-up to explain it: all this is horribly familiar. These are the habits of climate change deniers, against which the green movement has struggled valiantly, calling science to its aid. It is distressing to discover that when the facts don’t suit them, members of this movement resort to the follies they have denounced.

I find this topic revealing because anti-nuclear attitudes are typically held by liberals. Instead of using science and technology to find solutions to the problems of nuclear power, some liberals simply want it to go away. But these problems have solutions. For example, the Integral Fast Reactor was an experimental fourth-generation reactor program begun in 1984. It was cancelled ten years later by Democratic pressure, after it had proven that it was failsafe (it survived a loss of electrical power and loss of all coolant) and shown that it could reduce the waste leaving the facility to less than one percent that of conventional reactors.

The mothballing of the reactor cost more than letting the project conclude. Democrats can be as mindlessly ideological as Republicans.

While the U.S. civilian nuclear power industry has caused no deaths, the U.S. health burden from fossil fuel power generation is 30,000 to 50,000 premature deaths per year. Worldwide, the total is perhaps millions per year.

Breaking free

Some of the interviewees spoke of their change of mind. Mark Lynas said, “I was under no doubt that my whole career and my whole reputation as an environmental activist, communicator was at risk if I talked publicly about having changed my mind about nuclear power.”

Richard Rhodes said, “I came to realize [journalists] basically avoided looking at the whole picture. They only looked at the questions that seemed to prove to them that nuclear power was dangerous, as I had, too.”

I was most shocked at how little some of these environmentalists knew about nuclear power. They had their standard line—nuclear power of any type was bad—and they stuck with it. One career environmentalist admitted that he hadn’t known about natural background radiation from the ground, from space, and even from bananas. Natural potassium, of which bananas are a good source, is 0.012% potassium-40 (a radioactive isotope), and humans are more radioactive because of potassium than because of carbon-14.

Comparison with Christianity

Dr. Hellen Caldicott, the strident anti-nuclear activist, has a lot in common with Christian leaders. (Obviously, her opinion of religion isn’t the issue. I’m simply paralleling her actions with those of Christian leaders.)

  • Dogmatic. Caldicott is a charismatic speaker, and she has a ready audience eager to hear her message. She’s “the world’s foremost anti-nuclear campaigner” for a reason. She says that nuclear power is wicked just like a televangelist might say that same-sex marriage in America is wicked. She says that nuclear power of any type is bad, just a preacher might say abortion of any type is bad.
  • Confident and unchanging. Caldicott is well aware of this controversy and the fact that her figures are orders of magnitude greater than the most widely accepted data. Her position is at least grossly out of touch with reality and could even be called hysterical. But she uses this notoriety to her advantage, and I imagine her façade is as confident as ever.
  • Reputation. This is her livelihood and her identity, and she’s not likely to change. Like Harold Camping or John Hagee in the Christian domain, she can’t admit a big mistake. Some career environmentalists do change, though, as the film documents, and the soul-searching crisis that individual environmentalists endure parallels that of ex-Christians like Dan Barker, Bart Ehrman, or Matt Dillahunty. Leaving one’s identity in either domain means reinventing or even re-finding oneself, and former allies may ridicule or shun.
  • Embrace of science. Caldicott is like William Lane Craig and other apologists in that neither feels bound by science. They use science as it suits them. Caldicott is outraged that climate change deniers dismiss environmental dangers by ignoring or selecting their science, but then she does it herself. In the same way, William Lane Craig quotes cosmologists to defend the Big Bang (because he likes a beginning to the universe), but he ignores quantum physics when it says that events needn’t have causes (he’s desperate to find a cause for the universe).

I’m starting to worry that reason is an acquired taste.
— Sam Harris

Photo credit: Idaho National Laboratory

Reject the Scientific Consensus? How Do You Justify THAT?

What reason goes in that blank? What could possibly go in that blank? If you’re a layperson in a particular scientific field, how could you reject that field’s scientific consensus, where it exists, as the best provisional explanation of that aspect of nature?

People do reject the consensus—Creationists, for example—but I can’t begin to understand the thought process that justifies this conclusion. People reject radioisotope dating who aren’t geologists, reject the Big Bang who aren’t cosmologists, reject anthropogenic climate change who aren’t climate scientists, and reject evolution who aren’t biologists.

