Skip to main content
Log in

Abstract

I use modal logic and transfinite set-theory to define metaphysical foundations for a general theory of computation. A possible universe is a certain kind of situation; a situation is a set of facts. An algorithm is a certain kind of inductively defined property. A machine is a series of situations that instantiates an algorithm in a certain way. There are finite as well as transfinite algorithms and machines of any degree of complexity (e.g., Turing and super-Turing machines and more). There are physically and metaphysically possible machines. There is an iterative hierarchy of logically possible machines in the iterative hierarchy of sets. Some algorithms are such that machines that instantiate them are minds. So there is an iterative hierarchy of finitely and transfinitely complex minds.

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

References

  • Benacerraf, P. (1962), 'Tasks, Super-Tasks, and the Modern Eleatics', Journal of Philosophy 59, pp. 765–784.

    Google Scholar 

  • Blum, L., Cucker, F., Shub, M. and Smale, S. (1998), Complexity and Real Computation, NewYork: Springer.

    Google Scholar 

  • Boolos, G. and Jeffrey, R. (1980), Computability and Logic (2nd Edition), New York: Cambridge University Press.

    Google Scholar 

  • Boyce,W. and DiPrima, R. (1977), Elementary Differential Equations (3rd Edition), New York: John Wiley & Sons.

    Google Scholar 

  • Chalmers, D. (1996), 'Does a Rock Implement Every Finite-State Automaton?', Synthese 108, pp. 309–333.

    Google Scholar 

  • Cleland, C. (1993), 'Is the Church-Turing Thesis True?, Minds and Machines 3, pp. 283–312.

    Google Scholar 

  • Cleland, C. (1995), 'Effective Procedures and Computable Functions', Minds and Machines 5, pp. 9–23.

    Google Scholar 

  • Colburn, T. (1999), 'Software, Abstraction, and Ontology', The Monist 82, pp. 3–20.

    Google Scholar 

  • Copeland, B. J. and Sylvan, R. (1999), 'Beyond the Universal Turing Machine',' Australasian Journal of Philosophy 77, pp. 46–66.

    Google Scholar 

  • Copeland, B. J. (1996), 'What is Computation?', Synthese 108, pp. 335–359.

    Google Scholar 

  • Copeland, B. J. (1998a), Super-Turing Machines', Complexity 4, pp. 30–32.

    Google Scholar 

  • Copeland, B. J. (1998b), 'Even Turing Machines can Compute Uncomputable Functions',' in C. Calude, J. Casti and M. Dinneen, eds., Unconventional models of Computation, New York: Springer.

    Google Scholar 

  • Cresswell, M. J. (1972), 'The World is Everything That is the Case', in M. Loux, Ed., (1979) The Possible and the Actual, Ithaca, NY: Cornell University Press, pp. 129–145.

    Google Scholar 

  • Deutsch, D. (1985), 'Quantum Theory, the Church-Turing Principle and the Universal Quantum Computer', Proceedings of the Royal Society, Series A, 400, pp. 97–117.

    Google Scholar 

  • Devlin, K. (1991), The Joy of Sets: Fundamentals of Contemporary Set Theory, New York: Springer.

    Google Scholar 

  • Doyle, J. (1991), The Foundations of Psychology: A Logico-Computational Inquiry into the Concept of Mind' in R. Cummins and J. Pollock, eds., Philosophy and AI: Essays at the Interface, Cambridge, MA: MIT Press, pp. 39–78.

    Google Scholar 

  • Drake, F. (1974), Set Theory: An Introduction to Large Cardinals, New York: American Elsevier.

    Google Scholar 

  • Gandy, R. (1980), 'Church's Thesis and Principles for Mechanisms', in J. Barwise, H. Keisler and K. Kunen, eds., The Kleene Symposium, New York: North-Holland, pp. 123–148.

    Google Scholar 

  • Giunti, M. (1997), Computation, Dynamics, and Cognition, New York: Oxford University Press.

    Google Scholar 

  • Grunbaum, A. (1969), 'Can an Infinitude of Operations be Performed in a Finite Time?', British Journal of the Philosophy of Science 20, pp. 203–218.

    Google Scholar 

  • Hamkins, J. D. and Lewis, A. (2000),' Infinite Time Turing Machines', Journal of Symbolic Logic 65, pp. 567–604.

    Google Scholar 

  • Hogarth, M. (1994), 'Non-Turing Computers and non-Turing Computability', Philosophy of Science Association 1, pp. 126–138.

    Google Scholar 

  • Johnson, G. (2000), Untitled article. New York Times, Sunday, 31 December 2000, Sec. 4 (Week in Review), p. 4.

