<html><head><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=""><div class=""><div class="">Dear Karl,</div><div class="">This will be my last communication this week. I was not criticizing your structures. I was speaking of the use of formalism quite generally.</div><div class="">Goedelian limitations happen when the system contains its own logic and is rich enough to do number theory.</div><div class="">In mathematical practice we work with simpler systems that still embody limitations and may embody them even when embedded in larger systems with that logic.</div><div class=""><br class=""></div><div class="">A good example of this hands-on approach to limitations is the following argument.</div><div class="">Suppose that F_{1},F_{2},F_{3},… is any countable list of halting algorithms.</div><div class="">We can assume they are all written in some given programming language and each one can receive a natural number n </div><div class="">and produce the result F_{k}(n) in a finite amount of time.</div></div><div class="">With this, I can define a new algorithm by the formula F(n) = F_{n}(n) + 1 that also halts and F is not equal to any F_{k}.</div><div class="">Thus any list of halting algorithms is incomplete.</div><div class=""><br class=""></div><div class="">This result applies in any formal system or computer language that can have such algorithms and is what lies in back of Turing’s Theorem that the </div><div class="">Halting Problem is not decidable.</div><div class=""><br class=""></div><div class="">All these situations are part and parcel of the fact that formal systems show more than they can prove.</div><div class="">I do not expect to get beyond formal systems since they are just ways of writing down rules and assumptions that we intend to follow.</div><div class="">But we do need to realize that in actual mathematical and scientific practice one can be ready to not only work within certain formal systems, but one can</div><div class="">make new formal systems that are appropriate to the problems at hand.</div><div class=""><br class=""></div><div class="">There is no magic key to solving hard problems or predicting how structures might combine to form new structures.</div><div class="">At every junction we wait for new ideas or new distinctions that may occur through our experience, practice, observation or inspiration.</div><div class="">Best,</div><div class="">Lou</div><div class="">P.S. I will respond further next week.</div><div class="">PP.S. One of the things that we like to do is compare very different situations. Thus the incompleteness of infinite lists of algorithms resonates with the Euclid’s proof </div><div class="">of the incompleteness of finite lists of primes. Let p1,…, pk be a finite list of prime numbers . Let N = p1x p2 x … x pk +1. Then N is not divisible by and pi and so N is</div><div class="">either a new prime or a product of new primes. The Euclid argument does not apply to infinite products. But Euler gave a different proof of the infinitude of the primes via </div><div class="">the divergence of Sum_{n}(1/n) by writing it as an infinite product: Sum_{n}(1/n) = Product_{p}(1-1/p)^{-1} where the product runs over all the primes. The product cannot diverge unless it is an infinite product and so there must be infinitely many primes. I imagine that one of Euler’s thoughts in coming to his new proof over 2000 years after Euclid</div><div class="">was wondering how the finite products of Euclid’s proof might be related to the infinite structure of series and convergence and divergence. Mathematical imagination works</div><div class="">both outside and inside given formalities, looking for relationships. The formal systems are the fallout of such imaginings and give us platforms from which to make further </div><div class="">explorations. We live and think outside the formalities and we use the logic and formalities to structure, criticize and launch our own thought.</div><div class=""><br class=""></div><div class=""><br class=""></div><div class=""><br class=""></div><div class=""><br class=""></div><div class=""><br class=""></div><div class=""><br class=""></div><br class=""><div><blockquote type="cite" class=""><div class="">On Feb 6, 2024, at 8:54 AM, Karl Javorszky <<a href="mailto:karl.javorszky@gmail.com" class="">karl.javorszky@gmail.com</a>> wrote:</div><br class="Apple-interchange-newline"><div class=""><div dir="ltr" class=""><div dir="ltr" class=""><p class="MsoNormal"><span lang="DE-AT" class="">Dear Lou,<span class=""></span></span></p><p class="MsoNormal">(count = and ≠
differently, 2024 01 06)<span class=""></span></p><div class=""><span class=""> </span><br class="webkit-block-placeholder"></div><p class="MsoNormal">Thank you for the serious and elaborate restatement of
principles we agree on. Your politely implied question was: What use tabulating
numbers, we only see that as a result what we have set as rules? <span class=""></span></p><p class="MsoNormal">The sceptics is well-reasoned. It is on me to show that this
time it is different, that my shuffling of numbers is really, really something new
and produces spectacles like Nature does.<span class=""></span></p><div class=""><span class=""> </span><br class="webkit-block-placeholder"></div><p class="MsoNormal">I specifically enjoyed your sentence: <span class=""></span></p><p class="MsoNormal" style="margin-left:36pt"><i class="">it helps to look at the
simpler examples where one does not have the ideal that the formal system might
be rich enough to express everything including the logical structure of proofs
and demonstrations.<span class=""></span></i></p><p class="MsoNormal">because your ideas: a: start from simple, from bottom up, b.
