## Midcentury Monuments

Three serious books lie open before me. I had a variety of reasons for checking them out of the library, although they’re all related in one way or another to current goings on here at the Simons Institute:

• Donald O. Hebb’s The Organization of Behavior: A Neuropsychological Theory. This is the book that introduced a fundamental hypothesis about learning and memory, captured in the slogan “Neurons that fire together get wired together.”
• Norbert Wiener’s Cybernetics: or, Control and Communication in the Animal and the Machine, an eccentric and wide-ranging masterpiece with a crucial chapter on “Computing Machines and the Nervous System.”
• Claude Shannon’s The Mathematical Theory of Communication, the foundational document of information theory. (Shannon’s part of this work had appeared a year earlier in the Bell System Technical Journal; the book version includes an interpretive essay by Warren Weaver.)

When I got the three volumes home, I made a surprising discovery: They were all published at roughly the same time, in 1948 and 1949. What are the odds of that? Perhaps it means nothing—just the long arm of coincidence reaching out to tap me on the shoulder. On the other hand, maybe there was something in the air circa 1950, something that made the period unusually fertile for studies of information, communication, and computation in brains and machines.

I have done a little digging in library catalogues and Wikipedia, as well as in my own files, looking for other titles that might belong on this list of distinguished midcentury milestones.

It turns out that George Kingsley Zipf’s Human Behavior and the Principle of Least Effort was also published in 1949. (This is the one about the curious power-law distribution seen in rankings of word frequencies, city sizes, and so on.)

Gilbert Ryle’s The Concept of Mind is another 1949 title, though I’ve never read it. Also from 1949: Nicholas Metropolis and Stanislaw Ulam published the first open account of the Monte Carlo method.

Drifting forward into 1950, we find another cluster of notables. There is John Nash’s one-page paper introducing what we now call the Nash equilibrium. Elsewhere in game theory, 1950 was the debut year for prisoner’s dilemma, although Merrill Flood’s paper describing it did not appear until two years later. Richard Hamming published “Error Detecting and Error Correcting Codes” in 1950. (It’s another paper from the Bell System Technical Journal.) Finally, there’s Alan M. Turing’s famous essay on “Computing Machinery and Intelligence.”

Does the density of high-octane publications really make 1948–50 an exceptional season of intellectual history? I can’t offer any solid statistical support for that notion. In the first place, my criteria for inclusion on the list are way too vague. (“Subjects I find interesting” may be closest to the truth.) In the second place, I can’t offer any evidence that other intervals were not equally productive. As a matter of fact, in my bibliographic rummaging I came across a nexus of brilliance five years earlier:

• Warren S. McCollough and Walter H. Pitts, “A logical calculus of the ideas immanent in nervous activity,” 1943.
• John von Neumann and Oskar Morgenstern, Theory of Games and Economic Behavior, 1944.
• Erwin Schrödinger, What Is Life? The Physical Aspect of the Living Cell, 1944.
• Vannevar Bush, “As We May Think,” 1945.
• John von Neumann, “First Draft of a Report on the EDVAC,” 1945

I acknowledge a further reason for caution when I cite 1949 as a year of special distinction. It’s my year, the year of my birth.

Posted in uncategorized | 6 Comments

## Notes from JMM 2018

The annual Joint Mathematics Meeting always charges my batteries. Here are few items from this year’s gathering in San Diego.

### A Formal Affair

In 1994 a document called the QED Manifesto made the rounds of certain mathematical mailing lists and Usenet groups.

QED is the very tentative title of a project to build a computer system that effectively represents all important mathematical knowledge and techniques. The QED system will conform to the highest standards of mathematical rigor, including the use of strict formality in the internal representation of knowledge and the use of mechanical methods to check proofs of the correctness of all entries in the system.

The ambitions of the QED project—and its eventual failure—were front and center in a talk by Thomas Hales (University of Pittsburgh) on Formal Abstracts in Mathematics. Hales is proposing another such undertaking: A comprehensive database of theorems and other mathematical propositions, along with the axioms, assumptions, and definitions on which the theorems depend, all represented in a formal notation readable by both humans and machines. Unlike QED, however, these “formal abstracts” would not include proofs of the theorems. Excluding proofs is a huge retreat from the aims of the QED group, but Hales argues that it’s necessary to make the project feasible with current technology.

