James' World 2

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Here’s a math problem that everybody can solve: What is 1 − 1? 0. So far so good. If we then add a 1, the sum grows, but if we subtract yet another 1, we’re back at 0. Let’s say, we keep doing this forever:

1 – 1 + 1 – 1 + 1 – 1 + …

What is the resulting sum? The question seems simple, silly even, but it puzzled some of the greatest mathematicians of the 18th century. Paradoxes surround the problem because multiple seemingly sound arguments about the sum reach radically different conclusions. The first person to deeply investigate it thought it explained how God created the universe. Its resolution in modern terms illustrates that mathematics is a more human enterprise than sometimes appreciated.

Take a guess at what you think the infinite sum equals. I’ll give you multiple choices:

A. 0
B. 1
C. ½
D. It does not equal anything

The argument for 0 comes naturally if we include suggestive parentheses:

(1 – 1) + (1 – 1) + (1 – 1) + …

Recall that in mathematics, the order of operations dictates that we evaluate those inside parentheses before evaluating those outside. Each (1 − 1) cancels to 0, so the above works out to 0 + 0 + 0 +…, which clearly amounts to nothing.

Yet a slight shift of the brackets yields a different result. If we set aside the first 1, then the second and third terms also cancel, and the fourth and fifth cancel:

1 + (–1 + 1) + (–1 + 1) + (–1 + 1) + …

Again, all the parentheticals add up to 0, but we have this extra positive 1 at the beginning, which suggests that the whole expression sums to 1.

Italian monk and mathematician Luigi Guido Grandi first investigated the series (the sum of infinitely many numbers) in 1703. Grandi, whom this particular series is now named after, observed that by merely shifting around parentheses he could make the series sum to 0 or 1. According to math historian Giorgio Bagni, this arithmetic inconsistency held theological significance for Grandi, who believed it showed that creation out of nothing was “perfectly plausible.

The series summing to both 0 and 1 seems paradoxical, but surely option C (½) is no less troubling. How could a sum of infinitely many integers ever yield a fraction? Yet ultimately, Grandi and many prominent 18th-­century mathematicians after him thought the answer was ½. Grandi argued for this with a parable: Imagine that two brothers inherit a single gem from their father, and each keeps it in their own museum on alternating years. If this tradition of passing the gem back and forth carried on with their descendants, then the two families would each have ½ ownership over the gem.

As proofs go, I wouldn’t recommend putting the gem story on your next math test. German mathematician Gottfried Wilhelm Leibniz agreed with Grandi’s conclusion, but he tried to support it with probabilistic reasoning. Leibniz argued that if you stop summing the series at a random point, then your sum up to that point will be either 0 or 1 with equal probability, so it makes sense to average them to ½. He thought the result was correct but acknowledged that his argument was more “metaphysical than mathematical.”

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After the first year of being divorced, I knew my two kids needed their own phones. It was becoming increasingly difficult to connect with my elementary-aged kids while they were with the other parent.

Adjusting to sharing time with them week to week meant long gaps of time without talking to my kids. This was heartbreaking not only for me but also for my kids.

Whenever I asked my ex how the kids were doing and if I could talk to them, even for a few minutes, it started a huge argument. He gave me a time slot on certain days when I could call and talk to my children. I knew this situation wouldn’t improve, and I had to do something about it, so I got them both phones.

I would’ve waited longer had it not been because of the divorce

If we had stayed married, I might have waited a few more years to give my kids their first phones, but as a divorced parent, the phone was a must-get. I’ll never forget the relief I felt when my kids had their first phones.

My daughter got her first phone in first grade. My son was in fourth grade. Phones are not a one-size-fits-all situation, and I know the thought of phones and kids can be scary for a lot of parents, but for me, I only wished I had gotten my kid’s phones earlier.

Now, I can talk with my kids daily without a middleman. I could send them my love in a simple text with a sunflower or tulip emoji for my daughter or a soccer ball for my son. I could send pictures of a memory from the week before. I could share silly jokes to make them smile, and my caring words would be like a virtual hug.

We text all the time now

My daughter texts me about everything, and I’ll admit I love it. She texts me how she feels, when she has a tummy ache, and shares a video she saw on YouTube that is cute or funny.

She types longer messages than my son, who is now almost 12, and typically replies with a “Good” or just the letter “K.” He texts longer messages when he needs something, such as asking me to bring him 10 dollars or a new pair of socks when he is at a friend’s house. Or he’ll send a text about staying after school for an hour so he can hang out with his friends.

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The author gave her kids phones so she could stay in touch with them while they were at their dad’s house. FG Trade Latin/Getty Images

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At least one-quarter of people who have severe brain injuries and cannot respond physically to commands are actually conscious, according to the first international study of its kind.

