<div dir="ltr">Clinicians often call that "being oppositional".  <div><br></div><div>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:</div><div><br></div><div><p style="margin:0.5em 0px;color:rgb(32,33,34);font-family:sans-serif"><font size="1">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 <span class="gmail-texhtml" style="font-feature-settings:"lnum","tnum","kern" 0;font-variant-numeric:lining-nums tabular-nums;font-kerning:none;font-family:"Nimbus Roman No9 L","Times New Roman",Times,serif;line-height:1;white-space:nowrap"><i>a</i></span> and <span class="gmail-texhtml" style="font-feature-settings:"lnum","tnum","kern" 0;font-variant-numeric:lining-nums tabular-nums;font-kerning:none;font-family:"Nimbus Roman No9 L","Times New Roman",Times,serif;line-height:1;white-space:nowrap"><i>b</i></span>, because the <a href="https://en.wikipedia.org/wiki/Singly_and_doubly_even#Definitions" title="Singly and doubly even" style="text-decoration-line:none;color:rgb(11,0,128);background:none">valuation</a> (i.e., highest power of 2 dividing a number) of <span class="gmail-texhtml" style="font-feature-settings:"lnum","tnum","kern" 0;font-variant-numeric:lining-nums tabular-nums;font-kerning:none;font-family:"Nimbus Roman No9 L","Times New Roman",Times,serif;line-height:1;white-space:nowrap">2<i>b</i><sup style="line-height:1">2</sup></span> is odd, while the valuation of <span class="gmail-texhtml" style="font-feature-settings:"lnum","tnum","kern" 0;font-variant-numeric:lining-nums tabular-nums;font-kerning:none;font-family:"Nimbus Roman No9 L","Times New Roman",Times,serif;line-height:1;white-space:nowrap"><i>a</i><sup style="line-height:1">2</sup></span> is even, they must be distinct integers; thus <span class="gmail-texhtml" style="font-feature-settings:"lnum","tnum","kern" 0;font-variant-numeric:lining-nums tabular-nums;font-kerning:none;font-family:"Nimbus Roman No9 L","Times New Roman",Times,serif;line-height:1;white-space:nowrap">|<span class="gmail-nowrap" style="padding-left:0.1em;padding-right:0.1em">2<i>b</i><sup style="line-height:1">2</sup> − <i>a</i><sup style="line-height:1">2</sup></span>| ≥ 1</span>. Then<sup id="gmail-cite_ref-17" class="gmail-reference" style="line-height:1;unicode-bidi:isolate;white-space:nowrap"><a href="https://en.wikipedia.org/wiki/Square_root_of_2#cite_note-17" style="text-decoration-line:none;color:rgb(11,0,128);background:none">[17]</a></sup></font></p><dl style="margin-top:0.2em;margin-bottom:0.5em;color:rgb(32,33,34);font-family:sans-serif"><dd style="margin-left:1.6em;margin-bottom:0.1em;margin-right:0px"><span class="gmail-mwe-math-element"><font size="1"><span class="gmail-mwe-math-mathml-inline gmail-mwe-math-mathml-a11y" style="display:none;overflow:hidden;width:1px;height:1px;opacity:0">{\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}}},}</span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/641b9e87f603636755874eee6c5d85875f907483" class="gmail-mwe-math-fallback-image-inline" alt="{\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}}},}" style="border: 0px; vertical-align: -4.505ex; display: inline-block; width: 50.681ex; height: 8.509ex;"></font></span></dd></dl><p style="margin:0.5em 0px;color:rgb(32,33,34);font-family:sans-serif"><font size="1">the latter inequality being true because it is assumed that <span class="gmail-texhtml" style="font-feature-settings:"lnum","tnum","kern" 0;font-variant-numeric:lining-nums tabular-nums;font-kerning:none;font-family:"Nimbus Roman No9 L","Times New Roman",Times,serif;line-height:1;white-space:nowrap"><span class="gmail-sfrac gmail-nowrap gmail-tion" style="display:inline-block;vertical-align:-0.5em;text-align:center"><span