# Discussion Constructing The Counting Numbers

Many people claim to be interested in learning more
about mathematics, both content and methods. This
is the start of a series that is intended to bring
readers up to speed with how to think mathematically,
starting with things we know, and then progressing.

It starts with Constructing the Counting Numbers,
and this page is available for discussion.

## Comments from Snapey1979

 These are good comments, and I'll revise it shortly. Thanks.
I'm not exactly sure of your audience here, which caveats my thoughts. however, I like the idea of formalising what are usually taken as intuitive concepts. Also, it is always good to see a different angle for explaining maths - there is always someone who needs that having not "got on" with the way they were taught.

That said, here are some ideas:

- I think that, given you are trying to build a case from nothing, exercise 1 involves a rather technical understanding of "prove". I would number the lists of defining statements about equivalence relations above and add something after "prove" (i.e. use only the rules 1-3 above)

• Done that.
- Make clear that exercise 1(b) is not inheriting ANY of the rules of 1(a) above.
• Done that.
- Similarly with "There's a matching". I think you're saying - "if for two collections, we can "pair off" each member of one collection with a member from the other (which may or may not have anything in common) and there are none left over from either class". I know you are driving towards equivalence classes for each number. But I'm not sure the definition of the matching operation is completely clear. This makes it hard to do Exercise 2. For instance, for many people intuitively, I would say that they would see that {apple, orange, banana} can be matched with {orange, banana, apple}, but not with {philosophy, physics, politics}, whereas you want those to be matching. Even worse, they might think (because however hard we try we can't switch off our intuitive pattern matching) that {car, green, physics} would match {road, apple, gravity} but not {fish, cake, chips}
• Done that.
- In defining addition: "We've taken two specific sets and formed their union" Suggest: "we've taken two collections and formed a new one, consisting of all the members of the original collections. This is where we start talking about sets - we have taken two specific sets and formed their union"
• Done that.

 This page was open for editing by people in general, but it's been found by spammers, so I've had to lock it down. Shame, really, but that's why we can't have nice things.