Perceive this as that – Analogies, artificial perception, and category theory

Abstract

This paper formalizes and analyzes cognitive transitions between artificial perceptions that consist of an analogical or metaphorical transference of perception. The formalization is performed within a mathematical framework that has been used before to formalize other aspects of artificial perception and cognition. The mathematical infrastructure consists of a basic category of ‘artificial perceptions’. Each ‘perception’ consists of a set of ‘world elements’, a set of ‘connotations’, and a three valued (true, false, undefined) predicative connection between the two sets. ‘Perception morphisms’ describe structure preserving paths between perceptions. Quite a few artificial cognitive processes can be viewed and formalized as perception morphisms or as other categorical constructs. We show here how analogical transitions can be formalized in a similar way. A factorization of every analogical transition is shown to formalize metaphorical perceptions that are inspired by the analogy. It is further shown how structural aspects of ‘better’ analogies and metaphors can be captured and evaluated by the same categorical setting, as well as generalizations that emerge from analogies. The results of this study are then embedded in the existing mathematical formalization of other artificial cognitive processes within the same premises. A fallout of the rigorous unified mathematical theory is that structured analogies and metaphors share common formal aspects with other perceptually acute cognitive processes.

This is a preview of subscription content, log in to check access.

References

  1. [1]

    J.F. Allen, AI growing up — the changes and opportunities, AI Magazine 19(4) (1998) 13-23.

    Google Scholar 

  2. [2]

    Z. Arzi-Gonczarowski, Wisely non rational — a categorical view of emotional cognitive artificial perceptions, in: Papers from the 1998 AAAI Fall Symposium: Emotional and Intelligent — The Tangled Knot of Cognition, ed. D. Cañamero, Orlando, FL, October 1998 (AAAI Press, 1998) pp. 7-12.

  3. [3]

    Z. Arzi-Gonczarowski, Categorical tools for perceptive design: Formalizing the artificial inner eye, in: Computational Models of Creative Design IV, eds. J.S. Gero and M.L. Maher, Key Centre of Design Computing and Cognition, University of Sydney, Australia (1999) pp. 321-354.

    Google Scholar 

  4. [4]

    Z. Arzi-Gonczarowski and D. Lehmann, From environments to representations — a mathematical theory of artificial perceptions, Artif. Intell. 102(2) (1998) 187-247.

    MATH  MathSciNet  Article  Google Scholar 

  5. [5]

    Z. Arzi-Gonczarowski and D. Lehmann, Introducing the mathematical category of artificial perceptions, Ann. Math. Artif. Intell. 23 (3,4) (1998) 267-298.

    MATH  MathSciNet  Article  Google Scholar 

  6. [6]

    A. Asperti and G. Longo, Categories, Types, and Structures (MIT Press, Cambridge, MA, 1991).

    Google Scholar 

  7. [7]

    M. Barr and C. Wells, Category Theory for Computing Science (Prentice-Hall, Englewood Cliffs, NJ, 1995).

    Google Scholar 

  8. [8]

    J. Barwise and J. Seligman, Information Flow, Cambridge Tracts in Theoretical Computer Science, Vol. 44 (Cambridge University Press, 1997).

  9. [9]

    E.A. Bender, Mathematical Methods in Artificial Intelligence (IEEE, Los Alamitos, CA, 1995).

    Google Scholar 

  10. [10]

    M. Black, Metaphor. in: Philosophical Perspectives on Metaphor, ed. M. Johnson (University of Minnesota Press, Minneapolis, MN, 1981) pp. 63-82. A reprint of an article from 1955.

    Google Scholar 

  11. [11]

    S. Blackburn, Dictionary of Philosophy (Oxford University Press, 1996).

  12. [12]

    F. Borceux, A Handbook of Categorical Algebra (Cambridge University Press, 1993).

  13. [13]

    E. Boros, P.L. Hammer, T. Ibaraki, A. Kogan, E. Mayoraz and I. Muchnik, An implementation of logical analysis of data, Rutcor Research Report RRR 22-96, Rutgers University, New Brunswick, NJ (July 1996).

    Google Scholar 

  14. [14]

    H. Caygill, A Kant Dictionary (Blackwell Publishers, Great Britain, 1995).

    Google Scholar 

  15. [15]

    A. Clark, Being There, Putting Brain, Body, and World Together Again (MIT Press, Cambridge, MA, 1997).

    Google Scholar 

  16. [16]

    R.L. Crole, Categories for Types (Cambridge University Press, 1993).

  17. [17]

    M.A. Croon and F.J.R. Van de Vijver, eds., Viability of Mathematical Models in the Social and Behavioral Sciences (Swets and Zeitlinger B.V., Lisse, 1994).

    Google Scholar 

  18. [18]

    E.R. Doughherty and C.R. Giardina, Mathematical Methods for Artificial Intelligence and Autonomous Systems (Prentice-Hall, Englewood Cliffs, NJ, 1988).

    Google Scholar 

  19. [19]

    S. Eilenberg and S. Mac Lane, General theory of natural equivalences, Trans. Amer. Math. Soc. 58 (1945) 231-294.

    MATH  MathSciNet  Article  Google Scholar 

  20. [20]

    B. Falkenhainer, K.D. Forbus and D. Gentner, The structure-mapping engine, Artif. Intell. 41(1) (1990) 1-63.

    Article  Google Scholar 

  21. [21]

    G. Fauconnier, Mappings in Thought and Language (Cambridge University Press, 1997).

  22. [22]

    R.M. French, The Subtlety of Sameness — A Theory and Computer Model of Analogy Making, A Bradford Book, (MIT Press, Cambridge, MA, 1995).

    Google Scholar 

  23. [23]

    M. Genesereth and N.J. Nilsson, Logical Foundations of Artificial Intelligence (Morgan Publishers, San Mateo, CA, 1987).

    Google Scholar 

  24. [24]

    D. Gentner, The mechanisms of analogical learning, in: Similarity and Analogical Reasoning, eds. S. Vosniadou and A. Ortony (Cambridge University Press, 1989) pp. 199-241.

  25. [25]

    D. Gentner, Analogy in: A Companion to Cognitive Science (Blackwell, 1998) chapter II(1), pp. 107-113.

  26. [26]

    S.M. Glynn, R. Duit and R.B. Thiele, Teaching science with analogies: A strategy for constructing knowledge, in: Learning Science in the Schools: Research Reforming Practice, eds. S.M. Glynn and R. Duit (Lawrence Erlbaum, Mahwah, NJ, 1995) chapter 11.

    Google Scholar 

  27. [27]

    N. Goodman, Fact, Fiction, and Forecast (Harvard University Press, 4th edition, 1983).

  28. [28]

    S. Harnad, The symbol grounding problem, Phys. D 42 (1990) 335-346.

    Article  Google Scholar 

  29. [29]

    H. Herrlich and G.E. Strecker, Category Theory (Allyn and Bacon, 1973).

  30. [30]

    P.G. Hewitt, Conceptual Physics — The Highschool Physics Program (Addison-Wesley, Menlo Park, CA, 2nd edition, 1992).

    Google Scholar 

  31. [31]

    D. Hofstadter, Fluid Concepts and Creative Analogies — Computer Models of the Fundamental Mechanisms of Thought (Basic Books, 1995).

  32. [32]

    D. Holland and N. Quinn, eds., Cultural Models in Language and Thought (Cambridge University Press, Cambridge, MA, 1987).

    Google Scholar 

  33. [33]

    K.J. Holyoak and P. Thagard, Mental Leaps — Analogy in Creative Thought, A Bradford Book (MIT Press, Cambridge, MA, 1995).

    Google Scholar 

  34. [34]

    E. Hutchins, Cognition in the Wild (MIT Press, Cambridge, MA, 1995).

    Google Scholar 

  35. [35]

    B. Indurkhya, Metaphor and Cognition (Kluwer Academic, 1992).

  36. [36]

    B. Indurkhya, Metaphor as change of representation: an artificial intelligence perspective, J. Exper. Theor. Artif. Intell. 9(1) (1997) 1-36.

