Kantian Philosophy of Mathematics and Young Robots

  • Aaron Sloman
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5144)

Abstract

A child, or young human-like robot of the future, needs to develop an information-processing architecture, forms of representation, and mechanisms to support perceiving, manipulating, and thinking about the world, especially perceiving and thinking about actual and possible structures and processes in a 3-D environment. The mechanisms for extending those representations and mechanisms, are also the core mechanisms required for developing mathematical competences, especially geometric and topological reasoning competences. Understanding both the natural processes and the requirements for future human-like robots requires AI designers to develop new forms of representation and mechanisms for geometric and topological reasoning to explain a child’s (or robot’s) development of understanding of affordances, and the proto-affordances that underlie them. A suitable multi-functional self-extending architecture will enable those competences to be developed. Within such a machine, human-like mathematical learning will be possible. It is argued that this can support Kant’s philosophy of mathematics, as against Humean philosophies. It also exposes serious limitations in studies of mathematical development by psychologists.

Keywords

learning mathematics philosophy of mathematics robot 3-D vision self-extending architecture epigenetic robotics 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Mill, J.S.: A System of Logic, Ratiocinative and Inductive. John W. Parker, London (1843)Google Scholar
  2. 2.
    Rips, L.J., Bloomfield, A., Asmuth, J.: From Numerical Concepts to Concepts of Number. The Behavioral and Brain Sciences (in press)Google Scholar
  3. 3.
    Heyting, J.: Intuitionism, an Introduction. North Holland, Amsterdam (1956)MATHGoogle Scholar
  4. 4.
    Kant, I.: Critique of Pure Reason. Macmillan, London (1781); (translated by N.K. Smith, 1929)Google Scholar
  5. 5.
    Penrose, R.: The Emperor’s New Mind: Concerning Computers Minds and the Laws of Physics. Oxford University Press, Oxford (1989)Google Scholar
  6. 6.
    Frege, G.: The Foundations of Arithmetic: a logico-mathematical enquiry into the concept of number. B.H. Blackwell, Oxford (1950); (original, 1884)Google Scholar
  7. 7.
    Russell, B.: The Principles of Mathematics. CUP, Cambridge (1903)MATHGoogle Scholar
  8. 8.
    Russell, B.: Mysticism and Logic and Other Essays. Allen & Unwin, London (1917)Google Scholar
  9. 9.
    Lakatos, I.: Proofs and Refutations. CUP, Cambridge (1976)MATHGoogle Scholar
  10. 10.
    Sloman, A.: Necessary, A Priori and Analytic. Analysis 26(1), 12–16 (1965), http://www.cs.bham.ac.uk/research/projects/cogaff/07.html#701 CrossRefGoogle Scholar
  11. 11.
    Sloman, A.: Knowing and Understanding: Relations between meaning and truth, meaning and necessary truth, meaning and synthetic necessary truth. PhD thesis, Oxford University (1962), http://www.cs.bham.ac.uk/research/projects/cogaff/07.html#706
  12. 12.
    Chappell, J., Sloman, A.: Natural and artificial meta-configured altricial information-processing systems. International Journal of Unconventional Computing 3(3), 211–239 (2007), http://www.cs.bham.ac.uk/research/projects/cosy/papers/#tr0609 Google Scholar
  13. 13.
    Sloman, A., Chappell, J.: Computational Cognitive Epigenetics (Commentary on [32]). Behavioral and Brain Sciences 30(4), 375–386 (2007), http://www.cs.bham.ac.uk/research/projects/cosy/papers/#tr0703 CrossRefGoogle Scholar
  14. 14.
    Sloman, A.: The Computer Revolution in Philosophy. Harvester Press (and Humanities Press), Hassocks, Sussex (1978), http://www.cs.bham.ac.uk/research/cogaff/crp
  15. 15.
    Liebeck, P.: How Children Learn Mathematics: A Guide for Parents and Teachers. Penguin Books, Harmondsworth (1984)Google Scholar
  16. 16.
    Sauvy, J., Suavy, S.: The Child’s Discovery of Space: From hopscotch to mazes – an introduction to intuitive topology. Penguin Education, Harmondsworth (1974) (Translated from the French by Pam Wells)Google Scholar
  17. 17.
    Sussman, G.: A computational model of skill acquisition. Elsevier, Amsterdam (1975)Google Scholar
  18. 18.
    Liebeck, P.: Scores and Forfeits: An Intuitive Model for Integer Arithmetic. Educational Studies in Mathematics 21(3), 221–239 (1990)CrossRefGoogle Scholar
  19. 19.
    McCarthy, J., Hayes, P.: Some philosophical problems from the standpoint of AI. In: Meltzer, B., Michie, D. (eds.) Machine Intelligence 4, pp. 463–502. Edinburgh University Press, Edinburgh (1969), http://www-formal.stanford.edu/jmc/mcchay69/mcchay69.html Google Scholar
  20. 20.
    Sloman, A.: Interactions between philosophy and AI: The role of intuition and non-logical reasoning in intelligence. In: Proc 2nd IJCAI, pp. 209–226. William Kaufmann, London (1971), http://www.cs.bham.ac.uk/research/cogaff/04.html#200407 Google Scholar
