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]

The Reason Our Teeth Are So Sensitive to Pain

This woman's tooth pain is actually helping her avoid further damage.
This woman's tooth pain is actually helping her avoid further damage.
champja/iStock via Getty Images

On a good day, your teeth can chew through tough steak and split hard candy into pieces without you feeling a thing. But sometimes, something as simple as slurping a frosty milkshake can send a shock through your tooth that feels even more painful than stubbing your toe.

According to Live Science, that sensitivity is a defense mechanism we’ve developed to protect damaged teeth from further injury.

“If you eat something too hot or chew something too cold, or if the tooth is worn down enough where the underlying tissue underneath is exposed, all of those things cause pain,” Julius Manz, American Dental Association spokesperson and director of the San Juan College dental hygiene program, told Live Science. “And then the pain causes the person not to use that tooth to try to protect it a little bit more.”

Teeth are made of three layers: enamel on the outside, pulp on the inside, and dentin between the two. Pulp, which contains blood vessels and nerves, is the layer that actually feels pain—but that doesn’t mean the other two layers aren’t involved. When your enamel (which isn’t alive and can’t feel anything at all) is worn down, it exposes the dentin, a tissue that will then allow especially hot or cold substances to stimulate the nerves in the pulp. Pulp can’t sense temperature, so it interprets just about every stimulus as pain.

If you do have a toothache, however, pulp might not be the (only) culprit. The periodontal ligament, which connects teeth to the jawbone, can also feel pain. As Manz explains, that sore feeling people sometimes get because of an orthodontic treatment like braces is usually coming from the periodontal ligament rather than the pulp.

To help you avoid tooth pain in the first place, here are seven tips for healthier teeth.

[h/t Live Science]

Arrokoth, the Farthest, Oldest Solar System Object Ever Studied, Could Reveal the Origins of Planets

NASA/Johns Hopkins University Applied Physics Laboratory/Southwest Research Institute/Roman Tkachenko
NASA/Johns Hopkins University Applied Physics Laboratory/Southwest Research Institute/Roman Tkachenko

A trip to the most remote part of our solar system has revealed some surprising insights into the formation of our own planet. Three new studies based on data gathered on NASA's flyby of Arrokoth—the farthest object in the solar system from Earth and the oldest body ever studied—is giving researchers a better idea of how the building blocks of planets were formed, what Arrokoth's surface is made of, and why it looks like a giant circus peanut.

Arrokoth is a 21-mile-wide space object that formed roughly 4 billion years ago. Located past Pluto in the Kuiper Belt, it's received much less abuse than other primordial bodies that sit in asteroid belts or closer to the sun. "[The objects] that form there have basically been unperturbed since the beginning of the solar system," William McKinnon, lead author of one of the studies, said at a news briefing.

That means, despite its age, Arrokoth doesn't look much different today than when it first came into being billions of years ago, making it the perfect tool for studying the origins of planets.

In 2019, the NASA spacecraft New Horizons performed a flyby of Arrokoth on the edge of the solar system 4 billion miles away from Earth. The probe captured a binary object consisting of two connected lobes that were once separate fragments. In their paper, McKinnon and colleagues explain that Arrokoth "is the product of a gentle, low-speed merger in the early solar system."

Prior to these new findings, there were two competing theories into how the solid building blocks of planets, or planetesimals, form. The first theory is called hierarchical accretion, and it states that planetesimals are created when two separate parts of a nebula—the cloud of gas and space dust born from a dying star—crash into one another.

The latest observations of Arrokoth support the second theory: Instead of a sudden, violent collision, planetesimals form when gases and particles in a nebula gradually amass to the point where they become too dense to withstand their own gravity. Nearby components meld together gradually, and a planetesimal is born. "All these particles are falling toward the center, then whoosh, they make a big planetesimal. Maybe 10, 20, 30, 100 kilometers across," said McKinnon, a professor of Earth and planetary sciences at Washington University. This type of cloud collapse typically results in binary shapes rather than smooth spheroids, hence Arrokoth's peanut-like silhouette.

If this is the origin of Arrokoth, it was likely the origin of other planetesimals, including those that assembled Earth. "This is how planetesimal formation took place across the Kuiper Belt, and quite possibly across the solar system," New Horizons principal investigator Alan Stern said at the briefing.

The package of studies, published in the journal Science, also includes findings on the look and substance of Arrokoth. In their paper, Northern Arizona University planetary scientist Will Grundy and colleagues reveal that the surface of the body is covered in "ultrared" matter so thermodynamically unstable that it can't exist at higher temperatures closer to the sun.

The ultrared color is a sign of the presence of organic substances, namely methanol ice. Grundy and colleagues speculate that the frozen alcohol may be the product of water and methane ice reacting with cosmic rays. New Horizons didn't detect any water on the body, but the researchers say its possible that H2O was present but hidden from view. Other unidentified organic compounds were also found on Arrokoth.

New Horizon's flyby of Pluto and Arrokoth took place over the course of a few days. To gain a further understanding of how the object formed and what it's made of, researchers need to find a way to send a probe to the Kuiper Belt for a longer length of time, perhaps by locking it into the orbit of a larger body. Such a mission could tell us even more about the infancy of the solar system and the composition of our planetary neighborhood's outer limits.

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