I have a few other posts saved as drafts, and I want to get to them, especially one on technology, but I can’t pretend to have finished talking about that alternative decimal representation of the number one (0.999···) that carries with it an infinite chain of nines. To say that I’m comfortable with that representation of the number is almost as brash as claiming to have solved Zeno’s Paradox. Infinity is a funny thing. In fact, it would be more honest to say, infinities are funny things. After all there are lots of them. And there’s no reason they should all be the same—in fact, they are not.
But before we dive off the deep end, we should pause to think about what it means for finite numbers to be the same. Have you ever given much thought to statements like “5=5”? So, now, before you read on, turn to the nearest 8 year old you can find and ask her, “Does five equal five?” [That’s the easy part.] Now ask her, “How do you know?” and listen for a response. [That’s the medium-hard part.] If your 8 year old doesn’t sufficiently convince you that five is, indeed, equal to five, explain to her why it’s true in plain terms that no one could dispute. [That’s the hard part.]
When I was taking an abstract algebra course, my professor asked the class what the number three is. We all knew it was a trick, so we waited in nervous silence, each hoping that he wouldn’t start dead-calling members of the audience. After all, college math concentrators should know what the number three is. We were studying math, and numbers, I’m told, are an integral part of mathematics. So what was the Fields medalist‘s definition? He stole his answer from a six year old: the number three is three fire trucks without the fire trucks. Hold up, what? That’s a surprisinglyl useful way to think about number. I think that’s how Frege and Hume viewed number, and by some standards, they’re famous. So maybe this little kid is on to something.
Let’s pretend for a little while. You didn’t know this, but I have a pack of llamas. Every evening I feed them each one carrot for dessert at dinner time. The problem is, I can’t count. I have a bunch of carrots and a pack of llamas. How can I know if I brought out the same number of carrots as the llamas—without counting?
Well, I could feed the llamas each one carrot. If I had more carrots than llamas, then I’ll have carrots left over at the end. If there were more llamas, then I’ll run out of carrots before I finish feeding—the llamas hate that. But if the number of carrots and llamas equal, then after feeding time, each llama will have had exactly one carrot and no carrots would remain. That is, I would be able to put all the carrots in a one-to-one correspondence with the llamas. If I had more carrots than llamas, then some llamas would get the left-overs. That relationship is not one-to-one because some llamas get more than one carrot.
But we’ve got a problem, one-to-oneness is certainly necessary for there to be the same number of carrots and llamas, but it is not sufficient. If I had fewer carrots than llamas, every llama who gets a carrot gets only one. But because I run out of carrots before I run out of llamas, some llamas are left out. In order to know whether there are the same number of carrots and llamas, every llama needs to get exactly one carrot and there can’t be any carrots left over. This is tricky business.
If every llama gets at least one carrot, then we say that the pairing of carrots to llamas is onto. Ontoness is also a necessary condition, but like one-to-oneness it is not sufficient. Having more carrots than llamas leads to a pairing that is onto. Every llama gets one carrot and some get more. What we’re looking for is a matching of carrots and llamas that is both one-to-one and onto. Then for every carrot there is exactly one llama. Likewise, for every llama there is exactly one carrot. The size of the pack of llamas and the size of bunch of carrots is the same. Mathematicians like to have a standard way to talk to one another, so they call the number of elements in a group the group’s cardinality.
So, we’ve done it! We’ve found a way to determine whether two collections of things are the same size, if they have the same cardinality, without resorting to counting. In fact, we’ve secretly discovered a very powerful way of thinking. One thing we can do with our new-found friends, the one-to-one and onto functions, is define what it means to count. I don’t have the room to do it here. In fact, I don’t think we’re even going to get to infinity in this post. Instead, I’ll cop-out and refer you to some of my set theory notes. (That’s what we were doing: set theory.)
In my notes, I’ve asked just a number of questions. I wrote these questions for a practicum for a class it took this fall. And I used them on real, high school juniors in at Codman Academy. We had a little bit more time to go over each of the questions carefully and resources that allowed dynamic, colorful diagrams—which the students largely produced for me. I was only asking questions, it was up to them to answer them. But please, don’t wait until 11th grade to feed the llamas.
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