Please be kind to everyone. Thank you. -------------------- JUSTIN M COSLOR PO BOX 367 MOUNT VERNON, WA 98273 ------------------------------------------------------------------------------ ------------------------------------------------------------------------------ . Possibility Thinking: Explorations in Logic and Thought by Justin Coslor Possibility Thinking: Explorations in Logic and Thought by Justin Coslor *************** Table of Contents: *************** ---------------------- Book I: Patterns In Contexts ---------------------- ---------- Stuff that occurred to me while going through some of my old journal entries (about eight pages worth). ---------- Numbers & Patterns Across Contexts ---------- Property grouping axioms in cross-domain relations ---------- Properties ---------- Patterns In Contexts: Neural Nets As Priority Systems ---------- Patterns In Contexts: a computational model for representing information metaphorically through abdicative reasoning ---------- Augmenting Ideas: Generating New Perspectives on Information ---------- Epistemology Systems ---------- Some definitions for Patterns In Contexts Theory ---------- Cliff Partitions ---------- Object-oriented processing ---------- Simulated Models and Utility Axioms ---------- Operation Spaces continued - Tomographic Data Structures ---------- Operation Spaces: Grids V.S. Networks ---------- Key axioms and branch axioms in pattern collections ---------- Hypothetical Relation Highlighting in Undefined Data Sets ---------- Am I reinventing the wheel? ---------- Programming ---------- Programming Languages ---------- Patterns In Contexts Cognition Kernel ---------- Complexity ---------- Linker Patterns ---------- Patterns In Contexts Cognition ---------- Knowledge Mining ---------- Pattern Occurances ---------- ePIC Goal Representation ---------- Cross-Domain Relations in Analogical Relations ---------- Patterns In Context and Question Asking Systems for Object-Oriented Programming ---------- Complexity Progressions ---------- Metaphoric Operations on Patterns Across Contexts ---------- Information Theory Quotes ---------- Metaphoric Operations ---------- Visual Dictionaries and Axiomatic Abdicative Simulation ---------- Patterns In Contexts: 3D Engine ---------- Graphical Representation and Visual Heuristics ---------- Creativity & Understanding ---------- Concepts ---------- Measurement Systems ---------- Re-contextualized Patterns ---------- Observing patterns and differences ---------- Patterns Matching ---------- Remote-Controlled Contexts Via Pre-Processor Switchboards ---------- Definitions ---------- Geometric Abstractions ---------- Index of Topics ---------- Abstraction ---------- Analogical Recursions ---------- Implicit V.S. Explicit Knowledge ---------- Analogy, Metaphor, and Examples ---------- Sight ---------- Computer Vision ---------- Rules Are Behavioral Expectations ---------- Categories: Part 1 ---------- Hypothetical Relation Highlighting in Undefined Data Sets ---------- Some Thoughts on Information Theory ---------- Some Methods of Proof ---------- Axiom Notes ---------- Contexts ---------- Perception ---------- Perception -- continued from ----------.... ---------- artificial intelligence notes ---------- Mission Statement ------------------------ Book II: Networks of Questions ------------------------ ---------- Everything I know about questions ---------- Question Networks: option questions v.s. spectrum questions ---------- Question expectation templates and question context intersections ---------- Question asking systems ---------- Re-defining basic question thought forms ---------- Writing Tips ---------- Regarding Education ---------- Choice ---------- Creativity ---------- Intuitions ---------- Ideas and Probabilities --------------- Book III: Math Ideas: --------------- ---------- Infinity ---------- The upper limits of NP-Completeness ---------- Qualifying & Quantifying Dimensionality ---------- Spirals ---------- Light Spirals ---------- Light Spirals continued ---------- Sine Spiral Graphing ---------- Conical Satellite Orbit Graphing ---------- Justin Coslor Applications of Conical Hyperhemisphere Graphing When Combined With Sine Spiral Graphing ---------- Cross Domain Relations, for the Mathematics of Alternative Route Exploration ---------- Hierarchical Number Theory: Graph Theory Conversions ---------- Hierarchical Number Theory Applied to Graph Theory ---------- Odd and Even Prime Cardinality ---------- HOPS: Hierarchical Offset Prefixes ---------- Prime Breakdown Lookup Tables ---------- Prime divisor-checking in parallel processing pattern search ---------- Looking for ways to merge prime number perception algorithms ---------- Prime Sequence Matcher (to be made into software) ---------- Prime Sequence Matcher Algorithm ---------- Prime number patterns based on a ratio balance of the largest near-midpoint prime number and the non prime combinations of factors in the remainder ---------- Prime Numbers in Geometry continued . . . Modulo Binary ---------- Prime Numbers in Geometry ---------- Geometry of the Numberline: Pictograms and Polygons. ---------- Combining level.group.offset hierarchical representation ---------- N pictogram representation of numbers ---------- Optimal Data Compression: Geometric Numberline Pictograms ---------- Prime Inversion Charts ---------- Classical Algebra (textbook notes) ---------- Privacy ---------- A Simple, Concise, Encryption Syntax ---------- Automatic Systems ---------- How to combine sequences ---------- Notes from the book "Connections: The Geometric Bridge Between Art and Science" + some ideas ---------- Notes on three papers in the MIT Encyclopedias of Cognitive Science by Professor Wilfried Sieg: Formal Systems, Church Turing Thesis, and Godel's Theorems ---------- Concepts that I'll need to study to better understand logic and computation ---------------- Book IV: Invention Ideas ---------------- ---------- Penny Universities ---------- Inspiring book notes & some art product ideas ---------- Art Software For Making Symmetrical Design Patterns & Motifs ---------- Perspective Drawing Projects ---------- Robotics ---------- Video Game Idea ---------- Portable Operating Systems On Cross-Platform USB 2.0 Flash-Memory Devices ---------- Nanotechnology ---------- Product idea: seed kits for suburban gardens ---------- Bottled Water ---------- Camping Stuff ---------- Cooling Shirt ---------- Dune Shirt ---------- Alternative Energy System for homes that have an acre or more ---------- Knowledge lifespans and pictograph software ---------- Pictograph books for language adaptation ---------- Knowledge lifespans and pictograph software ---------- Industries, Priority Systems, and Adaptation ---------- Industry Creation ---------- Natural Gas Mining in Landfills, Using Grids of Vertical Living Bamboo Plant Tree Grass Life Tubes ---------- 3-Legged Walking Robot (It Gallops) ---------- Magneto-sonic Element Separator ---------- Diamagnetic Electromagnets ---------- PDA Kiosks ---------- PDA Virtual Reality/Augmented Reality System ---------- My eBike Shop: Eco-Bikes(TM)(R) ---------- Ram-Horn Handlebars for City Bikes ---------- Augmented Reality Goggles for doing X-Ray Vision and 3D Image Reconstruction ---------- New Kinds of Electric Generator/Electric Motors ---------- Batteries ---------- Diamagnetic Energy Generation Satellites ---------- Electromagnetism ---------- Magnetic Coordinates ---------- Magnetic Battery Idea ---------- Toroidal EM Fields ---------- Crystalline Memory Lattices, and Super-Cheap Gigantic Diamagnetic Electromagnets ---------- Supercomputer and Memory Technology ---------- Speculation about photodiodes, photovoltaics, and the nature of energy transmission through materials ---------- A New Kind of Potentiometer (which can be used as a tactile sensor) ---------- Highly sensitive touch-sensing/optical range-finding robotic finger sensor-pads ---------- A cheaper kind of touch-screen for computer kiosks for 2D operating system user interface haptics and 3D augmented reality user interface haptics ---------- Course Catalog and Skills Inventory (Software Idea) ---------- Portable Robotic Hole-Threading Tool for Building Robots and Stuff. ---------- Cottage Industry Explosion: Portable Inexpensive Manufacturing Robots for Intentional Communities in America ---------- Cottage Industry Stuff ---------- Holographic Rangefinding Orbiting Space Telescopes, for Holographic Geometric Reconstruction of Distant Celestial Objects and Celestial Systems (or point it at the Earth for Geographical Information Systems) ---------- Safety Skin for Robots ---------- Free public archive of great anonymous writing and poetry ---------- Precise EM field sensing nanocoil fabric ---------- Notes on an Intelligent Agents book ---------- 2D Wireframe Digitalization of Digital Images for Computer Vision ---------- Computer Vision Thought ---------- 3D Engine Idea ---------- Binary Space Partition Trees ---------- Data compression for 3D polygonal objects ---------- 3D compression scheme --------Re-write my paper on Chromodepth prism glasses and anaglyphs and how you can just snap regular photos in a black room that has a red, blue and green light sequentially lined up illuminating the objects front to back, for making 3D Chromodepth photos. ---------- ---------- Thoughts on Computer Vision: 2D Snapshots to 3D Wireframe ---------- More Computer Vision Thoughts ---------- Imaginative new perspectives on famous artworks ---------- More thoughts on my Thermoelectric Generator Cloth Invention ---------- Inventory of revolutionary inventions that I've invented (but not built) up to this date --------- Puzzles --------- World Medicine ---------------------- Book V: Philosophy & Quotes ---------------------- --------- Some topics I like that I think can really help people and the world --------- The Dreamworld --------- Violence quote --------- Ways To Help -- Topical Index of Ideas to Elaborate On: --------- Bubble quote --------- The Future --------- Homo-Sapien Evolution --------- Prizes --------- Interconnected Galaxies --------- Stargate (stay put) --------- A guess about Pulsars --------- Freedom in Sleep --------- Good Thinking V.S. Flawed Thinking :: Helpful V.S. Harmful --------- Hollywood quote "Do not watch television or movies." --------- Do magnets affect time? --------- Measuring Progress --------- Me --------- Expectations --------- New Logic Operators --------- Motion --------- Book Notes on "Connections", and "Telepathy, and the Etheric Vehicle" --------- Deep Thinking --------- Outside the library --------- The Soul --------- One possible order of developments leading up to friendly intelligence --------- Monality and Plurality --------- What I Believe --------- Life Skills :: Love and Forgiveness --------- More thoughts on the philosophy of love... --------- Excerpt from Justin's Journal --------- Appreciation --------- Ways to heighten interest and ways of inspiring motivation --------- On Motivation --------- Information Theory Quotes --------- Perls of Wisdom ---------------------------------------------------------------- ---------------------------------------------------------------- ---------------------------------------------------------------- The following copyrights and dates may or may not matter. ---------------------------------------------------------------- ---------------------------------------------------------------- ---------------------------------------------------------------- ---------------------------------------------------------------- ---------------------- Book I: Patterns In Contexts ---------------------- Stuff that occurred to me while going through some of my old journal entries (about eight pages worth). Analogies mimic patterns across contexts via cross-domain relations. That's the basis of Analogical Reasoning. Every pattern in every context is unique to the properties and axioms of the contexts they exist in. I've written this book without doing a lick of research or reading (except where indicated on a few entries), as an experiment to see if I could generate some new foundations of knowledge and understanding. Some experts say I succeeded. A symmetry is an example of an internal algebra. Unique symmetries are atomic repetitions, and are the simplest form of patterns, distinct from perceptually apparently random chaos. (I don't believe in ultimate randomness). Analogical mimicing results in similar, yet distinctly different patterns. All truth is but an approximation of a deeper truth. Understanding is subject to computational complexity of the perceiver and the data forms and content perceived. Knowledge is the quest of discovery, and understanding is the growth of the perceiver. It's how possibilities happen through careful navigation. There are no dead ends. Mark fundamental landmark differences in analogically mimiced patterns, for possible classification category augmentations (for navigation and data retrieval purposes). Beware of oversimplification of data streams in order to fit a pattern into a perceptual mold. Even if my ideas overlap with existing knowledge, they provide a new way of understanding that knowledge, and that is valuable because my ideas are not based on mimicry since I haven't studied topics related to them much (except a college philosophy course and K-13 math). These ideas exist for the most part in their own context. They can be no doubt eventually be linked to ideas in other contexts though. Lexicons can often be linked to external contexts. -------------------------------------- Numbers & Patterns Across Contexts Metaphorically speaking, prime numbers are injective and composite numbers are surjective, when translating functions from one context to another. Similarly, single-repetition patterns are injective and composite patterns are surjective, when translating relations from one context to another. This is an essential part of analogical reasoning. -------------------------------------- Property grouping axioms in cross-domain relations. (See diagram.) 1. All variables have properties. 2. All properties are independent of their variable's context(s). 3. All properties have some combination of qualitative relations, quantitative relations, existential locations, and existential conditions. 4. Every variable exists within a context and can vary from context to context. 5. Contexts are composed of networks of patterns, patterns are composed of networks of variables, and variables are composed of networks of properties. 6. Information can be represented as patterns in contexts, and in that way it can be represented metaphorically through analogical reasoning and abdicative reasoning. Relations of various kinds, location(s), and condition(s) (apon and of) exist at all of the various levels, and those are the data access points. ---------------------------------- Properties These are first-level definitions of some useful kinds of properties, any of which can be networked together to create relations and variables and patterns and contexts that may exist in physical and/or platonic reality: ---------------------------------- * qualitative identifiers: Categorical names and cross-references. * qualitative factors: Qualitative pieces of composite patterns. * quantitative identifiers: Cardinalities (orderings), scalers, and surjective equalities. * quantitative factors: Representational methods of measurement of dimension sets. * states: Observable distinct configurations that mark and increment step counts. * conditions: Dependencies that distinctly configure each state. * cycle counts: A tally that is increased with each repetition of a process. * recursions: A self-defined process or network (an internal algebra), or a function that calls itself. * repetitions: An algebra, atomic elements that repeat, composite patterns that repeat, or symmetries. * activity level: The number of cycles per step (positive, negative, random, or null). * step counts: A tally that is increased as conditions of each state transition is reached. * location: A place in a memory grid where identifiable data is stored. * positions: The sequence coordinates of variable in N-dimensional orderings. * orientations: The perspective that data maps are observed from: This may be contextual, or spatially framed position maps, and perspectives may even have translation conditions of their own. ------------------------ As you can see, activity level is just one kind of property, and priority systems such as neural nets can be based on that property. Other kinds of systems can be based on other properties. ------------------------ * Relations are the juxtaposition of infrastructures, which result in an output. ------------------------ Patterns In Contexts: Neural Nets As Priority Systems Neural nets are essentially priority systems for allocating and de-allocating priorities of networked elements such as variables on a grid. Each network can be considered a context, and can be said to be a network of patterns composed of variables and relations. If the patterns are functions, then the priority of each pattern determines the level of activity (cycles per step) of each pattern's function(s). Some priority level results in a random level of activity, and the other priority levels result in either positive levels of activity, negative (reverse) levels of activity, and an undefined priority setting results in no activity. Example: Context: ABCDEFG * R1R2R3R4R5R6R7R8 == a network of patterns (see diagram). Where each pattern == (a variable)(a relation Rn), and the level of activity of each pattern is: (undefined, -3, -2, -1, random, 1, 2. 3)(a variable)(a relation Rn), and the activity level determines how many cycles per step that the relation Rn operates on the variable, and the pattern it is linked to. These values could be anything, this is just an example. Activity level is one type of property of the variable. ***Variables are composed of networks of properties. ***Patterns are composed of networks of variables and relations. ***Contexts are composed of networks of patterns and relations to other contexts. Properties can be things like qualitative and quantitative identifiers and factors, states, conditions, cycle counts, recursions and repetitions, activity level (cycles per step), step counts, locations, positions, orientations, etc. Patterns in Contexts: a computational model for representing information metaphorically through abdicative reasoning. All ideas herein are Copyright by Justin Coslor on their respective dates. These notes are a progression of the concepts in the order they occured to me. This occured to me as a pseudo-sophomore at Carnegie Mellon University in Pittsburgh, PA. (It's a first draft so please forgive it's sketchyness.) All knowledge=information, which can be represented as metaphors. Metaphors are applied to specific contexts and general contexts=multiple contexts. A. All knowledge is metaphors applied to >= 1 context. B. A metaphor is a set of associations (links, patterns) that can or is applied to a context. A single context...A single specific context...general/nonspecific contexts. C. A context is a set of restrictions (restrictions on information, associations, links, patterns, sometimes even contexts). Therefore the statement A is equivalent to this statement: "Each piece of knowledge is a set of associations that can be applied to >= 1 set(s) of restrictions." ---------- The use of this I had in mind is to make a computer software that could understand and manipulate (and maybe even apply) metaphors. Many other ideas occurred to me today too, possibly due to doing yogi breathing and meditation exercises and taking vitamins since my health had been suffering. Epistemology Framework for artificial intelligence "Patterns in Contexts" continued... ---------- * A pattern is a collection of symmetries, where each partition section of data of every symmetry in the collection corresponds to another partition section of data in that collection, or sometimes corresponds to a piece or pieces of data in another collection (or other collections) which may or may not be part of a similar symmetry in that other collection. If data has recognizable features, it is a pattern. Repetition is what makes a symmetry, and is what makes a pattern's features recognizable. Unique partition sections of data are the atomic elements that a pattern's features are composed of. A symmetry is a type of repetition, but a repetition isn't always a symmetry (see metaphor definition below). * A context is a map of patterns within (thus bounded by) either a set or stream of data in which other patterns are ignored or are not apparent. Or a context bounded by a larger pattern than the map itself (which itself is a pattern that may or may not be part of the larger pattern), such as the ordinal of the map or a pattern larger than the boundaries of the scope. There can be many parallel streams, waves, or sets of data in, traveling through, across, or around the self-updating mapping of patterns which is chosen to be the context. Sometimes the corresponding partitions of data that make up a repetition are translated by some pattern with each iteration, such as in a methaphor. Yet similarity remains apparent (identifyable by some means). Again, I believe that information is patterns in contexts, and that information is metaphoric in nature. Tip: If confused by this write-up of my premise, try reading the sentence in reverse order then back through again. ------------ Information is a symphony of symbolism and symmetry. Information, by it's very nature, is a division. Yet it strives to become whole again, and at the very least, to become balanced. ------------ Category Theory: Abdicative context changing using identified metaphoric patterns. Some dimension additions for alternating or specializing the application of a pattern or set of patterns: - Location - Relative rate, relative timeline framework - Newly recognized relations found under sequential and parallelly recursive brute force and intuitively adaptive experimental logic search strategies -- Yields hypothetical considerations which can be temporally prioritized and recursively checked and updated from state to state and organized intelligently by current 1. depth, 2. branch size, 3. branch cardinality (alpha-numeric, etc), 4. task growth rate, and 5. average task completion rate (for scaling computability). ------------ When you figure out why a variable is a variable in a particular way, that understanding becomes a new relation to consider, which in effect and affect either increases or decreases the dimensionality of the variable's context. Some dimensions that are added usually increase task completion rate (such as specialization) other dimensions that ar added usually increase task growth rate (such as broadening the context or broadening the number of class categories to consider). Generalization can in some cases merge categories, classes, and/or contexts, or blur them for simplicity, and can increase or decrease completion rate. ----------- Generalization is useful for experimentation. All truth is but an approximation of a deeper truth. A pattern is like a function, and a context is like a field. Each has relations, variables (when thought of metaphorically), and often the potential for variations and unconsidered variables of dimensionality. ---------- A working definition of the mystery of consciousness might be ascribed to the interplay between 1. perspective, 2. priorities, 3. intentions, and 4. awareness; all of which depend on the flexibility, state, and