Hales has plenty of experience in this field. In 1998 he announced a proof of the Kepler conjecture—the assertion that the grocer’s stack of oranges embodies the densest possible arrangement of equal-size spheres in three-dimensional space. Hales’s proof was long and complex, so much so that it stymied the efforts of journal referees to fully check it. Hales and 21 collaborators then spent a dozen years constructing a formal, computer-mediated verification of the proof.

What’s the use of a database of mathematical assertions if it doesn’t include proofs? Hales held out several potential benefits, two of which I found particularly appealing. First, the database could answer global questions about the mathematical literature; one could ask, “How many theorems depend on the Riemann hypothesis?” Second, the formal abstracts would capture the meaning of mathematical statements, not just their surface form. A search for all mentions of the equation $$x^m - y^n = 1$$ would find instances that use symbols other than $$x, y, m, n,$$ or that take slightly different forms, such as $$x^m - 1 = y^n$$.

Hales’s formal abstracts sound intriguing, but I have to confess to a certain level of disappointment and bafflement. All around us, triumphant machines are conquering one domain after another—chess, go, poker, Jeopardy, the driver’s seat. But not proofs, apparently.

### Sperner’s Lemma

Am I the last person in the whole republic of numbers to learn that Sperner’s lemma is a discrete version of the Brouwer fixed-point theorem? Francis Su and John Stillwell clued me in.

The lemma—first stated in 1928 by the German mathematician Emanuel Sperner—seems rather narrow and specialized, but it turns up everywhere. It concerns a triangle whose vertices are assigned three distinct colors:

Divide the triangle into smaller triangles, constrained by two rules. First, no edge or segment of an edge can be part of more than two triangles. Second, if a vertex of a new small triangle lies on an edge of the original main triangle, the new vertex must be given one of the two colors found at the end points of that main edge. For example, a vertex along the red-green edge on the left side of the main triangle must be either red or green. Vertices strictly inside the main triangle can be given any of the three colors, without restriction.

The lemma states that at least one interior triangle must have a full complement of red, green, and blue vertices. Actually, the lemma’s claim is slightly stronger: The number of trichromatic inner triangles must be odd. In the augmented diagram below, adding a single new red vertex has created two more RGB triangles, for a total of three.

Su gave a quick proof of the lemma. Consider the set of all edge segments that have one red and one green endpoint. On the exterior boundary of the large triangle, such segments can appear only along the red-green edge, and there must be an odd number of them. Now draw a path that enters the large triangle from the outside, that crosses only red-green segments, and that crosses each such segment at most once.

One possible fate of this RG path is to enter through one red-green segment and exit through another. But since the number of red-green segments on the boundary is odd, there must be at least one path that enters the large triangle and never exits. The only way it can become trapped is to enter a red-green-blue triangle. (There’s nothing special about red-green segments, so this argument also holds for paths crossing red-blue and blue-green segments.)

So much for Sperner’s lemma. What do these nested triangles have to do with the Brouwer fixed-point theorem? That theorem operates in a continuous domain, which seems remote from the discrete network of Sperner’s triangulated triangle.

As the story goes (I can’t vouch for its provenance), L. E. J. Brouwer formulated his theorem at the breakfast table. Stirring his coffee, he noticed that there always seemed to be at least one stationary point on the surface of the moving liquid. He was able to prove this fact not just for the interior of a coffee cup but for any bounded, closed, and convex region, and not just for circular motion but for any continuous function that maps points within such a region to points in the same region. For each such function $$f$$, there is a point $$p$$ such that $$f(p) = p$$.

Brouwer’s fixed-point theorem was a landmark in the development of topology, and yet Brouwer himself later renounced the theorem—or at least his proof of it, because the proof was nonconstructive: It gave no procedure for finding or identifying the fixed point. John Stillwell argues that a proof based on Sperner’s lemma comes as close as possible to a constructive proof, though it would still have left Brouwer unsatisfied.

The proof relies on the same kind of paths represented by yellow arrows in the diagram above. At least one such path comes to an end inside a tri-colored triangle, which Sperner’s lemma shows must exist in any properly colored triangulated network. If we continue subdividing the triangles under the Sperner rules, and proceed to the limit where the edge lengths go to zero, then the path ends at a single, stationary point. (It’s the “proceed to the limit” step that Brouwer would not have liked.)

