What Is Infinity?


Albert Einstein famously said: “Two things are infinite: the universe and human stupidity. And I'm not sure about the universe.”

The notion of infinity has been pondered by the greatest minds over the ages, from Aristotle to German mathematician Georg Cantor. To most people today, it is something that is never-ending or has no limit. But if you really start to think about what that means, it might blow your mind. Is infinity just an abstract concept? Or can it exist in the real world?


Infinity is firmly rooted in mathematics. But according to Justin Moore, a math researcher at Cornell University in Ithaca, New York, even within the field there are slightly different uses of the word. “It's often referred to as a sort of virtual number at the end of the real number line,” he tells Mental Floss. “Or it can mean something too big to be counted by a whole number.”

There isn't just one type of infinity, either. Counting, for example, represents a type of infinity that is unbounded—what's known as a potential infinity. In theory, you can go on counting forever without ever reaching a largest number. However, infinity can be bounded, too, like the infinity symbol, for example. You can loop around it an unlimited number of times, but you must follow its contour—or boundary.

All infinities may not be equal, either. At the end of the 19th century, Cantor controversially proved that some collections of counting numbers are bigger than the counting numbers themselves. Since the counting numbers are already infinite, it means that some infinities are larger than others. He also showed that some types of infinities may be uncountable, as opposed to collections like the counting numbers.

"At the time, it was shocking—a real surprise," Oystein Linnebo, who researches philosophies of logic and mathematics at the University of Oslo, tells Mental Floss. "But over the course of a few decades, it got absorbed into mathematics."

Without infinity, many mathematical concepts would fall apart. The famous mathematical constant pi, for example, which is essential to many formulas involving the geometry of circles, spheres, and ellipses, is intrinsically linked to infinity. As an irrational number—a number that can't simply be expressed by a fraction—it's made up of an endless string of decimals.

And if infinity didn't exist, it would mean that there is a biggest number. "That would be a complete no-no," says Linnebo. Any number can be used to find an even bigger number, so it just wouldn't work, he says.


In the real world, though, infinity has yet to be pinned down. Perhaps you've seen infinite reflections in a pair of parallel mirrors on opposite sides of a room. But that's an optical effect—the objects themselves are not infinite, of course. "It's highly controversial and dubious whether you have infinities in the real world," says Linnebo. "Infinity has never been measured."

Trying to measure infinity to prove it exists might in itself be a futile task. Measurement implies a finite quantity, so the result would be the absence of a concrete amount. "The reading would be off the scale, and that's all you would be able to tell," says Linnebo.

The hunt for infinity in the real world has often turned to the universe—the biggest real thing that we know of. Yet there is no proof as to whether it is infinite or just very large. Einstein proposed that the universe is finite but unbounded—some sort of cross between the two. He described it as a variation of a sphere that is impossible to imagine.

We tend to think of infinity as being large, but some mathematicians have tried to seek out the infinitely small. In theory, if you take a segment between two points on a line, you should be able to divide it in two over and over again indefinitely. (This is the Xeno paradox known as dichotomy.) But if you try to apply the same logic to matter, you hit a roadblock. You can break down real-world objects into smaller and smaller pieces until you reach atoms and their elementary particles, such as electrons and the components of protons and neutrons. According to current knowledge, subatomic particles can't be broken down any further.


Black holes may be the closest we've come to detecting infinity in the real world. In the center of a black hole, a point called a singularity is a one-dimensional dot that is thought to contain a huge mass. Physicists theorize that at this bizarre location, some of the singularity's properties are infinite, such as density and curvature.

At the singularity, most of the laws of physics no longer work because these infinite quantities "break" many equations. Space and time, for example, are no longer two separate entities, and seem to merge.

According to Linnebo, though, black holes are far from being an example of a tangible infinity. "My impression is that the majority of physicists would say that is where our theory breaks down," he says. "When you get infinite curvature or density, you are beyond the area where the theory applies."

New theories may therefore be needed to describe this location, which seems to transcend what is possible in the physical world.

For now, infinity remains in the realm of the abstract. The human mind seems to have created the concept, yet can we even really picture what it looks like? Perhaps to truly envision it, our minds would need to be infinite as well.

How Long to Steep Your Tea, According to Science

The tea in your cabinet likely has vague instructions about how long to steep the leaves. Bigelow, for instance, suggests two to four minutes for black tea, and one to three minutes for green tea. According to Lipton, you should "try singing the National Anthem" while waiting for black tea leaves to infuse.

