Competing at the Limit

I participate from time to time at a site called Math StackExchange, where users ask and answer questions about mathematics.  Most often, the questions relate to a student's coursework, but there are some deeper questions as well.  It's one of a family of similar StackExchange sites devoted to a wide variety of topics, only some of which are academically inclined.

One question that comes up every now and then is the definition of a limit.  It looks like this:

limx→af(x)=L⇔∀ε>0,∃δ>0,∀x,0<|x−a|<δ⇒|f(x)−L|<ε

And it reads like this:

The limit of f(x) as x approaches a equals L, if and only if for every positive ε, there exists a positive δ such that whenever x is within δ of a (except possibly exactly at a), f(x) is within ε of L.

Understandably, to many math students starting introductory analysis, this looks like so much gobbledygook.  Textbooks typically try to aid understanding by drawing a picture of a function f(x) in the vicinity of some value x = a, showing that as x gets closer to a, f(x) in turn gets closer to its limiting value L (which might not in fact be f(a) itself, if that value even exists).

But what if the sticking point for students isn't always that notion of better and better approximations (central as that is to the definition of a limit)?  What if the sticking point is the interplay between the "for every" (symbolized by the upside-down A: ∀) and the "there exists" (symbolized by the upside-down E: ∃)?  The intent of this definition, first conceived of by the French mathematician Augustin-Louis Cauchy (1789–1857) and formalized by the Bohemian mathematician/philosopher Bernard Bolzano (1781–1848), is to ensure that we can always get as close as we want to the limiting value (without necessarily hitting it), simply by being as close as we need to be to the argument x = a.

We can represent this as a sort of (almost irredeemably nerdy) game between two players, the Verifier and the Falsifier.  The Verifier is trying to prove the limit is right by showing that everything near x = a maps to an f(x) that's close to L, while the Falsifier tries to disprove the limit by challenging the Verifier to get even closer to L.  For instance, if the function f(x) = 2x+3, the Verifier might be trying to demonstrate that the limit of f(x), as x approaches 5, is 13:

Falsifier.  I don't think it's true; I think the limit is not 13.

Verifier.  Well, if that's so, then you must think there's some neighborhood of 13 that I can't force f(x) to lie in.

Falsifier.  Right.  OK, I challenge you to get within 0.1 of 13.

Verifier.  Sure.  If x is within 0.05 of 5, then f(x) will be within 0.1 of 13: f(4.95) = 2×4.95+3 = 12.9, which is within 0.1 of 13, and f(5.05) = 2×5.05+3 = 13.1, which is also within 0.1 of 13.  [There is more to it than that, such as that f(x) is monotonically increasing, but we'll leave these details out for now.]

Falsifier.  All right, but can you get within 0.01 of 13?

Verifier.  Yes.  All I have to do is force x to be within 0.005 of 5: f(4.995) = 12.99 and f(5.005) = 13.01.  In fact, I can answer any neighborhood of 13 you challenge me with, simply by halving it to obtain my vicinity of x = 5.  If you want me to be within ε of 13, then all I have to do is be within δ = ε/2 of 5.  Then f(5–ε/2) = 2×(5–ε/2)+3 = 13–ε, and f(5+ε/2) = 2×(5+ε/2)+3 = 13+ε.  It's foolproof.

Falsifier.  Hmm, I guess you're right.  I'll have to concede that the limit is 13.

The exchange would have gone quite differently if Verifier had claimed that the limit was 12.  Then, for instance, when Falsifier challenged Verifier to get within, say, 0.1 of 12, Verifier would have been unable to choose a vicinity of x = 5 such that f(x) is between 11.9 and 12.1 over that entire vicinity, because any value of x very close to 5—close as we like—always has f(x) very close to 13, and that clearly doesn't fall between 11.9 and 12.1.  But if Verifier can always figure out the right vicinity to force the function to fall in Falsifier's neighborhood, then they can prove the limit to be correct.

This approach to proofs has much broader applicability; in game semantics, and in a kind of logic called independence-friendly logic, many demonstrations rely on this kind of interplay between a Falsifying universal quantifier (the "for every" ∀) and an existential quantifier (the "there exists" ∃).

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Now for a digression to something that will seem totally unrelated at first.

