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Topic: Rudin and nonsense
Replies: 29   Last Post: May 3, 2012 9:46 PM

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 Paul A. Tanner III Posts: 5,920 Registered: 12/6/04
Re: Rudin and nonsense
Posted: Apr 30, 2012 12:54 AM

Well, denying what Rudin actually wrote down on paper and published is clearly crank territory, yet we have people doing it here at math-teach.

Look at their denials all throughout the thread

"Just what is equality in mathematics, anyway?"

These denials included you evidently, when you said, "Proving the unprovable not is likely to be really successful."

That is, you evidently claimed the equality (a,0) = a for all real a,b to be unprovable even though I proved it to be simply a partial instantiation (substitution instance) of Rudin's Theorem 1.29, the equality (a,b) = a + bi for all real a,b. That is, simply replace b with 0, and we have for all real a,

(a,0) = a + 0i = a + 0 = a,

(a,0) = a.

There is no reason why this partial instantiation in the form of this last equality may not be viewed as a corollary to Theorem 1.29, yet we have people denying it. Truly amazing.

On Sun, Apr 29, 2012 at 10:34 PM, Wayne Bishop <wbishop@calstatela.edu> wrote:
> I have been busy and not following this thread but, from what little I have
>
> Wayne
>
> At 09:55 AM 4/29/2012, Paul Tanner wrote:
>
> On Sun, Apr 29, 2012 at 4:05 AM, Didacticus <ddidacticus@yahoo.com> wrote:

>> Nothing I wrote suggested Rudin's choice was *arbitrary* -- I simply
>> pointed out it was a *choice*.

>
> Well, then, the entire textbook that Rudin wrote is the result of
> choice. And so your talk of "choice" means nothing.
>
> And this "choice" gives us Theorem 1.29, the *equality* - and not mere
> equivalence - (a,b) = a + bi.
>

>>
>> The nonsense I alluded to is you going in circles ...
>>
>> ... and then accusing people of "denying" Rudin when they dare to point
>> out the silliness.

>
> Why do you think Rudin is "going in circles"?
>
> If I'm going in circles, then so is Rudin, since all I'm doing is
> laying out in detail what Rudin did, where the sequence of
> *equalities* - and not mere equivalences -
>
> (a,0) = a + 0i = a + 0 = a
>
> and so
>
> (a,0) = a
>
> is the end of what Rudin did with definitions and theorems 1.26 through
> 1.29.
>
> The equality (a,0) = a that follows from Theorem 1.29 is so trivially
> and clearly true as an application of Theorem 1.29, Rudin did not of
> course bother to actually write it out - he left that to us. And so
> the equality (a,0) = a may be viewed as simply an unwritten corollary
> to Theorem 1.29.
>
> You need to process what I in my prior post
>
> "Re: Rudin and nonsense"
> http://mathforum.org/kb/message.jspa?messageID=7807322
>
> actually wrote. Here is the most relevant part that proves that your
> thinking that Rudin is going in circles is pure silliness:
>
> Quote: "That is, taken as a whole, Rudin proved isomorphism in Theorem
> 1.26 by which (a,0) and a for all real a are at least equivalent (note
> that I did not say "equal"), but then by replacement and by Theorem
> 1.29, we find that not only are (a,0) and a equivalent, but that they
> are equal as well."
>
> Your thinking that Rudin is going in circles results from your mistake
> in thinking that equivalence is equality. It is not. Isomorphism is an
> equivalence relation. By definition, equivalence is necessarily a
> relation that is reflexive, symmetric, and transitive but it not
> necessarily antisymmetric, while by definition equality is necessarily
> all four.

Message was edited by: Paul A. Tanner III