Posted by: groupact | July 4, 2013

What Is A Number?

At PCMI this summer, the focus of our Reflecting On Practice class is mathematical tasks. What makes a task worthwhile? How can we adapt a task to make it more worthwhile for our classroom? What should we be considering as we implement the task in our classroom?

On the first day we had to answer the question, “What is the last cool thing you did in your classroom, and WHY?” For me, that last cool thing was simply devoting a full most of a class period to a discussion on “What is a number?” (I had students start the discussion at their table, followed by sharing an idea from each table, followed by a whole class discussion). I’ll follow up some time next week with what my goal was for this task and why I thought it was so cool. I already plan to use the prompt (with perhaps another structure to the discussion) in another class next year, but first I’d love to hear others answer this question. Once we can answer this question, we can answer Paul’s question of “What is a numeric illustration of the fact that a negative number multiplied by a negative number is a positive number?” So, please, fire away in comments or tweets.

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Responses

  1. I mentioned this on twitter, but I feel unequal to the question. It might be interesting and productive to think about this together, though, so here goes an attempt to make sense about this:

    I think that “cow” is a concept that’s real and true — that is, sentences about cows are true, and they’re *really* true. By *really* true, I mean that I’m not using some watered-down sense of truth like “Oh, when I say “Cows are large’ I mean that what I personally feel and mean when I utter the words ‘cows’…” None of that. Talk of cows is super-duper true.

    But what is a concept like “cow” anyway? Here’s a sloppy, first draft account: we go through the world and observe objects, and sometimes objects have common properties. A concept unifies objects under some commonality.

    (But wait, that can’t be right. “Unifies objects”? Like, you can’t have a concept that only applies to one object? I’ve just named the concept “MPershan” to describe someone exactly like me in every way. Is that not a proper concept? What about a concept that adheres to nothing at all? Philosophy is hard. I don’t know.)

    Anyway: you’ve got a bunch of objects, they have some stuff in common, and you name it. “Cow.” Bam.

    And I think of numbers similarly. You observe sets of objects. (What’s a set? Oh shoot. Philosophy is hard.) In order to unify them (i.e. a certain arrangement of stones; a certain herd of cattle; a certain collection of fruit) you apply a concept to them, something like “equality.”

    And from that, everything can follow. I connect this thought — or at least a better version of it — to someone called Frege, and what’s sometimes called Frege’s Theorem:

    “All truths of arithmetic follow logically from the principle—seemingly obvious, once one understands it—that the number of Fs is the same as the number of Gs if and only if the Fs are in oneone correspondence with the Gs.”

    And I think equality is just as valid a concept as “cow” or anything else. It serves a similar role in unifying our observations of the world, and if our thoughts make any sense at all, then these sorts of concepts are going to pervade them.

    • Or, more succinctly:

      Equality is a concept about certain collections of objects, and number emerges pretty easily out of talk of equality. So a “number” is a particular class of equivalent objects. (And though it’s been a few years, my memory is that this is pretty much what Frege says.)

  2. Thanks! I probably won’t be able to get to the Frege or Shapiro until after PCMI, but I will get to them. In the meantime, I think I understand the main idea. The key seems to be one-to-one correspondence. So the number ‘two’ describes that aspect that all sets which can be put in one-to-one correspondence with my hands have. Two eyes. Two feet. Two cows, etc.

    It seems this concept of number would not only include 0 (the equivalence class of the empty set), 1, 2, 3, 4, etc. but would also include (infinitely) many infinite numbers since not all infinite sets can be put in one-to-one correspondence with each other. I like that inclusiveness. Would this concept include one-half as a number? Certainly there are different things that we recognize something in common that we call one-half. One-half a pizza, one-half a dollar, one-half an inch. But I’m thinking those are really just one’s with a different unit in the sense that I could form a one-to-one correspondence that maps my half of a pizza to my mouth. I want to give this more thought, but I like the idea of one-to-one correspondence. It did not come up (at least in those terms) from my students responses, and I’m trying to think how I would incorporate that concept if it did. As I said, I promise to share more on those responses and my goals on Sunday. THANKS!

    • I like the “infinite numbers” thought. But I think actual infinity could be — at least potentially — problematic.

      All the sets that we referred to above were concrete, real ones. There are actually this many hands. There are actually this many toes. There are actually this many people in the world.

      But are there actually an infinite number of objects in the world that we can draw on to construct our collection of sets of infinite size? (This is similar to a problem that Bertrand Russell ran into during his writing of the Principia Mathematica.)

      You ask whether this concept would include one-half. I think that the easiest way to do this would be to construct ratios and rational numbers in terms of the integers, in the normal way. So that one-half refers to a sort of way of comparing 1 and 2.

      Though, this gets back to your original question — is there anything that unifies integers (as collections of one-to-one correspondences) and rational numbers? Just because something is made out of something doesn’t mean that it receives all the properties of what it’s made out of.

      By analogy: you make a cake out of flour, eggs, and sugar and other raw foods. That doesn’t mean that a cake is a raw food. So if rational numbers are constructed out of numbers, that doesn’t guarantee, on its own, that rational numbers are also numbers.

      So maybe a partial retreat is in order? Integers are one-to-one correspondences, and numbers are things that can be constructed out of those one-to-one correspondences?

      • I’m enjoying this conversation. I like the idea that it’s not clear that there is any concrete set which is infinite. The topic of infinity is one that my students love to talk about. I agree, though, that for now we should limit ourselves and the concreteness seems to be a reasonable justification for doing so.

        The next thing I think we must take some care to consider is negative integers. It seems to me that they are somewhat like rationals in the sense of being derived from our one-to-one correspondences, which are our basic building blocks which we might call natural numbers. I’m indifferent as to whether to include zero, a one-to-one correspondence with no objects, in this building block set of natural numbers.

        The idea of numbers as being things constructed out of those one-to-one correspondences is nicely inclusive. (I’m a big fan of inclusiveness) It would seem to include negatives, rationals, irrationals, and even complex numbers. It would also include infinites–they seem to keep popping up–as well as ordered paris (which in turn allows us to construct functions) as well as other sets. I think we either need to set some principles on what sorts of constructions are allowed, or perhaps take a different tact.

        It would certainly be reasonable to limit ourselves to saying that the natural numbers are the only true numbers (I want to say “real,” but that’s probably a bad idea) and other things that we sometimes call numbers, like rationals, are only called “numbers” sometimes out of convenience because they bear enough resemblance to the natural numbers. Even then, though, we could consider in what way(s) they resemble each other.

        As promised, I’ll talk about where I was going with this with students on Sunday, but I like that there are so many ways to go with this.


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