  • Koetsier, T. and Allis, V. (1997), 'Assaying Supertasks', Logique et Analyse 159, pp. 291–313.

    Google Scholar 

  • Lewis, D. (1973), 'Causation', Journal of Philosophy 70, pp. 556–567.

    Google Scholar 

  • Lewis, D. (1986), On the Plurality of Worlds, Cambridge, MA: Blackwell.

    Google Scholar 

  • Mandlebrot, B. (1978), Fractals: Form, Chance, and Dimension, San Francisco, CA: W. H. Freeman.

    Google Scholar 

  • Moravec, H. (1988), Mind Children: The Future of Robot and Human Intelligence, Cambridge, MA: Harvard University Press.

    Google Scholar 

  • Pendlebury, M. (1986), 'Facts as Truthmakers', The Monist 69, pp. 177–188.

    Google Scholar 

  • Poundstone, W. (1985), The Recursive Universe: Cosmic Complexity and the Limits of Scientific Knowledge, Chicago, IL: Contemporary Books, Inc.

    Google Scholar 

  • Putnam, H. (1988), Representation and Reality, Cambridge, MA: MIT Press.

    Google Scholar 

  • Quine, W. V. (1969), Ontological Relativity and Other Essays, New York: Columbia University Press.

    Google Scholar 

  • Quine, W. V. (1976), 'Wither physical objects?' Boston Studies in the Philosophy of Science 39, pp. 497–504.

    Google Scholar 

  • Quine, W. V. (1978), 'Facts of the matter', Southwestern Journal of Philosophy 9, pp. 155–169.

    Google Scholar 

  • Quine, W. V. (1981), Theories and Things, Cambridge, MA: Harvard University Press.

    Google Scholar 

  • Royce, J. (1899/1927), TheWorld and the Individual (First Series; Supplementary Essay), New York: Macmillan. Lectures originally delivered 1899.

    Google Scholar 

  • Royce, J. (1900/1976), The World and the Individual (Second Series), Gloucester, MA: Peter Smith. Lectures originally delivered 1900.

    Google Scholar 

  • Rucker, R. (1995), Infinity and the Mind, Princeton, NJ: Princeton University Press.

    Google Scholar 

  • Salmon, W. (1984), 'Causal connections', in J. Kim and E. Sosa, eds., (1999) Metaphysics, Malden, MA: Blackwell, pp. 444–457.

    Google Scholar 

  • Scheutz, M. (1999), 'When Physical Systems Realize Functions', Minds and Machines 9, pp. 161–196.

    Google Scholar 

  • Searle, J. (1990), 'Is the Brain a Digital Computer?', Proceedings and Addresses of the American Philosophical Association 64, pp. 21–37.

    Google Scholar 

  • Shagrir, O. (1997), 'Two Dogmas of Computationalism', Minds and Machines 7, pp. 321–344.

    Google Scholar 

  • Sieg,W. and Byrnes, J. (1999), 'An Abstract Model for Parallel Computations: Gandy's Thesis', The Monist 82, pp. 150–164.

    Google Scholar 

  • Siegelmann, H. (1996), 'The Simple Dynamics of Super Turing Theories', Theoretical Computer Science 168, pp. 461–472.

    Google Scholar 

  • Sloman, A. (1993), 'The Mind as a Control System', in C. Hookway and D. Peterson, eds., Philosophy and Cognitive Science, Royal Institute of Philosophy Supplement 34, New York: Cambridge University Press.

    Google Scholar 

  • Stillings, N., Weisler, S., Chase, C., Feinstein, M., Garfield, J. and Rissland, E. (1995), Cognitive Science: An Introduction (2nd Edition), Cambridge, MA: MIT Press.

    Google Scholar 

  • Takeuti, G. (1985), 'Proof Theory and Set Theory', Synthese 62, pp. 255–263.

    Google Scholar 

  • Thomas, G., Finney, R. and Weir, M. (1998) Calculus and Analytic Geometry (9th Edition), Reading, MA: Addison-Wesley.

    Google Scholar 

  • Thomson, J. (1954),' Tasks and Supertasks', Analysis 15, pp. 1–13.

    Google Scholar 

  • Toffoli, T. and Margolus, N. (1987), Cellular Automata Machines: A New Environment for Modeling, Cambridge, MA: MIT Press.

    Google Scholar 

  • Weyl, H. (1963), Philosophy of Mathematics and Natural Science, New York: Atheneum. (Original work in German 1927).

    Google Scholar 

  • Wilczek, F. (1999), 'Getting its from bits', Nature 397, pp. 303–306.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Steinhart, E. Logically Possible Machines. Minds and Machines 12, 259–280 (2002). https://doi.org/10.1023/A:1015603317236

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1015603317236

Navigation