mechanisms of belief held by the subject. ---------- My data symmetry section analysis technique for perception through patterns in contexts may be able to play a key role in automating axiom and theorem discovery for any given context (i.e. contexts such as the integers, the reals, wavefield analysis, map data, behavioral intention charts, language/speech modeling and representation, transform sequences, etc.). Any pattern discovered within a particular context can be applied to any of the known axioms and theorems of that context, and patterns that are discovered can sometimes be related to undiscovered axioms in that context. Anytime an axiom or theorem is discovered in a context, the entire context is redefined (as well as its subcontexts), and in doing so, its scope is narrowed. Choose -> Search -> Experiment -> Classify -> Test -> Prove. Choose/define context -> search for patterns -> searchfor patterns that relate discovered patterns -> postulate a classification for each discovered relation. For each relation, if a classification category does not exist that closely matches the relation, then further experimentation, context choosing (add and/or subtract context dimensions), and pattern searching must be done, starting with the characteristics of all partially matched categories, until an accurate or exact classification or category definition can be derived. After the relation's category is realized, search for more examples of that relation and derrive a proof of it. If the relation can be proven to be applicable to all patterns in a given context and all subsets of that context, it can be said to be an axiom of that context. ---------- Context can be thought of as a network as well as a shell that encompasses abstract nodes. The context of a set is merely its powerset, that is, until relations are applied. I don't believe in randomness, but I do believe that some contexts are larger or deeper than the scope of our perceptions. ---------- A context can also be thought of as a network of patterns, or even the network of relations that tlinks patterns. But when relations are applied to a context, it becomes an organism. An organism that is capable of translation (metaphoric operations), modification (adaptationn), division (duplication/reproduction/partitioning, and/or growth and association with other contexts. ---------- There are patterns, and they exist within and between/across contexts, and there are relations that act as reasoning engines that operate on the systems of patterns and contexts. Patterns can have analogue distortion, digital distortion, or metaphoric distortion. Contexts can be approximations of larger contexts, and elaborations or extensions of smaller contets or extensions of other contexts in general There can be relatively unique (somewhat unique, minimal commonality) patterns and contexts. Note: the word "commonality" is based on the greek root "monality", which is the "commonality" of the prime numbers. Each prime number is a "co-monality." This can be visualized in terms of geometry, to some extent. Every prime number is balanced, and is symmetrical, and contains a unique number of dimensions, which are also unique kinds of dimensions. Patterns and contexts and relations can also be symmetry pieces of other patterns and contexts and relations, regardless of whether or not they are distorted in any given state or piece or part or linkage. ---------- Every context is founded on its own set of axioms and theorems, and adding an axiom to or from a context's foundation fundamentally changes the context profoundly, yet some structures may remain un-affected. (*Note many of these notes may become invalid or ridiculous as you read more, so mental filtering may be necessary.) ---------- This is a quote from my journal. ""Metaphor" is a relational model of recursion, where the circular reasoning (in recursive definitions & recursive functions) cross-relates the elements of definitions & functions from multiple (or different) contexts. That is why cross-domain relations are so crucial to the metaphoric representation of knowledge and knowledge systems (logics)." ---------- "I also believe that information is metaphoric in nature (has algebraic interconnectivity), and that it can be represented as a composition of patterns in contexts, where the contexts themselves can be patterns, and the atomic elements of each pattern are composed of symmetry sections (partitiopns of data, where each partition is part of a local or dislocated repetition (a symmetry, and algebra)). And it is only through the repetition of a data section that part of a pattern can become recognizable from apparently random white noise. Randomness and white noise are probably patterns that are larger than the scope of our perceptions, so the data appears random. And I say that metaphors can be represented geometrically because all of the prime numbers (the balance points in the universe) are symmetrical when represented geometrically, and it is likely through primarily symmetrical sensory and cognitive structures that our minds can interpret information. And I think of metaphors not as A=B, but more like the similarity of the juxtaposition of A's elements in the context of B, and B's elements in the context of A, in terms of general systems theory. I equate truth with workable patterns that become more and more refined and defined as they get used. I believe that all truth that we are capable of perceiving is but a small approximation of the whole truth. And that the truth/patterns that we are capable of using is often subject to perception within varying contexts. But there seem to exist connections between information none-the-less, through whatever means. Possibly since (in my opinion) everything came from oneness)." ---------- Scope & Context -> Boundaries and Restrictions/Limitations Class -> Purpose Type -> Syntax Pattern Definitions -> Semantics Data Element Groups -> Configurations (Data Maps & Dependencies) ---------- Patterns in Contexts Cognition Kernal Database -> Metabase -> Context Rotator -> Experiment Application Field Expandable Adaptable Translatable Summarizable subjectively/objectively ======> Metaphoric Linkers Patterns Toolkit Augmentation Socket Parameters Analysis Scope Dimensionality of (Input/Internal perspective "eyes") Geometry & quantitative & qualitative properties of simultaneous interrupts and their instantaneous functional interrelations and interactions across multivariate sequence states (such as time & symmetry equivalencies). *Every set state is but an approximation of the possible combinatorial translations. ---------- Epistemology thoughts on Metaphor Abduction Metaphors hide cross-domain relations between generalized nouns, adjectives, and systems within a semi-subjective context of perspective. The descriptive mappings of metaphors and multi-layered metaphoric operations are generallymore foundational than their analogical counterparts, as the metaphoric objects and relational context is generalized (from set, type, and categorical specifics), which simplifies the computational complexity of the models' qualitative factors, and provides new bases for consideration and re-application of data, relations, and knowledge. Metaphor generation provides the architectural basis and objective of considering newe relations and data experimentatiopn for deriving and arriving at new models of understanding. data --> context unknown patterns --> hypothetical contexts relations --> categorical context parsing metaphoric relations --> cross-domain functions across contexts specific knowledge --> contextual scope focusing/narrowing analogies --> applies metaphoric relations to different examples of specific knowledge for partial transitivity new knowledge --> modifies existing contexts to incorporate new axioms. ---------- Prioritization and choice in decision systems (Part of a reasoning engine.) ---------------------- New action (such as prioritization or actual action) ^---^ Evaluaction ->criteria ^---^ outcome ^---^ choice ^---^ initiative factor(s) ^---^ prioritization ^---^ evaluation-->criteria ^---^ possibilities ---------------- ---------- Inventing industries with patterns in contexts What the world needs more of in order to support the ever rising population levels, is more industries. An entire industry can be created simply by developing a new kind of alagorithm, or an algorithm that creates a niche for people to fill with services or products. ********************** An algorithm can be developed by applying an axiom to a new context. ********************** This may require forming or describing a new context or kind of context, with intentions and expectations and attributes or properties in mind, as axoms are chosen and adapted to make that possible. Theorems can then be derived from those axioms, that are specific to that context, and when possible, they can be metaphorically related to theorems in other contexts. This is the basis for the patterns in contexts model for creating new information. It relates directly to abdicative reasoning, analogical reasoning, and cross-domain relations. Axioms depend on which dimensions they can exist in and apply to. For they are the links that connect different dimensions, parts of dimensions, and sets of dimensions, with the goal of unique lowest-terms representation. Usually they incorporate at least some implicit knowledge or material structure in their model. ----------- Property grouping axioms in cross-domain relations. (See diagram.) 1. All variables have properties. 2. All properties are independent of their variable's context(s). 3. All properties have some combination of qualitative relations, quantitative relations, existential locations, and existential conditions. 4. Every variable exists within a context and can vary from context to context. 5. Contexts are composed of networks of patterns, patterns are composed of networks of variables, and variables are composed of networks of properties. 6. Information can be represented as patterns in contexts, and in that way it can be represented metaphoricly through analogical reasoning and abdicative reasoning. Relations of various kinds, location(s), and condition(s) (apon and of) exist at all of the various levels, and those are the data access points. Today in a Cafe I was talking to my friend -------------- telling him about how I come up with ideas. Besides keeping an ever growing network of questions in the back of my mind, I take a topic or generate a topic by combining keywords, and then think about how that topic is typically represented, then I try to epistemologically dissect that representation and then rebuild the content using different, if not more foundational contextualization of those concepts. Then I go off on a tangent exploring the most interesting parts by associating other concepts, patterns, contexts, and operations to the new representation of the concepts in the original topic. It is often very valuable to have alternative representations of ideas and concepts and topics because each representation can yield a useful perception. If there is any word sense ambiguity, or use of metaphor, then each alternative representation can yield many perceptions, each of which could uncover previously unseen or unconsidered aspects of the topics, ideas, and concepts. So in the end, exploring and mapping out alternative representations of concepts, ideas, and topics is a way to augment their knowledge base, by generating new perspectives on the information, which can generate entirely new contexts, which can generate entirely new knowledge bases, by treating all information metaphorically. People are currently very good at metaphoric interpretation and analogical reasoning. Computer programs currently are not. It's the next step towards computational methods of abdicative (round-about scenic-route) reasoning. Anyway, my friend said I should make a program that does what I do, i.e.: a program that recontextualizes information from different perspectives of association, sort of like a choose-your-own-adventure story, but more like a choose-your-own-perspective program. Like a computer program that generates alternative representations of ideas, topics, and concepts. Or even more generally, a computer program that generates alternative representations of patterns (thoughtforms) in a variety of contexts (settings). ---------- Epistemology Systems Categories, and complete dictionaries as foundations. Quantified objects (and systems) can be juxtaposed into relations that balance alternative representations of objects and systems via a structural or syntactic methodology that acts as a transformation into some of the possible alternative representations of the quantified objects and systems. Algebras as alternative representations of information. Algebras can rename, or point to representations of information, as well as interconnect and dissect informational objects and systems. All objects and systems are named. Simulations, recontextualizations, and "polymachines" as alternative models of systems. Proof is contextual, in other words: proof is dependent on perspective and representation. In much larger contexts than the original context in which something was proven, most "proof" becomes incomplete or uncompatible, and sometimes even false if more foundational epistemological structures are found to have been overlooked. Proof is complete, logically consistent introspection of perceptions of concepts. Any given proof is only applicable to specific axiom sets. I.E. a proof based on one axiom set may be incomplete or uncompatible or even false in a context composed of a different set of axioms. Therefore concepts must be analogically translated into other contexts and their translations must be formed concurrently with their proof validity in their new context, as a best-fit categorical search procedure. The proof is a complete, concise system, so the proof in it's new context can be considered to be a polymachine, since it is an alternative representation of that system. A polymachine is a set of cross-domain relations that operate on analogically-matched patterns from an original context to a new context, and represents an alternative form of a system in a different context. Polymachines are created by inductive, deductive, or (in the case of analogically translated proofs) abdicative reasoning. Cross-domain relations are relations that analogically match the domain of a relation in one context to the domain of a relation in another context whose range approximates the same infrastructure and quantitative parameters while leaving the qualitative parameters categorically open-ended; they are a form of analogical reasoning. Input Devices->Internal model buffer->Association and repetition filter->Analysis/comparison engine->Perceptions on experience->Algebraic Conceptualization->Character sets and dictionaries, or number systems and axiom sets -> statements, arguments, inquiries, propositions, implications, operations, filtrations, combinations, exegesis, dissertation, assignments, contextualizations, templates, associations, compositions, dissections, introspections, modifications, adaptations, introductions, translations, transformations, distortion, refinement, recontextualization, proof, mapping, search, buffering, sorting, indexing, encoding, decoding, regulation, pattern formulation, trans-substantiation (joke), frollick. Copyright 5/23/2005 Justin Coslor Some definitions for patterns in contexts theory Metaphoric objects are informational objects defined by their relational properties. In relational contexts, sub-contexts of each property are independent of the application context. Qualitative factors are computed by mapping and defining a lexicon of their properties. Qualitative factors are reflective and algebraic usually. Quantitative factors are computed by counting and performing materialistic operations on them, and mapping them in that way. Quantitative factors are materialistic and geometric usually. Copyright 8/2/2005 to 8/3/2005 Justin Coslor Cliff Partitions Cliff partitions are perceptual references that distinguish deeply layered patterns from surface patterns, much like a cliff wall bordering the ocean. In the ocean, every couple of feet down an ocean wall is a new layer, much like how layers of pixel groups can be laid out on a visual canvas, with some stacked up several layers high on an edge. Cliff partitions are essential markers of where a topology has a steep slope that may or may not be an overhanging awning above a hidden hollow or cave. In topology, cliff partitions are useful for analyzing the depth perception of a view. In linguistics, cliff partitions may indicate a sentence that is placed in the wrong order, or it may indicate a sudden change of topic, or a jump from one perspective of a context to a deeper or more superficial depth of perspective of that same context. Cliff partitions in linguistics may also indicate the boundaries of a given context, where one context ends and another begins. Cliff partitions are only conceptual perceptual references in linguistic domains, as writers and speakers linearly paint a nonlinear picture with their words. Copyright 5/23/2005 Justin Coslor Object-oriented processing Grids (a.k.a. manifolds), networks, and gridded networks all can house patterns in contexts of information data sectors as the representation of knowledge (knowledge is information that contains meaning). Grids, networks, and gridded networks are materialistic operation spaces for knowledge representation, whereas the notion of "patterns in contexts" are the Platonic operation spaces that form the meaning behind the scenes on the materialistic operation spaces. Identifying the representation of knowledge in an operation space as "patterns in contexts" and specifying the details allows us to work with the information in an object-oriented manner. Copyright 5/27/2005 Justin Coslor Simulated Models and Utility Axioms If a network or grid is composed of N elements, then it is capable of simulating every possible permutation of those elements by forming internal networks and sub-networks (& grids). Grid networks allow for an infinite number of combinations to be simulated though, but only some simulations are of any use. Maybe there are utility axioms that can be defined to tell us what classes of models contain useful representations. It seems like some factors that might determine whether or not a model is useful would be: 1. Compatibility with existing useful models. 2. Novel representation or novel perspective. 3. Incorporation of new information. 4. Novel capability. 5. Ability to link two or more other models together. 6. Ability to prune other models. There may be many more factors directly related to evaluating the worthiness of a model. Simulation allows for recontextualization of models and problems and systems. Copyright 5/22/2005 Justin Coslor Operation Spaces continued - Tomographic Data Structures In the gridded network system, as described previously, a multidimensional array is built between selection of nodes in a network, where elements of this array can be used to build internal networks between the primary node anchors of the array, or between other nodes in other networks --as in cross-domain relations. This process can repeat to an infinite depth, in the order of network node to array anchor to array node to tomographic network to cross-domain relation network to array grid, cyclically. This is a way of creating tomographic data structures of an infinite depth and of infinite permutations, due to the potential for infinite depth, all without adding any extra primary nodes. Every array element and every node represents a relation to or between their anchors or parent nodes. Copyright 5/21/2005 Justin Coslor Operation Spaces: Grids V.S. Networks Rows and columns and layers are dimensions of a grid, but dimensions can also be parts of an N-dimensional array. Each of the dimensional intersections form a unique partition that relates or is categorized by it's parent sets' position along their own sequences. So in this way, elemenets on a grid (i.e. in an array) come from multiple parents, wheras elemenets in a network can often come from only one parent (an injective branch). However, in some networks, such as where a planar geometry can exist by the interconnection of more than two nodes, multiple parents can be a grid of subspaces between the nodes on the plane that they make, and in those subspaces multivariable position and quantized quality relations can be said to exist, that are anchored to multiple origin points (each vertex be treated as an origin, and angles between them only serve to define the partitioning of the planar grid). I'll call this kind of transformation of a network "a subspace grid of vertices". Maybe this combination of a grid network can enhance the operation space by making any nodes on a network able to be related to eachother, in grid format, between particular data sections on the subspace grid as well as between other primary nodes. The other kind of operation space is the Swiss cheese like structure that surrounds a subspace geometrized grid transformed network. The inner edge of that space is where one context ends and other contexts may begin to exist. Copyright 5/14/2005 Justin Coslor Key axioms and branch axioms in pattern collections. Patterns are composed of smaller parts, with the smallest parts being repetitions of unique elements in which no sub-patterns are apparent; also, these smallest parts exist and their repetitions make them algebraicly recognizable due to certain axioms, which act as fundamental truths (self-evident assumptions) for which no proof is said to be needed. This being said, we can say that all patterns that are unique in some manner must contain at least some unique axioms, and if we look at a collection of basic patterns and determine what is unique about each one and what is in common between them, and then figure out how those similarities and differences ar ordered on an axiomatic level, we may discover key axioms and branch axioms which can be represented in a nodal network graph. The value of this is that we can then understand, at the most basic level, what makes a pattern exist, what makes a pattern recognizable and similar to other patterns, and what makes a pattern unique. We can use that understanding to select axioms suitable to generate a set of patterns with a measurable degree of flexibility/adaptability, to use in constructing a system of perception, similar to a painter mixing paints on an artists pallete, while he mixes concepts in his mind's eye. Copyright 11/7/2004 Justin Coslor Hypothetical Relation Highlighting in Undefined Data Sets: If categorical names have been assigned to finite elements in a domain, the rest of the data in the set can be hypothetically considered to be relations or parts of relations (on those elements and elements not in that buffered data set). Or they may be elements of categories you don't yet recognize or know of yet. 9/23/2004 Justin Coslor Am I reinventing the wheel? Today while studying a diagrammatic map on "Can Computers Think?" that Seth Casana gave me I learned of work that has already been done in Artificial Intelligence that is very similar to some of the concepts that I came up with on my own. For instance, there has been work done in the area of making computer software that can understand "analogies". That is very similar to my concept of "metaphoric operations". Also, in 1989 in seems, Keith Holyoak and Paul Thagard created ACME, which is a connectionist network that discovers "cross domain analogical mappings." That soundsd just like my concept of "cross domain relations for alternative route mathematics", that I have written about prior to reading anything about it, and I came up with it all on my own earlier this year. Here are some Analogy Systems: Copycat - Douglas Hofstadter and Melanie Mitchell 1995. SME - Brian Falkenhaimer, K. Forbus, and D. Gentner, 1990. ACME - Keith Holyoak and Paul Thagard, 1989. 