how rich (flexible) is the logical language to express complicated interdependencies
(eg in AI or genetic) deal with central points of what we discuss here.<span class=""></span></p><div class=""><span class=""> </span><br class="webkit-block-placeholder"></div><p class="MsoNormal">Ad 1. Start from scratch<span class=""></span></p><p class="MsoNormal">We address the conflict that <i class="">(1,3) </i>is on <i class="">3<sup class="">rd</sup> </i>rank
in the sequence <i class="">(1,1),(1,2),(1,3),… </i><b class="">but </b><i class="">4<sup class="">th</sup>
</i>rank in sequence <i class="">(1,1),(1,2),(2,2),(1,3),…</i>(which
are sorts of <i class="">a,b). <b class="">Eq. 1</b></i><b class=""><span class=""></span></b></p><p class="MsoNormal">Like Pythagoras, we draw two axes in the sand and point out
place <i class="">x: 3, y: 4 </i>as a logical dissolution
of the conflict by giving it a solution in the next higher dimension.<span class=""></span></p><p class="MsoNormal" style="margin-left:36pt">Change in attitude required. So
far, we use the Sumerian concept of Unit, comparable to a playing card one side
saying “1” and the other side uniform. Up to today, we count the height of the
stack of elementary units as the content of the message. The invention turns
over the Sumerian playing cards and finds <i class="">136
</i>different symbols etched on the reverse side. We play a completely
different game, within the general game, based on secret signs that signify
belonging to cooperators, if circumstances arise.<span class=""></span></p><p class="MsoNormal">Do it your own way, don’t depend on others. Use only deictic
definitions. Create a cohort of pairs of <i class="">(a,b);
a </i><i class="">≤ b; a,b </i><i class="">≤
16. </i>We listen to chamber music performed by <i class="">136 </i>individuals. <span class=""></span></p><p class="MsoNormal">Ad 2. Flexibility of the language<span class=""></span></p><p class="MsoNormal">We live in the correct impression that the world can be
described by a language that uses unform units. The language itself does not
hinder us to express differences and contradictions. The inner controversy demonstrated
by <b class="">Eq. 1. </b>can easily be thematized
and narrated.<span class=""></span></p><p class="MsoNormal">Seen as a planar coordinate, the contradiction of <b class="">Eq. 1 </b>dissolves. The language is
flexible enough to express the observed facts, if only we would observe the
facts.<span class=""></span></p><p class="MsoNormal">My point in answering you:<span class=""></span></p><p class="MsoNormal">This is no joke. Neurology makes use of a numeric fact,
which lies in fine details of how we count (perceive) ≠ before a background of =, and = before a background
of ≠. If Nature
uses <i class="">two </i>basic descriptive properties
of a heap of input, we should try to do likewise. The argument that <i class="">0,1 </i>are also <i class="">two </i>reference systems, is void, because <i class="">0,1 </i>are symbols of an abstract nature, while ≠, = describe material
differences that are gradated and within limits and thresholds.<span class=""></span></p><p class="MsoNormal">It is nebbich a fact that one has more variants of space available
to accommodate that many material variants that <i class="">n </i>objects can be incorporating, <i class="">if
and only if the <b class="">objects number < 32
or > 97. </b></i>We may like or not like the fact, but it remains true that
the possibly existing variants of material properties of the objects will not
fit in the number of available spatial segments which the objects can produce, <i class="">if and only if the objects number <b class="">32 < n < 97. <span class=""></span></b></i></p><p class="MsoNormal">We do not invent anything Nature would not have discovered. The
syntax of the DNA is the best argument. Let your logical primitives exercise
and the first thing they do is building sequenced threesomes of one of among
four (restricted to one of two of two pairs),<span class=""></span></p><div class=""><span class=""> </span><br class="webkit-block-placeholder"></div><p class="MsoNormal">Dear Lou, thank you for your efforts to build bridges for me.
The relative isolation of this new family of algorithms lies in the
requirements you pointed out at the beginning. If something is that new, that
an update on a+b=c is necessary, it is necessarily self-contained and uses
deictic definitions. It is pure chance that the Eddington constants and the DNA
support the hair-raising ideas transmitted here.<span class=""></span></p><div class=""><span class=""> </span><br class="webkit-block-placeholder"></div><p class="MsoNormal">We have a complicated rhetorical situation here. Let us grow
to the challenge while we discuss that the central idea of information is that
something is <i class="">otherwise, </i>that in the
Sumerian system nothing ever can be <i class="">otherwise,
</i>therefore the rational language appears to be inadequate to describe, discuss
or express something that can not exist by its nature.<span class=""></span></p><p class="MsoNormal">No reason to be overly pessimistic, Some nephews of
Pythagoras have surely played with diverse toy soldiers, exercising them, and
some say to have heard Pythagoras curse under his beard: what a horrible spell
dampens my ability to tabulate and memorize! Elementary logical symbols allow
recognizing elementary logical patterns if we perform elementary logical operations
on them! I wish I could live in some 2600 years hence. I’d have computers and I’d
show them what is a pattern.<span class=""></span></p><div class=""><span class=""> </span><br class="webkit-block-placeholder"></div><p class="MsoNormal">Friendly greetings:<span class=""></span></p><p class="MsoNormal">Karl<span class=""></span></p><div class=""><span class=""> </span><br class="webkit-block-placeholder"></div><div class=""><span class=""> </span><br class="webkit-block-placeholder"></div></div></div>
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