In the late 11th century, into the 12th, there lived a Breton named Pierre le Pallet who was a precocious philosopher.  He was initially trained by William of Champeaux, but quickly grew capable of duelling wits with his teacher, and ended by starting a school of his own, against the advice of William.  By all accounts, he was a self-proud man, convinced simultaneously that he was brighter than anyone else and that no one else was giving him proper credit for this.  In his defense, he was generally regarded as one of the leading philosophers of his time, his specialty being logic, a tool that he wielded in an almost competitive spirit in defense of positions that were then considered heretical.  It was during his late adolescence that he took on the name that we know him by today, Peter Abelard.

As Abelard, his fame grew considerably, and people from all around sought his counsel.  One of these was a canon in Notre Dame named Fulbert, who wanted Abelard as a tutor for his niece.  She was then in her early twenties (we think—there is significant uncertainty about her birthdate), and had demonstrated herself to be remarkably capable in classical letters.  She had mastered Latin, and Greek, and Hebrew, and had applied these to a study of Christianity, to which she was devoutly dedicated.

Her name was Heloise d'Argenteuil, and she and her relationship with Abelard were in time to become famous.  Both of them found the other attractive, and in or around 1115, they started an affair just out of the watchful eye of her uncle.  Ostensibly, Abelard was tutoring her, but this would be interrupted periodically by a bout of lovemaking.  When they were separated, they would exchange personal messages on wax slate (parchment being too expensive even for billet doux that would have to be discarded or hidden).  A message would be incised on a layer of wax mounted to a wooden back; this message could then be read and the wax melted and smoothed over to be used again and again.

The two lovers could not necessarily deliver the messages personally without incurring Fulbert's suspicion, and so would have to rely on the discretion of messengers.  But as the messages were typically written in Latin or Greek, which the messengers couldn't read, teacher and pupil could exchange their letters under the apparent guise of lessons.  Abelard and Heloise apparently exchanged over a hundred letters this way, letters we have access to only because Heloise seems to have transcribed them onto a scroll (now lost) which was found centuries later by a French monk named Johannes de Vepria.

The affair progressed as far as Heloise bearing a son by Abelard, whom she called Astrolabe, after the astronomical instrument, and about whom we know almost nothing at all.  Around this time, Fulbert caught wind of it, and managed to force them to marry, although Abelard extracted a promise from Fulbert not to publicize the marriage, so as to protect Abelard's reputation.

Fulbert, however, had had his own reputation damaged by Abelard over other matters, and so he began spreading rumors of the marriage.  Abelard had Heloise installed at an abbey for her own protection, a gesture that Fulbert misunderstood as Abelard trying to wash his hands of her.  So Fulbert hired some henchmen, and one night, they went to Abelard's sleeping quarters, and castrated him.

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Abelard went into seclusion, and it is unclear that he ever saw Heloise again after this time.  However, about a decade or two later, they exchanged a sequence of seven or so longer letters, instigated when Heloise somehow got her hands on a letter that Abelard had written to a monk about his life story.  That letter included a retelling of her own story, and the two lovers were reintroduced to one another in this way.

Except that by this time, Abelard had decided to impose a sort of pious asceticism on himself that extended to any romantic feelings he might have had for his one-time wife.  Heloise, in turn, wrote him back, entreating him to concede those feelings, feelings she was sure he still retained.  In the last pair of letters, Heloise appears to have relented, and buried herself in her religious life, and Abelard seems to have praised and encouraged this.  But these letters are permeated through and through with an almost overwrought subtext.

So who convinced whom?  As if in honor of these two, whose story has become synonymous with medieval romance, the roles of the Falsifier and the Verifier are often personified by the love-denying Abelard, whose initial is a convenient mnemonic for the universal quantifier ∀, and by the love-asserting Heloise, whose name is sometimes spelled Eloise, whereby her initial is a convenient mnemonic for the existential quantifier ∃—symbols ineluctably entwined in the cherished logic of Abelard's youth.

Facing the Facts and Missing the Forest

Bill Nye, he of Science Guydom, recently had this to say about creationism and evolution, and the debates between them.

I have mixed feelings about his position on debating creationists.  I sort of agree with him that there should be debate—although I view most televised debates as theatre without much illumination—but I think he goes about it in an ineffective way.

