<html><head><meta http-equiv="Content-Type" content="text/html charset=utf-8"><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 Carlos,<div class="">You ignore the mathematical level when you attempt a distinction between theory and practice.</div><div class="">The mathematical level is practical.</div><div class="">You know why a^2 + b^2 = c^2 for any right triangle. You do not have to measure each one.</div><div class="">You know that Newton’s Laws apply to good approximation to any orbiting object and you can make use of them.</div><div class="">The notion that you can just look at things in a finite world is erroneous.</div><div class="">There are real mathematical problems associated with whether algorithms halt or whether theorems are true or false.</div><div class="">There is real interest in these problems.</div><div class="">Of course it is a matter of taste.</div><div class="">It is possible that you are a person who never got interested in any real mathematical questions.</div><div class="">If so,then at least its my duty to show you the sort of thing I mean.</div><div class=""><br class=""></div><div class="">Euclid proved that there are infinitely many prmes. His proof is summarized by </div><div class="">N_{k} = p_{1} x p_{2} x … x p_{k} + 1.</div><div class="">If p_{1}, p_{2}, … , p_{k} are the first k prime numbers, then N_{k} is either a prime number or it is divisible by new primes different from</div><div class=""> p_{1}, p_{2}, … , p_{k}. Ok. Now consider the sequence N_{1},N_{2}, N_{3}, … Are there infinitely many prime numbers in this sequence?</div><div class="">No one knows, and if you just use a computer it will not tell you the answer. </div><div class=""><br class=""></div><div class="">Goldbach Conjecture: Every even number greater than 4 appears to be the sum of two odd primes, usually in many ways.</div><div class="">This is something that you can explore with a computer. For example</div><div class=""><br class=""></div><div class=""></div><img name="Screen Shot 2024-02-17 at 1.57.16 AM.png" apple-inline="yes" id="236B8F42-F654-4F1E-BA84-EA43C97780A1" height="192" width="90" apple-width="yes" apple-height="yes" src="cid:CD451D2B-8558-4341-8462-62B762816B66" class=""><meta http-equiv="Content-Type" content="text/html charset=utf-8" class=""><meta http-equiv="Content-Type" content="text/html charset=utf-8" class=""><meta http-equiv="Content-Type" content="text/html charset=utf-8" class=""><div class=""></div><div class=""><br class=""></div><div class="">No one has managed to prove this result. This is a case where I am pretty much convinced by my computer that the Goldbach Conjecture is true.</div><div class="">It is then a great theoretical challenge to find a proof of that conjecture. Would a proof have practical consequences? I do not know, but I bet there would be</div><div class="">practical spinoffs, just as soon there will be very serious practical spinoffs to the Shor quantum factoring algorithm and its theoretical basis.</div><div class=""> </div><div class="">Collate Problem: Let C(n) = 1 if the following algorithm on n halts at 1. Let C(n) = 0 if it does not halt.</div><div class="">n —> n/2 if n even.</div><div class="">n —> 3n + 1 i n odd</div><div class="">Repeat until 1 is reached.</div><div class="">If 1 is reached, STOP.</div><div class="">If 1 is not reached keep computing.</div><div class="">e.g. 7 —> 22 —> 11—> 34 —> 17 —> 52 —> 26 —> 13 —> 40 —> 20 —> 10 —> 5 —> 16 —> 8 —> 4 —> 2 —> 1 , STOP.</div><div class="">No one knows if this algorithm halts, but it has been seen to halt for every n that has ever been tried.</div><div class="">In this case finite computer exploration seems to be one way to get possible insight into the problem.</div><div class=""><br class=""></div><div class="">Famous Example. A 2n x 2n checkerboard has opposite corner squares removed. Can it be paved with dominos with no overhangs?</div><div class="">Answer: No!</div><div class="">Discussion. For small n you can do a computer search and find the answer, but try that for even n= 10^{100} and you will be in trouble.</div><div class="">On the other hand, one can make the intelligent mathematical observation that the two opposite corner squares have the same color, and so there are a different number of </div><div class="">black and white squares on the deleted checkerboard. Since each domino covers one white square and one black square, this means that the deleted checkerboard cannot be covered by dominos with no overhangs.</div><div class=""><br class=""></div><div class="">At mathematical level we are searching for structural observations that will encompass an infinity of cases. </div><div class="">This is practical, because it creates an economy of thought and action.</div><div class="">Best,</div><div class="">Lou</div><div class=""></div>P.S. I sent a paper but it bounced. If anyone would like to read a paper that elaborates on the above lletter, send me an email to <div class=""><a href="mailto:loukau@gmail.com" class="">loukau@gmail.com</a></div><div class="">and I will send it directly to you.</div><div class=""><br class=""><meta http-equiv="Content-Type" content="text/html charset=utf-8" class=""><meta http-equiv="Content-Type" content="text/html charset=utf-8" class=""><meta http-equiv="Content-Type" content="text/html charset=utf-8" class=""><div class=""></div><div class=""><br class=""></div><div class=""><br class=""></div><div class=""><div class=""><blockquote type="cite" class=""><div class="">On Feb 16, 2024, at 4:16 PM, Carlos Gershenson <<a href="mailto:cgershen@gmail.com" class="">cgershen@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 style="overflow-wrap: break-word; -webkit-nbsp-mode: space; line-break: after-white-space;" class=""><div dir="auto" style="overflow-wrap: break-word; -webkit-nbsp-mode: space; line-break: after-white-space;" class="">Dear Lou,<div class=""><br class=""></div><div class="">Thank you for your examples. </div><div class=""><br class=""></div><div class="">The Kleene argument precisely shows the relevance of the difference between theory and practice. In theory, there will always be undecidability in “powerful enough” formal systems. In practice, we can arbitrarily try out all algorithms that we want for a finite time, and come up with a table of whether they halted or not. </div><div class=""><br class=""></div><div class="">Now, the question would be whether with such a pragmatic approach we can overcome the limits of formal systems that prevent us from representing properly e.g. evolutionary innovations, and changes in function in general. </div><div class=""><br class=""></div><div class="">But maybe we just need two levels: one where rules don’t change, and another where we can allow rules and meaning to change. The thing is that the way in which rules and meaning change cannot change… but then if we add another layer… To avoid turtles all the way down, could we have layer A changing layer B and vice versa? A coevolution of formal systems?</div><div class=""><br class=""></div><div class="">Best wishes,<br class=""><div class="">
<div style="letter-spacing: normal; orphans: auto; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: auto; word-spacing: 0px; -webkit-text-stroke-width: 0px; word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space;" class=""><div style="letter-spacing: normal; orphans: auto; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: auto; word-spacing: 0px; -webkit-text-stroke-width: 0px; word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space;" class=""><span class="Apple-style-span" style="border-collapse: separate; font-family: Helvetica; border-spacing: 0px;"><div style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space;" class="">Carlos</div></span></div></div>
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