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Topic: Non-Euclidean Arithmetic
Replies: 33   Last Post: Sep 21, 2012 2:48 PM

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 Paul A. Tanner III Posts: 5,920 Registered: 12/6/04
Re: Non-Euclidean Arithmetic
Posted: Sep 12, 2012 3:26 PM

On Wed, Sep 12, 2012 at 1:03 PM, Joe Niederberger <niederberger@comcast.net> wrote:
> PT III says:
>>Take a standard x-axis and y-axis in the Cartesian plane, and for sake of simplicity, name the points on these lines according to the fact they are each a real number line. Then:
> [etc. etc. etc.]
>
> But I just explained that most reals (with probability 100%) have no names. - there's a problem here.
>

The term can be used in more than one way. When I said "name" I meant, for sake of simplicity as I explicitly said, instead of saying (x,0) just say x. That's a type of naming of an arbitrary point on the x-axis.

There is therefore no problem here.

>
> Even for some that have "names' I just couldn't "home in" on them.

So what? You talk as if you have not heard of "without loss of generality" - look it up. It's standard in mathematics. (I used it a number of times when proving theorems during obtaining my math degree and when teaching geometry. By the way, what degree or degrees do you have? What's your training in mathematics?)

Look at it again - in fact, it's a proof of the following theorem using "without loss of generality": Given real number (point) 1 and any two positive real numbers (points) a and b, we can construct the exact location of real number (point) ab on the real number line:

Connect 1 on the x-axis to b on the y-axis, and, parallel to that drawn line segment, connect a on the x-axis to the y-axis, and this point on the y-axis is ab. That is, in terms of distance from 0 or magnitude or absolute value: ab is to b as a is to 1 - that is, written in terms of ratios or proportions: ab:b as a:1.

And via commutativity we have the other way as an alternative:

Connect point 1 on the x-axis to point a on the y axis, and, parallel to that drawn line segment, connect b on the x-axis to the y-axis, and this point on the y-axis is ab. That is, in terms of distance from 0 or magnitude or absolute value: ab is to a as b is to 1 - that is, written in terms of ratios or proportions: ab:a as b:1.

Note: Your "home in on" talk and your need to name things says that you really need to learn about "without loss of generality" before you speak on this again.

>
> It looked like a process, smelled liked a process, but there's just no carrying it out in general.
>

It's a process.

It is as I said: You are disallowing whatever uses of terms like "process" to satisfy your preconceived conclusions.