And it's sort of lampshaded immediately, in that link, in his explanation as to why people oppose evolution.  "I think humans have an unwillingness to face the facts...."  I think many of his answers are well considered and on point, but with that sentence right off the bat, you might as well dig trenches for everyone and tell them to settle in.  Nothing's going to make people more hostile to evolution than telling them they're only opposed to it because they're delusional.

One of the more illuminating ideas in Michael Shermer's Why Smart People Believe Weird Things is the notion that people generally do not arrive at their beliefs through a series of rational arguments.  They might construct rational arguments once they've arrived at a belief (and smart people are usually better at that, convincing not only others but themselves), but the order is usually belief first, rational argument second, and that goes for evolutionists and creationists alike.  Of course, we all have conclusions that we do reach through rational argument, and that tends to make us think that we do that as a matter of course—because rational argument is good, right?—but I got from Shermer the principle that that sort of situation is the part of the iceberg above the water, the part we're conscious of.  The part below the water, the unconscious part, is much, much larger.

There's no way to prove that, of course—it's a way to view the world, not really a testable statement (yet)—but I find it useful.  It explains a lot of things to me, such as why Nye in some ways got owned by Ken Ham in their debate: He tried to make the debate about facts.  To do so is reasonable at first blush, since the facts are on his side, I think, but wouldn't evolutionists suppose that if the debate were about facts, it would have been over a long time ago?  Nye claims that he, unlike previous scientists in his position, was not outcompeted or outmaneuvered, but I think he was, a bit, because he failed to adequately address what makes creationism compelling.  If Nye, and others, are genuinely interested in drawing people "across the fence" (or, at least, "off the fence"), they must convey something of the elegance and wonder of the scientific view.  Scientists often say that the scientific view is elegant and wonderful (and it is!), but it's not enough to say it; one must, I think, illustrate it in a vivid way that's not simultaneously a knock on creationism.

I think scientists in this debate fail to consider sufficiently what draws people to the creationist perspective in the first place.  What is creationism, after all, but a way to view the natural world?  There's a wealth of beauty and danger in the world; how did that all come about?  Creationism provides an explanation for the natural world that is straightforward, intuitive, and convincing, if only largely because most of its adherents were exposed to creationism before anything else.  With theories, as with people, first impressions matter, a lot.

In speaking with people who find creationism at least plausible, I sometimes ask them why they like it.  One common thread is that they like it because they enjoy the thought of the ubiquitous immanence of God (although they don't usually put it that way).  It reinforces the pleasure they get from believing in God; each biological oddity becomes yet another brick in the divine edifice, a reminder that God is once again just around the corner.  The constant refrain of God-did-it, far from being the vehicle of ridicule intended by evolutionists, is encouraging to creationists.  It proclaims God's pervasive influence in the universe.  If you're invested in believing in an all-powerful deity (and evolutionists generally say that they want to accommodate these people), honestly—why wouldn't you find that appealing?

Science has, to a large extent, not spoken loud enough on this matter.  In my more cynical moments, I feel that creationists intentionally draw evolutionists toward facts, not because they're too clueless to realize that their facts are wrong, but because they're insightful enough to recognize that it doesn't matter whether their facts are wrong.  Even if that's not the case, though, it behooves scientists to every now and then resist the siren call of facts, and speak more about why science is a beautiful way to view the world.

And it is beautiful, in my (admittedly biased) opinion!  If one thinks of creationism as a sequoia with God in the towering trunk and the various aspects of the natural world as branches going outward at every height, then science in general and evolution in particular is a web of unimaginable richness, with connections in every conceivable direction, splitting and rejoining and looping in almost infinite variety.  The strength of the sequoia is its enormous trunk, a monolithic invulnerability; that of the web is its deep interconnectedness, so that even if a few of its strands are found to be flawed (and they surely are, from time to time), the overall structure retains its integrity with room to spare.  A scientific theory can tie a wide array of observations together, and do it with a beauty that is utterly captivating.  Albert Einstein was famously convinced of the rightness of his general theory of relativity, not through observation, but through its elegance and beauty.

One of my favorite stories as a child was Aesop's fable of the fasces, so much so that I reference it all the time.  Maybe you know it: An old man demonstrates to his squabbling sons the strength of unity, of a single purpose, by showing how easy it is to snap a single stick, but how difficult it is to snap a lot of sticks bundled together.  I was struck by the moral when I first read it, but when I was rather older, I began wondering why one wouldn't simply split the bundle down its length.  As far as I could tell, one could put together sticks in a more connected way that would be much harder to break down significantly.  And I find in that a deep metaphor for how I see the world.