  21. 21.
    Glasgow, J., Narayanan, H., Chandrasekaran, B. (eds.): Diagrammatic Reasoning: Computational and Cognitive Perspectives. MIT Press, Cambridge (1995)Google Scholar
  22. 22.
    Sloman, A.: Architectural and representational requirements for seeing processes and affordances. Research paper, for Workshop Proceedings COSY-TR-0801, University of Birmingham, UK. School of Computer Science (March 2008), http://www.cs.bham.ac.uk/research/projects/cosy/papers#tr0801
  23. 23.
    Sloman, A.: Putting the Pieces Together Again. In: Sun, R. (ed.) Cambridge Handbook on Computational Psychology. CUP, New York (2008), http://www.cs.bham.ac.uk/research/projects/cogaff/07.html#710 Google Scholar
  24. 24.
    Sloman, A.: On designing a visual system (towards a gibsonian computational model of vision). Journal of Experimental and Theoretical AI 1(4), 289–337 (1989), http://www.cs.bham.ac.uk/research/projects/cogaff/81-95.html#7 CrossRefGoogle Scholar
  25. 25.
    Sloman, A.: Architecture-based conceptions of mind. In: The Scope of Logic, Methodology, and Philosophy of Science (Vol II). Synthese Library, vol. 316, pp. 403–427. Kluwer, Dordrecht (2002), http://www.cs.bham.ac.uk/research/projects/cogaff/00-02.html#57 Google Scholar
  26. 26.
    Sloman, A.: Beyond shallow models of emotion. Cognitive Processing: International Quarterly of Cognitive Science 2(1), 177–198 (2001)Google Scholar
  27. 27.
    Sloman, A.: Evolvable biologically plausible visual architectures. In: Cootes, T., Taylor, C. (eds.) Proceedings of British Machine Vision Conference, Manchester, BMVA, pp. 313–322 (2001)Google Scholar
  28. 28.
    Sloman, A.: Interacting trajectories in design space and niche space: A philosopher speculates about evolution. In: Deb, K., Rudolph, G., Lutton, E., Merelo, J.J., Schoenauer, M., Schwefel, H.-P., Yao, X. (eds.) PPSN 2000. LNCS, vol. 1917, pp. 3–16. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  29. 29.
    Sloman, A.: Diversity of Developmental Trajectories in Natural and Artificial Intelligence. In: Morrison, C.T., Oates, T.T., (eds.) Computational Approaches to Representation Change during Learning and Development, AAAI Fall Symposium 2007. Technical Report FS-07-03, Menlo Park, CA, pp. 70–79. AAAI Press (2007), http://www.cs.bham.ac.uk/research/projects/cosy/papers/#tr0704
  30. 30.
    Sloman, A., Chappell, J.: The Altricial-Precocial Spectrum for Robots. In: Proceedings IJCAI 2005. Edinburgh, IJCAI, pp. 1187–1192 (2005), http://www.cs.bham.ac.uk/research/cogaff/05.html#200502
  31. 31.
    Sloman, A.: The primacy of non-communicative language. In: MacCafferty, M., Gray, K. (eds.) The analysis of Meaning: Informatics 5 Proceedings ASLIB/BCS Conference, March 1979, pp. 1–15. Oxford, London (1979), http://www.cs.bham.ac.uk/research/projects/cogaff/81-95.html#43 Google Scholar
  32. 32.
    Jablonka, E., Lamb, M.J.: Evolution in Four Dimensions: Genetic, Epigenetic, Behavioral, and Symbolic Variation in the History of Life. MIT Press, Cambridge (2005)Google Scholar
  33. 33.
    Sloman, A.: Image interpretation: The way ahead? In: Braddick, O., Sleigh, A. (eds.) Physical and Biological Processing of Images (Proceedings of an international symposium organised by The Rank Prize Funds, London, 1982.), pp. 380–401. Springer, Berlin (1982), http://www.cs.bham.ac.uk/research/projects/cogaff/06.html#0604 Google Scholar
  34. 34.
    Berthoz, A.: The Brain’s sense of movement. Perspectives in Cognitive Science. Harvard University Press, London (2000)Google Scholar
  35. 35.
    Gibson, J.J.: The Ecological Approach to Visual Perception. Houghton Mifflin, Boston (1979)Google Scholar
  36. 36.
    Barrow, H., Tenenbaum, J.: Recovering intrinsic scene characteristics from images. In: Hanson, A., Riseman, E. (eds.) Computer Vision Systems. Academic Press, New York (1978)Google Scholar
  37. 37.
    Marr, D.: Vision. Freeman, San Francisco (1982)Google Scholar
  38. 38.
    Sloman, A.: Actual possibilities. In: Aiello, L., Shapiro, S. (eds.) Principles of Knowledge Representation and Reasoning: Proceedings of the Fifth International Conference (KR 1996), pp. 627–638. Morgan Kaufmann Publishers, Boston (1996)Google Scholar
  39. 39.
    Grush, R.: The emulation theory of representation: Motor control, imagery, and perception. Behavioral and Brain Sciences 27, 377–442 (2004)Google Scholar
  40. 40.
    Whitehead, A.N., Russell, B.: Principia Mathematica, vol. I – III. CUP, Cambridge (1910–1913)MATHGoogle Scholar
  41. 41.
    Feynman, R.: The Character of Physical Law. The 1964 Messenger Lectures. MIT Press, Cambridge (1964)Google Scholar
  42. 42.
    Lenat, D.B., Brown, J.S.: Why AM and EURISKO appear to work. Artificial Intelligence 23(3), 269–294 (1984)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Aaron Sloman
    • 1
  1. 1.School of Computer ScienceUniversity of BirminghamUK

Personalised recommendations