8/20/2004 Justin Coslor Programming In the preface to the introductory computer programming book "The Little Lisper" second edition ISBN 0-574-21955-2 it says: (that in LISP) "the primary programming activity is the creation of (potentially) recursive definitions." Now to me, that sounds like the main task (and goal) is to map out and/or define patterns that are either finite or infinite and to put them into a relational context that is capable of transforming incoming data patterns by relating them to stored data patterns, so that the output can be 1. represented, 2. stored, and 3. used/manipulated. I believe this because nothing is more recursive than a pattern (nothing is less recursive than a pattern as well, except that which is totally random). Patterns always exist within a context or contexts, otherwise they are not recognizable and look like random garbage (see Godel's Theorems). On page vii it also says that "LISP is the medium of choice for people who enjoy free style and flexibility. LISP was initially conceived as a theoretical vehicle for recursion theory and for symbolic algebra." (and likely Lambda Calculus & the EMACS environment for Artificial Intelligence)... LISP syntax looks very similar to my old nonlinear style of thought notation, with its parenthesis within parenthesis (which was good for scaling depth on tangents and concept descriptions). Copyright 8/4/2005 Justin Coslor Programming Languages "Programming languages are formal languages that have been designed to express computations." - How to Think Like a Computer Scientist - Java Edition In other words, programming languages are mappings of balanced processes. The flow of any kind of process can be mapped, if not only approximated by a systematic contextualization of patterns and relations involved in the process. Every system is like a state machine in motion, where the elements and operators are encapsulated by their interconnectivity via contextualization, which is a form of perspective of finite scope. Formal languages have fully defined axioms, and are consistent and complete in the mechanics of their methodology. But what is the methodology of mappings of balanced processes in general? The universality concept applies to them: they are consistent and complete because they are balanced about a tight contextualization, where the interconnectivity of the process's elements acts like a fulcrum (when thought of quantitatively), with no element left unconnected. That's why patterns in any context can be transformed through operations into different patterns, so long as there is a method of representing both sets of patterns. The balance comes from having multiple methods of representing each state of the elements in the process. The mapping comes from being able to contextualize the processes, which is only possible if the processes have finite scope, and are completely defined (thus interconnected), and must be systematic (thus logically consistent) in order to be precisely mappable with regularity throughout their states of operation. Copyright 9/22/2004 Justin Coslor Patterns In Context Cognition Kernel [Database]-> [Metabase]-> [Context Rotator]-> [Experiment Application Field] ----------------------- The following are *a. Subjectively and *b. Objectively 1. Expandable, 2. Adaptable, 3. Translatable, & 4. Summarizable: ------------------------ Metaphoric Linkers ------------------------ Pattern Toolkit ------------------------ Augmentation Socket Parameters ------------------------ *Considerations: ----------------- I. Analysis II. Scope Dimensionality (of input/internal perspective "eyes") III. Geometry & Quantitative & Qualitative properties of simultaneous interrupts and their instantaneous functional interrelations and interactions across multivariate sequence states (such as time & symmetry equivalences). **Every set state is but an approximation of the possible combinatorial translations. Copyright 6/13/2004 Justin Coslor Complexity Commercial or proprietary software is surjective or injective, but free open-source software is bijective. Part of the FRDCSA Tutorial (Free Research Database Cluster Study and Apply) on frdcsa.org says a blurb from an ACM paper about measuring the power of a set of axioms in order to measure the information contained within the set of theorems that can be deduced from those axioms. It says that one can only get out of a axiom sets what one puts in. The paper says something like: "If a set of theorems constitutes t bits of unique information, and the set of axioms that the theorems are based on contains less than t bits of unique information, then it is impossible to deduce those theorems from that set of axioms." My friend Andrew J. Dougherty of FRDCSA says that to understand the general necessity of having more software, simply replace "theorems" with "problems", and "axioms" with "programs", and "deduce" with "solve" in the previous statement. Doing that we get: "If a set of problems constitutes t bits of unique information, and a set of programs contains less than t bits of unique information, then it is impossible to solve these problems using just that set of programs. By "problems", I think he means "explicitly defined problems", because an explicitly defined problem is a program that has yet to be executed. Abduction may be necessary to define all of the elements and operators of a problem in the process of turning a problem into a program. I say, replace "theorems" with "context", and "axioms" with "patterns", and "solve" with "create". This yields: "If a set of contexts constitutes t bits of unique information, and the set of patterns that the contexts are based on contains less than t bits of unique information, then it is impossible to create those contexts from that set of patterns." Copyright ?/5/2004 Justin Coslor This is part of my method of knowledge representation for my epistemological representation of artificial intelligence through Patterns in Contexts. Contexts come from patterns that are combined. There can be patterns noticed in the cross-examination of different contexts, but those "patterns" are elements of a greater scope of context than any of the contexts being cross-examined, that is to say, when those cross-context patterns are not noticable when only examining any one of those contexts in relation to itself. This method of knowledge representation may hopefully prove to be useful in the abdicative search for new axioms within and across representable contexts. A context is represented by its systems of patterns (a.k.a. it's system of axioms). Copyright 6/5/2004 Justin Coslor New patterns can be discovered by experimenting with data sets: analyzing them in relation to metaphoric operations on other data sets. Metaphoric operations are operations that translate, juggle, predict/locate, and/or transform specified elements across specified contexts. Copyright 6/7/2004 Justin Coslor New metaphors can be discovered by combining axioms that come from multiple number sets, orderings, and/or algebras. Metaphors are esoteric relations. The application of a metaphoric operation on a data set sometimes results in the discovery of new axioms through the new perspective's set of relations brought about by the application of esoteric relations. Metaphoric perception is all about cross-domain relations. This is because the application of metaphors brings about both: 1. relations between the range of the metaphor and the range of all applicable operations (operations of applications) of the data set, and 2. new cross-domain relations (new domain perspectives) for both the operations of potentially all applications of the data set; and sometimes new cross-domain relations and new ranges for the system and set of relations that algebraically defines the metaphor (when applying the unmatched relations that are not bijective of the operations of applications of the data set) metaphorically (i.e. algebraicly to the metaphor). Copyright 6/5/2004 Justin Coslor Linker patterns Linker patterns require both an observation buffer (that is at least of equal size to the sum of the contexts to be linked), and linker patterns require an operation buffer that is at least as big as the observation buffer (though far larger is necessary for some observations, even though the amount of data that ends up in the operation buffer may be far less, in some instances, than the amount of data filtered out of the sum of the contexts into the observation buffer). Data gets filtered out of every applicable context by the linker pattern's "filter specifications", right into the linker pattern's observation buffer. Then the linker pattern's set of metaphoric patterns operates on the observation buffer one at a time or in parallel, but inside the operation buffer. The linker pattern contains a set of metaphoric patterns whose elements are referenced algebraically to the applicable data elements present in the observation buffer for every possible metaphoric pattern combination present in the linker pattern innately. Metaphors which are algebraically a complete set of elements to applicable/valid data elements are used in the observation buffer, then inside the operation buffer they perform their calculation (translating, juggling, and/or transforming of the data section by the metaphor) and the linker pattern then places the output in an organized form (so it can be referenced later), and those outputs are placed into a buffer called "the unified context" of the original contexts. This "unified context" includes the linker pattern's filter specifications and metaphor set that was used (i.e. the set that was computable). Linker patterns can duplicate themselves to divide up the work of applying their metaphoric pattern sets to the observation buffers' data (and they update each other with each successful operation). Each linker pattern is like a mobile set of operators that copies select groups of contexts and gives birth to unified contexts (which are new contexts). It is each linker pattern's unique set of filter specifications that differentiates one linker pattern from another. New axioms and theorems that are found elsewhere and within each operation are found and get added to the metaphor set after the valid discovered patterns are provably generalized. They are placed in all of the linker patterns. Linker patterns can also update each other's set of metaphoric patterns by sharing ones the other doesn't have, and copying new ones from the other. The observation buffer performs general quantifier type matching. Copyright 5/31/2004 Justin Coslor Patterns In Context Cognition A context is any specified number set, ordering, or system of numbers that is representative of something (symbolic). Take the desired outcome (the goal) and break it down into unique aspects. Treat each aspect as an element of a context that contains it, or as an element of several contexts that contain it. each element/aspect may have its own unique context at first. We will be striving to find the pattern or patterns that link all aspects of the goal into one context. A "linker pattern" can be a linker of the contexts that each of the elements of our goal exist within. Such a pattern links contexts together by assigning a system of translating, juggling, predicting/locating, and/or transforming the specified elements across their specified contexts. This "linker pattern" is metaphoric, and can act as the "unified context" in which we will search for the aspects of our goal, as well as search for alternate routes to each of these aspects (for optimization). After this experimental search has completed and an optimal cross-domain relation search for shorter routes to each aspect has been completed, we will have generated the optimal route map to our goal. Cross-domain relations can also be thought of as possible associations, or simply as patterns. They can very explicitly depict ambiguous relationships, such as when they are used with graph theory. Cross-domain relations are a little bit like surjective and bijective networks in logic but where two domains lead to the same range in a number set, even when the domains come from different contexts. They can also be thought of as alternative routes. Cross-domain relations can be searched for that relate aspects of our goal that are also aspects of goals that have different unified contexts than our goal. It's important to mark the optimal routes out of the cross-domain relations, but keep the other relations (possible routes) for use in future goal structures. By linking multiple goals in this manner, we expand our network of understanding. Copyright 9/7/2000 Justin Coslor Knowledge Mining Maybe amassing huge intelligent databases that can draw conclusions and make abstractions and predictions towards goals that can recognize & ask for specific data it needs to output one or more units of truth, which could help demystify fields of study and help major breakthroughs occur, if not by simply abstracting and relating so much specific data and general patterns in so many areas; to help bring everything to one's fingertips. A massively parallel search and correlation engine: The computer has to be able to understand a goal enough to figure out how to better understand that goal, so that it can design it's own searches (determine its own search criteria), and know what a conclusion would look like and would require to be complete enough to make an abstraction. What is the criteria of a conclusion? Is a conclusion just one particular perspective in every situation? How can the perspective be intelligently shifted and rotated in a search to generate and array of complimentary conclusions? At what point does the difference in goals generate opposing conclusions? (i.e. when do conclusions become apparently contradicting when using the same set of data...) Taking this into account, what difference in goals produces contradictory conclusions (perceptions) when searching (parsing) intersecting sets of data? Input something like a handbook of chemistry and physics, with a goal of making valid correlations that are not a listed part of that original data set. Start out with general patterns like input types, leading to language semantic patterns, leading to patterns of contextual settings, leading to metaphoric patterns between contexts, *leading to applications of the generalized raw data to the metaphoric patterns, leading to generalized predictions of the outcome of the previous step*, parsing the conclusions listed in the raw data and matching it to the metaphoric mold (the pattern and logic) that led the contained data elements (or equivalents) to that listed conclusion. . .in short: enable the software to understand how the data elements were led to the conclusion listed in the raw data, so that those patterns (metaphoric molds or logic operations) can be understood enough to be applied to the raw data in different permutations (ways) to uncover conclusions of previously unconsidered possibilities. Those patterns and derivation/discovery methods could also be used as a guide for designing new patterns built from recombining the old patterns with unique data. And since unique data almost always is unique due to its being composed of at least some unique patterns; parsing the old (known) patterns from new (unknown) patterns might make it easier to clarify what exactly the new pattern is or at least how it operates (or at the very least, its function). This is knowledge mining. . .One form of artificial intelligence. Circular Reasoning: I aught to look up the dictionary definition of a bunch of the key words in this.. Hey, why not all of the words? A number could correspond to the number of words the dictionary definition of each word had to reference (on every level of the tree of lookup words, each branch pausing when it ends up at it's own word (a loop)) until the parts of the world applicable to the context of the base word have been described (mapped)), until an upper limit has been reached on each word. The highest number out of all of the words will be the number of words in the applicable dictionary to the context of that paragraph (no repeats). It will be a complete system of circular reasoning. A complete system of circular reasoning is where every word in a dictionary is mapped to at least one other word in that dictionary. Some may be mapped to every word in that dictionary. A complete system of circular reasoning is one unit. It is aversion/perspective model of a truth. And different ones can be combined to build complex systems of truth. Like mitochondria building cells building structures. Copyright 7/13/2005 Justin Coslor Pattern Occurrences Some patterns are designed or brought about intentionally, and other patterns are brought about naturally, and others are brought about as an unintentional consequence of bringing about intentional patterns, such as in unintentional contexts that are created as a result of layering patterns, and grouping patterns, and modifying patterns. Some patterns occur naturally according to certain variable probabilities specific to their contexts, while others are subject to haphazard creation, randomness, and free will. Copyright 7/14/2005 Justin Coslor Since all patterns are composed of repetitions, and since the repetitions are what makes the parts recognizable, and since anything that is recognizable can be considered a pattern, the reference pieces for the parts of each pattern can be local, as part of the pattern's context, or the reference pieces can be remote, as part of other contexts that are accessible to the perception system. The reference pieces are instances of the repetitions that make the parts recognizable, and are usually cataloged by order of exposure to them, as well as by associations. When new patterns are encountered they are either recognized (thus categorizable), or they are unrecognizable (thus not categorizable) because their parts and properties are unknown, or they are partially recognized (thus potentially categorizable and partly referencable). If the pattern is new and it is recognized, then its parts are already known but are arranged in a new configuration and with potentially new properties due to the novel association of its parts. So basically, once the perception system is exposed to contexts, the pattern matching/classification system begins its task of dissecting new patterns into reference pieces, and classifying recognizable patterns into association contexts and utility contexts, and assigning priority ratings to everything so that the perception system can decide what to pay attention to. Priority ratings get constantly updated, and depend on how much bandwidth and processing power the perception system and reasoning engine have available. The reasoning engine does all of the heavy calculations, task and priority assignments, memory management, simulation modeling, and most of the decision making. Copyright 7/5/2005 Justin Coslor ePIC Goal Representation (ePIC = electronic Patterns In Contexts) A goal is an abstract construct, and the attainment of a goal is to fill in all of the details of the goal either: 1. in Platonic Reality (information space), or 2. in physical reality (matter configuration space). If the details have been filled in in Platonic Reality, then the result is a simulation. If the details have been filled in in physical reality, then the result is a working model. A prototype can be a preliminary model or preliminary simulation. The abstract construct of a goal is the starting point for changing your reality in some way. One need only be able to partially perceive of the abstraction to initiate the existence of the goal, but to fully specify it, a viable plan needs to be formulated. Usually there are unknown variables in every abstract goal, and specifying each variable becomes an iterative process. Often the abstract goal can be stated in the form of a question, and is the result of the questions that arose from some problem. Often times further questioning of the problem impetus is necessary to specify the goal and in doing so, the problem gets solved as the unknowns become decided or calculated. Many goals are qualitative/categorical subjective/objective priority system calculations, that rely on preference, perspective, universal truths, contextual restrictions, and contextual properties. However, all problems, goals, and solutions can be represented as patterns in contexts, such as undecided patterns in partially determined contexts, that evolve through storing and grouping of categorical, qualitative, and quantitative patterns across different contexts into an experimental buffer/model space towards the sufficiently representative construct or construction of networks of systems of patterns, that satisfy the objectives of the problem and goal, in the context of the final form of the problem and goal. The Patterns In Contexts concept is an epistemological language, which I strongly believe can be used to represent anything, any concept, and any information of any kind, including first person, second person, third person information in past present or future tense, and it adapts well into any other language. Copyright 6/3/2005 Justin Coslor Cross-Domain Relations in Analogical Relations A true cross-domain relation would have two domains that each lead to the same range. Analogical relations do something very similar to this, however not quite. In an analogical relation, the relation between the domain and range of one context is mimicked across a somewhat similar domain and range in a different context (only some properties need to be similar for the analogy to be formed, since a barely recognizable similarity needs to exist). The result is like having generalized an abstraction of the two domains and the relation, and using that abstraction to perform the abstracted relation on the second domain in the other context. Copyright 6/3/2005 Justin Coslor Patterns In Context and Question Asking Systems for Object-Oriented Programming The patterns in contexts model of knowledge representation and question asking systems based on forming networks of questions and networks of patterns in networks contexts can be used to make a profoundly sophisticated object-oriented programming system capable of doing analogical reasoning, deductive reasoning, as well as induction and recursions that are simply not representable in other systems. In this system there is a constant acceleration of computational complexity, all of which is progressively designed to simplify the system while augmenting abilities and understanding. Copyright 5/30/2005 Justin Coslor Complexity Progressions Every state of a complex pattern can be said to be the result of a progressive augmentation of the previous state or model/version by a new or repeated pattern, or by multiple patterns. That is, unless data loss has occurred due to random deletion or a random addition process. Copyright 5/31/2005 Justin Coslor Pattern Details & Randomness Every pattern is the iterative accumulation of modulations and augmentations of sub-patterns, right down to the atomic repetitions that are the first forms that are recognizable from randomness. Atomic repetitions may come in a wide variety of non-interoperable modes of partitioning, each of which is subject to a unique perception system that is capable of buffering and filtering its own particular spectrum of atomic repetitions that are partitioned from patterns and randomness that are unrecognizable to that mode. Randomness comes in two main forms: there is randomness that is compatible with the partition mode of a given perspective system, thus being countable or measurable via the mathematical comparison of the atomic repetitions of that mode (because it is just a randomization of those atomic repetitions); and the other kind of randomness is composed of randomized patterns that are partitioned in modes other than that which is compatible with the current perspective system. It is undecidable whether or not there exists a randomness that cannot be partitioned by any mode of perspective, i.e. a randomness that is not the randomization of some set of patterns or atomic repetitions. Copyright 9/7/2004 Justin Coslor Metaphoric Operations on Patterns Across Contexts I want to learn LISP and use it to make an intelligent agent capable of doing metaphoric operations on patterns across contexts. 