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.

Essential Science
What Is a GMO?

If you've followed the debate about GMOs with any sort of regularity, there's a strong chance you've come across a picture of a tomato stabbed by a giant syringe. That image, though a complete fiction, seems to perfectly capture what's preventing public acceptance of these foods: We don't really know what makes something a GMO.

GMOs aren't made with syringes and, at the moment, they aren't even made with tomatoes, at least not commercially. But that false image is everywhere, and surveys indicate consumers fear GMOs without knowing much about them.

So what exactly is a GMO?


The initialism stands for "genetically modified organism," but it's a term lacking scientific precision. Moreover, it's hard to find an organism in any way connected to humans that hasn't been genetically modified, says Alison Van Eenennaam, a geneticist at UC-Davis who specializes in animal biotechnology. "I might argue that a great Dane or a Corgi are 'genetically modified' relative to their ancestor, the wolf," she tells Mental Floss. "'GMO' is not a very useful term. Modified for what and why is really the more important question.”

GMOs are often described as if they were a recent invention of our industrial food system, but genetic modification of food isn't new at all. It's been happening for many millennia: As long as farmers have been saving high-performing seeds for future harvests, we've had GMOs. Perhaps the earliest known example of a GMO is the sweet potato, which scientists believe became modified when wild sweet potatoes became infected, quite naturally, by a particular kind of soil bacteria. Realizing these sweet potatoes were edible, people began saving the seeds and cultivating them for future harvests. That was about 8000 years ago.

These days, when people say "GMO," they tend to mean one particular modification method that scientists refer to as transgenesis. As Van Eenennaam explains, transgenesis is "a plant-breeding method whereby useful genetic variation is moved from one species to another using the methods of modern molecular biology, also known as genetic engineering."

Transgenic crops and animals have been modified with the addition of one or more genes from another living organism, using either a "gene gun," Agrobacteria—a genus of naturally occurring bacteria that insert DNA into plants—or electricity, in a process called electroporation.

The first commercial transgenic crops debuted in the early 1990s: a virus-resistant tobacco in China [PDF] and the Flavr-Savr tomato in the U.S., which was genetically altered to not get "squishy." (It's no longer on the market.)

As to the health risks of GMO foods, the scientific consensus is clear: Transgenic crops are no riskier than other crops. Van Eenennaam points to a 20-year history of safe use that includes "thousands of studies, eleven National Academies reports, and indeed [the consensus of] every major scientific society in the world."


Today, the most ubiquitous transgenic crops in the U.S. food system are cotton, soybeans, and corn, including those modified to resist the effects of the herbicide Roundup. Branded "Roundup Ready," these crops have been modified so that farmers can apply the herbicide directly to crops to control weeds without killing the crops themselves.

For farmers, the result was better weed control and higher yields. For critics of GMOs, these crops became their smoking gun. These opponents argue they're bad for the planet and bad for our health.

There's no question that use of glyphosate, the active ingredient in the herbicide Roundup, has increased since the introduction of GMOs, but measuring its environmental impact is a far more complex equation. For example, as glyphosate use has increased, so has the prevalence of conservation tillage, a beneficial agricultural approach that helps sequester carbon in the soil and mitigate the impacts of climate change.

Bt crops—transgenic crops modified with genes from the all-natural bacterial toxin Bt, short for Bacillus thuringiensis—have also reduced the use of insecticide, according to a 2016 National Academies of Science report.

And though evidence suggests herbicide use has increased since Roundup Ready GMOs were first commercialized in the U.S., herbicide use has increased amongst some non-GMO crops, too. Glyphosate also replaced more toxic herbicides on the market and, if farmers were to stop using it, many would likely replace glyphosate with another herbicide, possibly one that's more toxic. Glyphosate-resistant weeds are a problem, but banning glyphosate, or glyphosate-resistant GMOs for that matter, wouldn't solve the problem.

In recent years, opponents of GMOs have increasingly aimed their fire at glyphosate. The source of many of these claims is a 2015 assessment [PDF] by the International Agency for Research on Cancer (IARC) to categorize glyphosate as "probably carcinogenic." That categorization has been hotly contested by many scientists, as other governmental agencies have concluded glyphosate does not pose a carcinogenic hazard. And, in June, reporting revealed that the lead researcher at IARC withheld important studies from the research group's consideration.