You might think that evolution is a remarkable claim. Perhaps even that the evidence isn’t there (though the fact that you’ve only dipped your toe into the water would scream out as the obvious explanation). What I can’t imagine is concluding, based in that “research,” that the theory of evolution is flawed. On what grounds could someone reject the consensus of the people who actually understand the evidencethe people who actually have the doctorate degrees and who actually do the work on a daily basis?

I often hear people handwaving justifications. I attended a debate in which the Creationist speaker demanded that we follow the evidence. That sounds like odd advice from someone who doesn’t have the consensus on his side until you translate that into what he meant: give yourself license to weigh scientific conclusions yourself and discard the ones you dislike.

When science and scripture conflict, Ken Ham of Answers in Genesis tells Christians that they shouldn’t go with the discipline with the remarkable track record for telling us about reality. Instead, discard that and go with faith.

We should be very wary of any idea about life that has a consensus among non-Christians. Sadly, many Christians listen to the ideas of secular scientists and try to add these to the Bible. This shows that they have the same problem as the non-Christian—they don’t want to submit to God’s Word.

These same people wouldn’t dream of telling a surgeon how best to approach a particular operation or a pilot how to fly a plane. But they seem quite happy to read a Creationist book and reject any consensus in biology, geology, or cosmology that steps on their theological toes. Can these science deniers possibly be so self-important as to pretend to be the Judge of All Science®, anointing the correct science and rooting out the false?

Creationists’ arguments often defeat themselves. Ask them how they make their conclusion and few will say that they’ve evaluated the scientific papers themselves. Rather, they will point to Creationist or ID “scientists” from the Discovery Institute, the Institute for Creation Research (ICR), Answers in Genesis (AIG), or similar organizations. Apparently, they think that the conclusions of competent scientists is important. Fine, then be objective and look at the consensus of experts on that subject.

You’d think that organizations that constrained their “scholars” with statements of faith, as both the ICR and AIG do, would be disqualified from scientific discourse for being biased. Similarly, it’s baffling that many Christian scholars like William Lane Craig feel comfortable pontificating on science

  • when they have no scientific credentials to do so,
  • when they celebrate the consensus and use it to support their argument when it suits them (Big Bang) and then reject the consensus when it doesn’t (evolution), and
  • because they, too, are bound by statements of faith.

When they say that evolution is nonsense, I always ask: is that the facts or your statement of faith talking? (More in my post “Can Christian Scholars be Objective?”)

Science deniers often raise three objections.

  • The scientific consensus is wrong sometimes. Yes, sometimes it is. The consensus is not immutable truth but a provisional approximation to the truth. This doesn’t change the fact that the consensus is the best guess at the truth that we laypeople have.
  • Demanding that laypeople accept the scientific consensus in all cases is the Bandwagon Fallacy. No, it isn’t. The Bandwagon Fallacy (argumentum ad populum) says that “a million Chevy owners can’t be wrong”—if the majority believes it, it must be true. Note the differences: I argue that the consensus is our best bet, not that it is invariably true. Also, the scientific community is a group of experts of which laypeople are not members.
  • Okay, then it’s the Argument from Authority fallacy. Wrong again. The scientific community is quite different from a single authority. (I’ve written more about this fallacy fallacy and the role of consensus here.)

Sorry, folks—science is not a democracy. When it comes to the scientific consensus, no one cares what we laypeople think. Creationists have done an impressive job in deluding Americans so that, 150 years after Darwin’s Origin of Species, 40% still think that “God created human beings pretty much in their present form at one time within the last 10,000 years or so.”

This rejection of science can be merely a personal tragedy as when Steve Jobs refused conventional medicine for his treatable cancer in favor of unproven techniques. It becomes a bigger problem when policy makers adopt it. Ben Carson rejects evolution and the Big Bang but assures us that his grasp of the big issues makes him presidential timber.

Rep. Paul Broun (R-Georgia) called evolution, Big Bang theory, and other disagreeable aspects of science “lies straight from the pit of hell.” Rep. Ralph Hall (R-Texas) said about climate change, “I don’t think we can control what God controls.” Todd Akin (R-Missouri) has famously odd views on conception and evolution—he declared that “legitimate rape” rarely leads to pregnancy. Each has training wheels on their understanding of science, but all three were members of the House Science and Technology Committee.

Science deniers get high marks in Machiavellianism 101, but the American public’s childish relationship with reality puts us in a poor position to handle the challenges of the 21st century.