But while it's true that tea brewed for 30 seconds is technically just as drinkable as a forgotten mug of tea that's been steeping for 30 minutes, drinkable shouldn't be your goal. Taste and—depending on the tea you're drinking—antioxidant and caffeine levels all depend on the amount of time the leaves are in contact with the water. So how early is too early to pluck out a tea bag, and how long can you leave it in before passing the point of no return?


To achieve the perfect timing, you first need to understand the chemical process at work when you pour hot water over tea leaves. Black, green, white, and oolong tea all come from the leaves and buds of the same plant, Camellia sinensis. (Herbal teas aren't considered "true teas" because they don't come from C. sinensis.) The teas are processed differently: Green and white tea leaves are heated to dry them, limiting the amount of oxidation they get, while black and oolong tea leaves are exposed to oxygen before they're dried, creating the chemical reactions that give the tea its distinct color and flavor. Damaging the tea leaves—by macerating them, rolling them gently, or something in between—helps expose the chemicals inside their cells to varying levels of oxygen.

Both green and black teas contain a lot of the same chemical compounds that contribute to their flavor profiles and nutritional content. When the leaves are submerged in hot water, these compounds leach into the liquid through a process called osmotic diffusion, which occurs when there's fluid on both sides of a selectively permeable membrane—in this case, the tea leaf. Compounds on the surface of the leaf and in the interior cells damaged by processing will diffuse into the surrounding liquid until the compounds in both the leaf and the water reach equilibrium. In other words, if given enough time to steep, the liquid in your mug will become just as concentrated with tea compounds as the liquid in your tea leaves, and the ratio will stay that way.

Osmotic diffusion doesn't happen all at once—different compounds enter the water at different rates based on their molecular weight. The light, volatile chemicals that contribute to tea's aroma and flavor profile dissolve the fastest, which is why the smell from a bag of tea leaves becomes more potent the moment you dunk it in water. The next group of compounds to infuse with the water includes the micronutrients flavanols and polyphenols, which are antioxidants, and caffeine. They're followed by heavier flavanols and polyphenols such as tannins, which are the compounds responsible for tea's bitter flavor. (They're also what make your mouth feel dry after drinking a glass of wine.) Tea also has amino acids like theanine, which can offset the sharpness of tannins.

Water temperature is another factor to take into consideration when steeping your tea. High water temperature creates more kinetic energy, which encourages the compounds to dissolve. "The heat helps you to extract the compounds out of the tea leaves," Shengmin Sang, a North Carolina A&T State University researcher who studies the chemistry of tea, tells Mental Floss. "If you put it into cold water or low-temperature water, the efficiency to extract these compounds out of the leaves will be much lower." But not all water is equal: Bigelow Tea recommends using water at a rolling boil for black tea, and barely boiling water for green tea.


Osmotic diffusion takes place whether you use loose leaves or tea bags, but there are some notable differences between the two. When given room to expand, loose tea leaves swell to their full capacity, creating more room for water to flow in and extract all those desirable compounds. Tea that comes prepackaged in a bag, on the other hand, only has so much room to grow, and the quality suffers as a result. This is why some tea companies have started selling tea in roomier, pyramid-shaped bags, though the size matters more than the shape.

But even before the tea touches the water, there's a difference in quality. Loose leaf tea usually consists of whole leaves, while most teabags are filled with broken pieces of tea leaves called dust or fannings, which have less-nuanced flavors and infuse fewer antioxidants than whole leaves, no matter how long you let them steep.

So if you have a choice, go with loose leaf. But if tea bags are all you have on hand, don't bother adjusting your brewing method: The difference in taste and antioxidants isn't something that can be fixed with a few extra minutes, and according to Sang, you should follow the same steeping times for both tea bags and loose leaf.

To calculate the perfect brew times for what's in your mug, first consider what you want most out of your drink.


Suggested steeping time: 2 minutes, 30 seconds to 5 minutes

Tea leaves are packed with beneficial compounds. Research indicates that flavanols such as catechins and epicatechins, found in both green and black teas, help suppress inflammation and curb plaque build-up in arteries. Drinking tea may improve vascular reactivity, which dictates how well blood vessels adjust to stress. According an analysis of multiple tea-related studies published in the European Journal of Epidemiology in 2015, drinking three cups of tea a day reduces your risk of coronary heart disease by 27 percent, cardiac death by 26 percent, and total mortality by 24 percent. Polyphenolic antioxidants in tea may also protect against diabetes, depression, and liver disease.

Past research has shown that it takes 100 to 150 seconds to extract half the polyphenol content from green and black tea leaves. According to a study published in 2016 in the journal Beverages, you can get more polyphenols into your drink if you allow the leaves more time to steep. However, the returns may not be worth the extra effort: Most of the compounds the researchers measured after 10 minutes of steeping were extracted in the first 5 minutes.