I do not know which of these views of the world is more beautiful in any inherent sense.  It's entirely possible that the guiding principle here, as elsewhere, is de gustibus non disputandum est.  But I feel quite confident that there is little to be gained with continued debate that does not engage, at a significant level, the profound and abiding manner in which the human mind cleaves to beauty and elegance.  It's why many of us became scientists, and it seems a shame that we don't spend more time sharing it.

One Language, Under Force

I watched the Super Bowl.  Well, "watched" might be putting it a bit strongly.  I watched the first part, a very short part, in which the Broncos seemed as though they might have had a decent chance to win.  After that, I watched mostly to see what other parts could fall off the Denver bandwagon.  Congratulations to Seattle; they thoroughly outclassed their opponents.

That left the halftime show and the commercials.  I have to say that I didn't even watch those very assiduously, though I find the idea that they aren't as good as they used to be to be about a step or two shy of yelling at the neighborhood kids to get off the lawn.  It's Cranky Old Geezer time!

But even through my haze of disappointment in the football game, I did manage to get a look at the Coca-Cola "America the Beautiful" commercial.

The one-minute spot consists of a sequence of short video vignettes of the broad span of Americana, against which is sung "America the Beautiful."  There's nothing at all contentious about that, as far as it goes, of course.  What seems to have gotten lots of people in a lather is the fact that, except for the first and last phrase, the song is sung in several different languages.  No doubt Coca-Cola wanted to evoke the idea that part of what makes America beautiful is the wide variety of people that make it up, and that's what the commercial does.  In fact, Coca-Cola went so far as to follow the commercial up with a tweet, just in case someone missed the point:

Apparently, that's not the message that many people got.  I imagine the reaction of Coca-Cola to the some of the retweets ranged from bemused concern to horrified astonishment.  (Or maybe they're more cynical than that; it's quite plausible.)  I don't have the patience to drag them all out, obviously, so I'll just link to a collection of some of them here.

As might be expected, there's also been a backlash against those reactions, lambasting them as racist or ignorant or condescending, or who knows what.  I won't attempt to characterize them one way or the other; as I like to say, people feel what they feel, and it's pointless to tell them they're "wrong" to feel that way.  But I think it is interesting to try to suss out just why they feel that way.  What is it about diversity, in what seems like such a harmless context, that spooks some people?  Is our sense of national pride so fragile that it relies upon the exclusive use of a language that was brought forth onto this continent for the first time not half a millennium ago?  I have no idea whether one of the languages used in the commercial was an Amerind tongue (maybe someone can tell me), but I wonder what the reaction to that would be.

The melting-pot metaphor used to be a point of pride for us; it's a central point of one of those Schoolhouse Rock shorts, for those of you who remember those.  I don't remember anyone lashing back at those a few decades ago.  Shall we say to those who object to singing "America the Beautiful" in anything other than English that they are simply being too sensitive?

In connection with that possibility, let me introduce another commercial, which aired last year (and was brought back to mind by a friend of mine):

Notice anything out of the ordinary?  I have to admit that the first time I watched this, I didn't.  Then, my friend pointed out, "Look at who's in the box."  My reaction to this was, "Oh, PoCs in a box," since the forward-thinking hotel guest (who incidentally spouts some meaningless marketing mumbo-jumbo, but that's neither here nor there) is a white male, and all the persons of color, along with a white male or two (for variety I suppose), are in the box.  Even the guy who thinks to venture out before scurrying back to the safety of the box is a white male.  (I must say that the look of relief on the woman next to him is hilarious.  She should get a Cleo for that.)

A "natural" reaction by some people, in response to such a comment, might well be "Oh, you're being too sensitive.  They had to put someone outside the box; it just so happens it was a white guy."  In isolation, that point might be arguable.  However, it happens too frequently for it to be just random chance.  The vast majority of business travellers I work with are white males, and it's not surprising that they (the primary target of the commercial, after all) would prefer to see someone like themselves as the hero of the story.

