Some Mathematicians Think the Equal Sign is On Its Way Out

Paperkites/iStock via Getty Images
Paperkites/iStock via Getty Images

A growing number of mathematicians are skeptical that the equal sign, traditionally used to show exact relationships between sets of objects, holds up to new mathematical models, WIRED reports.

To understand their arguments, it’s important to understand set theory—a theory of mathematics that’s been around since at least the 1870s [PDF]. Take the classic formula 1+1=2. Say you have four pieces of fruit—an apple, an orange, and two bananas—and you put the apple and the orange on one side of a table and the two bananas on the other. In set theory, that’s an equation: One piece of fruit plus one piece of fruit on the left side of the table equals two pieces of fruit on the right side of the table. The two sets, or collections of objects, are the same size, so they’re equal.

But here’s where it gets complicated. What if you put an apple and a banana on the left side of the table and an orange and a banana on the other side? That’s clearly different from the first scenario, but set theory writes it as the same thing: 1+1=2. What if you switched the order of the first set of objects, so instead of having an apple and an orange, you had an orange and an apple? What if you had only bananas? There are potentially infinite scenarios, but set theory is limited to expressing them all in only one way.

“The problem is, there are many ways to pair up,” Joseph Campbell, a mathematics professor at Duke University, told Quanta Magazine. “We’ve forgotten them when we say ‘equals.’”

A better alternative is the idea of equivalence, some mathematicians say [PDF]. Equality is a strict relationship, but equivalence comes in different forms. The two-bananas-on-each-side-of-the-table scenario is considered strong equivalence—all of the elements in both sets are the same. The scenario where you have an apple and an orange on one side and two bananas on the other? That’s a slightly weaker form of equivalence.

A new wave of mathematicians is turning to the idea of category theory [PDF], which is based in understanding the relationships between different objects. Category theory is better than set theory at dealing with equivalence, and it’s also more universally applicable to different branches of mathematics.

But a switch to category theory won’t come overnight, according to Quanta. Interpreting equations using equivalence rather than equality is much more complicated, and it requires relearning and rewriting everything about mathematics—even down to algebra and arithmetic.

“This complicates matters enormously, in a way that makes it seem impossible to work with this new version of mathematics we’re imagining,” mathematician David Ayala told Quanta.

Several mathematicians are at the forefront of category theory research, but the field is still relatively young. So while the equal sign isn’t passé just yet, it’s likely that an oncoming mathematical revolution will change its meaning.

[h/t Wired]

Study Finds One in Seven Danish Children Will Be Diagnosed with Mental Illness

Kerkez/iStock via Getty Images
Kerkez/iStock via Getty Images

As researchers continue to investigate the origins of depression, anxiety, and other mental disorders, a new study in JAMA Psychiatry has found that one in seven children in Denmark will develop some form of mental illness before they turn 18.

The paper, by researchers at Aarhus University and other institutions, looked at a database of health information collected from 1.3 million Danish children from age 0 to 18. Boys had a 15.5 percent chance, and girls a 14.6 percent chance, of being clinically diagnosed with a mental illness before age 18. Attention-deficit/hyperactivity disorder (ADHD) was the most common disorder among boys, while anxiety was most prevalent among girls. Depression and schizophrenia were also present.

Researchers also examined when children were diagnosed. Boys tended to be labeled with ADHD as young as 8 years old, while girls received the same diagnosis more frequently at 17. Boys were also diagnosed with other illnesses earlier overall.

The study was limited to Denmark, and socioeconomic factors that may influence ailments and diagnoses can vary by country. Still, researchers said these statistics may help mental health professionals prepare earlier intervention and provide young people the help they need.

[h/t Independent]

First-Ever Map of Titan Reveals That Saturn’s Moon Is a Lot Like Earth

NASA/JPL-Caltech/Univ. Arizona/Univ. Idaho
NASA/JPL-Caltech/Univ. Arizona/Univ. Idaho

If there's any life in this solar system outside Earth, we likely won't find it on Mars or even on another planet. Saturn's moon Titan is the place in our celestial neighborhood that's most similar to our own home, and it's where scientists think we have one of the best chances of discovering life. Now, as Nature reports, newly visualized data shows just how much Titan has in common with Earth.

Between 2004 and 2017, the NASA spacecraft Cassini performed more than 100 fly-bys of Saturn's moon. Titan is unique in that it's the only moon in the solar system with clouds and a dense, weather-forming atmosphere. This has made it hard to study from space, but by flying close to the surface, Cassini was able to capture the landscape in an unprecedented level of detail.

Map of Titan.
The first global geologic map of Titan.
NASA/JPL-Caltech/ASU

NASA's new map of Titan, published in the journal Nature Astronomy, reveals a varied world of mountains, valleys, plains, and sandy dunes that starkly contrast with the desolate wastelands we've seen on neighboring planets. It's also home to seas and lakes, making it the only place in the solar system other than Earth with known bodies of liquid. But instead of water, the pools mottling the moon's surface consist of liquid methane.

Even with its Earth-like geology and atmosphere, chances of finding life on Titan are still slim: Temperatures on the surface average around -300°F. If life does exist there, it's likely limited to microbes in the moon's craters and icy volcanoes.

It will be a while before NASA is able to study Titan up close again: NASA's next drone mission to the body is set for 2034. Until then, scientists have plenty of data recorded by Cassini to teach them more about how the moon formed and continues to change.

[h/t Nature]

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