Sang makes another argument for not waiting too long to drink your tea. Antioxidants are slightly unstable, which means they will eventually break down and lose their healthy properties after infusing with water. “After you extract the compounds from the tea bag, you can not keep the solution for too long,” he says. “Because these compounds are not stable, they will be oxidized. So if you brew it in the morning, then you drink it in the afternoon, that's not good.” This oxidation can occur even after the tea leaves are removed from the cup, so if your tea has been sitting out for a few hours, it's better to brew a new batch than to pop it in the microwave.


Suggested steeping time: 3 to 5 minutes

Though less potent than its rival coffee, a properly brewed cup of tea packs a caffeine punch. According to a 2008 study published in the Journal of Analytical Toxicology [PDF], letting your tea brew for at least a few minutes has a big impact on the caffeine content. The study found that after brewing for one minute, a cup of regular Lipton black tea had 17 milligrams of caffeine per 6 ounces of water, 38 milligrams per 6 ounces after three minutes, and 47 milligrams per 6 ounces after five. (The nutritional information for Lipton black tea says a serving contains 55 milligrams of caffeine per 8 ounces, so it's pretty accurate.)

Some people may use those numbers as an excuse to steep their tea past the five-minute mark in an attempt to reach 100 percent dissolution. But a longer brewing time doesn't necessarily equal a stronger caffeine kick. Yes, more caffeine molecules will enter the tea, but so will other compounds like thearubigins. Caffeine works because it's perfectly shaped to bind to certain neuroreceptors in your brain, thus blocking the chemicals that tell you to feel tired. But caffeine is the right shape to bind to thearubigins as well, and if that happens first, less caffeine will get to those neuroreceptors. So if you're looking for a highly caffeinated cup of tea, you should remove the leaves after most of the caffeine has been extracted—after about three to five minutes—rather than waiting for every last milligram of caffeine to dissolve.


Suggested steeping time: 1 to 3 minutes

There's nothing wrong with enjoying a cup of tea for taste alone. Flavor is the most subjective factor influenced by steeping times, but for the sake of simplicity, let's assume you prefer a pronounced tea taste that's not overshadowed by bitterness. To extract those more delicate flavors, you don't need to steep your tea leaves for very long at all. Some of the first volatile organic compounds to break down in tea are geraniol and phenylacetaldehyde, tied to a tea's floral aroma, and linalool and linalool oxide, which give tea its sweetness.

The other compounds we associate with tea's distinctive taste are tannins. They're the difference between an aromatic, fruity cup of tea and a bitter cup that needs to be diluted with milk before it's palatable. But tannins aren't all bad: Some people prefer their tea to have a bracing astringency. Because tannins are some of the last molecules to dissolve into tea, if you want to add some bitter complexity to your drink, steep your tea for a minute or two longer than you normally would. A good way to keep track of the strength of your tea is to look at the color: Like tannins, pigments are heavy compounds, so if you see your tea getting darker, that means it's getting stronger as well.

And what about herbal teas? Feel free to leave the leaves in as long as you like. Because herbal teas are high in aromatic compounds and low in tannins, drinkers can be more liberal with their steep times without worrying about getting that astringent taste. Some teas, like rooibos and chamomile, also contain antioxidants, which is another reason to take your time.

And if you're new to the world of tea and aren't sure what your preferences are, put a kettle on the stove and start experimenting.

What Is Death?

The only thing you can be certain about in life is death. Or is it? Merriam-Webster defines death as "a permanent cessation of all vital functions." The Oxford English dictionary refines that to "the permanent ending of vital processes in a cell or tissue." But determining when someone is dead is surprisingly complicated—the medical definition has changed over the centuries and, in many ways, is still evolving.


For most of human history, doctors relied on basic observations to determine whether or not a person had died. (This may be why so many feared being buried alive and went to great lengths to ensure they wouldn't be.) According to Marion Leary, the director of innovation research for the Center for Resuscitation Science at the University of Pennsylvania, "If a person wasn't visibly breathing, if they were cold and bluish in color, for example, they would be considered dead."

As time went on, the markers for death changed. Before the mid-1700s, for example, people were declared dead when their hearts stopped beating—a conclusion drawn from watching traumatic deaths such as decapitations, where the heart seemed to be the last organ to give up. But as our understanding of the human body grew, other organs, like the lungs and brain, were considered metrics of life—or death.

Today, that remains true to some degree; you can still be declared dead when your heart and lungs cease activity. And yet you can also be declared dead if both organs are still working, but your brain is not.

