Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Hostname: page-component-8448b6f56d-gtxcr Total loading time: 0 Render date: 2024-04-19T11:50:16.017Z Has data issue: false hasContentIssue false

15 - On Attempting to Model the Mathematical Mind

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

Published online by Cambridge University Press:  05 March 2016

By
Roger Penrose
Edited by
S. Barry Cooper  and
Andrew Hodges
Roger Penrose
Affiliation:
University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK
S. Barry Cooper
Affiliation:
University of Leeds
Andrew Hodges
Affiliation:
University of Oxford
Get access

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.

Type
Chapter
Information
The Once and Future Turing
Computing the World
 , pp. 361 - 378
Publisher: Cambridge University Press
Print publication year: 2016

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

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.Google Scholar
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.Google Scholar
Deutsch, D. (1985). Quantum theory, the Church–Turing principle and the universal quantum computer. Proc. Roy. Soc. Lond. A 400, 97–117.Google Scholar
Feferman, S. (1962). Transfinite recursive progressions of axiomatic theories. J. Symb. Log. 27, 259–316.CrossRefGoogle Scholar
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.Google Scholar
Hameroff, S.R. and Penrose, R. (1996). Conscious events as orchestrated space–time selections. J. Consciousness Studies 3, 36–63.Google Scholar
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).Google Scholar
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.Google Scholar
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.Google Scholar
Penrose, R. (2000). Reminiscences of Christopher Strachey. Higher-Order Symb. Comp. 13, 83–84.CrossRefGoogle Scholar
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.Google Scholar
Penrose, R. (2014). On the gravitization of quantum mechanics 1: quantum state reduction. Found. Phys. 44, 557–575.CrossRefGoogle Scholar
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.Google Scholar
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.Google Scholar
Tegmark, M. (2000). Importance of quantum coherence in brain processes. Phys. Rev. E 61,4194–4206.Google Scholar
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.Google Scholar
Turing, A.M. (1937b). Computability and λ-definability. J. Symb. Log. 2, 153–163.Google Scholar
Turing, A.M. (1939). Systems of logic based on ordinals. Proc. Lond. Math. Soc. 45(2), 161–228.CrossRefGoogle Scholar
Turing, A.M. (1950). Computing machinery and intelligence. Mind 59, 236.Google Scholar

Save book to Kindle

To save this book to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×