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Solution: ‘The Bulldogs That Bulldogs Fight’

Fri, 2019-05-17 00:50

Our April Insights puzzle explored the magical concept of recursion, a self-referencing process that can create unending complexity from simple beginnings. Recursion, which commonly appears in mathematics, computer science and linguistics, seems especially head-spinning and brain-straining when identical elements are involved, as in our first problem.

Puzzle 1: The Fighting Bulldogs

One type of recursive sentence uses a grammatical structure called “center-embedding.” An example is the sentence, “The dog the man the maid married owned died.” You can make this somewhat easier to understand by inserting the linking words: “The dog that the man whom the maid married owned died.” Most people can understand who did what here, but it’s already getting tough. Now consider the Yale football cheer “Bulldogs bulldogs bulldogs fight fight fight.” The Rutgers University cognitive philosopher Jerry Fodor once pointed out that this is a grammatically correct triple center-embedded sentence. Your challenge is to try to understand how this cheer works as a real sentence. To make it more specific, imagine that the first set of bulldogs is red, the second brown and the third white. Try to answer the following questions:

  1. Whom do the red bulldogs fight?
  2. What color bulldogs do the brown bulldogs fight?
  3. Which bulldogs fight the brown bulldogs?
  4. What color bulldogs do the white bulldogs fight?

This puzzle can make your brain feel like the bruised and battered bulldogs so wonderfully rendered in Dan Page’s illustration. The best way to deal with recursion while minimizing brain strain is as follows:

  1. Start as simply as possible.
  2. Build on the recursion one element at a time, looking for a pattern.
  3. Once you find the pattern, let the pattern do the work.

Let’s apply these techniques to the fighting bulldogs. The simplest possible sentence you can start with by removing the center-embedding is: Bulldogs fight.

Who do they fight? This is not specified — fight is an intransitive verb here, a verb without an object. So the meaning conveyed here is that these bulldogs fight generally or among themselves.

Following our color code, these were the red bulldogs, so the red bulldogs fight generally.

Now let’s add the center-embedding of the second set of bulldogs: Bulldogs fight. The simplest sentence, adding in the colors, becomes: The (red) bulldogs fight.

The verb fight is transitive here and does have an object. The brown bulldogs fight the red bulldogs specifically. The red bulldogs, as far as we know, don’t change their character. They continue to fight generally.

Okay, let’s add the third set: Bulldogs fight. With the colors, we have this sentence: The (red) bulldogs fight.

The verb fight is again transitive here, and its object is the brown bulldogs. The white bulldogs fight the brown bulldogs specifically. Again, the description of the first two sets of bulldogs is not changed in any way.

The answers to the questions are:

  1. The red bulldogs fight generally.
  2. The brown bulldogs fight the red bulldogs.
  3. The white bulldogs fight the brown bulldogs.
  4. The white bulldogs fight the brown bulldogs.

(You may object that since the brown bulldogs fight the red bulldogs, the red bulldogs must fight them too. But this inference is not necessarily true: The red bulldogs might just keep fighting generally or among themselves, without paying any attention to the brown bulldogs, for all we know.)

As you can see, we arrive at this pattern: Group one fights generally, group two fights group one, group three fights group two, and so on. Thus, we can construct a recursive center-embedded sentence consisting of any number of bulldog groups and very easily determine who fights whom. A hypothetical group four would fight group three, group five would fight group four, and so on. Finding the pattern enables you to correctly answer the questions by offloading the work to the pattern or algorithm, without having to tax your brain’s short-term memory.

This is similar to how mathematicians deal with higher dimensions. No one can visualize the fourth dimension and beyond in the way we can visualize the first three. Not even Einstein could visualize higher dimensions. You have to find patterns or algorithms that generalize to the higher dimensions. Then you can accurately answer questions about them.

Several readers got the right answers, including boymeetswool, plouf, Nqabutho and Danielle Schaper. I enjoyed boymeetswool’s diagram and explanation of the sentence tree — it is perfectly correct. Danielle Schaper also produced some very nice diagrams based on the idea that subsets of bulldogs of each color are being referred to. This is one way to interpret the sentence, but it involves an additional assumption that is not specified explicitly. The most parsimonious approach is to assume that all the bulldogs of a given color do the same thing.

Puzzle 2: The Lying Legislators

Our second problem is briefly summarized below. For the full description, see the original puzzle column.

On a faraway planet with perfectly logical beings, there was a legislative assembly consisting of 100 members who met every day. Some among them were pathological liars, or “pathos.” A patho was perceived as having a long nose by everybody else, including other pathos. However, a patho was completely unaware that he or she had a long nose and couldn’t hear about it from anyone because of a taboo that prevented others from pointing out or saying anything about anybody else’s long nose. Any patho who logically inferred that he or she was a patho was compelled to resign before the end of the business day. One day an alien leader not bound by the taboo addressed the assembly and said, “I perceive that at least one of you has a long nose.” Many days later, about half of the legislators resigned en masse.

Question 1 (parallel universe situation): What could have caused this strange pattern of resignations?

Let us apply the recursion-solving rules stated above and start small. Suppose that there was only one person in the assembly who had a long nose. Then the alien leader’s statement would allow that one person to infer that he or she was the one with the long nose and cause him or her to resign. Let’s say this happens on day one (more on this choice below).

If there were two people with long noses (let’s say a male, A, and a female, B), each of them would see one person with a long nose. Legislator A infers that B would see nobody with a long nose if he, A, doesn’t have one, or she would see one person if he, A, does. In the first case, B would be compelled to resign on day one. But that wouldn’t happen because she does see A’s long nose, and by an identical reasoning process, she waits to see if he resigns on day one, which would mean that she, B, did not have a long nose. Thus, neither A nor B resign on day one. On day two, both A and B realize that they have long noses and resign.

If there were three people with long noses, each would see two pathos, and they would have to wait two days to see if both resign on day two, which would mean they themselves were not pathos. When that doesn’t happen, then they would all be forced to resign on day three.

