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?