Zeno's Paradox & Wall-E

Zeno's  Dichotomy Paradox is an ancient thought experiment that seems blatantly fallacious, but it's deceptively difficult to untangle.  It goes something like this:

Suppose that you wish to walk from here to there (doesn't matter where, as long as here & there aren't the same place).  You begin, and eventually you arrive at the halfway point between here and there.  And you continue, and you reach the halfway point between the first halfway point and the end point.  And you continue walking, and then reach a halfway point between the second halfway point and the ending.  And this process continues infinitely, because between any two locations there is a midpoint which must be traversed, meaning there are an infinite number of midpoints you must cross before you reach the end of your journey, which would take an infinite amount of time.  Therefore movement is impossible and displacement in space is an illusion.

I mention Zeno's Paradox because I think it mirrors the thinking demonstrated by many people I've met toward the modern sci-fi classic, Wall-E.  They've seen the movie, they've appreciated the references to the ways that the sedentary humans who live in an orbital space colony parrot some of the lazier aspects of our modern technological society, and they dismiss it as an amusing spectacle that might lurk in an impossibly distant future.  In a sense, they feel that the number of steps in human social and technological development that lie in between the status quo and the world of Wall-E is so tremendously large that the scenario doesn't bear serious consideration.  Just like Zeno suggests that we can never arrive at any destination.  In both cases, the reasoning is fatally flawed.

Converging Infinities

Let's start with Zeno.  The key to untangling his puzzle is an interrogation of the notion of infinity.  He's quite correct that numbers are infinitely divisible, so that any person traveling anywhere has to traverse an infinite number of distance increments.  The issue is that Zeno takes all infinities to be functionally equivalent, which is certainly not the case.  Infinities come in two zesty flavors: divergent infinities (like 1+2+3+.... a series that continuously grows by larger and larger values), and convergent infinities (like 1/2 + 1/4 + 1/8 + ... which grows by smaller and smaller chunks).  If you take a calculus class, you'll learn the rules for calculating the values of such infinite series and determining whether they converge or not, and if so what value they converge on.  But long story short, the kinds of infinite series that grow by diminishing amounts eventually reach a maximum value.  That's why you can, in fact, move from one place to another (or if you can't, it isn't because Zeno's Dichotomy Paradox).

Accelerating Toward Destruction

The same mathematical logic yields a horrifying conclusion when we use it to analyze the social evolution that tracks the course from early homo sapiens to the hypothetical future humans depicted in Wall-E.

If estimates from modern hunter-gatherers function as a viable comparison, early humans probably walked anywhere from eight to twelve miles a day on average.  On the other hand, the Wall-E people, who move exclusively via hover-chair, probably walk close to zero miles per day.  That puts humans in modern America pretty close to the halfway point, according to Catrine Tudor-Locke, head of the Pennington Biomedical Research Group, who estimates that we walk between five & six miles per day (pretty appropriate for the Zeno comparison, right?).  So that makes the real question whether our path to the hover chair follows a course of a convergent or divergent infinity.

Considering the rapidly accelerating pace of technological development, it seems like a given that our evolution towards a totally sedentary state follows the convergent pattern.  That means that every step we take towards a future where our bodies become rarely-utilized accessories hastens our progress down a potentially irreversible path.  Think about that every time you take the elevator instead of the stairs; every time you troll the parking lot for a premium space so that you don't have to make the hike; every time you longingly wish for a teleportation device to expedite your daily lives.

We are the stewards of all future human society, and if we don't actively choose to embrace and preserve our biological corpus, we'll soon be living without it.