Related post: You Don’t Like the Scientific Consensus? Ignore it Away.

Evolution is one of the most robust and widely accepted
principles of modern science.

— American Association for the Advancement of Science

(This is an update of a post that originally appeared 12/19/12.)

Scientist Thinking vs. Lawyer Thinking

Consider the difference between a lawyer in the courtroom and a scientist in the university. The scientist is encouraged to share research with colleagues and ask for advice. Unless the scientist is working on confidential research, openness and camaraderie are essential parts of scientific research. Scientists must follow the evidence, even if that evidence leads them away from a cherished hypothesis.

Contrast that with how the lawyer works. The courtroom is an intentionally adversarial environment. There is no collegial give and take between the opposing sides of the issue, no meeting of the minds, no compromise. If I’m paying someone to represent me in court, that lawyer must present just one side of the case—mine. I want that lawyer to be biased and argue very effectively about just one side of the issue. Evidence is valuable only to the extent that it supports the lawyer’s side of the argument.

The opposition does the same thing from the opposite viewpoint, and a judge or jury decides the merits of the two cases.

The concept of double jeopardy (someone can’t be tried twice for the same crime) applies in the domain of lawyers, but there’s no equivalent within science. A hypothesis or theory is always provisional. Any claim can be revisited.

The thinking of the scientist contrasted with that of the lawyer is brainstorming vs. winner take all. It’s forum vs. battleground. It’s a search for the truth vs. a presupposition of one’s rightness. Scientists should be open to changing their mind and backing another hypothesis, while courtroom lawyers are obliged to stick with their position regardless of the evidence.

There’s nothing wrong with lawyer thinking in its place. But let’s not confuse it with scientist thinking. The mistake is using one when we imagine we’re using the other.

I once attended a debate in which an atheist and a Lutheran pastor debated “Does God exist?” At one point, the pastor, an old-earth Creationist, turned to the topic of evolution and demanded that we follow the evidence. This sounded bizarre given his rejection of the scientific consensus about evolution, but this was his way of giving himself license to make conclusions himself. He has no doctorate in biology, of course, but he pretends that as an armchair biologist he’s entitled to weigh the evidence and reject the results of 150 years of scientific research and the consensus of tens of thousands of people who actually are practicing biologists.

This is why I have little patience for philosopher William Lane Craig prancing around in an imaginary lab coat and playing make-believe as a scientist (more on that here). Or take the Reasons to Believe ministry, whose mission is “to show that science and faith are, and always will be, allies, not enemies.” They cherry-pick the science to support their preconceptions. This is lawyer thinking, not scientist thinking.

A popular Christian apologist is J. Warner Wallace. He’s a cold-case homicide detective, and he does his best to argue that the courtroom is a great analogy to how we should evaluate the evidence for the claims of Christianity. It isn’t. The courtroom is precisely where we shouldn’t go for an analogy.

Note that the lawyer’s approach isn’t necessarily unjustifiable outside the courtroom. One might wonder, “If we applied some smart minds to the God question, what supporting arguments could they find?” That’s a reasonable question as long as we don’t pretend that it’s a quest for the best explanation.

Lawyer thinking feels natural. Our imperfect brains are saddled with biases—one example is confirmation bias (noticing only the evidence that confirms what we already believe). These biases make lawyer thinking a natural rut for us to fall into, but it begins with the conclusion rather than following the evidence where it leads.

Lawyer thinking is like religious thinking, and they’re both like advertising. You focus on the good points in your case and ignore any bad ones. Admit to yourself when you adopt this view, and don’t pretend to be thinking like a scientist. Too often, these two positions are conflated or confused.

We all have biases. None of us enjoys being wrong, and each of us probably needs to rein in our lawyer thinking. That goes for me as well.

Too often I’ll read something with the unstated theme, “Here’s how I interpret the facts to support my presupposition.” More useful would be, “Here’s where the facts lead.”

In science it often happens that scientists say, 
“You know that’s a really good argument; my position is mistaken,”
and then they actually change their minds 
and you never hear that old view from them again. 
They really do it. 
It doesn’t happen as often as it should, 
because scientists are human and change is sometimes painful. 
But it happens every day. 
I cannot recall the last time something like that 
happened in politics or religion. 
— Carl Sagan

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

Photo credit: Wikimedia

Are We Just Molecules in Motion?

