some thoughts I considered for book 3 of possibility thinking explorations in logic and thought 
[Dec. 3rd, 200709:31 am]
justincoslor

 These are some thoughts that I considered for book 3 of possibility thinking explorations in logic and thought and many of them are probably flawed so the burden of understanding lies entirely on the reader and gossip is not allowed.  This is an unfinished writing and I disclaim all liability.  Copyright 2/1/2005 Justin Coslor Hierarchical Number Theory: Graph Theory Conversions Looking for patterns in this: Prime odd and even cardinality on the natural number system (*See diagram). First I listed out the prime numbers all in a row, separated by commas. Then above them I drew connecting arcs over top of every other odd prime (of the ordering of primes). Over top of those I drew an arc over every two of those arcs, sequentially. Then over top of every sequential pair of those arcs I drew another arc, and so on. Then I did the same thing below the listing of the numbers, but this time starting with every other even prime. Then I sequentially listed out whole lot of natural numbers and did the same thing to them down below them, except I put both every other even and every other odd hierarchical ordering of arcs over top of one another, down below the listing of the natural number system. Then over top of the that listing of the natural number system I transposed the hierarchical arc structures from the prime number system; putting both every other even prime and every other odd prime hierarchically on top of each other, as I previously described. *Now I must note that in all of these, in the center of every arc I drew a line going straight up or down to the center number for that arc. (See diagram.) In another example, I took the data, and spread out the numbers all over the page in an optimal layout, where no no hierarchical lines cross each other, but the numbers act as nodal terminals where the hierarchical arches sprout out of. (See Diagram) This made a very beautiful picture which was very similar to a hypercube that has been unfolded onto a 2D surface. Graph Theory might be able to be applied to hierarchical representations that have been realigned in this manner, and in that way axioms from Graph Theory might be able to be translated into Hierarchical Number Theory. The centerpoles are very significant because when I transposed the prime number structures onto the natural number system there is a central nonprime even natural number in the very center directly between the centerpoles of the sequential arc structures of the every other even prime and every other odd prime of the same hierarchical level and group number. The incredibly amazing thing is that when dealing with very large prime numbers, those prime numbers can be further reduced by representing them as an offset equation of the central number plus or minus an offset number. The beauty of is, that the since the central numbers aren't prime, they can be reduced in parallel as the composite of some prime numbers, that when multiplied together total that central number; and those prime composite numbers can be further reduced in parallel by representing each one as their central number (just like I previously described) plus or minus some offset number, and so on and so on until you are dealing with very managably small numbers in a massively parallel computation. The offset numbers can be similarly crunched down to practically nothing as well. This very well may solve a large class of NP completeness problems!!! Hurray! It could be extremely valuable in encryption, decryption, heuristics, pattern recognition, random number testing, testing for primality in the search for new primes, several branches of mathematics and other hard sciences can benefit from it as well. I discovered pretty much independently, just playing around with numbers in a coffee shop one day on 1/31/2005, and elaborated on 2/1/2005, and it was on 2/4/2005 when describing it to a friend who wishes to remain anonymous that I realized this nifty primenumber crunching technique, a few days after talking with the Carnegie Mellon University Logic and Computation Grad Student Seth Casana, actually it was then that I realized that prime numbers could be represented as an offset equation, and then I figured out how to reduce the offset equations to sets of smaller and smaller offset equations. I was showing Seth the diagrams I had drawn and the patterns in them. He commented that it looked like a Friege lattice or something. I think After I pointed out the existance of central numbers in the diagrams Seth told me that sometimes people represent prime numbers as an offset, and that all he could think of was that they could be some kind of offset or something. He's a total genius. He's graduating this year with a thesis on complexity theory and the philosophy of science. He made a bunch of Flash animations that teach people epistemology. Copyright 2/1/2005 Justin Coslor Rough draft typed 3/19/2005. This is an entirely new way to perceive of number systems. It's a way to perceive of them hierarchically. Many mathematical patterns may ready become apparent for number theorists as larger and larger maps in this format are drawn and computed. Hopefully some will be in the prime number system, as perceived through a variety of other numbering systems and forms of cardinality. (See photos.) Copyright 3/25/2004 Justin Coslor Hierarchical Number Theory Applied to Graph Theory When everyothernumber numerical hierarchies are converted into dependency charts and then those dependency charts are generalized and pattern matched to graphs and partial graphs of problems, number theory can apply to those problems because the hierarchies are based on the number line of various cardinalities. I had fun at Go Club yesterday, and while I was at the gym I thought of another math invention. It was great. I figured out how to convert a graph into a numerical hierarchy which is based on the number line, so number theory can apply to the graph, and do so by pattern matching the graph to the various graphs that are generated by converting numerical hierarchical representations of the number line into dependency charts. I don't know if that will make sense without seeing the diagrams, but it's something like that. The exciting part is that almost any thing, concept, game, or situation can be represented as a graph, and now, a bunch of patterns can be translated into being able to apply to them. Copyright 1/31/2005 Justin Coslor Odd and Even Prime Cardinality First twenty primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73.  *See the photo of the digram I drew on the original page. What properties and relations are there between the odd primes? First ten odd primes: 2, 5, 11, 17, 23, 37, 43, 53, 61, 71. First five odd odd primes: 2, 11, 23, 43, 61. First five odd even primes: 5, 17, 37, 53, 71. First ten even primes: 3, 7, 13, 19, 29, 41, 47, 59, 67, 73. First five even even primes: 7, 19, 41, 59, 73. First five even odd primes: 3, 13, 29, 47, 67.  prime^(odd^4) = prime^(odd)^(odd)^(odd)^(odd) = 2, 61, . . prime^(odd^3) = prime^(odd)^(odd)^(odd) = 2, 23, 43, 61, . . . prime^(odd^2) = prime^(odd)^(odd) = 2, 11, 23, 43, 61, . . . prime^(odd) = prime^(odd) = 2, 5, 11, 17, 23, 37, 43, 53, 61, 71, . . . prime^(odd)^(even) = 5, 17, 37, 53, 71, . . . prime^(even)^(odd) = 3, 13, 29, 47, 67, . . .  