Weighing criticisms of glyphosate against its benefits certainly brings up complex issues in our agricultural system, but ultimately these issues are not unique to GMOs nor would they magically disappear if transgenic technology were eliminated altogether.


Most consumers probably can't name all the different methods of genetic modification, but there's a good chance they've eaten foods modified by one of these methods all the same. Layla Katiraeea human molecular geneticist at Integrated DNA Technologies and a science communicator, has written about these methods to illustrate why it makes little sense to single out transgenic crops. Examples include polyploidy, which gave us the seedless watermelon, and mutagenesis, which scientists used to engineer a brightly colored grapefruit. As Katiraee points out, sometimes two different methods can even create a very similar end result. For example, the non-browning Opal apple was developed using traditional cross-breeding, while the non-browning Arctic apple uses transgenic methods to silence the genes that control browning.

Katiraee says the most common objections to GMOs aren't exclusive to transgenic crops: “Don't like ‘Big Ag'? They use all methods of crop modification. Don't like herbicide-tolerant crops? They've been made with other methods. Don't like patents? Crops modified by all methods are patented. If you go through the list, you won't find one [objection] that applies exclusively to transgenesis.”

Katiraee's arguments illustrate why it doesn't make sense to label transgenic crops "GMO" while omitting the non-browning opal apple or a seedless watermelon. And the non-GMO label can often be misleading. Van Eenennaam points to one of the more ridiculous examples: non-GMO salt. "Salt doesn't contain DNA, so salt cannot be genetically engineered," she says. "All salt is 'non-GMO' salt."


The noisy GMO debate has often overshadowed the successes of lesser known, disease-resistant GMOs. Van Eenennaam argues that no one should object to these crops since protecting “plants and animals from disease aligns with most everyone's common interest in decreasing the use of chemicals in agricultural production systems, and minimizing the environmental footprint of food production." Examples include ringspot virus–resistant papaya in Hawaii [PDF] and the American chestnut, both rescued from the devastating effects of lethal plant viruses.

Disease-resistant crops often face an uphill battle for approval. In Uganda, scientists developed a disease-resistant banana that then faced difficult regulatory obstacles until a new law was finally approved in October by the country's Parliament. In Florida, where the disease called citrus greening has caused widespread crop damage and loss to the citrus industry, orange trees have been modified with a spinach gene to help crops resist the virus. But orange juice manufacturers will have to persuade consumers to buy it. 

Scientists have used transgenic modification to address health concerns too. For example, some variations of the wilt-resistant banana also include a boost of vitamin A. Scientists are working on a form of wheat that would be safe for people with celiac disease.

Van Eenennaam fears the controversy over GMOs has meant that, over the years, the public has missed out on important technologies. In the field of animal biotechnology, for example, animals have been produced that are resistant to disease, "that produce less pollution in their manure, [and] that have … elevated levels of omega-3 fatty acids," but none of these have been commercialized in the U.S.

Given that these crops and animals have a 20-year history of safe use, Van Eenennaam argues there's no reason that "fungus-resistant strawberries, disease-resistant bananas, and virus-resistant animals [should] sit on the shelf" unused.

Editor's note: This post has been updated. 

Essential Science
What Is Antibiotic Resistance?

The news is full of terms like "superbug," "post-antibiotic era," and an alphabet soup of abbreviations including NDM-1, MCR-1 (both antibiotic resistance genes), MRSA (a type of antibiotic-resistant bacteria), and others. These all refer to various aspects of antibiotic resistance—the ability of bacteria to out-maneuver the drugs which are supposed to kill them and stop an infection.

Now, there is concern that we could move back into a situation like that which existed in the early 20th century—a post-antibiotic era. Mental Floss spoke to Meghan Davis, a veterinarian and assistant professor of epidemiology at Johns Hopkins University, about some of the potential outcomes of losing antibiotics. "We have generations of recorded history that identify the risks to human society from infectious diseases that we are unable to treat or prevent," Davis warns.


