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<div class="moz-cite-prefix">Of course, Lou, you are free to send
your seminar invitation to the list.</div>
<div class="moz-cite-prefix">Remember that the limit of 2 messages
per week was slightly extended to 3.</div>
<div class="moz-cite-prefix">I mean from <><> to
<><><>.</div>
<div class="moz-cite-prefix">Best--Pedro</div>
<div class="moz-cite-prefix"><br>
</div>
<div class="moz-cite-prefix">El 01/02/2025 a las 6:06, Louis
Kauffman escribió:<br>
</div>
<blockquote type="cite"
cite="mid:D262B130-6F23-48C6-AD1F-9EAD2B46F993@gmail.com">
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Dear Folks,
<div class="">Thanks for this great month of exchanges! I see no
need to make a summary of the whole exploration. A bit of
mathematics did come out in relation to this discussion. We
talked about it at the Laws of Form meeting earlier today. I
discovered a “boundary arithmetic” similar to 3-valued logic
that could be used to find out simple (but hard otherwise to
determine) topological information about knots and links. The
arithmetic itself is an extension of Laws of Form where one has
the marked state <> and the rules <> <> =
<> (calling) and <<>> = (crossing). In
crossing, one crosses from the marked state and arrives at the
unmarked state. In this notation the unmarked state is indicated
by a blank space. To this we add a name for the unmarked state E
and another name O which is a state where < O > = O so
that crossing from O returns to O. O is neither marked nor is
it unmarked. You can think of O as the boundary between the
inside and the outside of a first distinction. In logic it would
be the third value, neither true nor false, in a three valued
logic. We also assume that markedness dominates the third value
so that O <> = <> O = <>. Now to get a
structure that unfolds into the well-known Luaksiewcz three
valued logic, one takes OO = O. BUT for “my” boundary arithmetic
I take OO = . That is O interacts with itself to disappear!
You can think of this as a boundary joining along another
boundary line and the seam between them becomes invisible. You
can also think of it as analogous to Odd + Odd = Even in the
sense that the sum of two odd numbers is even.</div>
<div class="">So we have an arithmetic with the rules</div>
<div class=""><><>=<></div>
<div class=""><<>> = </div>
<div class="">OO=</div>
<div class=""><> O = <></div>
<div class=""><O> = O.</div>
<div class="">It turns out that one can use this arithmetic to
calculate whether certain woven networks (that occur in DNA
recombination e.g.) have one loop (odd) or two loops (even). I
won’t repeat this here, but more information will be sent about
this that could be downloaded.</div>
<div class=""><br class="">
</div>
<div class="">A logicians point about such arithmetics is that
they do not satisfy the “Law of the Excluded Middle” which in
this notation would be P<P> = <>.</div>
<div class="">Thus here we have O<O> = OO = , so that it
is unmarked rather than marked. And in the usual three valued
logic we have O<O> = OO = O, and again it is </div>
<div class="">not marked.</div>
<div class=""><br class="">
</div>
<div class="">I have a question for Pedro. Is it ok to send the
Laws of Form announcement for my Friday seminar to fis? If not,
then fis people who want the announcement should write me an
email, and I will put you on the list. I’ll talk about the
boundary arithmetic a bit more next Friday and Karl Jaworsky
will give a talk about his theory. </div>
<div class=""><br class="">
</div>
<div class="">Getting back to the theme, with the boundary
arithmetic we see how a very abstract, somewhat philosophical
“logical arithmetic” can give us topological network
information. This is quite interesting to contemplate as it
shows how very macroscopic situations can be understood by using
simple mathematical patterning.</div>
<div class="">Here it is happening in a mathematical/logical
distinction domain. In our original biologic examples we saw how
the mathematical domain interacted with the microscopic
molecular biological domain in terms of both theory and
experiment. All of this suggests further exploration of our
patterned thoughts in relation to sensing and understanding
natural physical and biological worlds.</div>
<div class="">Very best,</div>
<div class="">Lou </div>
<div class=""><br class="">
</div>
<div class="">
<div><br class="">
<blockquote type="cite" class="">
<div class="">On Jan 31, 2025, at 3:22 PM, Pedro C. Marijuán
<<a href="mailto:pedroc.marijuan@gmail.com"
class="moz-txt-link-freetext" moz-do-not-send="true">pedroc.marijuan@gmail.com</a>>
wrote:</div>
<br class="Apple-interchange-newline">
<div class="">
<meta http-equiv="content-type"
content="text/html; charset=UTF-8" class="">
<div class="">
<p class="">Dear All,</p>
<p class="">As is tradition in this List, the beginning
of Chinese New Year announces the coming end of the
FIS New Year Lecture. <br class="">
</p>
<p class="">Thanks are given to Lou for his impressive
posting work and for the high quality of literary
contents (some concluding comments to be added?). </p>
<p class="">The related discussions may continue, but
now there are no chairing privileges--the iron rule of
2 - 3 mssgs per week applies to everybody.</p>
<p class="">And so, let us also wish our Chinese
Colleagues a very Happy New Year!</p>
<div align="center" class="">
<pre class="tw-ta tw-data-text tw-text-large"
data-placeholder="Traducción" id="tw-target-text"
style="text-align:left"
aria-label="Texto traducido: 新年快乐"
data-ved="2ahUKEwiCz9qv8aCLAxVBBdsEHfNIGm4Q3ewLegQIChAU"><span
class="Y2IQFc" lang="zh-CN"><font size="6" class="">新年快乐</font></span></pre>
</div>
<p class="">All the best,</p>
<p class="">--Pedro<br class="">
</p>
</div>
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</blockquote>
</div>
<br class="">
</div>
</blockquote>
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