Copyright 6/10/2005 Justin Coslor HOPS: Hierarchical Offset Prefixes For counting hierarchically, prefix each set by the following variables: parity, level, and group (group starting number). Then use that group starting number as the starting position, and count up to the number from zero placed at that starting position for representation of a number prior to HOP computation. I need to develop a calculation method for that representation. Have a highlevel index which lists all of the group starting numbers for one of the highest rows, then the rest of the number's group numbers can be derived for any given level above or below it. All calculations should access this index. If I was to look for the pattern "55" in a string of numbers, for example, I might search linearly and copy all twodigit locations that start with a "5" into a file, along with the memory address of each, then throw out all instances that don't contain a "5" as the second digit. That's one common way to search. But for addresses with a log of digits, such as extremely large numbers, this is impractical and it's much easier to do hierarchical level math to check for matches. The simplest way to do it is a hierarchical parity check + level check + group # check before proceeding to check both parities of every subgroup on level 1 of that the offset number. The offset begins at zero at the end of the prefix's group number, and a microhierarchy is built out of that offset. For large numbers, this is much faster than using big numbers for everything. Example: Imagine the number 123,456,789 on the number line. We'll call it "N". N = 9 digits in decimal, and many more digits in binary. In HOP notation, N = parity.level.group.offset. If I had a comprehensive index of all the group numbers for a bunch of the levels I could generate a prefix for this # N, and then I'd only have to work with a tiny number that is the difference between the closest highest group and the original number, because chances are the numbers I apply it to are also offset by that prefix or a nearby prefix. The great part about hierarchical offset prefixes is that it makes every number very close to every other number because you just have to jump around from level to level (vertically) and by group to group (horizontally). I'll need to ask a programmer to make me a program that generates an index of group numbers on each level, and the program should also be able to do conversions between decimal numbers and hierarchical offset prefixes (HOPs). That way there are only four simple equations necessary to add, subtract, multiply, divide any two HOP numbers: just perform the proper conversions between the HOPs' parity, levels, groups, and offsets. Parity conversions are simple, level conversions are just dealing with powers of 2, group conversions are just multiples of 2 + 1, and offset conversions just deal with regular mathematics using small numbers. Copyright 7/7/2005 Justin Coslor Prime Breakdown Lookup Tables Make a lookup table of all of the prime numbers in level 1 level.group.offset notation, and calculate values for N levels up from there for each prime in that same level.group.offset notation using the level 1 database. 2^n = distance between prime 2^(n + m) and prime 2^(n + (m + 1)). Center numbers are generated by picking another prime on that same level somehow (I'm not positive how yet), and the number inbetween them is the center number. Center number factoring can be done repeatedly so that, for example, if you wanted to multiply a million digit number by a million digit number, you could spread that out into several thousand small number calculations, and in that way primes can be factored using center numbers + their offsets. Also, prime number divisor checking can be done laterally in parallel by representing each divisor in level.group.offset notation and then converting the computation into a set of parallel processed center number prime breakdown calculations, which would be significantly faster than doing traditional divisor checking, especially for very large divisors, assuming you have a parallel processor computer at your disposal, or do distributed computing, and do multiprocessing/multi threading on each processor as well. Copyright 10/7/2004 Justin Coslor Prime divisorchecking in parallel processing pattern search. *I assume that people have always known this information. Prime Numbers are not: 1. Even > Add all even numbers to the reject filter. 2. Divisible by other prime numbers > Try dividing all numbers on the potentially prime list by all known primes. 3. Multiples of other prime numbers > Parallel process: Map out in parallel multiples of known primes up to a certain range for the scope of the search field, and add those to the reject filter for that search scope. When you try to divide numbers on the potentially prime list, all of those divisions can be done in parallel where each prime divisor is granted its own process, and multiple numbers on the potentially prime list for that search scope (actually all of the potentials) could be divisorchecked in parallel, where every number on the potentially prime list is granted its own complete set off parallel processes, where each set contains a separate parallel process for every known prime. So for less than half of the numbers in the search scope will initially qualify to make it onto the potentially prime list for divisor checking. And all of the potentially prime numbers will need to have their divisor check processes augmented as more primes are discovered in the search scope. The Sieve of Eratosthenes says that the search scope is in the range of n^2, where n is the largest known prime. Multiple search scopes can be running concurrently as well, and smaller divisor checks will always finish much sooner than the larger ones (sequentially) for all numbers not already filtered out. 12/24/2004 Justin Coslor Look for Ways to Merge Prime Number Perception Algorithms I don't yet understand how the Riemann Zeta Function works, but it might be compatible with some of the mathematics I came up with for prime numbers (sequential prime number word list heuristics, active filtering techniques, and every other number groupings on the primes and on the natural number system). Maybe there are lots of other prime number perception algorithms that can also be used in conjunction with my algorithms. ???  Try applying my algorithm for greatly simplifying the representation of large prime numbers to the Riemann Zeta function. My algorithm reduces the complexity of the patterns between sequential prime numbers to a fixed five variable word for each pair of sequential primes, and there are only 81 possible words in all. So as a result of fixing the pattern representation language to only look for certain qualities that are in every sequential prime relationship, rather than having infinite possibilities and not knowing what to look for, patterns will emerge after not to long into the computer runtime. These patterns can then be used to predict the range of the scope of future undiscovered prime numbers, which simplifies the search for the next prime dramatically, but even more important than that is that my algorithm reduces the cardinality complexity (the representation) of each prime number significantly for all primes past a certain point, so in essence, this language I've invented is a whole new number system, but I'm not sure how to run computations on it. . .though it can be used with a search engine as a cataloging method for dealing with extremely large numbers. My algorithm is in this format: The Nth prime (in relation to the prime that came before it) = the prime number nearest to [the midpoint of the Nth prime, whether it be in the upper half or the lower half] : in relation to the remainder of that "nearmidpointprime" when subtracted from the Nth prime. The biggest part always gets listed to the left of the smaller part (with a ratio sign separating them), and if for the N 1th prime if the prime prime part got listed on one side and