The pattern is already becoming clear. If n people in the assembly are pathos, then:

  • The mass resignations will happen on day n.
  • Everyone’s resignation day is one more than the number of pathos they see. Pathos see n-1 pathos, so their resignation day is n, while all nonpathos see n pathos, so their resignation day (if others don’t resign first) is n+1.

All of the pathos are hoping that the mass resignations happen on day n–1, but when that doesn’t happen, they realize they are pathos themselves and resign next day. Nonpathos see n people with long noses, and they are hoping that the resignations happen on day n, which indeed happens, confirming that they are not pathos.

So if about half of the legislators in the parallel universe resigned, then about half of them must have been pathos to begin with.

We know that the pathos’ days as legislators are numbered, but do we number them starting from zero or one? This point came up in the long discussion between plouf and Sunil Nandella.

We decided above that we should call the day the lone patho resigns “day one.” If we are thinking in terms of days after the alien leader’s visit, then the alien leader must have addressed the assembly after the day’s work was done (say, at an after-dinner speech), so a lone patho would have to resign the next day. On the other hand, if the alien leader addressed the assembly in a morning gathering, then a lone patho would be compelled to resign the same day, which would be day zero.  This would change the number of days by 1. This is the basis of the very common off-by-one-error (OBOE or OB1) that can cause bugs in computer software. Whether you want to use the index 1 or 0 for the first element of an array of numbers or first step of a recursion is an arbitrary choice that has to be made initially and applied consistently. Some computer languages force you to use one or the other (usually 0), whereas others allow you to choose, using a declaration such as “Option Base 0” or “Option Base 1”). The first option is commonly used by software engineers; the second is the one that all of us use naturally when counting, and that’s what we used here.

The above parallel universe scenario can take place only if all the legislators meet all the others every day and no one is ever absent. The Quanta universe scenario (questions 2 through 6) considers what happens in a messier situation where people may be absent temporarily (lost at sea and later found), absent indefinitely (slip into a coma) or where a nonpatho can change into a patho. Specifically, the puzzle stated that:

a.) On the 35th day, a nonpatho changes into a patho irreversibly.
b.) On the 43rd day, three legislators, including one patho, were lost at sea. On the 46th evening, one of the pathos was found and rejoined the assembly the next day.
c.) On the 45th day, a legislator who was a patho slipped into a coma.
d.) On the 49th day, there was a mass resignation of a large number of legislators.

Before we answer the remaining questions, let us consider how the absence of a patho from the assembly (as happens in items b and c) should be handled: Should the remaining pathos take the absence into account and resign one day earlier, or should they stick to the original plan? Let’s examine a simple situation.

Assume that there are three pathos in the group. After day one, one of them goes missing, so that as above, only a male A and female B remain. Should they resign on day two now, because there are only two of them actually present (modified plan), or should they follow the original plan and resign on day three?

On day two, legislator A will reason as follows: “Either I have a long nose, or I do not. If I do not, then the only long-nosed person B would have seen is C, and she can assume that the alien leader was referring to C as the one with the long nose. So there is nothing to compel her to resign today. If I do have a long nose, then B will reason in the same way about me that I am reasoning about her. So I cannot be certain that I don’t have a long nose, and neither can she.” Hence neither A nor B can be fully certain that they are pathos, and so neither can logically resign on day two. The modified plan does not work.

On the other hand, everything works if they adhere to the original plan and wait until day three, imagining that C is watching everything from some virtual place and would have followed the plan too if he were present (even if he is actually dead, for all they know). All three pathos had originally seen two people with long noses, and nobody resigned on day two. So they must resign on day three.

What about the addition of a newly created patho, as happens in item a above? In this case, from the point of view of the person who has become a patho, nothing has changed because she never knew whether she was a patho anyway: She still sees the original number of n pathos. The pathos now also see n pathos, and the nonpathos see n+1. This calls for everyone except the affected person to modify their resignation plan to one day later than it originally was. The pathos’ resignation day is now n+1, and the nonpathos’ is n+2. You can see that this situation is indistinguishable from having n+1 pathos from the beginning, provided that all the legislators except the changed one know about the change, which is true in this instance.

So the principle for absences is: Stick to the original plan, and the resignation day remains the same. For additions such as the universally observable change (except to the affected person) of a nonpatho into a patho, add 1 to the resignation day.

With this in mind, let’s answer the remaining questions:

2. Given all the events described, how many pathos were in the assembly originally and how many resigned on the 49th day?

There were 48 pathos originally, and 48 resigned on the 49th day (with the base day understood as day one). The 48 pathos would have all resigned on day 48, but the addition of the new patho in plain sight to everyone except the affected legislator forced them to change the resignation day to day 49. The comatose person was the missing 49th patho. (If the base day was understood as day zero, add 1 to all the answers.)

3. Was the originally truthful legislator who became a patho on the 35th day among those who resigned?

Yes. He had always seen 48 other pathos, so his resignation day was always 49.

4. The legislator who had slipped into a coma recovered and returned to the assembly on the 50th day. He was briefed about everything that had happened. Did he resign?

Yes. When he returned, he asked for the names of the people who had resigned. He saw the name of the legislator who had changed to a patho on the 35th day. He went down the list, desperately scouring it for a name that he knew to have been that of a nonpatho as of the 44th day. He didn’t find any. He resigned.

5. What would have happened if the legislator who had been lost at sea had been found a day later?

Nothing. All the resignations would still have happened on the 49th day. Absences do not affect the set plan.

6. Why was the Truon leader’s visit necessary for the mass resignations to occur? After all, all of the legislators always knew that at least one of them had a long nose.

This question holds the key to this puzzle. This is what is known as a “common knowledge” problem. Common knowledge, in logic, is defined as the knowledge of some truth, p, in a group, such that everybody in the group knows p,  they all know they know p, they all know that they all know that they know p, and so on ad infinitum. It is this common, infinitely recursive, web of knowledge that the alien leader’s words impart to the legislators. This web of recursive knowledge remains intact in the parallel universe because the legislators are present every day and are able to carry the recursive reasoning forward one step at a time. Thus, all the pathos are able to reach the same conclusion on the same day. When absences and additions are present, the original plan may need to be modified, and this can be done logically but only among the original legislators who were present at the declaration and therefore knew that everyone knew what they knew. The alien leader’s declaration also gives a common base to count the days starting on the day the declaration was made and the web of common knowledge was constructed. As the days pass, this web is recursively unraveled from the outside in, until resignation day arrives and everyone can logically infer their true nature.