class="gmail-num" style="display:block;line-height:1em;margin:0px 0.1em"><i>a</i></span><span class="gmail-slash gmail-visualhide" style="width:1px;height:1px;overflow:hidden">/</span><span class="gmail-den" style="display:block;line-height:1em;margin:0px 0.1em;border-top:1px solid"><i>b</i></span></span> ≤ 3 − <span class="gmail-nowrap">√<span style="border-top:1px solid;padding:0px 0.1em">2</span></span></span> (otherwise the quantitative apartness can be trivially established). This gives a lower bound of <span class="gmail-texhtml" style="font-feature-settings:"lnum","tnum","kern" 0;font-variant-numeric:lining-nums tabular-nums;font-kerning:none;font-family:"Nimbus Roman No9 L","Times New Roman",Times,serif;line-height:1;white-space:nowrap"><span class="gmail-sfrac gmail-nowrap gmail-tion" style="display:inline-block;vertical-align:-0.5em;text-align:center"><span class="gmail-num" style="display:block;line-height:1em;margin:0px 0.1em">1</span><span class="gmail-slash gmail-visualhide" style="width:1px;height:1px;overflow:hidden">/</span><span class="gmail-den" style="display:block;line-height:1em;margin:0px 0.1em;border-top:1px solid">3<i>b</i><sup style="line-height:1">2</sup></span></span></span> for the difference <span class="gmail-texhtml" style="font-feature-settings:"lnum","tnum","kern" 0;font-variant-numeric:lining-nums tabular-nums;font-kerning:none;font-family:"Nimbus Roman No9 L","Times New Roman",Times,serif;line-height:1;white-space:nowrap">|<span class="gmail-nowrap" style="padding-left:0.1em;padding-right:0.1em"><span class="gmail-nowrap">√<span style="border-top:1px solid;padding:0px 0.1em">2</span></span> − <span class="gmail-sfrac gmail-nowrap gmail-tion" style="display:inline-block;vertical-align:-0.5em;text-align:center"><span class="gmail-num" style="display:block;line-height:1em;margin:0px 0.1em"><i>a</i></span><span class="gmail-slash gmail-visualhide" style="width:1px;height:1px;overflow:hidden">/</span><span class="gmail-den" style="display:block;line-height:1em;margin:0px 0.1em;border-top:1px solid"><i>b</i></span></span></span>|</span>, yielding a direct proof of irrationality not relying on the <a href="https://en.wikipedia.org/wiki/Law_of_excluded_middle" title="Law of excluded middle" style="text-decoration-line:none;color:rgb(11,0,128);background:none">law of excluded middle</a>; see <a href="https://en.wikipedia.org/wiki/Errett_Bishop" title="Errett Bishop" style="text-decoration-line:none;color:rgb(11,0,128);background:none">Errett Bishop</a> (1985, p. 18). This proof constructively exhibits a discrepancy between <span class="gmail-texhtml" style="font-feature-settings:"lnum","tnum","kern" 0;font-variant-numeric:lining-nums tabular-nums;font-kerning:none;font-family:"Nimbus Roman No9 L","Times New Roman",Times,serif;line-height:1;white-space:nowrap"><span class="gmail-nowrap">√<span style="border-top:1px solid;padding:0px 0.1em">2</span></span></span> and any rational.</font></p></div></div><br><div class="gmail_quote"><div dir="ltr" class="gmail_attr">On Thu, May 21, 2020 at 10:50 AM Steve Smith <<a href="mailto:sasmyth@swcp.com">sasmyth@swcp.com</a>> wrote:<br></div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex"><br>
On 5/21/20 10:32 AM, uǝlƃ ☣ wrote:<br>
> 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.<br>
I KNOW! I know just what you mean!<br>
<br>
<note to Frank...  one of the species of animal in this group is "the<br>
Contrarian", but you probably already guessed that><br>
<br>
<br>
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</blockquote></div><br clear="all"><div><br></div>-- <br><div dir="ltr" class="gmail_signature">Frank Wimberly<br>140 Calle Ojo Feliz<br>Santa Fe, NM 87505<br>505 670-9918</div>