in the next if it's on the opposite side we take note of that. Next we find the difference in the two parts and note if it is positive or negative, even or odd, and lastly we compare it to the N1th difference to see if it is up, down, the same, or if N1's difference is greater than 1 and N's difference is 1 then we say it has been "reset". If the difference jumps from 1 to a larger difference in N's difference we say it's "undo reset". Also, the difference is the absolute value of the "nearmidpointprime" minus the remaining amount between it and the Nth prime. Now each of these qualities can be represented by one letter and placed in one of four sequential places (categories) to make a four character word. Numbers could even be used instead of characters, but that might confuse people (though not computers). ******************* "Prime Sequence Matcher" (to be made into software) ******************* This whole method is Copyright 10/25/2004 Justin Coslor, or even sooner (most likely 10/17/2004, since that's when it occurred to me. I thought of this idea to help the whole world and therefore must copyright it to ensure that nobody hordes or misuses it. The algorithms behind this method that I have invented are free for academic use by all United Nations member nations, for fair good intent only towards everyone.  Download a list of the first 10,000 prime numbers from the Internet, and consider formating it in EMACS to look something like this: 12 23 35 47 5 11 6 13 . . . 10,000 ____ and name that file primelist.txt  Write a computer program in C or Java called "PrimeSequenceMatcher" that generates a file called "primerelations.txt" in the following format based on calculations done on each of line of the file "primelist.txt". primelist.txt>PrimeSequenceMatcher>primerelations.txt file: primerelations.txt 2 3 2:1 diff 1 left, pos, odd, same 3 5 3:2 diff 1 left, pos, even, up 4 7 5:2 diff 3 left, pos, even, same 5 11 7:4 diff 3 LR, neg, even, down(or reset) 6 13 6:5 diff 1 right, pos, even, up(or undo reset) 7 17 10:7 diff 3 . . . N __ __:__ diff __ For the C program see pg. 241 to 251 of Kernigan and Ritchie's book, "The C Programming Language", for functions that might be useful in the program. See the scans of my journal entries from 10/17/2004, 10/18/2004, and 10/24/2004 for details on the process (*Note, there may be a few errors, and the paperwork is sort of sloppy for those dates...), and turn it into an efficient explicit algorithm. **2/22/2005 Update: I wrote out the gist of the algorithms for the software in my 10/26/2004 journal entry. The point of the generating the file primerelations.txt is to run the file through pattern searching algorithms, and build a relational database, because since the language of the primes's representation in my method is severely limited, patterns might emerge. Nobody knows whether or not the patterns will be consistent in predicting the range that the next primes will be in, but I hope that they will, and it's worth doing the experiment since that would be a remarkable tool to have discovered. The patterns may reveal in some cases which is larger: the nearesttomidpoint prime or it's corresponding additive part. Where the sum equals the prime. That would tell you a general range of where the next prime isn't at. Also the patterns may in some cases have a predictable "diff" value, which would be immensely valuable in knowing, so that you can compare it to the values of the prime that came before it, which would give a fairly close prediction of where the next prime may lye. By looking at the pattern of the ordering of sentences, we can possibly tell which side of the ratio sign the nearesttomidpoint prime of the next prime we are looking for lies on (and thus know whether it is in the upper half or the lower half of the search scope). The search scope for the next prime number is in the range of the largest known prime squared. We might also be able to in some cases determine how far from the absolute value of the difference between the nearestto midpoint prime and the prime number we are looking for, that the prime number that we are looking for is. Copyright 10/26/2004 to 10/27/2004 Justin Coslor I hereby release this idea under The GNU Public License Agreement (GPL). ************************* Prime Sequence Matcher Algorithm ************************* (This algorithm is to be turned into software. See previous journal entries that are related.) Concept conceived of originally on 10/17/2004 by Justin Coslor Trends in these sequential prime relation sentences might emerge as lists of these sentences are formed and parsed for all, or a large chunk of, the known primes.  The following definitions are important to know in order to understand the algorithm: nmp = the prime number nearest to the midpoint of "the Nth prime we are representing divided by 2" aptnmp = adjacent part of the nmp = prime number we are representing minus nmp prime/2 = (nmp+aptnmp)/2 = the midpoint of the prime nmp = (2 * midpoint)  aptnmp aptnmp = (2 * midpoint)  nmp prime = 2 * midpoint We take notice of whether nmp is greater than, equal to, or less than aptnmp. diff = nmp  aptnmp N prime = nmp:aptnmp or aptnmp:nmp, diff = nmp  aptnmp ___________________________________  a  b  c  d   left  pos  even  up   right  neg  odd  down   LR  null   same   RL    reset      undoreset   Each possible word can be abbreviated as a symbolic character or symbolic digit, so the sentence is shortened to the size of a four character word or four digit number. *Note: "a" only = "same" when prime = 2 (.....that is, when N = 1) **Note: If "c" ever = "same", then N is not prime, so halt. "abcd" has less than or equal to 100 possible sequential prime relation sentences (SPRS)'s, since the representation is limited by the algorithms listed below. Generate a list of SPRS's for all known primes and do pattern matching/search algorithms to look for trends that will limit the search scope. The algorithms might even include SPRS orderings recursively.  Here are the rules that govern abcd: If nmp > aptnmp, then a = left. If nmp < aptnmp, then a = right. If nmp = aptnmp, then a = same. If N  1's "a" = left, and N's "a" = right, then set N's "a" = LR. If N  1's "a" = right, and N's "a" = left, then set N's a = RL. If N's nmp  (N  1)'s nmp > 0, then b = pos. If N's nmp  (N  1)'s nmp < 0, then b = neg. If C = same, then b = null. Meaning, if N's nmp  (N1)'s nmp = 0, then b= null. If N's nmp  (N1)'s nmp is an even integer, then c = even. If N's nmp  (N  1)'s nmp is an odd integer, then c = odd. If N's diff > (N  1)'s diff, then d = up. If N's diff < (N  1)'s diff, then d = down. If N's diff = (N1)'s diff, then d = same. If (N  1)'s diff > 1 and N's diff = 1, then d = reset. If (N  1)'s diff = 1 and N's diff > 1, then d = undoreset. [......But maybe when (N  1)'s diff and N's diff = either 1 or 3, then d would also = up, or d = down.] If a = left or RL, then N prime = nmp:aptnmp, diff = nmp  aptnmp If a = right or LR, then N prime = aptnmp:nmp, diff = nmp  aptnmp If a = same, then N prime = nmp:nmp, diff = nmp  aptnmp, but only when N prime = N.  