[FRIAM] anonymity/deniability/ambiguity

[Frank Wimberly]

Clinicians often call that "being oppositional".

You say that I've known authorities.  I was just talking to John Baez about
my advisor Errett Bishop, often called the inventor of constructive
mathematics.  Here is a constructive proof, with no use of the excluded
middle, of the irrationality of sqrt(2) that I found in Wikipedia.
Apologies to those who don't care:

In a constructive approach, one distinguishes between on the one hand not
being rational, and on the other hand being irrational (i.e., being
quantifiably apart from every rational), the latter being a stronger
property. Given positive integers *a* and *b*, because the valuation
<https://en.wikipedia.org/wiki/Singly_and_doubly_even#Definitions> (i.e.,
highest power of 2 dividing a number) of 2*b*2 is odd, while the valuation
of *a*2 is even, they must be distinct integers; thus |2*b*2 − *a*2| ≥ 1.
Then[17] <https://en.wikipedia.org/wiki/Square_root_of_2#cite_note-17>
{\displaystyle \left|{\sqrt {2}}-{\frac {a}{b}}\right|={\frac
{|2b^{2}-a^{2}|}{b^{2}\left({\sqrt {2}}+{\frac {a}{b}}\right)}}\geq {\frac
{1}{b^{2}\left({\sqrt {2}}+{\frac {a}{b}}\right)}}\geq {\frac
{1}{3b^{2}}},}[image:
{\displaystyle \left|{\sqrt {2}}-{\frac {a}{b}}\right|={\frac
{|2b^{2}-a^{2}|}{b^{2}\left({\sqrt {2}}+{\frac {a}{b}}\right)}}\geq {\frac
{1}{b^{2}\left({\sqrt {2}}+{\frac {a}{b}}\right)}}\geq {\frac
{1}{3b^{2}}},}]

the latter inequality being true because it is assumed that *a*/*b* ≤
3 − √2 (otherwise
the quantitative apartness can be trivially established). This gives a
lower bound of 1/3*b*2 for the difference |√2 − *a*/*b*|, yielding a direct
proof of irrationality not relying on the law of excluded middle
<https://en.wikipedia.org/wiki/Law_of_excluded_middle>; see Errett Bishop
<https://en.wikipedia.org/wiki/Errett_Bishop> (1985, p. 18). This proof
constructively exhibits a discrepancy between √2 and any rational.

On Thu, May 21, 2020 at 10:50 AM Steve Smith <sasmyth at swcp.com> wrote:

>
> On 5/21/20 10:32 AM, uǝlƃ ☣ wrote:
> > Don't be fooled. "The problem with communication is the illusion that it
> exists." Or ie I believe in a stronger form of privacy than you believe in.
> I KNOW! I know just what you mean!
>
> <note to Frank...  one of the species of animal in this group is "the
> Contrarian", but you probably already guessed that>
>
>
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-- 
Frank Wimberly
140 Calle Ojo Feliz
Santa Fe, NM 87505
505 670-9918
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