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Re: Non-Euclidean Arithmetic
Posted:
Sep 19, 2012 11:57 AM
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On Tue, Sep 18, 2012 at 12:04 PM, Joe Niederberger <niederberger@comcast.net> wrote:
> Paul Tanner II says:
>>Repeated addition can compute products only if the products are computable - and since almost all real numbers are noncomputable, almost all real number multiplication is not computable.
>
> Is it possible you are getting somewhere Paul?
> That's exactly right. Neither you, nor Devlin, can multiply two real numbers in general.
>
> You can't use a "scaling process" to do so,
> nor can you do so through "repeated adamant insistance".
>
> You can assert the product exists, but that's a different game.
>
First:
You are again trying to limit what a term can mean, when you say that we cannot multiply any two real numbers.
That is, you are claiming that the verb "multiply" can be used legitimately only if we *compute* the product.
Of course we can multiply any two numbers or abstractly speaking any two elements of a set, for that matter, since this usage of the term is common.
To see that we do not have to actually be able to compute the product to legitimately use the term "multiply", for the abstract context enter such as
"multiply" group element
"multiply" ring element
"multiply" group elements
"multiply" ring elements
with the quotation marks included at Google, and get tens of millions of hits each time. And when we restrict the usage just to the real numbers or similar such numbers that find their way into the Google hits, try
"multiply" real numbers
and see about four million hits, including some on multiplying imaginary numbers.
Second:
The term "scaling" has been used by me or Devlin only in the sense of modeling real number multiplication, not in the sense of *being* real number multiplication.
You still seem to argue as if the models *are* what they model. I and he are trying to get people to see that this is not true, regardless of the model, scaling and repeated addition included, and regardless of whether whether the model is one in which we can compute.
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