2/17/2005 Update by Justin Coslor: I guess now, most Artificial Intelligence Programming is starting to be done in Java since it is cross-platform and simple to use. Copyright 9/8/2004 Justin Coslor To do this, the sub-agents will need to be able to research raw data configuration sets to look for algebraic repetition that can be considered to be patterns in the sea of cached buffered inputed/observed/recognized elements. In order to recognize something, it will have to have a known set of basic recursions (repetitions) to begin with. The prime numbers ar a good source to start out with (since they are the natural balance points in the universe). Then it will need to try to describe each data configuration set (data map) whose atomic repetition symmetries can be characterized or parsed. This description will be known as the pattern's type, and patterns with similar types will be grouped into classes. To describe a metaphoric operation will require generalizing the differences between each type description in a particular class, and then mapping the observed relations (between each of these types) in the form of a nodal network. That nodal network will be a metaphoric object that summarizes the class. Create metaphoric objects for all of the classes in the context (aka scope) of your total original cached raw data. Then form relations between different metaphoric objects, and combine and reconfigure different metaphoric objects, with the original metaphoric objects being treated as axioms of that particular context. The metaphoric object relations can be treated as templates for filtering other raw data contexts in the search for known patterns which contain their own distinct uniqueness, that will will warrant the generation of new types in new classes. Every context has its own types in their own classes. In other words, every raw data set has its own patterns in their own contexts. This is how to relate different raw data sets to extract their relational axioms, and the combination and reconfiguration of different contexts' axiom sets (metaphoric objects) is what I call metaphoric operations. Copyright 8/23/2004 by Justin Coslor. Information Theory Quotes ""Metaphor" is a relational model of recursion, where the circular reasoning (in recursive definitions & recursive functions) cross-relates the elements of definitions & functions from multiple (or different) contexts. That is why cross-domain relations are so crucial to the metaphoric representation of knowledge and knowledge systems (logics)." Copyright 8/26/2004 by Justin Coslor: "I also believe that information is metaphoric in nature (has algebraic interconnectivity), and that it can be represented as a composition of patterns in contexts, where the contexts themselves can be patterns, and the atomic elements of each pattern are composed of symmetry sections (partitions) of data, where each partition is part of a local or dislocated repetition (a symmetry, an algebra). And it is only through the repetition of a data section that part of a pattern can become recognizable from apparently random white noise. Randomness and white noise are probably patterns that are larger than the scope of our perceptions, so the data appears random. And I say that metaphors can be represented geometrically because all of the prime numbers (the balance points in the universe) are symmetrical when represented geometrically, and it is likely through primarily symmetrical sensory and cognitive structures that our minds can interpret information. And I think of metaphors not as A = B, but more like the similarity of the juxtaposition of A's elements in the context of B, and B's elements in the context of A, in terms of general systems theory. I equate truth with workable patterns that become more and more refined and defined as they get used. I believe that all truth that we are capable of perceiving is but a small approximation of the whole truth. And that the truth/patterns that we are capable of using is often subject to perception within varying contexts. But there seem to exist connections between information none-the-less, through whatever means. Possibly since (in my opinion) everything came from oneness)." Here's another quote from my journal Copyright 11/24/2003 by Justin Coslor: "Information is a symphony of symbolism and symmetry." Here's another journal entry Copyright 12/23/2003 by Justin Coslor: "Information, by it's very nature, is a division. Yet it strives to become whole again, and at the very least, to become balanced." 12/25/2004 ::Metaphoric Operations:: Metaphors are geometrical, in a sense, that is to say they follow mathematical geometries. That is to say, metaphors can be thought of in terms of geometrical patterns and systems. This is because metaphors can be diagrammed, and diagrams have a relative/nodal/graph-theoretic logic about them, and through the logic of their patterns and systems they can be recognizable as having a sort of relational geometry (at least in the unseen Platonic-reality). Patterns have an algebraic repetition at their foundation, and the most basic repetitions are symmetries. The most basic symmetries can be perceived through prime numbers, in that they are the fundamental building blocks of more complex symmetrical and a-symmetrical structures. A-symmetrical structures are constructed out of symmetrical structures, just as non-prime numbers (composite numbers) are constructed out of prime numbers and relations/functions. Metaphoric operations are relational templates that are axiomatic, adaptable, reconfigurable, and versatile. This is because they are collections of relations whose options have been generalized to optimize those qualities for relating similar, and different qualitative domains across contexts. Each context's qualities' relations are unique to that context's set of axioms. One might say that domains are qualitative, while the domain's ranges can be qualitative OR quantitative. Metaphoric operations relate different consistent, recursively complete, contexts by copying or moving elements from each context into the separate but more versatile context of the metaphoric template. Once the elements have leave their original context, their original context's axiom set(s) may be altered, as well as some of the context itself (and sub-contexts, if any are relevant). Metaphoric contexts may only need to copy some of the axioms of their element's original contexts, because they have axioms of their own that help to allow for the relation of the axioms that are buffered in from multiple other contexts. The metaphoric template's original axioms also help to relate the qualitative elements which are the current primary focuses, that were constructed out of relevant axioms from the external contexts. New knowledge is created when the qualitative elements' axioms sets are adapted to form new qualitative elements and relations out of the augmentation of the metaphoric template's context's axiom sets, by the elements' external axiom sets. With metaphors, anything that isn't explicitly cross-domain related is ignored. Qualities are itemized as they are noticed or as they are deemed relevant. Personally, I feel like the diagram of metaphoric operations is a lot prettier than the description... Copyright 12/25/2004 by Justin M. Coslor Copyright 1/9/2005 Justin Coslor Visual Dictionaries and Axiomatic Abdicative Simulations Maybe as part of building the logical framework for a systematic visual dictionary, we could try representing each image both as a set of angular or situational perspectives; but also I think it's important to try to axiomatize the image properties into contexts, and by doing so we can do abdicative creative constructions (and abstractions of those to some approximate goals), such as by perceiving each image as a series of nodes (graph theory vertices) and connections that are all linked together both contextually in the physical space, and conceptually in the historical-timeline/platonic interaction space. By doing this, the heuristic (guess-work) training can be semi-automized and the intelligence data on the scenario objects can have a far deeper meaning and farther reaching applications. Deepening the understanding of content and its abdicative recombinations and metaphoric transcombinations, both increases the potential for creating new applications and tools, and increases the versatility and effectiveness of existing tools and applications. Deepening understanding of content creates new contexts and reconceptualizes stuff by augmenting axiom sets that the contexts are based on. Copyright 1/21/2005 Justin Coslor Axiomatic Visual-Layer Interpretation Forming stronger linkages between axiom sets deepens the meaning of content of all structures that are based on those axioms. It can also complicate things by cluttering the contexts that those structures act within if the linkages are formed sub-optimally. Such is the case of an image with ambiguous layering. This has applications to steganography, computer vision, and virtual-reality educational environments, etc. Copyright 8/7/2005 Justin Coslor Patterns In Contexts: 3D Engine I think Java3D, combined with some inexpensive virtual reality equipment, will be the ideal environment for exploring Patterns In Contexts theory visually. Critical to that is a software that is able to parse video data into 2D objects and build 3D geometric reconstructions of those objects along with the parameters of their observable range of motion, and do heuristic guessing at the backsides of the objects that are hidden from view or just make the 2D objects into 3D avatars that always face you regardless of which side of theme you are on. That way, video data can be geometerized and represented as Patterns In Contexts, and 3D worlds can much more easily be created by mixing together objects and behaviors from an enormous archive of experience (from video sources) that is all parsed and sorted categorically by a visual dictionary that maps adjectives and nouns and verbs to pattern properties such as qualitative geometric relations, axiomatically defined variables and operations, and contextually associated references of objects and their pattern groupings. Each entry of the visual dictionary will contain an up-to-date list of all objects in the pattern archive that contain the geometric or otherwise visual property defined by that visual dictionary entry. Scale, color, orientation, state, position, and quantitative data in many cases can be ignored by the visual dictionary, unless the entry is directly intended to describe one or more of those properties. Copyright 1/9/2005 Justin Coslor Graphical Representation and Visual Heuristics Make a website loaded with graphs, diagrams, flowcharts, and simplified geometric reconstructions of stuff, events, places, flows, tools, intellectual understanding, interpretations and translations, programs, systems, etc. Call it "mapworld" or "graphworld." Make webcrawling intelligent agents that generate extensive thorough, and systematic visual dictionaries online. Similarly, there should be a webpage utility where you can enter the URL (Uniform Resource Locator = website address) to some text or copy/paste in some text directly and it could try to abstract visual perceptions of the text content's meaning and represent it in the form of a diagram or graph, etc. It could also try researching images on the internet that are related to the text. I realize that the second part might be difficult, since there aren't very many visual dictionaries in existance yet, and computer vision and machine learning technologies may not be that advanced yet (but maybe they are...?). Heuristics (guess-work on visual data and in language processing is just a matter of logical deduction, manual training of Bayesian statistical and Connectionist techniques, and metaphoric/analogical/cross-domain-relational mappings across contexts, to bridge systems not yet adapted to each other. Copyright 7/20/2005 Justin Coslor Creativity & Understanding Language is permutations of semantics, governed by syntax and context, with meaningful intention. So.... What is the language of creativity? What are its semantics? What is its syntax? What is its context? The language of creativity always contains either: 1. new semantics or new permutations of semantics, and/or 2. new syntax, and/or 3. new context. *Creativity does not always convey meaningful intention. The semantics of creativity are new patterns and/or old patterns thought of in new ways (recontextualized patterns). The syntax of creativity is either internally defined by the language of the format (if the format is known), or else (if the format is new) it is externally defined by the naturally occurring partitions and connections of the organic objects and systems of developments of the natural universe, or by the connections and partitions present in the diagonalizations of synthesized patterns juxtaposed through a relational operator or operation, and/or the diagonalization of the juxtaposition of synthesized patterns and natural patterns juxtaposed through a relational operator or operation, and/or the diagonalization of the juxtaposition of natural patterns juxtaposed through a relational operator or operation. The context of creativity is always at least partially new. Creative expressions composed entirely of entirely new patterns (not just modified ones) in entirely new contexts with external syntax that has never before been known of and that is unrelatable to known syntax will always appear random and entirely undecipherable unless the person or interpretation program is capable of analogical abdicative reasoning. However there will be no conclusive proof that the analogies drawn will be correct. The analogies may be qualitatively correct in the metaphoric sense, but they will never be proven quantitatively correct to the knowledge of the analyst. There has to be some decoding method, key, or common ground known to the analyst in order to decipher such a creative expression. Copyright 7/17/2005 Justin Coslor Concepts Layers of states and states of layers (As in "finite element state machines" and similar systems): ^^^^^^^^^^^^^^^^^^^^ Art Video Writing Talking Scent Taste Touch ------------------ Mathematics -> connections and differences in maps of possibilities Philosophy -> depth of possibility maps models of truth progressing Science and Technology -> exploration of possibilities through careful experimentation and adaptation to discoveries ------------------------------- Can you think of more? It is definitely possible. Look for stuff like those descriptions. Juxtapose operators and abdicatively reason into applications. Copyright 7/8/2005 Justin Coslor Measurement Systems In measurement, two or more quantities or qualities are compared to one another, such as a unit of measure applied to a starting point and ending point of another object. When a unit of measure is undefined, you look for the minimum unit(s) of commonality between the objects and mark the overlap points and the center-points between the starting and ending points, and the center points between those points, etc. If any number systems or other patterns are used as division or counting units (such as prime numbers), as well as center-point binary tree parsing, we realize that "all measurement is really comparison by parsing or partitioning". The units of the partitioning or parsing can be native common denominators of the observer's perception system and the object. The intersection of the juxtaposition of multiple objects is another native unit of parsing, which itself can be parsed into smaller units via a number system or other pattern. Common ground or compatibility is necessary for comparison, and since measurement is a form of comparison, measurement is an act of perception adaptation via parsing or partitioning. It's the act of trying to perceive of an object via the perception system of something else, and often times perception systems miss a lot because there are often lots of valid ways for a particular observer to perceive of things, and it's an undecidable problem about whether a perception system is not recognizing other undefined potential aspects of the object, let alone know what it is not perceiving through its axioms and atomic units of partitioning, and methods of parsing and grouping, and methods of determining anchor points, interpretation, starting and ending points, edge detection, pattern layering, and buffer sizes and contextualization, etc. Active measuring is when ea system's partitioning structure and methodology/reasoning system is constantly updated as something is being measured. An example of active measuring is a system capable of learning, such as an adaptive or evolutionary perception system, such as an artificially intelligent reasoning system or human being. A perception system that is merely adaptive but not evolutionary is autonomous or semi-autonomous, but not intelligent, since it only knows the context that it currently exists in. By storing perception systems adapted to multiple contexts, a system can then often map out the commonalty and differences between each context and form a general common-sense perception system which can be analyzed inductively, deductively, and abdicatively by its reasoning engine. Analysis via comparison of the domains and ranges of functions that exist in different contexts is an abdicative reasoning process since it is a form of analogical reasoning. Once again common ground must be mapped between the functions being compared or else an external perception system will have to artificially map its units onto both functions so that compatible parsing and partitioning can proceed in a measurable, if not blind (thus artificially simulated) representation. Passive measuring is when the measurement and perception system's reasoning engine is not updated by internal induction, deduction, or abduction during measurement, nor after measurement. Passive measurement is merely mechanical and not adaptive or evolutionary. Copyright 7/2/2005 Justin Coslor Re-contextualized Patterns It's interesting how patterns and their implications change as their raw data is re-contextualized and/or perceived from different perspective systems and contexts. The parameters of each context shapes the possibilities of its patterns' applications, implications, and recognized states of existence. Often times the possibilities of contexts overlap, and are subjective in the sense that there may exist several possible ways to perceive of and interpret a context, where each way may have equal or varying levels of probable truth in its systems, depending on the perspective system and intentions/expectations of the observer and/or the controller. Copyright 6/25/2005 Justin Coslor Observing patterns and differences Combining my poem about "Sight" with my poem about "Reasoning Engines", leaves me thinking about the line "from color comes shape" and the line about "thinking as storing and grouping knowledge", and how it takes a pattern to perceive of a pattern, such as one colored shape outlining or juxtaposing against another colored shape, and how each of these shapes (and color information) gets stored as a piece of knowledge (a pattern), and how both are grouped together by their situational context. The differences between them are patterns not origininally apparent in either piece of knowledge prior to their comparison, unless those patterns are stored in the perceiver's virtual knowledge base from prior experience or innate programming. So you can try grouping every atomic pattern with every other atomic pattern (time allowing), and as long as you're working with more than a one-dimensional medium, the differences between each atomic pattern being compared one-to-one will constitute a unique atomic pattern. This sort of comparison is one way of coming up with new knowledge in mediums that exist in two-dimensional (or greater) qualitative and/or quantitative and/or conditional mediums, and mediums that combine different types of properties. * Comparing unequivalent objects always creates partitions in either one or both of the objects. The remainder partitions are sometimes entirely knew but virtual objects. * ------------------ "Sight" From color comes shape, and from shape comes size, we triangulate images that come into our eyes. ----------------- "Reasoning Engines" 1. Knowledge as patterns in contexts. 