I see too frequently, however, the objection that people of color have an inferiority complex, that they play the race card too readily, that they are too comfortable in the victim role.  Does it really make sense that a group of people who are actually empowered would feel that way, that instead of doing what they're capable of, they would rather lie down and cry foul?  I'm not sure that there's been a significant group of people like that in the history of ever.  Regardless of whether that group is a victim of discrimination as they claim, or for some constitutional reason is less capable, or both, it's utterly implausible to me that they would rather blame someone else than have more power.  Blame may be a salve for what ails them, but equal power is the cure.  A small number of them might miss that, but not the whole group.

Could something similar be at work with the reaction to the Coca-Cola commercial?  Is it that people feel upset about the commercial because it represents a situation they have little or no immediate control over?  Undoubtedly that's part of it.  After all, the tweets are rife with threats to boycott Coke, but these threats would have essentially no impact on Coke's bottom line even were they credible.  As it is, I suspect the vast majority of those would-be boycotters will be back to drinking Coke before the month is out.  Inexpensive habits can be terribly hard to break.  And at any rate, Coca-Cola is here serving only as a proxy for what some evidently see as a distressing trend toward inclusiveness.

Isn't it provocative, though, that each side sees a given cultural portrayal as betraying an awful truth, and often speaks out vigorously against it—something that the other side views as oversensitive and tiresome?  And this may be the crux of the matter: that there is a kind of massive joint cognitive dissonance between the way that the various groups perceive the current cultural situation, and the way that the various groups think the situation should be.  This dissonance is made all the more contentious by the striking symmetry between the views.

There is one thing, however, that distinguishes the two cases, as exemplified by these commercials, and that is the distinction between equality and uniformity, something that has stuck with me ever since it was first explained to me in stark simplicity in Madeleine L'Engle's A Wrinkle in Time.

What is at the root of the desire for uniformity (for I see no other way to describe, as succinctly, the demand that people sing this song in English) that the Coca-Cola tweets share?  It seems to me that it aims for a feeling of security, that if we only trust those people who cleave to the majority culture, then all will be well in this world gone mad.  But if that's so, is it necessary to demand uniformity?  Can't we feel secure without insisting on the elimination of the traces of other cultures?  Why not cut the middle man of uniformity out of the picture entirely?

I fear, though, that this is not likely until people see that this kind of uniformity not only isn't the end goal, but is actually counter-productive as far as any real kind of security is concerned.  I like to say that religion is a laser of the people, by which I mean that it moves people to behave and operate in unison, almost as though they constituted a single being, which can do certain things that the individuals couldn't do, separately.  But that same uniformity has a cost, because if all the individuals uniformly have a weakness, that weakness is passed onto the group as a whole, and is not amortized, so to speak.  I'm reminded of the old Aesop's fable in which an old man, near the end of his life, demonstrates to his sons the value of unity by tying together a bundle of sticks.  That bundle, of course, could not be broken by vigorous effort, even as the individual sticks were easily snapped.  It's ironic to think, though, that one could quickly slice through the bundle if one were to cut lengthwise.

The amortization of weaknesses is what makes diverse groups so robust.  It's why a farm made up of a single strain of high-yield crops is not a good long-term strategy.  It's why a diversified investment portfolio is safer than one that relies on a single kind of asset.  And why shouldn't the same kind of reasoning apply just as well to people as to crops or funds?  Yes, uniformity is good in moderate doses, for it enables feats that could not be achieved otherwise, but in doses large enough to dominate an entire country, it's dangerous.  It's dangerous not only because it makes the country more vulnerable, but also because it is such an appealing dogma.

Who knows if there will come a time when ads like Coca-Cola's will not produce such a strong negative reaction.  But if it does, it will be because people understand, viscerally, the value of diversity, and do not see it for the demise of national security.

The Limitations of Sense

As I've mentioned previously, I lived in the dorms in college.  In addition to balky vending machines, the dorms also had a number of lounges—common areas on selected floors for people to gather for the purpose of studying (if they didn't mind a bit of noise), watching TV, or generally screwing around.  And, from time to time, there was the occasional Bible study group.

I hasten to emphasize that the study group people (who generally lived in the dorms themselves) were very reasonable about their use of the lounge.  They were perfectly willing to wander around in search of a mostly unused lounge, and they asked the others instead of just plopping themselves down and using the space.  In my own turn, I was perfectly willing to move over to defrag the chairspace in the lounge and allow them their own section.

