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 <[email protected]> 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> > > > -- --- .-. . .-.. --- -.-. -.- ... -..-. .- .-. . -..-. - .... . -..-. . > ... ... . -. - .. .- .-.. -..-. .-- --- .-. -.- . .-. ... > FRIAM Applied Complexity Group listserv > Zoom Fridays 9:30a-12p Mtn GMT-6 bit.ly/virtualfriam > un/subscribe http://redfish.com/mailman/listinfo/friam_redfish.com > archives: http://friam.471366.n2.nabble.com/ > FRIAM-COMIC <http://friam.471366.n2.nabble.com/FRIAM-COMIC> > http://friam-comic.blogspot.com/ > -- Frank Wimberly 140 Calle Ojo Feliz Santa Fe, NM 87505 505 670-9918
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