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]

10 Rad Gifts for Hikers

Greg Rosenke/Unsplash
Greg Rosenke/Unsplash

The popularity of bird-watching, camping, and hiking has skyrocketed this year. Whether your gift recipients are weekend warriors or seasoned dirtbags, they'll appreciate these tools and gear for getting most out of their hiking experience.

1. Stanley Nesting Two-Cup Cookset; $14

Amazon

Stanley’s compact and lightweight cookset includes a 20-ounce stainless steel pot with a locking handle, a vented lid, and two insulated 10-ounce tumblers. It’s the perfect size for brewing hot coffee, rehydrating soup, or boiling water while out on the trail with a buddy. And as some hardcore backpackers note in their Amazon reviews, your favorite hiker can take the tumblers out and stuff the pot with a camp stove, matches, and other necessities to make good use of space in their pack.

Buy it: Amazon

2. Osprey Sirrus and Stratos 24-Liter Hiking Packs; $140

Amazon

Osprey’s packs are designed with trail-tested details to maximize comfort and ease of use. The Sirrus pack (pictured) is sized for women, while the Stratos fits men’s proportions. Both include an internal sleeve for a hydration reservoir, exterior mesh and hipbelt pockets, an attachment for carrying trekking poles, and a built-in rain cover.

Buy them: Amazon, Amazon

3. Yeti Rambler 18-Ounce Bottle; $48

Amazon

Nothing beats ice-cold water after a summer hike or a sip of hot tea during a winter walk. The Yeti Rambler can serve up both: Beverages can stay hot or cold for hours thanks to its insulated construction, and its steel body (in a variety of colors) is basically indestructible. It will add weight to your hiker's pack, though—for a lighter-weight, non-insulated option, the tried-and-true Camelbak Chute water bottle is incredibly sturdy and leakproof.

Buy it: Amazon

4. Mappinners Greatest 100 Hikes of the National Parks Scratch-Off Poster; $30

Amazon

The perfect gift for park baggers in your life (or yourself), this 16-inch-by-20-inch poster features epic hikes like Angel’s Landing in Zion National Park and Half Dome in Yosemite National Park. Once the hike is complete, you can scratch off the gold foil to reveal an illustration of the park.

Buy it: Amazon

5. National Geographic Adventure Edition Road Atlas; $19

Amazon

Hikers can use this brand-new, updated road atlas to plan their next adventure. In addition to comprehensive maps of all 50 states, Puerto Rico, Canada, and Mexico, they'll get National Geographic’s top 100 outdoor destinations, useful details about the most popular national parks, and points on the maps noting off-the-beaten-path places to explore.  

Buy it: Amazon

6. Adventure Medical Kits Hiker First-Aid Kit; $25

Amazon

This handy 67-piece kit is stuffed with all the things you hope your hiker will never need in the wilderness. Not only does it contain supplies for pain, cuts and scrapes, burns, and blisters (every hiker’s nemesis!), the items are organized clearly in the bag to make it easy to find tweezers or an alcohol wipe in an emergency.

Buy it: Amazon

7. Hiker Hunger Ultralight Trekking Poles; $70

Amazon

Trekking poles will help increase your hiker's balance and stability and reduce strain on their lower body by distributing it to their arms and shoulders. This pair is made of carbon fiber, a super-strong and lightweight material. From the sweat-absorbing cork handles to the selection of pole tips for different terrain, these poles answer every need on the trail. 

Buy it: Amazon

8. Leatherman Signal Camping Multitool; $120

Amazon

What can’t this multitool do? This gadget contains 19 hiking-friendly tools in a 4.5-inch package, including pliers, screwdrivers, bottle opener, saw, knife, hammer, wire cutter, and even an emergency whistle.

Buy it: Amazon

9. RAVPower Power Bank; $24

Amazon

Don’t let your hiker get caught off the grid with a dead phone. They can charge RAVPower’s compact power bank before they head out on the trail, and then use it to quickly juice up a phone or tablet when the batteries get low. Its 3-inch-by-5-inch profile won’t take up much room in a pack or purse.

Buy it: Amazon

10. Pack of Four Indestructible Field Books; $14

Amazon

Neither rain, nor snow, nor hail will be a match for these waterproof, tearproof 3.5-inch-by-5.5-inch notebooks. Your hiker can stick one in their pocket along with a regular pen or pencil to record details of their hike or brainstorm their next viral Tweet.

Buy it: Amazon

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What Really Happens When Food Goes Down the 'Wrong Pipe'?

The dreaded 'wrong pipe' calamity can strike at any time.
The dreaded 'wrong pipe' calamity can strike at any time.
Photo by Adrienn from Pexels

Your average person isn’t expected to be well-versed in the linguistics of human anatomy, which is how we wind up with guns for biceps and noggins for heads. So when swallowing something is followed by throat irritation or coughing, the fleeting bit of discomfort is often described as food “going down the wrong pipe.” But what’s actually happening?

When food is consumed, HuffPost reports, more than 30 muscles activate to facilitate chewing and swallowing. When the food is ready to leave your tongue and head down to your stomach, it’s poised near the ends of two "pipes," the esophagus and the trachea. You want the food to take the esophageal route, which leads to the stomach. Your body knows this, which is why the voice box and epiglottis shift to close off the trachea, the “wrong pipe” of ingestion.

Since we don’t typically hold our breath when we eat, food can occasionally take a wrong turn into the trachea, an unpleasant scenario known as aspiration, which triggers an adrenaline response and provokes coughing and discomfort. Dislodging the food usually eases the sensation, but if it’s enough to become stuck, you have an obstructed airway and can now be officially said to be choking.

The “wrong pipe” can also be a result of eating while tired or otherwise distracted or the result of a mechanical problem owing to illness or injury.

You might also notice that this happens more often with liquids. A sip of water may provoke a coughing attack. That’s because liquids move much more quickly, giving the body less time to react.

In extreme cases, food or liquids headed in the “wrong” direction can wind up in the lungs and cause pneumonia. Fortunately, that’s uncommon, and coughing tends to get the food moving back into the esophagus.

The best way to minimize the chances of getting food stuck is to avoid talking with your mouth full—yes, your parents were right—and thoroughly chew sensible portions.

If you experience repeated bouts of aspiration, it’s possible an underlying swallowing disorder or neurological problem is to blame. An X-ray or other tests can help diagnose the issue.

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