On One Hand But Then on the Other

By Bobby Neal Winters

I was approached yesterday by a friend of mine at church in Opolis in the following way.

“You’re a math professor, so I got a question for you,” he said.  “It’s from my grandson.”

I remained calm outside, but on the inside I had the reaction a gunfighter in the Old West had whenever someone said, “They say you’re pretty fast.”  You never know when the person asking the question might be faster.

I decided to take my chances and listen.  He held out his right hand and began to count extending his fingers one by one:

“One, two, three, four, five,” he said, extending his pinky last. “And five is ten,” he said extending all of the fingers of his left hand at once.

I nodded because I knew we weren’t at the hard part.

He then held out the fingers of his left hand and started extending them thumb first:

“Ten, nine, eight, seven, six,” he said, giving emphasis to the six. “And five is eleven, so you got eleven fingers.  How can I explain to my grandson why this is wrong.”

And my friend knows it’s wrong.  But knowing it’s wrong and being able to explain why are two different things.  And I could go on to a digression about politics here, but my point is math, or at least the explaining of math.

My friend knows this is wrong because we have ten fingers and ten does not equal eleven, no matter how fast you talk.

Let’s first analyze how this is presented because that is very important to how the confusion comes in.  We start off with something that is true: one, two, three, four, five, and five is ten.   What has happened there?  We’ve listed off the names of the first five numbers that we learned when we learned how to count.  We’ve set up a correspondence between those names and our fingers.  

The next part is where something subtle is done which sets us up for the confusion. When he says, “And five makes ten,” he’s made a very subtle shift.  He’s using the name five to refer to the quantity of fingers on his left hand.”  The five he said when he held out his pinkie was referring to a place in order; the five he said when he held out all his fingers at once was referring to the quantity of fingers.  Mathematics call the first one ordinality and the second cardinality; think of these as order and quantity.

In listing the numbers in standard order, “one, two, three, etc,” the name of the last number listed is also the name of the quantity of the items in the list.

This does not work when you count backwards, and that is one of the things that the example my friend brought me illustrates. In counting backwards from ten, there is no point at which the number counted with also be the quantity of things counted. (Ironically, if he’d started counting at eleven it could be made to work, but you can’t do that because you have ten fingers. It is the center of this trick that you start counting backwards at ten because you know there are ten fingers.)   

When you hold out the pinkie on your left hand as you count backwards from ten until you get to six, you are giving its order as if all the fingers were counted beginning with the other hand.  It can refer to a quantity, but that quantity would be the pinkie itself and the fingers on the left  hand.

Now, you must understand that I didn’t tell my friend all this.  I said, “It’s a confusion between cardinality and ordinality.  Don’t let your grandson play poker with Bill.”

Bill is a man who’s given me poker lessons. After $15 worth, I learned: don’t play cards will Bill. But that is a different story.