Mathematicians don’t always agree on everything, but sometimes there are enough who agree on one thing to make it noteworthy. It seems as if many of those mathematicians are now saying that the equal sign, which has traditionally been used to show the exact relationships between objects may be on its way out. According to WIRED, they feel that it no longer holds up to newer mathematical models.
One important factor to understand when considering their arguments is set theory. This is a mathematical theory that has been around since the 1870s and perhaps even earlier. As an example, consider the 1+1=2 mathematical formula. To break it down even further, let’s say you have four items, a piece of celery, a tennis ball, and two oranges. If the celery and tennis ball are put on one side of the table and the two oranges are on the other side of the table, it’s an equation. Two different objects on one side of the table is equal to two of the same objects on the other side of the table. The collection of objects is the same, ergo they are equal.
Things start to get complicated when you put a piece of celery and an orange on one side of the table and the tennis ball and the other orange on the other side. It’s different than the first scenario but it is still written out using the same equation. You could continue to debate this in so many different ways, because the objects could continually be swapped out to an infinite number of scenarios.
“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.’”
Some mathematicians are now saying that the concept of equivalence is a better idea. Equality is very strict but equivalence can come in different forms. For example, having two oranges on both sides of the table is a matter of strong equivalence because everything is the same. The scenario with a tennis ball and piece of celery on one side and two oranges on the other side is still a matter of equivalence but it is weaker.
Category theory is another concept that many mathematicians are considering. It tends to complicate matters by considering the relationships between different objects, but it may be, strictly speaking, accurate. Category theory is certainly more accurate than set theory when dealing with matters of equivalence. It can also be applied to many branches of mathematics.
Quanta also reported that switching to category theory is not something that will be done with a snap of a finger or a flip of a switch. Interpreting equations using equivalence and dropping equality altogether is complicated. It requires people to relearn things that they already know and everything that uses mathematics, down to basic 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.
A number of mathematicians are taking the lead in research of category theory but the field is still in its infancy. We are not going to see the equal sign disappear overnight, but it may be on the horizon along with a new mathematical revolution.