2. Thinking as storing and grouping knowledge. ----------------- * Language contextualizes perceptions. The language used in each perception identifies and indicates patterns that have been parsed through comparison. * Copyright 6/24/2005 Justin Coslor Pattern Matching Previously I've written about how if you divide a circle into a bunch of equiangled sectors and if there is a prime number of sectors then no symmetrical alternating coloring patterns exist, but if there is a non-prime number of these equiangled sectors, then you can color in alternating sectors or groups of sectors to form symmetrical patterns that correspond to each of the composite number pieces. To apply this to pattern matching, simply cut the circle so that its sectors lie in a straight line and then look at the coloring patterns to match pieces of that linear pattern to strings of numbers, where each color might be a particular number, or just do it in binary. In this manner you can make numerical landmarks in raw data streams to look for patterns within potentially random data. When only a piece of one of these composite number symmetry patterns shows up in a linear data stream, that may indicate that other layers of patterns may be overlapping it. The thing that makes these patterns recognizable from randomness is the juxtaposition of their unique alternating prime partition patterns. An individual prime partition pattern piece that has been linearized is indistinguishable from any other linear prime partition pattern piece unless you know for sure that you're seeing the whole thing. But when you juxtapose two or more of these patterns together in the form of a composite symmetry pattern even a fragment of that pattern can dramatically narrow down the possibilities of its origin. Copyright 6/20/2005 Justin Coslor Remote-Controlled Contexts Via Preprocessor Switchboards (See Diagrams) Instead of having injective, surjective, and bijective, maybe there could be a preprocessor module that is bijective that goes in front of all surjective and injective relations. For a surjective relation: P1 = surjective ARP1 = bijective ARB = surjective = ACP1B For an injective relation: ARP2 = bijective P2RB = injective ARB = injective = ACP2RB A and B are domains P1 and P2 are preprocessors R is a relation, C is a cross-domain relation. In effect, the preprocessor becomes a duplicate of the domain element in A, but independent of the context of A. So since the preprocessing is done outside of A, you can have single-line inputs from A, and you can take several domain elements out of their contexts and perform their relations via remote control. In the second diagram, the cross-domain relation BCP2 is turned off, so context D doesn't contain its relation (P2RD) unit P2 gets turned back on. In that diagram, D is a remote-controlled context via the preprocessor operations switchboard S. A, B, and C are each in their own contexts and they combine in context D. The preprocessor modules allow for simple remote control like an operations switchboard. Copyright 6/12/2005 Justin Coslor Definitions Defining something by cataloging it's properties and relations is blind unless you specify the particular context of the thing, and the sub-contexts of the properties and relations it is composed of. Context is both an exoskelletal structure as well as an endoskelletal structure. Context is is defined by both the external limits as well as the internal limits. Copyright 6/5/2005 Justin Coslor Geometric Abstractions When doing abstraction on geometries and photos of patterns (symmetry formations, repetitions of patterns, and that which is recognizable from randomness), maybe all that is needed is a map of intersection points for each level of connectivity: i.e. a map of all points where two lines intersect, a map of all points where three lines intersect, etc. The union of all of those maps should form a sufficient geometric abstraction to recreate a recognizable approximation of the original model of photo patterns. Copyright 6/4/2005 Justin Coslor Index of Topics *(Remember to finish adding topics to this index, as it is only a partial list of ePIC-related topics I've written about so far.) choice creativity patterns contexts variables properties relations: quantitative, qualitative, cross-domain, analogical abstraction models simulations axioms: key, branch knowledge: implicit, explicit, representation intuitions complexity progressions pattern details randomness analogical recursions question asking systems question expectation templates object-oriented processing operation spaces: grids v.s. networks analogy metaphor examples Copyright 6/4/2005 Justin Coslor Abstraction Abstract relations are relations described by descriptions that are the simplified form of lexicons, where the details have been stripped and only the categorical data remains, along with some quantitative data (possibly. . .I'm not sure yet....), such as the dimensions and data types. Relations are fairly easy to abstract because you can just build an itemized list of the operators and verbs used on or in the general context of the domains that use them. Copyright 6/3/2005 Justin Coslor (See example diagrams) If a domain A is cross-domain related to a domain B analogically, that relation can be injective, or subjective; or if it is bijective, even if it's bijective to another element in the domain than the starting point, then we can say that the relation is recursive. This is an example of analogical recursions, because since all bijective relations are recursive, and analogical reasoning deals primarily with cross-domain relations, then all cross-domain relations that are bijective are analogical recursions. Another form of cross-domain analogical recursions comes from alternating back and forth through a set of relations between two or more domains, where the active element in the active domain is determined by some function on the ordering of the elements in that domain (a sequence function on the cardinality). Injective analogical recursions can also exist in a back and forth system that ultimately loops between the various domains of two or more contexts. Copyright 5/25/2005 Justin Coslor Implicit V.S. Explicit Knowledge In knowledge bases, facts and data are stored in patterns and contexts explicitly, but that same information may also belong to other contexts, and can be arranged into different patterns and may have unidentified relations to patterns in that data set and/or to patterns not in that data set. Often times there are multiple hierarchical levels and recursions of patterns in contexts and sub-contexts in patterns, and bridging across these levels are more of the same in many cases. Data that is implied can be treated as though it is hidden, though its role may be very important in the context of the data that depends on it. In the perception of questions, lots of implicit patterns and contexts are necessary to generate and adapt simulated models of the knowledge that is involved with the possible ways to represent the meaning of the question, as well as for generating models of the expectation parameters of the context templates involved with goal search, answer retrieval, and answer formulation (for discovering or constructing suitable content of the right level of detail). This is because every question is the intersection of multiple contexts, or rather every question is an attempt at adapting multiple contexts into compatibility, and thus unknowns must be declared. Copyright 5/16/2005 Justin Coslor Analogy, Metaphor, and Examples Now due to my lack of a dictionary on hand I'll create some of my own definitions (the names can be changed later). An analogy is like half of a metaphor. An analogy gives an elaborated example of a relation, whereas a metaphor gives an example of a relation across multiple contexts (a cross-domain relation). An example of an analogy is like saying: An apple is like an onion. Both rot, and are edible. An example of a metaphor is: Apple is to onion as postman is to salesman. An analogy is essentially a simile plus a moral or explanation/elaboration. A metaphor may describe the same relation(s) as an analogy in that it juxtaposes two or more pieces of information. This is similar to generating a unique diagonal length from a box generated by using one sequence or variable quantification as the x axis and another sequence or variable quantification as the y axis to produce a unique qualitative variable or sequence... Add more dimensions to the diagonalization to combine more variables or sequences or functions... Then just rotate the diagonal axis until it is horizontal. But metaphor goes a step farther and presents another example of that relation, but in a different context. *Examples are contextualizations of patterns. A relation between qualitative variables is thus a diagonalization of their quantitative mappings. In this way, qualitative mappings can be represented geometrically. Patterns are composed of variables and relations between variables. **Variables are usually qualitative property sets that have been quantitatively mapped into juxtaposition with their enumerated algebraic repetitions. Juxtaposition via diagonalization is a form of an operator. ***Operators are forms of juxtaposition of variables and patterns. Addition sequentially juxtaposes variables and patterns on a grid. Subtraction is the opposite of addition, as it removes variables and patterns from a grid. Multiplication sequentially adds to columns of categories, one category at a time. Division de-references and parses columns of categorical values, and is the opposite of multiplication. Addition, subtraction, multiplication, division, 2D geometry, trigonometry, algebra, calculus, etc, ... all are operations that can be performed on a grid. Change the operation space (i.e. change the context), and the axioms that these operations are based on may no longer apply; but some may, and those are the axioms we want to collect for a wide range of adaptability, and can be used in forming general systems theory grids and networks. As far as I know about operation spaces, there are grids and there are networks. Each can be within each, they can come in many different forms, and translations are possible between them, but the translation between a grid and a network always relies on a core set of axioms that are in common between the two data structures. Copyright 5/4/2005 Justin Coslor (Based on a theory I had around the year 2000) Sight -------- From color comes shape, and from shape comes size, we triangulate images that come into our eyes. --------------------------- Fall 2001 to 4/25/2003 Justin Coslor My fundamental theorem of Computer Vision: I believe that from color comes shape and from shape comes size; comparatively/relatively/contexually. I'll have to read about the cognition of vision to fill in the details and check out software and plasticware/firmware/hardware models of visual perception. Learn known mathematical techniques. Copyright 5/6/2005 Justin Coslor Rules Are Behavioral Expectations Here are some types of rules: Laws, priorities, environmental limitations (physics), trends, norms, common sense, personal limitations, societal beliefs, personal beliefs, lazy tendencies & optimizations, conditions, terms of use or license agreement, policy, ethics, morals, probability judgments, priority judgments, game theoretic strategy, preemptive negotiation, real-time negotiation, post hoc proc negotiation, design considerations, navigational control, pattern guidelines, pattern maps, mathematical modeling and calculation, combination possibilities, case-by-case possibilities; forum dimensionality, axioms, theorems, and restrictions; units and parsing and sorting methods and requirements; activation, deactivation, and flow control theory, network access methods, network exchange methods, network dynamics. Here are some qualifiers for those kinds of rules: Global, situational, regional, local, continuous, temporal, static, dynamic, linear, parallel, hierarchical, symmetric, independent, context specific, general, intentional, unintentional, conscious/unconscious, automatic, manual, modal, type, categorical. Find an ontology that lists concepts related to a given concept, in a hyper-linked format. Similar to encyclopedia references (see Wikipedia.org) or book topical references in the public library's card catalog. Copyright 1/7/2005 Justin Coslor Categories: Part 1 Even if categories get proven to be inaccurate (*Are accuracy proofs based on any subjective information?), then useful information about the compatibility of the data elements can be discovered as parameters get refined. Ultimately, it is the compatibility of the elements, both in and between data sets, that makes the fundamental definitions of the categories. Copyright 11/7/2004 Justin Coslor Hypothetical Relation Highlighting in Undefined Data Sets: If categorical names have been assigned to finite elements in a domain, the rest of the data in the set can be hypothetically considered to be relations or parts of relations (on those elements and elements not in that buffered data set). Or they may be elements of categories you don't yet recognize or know of yet. Guessing about Neural Architectures... This is a journal entry, Copyright 9/12/2004 by Justin Coslor. I could be totally wrong about this, but it is currently presumed, by me at least, that neural architectures tune to, receive, translate, and transmit various wavelengths of patterned energy configurations. The tuning functions may be in one unit, the receivers/input devices may be in another unit; the translation/manipulation apparatus may be in another unit; the translation/manipulation apparatus may operate in a unit of its own, and the transmission/re- communication apparatus may be in a unit of its own as well. There is likely data loss in the imperfections and limitations of the tuning apparatus, the receiving apparatus, and the re-transmission apparatus successively; however, the translation/manipulation apparatus may apply experience-based heuristics to fill in the holes and sharpen or simplify the distortions and puzzles in the data field. Each cluster of nodes, as well as the relation nodes themselves sometimes perform negotiations for syntactic and semantic consistency. Such negotiations are likely interfaces composed of multi-purpose reconfigurable general cellular nodes. Meaning might be derived from information streams by creating translations and equivalence representations in other classes and other contexts, and by defining and rating utility functions and organizing them in such a way that their priority can easily be determined relevant to the general function of the class of relations they belong to in generalized/easily-specialized contexts. The neat thing about information, rather than cause and effect, is that it can be re-conceptualized and re-contextualized and re-framed/re-patterned. 10/22/2004 Justin Coslor (after reading pg. 11 “Modern Algebra” by Gilbert and Vanstone) Some methods of Proof: - Assumptions (context) - Examples of problems or experience - Critical questions of interest - Representative language choice -Translation/Mapping -> same or different context? - Inventory of context axioms - Define critical question's search scope - Assume all questions are somewhat answerable - Convert other knowledge into current representation and abstract relationships without regard to hierarchical depth - Group compatible relationships - Mark partial compatibilities as overlapping sub-contexts - Hypothesize mappings that assume each relationship to be the answer to a series of questions - Look for hypothesized questions similar to questions of interest - If found, remap original examples in terms of those similar mappings of hypothesized questions - Define inconsistencies and address them - Represent conclusion - Explore relations of conclusion to other contexts - Blah blah blah, I should study more. 8/12/2004 Justin Coslor Axiom Notes (Here are some note I took at the public library today.) Structuring XML Documents / David Megginson CLP MAIN SCI&TECH QA76.76H94 M44 1998 The National Strategy To Secure Cyberspace February 2003 http://www.gpoaccess.gov - Perhaps people and machines should be trying to prove the limits of proof. - There are many shapes of non-Euclidean geometric reality. - Perhaps quanta of energy is a form of matter that exists on non-Euclidean spiral and tubular planes? Maybe quanta breaks off from matter and electrons that exist on non-Euclidean spherical planes during orbit changes and altitude changes? I read part of the end of the book Thinking about [TLC] LOGO: A graphic look at computing with ideas. pg. 206&207 ISBN 0-03-064116-0 Each set of axioms is based on a unique working model of the universe. (Regardless of the completeness of the model.) In many cases, there is some overlap between different sets of axioms, because many contexts have some properties and/or patterns that are in common. Metaphoric operations describe the relations between the properties and/or patterns that are in common between unique contexts. More than that, each set of axioms attempts to define a working model of the universe, and that no model of the universe is complete (hence it is a model) other than the universe itself; and from within the universe, a model of the universe can only be approximated, and to a varying degree of accuracy and/or applicability at that. 8/20/2004 Justin Coslor Update: So essentially, a set of axioms is only as good as the model they attempt to describe. Copyright 6/8/2004 Justin Coslor Contexts A context is a relation that defines a group of patterns. A pattern that is not related to any other patterns is isolated, and can for the most part be considered "invisible" to other contexts. A context can also be considered to be a pattern, and can sometimes also be considered as subject to this "isolation" concept. Patterns that exist within networks of contexts are the most easily located, since cross-domain relation-based experimental search and discovery methods need not be applied to locate or define them, as is necessary in many cases to find isolated patterns (i.e. island knowledge). Networks can consist of relations (surjective, injective, and bijective ) and cross-domain relations (which are potentially multi-node route reverse-surjective relations). Data turns into knowledge as the patterns and contexts and networks of contexts are mapped out. Copyright 6/5/2004 Justin Coslor Perception Every multi-state organization or cognitive organism exists on a higher plane than it is capable of perceiving, because nothing can monitor every aspect of itself (unless every cell is symmetrically identical) since the monitoring devices (sensors, etc), even when recursive, cannot monitor every aspect of themselves. This is because in order to perceive of something we must classify it in terms of something else we have perceived, and since we were born in motion, our consciousnesses pass forward from state to state, processing information (perceiving of things in terms of the physical universe) until parts, or the entirety of our bodies have fully ceased to move (i.e. until the breakdown of the subparts). As Godel's theorem implies: "no set can map its powerset". After some developed mental subparts have broken down, the structures of the consciousness that they were physically translating may continue to operate outside of the rest of the brain's physical time-frame. The latency of the various cognitive architectures in the brain may have a great deal to do with the relativistic self-observations of multi-sensory experiences. Since after all, some parts of the mind/body connection and mind/brain connection operate at near the speed of light (as electrons flow between the parts of each cell). Copyright 8/10/2005 Justin Coslor Perception -- continued from 6/5/2004.... On 6/5/2004 I wrote that "The latency of the various cognitive architectures in the brain may have a great deal to do with the relativistic self-observations of multi-sensory experiences." In other words, people think at different rates and depths from time to time, and that can create recall and encoding obstacles in grouping and interfacing memories between different cognitive states. However, those kinds of qualitative and quantitative differences between the contents of memorized perceptions can create bridges into depthier re-perceptions for recognition into fine- tuned contexts. 5/17/2003 Justin Coslor A.I. Notes Today I did a http://Google.com search on OpenCYC Thought Treasure V.S. OpenCyc came up. I guess both are major knowledge base ontology management systems, i.e. Reasoning Engines. Thought Treasure seems to have more stuff for Natural Language Processing than OpenCyc, but it is only free for noncommercial use. The Cyc technology though is the world's largest and most complete general knowledge base and commonsense reasoning engine. The CIA uses it, and did about 500 man-years worth of data-entry into to. OpenCyc is a much smaller subset of Cyc, and is open-source. Cycorp runs opencyc.org, and also makes ResearchCyc for R&D in academia and industry. Dependencies: none Languages: CycL, SubL, Java (other API's on the way) Platforms: Linux (Win32 coming soon) Sites: http://opencyc.org foundry.ai-depot.com/Project/OpenCyc /Amygdala /Fear /GAUL /Joone /LogicMoo /OpenAI /SigmaPi /Simbrain 10/20/2004 Justin Coslor Mission Statement Free open-source software is quite possibly the best hope, in conjunction with the freely accessible Internet, to give the common citizens a fighting chance at building foundations for their decendents in the midst of the mechanized empires of greed that thwart and encroach on their liberties and livelihoods in their attempts to squeeze and control the creative potential of supposedly free individuals who might otherwise be nurtured to blossom as citizens of a humbly selfless and harmonious planet Earth, that we all know can happen. ------------------------ Book II: Networks of Questions ------------------------ ======================= Everything I know about questions. ======================= By Justin Coslor justincoslor@gmail.com (These ideas are all copyright by Justin Coslor on their respective dates. I very much want to share these ideas, but I want to work in collaboration on related projects, so I withhold all intellectual property rights to this material. The work in this document is just a small selection of my ideas. Please do not steal my work via Tempest equipment or by any other means. If you are interested (due to tempest surveillance since I haven't shown anyone this,) and wish to collaborate, feel free to contact me at the above email address, and I will keep your technological secrets as long as you don't exploit me. Also, I'm egalitarian and I don't build weapons. Let's get that strait. Realize, I'm living on food stamps and measley disability stipend that barely pays the rent.) 2/28/98 Standard Inferences About actions: who what happened why when where how. Most everything else just uses one or a couple of these inferences. 10/6/2003 Query As far as questions are concerned... It seems like there are yes/no questions, option questions, spectrum questions, ind-depth (and short) descriptive questions, computational questions, essay questions, etc. Some are subjective (of opinion), some are neutral, some are definitive, some are explanatory, some are geometrical/visual, some are mathematical, some are of finite domain, some are impossible, some are biased, some are traps, some are falsely/inaccurately stated, some are open-ended, some are meaningless/pointless, some are direct, some are indirect, some are of infinite domain (a snapshot), some are time/space sensitive, some are quantitative, some are qualitative, some are recursive. 8/1/2004 Questions (a re-write) Questions can be used to define agendas; or indicate knowledge; gaps; or inquire about attributes, associations, and relations; or speculate; or to demonstrate something or make a statement; or be used for introspection or inspection; or to infer, deduce, or search for the elements of a pattern or context or its system(s) or relations; or to map assumptions (an assumption being the context that defines a set of beliefs); or to analyze and re-analyze data and information. 3/15/2005 Networks of Questions In considering ideas and information that is new to me, I ask networks of questions. The questions can be framed as dependency charts. Now what is a good way to understand dependency charts? List out the major nodes (most well-connected nodes) as open-ended definitions, and form lexicons out of the interconnected definitions. Next map out the rest of the nodes axiomatically using those definition structures. Turn this into a software for common sense perception. Maybe make a web crawler cognition engine that can learn the meaning behind things so that it can solve problems by figuring out new ways of thinking about things (adapting the context of question/perception networks). 3/15/2005 Experimentation Sometimes experimentation is necessary to solve problems and answer questions, because some nodes of information or questions or contextual perception networks are otherwise unreachable, and often entirely unknown to exist. 