Copyright 10/24/2004 Justin Coslor Prime number patterns based on a ratio balance of the largest nearmidpoint prime number and the nonprime combinations of factors in the remainder: An overlay of symmetries describe prime number patterns based on a ratio balance of the largest near midpoint prime number and the nonprime combinations of factors in the remainder. This is to cut down the search space for the next prime number, by guessing at what range to search the prime in first, using this data. For instance, we might describe the prime number 67 geometrically by layering the prime number 31 under the remainder 36, which has the modulo binary symmetry equivalency of the pattern 2*2*3*3. We always put the largest number on top in our description, regardless of whether it is prime or nonprime, because this ordering will be of importance in our sentence description of that prime. We describe the sentence in relation to how we described the prime number that came before it. For instance, we described 61 as 61=31:2*3*5 ratio (the larger composite always goes on the left of the ratio symbol, because it will be important to note which side the prime number ends up on), difference of 1 (difference shows how far from the center the nearmid prime lies. 3130=1), right>left (this changing of sides is important to note because it describes which side of the midpoint of the prime that the nearesttomidpoint prime lies on or has moved to, in terms of the ratio symbol) odd same (this describes whether the nearesttomidpoint primes of two prime numbers have a difference that is even, odd, or if they have the same nearesttomidpoint primes.) 67=2*2*3*3:31 ratio, difference of 5, left>right same undo last reset. By looking at the pattern in the sentence descriptions (180 possible sentences), we can tell which side of the ratio sign that the next prime's nearesttomidpoint prime lies on, which tells you which half of the search scope the next prime lies in, which might cut the computational task in finding the next finding that next prime number in half or more. A computer program to generate these sentences can be written for doing the pattern matching. In the prime number 67 example, the part that says "same", refers to whether the nearestto midpoint primes of two prime numbers have a difference that is even, odd, or if they have the same nearesttomidpoint primes. I threw in the "reset to 1" thing just because it probably occurs a lot, then there's also the infamous "undofromlastreset" which it brings the difference from 1 back to where it was previously at. Copyright 10/5/2004 Justin Coslor Prime Numbers in Geometry continued . . . Modulo Binary I think that if prime numbers can be expressed geometrically as ratios there might be a geometric shortcut to determining if a number is prime or maybe nonprime. Prime numbers can be represented symmetrically, but not with colored partitions. (*See diagrams.) Here's a new kind of binary code that I invented, based on the method of partitioning a circle and alternately coloring and grouping the equiangled symmetrical partitions of nonprime partition sections. (*Note, since prime numbers don't have symmetrical equiangled partitions, use the centernumber + offset converted into modulo binary (see my 2/4/2005 idea and the 2/1/2005 diagram I drew for prime odd and even cardinality and data compression on the prime numbers)). Modulo binary: *Based on geometric symmetry ratios. **I may not have been very consistent with my numbering scheme here, but you should be in final draft version. 1=1 2=11 3=111 4=1010 5=11111 6=110110 or 101010 7=1111111 8=10101010 or 11101110 9=110110110 10=1010101010 11=11111111111 12=110110110110 13=1111111111111 14=10101010101010 15=10110,10110,10110 16=1010,1010,1010,1010 Find a better way of doing this that might incorporate my prime center number + offset representation of the primes and nonprimes. This is an entirely new way of counting, so try to make it scalable, and calculatable. Secondary Levels of Modulo Binary: (*This is just experimental. . .I based these secondary levels on the first level numbers that are multiples of these.) 0=00 1=1 2=10 3=110 4=2+2=1010 5=10110 6=3+3=110110 or 111000 or 101101 7= 8=4+4=10101010 9=3+3+3=110110110 10=1010101010 11= 12=3+3+3+3=110110110110 13= 14=10101010101010101010 15=5+5+5=101101011010110 16=4+4+4+4=1010101010101010 Draw a 49 section and 56 section circle, and look for symmetries to figure out how best to represent the number 7 in the secondary layer of modulo binary. There needs to be a stop bit too. Maybe 00 or something, and always start numbers with a 1. The numbers on through ten should be sufficient for converting partially from base 10. Where calculations would still be done in base 10, but using modulo binary representations of each digit. For encryption obfuscation and stuff. It seems that for even numbers, the halfcircle symmetries rotate between 0,0 across the circle for numbers that are odd when divided by two, and the numbers that are odd when divided by two have alternatehalf 0,0 symmetry. But numbers that are prime when divided by two have middle across 0,1 symmetry. Copyright 9/30/2004 Justin Coslor Prime Numbers in Geometry *Turn this idea into a Design Science paper entitled "Patterns in prime composite partition coloring structures". In the paper, relate these discoveries to the periodic table. (All prime numbers can be represented as unique symmetries in Geometry.) 1/1 = 0 division lines 1/2 = 1 division lines 1/3 = 3 division lines 1/4 = 2 division lines 1/5 = 5 division lines 1/6 = 5 division lines = one 1/2 division line and two 1/3 division lines on each half circle. 1/7 = 7 division lines 1/8 = 4 division lines 1/9 = _____ division lines . . . Or maybe count by partition sections rather than division lines. . . How do I write an algorithm or computer program that counts how many division lines there are in a symmetrically equiangled partitioning of a circle, where if two division lines that meet in the middle (as all division lines do) form a straight line they would only count as one line and not two? Generate a sequential list of values to find their number of division lines, and see if there is any pattern in the nonprime division line numbers (i.e. 1/4, 1/6, 1/8, 1/9, 1/10, 1/12, ...) that might be able to be related to the process of determining or discovering which divisions are prime, or the sequence of the prime numbers (1/2, 1/3, 1/5, 1/7, 1/11, 1/13, 1/17, ...). 10/5/2004 Justin Coslor As it turns out, there is a pattern in the nonprime division lines that partition a circle. The equiangled symmetry partition patterns look like stacks of prime composites layered on top of one another like the Tower of Hanoi computer game, where each layer's nonprime symmetry pattern can be colored using it's own colors in an onoff configuration around the circle (See diagrams.). Prime layers can't be colored in an onoff pattern symmetrically if the partitions remain equiangled, because there would be two adjacent partitions somewhere in the circle of the same color, and that's not symmetrical. Copyright 7/25/2005 Justin Coslor Geometry of the Numberline: Pictograms and Polygons. (See diagrams) Obtain a list of sequential prime numbers. Then draw a pictogram chart for each number on graph paper, with the base 10 digits 1 through 10 on the Yaxis, and on the Xaxis of each pictogram the first column is the 1's column, the second column is the 10's column, the third columns is the 100's column, etc. Then plot the points for each digit of the prime number you're representing, and connect the lines sequentially. That pictogram is then the exact unique base10 geometrical representation of that particular prime number (and it can be done for nonprime numbers too). Another way to make the pictogram for a number is to plot the points as described, but then connect the points to form a maximum surface area polygon, because when you do that, that unique polygon exactly describes that particular number when it's listed in its original orientation. inside the base10 graph paper border that uses the minimum amount of Xaxis boxes necessary to convey the picture, and pictograms are always bordered on the canvas 10 boxes high in base 10. Other bases can be used too for different sets of pictograms. What does the pictogram for a given number look like in other