    Article  Google Scholar 

  37. [37]

    I. Kant, Grounding for the Metaphysics of Morals (Hackett Publishing Company, Inc., Indianapolis/Cambridge, 1992). Translated by J.W. Ellington. Originally published in 1785 as “Grundlegung zur Metaphysik der Sitten”.

    Google Scholar 

  38. [38]

    J. Kolodner, Case-Based Reasoning (Morgan Kaufman, San Mateo, CA, 1993).

    Google Scholar 

  39. [39]

    G. Lakoff, The death of dead metaphor, Metaphor and Symbolic Activity 2(2) (1987) 143-147.

    Article  Google Scholar 

  40. [40]

    G. Lakoff, Women, Fire, and Dangerous Things — What Categories Reveal About the Mind (The University of Chicago Press, 1987).

  41. [41]

    G. Lakoff and M. Johnson, Metaphors We Live By (The University of Chicago Press, 1980).

  42. [42]

    F.W. Lawvere, Tools for the advancement of objective logic: Closed categories and toposes, in: The Logical Foundations of Cognition, eds. J. Macnamara and G.E. Reyes, (Oxford University Press, 1994) pp. 43-55.

  43. [43]

    F.W. Lawvere and S.H. Schanuel, Conceptual Mathematics (Cambridge University Press, 1997).

  44. [44]

    D. Leake, ed., Case Based Reasoning: Experiences, Lessons, and Future Directions (AAAI Press, Menlo Park, CA, 1996).

  45. [45]

    J. Lukasiewicz, Many-valued systems of propositional logic (1930), in: Polish Logic, ed. S. McCall (Oxford University Press, 1967).

  46. [46]

    J. Lukasiewicz, On 3-valued logic (1920), in: Polish Logic, ed. S. McCall (Oxford University Press, 1967).

  47. [47]

    S. MacLane, Categories for the Working Mathematician (Springer-Verlag, 1972).

  48. [48]

    F. Magnan and G.E. Reyes, Category theory as a conceptual tool in the study of cognition, in: The Logical Foundations of Cognition, eds. J. Macnamara and G.E. Reyes (Oxford University Press, 1994) pp. 57-90.

  49. [49]

    M. Minsky, ed., Semantic Information Processing (MIT Press, Cambridge, MA, 1968).

  50. [50]

    M. Mitchell, Analogy Mapping as Perception — A Computer Model, A Bradford Book (MIT Press, Cambridge, MA, 1993).

    Google Scholar 

  51. [51]

    D.J. Moser, Sze-chuan pepper and Coca-Cola: The translation of Gödel, Escher, Bach into Chinese, Babel 37(2) (1991) 75-95.

    Google Scholar 

  52. [52]

    B.C. Pierce, Basic Category Theory for Computer Scientists (MIT Press, Cambridge, MA, 1991).

    Google Scholar 

  53. [53]

    D. Ross, Metaphor, Meaning and Cognition (Peter Lang, 1993).

  54. [54]

    A. Sloman, Architectural requirements for human-like agents, both natural and artificial (what sorts of machines can love?), in: Human Cognition and Social Agent Technology, ed. K. Dautenhahn, Advances in Consciousness Research (John Benjamins Publishing, forthcoming).

  55. [55]

    P. Thagard, Mind (MIT Press, Cambridge, MA, 1996).

    Google Scholar 

  56. [56]

    S. Vosniadou and A. Ortony, eds., Similarity and Analogical Reasoning (Cambridge University Press, 1989).

  57. [57]

    R.F.C. Walters, Categories and Computer Science (Cambridge University Press, 1991).

  58. [58]

    A.J. Wells, Turing's analysis of computation and theories of cogitive architecture, Cognitive Sci. 22(3) (1998) 269-294.

    Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Arzi-Gonczarowski, Z. Perceive this as that – Analogies, artificial perception, and category theory. Annals of Mathematics and Artificial Intelligence 26, 215–252 (1999). https://doi.org/10.1023/A:1018963029743

Download citation

Keywords

  • Metaphor
  • Category Theory
  • Cognitive Transition
  • Boolean Combination
  • Categorical Construct