4/15/2005 Question Networks, continued... Abdicative reasoning: When does a function discover or prove an axiom? What metric makes analogy recognizable? Do recursive lexicons have the potential for infinite macro-scale growth? Do they have the potential for infinite micro-scale growth? Or will they all be governed primarily by the initial categories? Form networks of questions to gain valuable perspectives on topical and problem data. Model question engines in a careful evolutionary goal manner with substainability and capability and necessity as the primary objectives. Map the spectrum of inquiry. Expectations->Intentions->Experimentation (scientific combinations)->Dependency Chart Gaps and Representation Gaps->Formulation of questions incorporating "known" data. 5/4/2005 Answers We're surrounded by answers, but they're all meaningless and often impossible to even detect without knowing at least some of the questions that they are derived from. Without this question/answer connection, there is no consciousness, awareness would not exist. Copyright 5/10/2005 Justin Coslor Re-defining basic question thought-forms: Why? = Is there a reason for how this came to be, and what is it? What? = The existence of this shall be called by a name that needs to be defined, and we are inquiring about that. When? = This occurred or shall occur at what time and day? How? = By what process does this function? Notice that I had to use "what" in every one of these (except I tried not to use "what" in it's own definition, which was difficult). Therefore "what" is the most important thought form to focus on. "What" is the algebraic domain of the relation "that", or "this", or "these", or "those"; and the range is unknown, and is entirely unbounded. "That", is a pointer to a specific instance of something in existence (whether it be in physical or Platonic reality). The difference between Platonic objects and physical objects is that Platonic objects are just pointers to other pointers, whereas physical objects are pointers that point to themselves in a loop. Physical objects can bound and/or link (like a chain) other physical objects because if we geometrize the representation of the pointers we have physics, since the pointers loop. We exist partially in physical reality, and partially in Platonic reality since we can make conversions between the two. It's like the difference between particles and waves. Copyright 6/17/2005 Justin Coslor Question Networks: Option questions v.s. spectrum questions A lot of ambiguity is in every question. For instance, if you asked a question today you'd get one answer most likely, but if you asked that same question 10,000 years from now you'd probably get a much different answer. The scope of a question can be narrow or it cana be wide. With a narrow scope, a question might be a basic question that can be modified by many options, or it can be a bunch of cases, as in specific questions. Those are option questions. They lay out perspective question options. When the scope is very general or comprehensive of a lot of possibilities, then it is a spectrum question because it is intended to explore a range of possibilities that are within the same domain. Option questions seek to explore multiple questions that are not necessarily of the same domain. Copyright 5/25/2005 Justin Coslor Question expectation templates and question context intersections Questions contain an expectation template of the kind or class of answers that thet inquisitor is looking for. Often times though, the answers that are found or generated, or the answers that are of the most use, do not match the question's expectation template. Often times, answers that ar suitable cannot be derived or located until the question is elaborated, generalized, or otherwise modeled using different representation, such as analogical equivalencies of its objects, objectives, contexts, and relations. Every question is the intersection of several contexts, where behind the scenes, each context has its own unique expectation template; such as: - 1. The type of question: who, what, where, when, why, how, and its structure and methods. - 2. The semantic purpose of the question indicated by the question's structure: to define a context, to define a pattern or variable property, to state an open-ended knowledge structure and indicate the unknowns and data access points, to explore a domain or a range, etc. Or to state facts alongside the question, to indicate expectation parameters of the answer(s). - 3. The setting of the question is object(s) and relation(s). Associated objects and relations can be explicit or implied, but does result in expectation parameters. - 4. The existential time frame or solidity or transitory frame cycle of the question's objects and relations in regard to their setting is another context involved in definitive consideration of answer expectation parameters. - 5. - 6. . . etc. There may be many more foundational contexts that intersect in the formation and existence of every question. Copyright 5/24/2005 Justin Coslor Question asking systems Question -> Perception of meaning of question -> Search and answer retrieval -> answer formulation -> interpretation of answer. In short, QUESTION -> ANSWER(S). The more methods of knowledge representation that are available to model the perception of the meaning of the question, the more depth and breadth the search scope will have in the answer retrieval process, and the experimental data combining buffer will have more possibilities to form experimental combination answers with. If the answers are experimental, then they may need to be tested or proven valid. Not all valid answers are useful though, and not all questions can be modeled accurately. Question-Answer systems and Question asking systems are subject to priority systems, in their exploration of patterns in contexts. Every question or series of questions is a rough draft of the question that will produce or point to the desired answer that contains the right level of detail of suitable content. Copyright 5/10/2005 Justin Coslor Re-defining basic question thought-forms: Why? = Is there a reason for how this came to be, and what is it? What? = The existence of this shall be called by a name that needs to be defined, and we are inquiring about that. When? = This occurred or shall occur at what time and day? How? = By what process does this function? Notice that I had to use "what" in every one of these (except I tried not to use "what" in it's own definition, which was difficult). Therefore "what" is the most important thought form to focus on. "What" is the algebraic domain of the relation "that", or "this", or "these", or "those"; and the range is unknown, and is entirely unbounded. "That", is a pointer to a specific instance of something in existence (whether it be in physical or Platonic reality). The difference between Platonic objects and physical objects is that Platonic objects are just pointers to other pointers, whereas physical objects are pointers that point to themselves in a loop. Physical objects can bound and/or link (like a chain) other physical objects because if we geometrize the representation of the pointers we have physics, since the pointers loop. We exist partially in physical reality, and partially in Platonic reality since we can make conversions between the two. It's like the difference between particles and waves... Copyright 5/11/2005 Justin Coslor Writing Tips When I want to write, to figure out what to write I try to figure out a priority system, where I aim to invent the most important new idea that I'm either interested in (topically) or that is very important but that is only interesting enough to write down for somebody else to explore. Once I've focused on a topic I start asking questions and map it out and associate it to other areas and build networks of questions and answers and arbitrary data. I have to feel like writing and be relaxed and well hydrated (slightly caffeinated helps), and it's best when I'm thinking at my best (not bogged down by emotions) I don't consider this entry an "idea" since I feel awful and am terribly lonely and depressed and have heartburn at the moment. I write best when I'm either really really happy (a bipolar high), or really really depressed (a bipolar low). 10/20/2004 Justin Coslor Regarding Education If people could be taught first how to learn on their own, and next how to find and update their sources of information and resources, and then be taught creative and consistent logic and how to interact in their chosen forums of discourse, along with some skills as to how they can negotiate their insights and nurtured talents for wellbeing and non-harmful prosperity to an extent limited to what they can successfully and comfortably manage without bloating into the arena of greed or ill- intent; then they would have a great potential for doing a lot of good in the world and in the happiness of their daily lives. Lives lived purposefully. 10/28/2004 Justin Coslor It's important to be semi-autonomous rather than being just another domino. Copyright 6/1/2005 Justin Coslor Choice Choice depends on having recognized options to choose from. The availability of options depends on a system of awareness, and the explorative mapping of solution spaces and/or experimentally created generation. Also an evaluation system or metric is necessary for the selection process to pick a suitable option. The initiation of choice can be voluntary or forced; and it can be activated out of necessity, or independently. Copyright 5/31/2005 Justin Coslor Creativity Creative processes start out with a chosen medium, and then a random process is activated that generates a lot of possibilities, then it is a matter of choosing the best or most desirable possibilities and elaborating on them and linking them. That is what creativity means to me. The creator may also have several goals or requirements in mind, and be aware of juggling prerequisites. Copyright 5/28/2005 Justin Coslor Intuitions Intuitions are sequences of inspiration, and are part possibility thinking, part logical thinking, and part random chance phenomenon. Intuitive truth requires proof. Copyright 8/13/2005 Justin Coslor Ideas and Probabilities Ideas change probabilities, and ideas come from experimentation: search, sort, and shuffle (grouping and storing patterns in contexts). Ideas guide courses of action and affect expectations, of which reactions are based on, and those are some of the probabilities ideas can change, but they can also pave the way for new systems and developments. Intentions guide experimentation, and logic guides methodology. Axioms define logic and modes of proof. Innate programming (instinct) and use of methodology in real-time guides intentions. Useful ideas change contexts dramatically, and in doing so, many of the patterns in those contexts are able to form lots of new associations and possibilities. The most sweeping ideas are at the axiom level, and as new axioms are added, contexts expand and develop greater depth and greater interconnectivity. --------------- Book III: Math Ideas: --------------- Copyright 9/13/2004 Justin Coslor Infinity Something that is infinite in one context may be finite in another context. For the re-definition of "infinite" is something that goes on forever along the dimensional framework of a given context. But once new axioms are applied to the context where that something went on forever, the context is changed, and thus so the definition of many if not all things that existed in the former context, and in many cases infinite objects may become quitet definable (finite). ---------------------------------- 3/2/2005 update by Justin Coslor Also, it's important to not that perspectives changes (such as recontexualizations), may come with different axiom sets than the original context. 10/19/2004 Justin Coslor Public Domain, free for well intended use only. The upper limits of NP-Completeness Polynomial time computations' upper limit can be described by saying "infinity^x", and that has finitely many dimensions of context, but infinite scope along those dimensions. Non-polynomial time computations can be described by "x^infinity", and that has finite scope, but infinitely many dimensions of context. As you can see, cannot exactly equal np, however, it can approximate an incomplete abstract summary of some parts of np, using part of p's scope. This is because exponents stand for the number of perpendicular or symmetrical dimensions that the variable exists in. So saying that p=np is like trying to say that infinity^x=x^infinity, which it clearly is not; but p can be composed of a selection of np's dimensions, as long as they have a common base for forming selective perspective. Copyright 10/17/2004 Justin Coslor Qualifying & Quantifying Dimensionality In equations such as AnX^n + A{n-1}X^n-1 + . . . + A1X + A0 = 0, the coefficients (An to A1) can be considered to be quantifiers, and X^n to X or Y's etc, can be considered to be qualitative variables. When the variables X, Y, etc have exponents or are multiplied together, each combination of exponents and variables defines the dimensionality of the planes that the equation is holding in relation to one another, and the coefficients define the size or length or quantity or magnitude of each dimensional/qualitative structure in the equation that is held in relation to each other dimensional/qualitative structure in the equation. Now some dimensional structures are best described by equations that have more than two sides to the equal sign, such as those that exist on higher prime and prime composite levels of balance than most of current mathematics is based on. So we can only approximate descriptions of those structures in a duality format if at all. I guess a computer array or database or arrays of arrays can be used to depict higher dimensionality, but past a certain number of dimensions it surpasses the human brain's neuro- hardware's ability to visualize the relations and dimensional complexity. Arrays can be used to list out infinitely many dimensions categorically and quantitatively. However, nobody as of yet has discovered a way to think of a way to bound the classification of objects or situations using more than two extremes, using dualities such as maximum and minimum to balance an equation. Triality, or quintality, etc, along the prime numbers may indeed be possible, though our brains don't seem to interpret the universe along those dimensions as of yet. Perhaps eventually we will learn to adapt higher logical foundations. Copyright 10/4/2004 Justin Coslor Spirals (See the photos of the pictures depicted by this text on this date.) A number that has exponents contains one perpendicular or symmetrical dimension per exponent , so f^5 in this equation might look something like the multidimensional picture of a spiral within a spiral within a spiral within a spiral within a spiral. This is how my math invention "Sine Spiral Graphing" applies to the discovery I made about dimensionality (see journal entry dated 7/10/2004 Justin Coslor). The line going through the center of the spiral might actually be a spiral, a circle, an elliptical loop, a curve, or some combination of those. This kind of visual notation ("Exponential Sine Spiral Graphing" I call it) can be used in conjunction with conical orbit graphing I call it) can be used in conjuntion with conical orbit graphing to simplify the interaction visualizations of multiple spinning and/or orbiting bodies that have at least one plane of rotation in common. ------------------------------ Update: Copyright 2/10/2005 Justin Coslor The optimal structure of nanotechnology parallel-processing supercomputer memory structure could be something like this f^5 composite exponential spiral, except with ribbons of memory units and have vertical pipelines interconnecting each exponential layer of the composite spiral, and have a brick made out of short columns of these f^4 or f^5 or f^n spirals that are laterally connected on the ends of each column and stack multiple columns on top of each other in sheets of intensely interconnected spirals, like slices of a tree trunk. 7/10/2004 by Justin Coslor Light Spirals For several weeks now I have believed that light (and other emissions of convecting energy) particles/packets/quanta travel not in waves, but in spirals and flocks of spirals. I came to this conclusion after figuring out how to visualize Balmer's frequency equation (the one with the Rydberg constant and electron shell radiuses: f=R(1/Nf^2 - 1/Ni^2) where Nf is the outermost shell and Ni is the initial shell) in terms of sine-spiral graphing (Sine-spiral graphing is something of my own invention, and is a 3D resentation of circular motion, where the sine-waves or cosine waves represented for all points in time as a spiral (cosine of a point is X, sine at that point is Y, and time at that point is Z in the 3D coordinate system....remember the unit circle?) through time (or through a 3rd dimension if time is irrelevant or instantaneous or if motion is uniform)). See pg 67 of the comic book "Introducing Quantum Theory" by J.P. McEvoy and Oscar Zarate - Copyright 1996 (2003 reprint) ISBN: 1840460571 for Balmer's frequency equation. *Note: Waveforms only look like that from a perpendicular side-view, and I think this because, interestingly enough, 3D spirals look exactly like that when they are looked at from a perpendicular side view, which essentially is a 2D perspective. That is part of the basis of my sine-spiral graphing methods (I came up with the math for it when I got way behind in 10th grade Math-Analysis class). 7/11/2004 update by Justin Coslor Light Spirals Continued To visualize it I juggled the equation around a little, and figured out the intent that went into creating the algorithm. In Nf^2 and Ni^2, f^2 and i^2 just means that the variable f exists in a two- dimensional plane where one f axis is perpendicular or symmetrical to every other variable in the composite of the multiplicative parts; and when numbers or values get plugged into those variables, the visualization depicts a specific graph within the context of that combined dimensionality. That is why multiplication is used in algorithms to combine variables that are proportional to each other. *Multiplication shows that they have a proportional relationship. **Multiplication can also show that variables' dimensionality can share the same space, by perceiving of them in the broader context of their dimensions' combined contexts (whether it be symbolic, semantic, algebraic, or geometrical). ***One variable=1 dimensional representation. Two variables=2 dimensional representation. Three variables=3D . . . There is a limit to our neuro hardware's dimensional ability. ****If a variable is squared it exists within a two-dimensional context, if it is cubed, it exists within a 3D context, etc. Copyright 5/6/1997 Justin Coslor Sine Spiral Graphing A new method of graphing motion called "Sine Spiral Graphing" was developed by me when I was 16. It allows for simultaneously graphing the sine and cosine curves of an object in motion, three-dimensionally. Sine and cosine, when graphed simultaneously in two dimensions, look like two staggered intersecting waves traveling in the same general direction. (Fig. 1) There has been a need for developing better methods of graphing an object's two-dimensional (flat) motion through space over a period of time that more clearly shows the progression of travel. At present, mapping three-dimensional motion using different variables is more complicated, but could be a further application of the principles presented in the "Sine Spiral Graphing" method. The "Sine Spiral" is based on the spiral shape of two-dimensional circular motion graphed in three dimensions using this new graphing technique. The name is derived from the general name of the sine wave combined with what the actual 3D graph looks like: a spiral. This technique could be helpful for scientists and students alike in many applications. Some possible application for the Sine Spiral could be: - Plotting the motion of a bead in a hula hoop as it spins around one's waist. - Calculating the position of various atomic/subatomic particles moving in relation to each other over time. - Plotting the velocity and position of a point on an automobile wheel as sit spins down a runway or curvy hilly road. - Plotting the motion of a baseball spinning through the air as it travels forward to the catcher over a period of time. - Calculating the motion of a point on a bowling ball as it rolls down the lane over time. - Calculating the speeds and positions of a set of points, on various gears at work, in a clock in relation to each other over time. - Calculating the motion of a point on a rocket ship, or of a point on a space satellite as it orbits a planet. - Plotting the movement of a chicken in a tornado. All of these examples listed present graphing difficulties when depicted on a normal graph. The motions in these examples could be calculated on a computer and represented in a simulated fashion to show the actual movement in space for one point in time at a time. Concurrent Sine Spiral graphs can also be drawn for comparison of points on multiple moving objects. However, it would be difficult to graphically represent these motions for all points in time all at once. A simulation could be like a video, where one can only view one place on the video at a time. Viewing forward and reverse at the same time is not logistically possible on a video. However, when motion is three-dimensionally graphed on a computer using a Sine Spiral, it is possible to view these motions for all points in time all at once. A very effective way to manipulate and browse three-dimensional graphs (such as a Sine Spiral) on a computer is with Virtual Reality equipment. With Virtual Reality equipment, the perspective of the viewer can freely move around in space (on the graph) and see the 3D objects in one's graph from any perspective. In a Virtual Reality graph, the user can have total control over what is viewed and how it is viewed. Understanding the trigonometric functions of sine, cosine, tangent, and their inverse counterparts is a necessity for understanding Sine Spiral Graphing. Trigonometric functions of real numbers, called "Circular Functions" (or Wrapping Functions), can be defined in terms of the coordinates of points on the unit circle with the equation x^2+y^2=1 having its center at the origin and a radius of 1. (Fig. 2) There are three elements in a two-dimensional trigonometric function: the angle of rotation (sigma), the radius of the rotation r, and the (x,y) position of the point at that angle and radius. As can be seen in Figure 3, the x and y portions of the graph are always perpendicular to each other. Thus a right triangle is formed between the x, y, and radius sides. Right triangle rules can therefore be applied to this point in space (Brown/Robbins 190). Such trigonometric functions as sine and cosine can be applied to the triangle formed by rotation. These functions, sin and cos, are of fundamental importance in all branches of mathematics. One can use points other than those on the unit circle to find values of the sine and cosine functions. (Fig. 4) If a point Q has coordinates (x,y), and it is at angle sigma in reference to the origin, (cos sigma) = x/r and (sin sigma) = y/r. To obtain a rough sketch of a sine wave, plot the points (t, sin t) (Fig. 5), then draw a smooth curve through them, and extend the configuration to the right and left in periodic fashion. This gives the portion