Results 1  10
of
11
RELATIVISTIC COMPUTERS AND THE TURING Barrier
, 2006
"... We examine the current status of the physical version of the ChurchTuring Thesis (PhCT for short) in view of latest developments in spacetime theory. This also amounts to investigating the status of hypercomputation in view of latest results on spacetime. We agree with Deutsch et al [17] that PhCT ..."
Abstract

Cited by 22 (10 self)
 Add to MetaCart
We examine the current status of the physical version of the ChurchTuring Thesis (PhCT for short) in view of latest developments in spacetime theory. This also amounts to investigating the status of hypercomputation in view of latest results on spacetime. We agree with Deutsch et al [17] that PhCT is not only a conjecture of mathematics but rather a conjecture of a combination of theoretical physics, mathematics and, in some sense, cosmology. Since the idea of computability is intimately connected with the nature of Time, relevance of spacetime theory seems to be unquestionable. We will see that recent developments in spacetime theory show that temporal developments may exhibit features that traditionally seemed impossible or absurd. We will see that recent results point in the direction that the possibility of artificial systems computing nonTuring computable functions may be consistent with spacetime theory. All these trigger new open questions and new research directions for spacetime theory, cosmology, and computability.
Alan Turing and the Mathematical Objection
 Minds and Machines 13(1
, 2003
"... Abstract. This paper concerns Alan Turing’s ideas about machines, mathematical methods of proof, and intelligence. By the late 1930s, Kurt Gödel and other logicians, including Turing himself, had shown that no finite set of rules could be used to generate all true mathematical statements. Yet accord ..."
Abstract

Cited by 4 (2 self)
 Add to MetaCart
Abstract. This paper concerns Alan Turing’s ideas about machines, mathematical methods of proof, and intelligence. By the late 1930s, Kurt Gödel and other logicians, including Turing himself, had shown that no finite set of rules could be used to generate all true mathematical statements. Yet according to Turing, there was no upper bound to the number of mathematical truths provable by intelligent human beings, for they could invent new rules and methods of proof. So, the output of a human mathematician, for Turing, was not a computable sequence (i.e., one that could be generated by a Turing machine). Since computers only contained a finite number of instructions (or programs), one might argue, they could not reproduce human intelligence. Turing called this the “mathematical objection ” to his view that machines can think. Logicomathematical reasons, stemming from his own work, helped to convince Turing that it should be possible to reproduce human intelligence, and eventually compete with it, by developing the appropriate kind of digital computer. He felt it should be possible to program a computer so that it could learn or discover new rules, overcoming the limitations imposed by the incompleteness and undecidability results in the same way that human mathematicians presumably do. Key words: artificial intelligence, ChurchTuring thesis, computability, effective procedure, incompleteness, machine, mathematical objection, ordinal logics, Turing, undecidability The ‘skin of an onion ’ analogy is also helpful. In considering the functions of the mind or the brain we find certain operations which we can express in purely mechanical terms. This we say does not correspond to the real mind: it is a sort of skin which we must strip off if we are to find the real mind. But then in what remains, we find a further skin to be stripped off, and so on. Proceeding in this way, do we ever come to the ‘real ’ mind, or do we eventually come to the skin which has nothing in it? In the latter case, the whole mind is mechanical (Turing, 1950, p. 454–455). 1.
General relativistic hypercomputing and foundation of mathematics
"... Abstract. Looking at very recent developments in spacetime theory, we can wonder whether these results exhibit features of hypercomputation that traditionally seemed impossible or absurd. Namely, we describe a physical device in relativistic spacetime which can compute a nonTuring computable task, ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
Abstract. Looking at very recent developments in spacetime theory, we can wonder whether these results exhibit features of hypercomputation that traditionally seemed impossible or absurd. Namely, we describe a physical device in relativistic spacetime which can compute a nonTuring computable task, e.g. which can decide the halting problem of Turing machines or decide whether ZF set theory is consistent (more precisely, can decide the theorems of ZF). Starting from this, we will discuss the impact of recent breakthrough results of relativity theory, black hole physics and cosmology to well established foundational issues of computability theory as well as to logic. We find that the unexpected, revolutionary results in the mentioned branches of science force us to reconsider the status of the physical Church Thesis and to consider it as being seriously challenged. We will outline the consequences of all this for the foundation of mathematics (e.g. to Hilbert’s programme). Observational, empirical evidence will be quoted to show that the statements above do not require any assumption of some physical universe outside of our own one: in our specific physical universe there seem to exist regions of spacetime supporting potential nonTuring computations. Additionally, new “engineering ” ideas will be outlined for solving the socalled blueshift problem of GRcomputing. Connections with related talks at the Physics and Computation meeting, e.g. those of Jerome DurandLose, Mark Hogarth and Martin Ziegler, will be indicated. 1
The Narrational Case against Church's Thesis
 Journal of Philosophy
, 1998
"... this paper presented at the 1993 Eastern APA meetings in Atlanta  comments which I incorporate in my response to Mendelson's response. I'm grateful to Michael McMenamin for providing, in his unpublished "Deciding Uncountable Sets and Church's Thesis," an excellent objection to my attack on Church ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
this paper presented at the 1993 Eastern APA meetings in Atlanta  comments which I incorporate in my response to Mendelson's response. I'm grateful to Michael McMenamin for providing, in his unpublished "Deciding Uncountable Sets and Church's Thesis," an excellent objection to my attack on Church's Thesis (which I rebut below).