bases? We can connect the dots to make a polygon too, that is exactly the specific representation in its proper orientation of that particular unique number represented in that base. Also I wonder what the pictograms and polygon pictograms look like when represented in polar coordinates? These pictogram patterns might show up a lot in nature and artwork, and it'd be interesting to do a mathematical study of photos and artwork, where each polygon that matches gets bordered by the border of it's particular matching pictogram polygon in whatever base it happens to be in, and pictures might be representable as layers of these numerical pictograms, spread out all over the canvas overlapping and all, and maybe partially hidden for some. You could in that way make a coordinate system in which to calculate the positions and layerings of the numerical pictograms that show up within the border of the photo or frame of the artwork, and it could even be a form of steganometry when intentionally layered into photos and artwork, for cryptography and art. Summing multiple vertexes of numerical polygon pictograms could also be used as a technique that would be useful for surjectively distorting sums of large numbers. That too might have applications in cryptography and computer vector artwork. See the diagram of the base 10 polar coordinate pictogram representation of the number 13,063. With polar notation, as with Cartesian Coordinate System notation of the pictograms, it's important to note where the reference point is, and what base it's in, and whether it's on a polar coordinate system or Cartesian Coordinate System. In polar coordinates, you need to know where the center point is in relation to the polygon. . .no I'm wrong, it can be calculated s long as no vertexes lie in a line. In all polygon representations, the edge needs to touch all vertexes. Copyright 7/27/2005 Justin Coslor Combining level.group.offset hierarchical representation with base N pictogram representation of numbers (See diagrams) level.group offset notation is (baseN^level)*group+offset Pictogram notation is as described previously. If you take the pictogram shape out of context and orient it differently it could mean a lot of different things, but if you know the orientation (you can calculate the spacing of the vertexes in different orientations to find the correct orientation, but you know must also know what base the number is in to begin with) then you can decipher what number the polygon represents. You must know what the base is because it could be of an enormous base. . .you must also know an anchor point for lining it up with the XY border of the number line context in that base because it could be a number shape floating inside a enormous base for all anyone knows, with that anchor point. Also, multiple numbers on the same straight line can be confusing unless they are clearly marked as vertexes. If multiple polygons are intersecting, then they could represent a matrix equation of all of those numbers. Or if there are three or four polygons connected to each other by a line or a single vertex, then the three pictograms might represent the three or four parts of a large or small level.group.offset number in a particular base. Pictograms connected in level.group offset notation would still need to be independently rotated into their correct orientation, and you'd need to know their anchor points and base, but you could very simply represent an unfathomably enormous number that way in just a tiny little drawing. Also, numbers might represent words in a dictionary or letters of an alphabet. This is literally the most concise way to represent unfathomably enormous numbers that possibly anyone has ever imagined. Ever. You could write a computer program that would draw and randomize these drawings as a translation from a dictionary/language set and word processor document. They could decoded in the reverse process by people who know the anchor point keys and base keys for each polygon. You can make the drawings as a subtle offwhite color blended into the white part of the background of a picture, and transmit enormous documents as a single tiny little picture that just needs some calculating and keys to decode. Different polygon pictograms, which each could represent a string of numbers, which can be partitioned into sections that each represents a word or character, could each be drawn in a different color. So polygons that are in different colors and different layers in a haphazard stack, could be organized, where the color of multiple polygons, means they are part of the same document string, and the layering of the polygons indicates the order that the documents are to be read in. Copyright 7/28/2005 Justin Coslor Optimal Data Compression: Geometric Numberline Pictograms If each polygon is represented using a different color, you don't even need to draw the lines that connect the vertexes, so that you can cram as many polygons as possible onto the canvas. In each polygon, the number of vertexes is the number of digits in whatever base it's being represented in. Large bases will mean larger image dimensions, but will allow for really small representations of large numbers. Ideally one should only use a particular color on one polygon once. For optimal representation, one should represent each number in a base that is as close to the number of digits in that base as possible. If you always do that, then you won't have to know what base the polygon is represented in to begin with (because it can be calculated). However, you will still need to know the starting vertex or another anchor point to figure out which orientation the polygon is to be perceived of in. On polar coordinate polygon pictograms, you will just need to know the center point and a reference point such as where the zero mark is, as well as what base the polygon is represented in (in most cases). Hierarchical level.group.offset data compression techniques or other data compression techniques can also be used. Copyright 7/24/2005 Justin Coslor Prime Inversion Charts (See diagram) Make a conversion list of the sequential prime numbers, where each number (prime 1 through the N'th prime) is inverted so that the least significant digit is now the most significant digit, and the most significant digit is now the least significant digit (ones column stays in the ones column, but the 10's column gets put in the 10ths column on the other side of the decimal point, same with hundreds, etc.). So you have a graph that goes from 0 through 10 on the Yaxis, and 0 through N along the X axis, and you just plot the points for prime 1 through the N'th prime and connect the dots sequentially. Also, you can convert this into a binary string by making it so that if any prime is higher up on the Yaxis than the prime before it, it becomes a 1, and if it is less than the prime before it, it becomes a 0. Then you can look for patterns in that. I noticed many recurring binary string patterns in that sequence, as well as many pallendrome string patterns in that representation (and I only looked at the first couple of numbers, so there might be something to it). 