Once, though, they did manage to irritate me.

I had settled in with my Walkman, listening to an album.  (For the benefit of those of you who were born in this millennium: Songs used to be sold on physical media, called "vinyl" or "records."  These records could be "singles," or they could be multiple songs sold on one "album."  We had this innovation—developed by Sony, a company that existed even then—called a Walkman, which played "tapes," on which songs could be transferred from the record.  It was called a Walkman because you could walk around with it.  You could listen to a whole entire album and not be tethered to your "component stereo system," which was a collection of devices used to play music at a time when computers had memory sizes measured in kilobytes.  We thought it was great.)

Anyway, the Bible group came in and said they wanted to use the lounge and they promised not to be too loud.  Since I was the only other one in the room and I didn't want to be a complete jackass, I cheerfully agreed and moved over to the other side.  But in doing so, I took off my headphones.  And so, as they began discussing the Bible, I listened to them.  It was interesting, after all.

After some time, however, I guess it became increasingly evident that I was listening to them, and since it was apparently one of their objectives to spread the word to as many people as they could, they began working on me.  Now, I was brought up without any religious background.  (Oddly, I do recall that we had a napkin holder that had some strange incantation on it about "daily bread," although that was never explained to me.  I had to find out about it on my own.  But that's a story for another time.  Essentially, there was no religion in my upbringing, at all.)

What's more, I had by this time become fascinated by science, and the scientific method.  I didn't have a firm idea, perhaps, of how science got done, exactly, but I did have the notion that people were fallible, and experiments were conducted so that we could find things out without relying solely on fallible humans.  And it seemed to me that the more fantastical stories in the Bible (as opposed to the moral precepts, say) simply would not stand up to any kind of scientific inquiry.  I did not believe that there existed anything like the Christian god.  And I'm sorry to say that, somehow, that came out.

Well, the floodgates opened up after that.  And I just could not get them closed back up.  For some reason, I was made to answer for the slightest failing or shortcoming of science as it pertained to anything, and I mean anything, in the Bible.  To be sure, I was not blameless in this; at that age, I had not learned to adopt the sort of detached self-doubt that I can effect these days, and I was unfoundedly certain about the points I made, which landed me in some hot water.

I don't remember how I managed to extricate myself from the "discussion," but I do know that it took a couple of hours, after which I went to my room and lay down.  I was exhausted.

A few of them came up to me the next day, and apologized for their aggressiveness.  I said I understood, and apologized for my unseemly certainty.  But it set me to thinking: I did feel pretty certain about my atheism.  Why?  What made me feel so certain?  I had some vague sense that it had something to do with a kind of epistemological conservatism (though I wouldn't have known to put it in such a way)—the idea that one believes in as few things as is possible to understand the world—and the proposition that extraordinary claims require extraordinary evidence.

It took me some years, however, before I could fully work out what my situation was with regard to atheism, and agnosticism, and all that.  It came about like this:

Much later, I was talking to this fellow, and I mentioned some of this mess I got in with the Bible study group.  And so he asked me, what did make me so certain?  He thought that people who could feel so certain that there was no god were just as scientifically irresponsible as those who could feel certain that there was one.

Fortunately, by this time, I had read Wittgenstein (I'll bet that's the only time you'll hear anyone consider it fortunate to have read Wittgenstein, and by the way, he looks just about that crazy in every picture of him I've ever seen), and I knew he had, too, so I could express it a bit more concisely.  I said that I was about as certain that there was no god as Wittgenstein was that he had a hand.  What good ol' Wittgenstein—and I, by extension—meant by that was that the knowledge that one has a hand represents an upper limit of certainty: a limit imposed by our senses.  We know it not because it is logically proven beyond a shadow of a doubt, but because doubt itself is pointless in this regard.  In other words, the degree to which we know it is a milestone of certainty—in a very real sense, defines it.  In fact, I think Wittgenstein says as much, right at the very start of his final work, On Certainty:

If you do know that here is one hand, we'll grant you all the rest.

My friend was satisfied by that, I believe, and he walked away.  As he walked out, though it hit me that that was it—that the limitations of my senses were the basis of my "certainty" that there was no god.