Although these people could not, say, give a thumbs-up when prompted, they nevertheless repeatedly showed brain activity when asked to imagine themselves moving or exercising.

“This is one of the very big landmark studies” in the field of coma and other consciousness disorders, says Daniel Kondziella, a neurologist at Rigshospitalet, the teaching hospital for Copenhagen University.

The results mean that a substantial number of people with brain injuries who seem unresponsive can hear things going on around them and might even be able to use brain–computer interfaces (BCIs) to communicate, says study leader Nicholas Schiff, a neurologist at Weill Cornell Medicine in New York City. BCIs are devices implanted into a person’s head that capture brain activity, decode it, and translate it into commands that can, for instance, move a computer cursor. “We should be allocating resources to go out and find these people and help them,” Schiff says. The work was published on August 14 in The New England Journal of Medicine.

The study included 353 people with brain injuries caused by events such as physical trauma, heart attacks, or strokes. Of these, 241 could not react to any of a battery of standard bedside tests for responsiveness, including one that asks for a thumbs-up; the other 112 could.

Everyone enrolled in the study underwent one or both of two types of brain scan. The first was functional magnetic resonance imaging (fMRI), which measures mental activity indirectly by detecting the oxygenation of blood in the brain. The second was electroencephalography (EEG), which uses an electrode-covered cap on a person’s scalp to measure brain-wave activity directly. During each scan, people were told to imagine themselves playing tennis or opening and closing their hand. The commands were repeated continuously for 15 to 30 seconds, then there was a pause; the exercise was then repeated for six to eight command sessions.

Of the physically unresponsive people, about 25 percent showed brain activity across the entire exam for either EEG or fMRI. The medical name for being able to respond mentally but not physically is cognitive motor dissociation. The 112 people in the study who were classified as responsive did a bit better on the brain-activity tests, but not much: only about 38 percent showed consistent activity. This is probably because the tests set a high bar, Schiff says. “I’ve been in the MRI, and I’ve done this experiment, and it’s hard,” he adds.

This isn’t the first time a study has found cognitive motor dissociation in people with brain injuries who were physically unresponsive. For instance, in a 2019 paper, 15 percent of the 104 people undergoing testing displayed this behavior. The latest study, however, is larger and is the first multi-center investigation of its kind. Tests were run at six medical facilities in four countries: Belgium, France, the United Kingdom and the United States.

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It turns out that a long, happy marriage resembles a slow-moving rom-com, one that plays out over decades.

The first few years of a marriage are rife with conflicts, but the emotional weather eventually changes, according to a 2018 study by psychology researchers at UC Berkeley. In time, humor—friendly teasing, jokes, and silliness—becomes more prevalent, and bickering and criticisms decline.

These findings, which must be among the sweetest to enter the crowded field of relationships research, were reached after psychologists analyzed videotaped interactions of 87 couples who had been married 15 to 35 years, and followed them over 13 years.

The study’s conclusions contradict an existing theory that positive emotions fade over time in a long relationship, point out the co-lead authors, Robert Levenson, a UC Berkeley psychology professor, and Alice Verstaen, a postdoctoral fellow at the VA Puget Sound health center. However, they align nicely with other recent longitudinal studies that show a U-shaped pattern of happiness in lengthy marriages. The questions of how unions change, and what triggers different twists and turns, are not settled, they write.

Importantly, jokes and gentle humor were not the only heroic behaviors that showed up in greater abundance in the marriages they followed. All the positive ways we can behave toward someone became more evident as the years passed, but primarily humor, enthusiasm, and validation (actively listening to and understanding your partner). Criticisms dropped off, as did the truly toxic, divorce-courting habits like stonewalling. Men demonstrated less anger, and women less contempt.

One outcome of the study was more in keeping with grimmer perceptions of marriages: Older couples were not more affectionate with each other. They either exchanged about the same number of caring statements and compliments through the years, or, wives offered fewer of them. But there was a silver lining here, too: Those trajectories, the researchers write, “offer support for the idea of love evolving as adults age.” Psychological studies—and, I’m guessing, anecdotal evidence from the long-married couples you know—have proposed that couples start off with a sense of passionate love that morphs into “companionate love” in time. Humor, they say, is arguably an expression of the second kind of devotion.

It’s worth keeping in mind that the study had several limitations, including a relatively small sample size, and a limited representation of marriage. It did not include same-sex marriages, for instance, or couples going to marriage counseling. It also only dealt in averages.

And of course, companionable humor is no guarantee of a relationship’s longevity. Anecdotally, it’s easy to point to couples that survived decades without a hint of shared laughter or goofiness, or ones that ended despite a healthy quotient of humor and compassionate behavior.