10/8/2004 Justin Coslor Classical Algebra (textbook notes) Pg. 157 of Classical Algebra fourth edition says: The Prime Number Theorem: In the interval of 1 through X, there are about X/LOGeX primes in this interval. P=X/LOGeX scope: (1,X) or something. The book claims that they cannot factor 200 digit primes yet. In 1999 Nayan Hajratwala found a record new prime 2^6972593  1 with his PC. It's a Mersenne Prime over 2 million digits long. This book deals a lot with encryption. I believe that nothing is 100% secure except for the potential for a delay. On pg. 39 it says "There is no known efficient procedure for finding prime numbers." On pg. 157 it directly contradicts that statement by saying: "There are efficient methods for finding very large prime numbers." The process I described in my 10/7/2004 journal entryis like the sieve of Eratosthenes, except my method goes a step farther in making a continuously augmented filter list of divisor multiplicants not to bother checking, while simultaneously running the Sieve of Eratosthenes in a massive synchronously parallel computational process. Prime numbers are useful for use in pattern search algorithms that operate in abductive and deductive reasoning engines (systems), which can be used to explore and grow and help solve problems and provide new opportunities and to invent things and do science simulations far beyond human capability. (Pg. 40) Theorem: An integer x>1 is either prime or contains a prime factor <=sqrt(x). Proof: x=ab where a and b are positive integers between 1 and x. Since P is the smallest prime factor, a>=p, b>=p and x=ab>=p^2. Hence p<=sqrt(x). Example: If x=10 a=2 and b=5. p=3 p^2=9 so 10=2*5>=9. So factors of x are within the scope of (2, sqrt(x)) or else it's prime. a^2>=b^2. x^2>=p^4. x^2/p^4=big. Try converting Fermat's Little Theorem and other corollaries into geometry symmetries and modulo binary format. The propositions in Modern Algebra about modulo might only hold for two dimensional arithmetic, but if you add a 3rd dimension the rotations are countable as periods on a spiral, which when viewed from a perpendicular sideview looks like a 2dimensional waveform. 9/26/2004 Justin Coslor Privacy True privacy may not be possible, but the best that we can hope for is a long enough delay in recognition of observations to have enough time and patience to put things intot the perspective of a more understanding context. Copyright 9/17/2004 Justin Coslor A Simple, Concise, Encryption Syntax. This can be one layer of an encryption, that can be the foundation of a concise syntax. *Important: The example does not do this, but in practice, if you plan on using this kind of encryption more than once, then be sure to generate a random unique binary string for each letter of the alphabet, and make each X digits long. Then generate a random binary string that is N times as long as the length of your message to be sent, and append unique sequential pieces (of equal length) of this random binary string to the right of each character's binary representation. The remote parts should have lots of securely acquired random unique alphabet/random binary string pairs, such as on a DVD that twas delivered by hand. In long messages, never use the same alphabet's character(s) more than once but rotate to the next binary character representation on the DVD sequentially. Here's the example alphabet (note that you can of course choose your own alphabetic representation as long as it is logically consistent): a 010101 b 011001 c 011101 d 100001 e 100101 f 101001 g 110001 h 110101 i 111001  j 010110 k 011010 l 011110 m 100010 n 100110 o 101010 p 110010 q 110110 r 111010  s 010111 t 011011 u 011111 v 100011 w 100111 x 101011 y 110011 z 110111 space 111011  EXAMPLE: "peace brother" can be encoded like this using that particular alphabet: 011011101001011000101101011100100110010110010101110111011010010110111011010110 0101111010101010100001101110110101111001010111101001  2/18/2005 Update by Justin Coslor Well, I forgot how to break my own code. Imagine that! I think it had something to do with making up a random string that was of a length that is divisible by the number of letters in the alphabet, yet is of equal bitlength to the bittranslated message, so that you know how long the message is, and you know how many bits it takes to represent each character in the alphabet. Then systematically mix in the random bits with the bits in the encoded message. In my alphabet I used 27 characters that were each six bits in length; and in my example, my message was 13 characters long, 11 of which were unique. I seriously have no idea what I was thinking when I wrote this example, but at least my alphabet I do understand, and it's pretty concise, and sufficiently obscured for some purposes. Copyright 6/30/2005 Justin Coslor Automatic Systems (See Diagram) There is 2D, and there are 3D snapshots represented in 2D, and there is the modeltheory approach of making graphs and flowcharts, but why not add dimensional metrics to graph diagrams to represent systems more accurately?  Atomic Elements > Mixing pot > Distillation/Recombination: A>B>C>D>E > State Machine Output Display (Active Graphing = realtime) > Output Parsing and calculation of refinements (Empirical) > Set of contextually adaptive relations: R1>A, R2>B, R3>C, R4>D, R5>E.  Copyright 5/11/2005 Justin Coslor How to combine sequences: Draw a set of Cartesian coordinate system axis, and on the x axis mark off the points for one sequence, and on the y axis mark off the points for the sequence you want to combine with it (and if you have three sequences you want to combine, mark off the third sequence on the zaxis. ...for more than 3 sequences, use linear algebra). Next draw a box between the origin and the first point on each sequence; then calculate the length of the diagonal. Then do the same for the next point in each sequence and calculate the length of the diagonal. Eventually you will have a unique sequence that is representative of all of the different sequences that you combined into one in this manner. For instance, you could generate a sequence that is the combination of the prime numbers and the Fibonacci Sequence. In fact, the prime numbers might be a combination of two or more other sequences in this manner, for all I know. 1/4/2005 Justin Coslor Notes from the book "Connections: The Geometric Bridge Between Art and Science" + some ideas. In a meeting with Nehru in India in 1958 he said "The problem of a comprehensive design science is to isolate specific instances of the pattern of a general, cosmic energy system and turn these to human use." The topic of design science was started by architect, designer, and inventor Buckminster Fuller. The chemical physicist Arthur Loeb, who considers design science to be the grammar of space. Buy that book, as well as the book "The Undecidable" by Martin Davis. Chemist Istvan Hergittai edited two large books on symmetry. He also edits the journals "symmetry" and "space structures" where I could submit my paper on the geometry of prime numbers and patterns in composite partition coloring structures. *Also, send it to Physical Science Review to solicit scientific applications of my discovery. Send it to some math journals too. Again, the paper I want to write is called "Patterns in prime composite partition coloring structures", and it will be based on that journal entry I had about symmetrically dividing up a circle into partitions, then labeling the alternating patterns in the symmetries using individual colors for each primary pattern in the stack, similar to that game "The Tower of Hanoi". Study the writings of Thales (Teacher of Pythagoras), who is known as the father of Greek mathematics, astronomy, and philosophy, and who visited Egypt to learn its secrets [Turnbull, 1961 "The Great Mathematicians], [Gorman, 1979 Pythagoras  A Life]  Connections page 11. Figure 1.7 The Ptolemaic scale based on the primes 2, 3, and 5. C=1, D=8/9, E=4/5, F=3/4, G=2/3, A=3/5, B=8/15, C=1/2.  Figure 1.6 The Pythagorean scale derived from the primes 2 and 3: C=1, space=8/9, D=8/9, space=8/9, E=64/81, space=243/256, F=3/4, space=8/9, G=2/3, space=8/9, A=16/27, space=8/9, B=128/243, space=243/256, C'=1/2, space=8/9, D'=4/9, space=8/9, E'=32/81, space=243/256, F'=3/8, space=8/9, G'=1/3, space=8/9, A'=8/27, space=8/9, B'=64/243, space=243/256, C"=1/4.  *1/4/2005 Project: Someday try writing an electronic music song that makes vivid use of parallel mathematical algorithms based on the prime numbers, actually come to think of it, this concept was presented in an episode of Star Trek Voyager.  8/26/2004 Justin Coslor Notes (pg. 1) These are my notes on three papers contributed to the MIT Encyclopedias of Cognitive Science by Wilfried Sieg in July 1997: Report CMUPHIL79, Philosophy, Methodology, Logic. Pittsburgh, Pennsylvania 152133890.  Formal Systems  Church Turing Thesis  Godel's Theorems  Notes on Wilfried Sieg's "Properties of Formal Systems" paper: Euclid's Elements > axiomaticdeductive method. Formal Systems = "Mechanical" regimentation of the inference steps along with only syntactic statements described in a precise symbolic language and a logical calculus, both of which must be recursive (by the ChurchTuring Thesis). Meaning Formal Systems use just the syntax of symbolic word statements (not their meaning), recursive logical calculus, and recursive symbolic definitions of each word. Frege in 1879: "a symbolic language (with relations and quantifiers)" + an adequate logical calculus > the means for the completely formal representation of mathematical proofs. Fregean frame > mathematical logic >Whitehead & Russell's "Principia Mathematica" > metamathematical perspective < Hilbert's "Grundlagen der Geometrie" 1899 *metamathematical perspective > Hilbert& Bernays "Die Prizipien der Mathematik" lectures 1917 1918 > first order logic = central language + made a suitable logical calculus. Questions raised: Completeness, consistency, decidability. Still active. Lots of progress has been made in these areas since then. **Hilbert & Bernays "Die Prizipien der Mathematik" lectures 19171918 > mathematical logic. Kinds of completeness: Quasiempirical completeness of Zermelo Fraenkel set theory, syntactic completeness of formal theories, and semantic completeness = all statements true in all models.  Sentential logic proved complete by Hilbert and Bernays (1918) and Post (1921).  First order logic proved complete by Godel (1930). "If every finite subset of a system has a model, so does the systems." But first order logic has some nonstandard models. Hilbert's Entsheidungsproblem proved undecidable by Church & Turing. It was the decision problem for first order logic. So the "decision problem" proved undecidable, but it lead to recursion theoretic complexity of sets, which lead to classification of 1. arithmetical, 2. hyperarithmetical, and 3. analytical hierarchies. It later lead to computational complexity classes. So they couldn't prove what could be decided in first order logic, but they could classify the complexity of modes of computation using first order logic. In first order logic, one can classify the empirical and computational complexity of syntactic configurations whose formulas and proofs are effectively decidable by a Turing Machine. I'm not positive about this next part. ...but, such syntactic configurations (aka software that eventually halts) are considered to be formed systems. In other words, ,one cannot classify the empirical and computational complexity of software that never halts (or hasn't halted), using first order logic. The Entsheidungsproblem (First order logic Decision Problem) resulted in model theory, proof theory, and computability thoery. It required "effective methods" of decision making to be precisely defined. Or rather, it required effective methods of characterizing what could or couldn't be decided in firstorder logic. The proof of the completeness theorem resulted in the relativity of "being countable" which in turn resulted in the Skolem paradox. ***I believe that paradoxes only occur when the context of a logic is incomplete or when it's foundations scope is not broad enough. Semantic arguments in geometry yielded "Relative Consistency Proofs". Hilbert used "finitist means" to establish the consistency of formal systems. Ackerman, von Neumann, and Herbrand used a very restricted induction principle to establish the consistency of number theory. Modern proof theory used "constructivist" means to prove significant parts of analysis. Insights have been gained into the "normal form" of proofs in sequent and natural deduction calculi. So they all wanted to map the spectrum of unbreakable reason. Godel firmly believed that the term "formal system' or 'formalism' should never be used for anything but software that halts.  9/1/2004 Justin Coslor Notes on Wilfried Sieg's "ChurchTuring Thesis" paper: Church redefined the term "effective calculable function" (of positive integers) with the mathematically precise term "recursive function". Kleen used the term "recursive" in "Introduction to Metamathematics, in 1952. Turing independently suggested identifying "effectively calculable functions" as functions whose values can be computed (mechanically) using a Turing Machine.Turing & Church's theses were, in effect, equivalent, and so jointly they are referred to as the ChurchTuring Thesis. Metamathematics takes formally presented theories as objects of mathematical study (Hilbert 1904), and it's been pursued since the 1920's, which led to precisely characterizing the class of effective procedures, which led to the Entsheidungsproblem, which was solved negatively relative to recursion (****but what about for nonrecursive systems?). Metamathematics also led to Godel's Incompleteness Theorems (1931), which apply to all formal systems, like type theory of Principia Mathematica or ZermaloFraenkel Set Theory, etc. Effective Computability: So it seems like they all wanted infallable systtems (formal systems), and the were convinced that the way to get there required a precise definition of effective calculability. Church and Kleen thought it was equivalent to lambdadefinability, and later prove that lambdadefinability is equivalent to recursiveness (19351936). Turing thought effective calculability could be defined as anything that can be calculated on a Turing Machine (1936). Godel defined the concept of a (general) recursive function using an equational calculus, but was not convinced that all effectively calculable functions would fall under it. Post (*my favorite definition...*) in 1936 made a model that is strikingly similar to Turing's, but didn't provide any analysis in support of the generality of his model. But Post did suggest verifying formal theories by investigating ever wider formulations and reducing them to his basic formulation. He considered this method of identifying/defining effectively calculable functions as a working hypothesis. Post's method is strikingly similar to my friend Andrew J. Dougherty's thesis of artificial intelligence, which is that at a certain point, the compactness of a set of functions is maximized through optimization and at that point, the complexity of their informational content plateaus, unless you keep adding new functions. So his solution to Artificial Intelligence is to assimilate all of the known useful functions in the world, and optimize them to the plateau point of complexity (put the information in lowest terms), and to then use that condensed information set/tool in exploring for new functions to add, so that the rich depth of the problem solving and information seeking technology can continually improve past any plateau points. (in 1939) Hilbert and Bernays showed that deductively formalized functions require that their proof predicates to be primitive recursive. Such "reconable" functions are recursive and can be evaluated in a very restricted number of theoretic formalism. Godel emphasized that provability and definability depend on the formalism considered. Godel