A recent note to amte had a statement that caught my eye: <<<<<<<<<<<<<<<<<<< However, if you take "plus" and "minus", there you start running into ambiguities. "Plus" can be a binary operator of addition, or it can be a unary operator of direction in the case of defining the integers. >>>>>>>>>>>>>>>>>> Let's not go overboard. Plus (+) refers only to the operation of addition, there is no unary operator of positive direction. Just look at your calculator. There is no + put in front of positive numbers (it would be redundant), the button that has + on it is for addition and that alone. There is a button with plus/minus which is used to change the sign of the number in the display (i.e. 3 goes to -3, or -7 goes to 7). It is enough of a problem that minus (-) is used in two quite different senses -- don't compound this by unnecessarily having two senses for +!
The questions about one/more meanings for subtraction and for multiplication seem to be forgetting history -- that both of these operations came into existence by generalizing/extending previous operations and only part of the original meaning could be carried along in the extended domain of the operations. This is typical. When you go from the reals R to the complexes C in order to solve more equations, you have to give up the possibility of having an order relation on C compatible with the rational operations (you can still use it on R of course). When you generalize multiplication from R or C to 2x2 square matrices, you retain many of the properties but you give up commutativity and you get zero-divisors! When you generalize multiplication to real-valued functions on some domain you keep commutativity but you get lots of zero-divisors, etc.
Originally subtracting 7 only applied to numbers bigger than 7 and meant taking away 7. Then negative integers were included with the usual order extension, and we extended the domain of subtraction to all integers so that 3 - 7 meant decrease 3 by 7 to get -4, and similarly for -10 - 7. But in fact we also extended addition (+) to apply to negative numbers and so we had to worry about 3+(-7) and whether that was the same as 3 - 7 (subtraction), and then 10 - (-7), etc. Eventually people checked this all out and by 1600 or before it was agreed that it seems to work consistently and efficiently -- which is what we need for modeling and applications.
Similarly for multiplication. Multiplication such as 5*39 may indeed have meant repeated addition with 5 summands each being 39. [And from counting rectangular arrays of objects we observe that we get the same result if we calculate 39*5 that way.] When multiplication is extended to cover negative integers as well, then we still can think of 5*(-23) as repeated adding of -23, but as the recent discussions of neg*neg=pos point out we can't reasonably do that for (-5)*(-23). And of course when we extend multiplication to rational fractions we also can't retain this repeated addition idea for all products (we can for some of course). That's the way it is. You give up simple meanings/explanations for more powerful tools/techniques. When you first start working in the extended domain you mostly proceed by analogy with the earlier situation (mathematicians used to talk a lot about retaining the same "form"). Then you practice a lot until it becomes automatic and you can start thinking about the important stuff -- how to use this newly acquired skill to model new phenomena and solve new problems.
Ladnor Geissinger Math Prof at UNC Chapel Hill & Math Chair at IAT email: firstname.lastname@example.org or email@example.com phone: 919-405-1925 address: Institute for Academic Technology 2525 Meridian Parkway, Suite 400 Durham NC 27713 USA IAT phone: 919-560-5031 IAT fax: 919-560-5047 IAT web home page: www.iat.unc.edu LEARN NC home page: www.learnnc.org Mathwright Library: ike.engr.washington.edu/mathwright/