Recently, I started reading The World is Flat by acclaimed New York Times columnist Thomas Friedman. Friedman's thesis is that a new phase of globalization has begun, driven by advances in communication technology and the fall of Communism

("Globalization 3.0" he calls it) in which it is becoming logistically possible, and economically mandatory, for any company in any country in the world to compete with any other for each contract and every person with every other person for each job.

Friedman's view is based on first-hand knowledge derived from interviewing various Indian captains of industry, Chinese technology workers, American software experts, etc. It's a relatively convincing framework for understanding the economic changes underlying things as divergent as outsourcing, the formation of the European Union, and the rise of Islamic terrorism.

It's horrifying to find in the book's first chapter, therefore, the horribly sloppy metaphorical reasoning that led him to this conclusion. Friedman's epiphany came after Nandan Nilekani, CEO of Indian tech giant Infosys, told him that the new information infrastructure made it possible for Indian companies to compete against Americans in almost every field: "Tom, the playing field is being leveled," he said. Friedman spun madly off from there:

"What Nandan is saying, I thought, is that the playing field is being flattened. . .Flattened? Flattened? My God, he's telling me the world is flat!"

This is just the worst kind of comparative reasoning, less metaphor than pure word association. Now, it's a nifty phrase and it does a lot of work for Friedman, encapsulating the differentiation of this new age of globalization from that begun by Columbus's discovery, as well as the reduction of traditional competitive hierarchies and spatial limitations brought about by the internet revolution. But, all these points strike discordancies with various aspects of the metaphor itself. Most importantly, the paradigm change involved with the round-world theory associated with Columbus had to do with it being *easier* to get from one place to another on a spherical than a flat world (the theoretical new route to India provided by such a geometry being particular apt in this argument). Also, the technological advances around the internet have been normally seen as eliminating physical space altogether (in favor of a new imagined "cyber" space which we create together) rather than merely "flattening" it.

It's the first of these two points I really want to go after here (I'll come back to the second one at the end). And I want to do it with a mode of argument as removed from (and arguably therefore as unfair to) Friedman's mushy metaphor as possible : the world of hard math.

To start, let's restate Friedman's claim in geometric terms. Here's what I take him to be saying: The average distance between two points on the surface of a sphere is greater than the average distance between two points in a plane. Well, let's look at a sphere and a plane, and get started:

We'll take the sphere first. The farthest apart two points can be on the surface of a sphere is half the circumference of a circular section of that sphere (once you've gone halfway, you start coming back, after all):

z - x = c

_{s}/ 2

where c_{s} is the circumference of the circular section. So, since the distance between two points on the continuous shell of the sphere ranges evenly between zero and half the sphere's circumference, the average distance between any two would be:

c

_{s}/ 4

Now, let's take the case of the "flat" world (just to simplify things, I'm going to treat the flat version as a two dimensional circle; I know we all tend to pictures planes (and maps) as rectangles, but this keeps the math *much* simpler. If you think this choice affects the outcome drop me a comment to let me know exactly how, the math there is way beyond me). To make things fair, we'll assume we've flattened the sphere down to a circle with the same radius. The farthest apart two points can be on this circle is its diameter:

z - x = d

_{c}

Again, the distance between any two points in the circle ranges evenly between zero and the diameter so the average distance between two points in a circle would be:

d

_{c}/ 2

So, now we can compare our two cases (taking M_{s} as the mean distance traveled within a sphere and M_{c} as that within a circle). First to restate, we know:

M

_{c}= d_{c}/ 2

M_{s}= c_{s}/ 4

From some basic laws of geometry we can say:

c

_{s}= 2πr

d_{c}= 2r

With this we've got enough to work out a comparison of our average distances:

M

_{c}= d_{c}/ 2

M_{c}= r

and

M

_{s}= c_{s}/ 4

M_{s}= 2πr / 4

M_{s}= (π/2)r

Since we assumed that both our circle and our sphere have the same radius, we've got our answer:

(π/2)r > r

implies (for radii greater than zero):

M

_{s}> M_{c}

In other words, the average distance between two points is greater on a sphere than a plane. Or, in terms of Friedman's metaphor, things actually are closer together in a "flat" world than a round one. He was right!

But, in addition to proving Friedman's metaphor accurate in the specifics of this case, this examination also shows just why Friedman, in his sloppiness, misses the bigger trend. It's not that the world has specifically gotten flatter, it's that space, no matter the shape, has become less important. The world has gotten to be less like a sphere, but also less like a flat plane, or even a one-dimensional line (which acts, in terms of the argument above about average distance, exactly like a plane). Instead, the world has gotten to be more like a single point, the "shape" without a shape that remains as the meaningful distance between each point in the world falls to zero. And this space without distance or geometry is exactly the virtual world of "cyberspace": the world where every point in physical space is equally connected and present to every other point regardless of external geometrical (or topographical) boundaries. Of course, this "cyberspace" is not neutral or un-shapely (so to speak), but has its own quirks, politics, and eccentricities.

Where Friedman's argument really starts to breakdown is where he leaves these eccentricities unexamined: his extremely brief gloss on the early and pre-history of the PC, his total lack of a detailed understanding of the meaning and effects of the world-wide dominance attained by Windows 95, and his complete avoidance of the issue of how old Communist-era rivalries have transfered themselves to the digital realm (just to name problems that show up on page 52 of my hard copy version).

Maybe all this means is that we'll get to see a future edition where Friedman admits his mistake. Maybe we'll read about a new epiphany found not at an Indian tech company, but in a virtual world: "What the giant butterfly creature with the computer monitor for a head is saying, I thought, is that it doesn't matter from where you log on, it's the online world that's the point. . .Doesn't matter? The Point? My God, he's telling me the world's not flat, it's pointy!"

Tagged: Friedman, The World Is Flat, globalization, outsourcing, geometry, math, proof, cyberspace