Online research in philosophy

Recent work on hypercomputation has raised new objections against the Church–Turing Thesis. In this paper, I focus on the challenge posed by a particular kind of hypercomputer, namely, SAD computers. I first consider deterministic and probabilistic barriers to the physical possibility of SAD computation. These suggest several ways to defend a Physical version of the Church–Turing Thesis. I then argue against Hogarth's analogy between non-Turing computability and non-Euclidean geometry, showing that it is a non-sequitur. I conclude that the Effective version of the Church–Turing Thesis is unaffected by SAD computation.

A. M. Turing has bequeathed us a conceptulary including 'Turing, or Turing-Church, thesis', 'Turing machine', 'universal Turing machine', 'Turing test' and 'Turing structures', plus other unnamed achievements. These include a proof that any formal language adequate to express arithmetic contains undecidable formulas, as well as achievements in computer science, artificial intelligence, mathematics, biology, and cognitive science. Here it is argued that these achievements hang together and have prospered well in the 50 years since Turing's death.

Accelerating Turing machines are Turing machines of a sort able to perform tasks that are commonly regarded as impossible for Turing machines. For example, they can determine whether or not the decimal representation of contains n consecutive 7s, for any n; solve the Turing-machine halting problem; and decide the predicate calculus. Are accelerating Turing machines, then, logically impossible devices? I argue that they are not. There are implications concerning the nature of effective procedures and the theoretical limits of computability. Contrary to a recent paper by Bringsjord, Bello and Ferrucci, however, the concept of an accelerating Turing machine cannot be used to shove up Searle's Chinese room argument.

We sketch the historical and conceptual context of Turing's analysis of algorithmic or mechanical computation. We then discuss two responses to that analysis, by Gödel and by Gandy, both of which raise, though in very different ways. The possibility of computation procedures that cannot be reduced to the basic procedures into which Turing decomposed computation. Along the way, we touch on some of Cleland's views.

There are various equivalent formulations of the Church-Turing thesis. A common one is that every effective computation can be carried out by a Turing machine. The Church-Turing thesis is often misunderstood, particularly in recent writing in the philosophy of mind.

Since the mid-twentieth century, the concept of the Turing machine has dominated thought about effective procedures. This paper presents an alternative to Turing's analysis; it unifies, refines, and extends my earlier work on this topic. I show that Turing machines cannot live up to their billing as paragons of effective procedure; at best, they may be said to provide us with mere procedure schemas. I argue that the concept of an effective procedure crucially depends upon distinguishing procedures as definite courses of action(- types) from the particular courses of action(-tokens) that actually instantiate them and the causal processes and/or interpretations that ultimately make them effective. On my analysis, effectiveness is not just a matter of logical form; `content' matters. The analysis I provide has the advantage of applying to ordinary, everyday procedures such as recipes and methods, as well as the more refined procedures of mathematics and computer science. It also has the virtue of making better sense of the physical possibilities for hypercomputation than the received view and its extensions, e.g. Turing's o-machines, accelerating machines.

Turing's (1936) analysis of effective symbolic procedures is a model of conceptual clarity that plays an essential role in the philosophy of mathematics. Yet appeal is often made to the effectiveness of human procedures in other areas of philosophy. This paper addresses the question of whether Turing's analysis can be applied to a broader class of effective human procedures. We use Sieg's (1994) presentation of Turing's Thesis to argue against Cleland's (1995) objections to Turing machines and we evaluate her proposal to understand the effectiveness of procedures in terms of their reliability and precision. A number of conditions for effectiveness are identified and these are used to provide a general argument against the possibility of a Leibnizian decision procedure.

This paper explores Church's Thesis and related claims madeby Turing. Church's Thesis concerns computable numerical functions, whileTuring's claims concern both procedures for manipulating uninterpreted marksand machines that generate the results that these procedures would yield. Itis argued that Turing's claims are true, and that they support (the truth of)Church's Thesis. It is further argued that the truth of Turing's and Church'sTheses has no interesting consequences for human cognition or cognitiveabilities. The Theses don't even mean that computers can do as much as peoplecan when it comes to carrying out effective procedures. For carrying out aprocedure is a purposive, intentional activity. No actual machine does, orcan do, as much.

The Church-Turing thesis makes a bold claim about the theoretical limits to computation. It is based upon independent analyses of the general notion of an effective procedure proposed by Alan Turing and Alonzo Church in the 1930''s. As originally construed, the thesis applied only to the number theoretic functions; it amounted to the claim that there were no number theoretic functions which couldn''t be computed by a Turing machine but could be computed by means of some other kind of effective procedure. Since that time, however, other interpretations of the thesis have appeared in the literature. In this paper I identify three domains of application which have been claimed for the thesis: (1) the number theoretic functions; (2) all functions; (3) mental and/or physical phenomena. Subsequently, I provide an analysis of our intuitive concept of a procedure which, unlike Turing''s, is based upon ordinary, everyday procedures such as recipes, directions and methods; I call them mundane procedures. I argue that mundane procedures can be said to be effective in the same sense in which Turing machine procedures can be said to be effective. I also argue that mundane procedures differ from Turing machine procedures in a fundamental way, viz., the former, but not the latter, generate causal processes. I apply my analysis to all three of the above mentioned interpretations of the Church-Turing thesis, arguing that the thesis is (i) clearly false under interpretation (3), (ii) false in at least some possible worlds (perhaps even in the actual world) under interpretation (2), and (iii) very much open to question under interpretation (1).

Horsten and Roelants have raised a number of important questions about my analysis of effective procedures and my evaluation of the Church-Turing thesis. They suggest that, on my account, effective procedures cannot enter the mathematical world because they have a built-in component of causality, and, hence, that my arguments against the Church-Turing thesis miss the mark. Unfortunately, however, their reasoning is based upon a number of misunderstandings. Effective mundane procedures do not, on my view, provide an analysis of ourgeneral concept of an effective procedure; mundane procedures and Turing machine procedures are different kinds of procedure. Moreover, the same sequence ofparticular physical action can realize both a mundane procedure and a Turing machine procedure; it is sequences of particular physical actions, not mundane procedures, which enter the world of mathematics. I conclude by discussing whether genuinely continuous physical processes can enter the world of real numbers and compute real-valued functions. I argue that the same kind of correspondence assumptions that are made between non-numerical structures and the natural numbers, in the case of Turing machines and personal computers, can be made in the case of genuinely continuous, physical processes and the real numbers.