To begin with: From time to time, some atheist wag will remark that we have no more evidence for the existence of the Christian god than we do for, say, the Flying Spaghetti Monster.  Which is true, so far as it goes, but it doesn't really establish atheism (the belief that there is no god) as it does agnosticism (the lack of a belief that there is a god).

So then, the hypothetical line of questioning goes, what would it take to establish the existence of a god in any kind of scientific way?  Because, as I tell others, if you take a position against something, then as a self-check, you must ask yourself what it would take to convince yourself you were wrong.  Because if there's no amount of evidence that would do it, then your position isn't a scientific one; it can't be falsified.

I thought about all the miracles that are said to be the work of some god or another, all the things that happened that could not be explained.  In most cases, I rather thought that these were evidence less for a god than for the selective ingenuity of humans: If people wanted to believe in something, they were remarkably ingenious about how they managed to assemble evidence in its favor.  But if they didn't want to believe it, that ingenuity mysteriously went away.  In other cases, I couldn't come up with a plausible explanation, except to say that the people who related these stories (thousands of years ago, remember) were either mistaken or, possibly, exaggerating.  That might not have satisfied anyone who was truly on the fence, but it satisfied me.

It boiled down, therefore, to what I could personally witness that would convince me I was wrong.  What could a supernatural being do that would sway me?  It quickly occurred to me that whatever evidence could possibly support the claim to existence of a god had to be much more extraordinary than the possibility that my senses were fallible.  When it came to the existence of a god, I could not grant that I had a hand.

We hear "Seeing is believing," but we see things all the time that, it later turns out, aren't true.  And so, not as an expression of any desire, but simply as an acknowledgement that my senses can fail, catastrophically at times, I flatly admit an incapacity to believe in a god, any god (as normally represented—I obviously don't mean just a super-powerful being, but someone who brought about the world).  It's a personal incapacity, not one that I could possibly extend to anyone else, but it's insuperable just the same.

Square Roots, Lasers, and Mobilization

I promised (threatened) that I would say more about square roots, and so I am. This is me, talking about square roots again. In typical fashion, though, I'm going to start with something else that will seem, for a time, completely unrelated.

Galileo, he of the telescope, the balls rolling down inclined planes (and probably not in actuality from the Tower of Pisa), the sotto voce thumbing of the nose at the Inquisition—Galileo also discovered, or more likely rediscovered, that pendulums mark out roughly even time, no matter how far they swing. It isn't perfectly even time, owing to friction and to the circular track of the pendulum bob (although both of those can be—and were—accounted for, starting with Huygens's employment of cycloid guides). But it's pretty close.

Since the pendulum keeps fairly even time, that must mean that if the pendulum swings in twice as big an arc, it must also be moving twice as fast, in order to keep beating out even time. Now, as it's defined in Newtonian physics, the kinetic energy of the pendulum bob—that is, the energy of the bob due to its motion—goes as the square of its velocity:

KE = ½ mv²

So, twice the arc, twice the velocity, four times the kinetic energy; three times the arc, three times the velocity, nine times the kinetic energy. And so on.

That swinging motion of the pendulum bob is an example of periodic or wave motion, so called by virtue of it swinging back and forth as a water wave swings up and down, if you were to watch it passing by a buoy. Wave motion is primarily characterized by two parameters: its frequency, which is how often it returns to its starting point; and its amplitude, which is how wide it swings. So the arc through which the pendulum bob swings is essentially its amplitude. (Actually, for historical reasons, the amplitude is defined as half of that arc, from the center point of the swing to either of its extremes, but this won't affect our discussion.) So we can say that the pendulum's energy is proportional to the square of its amplitude.


This turns out to be common to many different kinds of waves—including light waves. Light is a wave. (It's also a particle, in many ways, but we'll ignore that for now.) And being a wave, it has an amplitude, which is the extent to which the light oscillates. What is it that's oscillating, anyway? In the case of water waves, it's water, and in the case of sound, it's the molecules in the air. You can't have water waves without the water, and you can't have sound waves without the air; that's why sound doesn't travel in a vacuum. But light does travel in a vacuum, so what's waving the light, so to speak? Well, the answer is that the light itself is waving, or less opaquely (heh heh), the electromagnetic fields that permeate space are waving.

In any event, like other waves, light waves also carry energy that is proportional to the square of the light's amplitude. If you double the amplitude, you quadruple the energy; triple the amplitude, and the energy goes up nine-fold. And so on.