A natural axiomatization of Church’s thesis
, 2007
"... The Abstract State Machine Thesis asserts that every classical algorithm is behaviorally equivalent to an abstract state machine. This thesis has been shown to follow from three natural postulates about algorithmic computation. Here, we prove that augmenting those postulates with an additional requ ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
The Abstract State Machine Thesis asserts that every classical algorithm is behaviorally equivalent to an abstract state machine. This thesis has been shown to follow from three natural postulates about algorithmic computation. Here, we prove that augmenting those postulates with an additional requirement regarding basic operations implies Church’s Thesis, namely, that the only numeric functions that can be calculated by effective means are the recursive ones (which are the same, extensionally, as the Turingcomputable numeric functions). In particular, this gives a natural axiomatization of Church’s Thesis, as Gödel and others suggested may be possible.
Can new physics challenge “old ” computational barriers?
"... Abstract. We discuss the impact of very recent developments of spacetime theory, black hole physics, and cosmology to well established foundational issues of computability theory and logic. Namely, we describe a physical device in relativistic spacetime which can compute a nonTuring computable task ..."
Abstract
 Add to MetaCart
Abstract. We discuss the impact of very recent developments of spacetime theory, black hole physics, and cosmology to well established foundational issues of computability theory and logic. Namely, we describe a physical device in relativistic spacetime which can compute a nonTuring computable task, e.g. which can decide the halting problem of Turing machines or whether ZF set theory is consistent or not. Connections with foundation of mathematics and foundation of spacetime theory will be discussed. 1
From the Closed Universe to an Open World
"... Abstract. There are different aspects and spheres of unconventional computations. In this paper, we analyze philosophical and methodological implications of algorithmic issues of unconventional computations. At first, we describe how the algorithmic universe was developed and analyze why it became c ..."
Abstract
 Add to MetaCart
Abstract. There are different aspects and spheres of unconventional computations. In this paper, we analyze philosophical and methodological implications of algorithmic issues of unconventional computations. At first, we describe how the algorithmic universe was developed and analyze why it became closed in the conventional approach to computation. Then we explain how the new models of algorithms changed the algorithmic universe, making it open and allowing higher flexibility and superior creativity. As Gödel undecidability theorems imply, the closed algorithmic universe restricts essential forms of human cognition, while the open algorithmic universe eliminates such restrictions. 1
GUALTIERO PICCININI COMPUTATIONALISM, THE CHURCH–TURING THESIS, AND THE CHURCH–TURING FALLACY
"... ABSTRACT. The Church–Turing Thesis (CTT) is often employed in arguments for computationalism. I scrutinize the most prominent of such arguments in light of recent work on CTT and argue that they are unsound. Although CTT does nothing to support computationalism, it is not irrelevant to it. By elimin ..."
Abstract
 Add to MetaCart
ABSTRACT. The Church–Turing Thesis (CTT) is often employed in arguments for computationalism. I scrutinize the most prominent of such arguments in light of recent work on CTT and argue that they are unsound. Although CTT does nothing to support computationalism, it is not irrelevant to it. By eliminating misunderstandings about the relationship between CTT and computationalism, we deepen our appreciation of computationalism as an empirical hypothesis. Computationalism, or the Computational Theory of Mind, is the view that mental capacities are explained by inner computations. In the case of human beings, computationalists typically assume that inner computations are realized by neural processes; I will borrow a term from current neuroscience and refer to them as neural computations. 1 Typically, computationalists also maintain that neural computations are Turingcomputable, that is, computable by Turing Machines (TMs). The Church–Turing thesis (CTT) says that a function is computable, in the intuitive sense, if and only if it is Turingcomputable (Church 1936; Turing 1936–7). CTT entails that TMs, and any formalism equivalent to TMs, capture the intuitive notion of computation. In other words, according to CTT, if a function is computable in the intuitive sense, then there is a TM that computes it (or equivalently, it is Turingcomputable). 2 This applies to neural computations as well. Suppose that, as computationalism maintains, neural activity is computation, and suppose that the functions computed by neural mechanisms are computable in the intuitive sense. Then, by CTT, for any function computed by a neural mechanism, there is a TM that computes the same function. This is a legitimate argument for a technical version of computationalism, according to which neural computations are Turingcomputable, from a generic one, according to which neural processes are computations in the intuitive sense, via CTT. But should we believe CTT? The initial proponents of CTT, and most of CTT’s supporters, appeal to a number of intuitive