The Quanta prize for this puzzle goes to plouf, who answered both puzzles correctly. Plouf’s answers for the second puzzle counted the base day as zero, so they are 1 off from the ones given here, but they are correct for the assumed base day.

I hope you enjoyed this brain twister. If you need some time to untwist, you have a few weeks before we return next month with more Insights.

Categories: Science News

As Planet Discoveries Pile Up, a Gap Appears in the Pattern

Thu, 2019-05-16 22:00

After the sun formed, the dust and gas left over from its natal cloud slowly swirled into the eight planets we have today. Small, rocky things clung close to the sun. Gigantic gas worlds floated in the system’s distant reaches. And around countless stars in the galaxy, a version of this process repeated itself, forging plentiful planets in a spectrum of sizes — except, apparently, worlds just a tad bigger than Earth.

While NASA’s newest planet-hunting telescope, the Transiting Exoplanet Survey Satellite (TESS), steadily tallies more exoplanets, a mysterious gap in their sizes, first identified in 2017, has persisted. The gap shows that scientists need some new ideas to explain how planets are made, both in the broader cosmos and in our backyard.

Astronomers have used TESS to find hundreds of possible planets around the nearest stars since its launch in April 2018, including 24 confirmed worlds so far. The galaxy seems to host a lot of small planets, especially ones measuring between two and four times the size of Earth and others in Earth’s ballpark. But for some reason, planets with radii between 1.5 and two times that of Earth are rare.

The paucity of planets in that range, known as the “Fulton gap” after the lead author of the paper that pointed it out, first appeared in the findings of the Kepler Space Telescope, which hunted exoplanets for nearly a decade before passing the torch to TESS. While TESS doesn’t yet have enough planets in its statistics bin to confirm or disprove the Fulton gap, the trend has continued, and astronomers say they don’t expect the gap to disappear.

In an April paper in the Astrophysical Journal Letters, a team led by Diana Dragomir, an astronomer at the Massachusetts Institute of Technology who works with TESS data, reported the discovery of a star system harboring two planets on either side of the gap, for instance. One is a “mini-Neptune” around 2.6 times the radius of Earth, and the other a wee Earthlet about 90 percent as big as our planet. The latter is the first approximately Earth-sized world in the TESS catalog.

Dragomir said the radius gap points to possible rules both about how planets are formed, and what happens to them early on. Since a planet’s atmosphere can comprise a significant portion of its radius, many ideas center around what might happen to that atmosphere. One possibility, Dragomir said, is a reverse-Goldilocks scenario in which medium-sized rocky planets with atmospheres can’t last. “You are either going to be big enough to hold on to your atmosphere, or if you are intermediate in size, then you are probably not big enough and you are going to lose it all pretty quickly,” she said. “It’s like a tug of war; it’s really hard to stay in the middle.”

Though some kind of atmosphere loss is a reasonable guess, it is just one of three general ideas, said Sara Seager, an astronomer at MIT who is deputy science director for the TESS mission. Another theory holds that the gap results directly from planetary genesis, maybe because of the location or makeup of the gas and dust left over from the star’s birth. Or, as a third theory proposes, planets’ own cooling processes might cause their atmospheres to evaporate, an effect called “core-powered mass loss.” Akash Gupta and Hilke Schlichting of the University of California, Los Angeles demonstrated in research last year that as planets of certain sizes radiate heat from within into space, their atmosphere is blown away, which could send them to the other side of the radius gap.

The gap adds detail to emerging statistical patterns. In many exoplanet systems, as in our own backyard, astronomers are finding that smaller worlds tend to orbit close to their host stars, and bigger planets are more distant. Small planets’ proximity to their stars could be one reason they are small, Seager said. They could begin big like their far-flung brethren, but lose their atmospheres, and thereby lots of mass, to the searing heat and ultraviolet radiation of their stars.

Scientists think something like this happened to Mars. It started out with a thicker atmosphere, but once it lost its protective magnetic field, the sun was free to slowly blow that atmosphere away. Even Earth is still losing some of its hydrogen shell, Seager said.

“Some of these other systems might have even more severe early histories,” she said. “In the future we want to have a look at the atmospheres, and maybe that will give us some insight.”

As for the assorted exoplanets’ makeup, Seager said astronomers can’t tell yet what most of them are like inside. But people are trying. Planets two to four times Earth’s size, nicknamed super-Earths, or sometimes mini-Neptunes, are especially debated. Some astronomers think they are rock balls shrouded in thick atmospheres of hydrogen gas, while others argue they are shrouded in water, whether solid, liquid or vapor.

Last month, astronomers led by Li Zeng, a former student of Seager’s now at Harvard University, reported the results of computer simulations suggesting that these common planets are water worlds. Some could be up to 50 percent water, which would come in a variety of exotic forms. The water might be fluid all the way down, or compressed into high-pressure ices such as the newly discovered phase called “superionic ice” thousands of kilometers below the surface, Zeng said.

“These high-pressure ices are essentially like silicate rocks in Earth’s deep mantle, hot and hard,” Zeng wrote in an email. “These oceans are unfathomable, bottomless. They are different worlds compared to our own Earth.”

Zeng said these super-Earths or mini-Neptunes might be more common than the planets of our solar system, and there might indeed be no place like home. But Dragomir is more circumspect. She noted that Kepler had almost a decade to pick out patterns among its planet cornucopia, but TESS is just getting started. Whereas Kepler studied a small patch of sky in the constellation Cygnus, TESS will survey the whole sky, an area 400 times larger than Kepler’s field of view. And TESS will focus on bright, nearby stars, which will be possible to study with ground-based telescopes for follow-up observations.