In most countries, being brain dead—meaning the whole brain has stopped working and cannot return to functionality—is the standard for calling death, says neuroscientist James Bernat, of the Geisel School of Medicine at Dartmouth College in New Hampshire. "A doctor has to show that the loss of brain function is irreversible," he tells Mental Floss. In some cases, a person can appear to be brain dead if they have overdosed on certain drugs or have suffered from hypothermia, for example, but the lack of activity is only temporary—these people aren't truly brain dead.

In the U.S., all states follow some form of the Uniform Determination of Death Act, which in 1981 defined a dead person as "an individual who has sustained either (1) irreversible cessation of circulatory and respiratory functions, or (2) irreversible cessation of all functions of the entire brain, including the brain stem."

But that's not the end of the story. In two states, New York and New Jersey, families can reject the concept of brain death if it goes against their religious beliefs. This makes it possible for someone to be considered alive in some states and dead in others.


In the past, if one of a person's three vital systems—circulation, respiration, and brain function—failed, the rest would usually stop within minutes of each other, and there was no coming back from that. But today, thanks to technological advances and medical breakthroughs, that's no longer necessarily the case. CPR can be performed to restart a heartbeat; a person who has suffered cardiac arrest can often be resuscitated within a 20- to 30-minute window (in rare cases, people have been revived after several hours). And since the 1950s, machines have been used to take on the role of many of the body's vital functions. People who stop breathing naturally can be hooked up to ventilators to move air in and out of their lungs, for example.

While remarkable, this life-extending technology has blurred the line between life and death. "A person can now have certain characteristics of being alive and others of being dead," Bernat says.

People with severe, irreversible brain damage fall into this mixed category. Many lie in intensive care units where ventilators breathe for them, but because they have minimal reflexes or movements, they're considered alive, especially by their families. Medical professionals, however, may disagree, leading to painful and complex debates about whether someone is alive.

Take the case of Jahi McMath, whose tonsil surgery in 2013, at age 13, went terribly wrong, leaving her brain dead—or so doctors thought. Her family refused to believe she was dead and moved her from Oakland, California, to New Jersey, where she was provided with feeding tubes in addition to her ventilator. After several months, her mother began recording videos that she said were proof that Jahi could move different parts of her body when asked to. Additional brain scans revealed that although some parts of her brain, like her brain stem, were largely destroyed, the structure of large parts of her cerebrum, which is responsible for consciousness, language, and voluntary movements, was intact. Her heart rate also changed when her mother spoke, leading a neurologist to declare last year, after viewing many of her mother's videos, that she is technically alive—nearly four years after she was pronounced brain dead. By her mother's reckoning, Jahi turned 17 on October 24, 2017.

Organ donation adds another layer of complications. Since an organ needs to be transplanted as quickly as possible to avoid damage, doctors want to declare death as soon as they can after a person has been disconnected from a machine. The protocol is usually to wait for five minutes after a donor's heart and breathing have stopped. However, some believe that's not long enough, since the person could still be resuscitated at that point.

Bernat—whose research interests include brain death and the definition of death, consciousness disorders including coma and vegetative states, and ethical and philosophical issues in neurology—disagrees. "I would argue that breathing and circulation has permanently ceased even if it hasn't irreversibly ceased," he says. "It won't restart by itself."


As resuscitation technology improves, scientists may find new ways to reverse death. One promising approach is therapeutic hypothermia. Sometimes used on heart attack patients who have been revived, the therapy uses cooling devices to lower body temperature, usually for about 24 hours. "It improves a patient's chance of recovering from cardiac arrest and the brain injury [from a lack of oxygen] that can result from it," says Leary, who specializes in research and education relating to cardiac arrest, CPR quality, and therapeutic hypothermia.

One more out-there possibility—which had its heyday in the early 2000s but still has its proponents today—is cryonic freezing, in which dead bodies (and in some cases, just people's heads) are preserved in the hope that they can be brought back once technology advances. Just minutes after death, a cryonaut's body is chilled; a chest compression device called a thumper keeps blood flowing through the body, which is then shot up with anticoagulants to prevent blood clots from forming; and finally, the blood is flushed out and replaced with a kind of antifreeze to halt the cell damage that usually occurs from freezing.

The idea is highly controversial. "It makes a good story for a movie, but it seems crazy to me," Bernat says. "I don't think it's the answer." But even if cryogenics is out, Bernat does believe that certain types of brain damage now thought to be permanent could one day be subject to medical intervention. "There is currently a huge effort in many medical centers to study brain resuscitation," he says.

Genetics provides another potential frontier. Scientists recently found that some genes in mice and fish live on after they die. And even more surprisingly, other genes regulating embryonic development, which switch off when an animal is born, turn on again after death. We don't yet know if the same thing happens in humans.