### The Muffin Man

You have five muffins to share among three students; lets call the students April, May, and June. One solution is to give each student one whole muffin, then divide the remaining two muffins into pieces of size one-third and two-thirds. Then the portions are divvied up as follows:

This allotment is quantitatively fair, in that each student receives five-thirds of a muffin, but June complains that her two small pieces are less appetizing than the others’ larger ones. She feels she’s been given leftover crumbs. Hence the division is not envy-free.

There are surely many ways of addressing this complaint. You might cut all the muffins into pieces of size one-third, and give each student five equal pieces. Or you might give each student a muffin and a half, then eat the leftover half yourself. These are practical and sensible strategies, but they are not what Bill Gasarch was seeking when he gave a talk on the problem Saturday afternoon. Gasarch asked a specific question: What is the maximum size of the minimum piece? Can we do better than one-third?

The answer is yes. Here is a division that cuts one muffin in half and divides each of the other four muffins into portions of size seven-twelfths and five-twelfths. April and May each get $$\frac{1}{2} + \frac{7}{12} + \frac{7}{12}$$; June gets $$4 \times \frac{5}{12}$$.

Five-twelfths is larger than one-third, and thus should seem less crumby. Indeed, Gasarch and his colleagues have proved five-twelfths is the best result possible: It is the maximum of the minimum. (Nevertheless, I worry that June may still be unhappy. Her portion is cut up into four pieces, whereas the others get three pieces each; furthermore, all of June’s pieces are smaller than April’s and May’s. Again, however, these concerns lie outside the scope of the mathematical problem.)

A key observation is that the smallest piece can never be larger than one-half. This is thunderously obvious once you know it, but I failed to see it when I first started thinking about the problem.

Fair-division problems have a long history (going back at least as far as the Talmud), and cake-cutting versions have been proliferating for decades. A 1961 article by L. E. Dubins and E. H. Spanier (American Mathematical Monthly 68:1–17) inspired much further work. There are even connections with Sperner’s lemma. Nevertheless, the genre is not exhausted yet; the muffin problem seems to be a new wrinkle. Gasarch and six co-authors (three of them high school students) have prepared a 166-page manuscript describing a year’s worth of labor on the problem, with optimal results for all instances with up to six students (and any number of muffins), as well as upper and lower bounds on solutions to larger instances, and various conjectures on open problems.

Long-time readers of bit-player may remember that Gasarch has been mentioned here before. Back in 2009 he offered (and eventually paid) $$17^2$$ for a four-coloring of a 17-by-17 lattice such that no four lattice points forming a rectangle all have the same color. That problem attracted considerable attention both here and on Gasarch’s own Computational Complexity blog (conducted jointly with Lance Fortnow).

Note: In the comments Jim Propp points out that the muffin problem was invented by Alan Frank. The omission of this fact is my fault; Gasarch mentions it in his paper. The problem’s first appearance in print seems to be in a New York Times Numberplay column by Gary Antonick. Frank’s priority is acknowledged only in a footnote, which seems unfair. I apologize for again giving him credit only as an afterthought.

Posted in mathematics, problems and puzzles | 3 Comments

## Flipping Wyoming

Last week I spent five days in the driver’s seat, crossing the country from east to west, mostly on Interstate 80. I’ve made the trip before, though never on this route. In particular, the 900-mile stretch from Lincoln, Nebraska, across the southern tier of Wyoming, and down to Salt Lake City was new to me.

Driving is a task that engages only a part of one’s neural network, so the rest of the mind is free to wander. On this occasion my thoughts took a political turn. After all, I was boring through the bright red heart of America. Especially in Wyoming.

Based on the party affiliations of registered voters, Wyoming is far and away the most Republican state in the union, with the party claiming the allegiance of two-thirds of the electorate. The Democrats have 18 percent. A 2013 Gallup poll identified Wyoming as the most “conservative” state, with just over half those surveyed preferring that label to “moderate” or “liberal.”

The other singular distinction of Wyoming is that it has the smallest population of all the states, estimated at 579,000. The entire state has fewer people than many U.S. cities, including Albuquerque, Milwaukee, and Baltimore. The population density is a little under six people per square mile.

I looked up these numbers while staying the night in Laramie, the state’s college town, and I was mulling them over as I continued west the next morning, climbing through miles of rolling grassland and sagebrush with scarcely any sign of human habitation. A mischievous thought came upon me. What would it take to flip Wyoming? If we could somehow induce 125,000 liberal voters to take up legal residence here, the state would change sides. We’d have two more Democrats in the Senate, and one more in the House. Berkeley, California, my destination on this road trip, has a population of about 120,000. Maybe we could persuade everyone in Berkeley to give up Chez Panisse and Moe’s Books, and build a new People’s Republic somewhere on Wyoming’s Medicine Bow River.