Dragomir is waiting for TESS’s long-term observations of planets that orbit their stars at great distances. These worlds are harder to see because of simple geometry. TESS detects a planet’s presence by studying blips in a star’s brightness, which indicate something passing in front of it. Planets orbiting at great distances from their star take a long time to cross in front, creating a prolongued blip that is harder to pick up, and they dim the starlight less.

Drawing firm conclusions about which kinds of planets do and don’t form at this point, she said, “is like looking in 1 percent of the haystack and saying, ‘Oh, there’s no needle.’”

Categories: Science News

In Ecology Studies and Selfless Ants, He Finds Hope for the Future

Wed, 2019-05-15 21:50

No one else in biology has ever had a career quite like that of Edward O. Wilson. One of the world’s leading authorities on ants, an influential evolution theorist and an author who is at once prolific, bestselling and highly honored, E. O. Wilson — his first name comes and goes from bylines but the middle initial is ever-present — has over several decades been at the center of scientific controversies that spilled out of the journals and into wider public awareness. Among activists in the environmental movement, Wilson is the elder statesman, the intellectual patriarch whose writings are foundational to the campaign. Soon to celebrate his 90th birthday, he shows no sign of losing his enthusiasm for the fray.

“I’ll tell you something about Ed — he’s a bit of an intellectual grenade thrower,” observed David Sloan Wilson (no relation), an evolutionary biologist at Binghamton University in New York. “He likes to be a provocateur. That’s unusual in someone as established as he is.”

As a teenager, Edward Osborne Wilson began his career by identifying and classifying every ant species in Alabama, his home state. By the age of 29, Wilson had achieved tenure at Harvard University for his work on ants, evolution and animal behavior. Wider academic fame came to him in the 1960s, when he and the noted community ecologist Robert MacArthur developed the theory of island biogeography, which posited how life established itself on isolated, barren outcroppings of land in the mid-ocean. That study would become a pillar of the then-formative discipline of conservation biology.

In 1975, Wilson made waves with Sociobiology: The New Synthesis, a volume in which he took all he knew about insect behavior and applied it to vertebrates — humans among them. This work suggested that many of the social behaviors observed in people, including virtuous traits like altruism, could be attributed to natural selection. Wilson soon found himself accused of providing intellectual succor to racists and genetic determinists. Demonstrations in the streets of Cambridge demanded that Wilson be fired. The controversy muted only after Wilson won a Pulitzer Prize for nonfiction in 1979 for On Human Nature, his popularized version of Sociobiology.

Until that first Pulitzer, Wilson — a fluid and elegant writer — had mostly published for the academy. From then on, Wilson began addressing the popular audience, translating biology and his own research into an accessible form. Over the years, he’d win another Pulitzer for The Ants (1990), co-authored with the behavioral biologist Bert Hölldobler. He’d also produce a memoir, a novel and more than two dozen nonfiction works, many as contentious as Sociobiology.

Contentious or not, Wilson’s books have mostly addressed one theme: that we must know natural history and evolutionary theory to fully understand humanity’s future on the planet. In his 1986 manifesto Biophilia, for example, he suggested that humans have an innate biological need to be in nature and to “affiliate with other forms of life.” In Half Earth: Our Planet’s Fight for Life (2016), he offered his personal prescription to end the destruction of the world’s biodiversity: Governments should set aside half the planet as a nature reserve.

Two months ago saw the arrival of his latest work, Genesis: The Deep Origins of Society, an update and reconsideration of some ideas on evolution introduced in Wilson’s earlier books. Genesis, he insists, is “one of the most important books I’ve written.”

To discuss Genesis, and to learn Wilson’s thoughts on the new controversies the book might ignite, Quanta visited him last month at his home in Lexington, Massachusetts. An edited and condensed version of that three-hour conversation follows.

Is it true that you will have a 90th birthday in June?

Yes. And I can’t believe it! I feel like I’m about 35 or 45. I have the same enthusiasm, and I get out of bed in the morning with the same ease or difficulty I always had. I don’t know what happened. When I was 40, I just assumed that I would be doing the same things at 90. And I am.

I write a book a year. I’m still taking natural history trips. This past month I was to go to Gorongosa National Park in Mozambique to do fieldwork on my next book. However, there was this tragedy there, this typhoon that caused so many deaths and so much damage. My friends in Mozambique thought I should wait.

So here I am in Lexington, working on the book, my 32nd. Even if I can’t travel right at this moment, there’s quite a lot I can do on it from here.

What’s its focus?

Ecosystems. Last year, I was asked by the Massachusetts Institute of Technology to give a couple of lectures on ecosystems. In preparing my talks, I saw how little we know about them.

I sort of bumbled my way through, and I came to think that understanding ecosystems and what threatens their equilibrium is going to be the next big thing in biological science. To save the environment, we have to find out how to save the ecosystems.

You’re a bit of a workaholic, aren’t you?

Well, yes. I don’t think being a workaholic is a bad thing. When I was 13 years old, during the first year of the Second World War, there was a shortage of newspaper delivery boys in my hometown of Mobile, Alabama. The 18-year-olds were all at war. So I took a job delivering 420 copies of the Mobile Press Registrar each morning. I’d take all the papers I could, load them on my bike and deliver them. Then I’d go back to the house, get another stack and deliver those. I’d make it home by 7 a.m., eat breakfast and go to school.

I thought that was normal. I’ve always made it my custom to work long and hard. Doing something unusual requires hard work. I’ve written huge books. That’s hard work.

What do you consider your most significant achievements?

Do you want me to brag? OK, here goes: I created a couple of new ideas and disciplines. The theory of island biogeography became a foundation of modern conservation biology. And then I did things like break the chemical code of the ants where I worked out with chemists and mathematicians how ants speak to one another.

I invented the Encyclopedia of Life, putting out all the information on all known species. I invented, named and gave the first synthesis of sociobiology, which in turn gave birth to the field of evolutionary psychology.

It is said that one of your great contributions has been as a synthesizer of scientific ideas. Accurate or not?

I’d say I’ve been a synthesizer. I love to look at some aspect of nature, learn everything accessible, gather it all together and see if I can screen out something of relevance for a big question.

Give us an example of where you’ve done this.