If an individual becomes ill due to a bacterial infection, they typically see their physician for treatment. But in the years before antibiotics were discovered, people frequently died from scenarios we find difficult to fathom, including mere cuts or scratches that led to untreatable infections. Ear infections or urinary tract infections could lead to sepsis (bacteria in the blood). Arms or legs were surgically removed before an infected wound could lead to death.

When antibiotics were discovered, it's no surprise they were referred to as a "magic bullet" (or Zauberkugel in German, as conceived by medical pioneer Paul Ehrlich [PDF]). The drugs could wipe out an infection but not harm the host. They allowed people to recover from even the most serious of infections, and heralded a new era in medicine where people no longer feared bacteria.

Davis says the existence of antibiotics themselves has changed how we use medicine. Many medical procedures now rely on antibiotics to treat infections that may result from the intervention. "What is different about a post-antibiotic modern world is that we have established new patterns of behavior and medical norms that rely on the success of antimicrobial treatments," she says. "Imagine transplant or other major surgeries without the ability to control opportunistic infections with antibiotics. Loss of antibiotics would challenge many of our medical innovations."


One reason antibiotic resistance is difficult to control is that our antibiotics are derivatives of natural products. Our first antibiotic, penicillin, came from a common mold. Fungi, bacteria, parasites, and viruses all produce products to protect themselves as they battle each other in their microbial environments. We've taken advantage of the fruits of millions of years' worth of these invisible wars to harness antibiotics for our use. (This is also why we can find antibiotic resistance genes even in ancient bacteria that have never seen modern antibiotic drugs—because we've exploited the chemicals they use to protect themselves).

These microbes have evolved ways to evade their enemies—antibiotic resistance genes. Sometimes the products of these genes will render the antibiotic useless by chopping it into pieces or pumping it out of the bacterial cell. Importantly, these resistance genes can be swapped among different bacterial species like playing cards. Sometimes the genes will be useless because the bacteria aren't being exposed to a particular drug, but sometimes they'll be dealt an ace and survive while others die from antibiotic exposure.

And many of these resistance genes are already out there in the bacterial populations. Imagine just one in a million bacterial cells that are growing in a human gut have a resistance gene already in their DNA. When a person takes a dose of antibiotics, all the susceptible bacteria will die off—but that one-in-a-million bacterium that can withstand the antibiotic suddenly has a lot of room to replicate, and the population of bacteria carrying that resistance gene will dramatically increase.

If the person then transfers those resistant gut bacteria to others, resistance can spread as well. This is why it's important to keep control over antibiotic use in all populations—because someone else's use of the drugs can potentially make your own bacteria resistant to antibiotics. This is also why hand washing is important: You can unknowingly pick up new bacteria all the time from other people, animals, or surfaces. Washing your hands will send most of these passenger bacteria down the sink drain, instead of allowing them to live on your body.


Most importantly, never ask for antibiotics from your doctor; if you have a bacterial infection that can be treated by antibiotics, your doctor will prescribe them. Many illnesses are due to viruses (such as the common cold), but antibiotics only work against bacteria. It is useless to take antibiotics for a virus, and doing so will only breed resistance in the other bacteria living in your body, which can predispose you or others in your household and community to developing an antibiotic-resistant infection. Remember, those resistant bacteria can linger in your body—in your gut, on your skin, in your mouth and elsewhere, and can swap resistance genes from the mostly harmless bacteria you live with to the nasty pathogens you may encounter, further spreading resistance in the population.

Antibiotics are also used in animals, including livestock. Purchasing meat that is labeled "raised without antibiotics" will reduce your chance of acquiring antibiotic-resistant bacteria that are generated on the farm and can be spread via meat products.

Davis notes clients often requested antibiotics for their pets as well, even when it was an issue that did not require them. She explained to them why antibiotics were not necessary. She counsels, "Individuals can partner with their physician and veterinarian to promote good antimicrobial stewardship. Use of antibiotics carries risks, and these risks are related both to side effects and to promotion of resistance. Therefore, decisions to use antibiotics should be treated with caution and deliberation."