Let me quickly interject: This is a daydream, or maybe a nightmare, and not a serious proposal. Colonizing Wyoming for political purposes would not be a happy experience for either the immigrants or the natives. The scheme belongs in the same category as a plan announced by a former Mormon bishop to build a new city of a million people in Vermont. (Vermont has a population of about 624,000, the second smallest among U.S. states.)

Rather than trying to flip Wyoming, maybe one should try to fix it. Why is it the least populated state, and the most Republican? Why is so much of the landscape vacant? Why aren’t entrepreneurs with dreams of cryptocurrency fortunes flocking to Cheyenne or Casper with their plans for startup companies?

The experience of driving through the state on I-80 suggests some answers to these questions. I found myself wondering how even the existing population of a few hundred thousand manages to sustain itself. Wikipedia says there’s some agriculture in the state (beef, hay, sugar beets), but I saw little evidence of it. There’s tourism, but that’s mostly in the northwest corner, focused on Yellowstone and Grand Teton national parks and the cowboy-chic enclave of Jackson Hole. The only conspicuous economic activity along the I-80 corridor is connected with the mining and energy industries. My very first experience of Wyoming was olfactory: Coming downhill from Pine Bluffs, Nebraska, I caught of whiff of the Frontier oil refinery in Cheyenne; as I got closer to town, I watched the sun set behind a low-hanging purple haze that might also be refinery-related. The next day, halfway across the state, the Sinclair refinery announced itself in a similar way.

Still farther west, coal takes over where oil leaves off. The Jim Bridger power plant, whose stacks and cooling-tower plumes are visible from the highway, burns locally mined coal and exports the electricity.

As the author of a book celebrating industrial artifacts, I’m hardly the one to gripe about the presence of such infrastructure. On the other hand, oil and coal are not much of a foundation for a modern economy. Even with all the wells, the pipelines, the refineries, the mines, and the power plants, Wyoming employment in the “extractive” sector is only about 24,000 (or 7 percent of the state’s workforce), down sharply from a peak of 39,000 in 2008. If this is the industry that will build the state’s future, then the future looks bleak.

Economists going all the way back to Adam Smith have puzzled over the question: Why do some places prosper while others languish? Why, for example, are Denver and Boulder so much livelier than Cheyenne and Laramie? The Colorado cities and the Wyoming ones are only about 100 miles apart, and they share similar histories and physical environments. But Denver is booming, with a diverse and growing economy and a population approaching 700,000—greater than the entire state of Wyoming. Cheyenne remains a tenth the size of Denver, and in Cheyenne you don’t have to fight off hordes of hipsters to book a table for dinner. What makes the difference? I suspect the answer lies in a Yogi Berra phenomenon. Everybody wants to go to Denver because everyone is there already. Nobody wants to be in Cheyenne because it’s so lonely. If this guess is correct, maybe we’d be doing Wyoming a favor by bringing in that invasion of 125,000 sandal-and-hoodie–clad bicoastals.

One more Wyoming story. At the midpoint of my journey across the state, near milepost 205 on I-80, I passed the sign shown at left. I am an aficionado of continental divide crossings, and so I took particular note. Then, 50 miles farther along, I passed another sign, shown at right. On seeing this second crossing, I put myself on high alert for a third such sign. This is a matter of simple topology, or so I thought. If a line—perhaps a very wiggly one—divides an area into two regions, then if you start in one region and end up in the other, you must have crossed the line an odd number of times. Shown below are some possible configurations. In each case the red line is the path of the continental divide, and the dashed blue line is the road’s trajectory across it. At far left the situation is simple: The road intersects the divide in a single point. The middle diagram shows three crossings; it’s easy to see how further elaboration of the meandering path could yield five or seven or any odd number of crossings. An arrangement that might seem to generate just two crossings is show at right. One of the “crossings” is not a crossing at all but a point of tangency. Depending on your taste in such matters, the tangent intersection could be counted as crossing the divide twice or not at all; in either case, the total number of crossings remains odd.

In the remainder of my trip I never saw a sign marking a third crossing of the divide. The explanation has nothing to do with points of tangency. I should have known that, because I’ve actually written about this peculiarity of Wyoming topography before. Can you guess what’s happening? Wikipedia tells all.

