Talk:Multiplication

Not just repeated addition

I am moderating the language, "multiplication is defined by repeated addition". There are serious objections to "defining" multiplication in this way from the "more basic" operation addition, first among them being that with no modification, the statement is wrong. See the articles [1] and [2] by noted mathematician and expositor Keith Devlin for more about this. The second gives references to educational authorities who agree.

You would have been right to challenge "multiplication is defined by repeated addition" if that had been what was written. The wording "Multiplication is defined in terms of repeated addition" (not mine incidentally) actually means something different and rather more subtle.

I'm not happy with introducing non-commutativity into an article at this level in this way. If you begin by defining multiplication as the "binary mathematical operation of scaling one number or quantity by another (multiplying)" and one of the "basic operations in elementary arithmetic" then it is commutative without question. Matrix multiplication etc probably belong in a paragraph at the end about generalisations. If you want to introduce such relatively sophisticated concepts into an article about elementary arithmetic, then "The most general context in which a multiplication operation exists, encompassing all of the above examples, is that of the abstract ring" is not correct (or at least a pure matter of opinion) either. The most general context is a magma, a partial binary operation with no other properties. Richard Pinch 07:30, 17 December 2008 (UTC)

What is the subtlety? Does the original wording work when the multiplier is allowed to be irrational or a negative integer?

I was actually thinking of moving some of the stuff to an advanced page, but since matrices had already been mentioned, I just stuck it on the main page for now. Is it best to just list other types of multiplication in the intro, and then to describe them on their respective pages? I still think the words "of numbers" should appear next to the commutative property.

I don't view multiplication as a stand-alone binary operation. Perhaps its semantics, and perhaps I am misinformed, but in my experience, the word "multiplication" rarely occurs without some sort of associated "addition" operation to be distributive over. (Exception off of the top of my head: Cartesian product). Anything else is just called an "operation". Sure, in group theory, you can show that the multiplicative group of nonzero real numbers forms a group and ignore the additive structure, but you still know about addition in the back of your mind. I suppose some loosely say "product" when combining two elements of a magma, but that is for convenience. You still formally define "magma" by saying "binary operation", not "product" or "multiplication".

I hope there is some linguist who studies mathematical vocabulary and wrote up detailed etymologies and usages. I'd love to have such a book. All I have to go by is my experience, which is very limited -- not just Americocentric, but SouthernCaliforniacentric. "Product", "operator", and in a different vein, but new to me, "scholion". I give them precise meanings from experience, but they are probably not always the same as others' meanings.

I'll go ahead and remove the matrix bit from the commutativity, but leave in the stuff in the first paragraph.Barry R. Smith 16:18, 17 December 2008 (UTC)