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]

Science Finds a Better Way to Calculate 'Dog Years'

thegoodphoto/iStock via Getty Images
thegoodphoto/iStock via Getty Images

Anyone who has ever owned a pet is likely familiar with the concept of “dog years,” which suggests that one year for a dog is like seven years for a human. Using this conversion metric, a 2-year-old dog is akin to a high school freshman, while a 10-year-old dog is ready for an assisted living facility.

If that seems rather arbitrary, that’s because it is. But now, researchers at the University of California, San Diego have come to a more data-based measurement on dog aging through DNA.

The paper, published on the preprint server bioRxiv, based the finding on DNA methylation, a process in which molecules called methyl groups attach themselves to DNA and serve as an indicator of aging. Generally speaking, the older living beings get, the faster the rate of methylation. In the study, 104 Labrador retrievers were examined, with subjects ranging from 1 month to 16 years old. The results of their DNA methylation were compared to human profiles. While the rate of methylation tracked closely between the two—young and old dogs had similar rates to young and old people—adolescent and mature dogs experienced more accelerated aging.

Their recommended formula for comparing dog and human aging? Multiply the natural logarithm of a dog’s age by 16, then add 31. Or, just use this calculator. Users will see that a 2-year-old dog, for example, wouldn’t be the canine equivalent of a 14-year-old. It would be equivalent to 42 human years old and should probably start putting money into a 401(k). But because methylation slows considerably in mid-life, a 5-year-old dog is approximately a 57-year-old human, while a 6-year-old dog is nearing 60 in human years—a minor difference. Things level out as the dog gets much older, with a 10-year-old dog nearing a 70-year-old human.

Different breeds age at different rates, so the formula might not necessarily apply to other dog breeds—only Labs were studied. The work is awaiting peer review, but it does offer a promising glimpse into how our furry companions grow older.

[h/t Live Science]

Sssspectacular: Tree Snakes in Australia Can Actually Jump

sirichai_raksue/iStock via Getty Images
sirichai_raksue/iStock via Getty Images

Ophidiophobia, or fear of snakes, is common among humans. We avoid snakes in the wild, have nightmares about snakes at night, and recoil at snakes on television. We might even be born with the aversion. When researchers showed babies photos of snakes and spiders, their tiny pupils dilated, indicating an arousal response to these ancestral threats.

If you really want to scare a baby, show them footage of an Australian tree snake. Thanks to researchers at Virginia Tech, we now know these non-venomous snakes of the genus Dendrelaphis can become airborne, propelling themselves around treetops like sentient Silly String.

That’s Dendrelaphis pictus, which was caught zipping through the air in 2010. After looking at footage previously filmed by her advisor Jake Socha, Virginia Tech Ph.D. candidate Michelle Graham headed for Australia and built a kind of American Ninja Warrior course for snakes out of PVC piping and tree branches. Graham observed that the snakes tend to spot their landing target, then spring upward. The momentum gets them across gaps that would otherwise not be practical to cross.

Graham next plans to investigate why snakes feel compelled to jump. They might feel a need to escape, or continue moving, or do it because they can. Two scientific papers due in 2020 could provide answers.

Dendrelaphis isn’t the only kind of snake with propulsive capabilities. The Chrysopelea genus includes five species found in Southeast Asia and China, among other places, that can glide through the air.

[h/t National Geographic]

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