My fourth book, The Insect Societies, is one. In the 1960s, you had many dedicated entomologists working on understanding social insects — bees, wasps, ants. But we didn’t have a summary of all that was known and what, together, it meant. So in 1971, I published The Insect Societies, which was very successful. In fact, the book was a finalist for the National Book Award, which surprised me. Till then, I had never thought of what I was doing as literature. The book’s success led me to thinking I should next do a similar review of vertebrates — mammals, reptiles, amphibians, fishes.

At that time, you saw a lot of good biologists working on the social behavior of different types of vertebrates — people like Jane Goodall and Dian Fossey. I thought it time to incorporate their newer research into a more general theory, linking that to what I and others had developed for invertebrates. That synthesis, which was published in 1975 as Sociobiology, included new research on the social behavior of primates.

In fact, at the end of the book, I had an entire chapter on Homo sapiens, a primate that had gone through many steps of evolution. I suggested that a lot of human social behavior could be explained by a natural selection of certain activities and steps, leading to ever more complex group selection.

This wasn’t anything new. Darwin himself had introduced the idea with impeccable logic. What was new was that I was bringing modern population genetics and evolutionary theory into the study of human social behavior. I was seeking to bring the biological and social sciences together so that we could better understand human nature.

When you wrote that final chapter, did you realize you were stepping on a landmine?

At the time, not at all, no. I thought there would be accolades because it would add to the social sciences a new armamentarium of background information, comparative analysis, terminology and general conception that could illuminate previously unexamined aspects of human social behavior.

But the early 1970s, when the book was written, was a time of heated political controversy, much of it related to the war in Vietnam, civil rights and anger about economic inequality. At Harvard, some of my colleagues — I won’t mention their names here — had problems with the idea that there could be instincts in humans. They saw Sociobiology as dangerous, full of potential for racism and eugenics.

Now, my book had nothing to do with racism, but these people forged their own account of how the ideas might be used.

Did they think that Sociobiology could be used to support racist ideas about genetics?

I think you might describe their views that way. In any event, protests began. Things got very bad.

When I gave a talk at the Harvard Science Center on the subject, a mob gathered outside the front of the building. I had to be escorted by police in the back in order to get to the lecture room to give my lecture. When I appeared at a meeting of the American Association for the Advancement of Science (AAAS), some protesters took over the podium to shout their objections, and one of them came from behind me and dumped a pitcher of ice water on my head.

What did you do?

I dried myself off and continued without a break. That was the only thing I could do.

Though you don’t discuss your politics widely, one senses you to be a person of generally liberal beliefs. How did you feel about being characterized as this arch reactionary?

You want to know how I felt? I was afraid this might disturb my family, my wife and daughter. One day there was a mob in Harvard Square, stopping traffic and demanding that the university fire me because of my “racism.” It never did get to my family, though. I knew I was right. I knew I’d just have to weather the storm.

Sure enough, after a while the ideas of the book began to percolate: that genetics is an effective way to understand many aspects of evolutionary biology and behavior. With time, the notion that this book was harmful began to fade, and more scientists wrote favorably about the approach. Some even undertook it in their own work.

What really ended this was two years later when I received the National Medal of Science from President Jimmy Carter. I also wrote and published a book on sociobiology for a broader audience, On Human Nature. It won the Pulitzer Prize for General Nonfiction.

Your recently published book, Genesis, picks up on some of the ideas introduced in Sociobiology. Among the questions you revisit is, “What is human nature?” You also ask, “Did selfishness drive human evolution?” I’m curious: Why write this book now?

The history is that in the early 1960s, I met a British geneticist, William D. Hamilton. He had this brilliant idea that social behavior originated with what is called “kin selection,” or “inclusive fitness,” where individuals within a group behaved altruistically toward those they shared the most genes with.

In kin selection, an individual might sacrifice their possessions, or even their lives, for the benefit of the relative with whom they shared the most genes. Thus, an individual might be more likely to sacrifice for a sibling than a cousin or nonrelatives. The ultimate result of kin selection would be a kind of altruism, though it would be limited to your kin group.

This idea soon became gospel in the evolutionary biology world. I had helped promote Hamilton’s work, but as time went on, I developed my doubts about it.

Certainly, in my own research, I’d observed sophisticated societies that evolved through group selection, where individuals would be altruistic for the sake of their group’s survival. The ants are an example. In fact, when you think about it, the creatures that dominate the earth are cooperative — ants, termites, humans.

Meanwhile, Martin Nowak, a Harvard applied mathematician, was entertaining similar questions. He and his colleague Corina Tarnita had been preparing their own paper detailing their misgivings about kin selection. We dovetailed our efforts, eventually producing a paper for the journal Nature where we asserted that Hamilton’s theory was fundamentally flawed. We felt it could not explain how complex societies arose.

Your Nature article, published in 2010, kicked off yet another round of academic warfare. A few months after the paper appeared, more than 130 evolutionary biologists — your colleagues — sent a letter to the editor disputing your thesis. Did you think, “Oh no, here we go again?”

Well, Nature’s editors had a different view. Before publication, they’d sent over an editor from London, and we had a whole seminar going over the issues in our paper. They have pretty high standards, and afterwards, they were satisfied that this was a soundly reasoned article — maybe it was wrong in some places that weren’t obvious, but they decided to print it. In fact, they liked it so much, they made it a cover story.

So why the uproar?

I was canceling, or trying to replace, a body of theory that had gained quite a few followers who’d applied it to their Ph.D.s and their CVs. Their careers depended on it. They had written articles and books and given seminars on it.

So they didn’t like me. They said, “It’s so obvious that it’s true. How can you deny it?” We said, “We have math models. Take a look.”

With the publication of Genesis, you are reopening old wounds. Were you looking to go one more round with your critics?

Yes and no. I did want to settle the questions about group selection for once and for all. I thought it was important to put our theory on a firm mathematical and evidential basis. Either that or dispose of it.

Genesis turns out to be one of the more important books I’ve written. The book shows that group selection is a phenomenon that can be exactly defined. I show that it has occurred at least 17 times.

