Search All of the Math Forum:
Views expressed in these public forums are not endorsed by
Drexel University or The Math Forum.
|
|
|
|
Re: Just what is equality in mathematics, anyway?
Posted:
Apr 16, 2012 4:10 PM
|
|
On Mon, Apr 16, 2012 at 10:55 AM, Joe Niederberger
<niederberger@comcast.net> wrote:
> Let me add, Paul, I see you all the time trying to "prove" or settle such matters by mentioning say, theorems from algebra and applying to something in
> logic or some other such obliviousness to context
> and boundaries.
>
> I'm sure its very useful to compare and contrast the foundations of various branches of math, but the way you do it just seems confused to me.
>
> For instance, I was alluding to an entire chain(s) of constructions that lead from N up to C. Those games are played for certain purposes at hand. In *those* games, every complex number is an "ordered pair* of real numbers, and no real number is an ordered pair of real numbers.
>
> To suppose one can cut through that by invoking ring axioms, I'm afraid is to just not get it.
>
>>(1) (x,0) = x + 0i = x + 0 = x for all real x.
>
> I'll just say your mistake here is right after the first equal sign - *if* you are talking about a particular construction of C in which the elements of C are always
> ordered pairs of real numbers, then in *that* construction (x,0) = (x,0) always and NEVER (x,0) = x + 0i. There is NO "i" in that system!
>
> Of course you could write (x,0) = (x,0) + (0,0).
> Perhaps you can analyze that.
>
> Joe N
>
Although the above is almost a double post, I will still reply by
linking to my full reply in this thread, this reply found here (it
includes a fixed typo),
"Re: Just what is equality in mathematics, anyway?"
http://mathforum.org/kb/message.jspa?messageID=7769795
in which I show by citing Rudin that it is not a mistake.
|
|
|
|
