On Sun, Dec 30, 2012 at 1:06 PM, Joe Niederberger <email@example.com> wrote: ... > > I'll also add that Devlin with his view of "multiplication" as a *process* appears to be a bit confused between two opposing views. >
To provide the reader with some context so that he or she can see Keith Devlin meant, to see that he is not confused at all, here is part of what Keith Devlin actually wrote:
The concept I have of multiplication when I am working within mathematics is operational rather than ontological, and is built on the axiomatically defined, abstract binary function that constitutes one of the two fundamental operations in a field (or more generally a ring). I know the properties of multiplication, both obligatory (such as associativity) and conditional (such as commutativity) [this talk of a property under multiplication being obligatory or conditional is clearly a reference to the definition of rings], how it relates to other functions (e.g. distributivity), and whether or not a particular instance has an inverse. I am comfortable dealing with it. I don't ask what it is; all that matters are its properties.
I realize that my professional's concept of multiplication is abstracted from the everyday notion of multiplication that I learned as a child and use in my everyday life. But my abstract notion of multiplication misses many of the complexities that are part of my far more complex mental concept of multiplication as a cognitive process. That is the whole point of abstraction. Though many non-mathematicians retreat from the mathematicians' level of abstraction, it actually makes things very simple. Mathematics is the ultimate simplifier.
For instance, the mathematician's concept of integer or real number multiplication is commutative: M x N = N x M. (That is one of the axioms.) The order of the numbers does not matter. Nor are there any units involved: the M and the N are pure numbers. But the non-abstract, real-world operation of multiplication is very definitely not commutative and units are a major issue. Three bags of four apples is not the same as four bags of three apples. And taking an elastic band of length 7.5 inches and stretching it by a factor of 3.8 is not the same as taking a band of length 3.8 inches and stretching it by a factor of 7.5.
In fact, the nature of the units is a major distinction between addition and multiplication, and one of several reasons why it is not a good idea to suggest that multiplication is repeated addition, even in the one case where repeated addition makes sense, namely when you are dealing with cardinalities of collections. With addition, the two collections being added have to have the same units. You can add 3 apples to 5 apples to give 8 apples, but you cannot add 3 apples to 5 oranges. In order to add, you need to change the units to make them the same, say by classifying both as fruits, so that 3 fruits plus 5 fruits is 8 fruits. But for multiplication, the two collections are of a very different nature and necessarily have different units. With multiplication you have a multiplicand (written second) multiplied by a multiplier (written first). The unit for the multiplier has to be sets of the unit for the multiplicand. For example, if you have 3 bags each containing 5 apples, then you can multiply to give
[3 BAGS] x [5 APPLES PER BAG] = 15 APPLES Note how the units cancel: BAGS X APPLES/BAG = APPLES
In this example, there is a possibility of performing a repeated addition: you peer into each bag in turn and add. Alternatively, you empty out the 3 bags and count up the number of apples. Either way you will determine that there are 15 apples. Of course you get the same answer if you multiply. It is a fact about integer multiplication that it gives the same answer as repeated addition. But giving the same answer does not make the operations the same."
In addition, I add that all the properties of addition and all the properties of multiplication (including the property of multiplication as repeated addition and any asymmetry in the properties of addition and multiplication) merely from a few of the algebraic properties that are part of the definitions of a field or, close to most generally, of a ringoid. (See http://www.proofwiki.org/wiki/Definition:Ringoid or, where the convention is used of using the terms "addition" and "multiplication" for the two binary operations, http://mathworld.wolfram.com/Ringoid.html for more.
This means that one operation in a ringoid is not more fundamental than the other - that's right, in any ringoid, addition is not more fundamental than multiplication.
via the derivations of what I called "Very Little Ringoid Theorem", "Little Ringoid Theorem", and "Little Field Theorem" I demonstrated that we can derive the property of repeated addition merely from the relevant algebraic properties.
As for any asymmetry in a ringoid between addition and multiplication:
This asymmetry occurs in a ringoid (see the above definitions) in which the distribution property, the one that that "connects" the two binary operations, is asymmetrical, or is non-dual - that is, it goes in only one direction, meaning that one of the operations (conventionally called addition) distributes over the other (conventionally called multiplication) but not the other way around.
This asymmetry in the non-dual distributive property that "connects" the two binary operations of this type of ringoid fully explains the asymmetry between the binary operations we see in such examples of such a ringoid as the real numbers and its subsets like the integers.
There are ringoids in which the property that "connects" the two binary operations, the distributive property, is symmetrical - it goes in both directions. One example of such a symmetrical "connecting" property of the dual distributive property is what we see in such as Boolean Algebra. In such a ringoid, the asymmetry in question between the two binary operations disappears.