Skip to main content
SpringerLink
Log in

Kantian Philosophy of Mathematics and Young Robots

  • Conference paper
Intelligent Computer Mathematics (CICM 2008)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 5144))

Included in the following conference series:

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.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Mill, J.S.: A System of Logic, Ratiocinative and Inductive. John W. Parker, London (1843)

    Google Scholar 

  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. Heyting, J.: Intuitionism, an Introduction. North Holland, Amsterdam (1956)

    MATH  Google Scholar 

  4. Kant, I.: Critique of Pure Reason. Macmillan, London (1781); (translated by N.K. Smith, 1929)

    Google Scholar 

  5. Penrose, R.: The Emperor’s New Mind: Concerning Computers Minds and the Laws of Physics. Oxford University Press, Oxford (1989)

    Google Scholar 

  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. Russell, B.: The Principles of Mathematics. CUP, Cambridge (1903)

    MATH  Google Scholar 

  8. Russell, B.: Mysticism and Logic and Other Essays. Allen & Unwin, London (1917)

    Google Scholar 

  9. Lakatos, I.: Proofs and Refutations. CUP, Cambridge (1976)

    MATH  Google Scholar 

  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

    Article  Google Scholar 

  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. 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. 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

    Article  Google Scholar 

  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. Liebeck, P.: How Children Learn Mathematics: A Guide for Parents and Teachers. Penguin Books, Harmondsworth (1984)

    Google Scholar 

  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. Sussman, G.: A computational model of skill acquisition. Elsevier, Amsterdam (1975)

    Google Scholar 

  18. Liebeck, P.: Scores and Forfeits: An Intuitive Model for Integer Arithmetic. Educational Studies in Mathematics 21(3), 221–239 (1990)

    Article  Google Scholar 

  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. 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. Glasgow, J., Narayanan, H., Chandrasekaran, B. (eds.): Diagrammatic Reasoning: Computational and Cognitive Perspectives. MIT Press, Cambridge (1995)

    Google Scholar 

  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. 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. 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

    Article  Google Scholar 

  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. Sloman, A.: Beyond shallow models of emotion. Cognitive Processing: International Quarterly of Cognitive Science 2(1), 177–198 (2001)

    Google Scholar 

  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. 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)

    Chapter  Google Scholar 

  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. 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. 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. 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. 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. Berthoz, A.: The Brain’s sense of movement. Perspectives in Cognitive Science. Harvard University Press, London (2000)

    Google Scholar 

  35. Gibson, J.J.: The Ecological Approach to Visual Perception. Houghton Mifflin, Boston (1979)

    Google Scholar 

  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. Marr, D.: Vision. Freeman, San Francisco (1982)

    Google Scholar 

  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. Grush, R.: The emulation theory of representation: Motor control, imagery, and perception. Behavioral and Brain Sciences 27, 377–442 (2004)

    Google Scholar 

  40. Whitehead, A.N., Russell, B.: Principia Mathematica, vol. I – III. CUP, Cambridge (1910–1913)

    MATH  Google Scholar 

  41. Feynman, R.: The Character of Physical Law. The 1964 Messenger Lectures. MIT Press, Cambridge (1964)

    Google Scholar 

  42. Lenat, D.B., Brown, J.S.: Why AM and EURISKO appear to work. Artificial Intelligence 23(3), 269–294 (1984)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Computer Science, University of Birmingham, UK

    Aaron Sloman

Authors
  1. Aaron Sloman
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Serge Autexier John Campbell Julio Rubio Volker Sorge Masakazu Suzuki Freek Wiedijk

Rights and permissions

Reprints and permissions

Copyright information

© 2008 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Sloman, A. (2008). Kantian Philosophy of Mathematics and Young Robots. In: Autexier, S., Campbell, J., Rubio, J., Sorge, V., Suzuki, M., Wiedijk, F. (eds) Intelligent Computer Mathematics. CICM 2008. Lecture Notes in Computer Science(), vol 5144. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-85110-3_45

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-85110-3_45

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-85109-7

  • Online ISBN: 978-3-540-85110-3

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation