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  and  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)
- Oh, you already moved the more advanced stuff. Now this is in that "difficulty pipe" format that Wikipedians like to use, but I personally dislike. But as this page is aimed at university-educated folks, I don't think all of the stuff in the last paragraph needs to go. Certainly there should be some indication in the article that you can "multiply" things that aren't "numbers".Barry R. Smith 16:22, 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?" No, of course not and I don't think it was intended to. Multiplication can be defined in terms of repeated addition in the sense that multiplication by positive integers can be defined as repeated addition (if you choose) and multiplication by all other complex numbers can be defined building on that concept (which is the extra subtlety I refer to). For example, multiplication by negative integers as sign-change and multiplication by positive integers; multiplication by rational numbers as taking a common denominator and multiplication by integers; multiplication by real numbers as multiplications by elements of a Cauchy sequence; multiplication by complex numbers as ordered pairs of real numbers. Just saying multiplication is a "basic operation" does actually give any reason (beyond assertion) to suppose that any operation at all exists with the desired properties. To say that it can be defined in terms of some other operation gives a clue that it can be defined in such a way as to make the claimed properties hold and also a hint as to how that might be done.
- As for "multiplication as a stand-alone binary operation" -- you mentioned one, namely Cartesian product (which actually distributes over disjoint union). There's also cardinal multiplication and ordinal multiplication, which do not distribute sensibly over their respective additions. None of these form a ring. Richard Pinch 18:23, 17 December 2008 (UTC)
- I see your point with the "in terms of" vs. "by". Devlin's main point, and that of the sources he quotes, is that encouraging the idea from the outset that multiplication is a special case of addition causes subtle problems in many student's understanding further down the line. They are not problems with understanding multiplication computationally, per se, but problems understanding that the most useful multiplication for people who intend to use it (that the average "educated" person would want to understand, and that we should be teaching all students) is multiplication of real numbers (if they need complex numbers eventually, they will be sophisticated enough when it appears to add it to their repertoire). Teaching first that multiplication is repeated addition, and then changing that when you get to negative numbers, and then again with rational numbers and finally real numbers contributes to the idea that mathematics is a sequence of arbitrary rules that one has to memorize and then the hard part, figure out which to apply in which context. The idea that the earlier types of multiplication are precise specializations of the more general types is lost. (I think they neglect to emphasize that you can move in the reverse direction, showing that each more general type is defined "in terms of" earlier types, as you do). A substantial number of adults, when asked what multiplication is, respond "repeated addition". This limited understanding also contributes to misunderstandings about properties of multiplication that hold only for specialized types. For instance, many people assume from experience with positive integers that "multiplying a number makes it bigger". Devlin advocates teaching instead that multiplication and addition of real numbers are each basic and then showing or having people discover that there is a connection in the realm of positive integers.
- Leaving the article as it was, multiplication "in terms of" repeated addition, would not have been enlightening to those who do miss the underlying organization. I imagine the typical person who needs this page as being someone who can multiply small positive integers, but struggles with multiplication in other contexts. The struggle, I imagine, comes from a lack of understanding of the connections between multiplication of general and specialized types.
- I would be happy if your description of "in terms of" is inserted in an appropriate place. Or I would be happy if each type of multiplication and its reliance on earlier definitions are split up throughout the article. In any case, descriptions of multiplication of different types should be discussed in detail.
- Before this is done, an organizational plan should be agreed upon: is it preferable to build up from specialized types to more general types, as is the logical progression and as is often done in school? Or is it preferable to start with multiplication of real numbers, since it seems we are agreeing that that is the main context of this article, say by working with decimal expansions, and then show that there are simpler descriptions of this operation when multiplier is rational, or a negative integer. I am inclined to the latter, because I see the "imaginary reader" as someone who probably learned multiplication from the former perspective, has encountered already different types of multiplication, but now needs a better understanding of the underlying unity.
- Addendum: I never spent the time to learn about ordinal arithmetic, although I should. I retract my statement that "the most general context" where one uses multiplication is in a ring, although I don't retract that multiplication "rarely occurs" outside of this context.Barry R. Smith 19:11, 17 December 2008 (UTC)
- There might well be scope for an article explaining how these sets and operations can be built up from naive set theory (a technical term in this context). It was a 24-lecture course when I did it ... Richard Pinch 19:44, 17 December 2008 (UTC)
- If it takes 24 lectures to do in detail, perhaps it could take up two articles...I seem to remember that Halmos regretted using the word "naive", which he thought was a bona fide technical word meaning "non-axiomatic". I guess at the time, not everyone agreed.Barry R. Smith 20:13, 17 December 2008 (UTC)