also emphasized that recursiveness or computability have an absoluteness property not shared by provability or definability, and other metamathematical notions. My theory is a bottomup approach for pattern discovery and adaptive reconceptualization between the domains of different contexts, and can provide the theoretical framework for abductive reaasoning, necessary for the application of my friend Andrew J. Dougherty's thesis. Perhaps my theories could be abductively formalized? My theories do not require empiricism (deduction), to produce new elements that are primitiverecursive to produce new elements that are primitiverecursive (circularreasoningbased/symbolic/repetitionbased) predicates to be used in building and calculating statements and structures, that can add new information. To me, "meaning" implies having an "appreciation" for the information and functions and relations, at least in part; and that this "appreciation" is obtained through recognition of the information (and functions' and relations') utility or relative utility via use or simulation experience within partially defined contexts. I say "partiallydefined" contexts because by Godel's Incompleteness Theorems, an allencompassing ultimate context cannot be completely defined since the definition itself (and it's definer would have to be part of that context, which isn't possible because it would have to be infinitely recursive and thus never fully representable. Turing invented a mechanical method for operating symbolically. His invention's concepts provided the mechanical means for running simulations. Andrew J. Dougherty and I have created the concepts for mechanically creating new simulations to run until all possible simulations that can be created in good intention, that are helpful and fair for all, exceeds the number of such programs that can be possibly used in all of existence, in all time frames forever, God willing. Turing was a uniter not a divider and he demanded immediate recognizability of symbolic configurations, so that basic computation steps need not be further subdivided. *But there are limitations in taking input at face value. Sieg in 19944, inspired by Turing's 1936 paper formulated the following boundedness conditions and locality limitations of computors: (B.1) there is a fixed bound for the number of symbolic configurations a computor can immediately recognize; (B.2) there is a fixed bound for the number of a computor's internal states that need to be taken into account;  therefore he can carry out only finitely many different operations. These operations are restricted by the following locality conditions: (L.1) only elements of observed configurations can be changed. (L.2) the computor can shift his attention from one symbolic configuration to another only if the second is within a bounded distance from the first. *Humans are capable of more than just mechanical processes.  Notes on Wilfried Sieg's "Godel's Theorems" paper: Kurt Godel established a number of absolutely essential facts:  completeness of first order logic  relative consistency of the axiom of choice  generalized continuum hypothesis  (And relevant to the foundations of mathematics:) *His two Incompleteness Theorems (a.k.a. Godel's Theorems. In the early 20th century dramatic development of logic in the context of deep problems in the foundations in mathematics provided for the first time the means to reflect mathematical practice in formal theories. 1.  One question asked was: "Is there a formal theory such that mathematical truth is co extensive with provability in that theory?" (Possibly... See Russell's type theory P of Principia Mathematica and axiomatic set theory as formulated by Zermelo...)  From Hilbert's research around 1920 another question emerged: 2. "Is the consistency of mathematics in its formalized presentation provable by restricted mathematical, socalled finitist means? *To summarize informally: 1. Is truth coextensive with provability? 2. Is consistency provable by finitist means? Godel proved the second question to be negative for the case of formalizably finitist means. Godel's Incompleteness theorems:  If P is consistent (thus recursive), then there is a sentence sigma in the language of P, such that neither sigma nor its negation notsigma is provable in P. Sigma is thus independent of P. (Is sigma the dohnut hole of reason that fits into the center of the circular reasoning (into the center of, but independent from the recursion)?)  If P is consistent, then cons, the statement in the language of P that expresses the consistency of P, is not provable in P. Actually Godel's second theorem claims the unprovability of that second (meta) mathematical meaningful statement noted on pg. 7. Godel's first incompleteness theorem's purpose is to actually demonstrate that some syntactically true statements can be semantically false. He possibly did this to show that formal theories are not adequate by themselves to fully describe true knowledge, at least with knowledge that is represented by numbers, that is. It illustrates how it is possible to lie with numbers. In other words, syntax and semantics are mutually exclusive, and Godel's second Incompleteness Theorem demonstrates that. In other words the symbolically representative nature of language makes it possible to lie and misinterpret. Godel liked to explain how every consistently formal system that contains a certain amount of number theory can be rigorously proven to contain undecidably arithmetical propositions, including proving that the consistency of systems within such a system is nondemonstratable; and that this can all be proven using a Turing Machine. Godel thought "the human mind (even within the realm of pure mathematics) infinitely surpasses the power of any finite machine." **But what about massively parallel Quantum supercomputers? Keep in mind the boundary and limitation conditions that Sieg noted in his ChurchTuring Thesis paper of dimensional minds in relatable timelines... (Computors). 8/26/2004 Justin Coslor Concepts that I'll need to study to better understand logic and computation: Readings: Euclid's Elements Principia Mathematica Completeness: quasiempirical completeness, syntactic completeness, semantic completeness consistency decidability recursion theoretic complexity of sets classification hierarchies computational complexity classes modes of computation model theory proof theory computability theory relative consistency proofs consistency of formal systems consistency of number theory modern proof theory constructivist proofs semantic arguments in geometry analysis sequent and natural deduction calculi recursive functions Metamathematics Type Theory ZermeloFraenkel Set Theory effective computability Lambdadefinability investigating everwider formulations primitive recursive proof predicates provability and definability meaning: [11/11/2004 Justin Coslor  Meaning depends on goalsubjective relative utility. In other words, Experience leading up to perspective filters and perspective relational association buffers.] utility and relative utility simulation deductively formalized functions boundedness conditions locality limitations formalizably finitist means choice, continuum, foundations syntax & semantics incompleteness undecidable arithmetical propositions hierarchies: arithmetical, hyperarithmetical (is hyperarithmetical where all of the nodes' relations are able to be retranslated to the perspective of any particular node?), and analytical hierarchies hierarchical complexity computational complexity Graph Theory Knowledge Representation Epistemology Pattern Search, Recognition, Storage, and retrieval Appreciation  This is an unfinished writing and I disclaim all liability.  

