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  • Print publication year: 2016
  • Online publication date: March 2016

15 - On Attempting to Model the Mathematical Mind

from Part Five - Oracles, Infinitary Computation, and the Physics of the Mind

Summary

Abstract

In his important 1939 paper, Alan Turing introduced novel notions such as ordinal logics and oracle machines. These could be interpreted as possible ingredients of an approach to model human mathematical understanding in a way that goes beyond the conventional ideas of formal systems of axioms and rules of procedure. A hope appears to have been that in this way one might circumvent the limitations to formal reasoning that are revealed by Gödel's incompleteness theorems. In line with such aims, an idea of a cautious oracle device is here introduced (differing, in intention, from related ideas put forward by others previously), which is supposed to give accurate answers to mathematical questions whenever it claims to have an answer, but which may sometimes confess to being unable to provide an answer and sometimes continues trying indefinitely without success. Despite such devices seeming to be somewhat closer to human mathematical capabilities than appears to be provided by a standard Turing machine, or Turing oracle machine, they are still limited by being subject to a Gödel-type diagonalization argument. Although leaving open the question of what actual physical processes might underlie human mathematical insight, these arguments appear to indicate a significant constraint on any such hypothetical process.

Turing's ordinal logics

In early September 1955, I attended a lecture given by Max Newman on the topic of ordinal logic. I found the lecture to be one of the most fascinating that I ever attended. Alan Turing had died only a little over a year earlier and this talk was dedicated to him, being essentially based on Turing's 1939 paper on this topic. Newman also started his lecture by providing, as Turing had done in his paper, an introduction to Church's λ -calculus. It has been said that the somewhat limited initial impact that Turing's 1939 paper had on the mathematical community at that time may have been partly due to his phrasing the paper in terms of the λ -calculus, which is hard to employ in an explicit way and makes the reading difficult. Nonetheless, one of the things that did strike me particularly about Newman's lecture was the extraordinary economy of concept exhibited by Church's calculus.

Cooper, S.B. (1990). Enumeration reducibility, nondeterministic computations and relative computability of partial functions. In Recursion Theory Week, Oberwolfach 1989, K., Ambos-Spies, G., Müller, G. E., Sacks (eds.), Springer-Verlag, pp. 57–110.
Cooper, S.B. (2004). Computability Theory_a1. Chapman and Hall.
Copeland, B.J. (2002). Accelerating Turing Machines. Minds and Machines 12(2), 281– 300.
Copeland, B.J. and Proudfoot, D. (2000). What Turing did after he invented the universal Turing machine. J. Logic, Language and Information 9(4), 491–509.
Deutsch, D. (1985). Quantum theory, the Church–Turing principle and the universal quantum computer. Proc. Roy. Soc. Lond. A 400, 97–117.
Feferman, S. (1962). Transfinite recursive progressions of axiomatic theories. J. Symb. Log. 27, 259–316.
Feferman, S. (1988). Turing in the land of O(z). In The Universal Turing Machine: A Half- Century Survey, R, Herken (ed.), Kammerer and Unverzagt.
Hagan, S., Hameroff, S. and Tuszynski, J. (2002). Quantum computation in brain microtubules? Decoherence and biological feasibility. Phys. Rev. E 65, 061901.
Hameroff, S.R. and Penrose, R. (1996). Conscious events as orchestrated space–time selections. J. Consciousness Studies 3, 36–63.
Hameroff, S. and Penrose, R. (2014). Consciousness in the universe: a review of the ‘Orch OR’ theory. Phys. Life Rev. 11 (1), 39–78 (also 104–112).
Hodges, A.P. (1983). Alan Turing: The Enigma, Burnett Books and Hutchinson; Simon and Schuster.
Hodges, A.P. (1988). Alan Turing and the Turing Machine. In The Universal Turing Machine: A Half-Century Survey, R., Herken (ed.), Kammerer and Unverzagt.
Kleene, S.C. (1952). Introduction to Metamathematics. North-Holland.
Lucas, J.R. (1961). Minds, machines and Gödel. Philosophy 36, 120–124; reprinted in Alan Ross Anderson (1964), Minds and Machines, Prentice–Hall.
Nagel, E. and Newman, J.R. (1958). Gödel's Proof, Routledge and Kegan Paul.
Penrose, R. (1989). The Emperor's New Mind: Concerning Computers, Minds, and the Laws of Physics, Oxford University Press.
Penrose, R. (1994). Shadows of the Mind: An Approach to the Missing Science of Consciousness, Oxford University Press.
Penrose, R. (1996). Beyond the doubting of a shadow. Psyche 2(23), 89–129. Also available at http:psyche.cs.monash.edu.au/psyche-index-v2_1.html.
Penrose, R. (2000). Reminiscences of Christopher Strachey. Higher-Order Symb. Comp. 13, 83–84.
Penrose, R. (2011a). Gödel, the mind, and the laws of physics. In Kurt Gödel and the Foundations of Mathematics: Horizons of Truth, M., Baaz, C.H., Papadimitriou, H.W., Putnam, D.S., Scott, and C.L., Harper, Jr. (eds.), Cambridge University Press.
Penrose, R. (2011b). Mathematics, the mind, and the laws of physics. In Meaning in Mathematics, John Polkinghorne (ed.), Oxford University Press.
Penrose, R. (2011c). Uncertainty in quantum mechanics: faith or fantasy?Phil. Trans. Roy. Soc. A 369, 4864–4890.
Penrose, R. (2014). On the gravitization of quantum mechanics 1: quantum state reduction. Found. Phys. 44, 557–575.
Sahu, S, Ghosh, S, Ghosh, B, Aswani, K, Hirata, K, Fujita, D, and Bandyopadhyay, A. (2013a). Atomic water channel controlling remarkable properties of a single brain microtubule: correlating single protein to its supramolecular assembly. Biosens. Bioelectron 47, 141–148.
Sahu, S, Ghosh, S, Hirata, K, Fujita, D, and Bandyopadhyay, A. (2013b). Multi-level memory switching properties of a single brain microtubule. Appl. Phys. Lett., 102, 123701.
Tegmark, M. (2000). Importance of quantum coherence in brain processes. Phys. Rev. E 61,4194–4206.
Turing, A.M. (1937a). On computable numbers, with an application to the Entscheidungsproblem. Proc. Lond. Math. Soc. (ser. 2) 42, 230–265; a correction, 43, 544–546.
Turing, A.M. (1937b). Computability and λ-definability. J. Symb. Log. 2, 153–163.
Turing, A.M. (1939). Systems of logic based on ordinals. Proc. Lond. Math. Soc. 45(2), 161–228.
Turing, A.M. (1950). Computing machinery and intelligence. Mind 59, 236.