Abstract
The property of being the implementation of a computational structure has been argued to be vacuously instantiated. This claim provides the basis for most antirealist arguments in the field of the philosophy of computation. Standard manoeuvres for combating these antirealist arguments treat the problem as endogenous to computational theories. The contrastive analysis of computational and other mathematical representations put forward here reveals that the problem should instead be treated within the more general framework of the Newman problem in structuralist accounts of mathematical representation. It is argued that purely structuralist and purely functionalist accounts of implementation are inadequate to tackle the problem. An extensive evaluation of semantic accounts is provided, arguing that semantic properties are, unlike structural and functional ones, suitable to restrict the intended domain of implementation of computational properties in such a way as to block the Newman problem. The semantic hypothesis is defended from a number of recent objections.
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Notes
In particular, they can be argued to belong to a restricted subclass of mathematical dynamical systems. These are abstract structures represented by triplets of the form: \( < T,M,\{ g^{t} \} >, \) where T is the time set, M is the state space and {g t} is the set of state transitions \( \{ g^{t} \} :M \to M \). Abstract computational systems can be argued to belong to this class of mathematical structures. The following is an example of how this can be proved in the case of Turing machines (see Giunti 1995). The future behaviour of a Turing machine is determined when: (1) the state of the internal control unit, (2) the symbol on the tape scanned by the head and (3) the position of the head, are given. It is therefore possible to take the set of all triplets <state, head-position, tape-content> as the state space M of the system; the set of non-negative integers can be taken as the (machine) time set T; the set of quadruples of the machine is then used to form the set {g t} of state transitions. Each state transition function is such that {g 0} is the identity function on M, and \( g^{t + 1} (x) = g(g^{t} (x)) \). Similar considerations apply to all other computational structures.
Following a standard use, the term “implementation” here denotes specifically the realization of computational structures. When referring to the realization of generic mathematical structures (not necessarily computational), I shall use the term “instantiation”.
Attempts to ground the notion of implementation on that of step satisfaction (see for example Cummins 1989: 91–92), for instance, or on the digital vs. analogous distinction (see Haugeland 1981), or on the instantiation of fundamental vs. derivative physical laws (Block and Fodor 1972) have all been argued to fail to make a case for computational realism. For an extensive survey of the various proposals and of their shortcomings see Shagrir (1997).
The technical result, moreover, has been argued to be extendable to the claim that any open physical system implements any automaton. An argument to this effect can be found in Scheutz (1999).
See Melia and Saatsi (2006) for an extensive discussion of the various options.
Cantor's theorem is of course true also in Skolem's countable model.
This does not contradict the thesis that functional accounts are unsuitable, alone, to ground the notion of implementation. The essential role played by functional, mechanistic properties in individuating the relevant computational properties is, I argue, a practical, not a metaphysical one. Here we are concerned with necessary and sufficient conditions for the implementation of computational structures. See the conclusive remarks at the end of this paper for a discussion of the relation that obtains between the various accounts of implementations presented here.
For a detailed analysis of the problem of liberalism in functionalist accounts see Block 1980 and Weir (2001). Weir argues that functionalist accounts based on D. Lewis’ analysis fall prey of the chauvinistic horn of the dilemma, while other accounts fail to block excessively liberal implementations.
See the next section of this paper for a discussion of the philosophical implications of Marr’s theory of vision.
The discovery of this algorithm is due to J. Buntrock and H. Marxen.
The symbols R and L refer to the familiar actions of moving the read/write head one cell to the right or one to the left, respectively.
Indeed it could be argued that we acquire the representations of the numerals, and learn to manipulate them, long before we come to understand what the numerals are intended to represent.
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I am indebted to Andrea Bianchi, Peter Clark, Carl Hoefer, the editors of the LSE Philosophy Papers, and two anonymous referees, for useful comments and interesting discussions on previous drafts of this paper. An earlier version of this paper was presented at the LOGOS talks at the University of Barcelona (Spain), and at the Institito de Investigaciones Filosoficas of the Universidad Nacional Autonoma de México, thanks to the audience for their comments.
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Boccardi, E. Who’s Driving the Syntactic Engine?. J Gen Philos Sci 40, 23–50 (2009). https://doi.org/10.1007/s10838-009-9085-1
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DOI: https://doi.org/10.1007/s10838-009-9085-1