According to the table, they used about 80 cups of popcorn seed each of the remaining days of the week. They will need 80 x 5 or 400 cups of popcorn

seed for Sunday-Thursday.

Although it is interesting that they arrived at the 80 by averaging popcorn seed over the 4 days Monday-Thursday and rounding down. If I wanted to make sure I had enough seed I’d be inclined to round up for any rounding.

]]>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.

]]>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?

]]>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!

]]>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.)

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