How would light's amplitude be doubled, though? You might imagine that if you put two flashlights, the amplitude of the two together would be twice that of each individual flashlight, and the combined light output—the energy of the two together—would be four times that of each flashlight. But I think, intuitively, we know this to be false, that the combination is only twice as bright as each flashlight. And if you measure the light carefully, in a dark room, this turns out to be perfectly true.

What happened? Light waves, like other waves, have a secondary property, called phase. Two waves of the same frequency are said to be in phase if they swing in the same "direction" (in some not altogether well-defined sense); imagine two pendulums swinging in unison, so that when one swings left, the other does, too. They are out of phase if when one swings left, the other swings right, and vice versa. Or, they may be partly in phase, partly out of phase.

When you combine two light waves of the same frequency and the same amplitude, you get for all intents and purposes a single wave that is the two original waves added together. If they're in phase, the peaks get peakier and the valleys get, err, valleyier, and the amplitude of the waves is in fact doubled. On the other hand, if they're out of phase, the peaks of one get cancelled out by the valleys of the other (and vice versa), and the resultant wave has no amplitude at all.

More typically, though, the two waves are partly in phase and partly out of phase, and the resulting wave's amplitude is somewhere in between zero and two times the original. On average, one can show that the amplitude is the original times √2 . What's more, if you add three waves together at random phases, the amplitude of the sum is the original times √3 . And so on. Aha, the square root!

And since the energy of the final wave is the square of the amplitude, what comes out has two, three, or whatever times the original energy. Which is, of course, exactly what you'd expect. And good thing, too, because if it came out otherwise, we'd have a violation of the conservation of energy. Clearly, it takes n times as much energy to run n flashlights as it does to run one, and if their combined output were something other than n times the original, we'd have to seriously rethink our physics.

You might wonder if there isn't a way to get the waves to line up properly in phase so that the amplitudes do add up in the normal way, and you get a dramatic ramp up in energy. And there is; it's called a laser. A laser essentially gets n individual photons to line up in phase so that what comes out is a sort of super-photon (or super-wave, equivalently) with n² times the energy of any of the input photons. The physics-saving catch is that it takes more energy to line up, or lase, the light than you get as a result.

Nevertheless, that single photon or wave, coordinated as it is, can do things that you couldn't do with the individual photons separately. You can shine a bunch of flashlights at your eye and nothing will happen, other than a rather annoying afterimage and perhaps a headache. But even a modest laser can be used to reshape your cornea and render your eyeglasses superfluous. Of course, it should go without saying that it's not such a great idea to randomly shine lasers into your eye!

Or out, for that matter.

I see in this a kind of metaphor for human nature, and I hasten to say it's only that; as far as I know, one can't really take this and apply it rigorously in any scientific sense. But I think it's a useful metaphor all the same. I like to say that religion, among other things, is a laser of people. What on earth do I mean by that? A single human being can do a certain amount of work (in physics, work is defined as energy applied in furtherance of a force). What happens if you get two human beings together? Well, if they work against each other—if they're out of phase, in other words—less work gets done. Maybe none, if they spend all their time squabbling. Even if they're not exactly out of phase, if they're not particularly coordinated, their combined output is rather less than you might think, like the drunkard making slow and halting progress homeward because he can't put one foot directly in front of the other.

On the other hand, if they cooperate—if they're in phase—they can do twice the work. In fact, maybe they can get even more done, for there's no arguing that a coordinated combination of two people can do things that each individual person couldn't do, even adding their results together. Two people can erect a wall, for instance, that neither person could individually. Maybe, in some sense, those two people can do what it would take four people, working randomly, to achieve. And perhaps three coordinated people can do what it would take nine randomly working people to. And so on.

But it's pretty straightforward to get two or three people to work together, if they're of a mind to. But what about a hundred, or a thousand, or a million? That's where ideologies can be enormously effective; through them, a thousand can achieve what would otherwise require a million. And there may be no ideology better suited for the purpose than religion, although other ideologies—sociological, fiscal, even autocratical—may suffice. That's not to say that all that these various ideologies achieve is beneficial: for every great liberation, there may be a dozen pogroms. But they are part and parcel of a society's capacity for achievement; without them, we get only as far as a drunkard's walk will take us.