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]

15 Science Experiments You Can Do With Your Kids

These boys are about to demonstrate osmosis with gummy bears.
These boys are about to demonstrate osmosis with gummy bears.
G&J Fey/iStock via Getty Images

Parents and teachers across the internet have found fun ways to teach kids science, and have documented the experiments for the rest of us. Here are 15 hands-on science lessons that will stick in a kid’s brain far longer than anything they get from a textbook.

1. Make a lemon-scented volcano

Fun Quotient: It's like the classic baking soda and vinegar volcano experiment, but it smells a lot nicer.

Teaches: The baking soda base and the citric acid create an endothermic reaction while releasing carbon dioxide in bubble form. You have to look up endothermic reaction on your own.

Find it: Fun Littles

2. Set money on fire

Fun Quotient: Wait, what? You’re burning money? Why?!

Teaches: Combustion, the process behind fire. Rubbing alcohol is flammable, but the wet, cottony dollar isn't. The fire will go out once the alcohol has been consumed.

Find it: Barefoot in Suburbia

3. Create rock candy skewers

Fun Quotient: It makes pretty rocks you can eat.

Teaches: Water evaporates, but the sugar crystals don’t. The sugar precipitates meaning it separates from the supersaturated sugar water. Seed crystals form on your stick, attracting more sugar crystals, until finally, about a week later, you got yourself some tasty science.

Find it: Science Bob

4. Build an electromagnet

Fun Quotient: Kids get to use sharp things and electricity, which is Frankenstein-level cool.

Teaches: Electromagnets are everywhere. They make motors spin, CDs play, and most modern cars run. This experiment shows the difference between a permanent magnet (the ones on your fridge) and the kind that can be turned on and off at will. When turned on, the electricity forces the molecules in the nail to attract metal, even though the nail itself isn’t magnetic.

Find it: Science Bob

5. Write a message with invisible ink

Fun Quotient: Kids can pretend they're spies sending highly classified information (not recommended in real life).

Teaches: Oxidation, a.k.a. the process that creates rust. Lemon juice is acidic enough to resist oxidation in open air, but a little heat “rusts” it right up.

Find it: Scientific American

6. Walk on eggs

Fun Quotient: Like walking on hot coals, but not as painful.

Teaches: Structure matters. No matter how flimsy an egg shell is, its shape gives it amazing strength, as long as you put the weight in the right place.

Find it: Steve Spangler Science

7. Make a tea bag rocket

Fun Quotient: Every child enjoys watching things burst into flame and fly around the kitchen.

Teaches: Hot air rises and cooler air sinks. But it also demonstrates convection current, which is the force that makes it shoot into the air.

Find it: Physics Central

8. Discover how cornstarch and water can dance

Fun Quotient: Oobleck is a mix of cornstarch and water that can act like a liquid and a solid. By itself it’s fun, but add a sub-woofer and the glop will shimmy in its container.

Teaches: Sound waves. You can’t see them, but they exist, and they like to party.

Find it: Housing a Forest

9. Create an Ivory Soap monster

Fun Quotient: You get to nuke a bar of soap until it becomes a frothy cloud of 99 percent pure mess.

Teaches: When the gas molecules trapped in the soft pliable soap get hot, they need more space. They make a break for it and take the soap with them. As the temperature of the gas increases, so does its volume.

Find it: ThoughtCo

10. Launch marshmallows across the room

Fun Quotient: Weaponized marshmallows, hello!

Teaches: Force equals mass times acceleration. A little thing going very fast will hit you just as hard as a big thing going slow. That’s Newton’s second law.

Find it: Adventure Science Center

11. Poke a "magic" plastic bag

Fun Quotient: You'll find out it's possible to poke a pencil through a plastic bag of water without spilling it.

Teaches: How polymers work. Also, on a different level, why you’re not supposed to take the arrow out of a person after they get impaled in movies.

Find it: Tinkerlab

12. Make gummy bears change shape

Fun Quotient: Deform gummy bears by dunking them in a variety of potions.

Teaches: Osmosis, and which kinds of liquids do it best.

Find it: Sciencing

13. Design an optical illusion

Fun Quotient: Animate a cartoon the old-fashioned way.

Teaches: Your eyes aren’t entirely reliable. Optical illusions occur because our brains fill in the gaps for whatever our eye isn’t processing, so two pictures become one.

Find it: Science Sparks

14. Set up a chain reaction

Fun Quotient: It’s tough to get started, but the payoff is clatter and splatter.

Teaches: A demonstration of potential energy, kinetic energy, and chain reactions.

Find it: The Kid Should See This

15. Make Molecules Move

Fun Quotient: Slow and steady wins here; kids with an artistic streak will love creating designs as the colors move through liquids.

Teaches: Why oil and water don't mix. Kids will witness how molecules of water, fats, and proteins come together and move apart in different substances.

Find it: American Chemical Society

Take a Virtual Tour of One of the Quietest Places on Earth

Paul Robinson, Wikimedia Commons // CC BY-SA
Paul Robinson, Wikimedia Commons // CC BY-SA

Orfield Laboratories in Minneapolis, Minnesota, is notable for what it lacks. Inside the anechoic chamber, you won't hear the sounds of traffic, chatter, or even the hum of appliances. The soundproof room is one of the quietest places on Earth, and you can take a virtual tour of the space in the video below.

According to UPROXX, Orfield Laboratories is primarily used for research. The background noise inside the chamber clocks in at -9.4 dBA. For comparison, the quietest place the average person has access to has sound levels closer to 30 dBA. Without any outside noises to interfere, products tested for noise levels inside the space produce extremely accurate results. The room has other purposes as well, such as preparing astronauts for space missions and freaking out tourists willing to pay for a visit.

Experiencing sound levels lower than what's found in nature has odd effects on the human body. Your ears adapt to pick up sounds that are usually inaudible. With nothing else to fill the space, the sounds of your heart, stomach, and even lungs can become deafening. Dizziness, anxiety, and out-of-body-sensations are a few of the reported responses to spending time in the eerily silent chamber.

Though you won't be able to get the full experience from home, you can pop in a pair of headphones and get a taste of what the room is like in this video. The lab is currently closed to visitors, but when it's open it offers tours starting at $125 per person.

[h/t UPROXX]

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