Abstract
Wittgenstein saw a problem with the idea that ‘rule following’ is a transparent process. Here I present an additional problem, based on recent ideas about non-Turing computing. I show that even the simplest algorithm—Frege’s successor function, i.e. counting—cannot by itself determine the ‘output’. Specification of a computing machine is also required.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
Meaning ‘fools’. The ancient Athenians took a dim view of their neighbours in Boeotia.
This is arguably more than just a metaphor/simile. Computers employ space and time = spacetime = spacetime geometry. Thus if Geometry is two-sided, then it is not unreasonable to expect Computability to be two-sided too.
References
Davies EB (2001) Building infinite machines. Br J Philos Sci 52:582–671
Earman J, Norton J (1993) Forever is a day: supertasks in Pitowsky and Malament–Hogarth spacetimes. Philos Sci 5:22–42
Etesi G, Nemeti I (2002) Non-Turing computations via Malament–Hogarth space-times. Int J Theor Phys 41(2):341–370
Hogarth M (1992) Does general relativity allow an observer to view eternity in a finite time? Found Phys Lett 5:173–181
Hogarth M (1994) Non-Turing computers and non-Turing computability. In: Hull D, Forbes M, Okruhlik K (eds) PSA 1994, vol 1. Philosophy of Science Association, East Lansing, pp 126–138
Hogarth M (2004) Deciding arithmetic using SAD computers. Br J Philos Sci 55:681–691
Hogarth M (2009, forthcoming) Non-Turing computers are the new non-Euclidean geometries. Int J Unconv Comput
Nemeti I, David Gy (2006) Relativistic computers and the Turing barrier. J Appl Math Comput 178:118–142
Russell B (1996) An essay on the foundations of geometry. Routledge, London
Torretti R (1978) Philosophy of Geometry from Riemann to Poincaré. D. Reidel Publishing Co, Dordrecht
Wittgenstein L (1953) Philosophical investigations. Blackwell Publishing, Oxford
Author information
Authors and Affiliations
Corresponding author
Appendix
Appendix
The Fig. 1 shows representations of four computing devices. The vertical dimension is time, the horizontal space. A filled dot represents an event; an unfilled dot represents spacetime at ‘infinity’. The unattached filled dot is a typical event on a computer user’s worldline (though the worldline itself is not shown). A line is a worldline of a computer. In (i) the computer stops computing after a finite number of operations. This is the finite Turing machine or FTM. In (ii) the computer never stops computing (but the user can access only a finite number of computational steps). This is the ordinary Turing machine or OTM. In (iii) the computer is underpinned by a so-called Malament–Hogarth spacetime, which permits the user access to an infinite number of steps. This is called a SAD 1 computer (because it can decide arbitrary sentences in arithmetic with one quantifier). In (iv) is a SAD 2, that is a ‘string’ of SAD 1s.
Four different computers, as represented in spacetime
The numbers to the left of each computer are the numbers observed from time to time by a computer user. Of course the ‘numbers’ must be interpreted from the signal data. This holds for 1, 2, etc., but also, in (iii), for ω, which is the interpretation of the absence of a signal. Further details can be found in Hogarth (2004).
Rights and permissions
About this article
Cite this article
Hogarth, M. A new problem for rule following. Nat Comput 8, 493–498 (2009). https://doi.org/10.1007/s11047-009-9116-1
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11047-009-9116-1