I’m on vacation in Europe, so I’ll be posting from the archives for a few weeks.

How can humans be nothing more than molecules on motion? From only this, how could we get consciousness or morality or other high-level human traits?

Christian apologists object to what they see as the consequences of the naturalistic view—that humans are nothing more than machines. Can this mechanistic outlook explain the facts of consciousness and morality?

In the first place, the Heisenberg Uncertainty Principle makes clear that we’re not living in a clockwork universe where, if we could only understand the state of things precisely enough, we could in theory predict the future. Quarks are not like billiard balls. Heisenberg states that there is unavoidable randomness at the quantum level, and things are not deterministic. Naturalists agree that a deterministic, molecules-in-motion worldview doesn’t work.

This caricature refuses to go beyond the simple laws governing our molecules, but reality is a bit more complicated. Consciousness, morality, and other complex human traits don’t follow directly from fundamental quantum laws, but they are examples of emergent phenomena.

Emergent phenomena in water

Consider water molecules. At the quantum level, the forces are pretty straightforward—electrostatic forces hold things together and heat drives them apart; the strong force keeps the nucleus intact and the weak force governs when it breaks apart; and so on.

The diagram above shows a water molecule, with the bonding angle and distance between nuclei shown. Imagine conveying all this information about water to quantum physicists in another galaxy who don’t have water. They would know everything about water but only at the nano scale.

From this, could they deduce the existence of whirlpools, turbulence, erosion, or waves on the shore? Would they anticipate rainbows, clouds, snowflakes, fog, and frost? Could they predict that water self-ionizes or expands as it freezes? No single water molecule possesses the properties of wetness, fluidity, pH, salinity, or surface tension, but these and other properties emerge when trillions of trillions of water molecules are put together.

Wetness isn’t a property of a water molecule; it emerges. Individually it’s not there, but it appears collectively.

Other examples

Take alloys as another example. If you add water to alcohol, you get dilute alcohol. But if you add tin to copper, you get bronze. Add carbon to iron to get steel. The resulting alloys can be more useful than the elements that make them. Sometimes 1 + 1 = 3.

Take the human brain. Our brains have roughly 100 billion (that’s 1011) neurons. A single neuron doesn’t think 10–11 times as fast; it doesn’t think at all. Thinking is another emergent phenomenon.

Similarly, consciousness doesn’t reside in any one neuron or in any section of the brain; it emerges from an entire working brain. The same is true with morality. If there’s no need to imagine a mind separate from the brain to explain consciousness in a mouse, why would it be necessary for humans?

Compare a human brain with a lizard brain. A lizard doesn’t have a poorly developed sense of humor (or wit or irony), it has none at all. Humans’ extra brain cells don’t make us a super lizard, they makes us something completely new and different, something emergent.

Denigrate the brain with “we’re just molecules in motion” if you want, but consciousness is like wetness—it emerges out of the whole. There is no need to imagine an objective morality grounded outside the human brain, or a mind to provide consciousness, or a soul to provide personality. All evidence points to their simply being properties of the brain.

Clay is molded to make a pot, 
but it is in the space where there is nothing 
that the usefulness of the pot lies.
Cut out doors and windows to make a room, 
and it is in the spaces where there is nothing 
that the usefulness of the room lies.
— Chinese classic Tao Te Ching

(This is an update of a post that originally appeared 11/12/12.)

Photo credit: Wikimedia

Infinity—Nothing to Trifle With (2 of 2)

Infinity(See Part 1 for the beginning of this discussion.)

Puzzling aspects of infinity

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 I start with 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.

Craig has argued there is no actual infinite (to avoid an infinitely old universe), but he needs one for himself. His god is supposedly infinitely good, infinitely powerful, and so on. And since the universe has a finite age, for God to be either infinitely old or timeless, he must’ve crossed an infinite number of moments of time before he created the universe. Perhaps Craig is hoist by his own petard.

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.

More from William Lane Craig

Let’s conclude by revisiting 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.

Or, imagine catching the Counting Man at any point in his project. He’d tell you that he’d already counted an infinite number of numbers. So where’s the problem?

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, this may be just the ticket.

When the answer is known, science knows it.
When science doesn’t know it, neither does religion.
— Paraphrase of AronRa

(This is an update of a post that originally appeared 7/13/12.)

Infinity—Nothing to Trifle With

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.)