of the graph shown in Figure 5 (Swolowski 78). A cosine can be graphed in the same fashion by simply shifting the graph 90 degrees to the right. (Fig. 6) An object's circular motion can be described by either a sine wave or a cosine graphed in the same fashion. Such a wave is composed of the object's radius of rotation and the vperiod (number of degrees in on cycle) per unit of time that it rotates. Seeing an object's sine and cosine graph simultaneously greatly helps in visualizing the object's motion analytically compared how it found in real life. Watching an animation of an object spinning is the same as seeing the x and y coordinates (cosine and sine) of the object for each frame of the animation, one frame at a time. This is because one could see a scale view of its whole two-dimensional motion over a period of time. Visualizing an object's true motion in nature from merely looking at a graph of its sine or cosine can be difficult to conceptualize. For this reason, the Sine Spiral may be an improvement in current co-linear graphing (Fig. 7). Velocity over the period of one rotation on a sine curve can be measured by dividing the distance traveled in one rotation by the amount of time it takes to complete that one rotation. Velocity = change in distance/change in time + direction. Any change in velocity (a change in time) will change the distance between peaks of the spiral. The whole Z-axis around which the spiral revolves represents time passed. When the velocity is constant, the distance from peak to peak in the spiral is constant or each distance from one peak to another peak is the same. (Fig. 6) Therefore, if the distance from one peak to another changes somewhere in the spiral, this indicates that the velocity has changed at that point in time. Within the Sine Spiral, some of the variables that can change in the object's motion are velocity, radius of rotation, position of axis of revolution, and the scale upon which measurements are based. The shape of this spiral is an indication of any and all of these variables. The change in the shape of the spiral correlates to the change in one or more of these variables. (Fig. 7) Webster's Third New International Dictionary defines a spiral as "A three-dimensional curve (as a helix) with one or more turns around and axis." In current circular motion, the sine of the angle of rotation provides a Y value (Sine=Y/Radius of Rotation), while the cosine of that same angle provides and X value (Cosine=X/Radius of Rotation). These X and Y values are all that is needed to draw the two-dimensional models of rotation known as the sine curve and the two- dimensional models of rotation known as the sine curve and cosine curve (or sine wave). To my knowledge it has not been thought possible to graph this same motion in three dimensions though, because one needs an X, Y, and Z coordinate in order to graph in 3D. There can be an X and Y coordinate by finding the sine and cosine of a unit circle. All that is needed is a Z coordinate to make the circular motion graphable in three dimensions. That Z coordinate could be representable by time, or speed of rotation, or even the period of degrees it takes for one complete rotation. In a sine wave, the period is 360 degrees. Using the period of degrees in one rotation, one can find a constantly increasing Z coordinate by dividing the current number of degrees traveled by the period of degrees it takes to complete one rotation. In short, degrees/period. The period can be depicted by a set amount of time. Finding a ratio between something that can be used as a reference point (one second vs the number of degrees in one rotation) to one's current progress in that measurement scale (number of seconds that have passed vs number of degrees that have been traveled) determine where one is on the Z- axis. By dividing one's progress by a predetermined scale of reference, a new dimension can be generated in which to plot on a graph in order to illustrate this in three-dimensional fashion. This new dimension can be called the "Z-axis". Now that there is an X, Y, and Z dimension available, a three-dimensional model of an object's progress through its path of circular motion is possible. For 3D motion, one can draw three spirals over the same T axis and where two of the spirals intersect, plot a point. Connecting the dots between the points gives one a tri-spiral (a spiral or shape that represents 3D motion over time). One can continue plotting the points with several objects and where the tri-spirals intersect, the objects intersect. One can break down the tri-spiral to find out where the X, Y, and Z coordinates are in space and the time coordinates of the intersection. To use the Sine Spiral to map the 3D motion throughout time, one could mark the spiral with tags (or color code it) that tell one when and how far down the Z-axis it travels. Then to graph several objects to compare their motions and positions to each other, one can have a computer draw lines of the same color of the Z-tag, linking all of the objects that intersect on the two planes like the ZX plane, or the ZY plane. That way, one could identify when objects like planets line up on a plane or intersect. There is much to benefit from in being able to graph an object's progress at the same time as its position in space. One can see time from an outside perspective and also see how an object's motion, position, and speed relate to any point in time. In many circumstances, it may be very useful to finally be able to get to see the general shape of an object's travel through all points in time all at once. This new method of graphing circular motion in three dimensions is the "Sine Spiral". The graph forms a regularly spaced spiral whose axis is a straight line equidistant from the perimeter of the spiral. Changing the radius of rotation around a center axis changes the radius of the spiral around the Z axis. Changing the center of rotation in two-dimensional space (X, Y coordinates) makes the Z axis of the sign spiral curve up, down, or to the sides when graphed (instead of the normal straight line Z-axis). For instance, an air hockey puck pinning in place would have a regular sign spiral that represents a point on the puck's perimeter that is traveling in a circle. Now if the spinning puck were to be slid across an air hockey table, that same point (on the perimeter of the puck) would have an irregular sine spiral whose radius would be constant, but the Z-axis around which the graph spirals would instantaneously bend at a ninety degree angle. A computer can easily generate this three-dimensional picture of an object "N" at point "T" in time if the speed of travel is irregular (or at the ratio of degrees traveled to the period of one complete rotation if the speed is constant). (Fig. 8) Graphing any two-dimensional motion (motion that moves in any direction on a flat plane), or rotation in three-dimensions using time or progress as the third dimension allows one to look at time from an outside perspective. The Sine Spiral can be used to graph any such two-dimensional motion, or any number of combinations of such motion. It can be used to graph several objects moving around in 2D (flat) space on the same plane. The Sine Spiral can be used to graph an object which has a rotation within a rotation, and so on (Fig. 9). In this case, each next level of rotation is on an incrementally larger scale. To view some of the higher levels of rotation, one must graph the object's motion over a longer period of time. This concept can relate to complex motions of a longer period of time. This concept can relate to complex motions of a large scale found in, for example, the universe. Sine Spiral graphing can literally be used to graph the motion of every particle in perceivable universe for all points in observable time, simultaneously (by bending the Z-axis appropriately to accommodate changes is axis orientation). Using the Sine Spiral, graphing motion in the Z-axis, or time, requires one to employ a means to mark or reference the Sine Spiral in order to distinguish how deep down the Z-axis the motion has traveled. Without a Sine Spiral, one can only pick three-dimensions to see on a graph for all points in those dimensions. One could have X, Y, and Z coordinates on a 3D graph all at once, but only for one point in time per graph. Or one could illustrate motion in any two dimensions for all points in time using the Sine Spiral. Here are some of the dimensions from which one can choose: X, Y, Z, and Time. One can have four or more dimensions on a graph by selecting 3 variables form out of the X, Y, Z, and Time, as well as any number of descriptive, qualitative, categorical, computational, or other quantitative dimensions. These kinds of dimensions may appeal/apply to one's senses and could be described in "real" dimensions such as the Z-axis and others. With 3D applications using this concept (once improved methods of graphing 3D motion with the sine spirals are better developed), other more complex spirals can be mapped. Such 3D applications could include the universe in their motion through space throughout all time to see where certain ones meet or line up), and graphing the motion of particles of a sun during a supernova (the spiral would look similar to a tangent spiral as described below). The Sine Spiral may be an improvement in the graphing of nonlinear and linear motion. With the help of the recent Virtual Reality technology, most any computer can be used to build 3D models such as Sine Spirals. We can construct and view a Sine Spiral and have complete control over the graph, viewing it in 3D space as if it were physically here. There are many new math applications and theorems that may apply to this concept. Different types of spirals are possible with the general Sine Spiral method. Such shapes could include the Sine Tube (a sine spiral whose period is infinitely small), the Tangent Spiral (which uses a sine spiral whose period is infinitely small), the Tangent Spiral (uses the equation Tan sigma = (y/r)/ (x/r) for the x and y coordinates), and the secant spiral (uses sec sigma = 1/(x/r) for the coordinate and csc sigma = 1/(y/r) for the y coordinate). Also, in either two-dimensional or three-dimensional motion (when a graphing method is available), an object can be spinning in a circle within a circle (each level of rotation incrementally bigger than the previous), and this will make a very special type of Sine Spiral that looks like a spiral within a spiral within a spiral, etc., depending on how many levels of rotation are going on. More new math applications ar sure to be found that can apply to the Sine Spiral as it is used. Graphing three-dimensional motion with the Sine Spiral is more difficult to do, but can be done effectively. Graphing three-dimensional motion using the Sine Spiral needs further refinement at this time, but will hopefully be available for use in the near future. There are many new avenues that open up as people figure things out in science and math. The Sine Spiral may be another door in mathematics ready to be opened up and entered. Through this door may be a whole new way to look at things, a way to see objects in nonlinear motion from a standpoint outside of time. ---------------------------------- Works Cited: Brown R., and D. Robbins, "Advanced Mathematics: A Precalculus Course" Boston: Houghton Mifflin Company 1987. Fleenor C., M. Shanks, and C. Brumfiel. "The Elementary Functions". Boston: Addison-Wesley Publishing Company, 1973. Gove, P.B., ed. "Webster's Third New International Dictionary, Unabridged". Springfield, MA: Miriam-Webster, 1986. Manougian, M.N. "Trigonometry with Applications". Tampa, FL: Mariner Publishing Co., 1980. Swokowski, E.W. "Fundamentals of Trigonometry". Boston: Prindle, Weber & Schmidt, Incorporated, 1982. -------------------------------- Copyright 6/28/2003 Justin Coslor Conical Satellite Orbit Graphing (See Diagrams dated 10/4/2004, 3/1/2004, and 9/15/2001) I do think the conical satellite orbit graphing idea I thought of in winter 2001 (or the year before) could still be something valuable in detecting and calculating collisions and for 3D space junk detection. It's based on the hypothesis that if you compress a half-sphere into the shape of a cone, the 180 degree arcs become straight lines, and straight lines are easier to represent, interpret depth of, and run calculations on than arcs. Elliptical orbits would just re straight lines at an angle, each line representing the orbital path of an object in space. Where two or more lines intersect, a collision is possible at the point by either accelerating or de-accelerating the objects that the lines represent. Each object in a hemisphere cone is represented by a maximum and minimum altitude, and an angle representing the direction in which the object is traveling. There is one cone for each hemisphere. The neat thing about the conical format is that you can see how a bunch of objects, traveling in different directions at various altitudes, stack up along a common line of altitude protruding through the center of the planet, sun, moon, atom, galaxy, etc, and you can see how this line of altitude intersects each of those objects at two points in time (one for each hypersphere cone), along their various paths of travel. Conical orbit graphing is a way to group a set of satellites (or other objects in orbit) by a single line protruding through the center of the central mass out into space (with a longitude and latitude coordinate for each hemisphere from which the line emerges). All sorts of nifty computer software functions can be incorporated into this as well, such as having a 2D map of the central mass (such as a planetary map or electron orbital map) as a clickable image map that generates a unique pair of orbit cones for each coordinate (one for each hemisphere of the hypersphere for objects traveling 360 degrees or more around the planet). It would have a timing component as well and can be used as a multi-body gravitational clock, viewable with virtual reality equipment or a regular computer. There can also be a range component that highlights any possible collisions within a certain proximity of the satellites in focus (the satellites that intersect a common axis of altitude, have one pair of cones for each axis of altitude). The user should also be able to zoom in and out, rotate the cones, focus on different axis' of altitude, combine complex orbits with sine- spiral graphing techniques (see my paper on that), and watch the satellites travel along their path lines in real-time (at an adjustable rate) using live or recorded data collected from sensors and observational equipment. It would help if most modern satellites were equipped to detect space junk and satellites around them and relay it back to the ground so that the world has a constantly updated fairly accurate map of all of the objects and space junk in orbit around the earth, since space flight has been compared to flying through a high-speed shooting gallery. Ideally, some kind of Star Trek-like/Tesla Wardencliffe-tower-like shields or something are needed for the safety of that hazard (but not for use as a weapon), but a good 3D navigational map can't hurt. For each satellite the computer can run a conical orbit graphing collision detection test for each point in time along it's projected path of travel. The main use of conical orbit graphing as I see it, is for detecting collisions at points along a line of altitude, using one pair of cones for each point in time (or as a 3D interactive video). The user should be able to pick a time and x-y coordinate, see the satellites that intersect that line of altitude, then zoom in on the part of the path of the satellite that they are interested in, then click on a point in that path, and a new set of cones will be generated using that point as an altitude line in the center of the cones so that you can then see what possible collisions and path intersections there are for that point in the satellite flight path-time. All as straight lines so that it's easier to comprehend in complex situations. The computer calculations might even be quicker than calculating arcs. I'd assume elliptical arcs to be the most computationally intensive using traditional methods, but they too could probably be represented as straight lines in the software (going diagonally across the cones from one height to another height, and then the opposite for the other cone). It would be a 3D software tool for visualization and collision-interception calculation (and might be able to help protect all countries from incoming intercontinental thermonuclear ballistic missiles by combing this visualization method with a ground-based or space-based or airplane-based or reusable non-offensive missile based laser/maser anti-ballistic missile defense system. There might be many other beneficial uses for this visualization method that I haven't thought of yet (such as charting asteroids around Saturn or something; though hopefully it won't ever be used for, or even be useful for offensive purposes of any kind).. I haven't written any of the code yet or figured out much of the math yet to make it possible yet. Scholarly help is encouraged. Copyright 8/28/2005 Justin Coslor Applications of Conical Hyperhemisphere Graphing When Combined With Sine Spiral Graphing (See my papers dated 5/6/1997 and 6/28/2003.) A Sine Spiral graph can be used to depict how an object rotates in N dimensions as it moves from point to point in time (as though it were rotating in place through time without actually traveling forward along a path). Then those time coordinates can be linked to a conical orbit graph of the distance vectors that the object moves through along its path (or use a 3D Cartesian Coordinate grid of its path if it isn't going to travel a full orbit around the planet...or not...). This combination of graphing techniques works regardless of whether the object is below, on, or above the surface of the Earth, or other center of mass in space. For instance it could be used for mapping the path of a vessel that goes from under the ocean, up into the sky, and out into space into an orbit around the moon or something. ***************** Each layer of the hyperhemisphere cone is a polar grid of a different altitude. Elliptical and circular orbits are represented as straight lines going across a pair of cones and intersect with an axis of altitude line that goes vertically through the center of each cone, where the axis of altitude represents a line going through the center of the planet and out both sides into space. Elliptical orbits go diagonally across the cones in this fashion from one altitude to another, and back the opposite way in the cone that represents the other half of the hyperhemisphere. Circular orbits go straight across the cones at whatever altitude and declination they happen to be on. ***************** Space stations could use these mapping techniques to coordinate their motion and to dock incoming spacecraft, and it could be useful for spaceship navigation and satellite positioning, coordination, and communication routing too. Navy submarines could use these sine spiral + conical hyperhemisphere (or sine spiral + Cartesian or polar) graphs when planning and plotting routes through the oceans of the world through different depths and complex courses. Air-Force planes in perpetual (or merely long distance) flight could also use it to plan or plot their courses, so could airlines. It could simplify autonomous agent motion through extremely complicated environments, such as space, or for nanobots navigation in a chip or in colloidal fluid, or autopiloted aircraft in extremely crowded skies (such as autopiloted personal aircraft for overcrowded cities). Cross Domain Relations, for the Mathematics of Alternative Route Exploration Aside from the first order logic stuff, the ideas and depictions in this paper were originally conceived of and are Copyright 5/22/2004 by Justin M. Coslor, ALL RIGHTS RESERVED (Please contact me for conditions of use...). This Rough Draft was typed on 6/9/2004 in AbiWord on an X86-Compatible Personal Computer running GNU Sarge (a free Debian Linux Operating System), and was encouraged by the FRDCSA.ORG project. These ideas are intended to enhance the ability to discover and invent new routes in any field of study, and to aid in evaluating the relative utility of known routes, as well as to simplify some of the problems posed by computability theory. Figure 1. From the foundations of relational logic, we already know that: if a relation is xRy: X-->Y, then it is injective; or if xRy: X<-->Y, then it is bijective; or if xRy: X-->(y1, y2, ..., yp), then it is surjective; We also know that if a functional relation is xRy: y=f(x)=m where f(x) represents an arbitrary function of the domain X that yields a set of unique m's that are sub-ranges (y's) within the bounds of the range Y (a.k.a. the Class Y), where each m corresponds to a uniquely arbitrary domain x through the functional relation f(x). In this case, [f(x)]=R in the equation xRy. (*Remember for later that any mathematical operator (+, -, /, *, etc.) can be a relation. Any piece of computer software can also be treated as a relation, since software performs operations, and is basically a collection of algebraicly-tied operators.) But perhaps, we can broaden the scope of the Context in order to allow for more possibilities. This "broadening", may include metaphoric operations and metaphoric relations between the data type(s) of the functional relation(s) in focus and various specified number sets, orderings, and systems of numbers (including symbolic ones). (*We'll cover more on this later.) Let us introduce a new type of relation, that is a relation that relates relations, and let us call it a Cross-Domain Relation, and depict it as such: One goal of this paper is to show a system to accurately depict the following kinds of relation: xCy: (x1, x2, ..., xp)-->knEY (injective), or xCy: (x1, x2, ..., xp)<-->knEY (bijective), or xCy: (x1, x2, ..., xp)-->(k1, k2, ..., kn)EY (surjective); where every sub-range k that is an element of the range Y, has multiple domains that relate to each k in a unique way (through a unique route). Each sub-domain in the Domain X can come from different contexts and each sub- domain may operate under a different relation to specific sub-ranges in Y than other sub- domains relations to those same sub-ranges in Y. In these relations, some of the sub-domains may be injective, some may be bijective, and some may be surjective. In order to label and order each set of sub-domains that is part of a unique cross-domain relation, we introduce the ordering term "n". We can use the "n" component here to differentiate and/or order cross-domain relations, by combining the ordering of cross-domain-related sub-domains (i.e. nCx) with the individual relations between those sub-domains and any given sub-ranges (i.e. xRy), as such: nCx: N-->X (injective), or nCx: N<-->X (bijective), or nCx: N-->{x1, x2, ..., xp} (surjective), then nCxRy describes the cross-domain relation where n is an element in the cross-domain N such that x=f(n), and x is an element of a particular cross-domain subset n (as well as being a sub-domain of the domain X), where x has a relation (a.k.a. a route) to a particular sub-range y in Y.; (***Note: every x in X and/or every n in N can come from vastly different contexts, yet still lead to the same y(s) in Y.) where for each sub-range y, f(n) is every function in the cross-domain