Group selection is a big part of the great transitions of evolution, where life progressed from bacteriumlike organisms to cells with structures inside, and on to simple organisms that were collections of these cells, to the differentiated organisms forming groups and so on. I presented these transitions against the backdrop of group versus individual selection.

Now, there exists a succession of social behaviors that advanced society is based upon. With humans, our advancement was aided by the fact that we were bipedal, with free arms and grasping fingers, and that we first lived on the savanna, where frequent fires gave us precooked animals to eat. What’s more, we had a good long-term memory and a capacity for high levels of cooperation, with altruism being a strong motivating factor.

The Hamilton theory implies that a mechanism was going on when relatives got together and that they were more likely to form a group because of their shared genes. However, this explanation is filled with mathematical errors and difficulties. Some of our evolutionary success occurred because groups formed, and they tended to be altruistic. Genetic relationships or not, these groups often cooperated, which is part of why we Homo sapiens were successful.

Might you give us “the elevator pitch,” the summary conclusion of your theory?

It’s the way my colleague David Sloan Wilson puts it. He says that within groups, selfish individuals will defeat altruistic ones. However, in conflict, groups of altruistic individuals will defeat groups of selfish individuals.

You know, we’ve heard everything we can possibly hear about the destructive and negative aspects of human nature. There’s a lot of evidence that we evolved because of qualities we consider unifying and propitious for the future.

Dr. Wilson, in person, you are remarkably genial and polite. Why then are you a lightning rod for so much controversy?

Maybe it’s because I prefer ideas that are original over those that are just pleasing.

Your collaboration with Martin Nowak fascinates. Do you often partner with mathematicians?

I do. I think that mathematical models are a good way of thinking about complex quantitative and sometimes qualitative phenomena.

Mathematical models can predict these things with precision. Biological research tests those models. When I’m trying to build an exact testable theory, as I was in Genesis, I’ll give the applied mathematicians my input, and, with luck, they’ll take hold of a problem.

I find this approach exciting. Partly because of my work with Nowak, I’ve come to believe that a whole new science is emerging that will combine natural history in the field with mathematical modeling and experiments similar to those conducted in a laboratory.

This kind of science will be more interesting to the public and attractive to young people who wish to enter careers in science and technology. It will also give us a firmer base on which to save the natural world.

When you are considering a mathematician to collaborate with, what are the qualities you seek?

The same I would look for in a plumber or a building contractor. I want them to be the best at what they do.

Where else in your career have you entered such partnerships?

When I was working out a theory of pheromone transmission — how odors are transmitted among ants and moths — I collaborated with Bill Bossert, an applied mathematician who later received a named professorship at Harvard.

Earlier, I had gotten together with another brilliant mathematically trained ecologist, the late Robert MacArthur of Princeton University. Together we worked out the theory of island biogeography, which helped explain why there were certain numbers of species of different kinds of organisms on islands of different size.

Some of our data there had been collected years earlier when I’d gone to the South Pacific to study ant species. MacArthur was able to come in with the right model to figure out how my data could apply to the new question.

The theory of island biogeography made your career. But as your 90th birthday approaches, do you think about what you’d like most to be remembered for?

You know, I’ve never really tried to think about that, honestly.

Well, perhaps I’d like to be remembered for obtaining such a great age and staying productive to the end — I’d like to be remembered for those things I’ve put my efforts into. I’d certainly like to be remembered for having created several new disciplines and bodies of theory that had an impact on science.

I don’t wish to be insensitive, but I am wondering if you ever think about death?

Oh, I’ve learned to live with mortality. My favorite line from Darwin was his last line to his family. He said, “I am not in the least afraid of dying.”

And I’m not either. I look on life as a story. It’s a series of events that have occurred, some of them momentous to you and a few other people. You made it through OK, you did this and that. And it could be written as a story. That’s what a life means.

Too many people think of it as a waiting station for the next life up. Or maybe they’ll find a way to extend this life by another 10 percent or 20 percent. I don’t think that’s a very smart way to live.

So I’m not afraid. I’m just really anxious to finish this book I’m writing now on ecosystems. And to figure out how I’ll get to Mozambique to do the fieldwork.

Categories: Science News

How Feynman Diagrams Revolutionized Physics

Tue, 2019-05-14 20:55

As one of the most famous physicists of the 20th century, Richard Feynman was known for a lot. Early in his career, he contributed to the development of the first atomic bomb as a group leader of the Manhattan Project. Hans Bethe, the scientific leader of the project who won a Nobel Prize in Physics in 1967 (two years after Feynman did), has been quoted on what set his protégé apart: “There are two types of genius. Ordinary geniuses do great things, but they leave you room to believe that you could do the same if only you worked hard enough. Then there are magicians, and you can have no idea how they do it. Feynman was a magician.”

In his 1993 biography Genius, James Gleick called Feynman “brash,” “ebullient” and “the most brilliant, iconoclastic and influential physicist of modern times.” Feynman captured the popular imagination when he played the bongo drums and sang about orange juice. He was a fun-loving, charismatic practical joker who toured America on long road trips. His colleague Freeman Dyson described him as “half genius and half buffoon.” At times, his oxygen-sucking arrogance rubbed some the wrong way, and his performative sexism looks very different to modern eyes. Feynman will also be remembered for his teaching: The lectures he delivered to Caltech freshmen and sophomores in 1962 set the gold standard in physics instruction and, when later published as a three-volume set, sold millions of copies worldwide.

What most people outside of the physics community are likely to be least familiar with is the work that counts as Feynman’s crowning scientific achievement. With physicists in the late 1940s struggling to reformulate a relativistic quantum theory describing the interactions of electrically charged particles, Feynman conjured up some Nobel Prize-winning magic. He introduced a visual method to simplify the seemingly impossible calculations needed to describe basic particle interactions. As Gleick put it in Genius, “He took the half-made conceptions of waves and particles in the 1940s and shaped them into tools that ordinary physicists could use and understand.” Through the work of Feynman, Dyson, Julian Schwinger and Sin-Itiro Tomonaga, a new and improved theory of quantum electrodynamics was born.