Posted in mathematics, modern life, social science, technology | 1 Comment

## Sir Roger Penrose’s Toilet Paper

Twenty years ago, Kimberly-Clark, the Kleenex company, introduced a line of toilet paper embossed with the kite-and-dart aperiodic tiling discovered by Roger Penrose. When I first heard about this, I thought: How clever. Because the pattern never repeats, the creases in successive layers of a roll would never line up over any extended region, and so the sheets would be less likely to stick together.

Sir Roger Penrose had a different response. Apparently be believes the pattern is subject to copyright protection, and he also managed to get a patent issued in 1979, although that would have expired about the time of the toilet paper scandal. Penrose assigned his rights to a British company called Pentaplex Ltd. An article in the Times of London quoted a representative of Pentaplex:

So often we read of very large companies riding roughshod over small businesses or individuals, but when it comes to the population of Great Britain being invited by a multinational [company] to wipe their bottoms on what appears to be the work of a knight of the realm without his permission, then a last stand must be made.

Sir Roger sued. I haven’t been able to find a documented account of how the legal action was resolved, but it seems Kimberly-Clark quickly withdrew the product.

Some years ago I was given a small sample of the infamous Penrose toilet paper. It came to me from Phil and Phylis Morrison; a note from Phylis indicates that they acquired it from Marion Walter. Now I would like to pass this treasure on to a new custodian. The specimen is unused though not pristine, roughly a foot long, and accompanied by a photocopy of the abovementioned Times news item. In the photograph below I have boosted the contrast to make the raised ridges more visible; in real life the pattern is subtle.

Are you interested in artifacts with unusual symmetries? Would you like to add this object to your collection? Send a note with a U.S. mailing address to brian@bit-player.org. If I get multiple requests, I’ll figure out some Solomonic procedure for choosing the recipient(s). If there are no takers, I guess I’ll use it for its intended purpose.

I must also note that my hypothesis about the special non-nesting property of the embossed paper is totally bogus. In the first place, a roll of toilet paper is an Archimedian spiral, so that the circumference increases from one layer to the next; even a perfectly regular pattern will come into coincidence with itself only when the circumference equals an integer multiple of the pattern period. Second, the texture imprinted on the toilet paper is surely not a real aperiodic tiling. The manufacturing process would have involved passing the sheet between a pair of steel crimping cylinders bearing the incised network of kites and darts. Those cylinders are necessarily of finite diameter, and so the pattern must in fact repeat. If Kimberly-Clark had contested the law suit, they might have used that point in their defense.

Posted in mathematics, off-topic, uncategorized | 1 Comment

## The Threats to the Net

My first glimpse of the World Wide Web came in 1993 on a visit to Fermilab, the physics playground near Chicago. Tom Nash, head of the computing division, showed me a screenful of text with a few highlighted phrases. When he selected one of the phrases, the screen went blank for a moment, and then another page of text appeared. We had just followed a hyperlink. I asked Tom what the system was good for, and he said it was great for sharing software documentation. I was so unimpressed I failed even to mention this new tool in the article I was writing about scientific computing at Fermilab.

A year later, after the Mosaic browser came on the scene, my eyes were opened. I wrote a gushing article on the marvels of the WWW.

There have long been protocols for transferring various kinds of information over the Internet, but the Web offers the first seamless interface to the entire network . . . The Web promotes the illusion that all resources are at your fingertips; the universe of information is inside the little box that sits on your desk.

I was still missing half the story. Yes, the web (which has since lost its capital W) opened up an amazing portal onto humanity’s accumulated storehouse of knowledge. But it did something else as well: It empowered all of us to put our own stories and ideas before the public. Economic and technological barriers were swept away; we could all become creators as well as consumers. Perhaps for the first time since Gutenberg, public communication became a reasonably symmetrical, two-way social process.

The miracle of the web is not just that the technology exists, but that it’s accessible to much of the world’s population. The entire software infrastructure is freely available, including the HTTP protocol that started it all, the languages for markup, styling, and scripting (HTML, CSS, JavaScript), server software (Apache, Nginx), content-management systems such as WordPress, and also editors, debuggers, and other development tools. Thanks to this community effort, I get to have my own little broadcasting station, my personal media empire.

But can it last?