N that leads to multiple sub-domains in X that lead to the the multilateral result k (which is a specific singular sub-range y in the Class Y with multiple relations leading to it from the domain X) through the route: F[C{f(x)}]-->kEY (kEY means k is an Element of Y), where C{f(x)}=n and y=f(x) and x=f(n), and k represents any specific unique sub-range in Y that can be arrived at via multiple domains' functional relations, where each of the multiple domain's relations goes from any sub-domain x in X to the same specific sub- range y in Y; where F[N] is the set of all routes to all k's in Y, K is the set of all k's in Y, and F[n] is the a relation describing set of all routes to a specific k in Y. k is used to depict sub-ranges that have multiple ways to arrive at them; that is to say, ways that include origin variations, and intermediary relation combinations (middle-man combinations). In short: "If some unique X's yield some of the same unique y's through various relations, then those X's are said to have "cross-domain relations", because those domains have some relations whose end results have something in common." ^^^^^^^^^^^^^^^^This is what I was trying to draw and put into an equation format. I'm not sure if I succeeded, but probably. It seemed like there needed to be a formal word for what I was trying to depict as relations that relate different contexts' functions' domains by a representable equivalence or similarity in their ranges (when there is exists such a representable equivalence or similarity), so that's how I came up with the name "cross- domain relations". Computer software can essentially be treated as such functions, for which cross-domain relations that lead to alternative routes may exist for any given set or class of software functions. It's basically all about alternative routes. Such a mapping can be quite useful for exploring alternative, or previously unconsidered, or unknown possibilities and modalities. In Figure 1., X is a class that contains domains that lead to ranges within the class Y. There may be other classes that lead to those ranges, even if they do so indirectly through other classes by broadening the applicable context. By saying "lead to them" I mean "relate to them" in any "chosen" way(s). The route equations can get very complex the more classes and destinations you're analyzing when looking for and mapping cross-domain relations. In practice, the user ends up with a concise pack of cross-domain relation equations that summarizes the entire complexity of the known patterns in the contexts of any situation or model. The equation packs can also be used to represent the possible outlets to explore for new patterns based on perceived priority of their beginning class of categories, and perceived attainability/computability. . . mark off potentially infinite patterns and recursive loops accordingly, after exploring the first few layers only. Conclusion: Cross-domain relations can be used when depicting, predicting, finding, manipulating, creating, using, analyzing, backtracking, tracing, comparing, and reverse engineering alternative routes to anything, in any field of application. --------------------------------------------------------------------- --------------------------------------------------------------------- --------------------------------------------------------------------- --------------------------------------------------------------------- Examples: (*In the following examples I have defined the "underscore" character "_" to be the equivalent of the logical statement "OR", which is equivalent of the English language statement "and/or". I use the "_" character to link multiple routes to a sub-range, so that the patterns of the context of that sub-range can all be packaged into one continuous string. Such a string can then be parsed easily and sorted according to factors such as: route-scale (number of computable degrees or nodes v.s. potentially infinite possibilities), route category, route-size, relative route location, etc.) Example 1: In Figure 1., the domain x2 has an alternative route to y1 through the cross-domain relation n1Cx2Ry1_n1Cx1Cy1 where n1Cx2Ry1=Route k1, and n1Cx1Cy1=Route k2. ((In my examples I like to use C to represent bijective relations, and R to represent surjective or injective relations.) *Note that x1Cy1Cn1=n1Cx1Cy1 ) So in this example k1 and k2 are the known routes to y1, and since we know about more than one route to y1, we can call k1 and k2 cross-domain relations. Or we can simply reference that group of routes by the meta-name k1_k2. Figure 1. ----------------------------------------------------------------------- ----------------------------------------------------------------------- ----------------------------------------------------------------------- The following are Graph Theory Examples of Cross-Domain Relations: (**In the following examples, I use R to represent an injective or surjective route, and use C to represent the continuous directional flow of a bijective route. I use the symbol "$" to indicate that the routes on each side of the "$" have a bijective relationship. The "$" symbol is used when comparing two or more routes. The "=>" symbol means "directly implies".) Example 2: First have a look at Figure 2. (***Note: in ACB, (a1Cb1)$(b1Ca1), because BCA exists.) Figure 2. Alternative routes to A from C: ARC=a2Ra2 => a2 = ACCk1 ACBRC=a1Cb1Rb3_ a1Cb1Ca1Cb1Rb3 => ACCk2_ACCk3 ACBCDCC=a1Cb1Cb2Cd1Cd2_A$B$D$C=a1b2d2 => Routes ACCk4 through ACCk11 (****Many more complex routes beginning at A and terminating at C exist, and can be very explicitly depicted in this manner.) ---------------------------------------------------------------------- ---------------------------------------------------------------------- ---------------------------------------------------------------------- ---------------------------------------------------------------------- Example 3: In Figure 2, by entering each line's node relationship into a computer in a format such as: [ACB,BCD,DCC,BRC,ARC,ARF,FCE,FCC], (<----This is the Context.) (Next I'll describe the Patterns in that Context...) the computer can generate on the fly all of the possible routes from any given node to any other given node, including curtailed potentially infinite loop structures (by representing loop structures via the "$" operator, as noted earlier), and it can explicitly represent the optimal routes and rank the suboptimal routes using relation and cross-domain relation notation. Perhaps in some situations, one might even order the routes by largest perimeter of closed polygonal circuit region to smallest polygonal circuit perimeter, followed by largest open leg to smallest open leg, when declaring a context. (*****Where ";" is the character that indicates the parsing of each closed-circuit polygonal region or open leg in this notation variation.) This might look something like: [ARC_CCF_ARF;ARC_ACB_BRC;BCD_DCC_BRC;FCE] ...if the proportions were correctly represented in my diagram, that is... Figure 2. 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 re-aligned in this manner, and in that way axioms from Graph Theory might be able to be translated into Hierarchical Number Theory. The center-poles are very significant because when I transposed the prime number structures onto the natural number system there is a central non-prime even natural number in the very center directly between the center-poles 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 N-P 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 prime-number 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 every-other-number 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 high-level 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 two-digit 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 micro-hierarchy 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 in-between 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 divisor-checking 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 divisor-checked 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 "near-midpoint-prime" 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 N-1th difference to see if it is up, down, the same, or if N-1'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 "near-midpoint-prime" 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 nearest-to-midpoint 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 nearest-to-midpoint 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 nearest-to- 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 - (N-1)'s nmp = 0, then b= null. If N's nmp - (N-1)'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 = (N-1)'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 near-midpoint prime number and the non-prime 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 non-prime 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 non-prime, 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 near-mid prime lies. 31-30=1), right->left (this changing of sides is important to note because it describes which side of the midpoint of the prime that the nearest-to-midpoint prime lies on or has moved to, in terms of the ratio symbol) odd same (this describes whether the nearest-to-midpoint primes of two prime numbers have a difference that is even, odd, or if they have the same nearest-to-midpoint 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 nearest-to-midpoint 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 nearest-to- midpoint primes of two prime numbers have a difference that is even, odd, or if they have the same nearest-to-midpoint primes. I threw in the "reset to 1" thing just because it probably occurs a lot, then there's also the infamous "undo-from-last-reset" 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 non-prime. 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 non-prime partition sections. (*Note, since prime numbers don't have symmetrical equiangled partitions, use the center-number + 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 non-primes. 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 half-circle 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 alternate-half 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 non-prime 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 non-prime 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 non-prime symmetry pattern can be colored using it's own colors in an on-off configuration around the circle (See diagrams.). Prime layers can't be colored in an on-off 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 Y-axis, and on the X-axis 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 base-10 geometrical representation of that particular prime number (and it can be done for non-prime 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 base-10 graph paper border that uses the minimum amount of X-axis 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 off-white 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 Y-axis, 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 Y-axis 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 abdicative 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 side-view looks like a 2-dimensional 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 bit-length to the bit-translated 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 model-theory 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 = real-time) -> 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 z-axis. ...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 CMU-PHIL-79, Philosophy, Methodology, Logic. Pittsburgh, Pennsylvania 15213-3890. - Formal Systems - Church Turing Thesis - Godel's Theorems -------------------------------- Notes on Wilfried Sieg's "Properties of Formal Systems" paper: Euclid's Elements -> axiomatic-deductive 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 Church-Turing 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 1917-1918 -> mathematical logic. Kinds of completeness: Quasi-empirical 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 non-standard 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. hyper-arithmetical, 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 first-order 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 "Church-Turing Thesis" paper: Church re-defined 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 Church-Turing 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 non-recursive systems?). Metamathematics also led to Godel's Incompleteness Theorems (1931), which apply to all formal systems, like type theory of Principia Mathematica or Zermalo-Fraenkel 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 lambda-definability, and later prove that lambda-definability is equivalent to recursiveness (1935-1936). 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 bottom-up approach for pattern discovery and adaptive reconceptualization between the domains of different contexts, and can provide the theoretical framework for abdicative reaasoning, necessary for the application of my friend Andrew J. Dougherty's thesis. Perhaps my theories could be abdicatively formalized? My theories do not require empiricism (deduction), to produce new elements that are primitive-recursive to produce new elements that are primitive-recursive (circular-reasoning-based/symbolic/repetition-based) 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 "partially-defined" contexts because by Godel's Incompleteness Theorems, an all-encompassing 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, so-called finitist means? *To summarize informally: 1. Is truth co-extensive 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 not-sigma 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 non-demonstratable; 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 Church-Turing 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: quasi-empirical 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 Zermelo-Fraenkel Set Theory effective computability Lambda-definability investigating ever-wider formulations primitive recursive proof predicates provability and definability meaning: [11/11/2004 Justin Coslor -- Meaning depends on goal-subjective 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, hyper-arithmetical (is hyper-arithmetical 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 ---------------- Book IV: Invention Ideas ---------------- 10/28/2004 Justin Coslor Penny Universities I read in an article on the Internet that coffee houses used to be called "Penny Universities" in the mid 1600's in England, because it cost a penny for admission and a mug of coffee. "TIPS" is an acronym that was posted on a tin at the counter, which stood for "To Insure Prompt Service", and people would toss in a coin as a perk. It was a brilliant idea when the main branch of the Carnegie Library of Pittsburgh (nested between Carnegie Mellon University and the University of Pittsburgh) opened a coffee shop and free Internet access terminals this year in their library (terminals have been in place for several years). Their renovations are beautiful! Copyright 1/1/2005 Justin Coslor Inspiring book & some art product ideas. I've been reading the most amazing book called "Connections: The Geometric Bridge Between Art and Science" ISBN: 0-07-034250-4, and it's inspired me to want to read more books on the topic of Design Science. On page 264-265 of it it talks about the duality of Platonic polyhedra: namely The Inscribed Sphere. face<->vertex edge<->edge p<->q Face centroids of each platonic polyhedra are also vertex points for their duals, and so they lie equidistant from a common center. These pages in it talk also talk about Duality, in some of it's various forms: 1. in Bartók's music - 103 2. of maps - 125-127 3. of regular tilings - 177 4. of semiregular tilings - 181-182 5. of reciprocal figure - 224-230 6. of Platonic solids - 264-268 7. interpreting duals - 266-267 8. of convex polyhedra - 291-292 9. interpreting duals - 299-301 10. of Archimedean solids - 335-337 11. interpreting duals - 351 12. of networks - 362-368, 370-371 13. isometric vector matrix (IVM dual) - 370-371 ------------------------------------ Product Idea #1: Copyright 1/1/2005 Justin Coslor ERASABLE DOT MATRIX SKETCH PADS WITH AN ASSORTMENT OF VANISHING POINTS, AND THEIR CROSS-PLATFORM DO-IT-YOURSELF COMPUTER SOFTWARE EQUIVALENT, FOR 3D PERSPECTIVE DRAWING Here is a product idea I came up with today. *Look at figure 6.A.2 in the book "Projective Geometry" (pg. 248-253) That picture inspired the following idea. Here's a computer product I could sell both in computer form, and in paper form as an artists canvas tool: make a 3D dotted line matrix that has vanishing points along multiple parallel horizontal lines so artists can draw realistically spacial drawings. The dots close up are the biggest, and the dots get smaller and smaller the farther into the background you look. Then the artist can use those dots to realistically model 3D spacial representations on the paper, and when they have their sketch done they can just erase the dots, because the dots can be lightly printed on the page using erasable ink or erasable graphite. It'd cost fractions of a cent to print out each page of this kind of drawing paper, and the paper could come in a variety of perspectives of vanishing point angles. It'd be a great product to sell archive-quality drawing pads full of an assortment 3D dot matrix vanishing point angles printed on nice white drawing paper. On the cover of each pad there should also be a website address for where to download or order art software such as a software version of these different matrix patterns that a person can print out onto a page in very very light print (so light that it won't show up on a photocopy machine duplicate). Have some screenshots and a brochure-like visual demonstration and minimum system requirements shown on an advertisement page of how to use the software, and have that advertisement page as the first page of the sketch pads that get sold in stores all over the world. On the software brochure page (note, make the advertisement double-sided and on a perforated tear-out page that is scored for easy folding into the shape of a brochure, and use the UPC symbol as a discount coupon for $5 off of the the price of the software, when ordered online or through the mail. Include a self-addressed envelope and software payment form that can be torn out on the next page. I'd likely sell lots of software that way. On the brochure, suggest that they draw their lines on the computer paper lightly dotted matrix perspectives by hand, and that they should then either scan the page back into the computer and run it through a filter to take out the dots, or photocopy the original to take out the dots, since computer printer ink isn't erasable. This way, they wouldn't need to run to the store to buy new pads of paper. I think both of these products would be a fantastic product to sell all over the world, and I could probably get a patent on the pads. I don't agree with the notion of software patents though. I'd probably write the software using the Java3D API so that it's cross-platform. ---------------------------------- Product Idea #2: Copyright 1/1/2005 Justin Coslor N-DIMENSIONAL POLYHEDRAL MOTIF DESIGNING SOFTWARE FOR ARTISTS First look at some writings by the mathematician artists from the fifteenth century, such as: Alberti, Leonardo da Vinci, and Albrecht Dürer. Then write a little computer software that makes N-dimensional polyhedral motifs (fundamental patterns) and lets the user design their own and use them as wallpaper and skins for 3D objects. So people can make M.C. Escher-like artwork easily. It'll deal with symmetry design and manipulation very easily. Copyright 1/15/2005 Justin Coslor Art Software For Making Symmetrical Design Patterns & Motifs: Write a computer graphics software that allows the user to make precise symmetry design patterns and to be able to color and shade in the sections with the greatest of ease. It could also be good for making repeatable interlocking motifs. Copyright 1/1/2005 Justin Coslor Perspective Drawing Projects When I first wrote this I was reading the book "Basic Perspective" by Robert W. Gill - Library of Congress Card Number 79-64518. 1. Try drawing a top view of an object and draw a circle of frames (Boxes) around it, and in each box draw the 3D perspective side-view of the object as it would look to a person standing on a point in the center of each frame at that particular frame's perspective angle, where each frame is a 3D perspective drawing, complete with accurate shading. 2. Draw a 3D perspective drawing form the viewpoint of a cat, and include the cat's nose, whiskers and the tip of it's tail in the drawing. Include special lighting and coloring effects such as how the scene would look if it was a snapshot from the lens of a Kirlian photography camera, as cats see a different spectrum of colors than people, and possibly auras. Copyright 1-13-98 by Justin Coslor Robotics If one robot figures something out about it's surroundings it goes and tells the rest, that makes available more potential actions that the rest can do and have to work with and use as a tool. When one solves a problem it eliminates an obstacle which each robot would otherwise have to spend its time figuring out how to do because it communicates the solution. Example application: new, more efficient route planner. They could even learn teamwork too. If one robot solves part of a problem, but knows that the whole thing isn't solved and wants to completely solve it but not waste all day working on it, it can go and tell the other robots about the whole problem (denoting the part they've already solved) instead of just telling them part of the solution that they've solved. It would also tell the other robots the estimated "worth" (personal and societal ... sort of like a priority rating) of getting the problem completely solved. In a sense it "asks" the others for help. If the requested action's estimated priority is higher than the current actions priority rating, then the robots will go and help (fulfill the request) and in a sense perform a "favor" thus the robot in need would be making some "friends". ------------------------------------------------ Fundamental motivations: #1 stimulate/reinforce self (work towards own goals). #2 stimulate/reinforce others. The robot might stimulate/reinforce itself in order to stimulate/reinforce others, but not quite as much as stimulating or reinforcing itself directly, and it works on a "friends" and "favors" system socially. Do a favor and they are grateful and they are your friend (and doing #2 on a friend is equal to doing #1 and it is a random choice between the two when in question -- when asked to return a favor for a friend or do something for yourself). Each robot knows that it can do more faster with friends helping, so if it has something it needs help with or is bored it might think ahead and stop what it is doing and go out and get some friends by doing favors for some others and then later go and ask everyone for help. It would ask even those who aren't friends because some of them may help and after a few exchanges of favors with them, they too might become friends. With a small group (approximately six robots), in order to have any of then get any of their own stuff done you must either build a "friend removal" system, where robots are no longer considered a friend after a certain amount of rejections