Feynman’s lines and squiggles, which became known as Feynman diagrams, have since “revolutionized nearly every aspect of theoretical physics,” wrote the historian of science David Kaiser in 2005. “In the same way that computer-enabled computation might today be said to be enabling a genomic revolution, Feynman diagrams helped to transform the way physicists saw the world, and their place in it.”

To learn more about Feynman diagrams and how they’ve changed the way physicists work, watch our new In Theory video:

Decades later, as Natalie Wolchover reported in 2013, “it became apparent that Feynman’s apparatus was a Rube Goldberg machine.” Even the collision of two subatomic particles called gluons to produce four less energetic gluons, an event that happens billions of times a second during collisions at the Large Hadron Collider, she wrote, “involves 220 diagrams, which collectively contribute thousands of terms to the calculation of the scattering amplitude.” Now, a group of physicists and mathematicians is studying a geometric object called an “amplituhedron” that has the potential to further simplify calculations of particle interactions.

Meanwhile, other physicists hope that emerging connections between Feynman diagrams and number theory can help identify patterns in the values generated from more complicated diagrams. As Kevin Hartnett reported in 2016, understanding these patterns could make particle calculations much simpler, but like the amplituhedron approach, this is still a work in progress.

“Feynman diagrams remain a treasured asset in physics because they often provide good approximations to reality,” wrote the Nobel Prize-winning physicist Frank Wilczek three years ago. “They help us bring our powers of visual imagination to bear on worlds we can’t actually see.”

If you liked the fifth and final episode from season two of Quanta’s In Theory video series, you may also enjoy our previous videos on universalityquantum gravityemergence and turbulence.

Categories: Science News

Out of a Magic Math Function, One Solution to Rule Them All

Mon, 2019-05-13 22:20

Three years ago, Maryna Viazovska, of the Swiss Federal Institute of Technology in Lausanne, dazzled mathematicians by identifying the densest way to pack equal-sized spheres in eight- and 24-dimensional space (the second of these in collaboration with four co-authors). Now, she and her co-authors have proved something even more remarkable: The configurations that solve the sphere-packing problem in those two dimensions also solve an infinite number of other problems about the best arrangement for points that are trying to avoid each other.

The points could be an infinite collection of electrons, for example, repelling each other and trying to settle into the lowest-energy configuration. Or the points could represent the centers of long, twisty polymers in a solution, trying to position themselves so they won’t bump into their neighbors. There’s a host of different such problems, and it’s not obvious why they should all have the same solution. In most dimensions, mathematicians don’t believe this is remotely likely to be true.

But dimensions eight and 24 each contain a special, highly symmetric point configuration that, we now know, simultaneously solves all these different problems. In the language of mathematics, these two configurations are “universally optimal.”

The sweeping new finding vastly generalizes Viazovska and her collaborators’ previous work. “The fireworks have not stopped,” said Thomas Hales, a mathematician at the University of Pittsburgh who in 1998 proved that the familiar pyramidal stacking of oranges is the densest way to pack spheres in three-dimensional space.

Eight and 24 now join dimension one as the only dimensions known to have universally optimal configurations. In the two-dimensional plane, there’s a candidate for universal optimality — the equilateral triangle lattice — but no proof. Dimension three, meanwhile, is a zoo: Different point configurations are better in different circumstances, and for some problems, mathematicians don’t even have a good guess for what the best configuration is.

“You change the dimension or you change the problem a little bit and then things may be completely unknown,” said Richard Schwartz, a mathematician at Brown University in Providence. “I don’t know why the mathematical universe is built this way.”

Proving universal optimality is much harder than solving the sphere-packing problem. That’s partly because universal optimality encompasses infinitely many different problems at once, but also because the problems themselves are harder. In sphere packing, each sphere cares only about the location of its nearest neighbors, but for something like electrons scattered through space, every electron interacts with every other electron, no matter how far apart they are. “Even in light of the earlier work, I would not have expected this to be possible to do,” Hales said.

“I’m very, very impressed,” said Sylvia Serfaty, a mathematician at New York University. “It’s at the level of the big 19th-century mathematics breakthroughs.”

A Magic Certificate

It might seem strange that dimensions eight and 24 should behave differently from, say, dimension seven or 18 or 25. But mathematicians have long known that packing objects into space works differently in different dimensions. For instance, consider a higher-dimensional sphere, defined simply as the collection of points some fixed distance from a center point. If you compare the sphere’s volume to that of the smallest cube that fits around it, the sphere fills up less and less of the cube as you go up in dimension. If you wanted to ship an eight-dimensional soccer ball in the smallest possible box, the ball would fill less than 2 percent of the box’s volume — the rest would be wasted space.

In each dimension higher than three, it’s possible to construct a configuration analogous to the pyramidal orange arrangement, and as the dimension increases, the gaps between the spheres grow. When you hit dimension eight, there’s suddenly enough room to fit new spheres into the gaps. Doing so produces a highly symmetric configuration called the E8 lattice. Likewise, in dimension 24, the Leech lattice arises from fitting extra spheres into the gaps in another well-understood sphere packing.

For reasons mathematicians don’t fully understand, these two lattices crop up in one area of mathematics after another, from number theory to analysis to mathematical physics. “I don’t know of a single root cause for everything,” said Henry Cohn, of Microsoft Research New England in Cambridge, Mass., one of the new paper’s five authors.

For more than a decade, mathematicians have had strong numerical evidence suggesting that E8 and the Leech lattice are universally optimal in their respective dimensions — but until recently they had no idea how to prove it. Then in 2016, Viazovska took the first step by proving that these two lattices are the best possible sphere packings (she was joined, for the Leech lattice proof, by Cohn and the other three authors of the new paper: Abhinav Kumar, Stephen Miller of Rutgers University and Danylo Radchenko of the Max Planck Institute for Mathematics in Bonn, Germany).

While Hales’ proof for the three-dimensional case filled hundreds of pages and required extensive computer calculations, Viazovska’s E8 proof came in at just 23 pages. The core of her argument involved identifying a “magic” function (as mathematicians have come to call it) that provided what Hales called a “certificate” that E8 is the best sphere packing — a proof that might be hard to come up with, but that once found carries instant conviction. For example, if someone asked you whether any real number x makes the polynomial x2 – 6x + 9 negative, you might be hard-pressed to reply. But if you realized that the polynomial equals (x – 3)2, you would immediately know that the answer is no, since squared numbers are never negative.