However, the study’s strengths, compared to similar research, give it credence too: The couples’ interactions were coded by an observer, so the results didn’t rely on self-reported measures. Also, the team quantified behaviors, not subjective constructs like “satisfaction” or “happiness.”

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Try jeering from the sidelines. Photo from Reuters/ Baz Ratner.

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Thanks to the development of modern agriculture, the germ theory of disease and pasteurization, as well as the advent of freezers, electric ovens, and fridges, millions of people can now access safe, disease-free food in many parts of the world. But despite these advances, foodborne illnesses endure. The Centers for Disease Control and Prevention estimates that around 48 million Americans—one in seven—get sick from food each year.

Harmful, illness-causing pathogens lurk and fester in many different foods, from salad greens, fruit, and vegetables to meat, eggs, rice, and seafood. Improper food preparation and storage, lack of hand hygiene, general unsanitary conditions, and insufficient cooking or reheating can all lead to food contamination. A lot of foodborne illnesses are caused by such improper handling. “The vast majority will be sporadic cases,” says Martin Wiedmann, a food scientist at the Cornell University College of Agriculture and Life Sciences. “One person gets sick because one thing went wrong.”

It is possible for contamination to occur during growth of crops, animal agriculture, and production procedures, however. Strict food hygiene laws and monitoring agencies (including the CDC, the Food and Drug Administration, and the U.S. Department of Agriculture’s Food Safety and Inspection Service) ensure that the U.S. has one of the best food safety records in the world. But unexpected events or mistakes on farms or in factories can still lead to foodborne pathogen contamination. When outbreaks of foodborne illnesses do happen, experts can use DNA fingerprinting of bacteria to quickly identify the origin and recall any food that might be contaminated, helping to contain the spread.

Common symptoms of foodborne illness include upset stomach, nausea, vomiting, diarrhea, fatigue, and fever. In severe cases, people can be hospitalized and die. Pregnant women, elderly adults, immunocompromised individuals, and young children are more susceptible to foodborne illnesses. Around 30 percent of all foodborne illness deaths worldwide occur in children under five years old.

The CDC recognizes 31 pathogens as common sources of foodborne illness. Bacteria, viruses, chemicals and even parasites (such as tapeworms) can all be culprits. These are some of the major microbes you should especially watch out for.

E. coli

Escherichia coli bacteria normally live peacefully inside your intestines without harming you. If ingested, however, these bacteria can infect other areas of the body, causing diarrhea, vomiting, and stomach pain. Some severe strains (such as those that produce so-called Shiga toxin) can lead to life-threatening dehydration and serious kidney damage.

E. coli can be expelled from the intestines in feces, meaning that unsanitary bathroom conditions and poor hygiene habits can all cause contamination of food. If a person gets such bacteria on their hands from direct or indirect contact with fecal matter, and then they go to prepare food or handle kitchen utensils with unwashed hands, for example, the E.coli can easily spread and become foodborne. Unclean water and unpasteurized beverages (such as raw milk) can also spread E. coli. Farmed produce also carries an E. coli risk – particularly if wildlife or livestock feces come into contact with the bacterium.

“E. coli is often associated with cattle and other ruminants,” says Martin Bucknavage, food safety and quality specialist at the Pennsylvania State University’s Penn Extension, which focuses on agriculture. “It could be cows, sheep, even white-tailed deer—it’s in their intestinal tract.” Contamination can occur if an infected animal’s feces get into water supplies and crops. This route is thought to have caused a number of E. coli outbreaks in produce, such as one in spinach in 2006. But this type of pathogen exposure has decreased in recent years, partly because farmers have been better at controlling livestock waste, Bucknavage says. “The meat industry has done a lot of work to try to minimize it.”

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You don’t need a quiet, harmonious household to raise the next Steve Jobs or Frida Kahlo.

Kids who grow up with parents who regularly disagree — in a constructive fashion — can become more creative adults, Adam Grant, an organizational psychologist at the Wharton School of the University of Pennsylvania recently told the “What Now? With Trevor Noah” podcast.

Such children can also become more mentally resilient, Grant wrote in a 2017 New York Times essay — a skill that highly successful adults often develop early in life, experts say.

By arguing, Grant doesn’t mean yelling and screaming. Instead, the idea is to model productive discussions for your children, ones in which both parties engage in conversation, hear each other out, and, ideally, reach a healthy consensus.

Growing up in a household with productive tension can show children that arguments don’t necessarily create lasting conflict, and can lead to creative ways of solving problems, said Grant.

“Instead of just defaulting or deferring to whatever an authority figure tells you, you realize, ’Well, there are two different authority figures … and they don’t agree,” he said during the podcast episode, which published on August 15. ”[It can] lead to cognitive complexity, but it can also lead to more courage when it comes to challenging the status quo because there’s not just one right answer.”

How constructive disagreements can foster creativity

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