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• Some mathematicians think the equals sign may become passé due to its limits with regard to describing the relationship between different objects. I.e. 1+1=2 doesn’t fully capture what is being added up. For example, we know that one object plus another object equals two objects, but it doesn’t tell us if the objects being added up are the same object or if they are different. It also doesn’t factor in the order of the objects. It’s limited by what is called “set theory” which only factors in the number of objects that are present. It doesn’t factor in the kind of objects that are present.

The mental floss article uses the example of fruit. One orange plus one apple equals two pieces of fruit. But is one orange plus one apple the same as one orange plus one banana? What about two bananas or two oranges? Or one grape and one melon? They all equal to two amounts of fruit, but are they really the same? 1+1=2 doesn’t fully capture what is being added up here.

What some mathematicians are hypothesizing is that what could eventually replace set theory is what is called “category theory” which is all about understanding how different objects relate to each other. Going back to our fruit example, rather than saying one orange plus one apple “equals” two pieces of fruit, we could say one orange plus one apple is “equivalent” to two pieces of fruit. Equality  is too strict, black and white, and final. Whereas equivalency has more flexibility and has more expansive possibilities.

Will category theory push set theory out the door? Not in the near future, but it is possible that overtime, it will. The big challenge will be getting us to rethink mathematics in a whole new way. Reprogramming our brains to think of things in terms of category theory as opposed to set theory, which has dominated mathematical thought since the 1870s.

I’m curious to get your thoughts. Especially those who have a strong background in math. Does any of this make sense to you guys?

• I’m wondering why the sign itself needs to go? I must be ahead of the curve because I’ve always taught my kids that the two sides of the equation are equivalent. They aren’t the same.

I guess what they’re getting at is possibly the need for a stronger sign or a weaker sign to distinguish “equivalent but not equal” from “equal.” For example, I would casually write 1 papaya = 10 oz to organize my thoughts, but that’s really distinct from saying 4 oz + 6 oz = 10 oz.

We already have a squiggly equals sign to indicate approximate equality (equivalency?). Maybe we need a circled equals sign to indicate a stronger relationship.

• Great thinking! Seems like you are indeed ahead of the curve on this. Rather than kissing the equals sign goodbye, why not modify it like you said?

• I’m curious to get your thoughts. Especially those who have a strong background in math. Does any of this make sense to you guys?

I haven't followed through to the original source of that Mental Floss articles, but I'm having some doubts about that.

First of all, this isn't really how mathematicians do things. They don't come up with some weird symbol first ("hey, why don't we draw two horizontal lines?"), and then try to add some sense to that afterwards. Mathematical symbols aren't the beginning of the story, but rather the end of it:

In the case we're talking about, there's a whole bunch of ideas about what it might mean when we say that two things, whether that is sets, or sides of a formula, or anything else, "are equal". Only once these ideas were formed, someone came up with a shorthand for that, which is our equal sign.

As long as the idea of two things being equal continues to have some merit, the symbol will continue to exist. This is the case in many contexts - for example, in the original example of having one object and another, indistinguishable object, this is equal to having two objects: 1+1=2. I'm sure that no one is going to change this symbol in elementary school education anytime soon.

Even beyond that, the equal sign will continue to exist. The article is literally comparing apples and oranges, but even in that context we can make sense of the idea that something is equal to something else: if we only care about the type of fruit, then one basket with one apple and one banana is equal to another basket with one apple and one banana - and both are not equal to a third basket that contains an orange and a banana.

Second, if the symbol "=" is defined as "is equal to", it follows exactly what @amacbean16 already suggested. If there's another relation that makes sense (like "is equivalent to"), we can just come up with a different symbol and use it where it is appropriate, without poor equal sign becoming obsolete.

As has been mentioned, there's the "squiggly equal sign" meaning "approximately equal", but there's also three horizontal line often meaning "is identical to", or one squiggly line meaning proportionality, and so on. The last one, by the way, would probably be the most appropriate one for the papaya vs. oz example. If "1 papaya ~ 10 oz", then "2 * 1 papaya ~ 2 * 10 oz", or simply "2 papaya ~ 20oz". ;)

Last but not least, I don't think the article does a good job defining what "is equivalent to" actually means. They state something along the lines of both "apple, orange" and "orange, banana" being equivalent to "two fruit" - but if two things both are equivalent to a third, does it mean that they are also equivalent to each other? Is the statement "an apple and an orange is equivalent to an orange and a banana" something that you'd consider to be true? Why or why not?

The answer, again, is context. At a buffet where every piece of fruit costs a fixed amount of money, both examples actually are equivalent. In a different context, for example if you're concerned about your vitamin intake, or if you're allergic to apples but not to oranges, they are not.

• I think you hit the nail on the head. Context is what's key here and there are already are different symbols in mathematics that help to convey these different kind of relationships. I think what you're getting at, correct me if I'm wrong, is that the author of the mental floss article is over-thinking this.

So long as the concept of equals has a room in mathematics, I don't see it going away. Especially in traditional math classes ranging from Algebra I to Calculus and Statistics. Perhaps we'll see more uses for equivalency down the road, but that doesn't mean the equals sign will go away. That's my takeaway from this.

• I think what you're getting at, correct me if I'm wrong, is that the author of the mental floss article is over-thinking this.

Exactly. I just don't know if this act of over-thinking happened accidentally, or deliberately because "damn intellectuals want to make math even harder for our kids!!!" generates more clicks than "in math, there are different valid concepts all roughly similar to equality". ;)