<html><head><meta http-equiv="Content-Type" content="text/html charset=utf-8"><meta http-equiv="Content-Type" content="text/html charset=utf-8"></head><body style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space;" class="">Dear Joseph,<div class=""><br class=""></div><div class="">(I am sending this again without graphics so it can end up on fis.) </div><div class=""><br class=""><div class="">The RD construction is very general, but we have articulated it with particular models. </div><div class=""><br class=""></div><div class="">One starts with a domain where distinctions can be made and there is a notion of locality.</div><div class="">In that domain one makes distinctions about given entities </div><div class="">and replaces the entities by the corresponding signs of distinction. </div><div class="">These signs amalgamate to become new entities to be distinguished at the next round. </div><div class=""><br class=""></div><div class="">Lest this seem abstract, look again at the model.</div><div class=""><img apple-inline="yes" id="76CE295D-901D-4180-8A16-241547BC1538" height="72" width="339" apple-width="yes" apple-height="yes" class="" src="cid:2123B3A7-E19D-4E71-BAA0-97EFC8933166"></div><div class="">The sign = is an amalgamation of empty word on left and empty word on rght.</div><div class="">The sign ] is amalgamation of empty word on left and vertical bar | on rght.</div><div class="">The sign [] is amalgamation of vertical bar | on left and empty word on rght.</div><div class="">The sign O is meant to be a box and so is an amalgamation of vertical bar on left and right.</div><div class=""><br class=""></div><div class="">The point is that the alphabet arises by the amalgamations of previous distinctions and so new symbolic entities arise from the process of distinguishing by these rules.</div><div class="">After the body of this email, I will show you how the two dimensional RD alphabet arises.</div><div class=""><br class=""></div><div class="">This level of model includes notions of language and description in a very elementary formal framework. </div><div class="">I have not included a logical language or a language that even begins to</div><div class="">have self-reference and ordinary reference. Thus meta-levels and thinking about thinking are not fully reflected in this small model. </div><div class="">But the action of the model gives one the opportunity to reflect on these</div><div class="">larger issues. I am in the business of finding significant minimal models. </div><div class="">These models are not going to be complete articulations of the whole situation of thinking about thinking or</div><div class="">reflecting on reflecting. They are intended as ways to help thinking about that. </div><div class=""><br class=""></div><div class="">Note also that the RD above engages in “mitosis” or “self-replication”. It does so without a reflective level. This is of interest</div><div class="">for thinking about coding in biology and in thinking about what coding would mean in physical situations “below” biology.</div><div class=""><br class=""></div><div class="">Let me give here another example: the audio-active sequences studied by John Horton Conway.</div><div class="">1</div><div class="">11 - read “one one” and it describes the line above.</div><div class="">21- read “two ones” and it describes the line above.</div><div class="">1211- read “one two, one one” and it describes the line above.</div><div class="">111221- read “three ones, two two’s, one one” and it describes the line above.</div><div class="">312211- read “one three, two two’s, one one” and it describes the line above.</div><div class="">…</div><div class="">This recursion is an example of "describing describing" and the coding issues are somewhat different from the RD.</div><div class="">We retain locality of interaction and have a fixed alphabet that is less iconic. Counting is needed at the descriptive level.</div><div class="">This recursion requires more structure to run, but it is very very interesting.</div><div class="">Note that 22 is the only self-referential sequence.</div><div class=""><br class=""></div><div class="">Note also that we could write nx —> xx…x (n x’s) meaning that nx describes xx…x. </div><div class="">In the audio active line we would have</div><div class="">xxxx…x</div><div class="">nx</div><div class=""><br class=""></div><div class="">as in</div><div class="">777</div><div class="">37.</div><div class=""><br class=""></div><div class="">So we have nx —> xx…x</div><div class="">and 2x —> xx</div><div class="">and so</div><div class="">22 —> 22.</div><div class=""><br class=""></div><div class="">This pattern fits into the whole of 20th century logic since Russell.</div><div class="">Let me tender persuasions.</div><div class="">Russell: Rx = ~xx</div><div class="">This is the definition of the Russell set if you take AB to mean “B is a member of A” and ~ is “not”.</div><div class="">Then: RR = ~ RR is the Russell paradox.</div><div class="">In this formalism, the Russell paradox becomes the production of a fixed point for negation.</div><div class=""><br class=""></div><div class="">Church and Curry generalized this to an abstract formalism (lambda calculus) where they could write</div><div class="">gx = F(xx) for an arbitrary F.</div><div class="">Then</div><div class="">gg = F(gg) and we produce a fixed point for F.</div><div class="">This is exactly how we got to the self reference of 22 by</div><div class="">2x —> xx</div><div class="">22 —> 22 </div><div class="">where here equality is replaced by reference.</div><div class=""><br class=""></div><div class="">The notion of replacement of reference is integral to Goedelian self-refefrence.</div><div class="">g ——> P(x) now interpreted as “g is the Goedel number of the proposition P(x) with free variable x.</div><div class="">Then let #g be the Goedel number of P(g) so that </div><div class="">#g —> P(g).</div><div class="">Let the language be rich enough so that # is an operation in the language.</div><div class="">h —> P(#x)</div><div class="">Then</div><div class="">#h—> P(#h)</div><div class="">and P(#h) talks about its own Goedel number.</div><div class="">Then with Goedel let ~B(x) mean that “there is no proof of the statement with Goedel number x.</div><div class="">Then with </div><div class="">h —> ~B(#x)</div><div class="">#h —> ~B(#h)</div><div class="">and we have produced the proposition ~B(#h) that asserts its own unprovability!</div><div class="">This is the core of Goedel’s incompleteness Theorem.</div><div class=""><br class=""></div><div class="">I hope you see that we have arrived Goedel’s Theorem on a track leading directly from the self-reference of 22, by way of the Russell Paradox.</div><div class="">All this comes from the capacity of language to speak about language and thinking to think about thinking.</div><div class="">Goedel in fact shows us that all formalisms that we build that are rich enough and are consistent will be incomplete.</div><div class="">So there is no intent here to create complete formalisms.</div><div class="">I want to look at how the simplest non trivial formalisms behave, and how even very similar ones such as the RD and the audio activity are related to each other and to </div><div class="">the larger issues of reference, self reference and the generation of such dialogues at all levels.</div><div class=""><br class=""></div><div class="">Note also how small formalisms can summarize wide ideas.</div><div class="">The von Neumann Universal Building Machine B acts as follows.</div><div class="">B,x —> X,x.</div><div class="">Give B a blueprint x and B will produce X (the entity described by the blueprint).</div><div class="">Hence if b is the blueprint for B, then B can build itself!</div><div class="">B,b —> B,b.</div><div class="">This is another variant of 2x —> xx so 22—> 22.</div><div class=""><br class=""></div><div class="">Another thing that happened in the 20th century is that everyone got frightened about Rx =~xx giving RR = ~ RR,</div><div class="">and the story I am telling here of the central role of reference and self-reference is still not fully appreciated by mathematical practitioners such as</div><div class="">economists and even physicists.</div><div class="">Very best,</div><div class="">Lou</div><div class=""><br class=""></div><div class=""><br class=""></div></div></body></html>