I am a father of two young boys and I looks forward to exploring mathematics with them for as long as they will let me :-). I would really like for them to have a deeper understanding of mathematics than what I had when I was a young student. As I think about how I might approach some of the topics, there is one that remains particularly unclear to me to this day - the multiplication operation. Now I do not have a strong background in mathematics (e.g. never had a course in abstract algebra), so please forgive me if some things that I say are off - maybe even way off.
I have seen that there have been debates online as to what multiplication is, and how to teach it to students. Often the discussion turns into interpretations of multiplication (e.g. repeated addition, scaling, etc.) but the discussions/debate from this approach seem to be fruitless. Other times properties of multiplication are discussed, but often the properties are the same as those found under different types of operations. Integer multiplication may be associative, but so is integer addition - leaving me no more informed about the unique and universal thread for the concept of multiplication.
From my perspective, I am most confused by the many definitions for the multiplication operation depending on the type of objects of interest (real numbers, complex numbers, matrices, etc). I always think to myself, "why would mathematics allow the same name to be associated with multiple definitions?". It seems like there must be something that all the definitions must have in common. Surely, not just any binary operation on a set of objects can be labeled multiplication on a whim...or can it? So this is my question, is there a characterization of the multiplication operation that holds true for all operations labeled multiplication, that it is agreed on within the academic community, and is unique enough to be able to distinguish it from other operations (namely addition)? If so, please do share. And if not, how would you explain why the same term has various definitions in mathematics to students learning about operations like multiplication?
From my limited mathematical knowledge, it appears that the only thing in common with different definitions of multiplication on different objects is that that they all rely on the use of the addition operation in their construction. So perhaps the term addition is used to reference an operation for a set of objects that is considered to be the simplest method for combining/connecting two objects in a set, and multiplication is a more complex method for doing so (perhaps based on the use of simpler operations, like addition, already defined for the set). But, I would prefer that my discussion with my sons not rely on my experience. Hence, the reason for the post. Many thanks for taking the time to review my write up and I look forward to any insight that may be offered.