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Re: Non-Euclidean Arithmetic
Posted:
Sep 16, 2012 2:34 PM
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On Fri, Sep 14, 2012 at 11:40 AM, Joe Niederberger
<niederberger@comcast.net> wrote:
> PT II says:
> - ------------------------------------------
> A proof is a process, especially a constructive proof like the one I gave in
>
> http://mathforum.org/kb/message.jspa?messageID=7889634
>
> of how to construct the location of product ab on the real number line
> from being given the locations of 1, a, and b on the real number line,
> where a and b are arbitrary reals.
> - ---------------------------------------------
>
> Oh for heavens sake Paul, you didn't prove anything in any meaningful sense of the word. You described a rather standard construction that illustrates the relationships between line lengths and real number products. If you want to call your describing thereof a "process", I don't care.
>
> But the process we were discussing is supposedly one that takes any two real numbers in general and produces an answer. Any grade school kid knows what that means for small integers: What is 2 times 2? Answer is 4. What is 3 times 7? Answer: "it exists!" -- uh, no, 21; detention for you Johnny. They could even use various processes to get to these answers.
>
> So for this one aspect, and a very important one, there is this vast gulf between multiplication viewed as something that takes two numbers as input and produces a number as output, and real number multiplication for which no such process exists. For other aspects, there is common ground.
>
> Perhaps you should brush up on grade 2 concepts.
>
I in the post
http://mathforum.org/kb/message.jspa?messageID=7889634
proved the claim that we can find the location of product ab on the real number line given the locations of 1, a, and b on the real number line for all positive real numbers a and b. Look at what I wrote in that post above, and this is the claim being proved by the construction.
Illustrating in one way or another the relationships between the line lengths of numbers a and b and the line length of their product ab is exactly what Devlin was getting at when he said that we should give a model of multiplication to kids that holds up all the way through the reals, namely scaling.
You seem to be stuck on the idea that a model of an arithmetic operation is no good if it's not a computation that yields a computable result.
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