In the U.S., the immediate threat to the web is the repeal of net-neutrality regulations. Under the new rules (or non-rules), Internet service providers will be allowed to set up toll booths and roadblocks, fast lanes and slow lanes. They will be able to expedite content from favored sources (perhaps their own affiliates) and impede or block other kinds of traffic. They could charge consumers extra fees for access to some sites, or collect back-channel payments from publishers who want preferential treatment. For a glimpse of what might be in store, a New York Times article looks at some recent developments in Europe. (The European Union has its own net-neutrality law, but apparently it’s not being consistently enforced.)

The loss of net neutrality has elicited much wringing of hands and gnashing of teeth. I’m as annoyed as the next netizen. But I also think it’s important to keep in mind that the web (along with the internet more generally) has always lived at the edge of the precipice. Losing net neutrality will further erode the foundations, but it is not the only threat, and probably not the worst one.

Need I point out that the internet lost its innocence a long time ago? In the early years, when the network was entirely funded by the federal government, most commercial activity was forbidden. That began to change circa 1990, when crosslinks with private-enterprise networks were put in place, and the general public found ways to get online through dial-up links. The broadening of access did not please everyone. Internet insiders recoiled at the onslaught of clueless newbies (like me); commercial network operators such as CompuServe and AmericaOnline feared that their customers would be lured away by a heavily subsidized competitor. Both sides were right about the outcome.

As late as 1994, hucksterism on the internet was still a social trangression if not a legal one. Advertising, in particular, was punished by vigorous and vocal vigilante action. But the cause was already lost. The insular, nerdy community of internet adepts was soon overwhelmed by the dot-com boom. Advertising, of course, is now the engine that drives most of the largest websites.

Commerce also intruded at a deeper level in the stack of internet technologies. When the internet first became inter—a network of networks—bits moved freely from one system to another through an arrangement called peering, in which no money changed hands. By the late 1990s, however, peering was reserved for true peers—for networks of roughly the same size. Smaller carriers, such as local ISPs, had to pay to connect to the network backbone. These pay-to-play arrangements were never affected by network neutrality rules.

A patch panel in a “meet me room” allows independent nework carriers to exchange streams of bits. Some of the data transfers are peering arrangements, made without payment, but others are cash transactions. The meet-me room is at the Summer Street internet switching center in Boston.

Express lanes and tolls are also not a novelty on the internet. Netflix, for example, pays to place disk farms full of videos at strategic internet nodes around the world, reducing both transit time and network congestion. And Google has built its own private data highways, laying thousands of miles of fiber optic cable to bypass the major backbone carriers. If you’re not Netflix or Google, and you can’t quite afford to build your own global distribution system, you can hire a content delivery network (CDN) such as Akamai or Cloudflare to do it for you. What you get for your money: speedier delivery, caching of static content near the destination, and some protection against malicious traffic. Again the network neutrality rules do not apply to CDNs, even when they are owned and run by companies that also act as telecommunications carriers and ISPs, such as AT&T.

In pointing out that there’s already a lot of money grubbing in the temple of the internet, I don’t mean to suggest that the repeal of net neutrality doesn’t matter or won’t make a difference. It’s a stupid decision. As a consumer, I dread the prospect of buying internet service the way one buys bundles of cable TV channels. As a creator of websites, I fear losing affordable access to readers. As a citizen, I denounce the reckless endangerment of a valuable civic asset. This is nothing but muddy boots trampling a cultural treasure.

Still and all, it could be worse. Most likely it will be. Here are three developments that make me uneasy about the future of the web.

Dominance. In round numbers, the web has something like a billion sites and four billion users—an extraordinarily close match of producers to consumers. For any other modern medium—television stations and their viewers, newspaper and their readers—the ratio is surely orders of magnitude larger. Yet the ratio for the web is also misleading. Three fourths of those billion web sites have no content and no audience (they are “parked” domain names), and almost all the rest are tiny. Meanwhile, Facebook gets the attention of roughly half of the four billion web users. Google and Facebook together, along with their subsidiaries such as YouTube, account for 70 percent of all internet traffic. The wealth distribution of the web is even more skewed than that of the world economy.

It’s not just the scale of the few large sites that I find intimidating. Facebook in particular seems eager not just to dominate the web but to supplant it. They make an offer to the consumer: We’ll give you a better internet, a curated experience; we’ll show you what you want to see and filter out the crap. And they make an offer to the publisher and advertiser: This is where the people are. If you want to reach them, buy a ticket and join the party.

If everyone follows the same trail to the same few destinations, net neutrality is meaningless.