/ Pratik Mallya
My experience in childhood with learning was one of accepting what authority figures said as Gospel and then relying on those as the laws of nature. The circle of authority started with parents, then extended to teachers, then to books etc. I distinctly remember the first time I read a book on something I wanted to learn more about (this was in an unimaginable world before internet and cell phones) and the stupid realization that … what if there were books that could explain anything? That seemed too good to be true. But you can explain an awful lot of stuff with the existing theories of things and it took quite a long time for me to hit the frontiers of human knowledge (e.g. what is life? Is P = NP? etc.).
One serious drawback that left me very confused for the longest time was what did people mean when they say “its not defined”. In my simple worldview, everything that could be known was either known or was being worked out as we speak. My mental model couldn’t understand what it meant when division by zero was “not defined”. What did that even mean? Part of the confusion was a somewhat strange belief that mathematical theory had to have some relation to the physical world. I understood the number 3 because I understood the concept of 3 apples (i.e. the abstraction represented by the symbol 3 made sense). So -3 didn’t make any sense to me. What does it mean, negative three apples?
Natural numbers are a theory which does have basis in the “physical world”, or can be visualized as such. Negative integers? They may not, but that gives us the opportunity to accurately define it based on certain properties. If that definition and understanding is shared among everyone, it becomes a useful concept and system. (One way of defining negative integers is as additive inverses of natural numbers, where
additive inverse is yet another construct in this theory). So “not defined” means exactly that: in the theory that we’re describing, an entity or operation that is “not defined” is one that we explicitly decide is an exception, or doesn’t follow, or simply cannot be accounted for, so is explicitly called out as something that is “not defined”. The entity/operation could have multiple results, no results, results that are “not defined” etc. Division by zero is not defined. Square root of -1 is not defined in certain systems, but is defined in the system of so called complex numbers.
So that’s what “not defined” means. It means exactly what it says it is. I just didn’t believe it for the longest time.