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Intelligent Patterning and problem solving Tom Carter CSU Stanislaus December 27, 2018

Intelligent Patterning and problem solving Tom Carter CSU Stanislaus December 27, 2018. Intelligent Patterning. 1 / 15. 1 General problem solving. Pattern recognition. 2. Brief overview. Symbols and signs. 3. Intelligent patterning. 4. Some history. 5.

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Intelligent Patterning and problem solving Tom Carter CSU Stanislaus December 27, 2018

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  1. IntelligentPatterning and problemsolving TomCarter CSUStanislaus December 27,2018 IntelligentPatterning December 27,2018 1/15

  2. 1 General problemsolving Patternrecognition 2 Briefoverview Symbols andsigns 3 Intelligent patterning 4 Somehistory 5 6 Making things lookright What’s wrong in computingtoday 7 The intelligent mathematicalassistant 8 IntelligentPatterning December 27,2018 1/15

  3. General problemsolving Understanding theproblem Problem context and statement of theproblem Solving the right problem (ill-posed and ill-conditioned problems) Preconceptions Language and restating theproblem 1 General problemsolving 2 3 4 The role ofexperience Similar problems and analogy Appropriatetools Specificexperience 1 2 3 IntelligentPatterning December 27,2018 1/15

  4. General problemsolving Plug and grind Guess and prove Look itup 1 Three basicmethods 2 3 Hypothesis generation andtesting Flexibility and freedom — willingness to try and fail Recognizing blind alleys, and the value of exploring Appropriatehypotheses Lateralthinking 1 2 3 4 IntelligentPatterning December 27,2018 1/15

  5. General problemsolving Recognizingsolutions “A” solution vs. “the” solution Usefulsolutions When a “solution” solves an un-posed, but more significantproblem 1 2 3 IntelligentPatterning December 27,2018 1/15

  6. Patternrecognition Images (“visual patterns”) vs. “syntactic”patterns Symbols as patterns, and symbols as patternlabels Patterns ofsymbols Hierarchies of patterns, and symbols as tools for recognizingpatterns Patternmanipulation Learning to recognize patterns, and pattern recognition aslearning Patternrecognition IntelligentPatterning December 27,2018 1/15

  7. Patternrecognition What number comes next in the sequence? 1, 1, 2, 3, 5, 8, 13, . .. Pattern recognitionexamples IntelligentPatterning December 27,2018 7 /15

  8. Patternrecognition What number comes next in the sequence? 1, 1, 2, 3, 5, 8, 13, . .. What number comes next in the sequence? 8, 5, 4, 9, 1, 7, 6, 3, . .. Pattern recognitionexamples IntelligentPatterning December 27,2018 7 /15

  9. Patternrecognition What number comes next in the sequence? 1, 1, 2, 3, 5, 8, 13, . .. What number comes next in the sequence? 8, 5, 4, 9, 1, 7, 6, 3, . .. What letter comes next in the sequence? E, T, A, O, I, N, S, H, . .. Pattern recognitionexamples IntelligentPatterning December 27,2018 7 /15

  10. Patternrecognition What number comes next in the sequence? 1, 1, 2, 3, 5, 8, 13, . .. What number comes next in the sequence? 8, 5, 4, 9, 1, 7, 6, 3, . .. What letter comes next in the sequence? E, T, A, O, I, N, S, H, . .. In which row does Zgo? A, E, F, H, I, K, L, M, N, T, V, W, X,Y B, C, D, G, J, O, P, Q, R, S,U Pattern recognitionexamples IntelligentPatterning December 27,2018 7 /15

  11. Patternrecognition What number comes next in the sequence? 1, 1, 2, 3, 5, 8, 13, . .. What number comes next in the sequence? 8, 5, 4, 9, 1, 7, 6, 3, . .. What letter comes next in the sequence? E, T, A, O, I, N, S, H, . .. In which row does Zgo? A, E, F, H, I, K, L, M, N, T, V, W, X,Y B, C, D, G, J, O, P, Q, R, S,U What letter comes next in the sequence? W, L, C, N, I, T, . .. Pattern recognitionexamples IntelligentPatterning December 27,2018 7 /15

  12. Symbols andsigns The utility and power of symbols Choosing symbols, naming and pointing Symbols as “chunking”tools When to usesymbols Symbols andsigns The importance of anonymity (e.g., the lambdacalculus) Place holders (variables) Temporary and tentativesymbols 1 2 3 Signs, symbols, content andmeaning IntelligentPatterning Tom Carter (CSUStanislaus) December 27,2018

  13. Intelligentpatterning Creativity andArt Knowing when to pattern Symbol attachment andcreation; patterns/symbols as revealers and concealers Levels ofpatterning 1 2 Intelligentpatterning 3 Multiple patterns and selection (x−1)(x−2)(x−3)−6 x3−6x2+11x−12 (x−4)(x2−2x+3) Adaptive pattern recognition Are the patterns reallythere? IntelligentPatterning Tom Carter (CSUStanislaus) December 27,2018

  14. Somehistory Physics Philosophy (theory of knowledge) Mathematics Somehistory Matrix manipulation Topology Algebra Liegroups Manifolds and relativitytheory Algebraictopology 1 2 3 4 5 6 IntelligentPatterning Tom Carter (CSUStanislaus) December 27,2018

  15. Making things lookright Consider this piece of mathematics (here, and the next page). How do we get this typeset? See the following page for LATEX . .. We have the map bn : Σ2U(n) → SU(n +1) Making things lookright: given by bn(g,r,s)=[i(g),vn(r,s)] where i (g ) is the inclusion, [g, h] =ghg−1h−1 and IntelligentPatterning Tom Carter (CSUStanislaus) December 27,2018

  16. Making things lookright vn(r,s)=   0 1 2 α 0 0 0 ··· α0 ··· β(−α)0β α ··· ... 0 0 0 β(−α) β(−α) β(−α) β(−α)β   β(−α)1β   . . . . .   . . . . . . .. . .   . . . . . . . . . .   . . . ··· ··· ··· ··· . αβ(−α) .    n n−1 β(−α) β β(−α)n−2β −(−α)nβ −(−α)n−1β −(−α)0β −(−α)n where α=α(r,s)=cos(πr)+isin(πr)cos(πs) β=β(r,s)=isin(πr)sin(πs) IntelligentPatterning Tom Carter (CSUStanislaus) December 27,2018

  17. Making things lookright IntelligentPatterning Tom Carter (CSUStanislaus) December 27,2018

  18. What’s wrong in computingtoday Not enough resolution on displays (seems mostly solved . . . :-) Not enough processing power andmemory Not enoughparallelism Software tools are (largely) “flat” and sequential rather thanhierarchical What’s wrong in computingtoday IntelligentPatterning Tom Carter (CSUStanislaus) December 27,2018

  19. The intelligent mathematicalassistant Adaptive symbolic input andoutput Strong basic skills (all of arithmetic through college calculus and elementary discretestructures) First order logiccapabilities Adaptive “patterning” and“symboling” Elementary hypothesis generation andtesting The intelligent mathematicalassistant IntelligentPatterning Tom Carter (CSUStanislaus) December 27,2018

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