Viazovska’s magic function method was powerful — almost too powerful, in fact. The sphere-packing problem only cares about interactions between nearby points, but Viazovska’s approach seemed as if it might work for long-range interactions as well, like those between distant electrons.

High-Dimensional Uncertainty

To show that a configuration of points in space is universally optimal, one must first specify the universe in question. No point configuration is optimal with respect to every single goal: For instance, when an attractive force acts on the points, the lowest-energy configuration will not be some lattice but just a massive pileup, with all the points at the same spot.

Viazovska, Cohn and their collaborators restricted their attention to the universe of repulsive forces. More specifically, they considered ones that are completely monotonic, meaning (among other things) that the repulsion is stronger when points are closer to each other. This broad family includes many of the forces most common in the physical world. It includes inverse power laws — such as Coulomb’s inverse square law for electrically charged particles — and Gaussians, the bell curves that capture the behavior of entities with many essentially independent repelling parts, such as long polymers. The sphere-packing problem sits at the outer edge of this universe: The requirement that the spheres not overlap translates into an infinitely strong repulsion when their center points are closer together than the diameter of the spheres.

For any one of these completely monotonic forces, the question becomes, what is the lowest-energy configuration — the “ground state” — for an infinite collection of particles? In 2006, Cohn and Kumar developed a method for finding lower bounds on the energy of the ground state by comparing the energy function to smaller “auxiliary” functions with especially nice properties. They found an infinite supply of auxiliary functions for each dimension, but they didn’t know how to find the best auxiliary function.

In most dimensions, the numerical bounds Cohn and Kumar found bore little resemblance to the energy of the best-known configurations. But in dimensions eight and 24, the bounds came astonishingly close to the energy of E8 and the Leech lattice, for every repulsive force Cohn and Kumar tried their method on. It was natural to wonder whether, for any given repulsive force, there might be some perfect auxiliary function that would give a bound exactly matching the energy of E8 or the Leech lattice. For the sphere-packing problem, that’s exactly what Viazovska did three years ago: She found the perfect, “magic” auxiliary function by looking among a class of functions called modular forms whose special symmetries have made them objects of study for centuries.

When it came to other repelling-point problems, such as the electron problem, the researchers knew what properties each magic function would need to satisfy: It would have to take on special values at certain points, and its Fourier transform — which measures the function’s natural frequencies — would need to take on special values at other points. What they didn’t know, in general, was whether such a function actually exists.

It’s usually easy to construct a function that does what you want at your favorite points, but it’s surprisingly tricky to control a function and its Fourier transform at the same time. “When you impose your will on one of them, the other one does something that’s totally different from what you wanted,” Cohn said.

In fact, this persnicketiness is none other than the famous uncertainty principle from physics in disguise. Heisenberg’s uncertainty principle — which says that the more you know about a particle’s position, the less you can know about its momentum, and vice versa — is a special case of this general principle, since a particle’s momentum wave is the Fourier transform of its position wave.

In the case of a repulsive force in dimension eight or 24, Viazovska made a daring conjecture: that the limitations the team wanted to place on their magic function and its Fourier transform lay precisely on the border between the possible and the impossible. Any more limitations, she suspected, and no such function could exist; fewer limitations, and many functions could exist. In the situation the team cared about, she proposed, there should be exactly one function that fit.

“This is, I think, one of the great things about Maryna,” Cohn said. “She’s very insightful and also very bold.”

At the time, Cohn was skeptical — Viazovska’s guess seemed too good to be true — but the team eventually proved her right. Not only did they show that there exists exactly one magic function for each repulsive force, but they gave a recipe for how to make it. As with sphere packing, this construction provided an immediate certificate of the optimality of E8 and the Leech lattice. “It’s kind of a monumental result,” Schwartz said.

The Triangle Lattice

Beyond settling the universal optimality problem, the new proof answers a burning question many mathematicians have had since Viazovska solved the sphere-packing problem three years ago: Just where did her magic function come from? “I think many people were puzzled,” Viazovska said. “They asked, ‘What is the meaning here?’”

In the new paper, Viazovska and her collaborators showed that the sphere-packing magic function is the first in a sequence of modular-form building blocks that can be used to construct magic functions for every repulsive force. “Now it has many brothers and sisters,” Viazovska said.

It still feels somewhat miraculous to Cohn that everything worked out so neatly. “There are some things in mathematics that you do by persistence and brute force,” he said. “And then there are times like this where it’s like mathematics wants something to happen.”

The natural next question is whether these methods can be adapted to prove universal optimality for the only other clear candidate out there: the equilateral triangle lattice in the two-dimensional plane. For mathematicians, the fact that no one has come up with a proof in this simple setting has been a “big embarrassment for the whole community,” said Edward Saff, a mathematician at Vanderbilt University in Nashville.

Unlike E8 and the Leech lattice, the two-dimensional triangular lattice shows up all over the place in nature, from the structure of honeycombs to the arrangement of vortices in superconductors. Physicists already assume this lattice is optimal in a wide range of contexts, based on a mountain of experiments and simulations. But, Cohn said, no one has a conceptual explanation for why the triangular lattice should be universally optimal — something a mathematical proof would hopefully provide.

Dimension two is the only dimension other than eight and 24 in which Cohn and Kumar’s numerical lower bounds work well. This strongly suggests that there should be magic functions in dimension two as well. But the team’s method for constructing magic functions is unlikely to carry over to this new domain: It relies heavily on the fact that the distances between points in E8 and the Leech lattice are especially well-behaved numbers, which is not the case in dimension two. That dimension “seems beyond humanity’s abilities right now,” Cohn said.

For the time being, mathematicians are celebrating their new insight into the strange worlds of eight- and 24-dimensional space. It is, Schwartz said, “one of the best things I’ll probably see in my lifetime.”

Categories: Science News