Devlin's Right Angle, Part IV

Oh, gosh, I may have to go back on my promise to make this my last post on the topic.

A gentleman by the name of Joe Niederberger has been dominating the comment thread at Let's Play Math. His first comment is uninspiring:

Devlin unfortunately makes the mistake of thinking of multiplication as one “thing.” It’s true multiplication of any two real numbers cannot be simply reduced to repeated addition, however, the multiplication of any two integers *can* always be reduced (or thought of, or defined by) repeated addition.

Very well, then, Mr. Niederberger. Solve for a:

a + a + a + a = 8
If it is indeed true that the "multiplication of any two integers *can* always be reduced [to] (or thought of, or defined by) repeated addition, then it should be a snap to solve this repeated addition problem without recourse to multiplication (or its inverse, division). I certainly could be wrong, but it seems to me there are two choices if you actually want to solve this problem within the restrictions given (and without guessing or modeling) rather than simply define it: (a) you can subtract a from 8 four times, or (b) you can subtract 4 from 8 a times, which wouldn't make any sense, given that we don't know about multiplication. Either way, you're up a creek without a Peano (or with one).

Update: A couple of astute readers--Bonnie and Steven--have E-mailed me good solutions to the problem above that do not involve multiplication. Hey, I am always willing to admit my mistakes when they are pointed out to me. (Keep in mind however that the solutions sent to me were, nevertheless, not "a snap" by any means. Far from it.)

Here's Joe again, on the comment thread:

I’d like to repeat a key point that I make: even Peano in his axiomatic defintion needs to define multiplication of whole numbers with a recursive (repeated) addition definition.

That’s just the way it is and Devlin has not offered an alternative.

Niederberger ignores a great deal here. As I alluded to before, just because you can define something in terms of something else, does not make it mathematically useful, which is why I presented my algebra challenge above. I think it's wonderful that Joe can call upon Peano to define multiplication:

You can refer to a previous most of mine that gives the Peano recursive definition of multiplication that uses addition. You can do it more informally by saying that MxN (M,N integers) denotes a function whose definition is given as MxN = M+…+M (N times). That’s all anybody means when they say that.

But if the only way Joe can use that definition is to rewrite multiplication expressions as repeated-addition expressions, then it's not very useful in our present discussion. And it certainly isn't very useful in solving mathematical problems. So why should we think of it as something necessary to plant into our kids' brains?

By the way, there IS a way to define the operation of multiplication on natural numbers that makes no use of the concept of repeated addition. Next time. Got some pacing to do.

Part I | Part II | Part III | Part IV | Part V | Part VI | Finale