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Predictability, Computability and Spacetime
, 2002
"... thesis is the result of the author’s own work and includes nothing which is the outcome of work done in collaboration. To my Mum and Dad, who succeeded in violating Larkin’s Law. And to my sister Lyn, who recently stopped pulling my hair. Acknowledgements The following have personally helped to shap ..."
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thesis is the result of the author’s own work and includes nothing which is the outcome of work done in collaboration. To my Mum and Dad, who succeeded in violating Larkin’s Law. And to my sister Lyn, who recently stopped pulling my hair. Acknowledgements The following have personally helped to shape the ideas in the thesis: Gordon Belot,
Gödel on computability
"... Around 1950, both Gödel and Turing wrote papers for broader audiences. 1 Gödel drew in his 1951 dramatic philosophical conclusions from the general formulation of his second incompleteness theorem. These conclusions concerned the nature of mathematics and the human mind. The general formulation of t ..."
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Around 1950, both Gödel and Turing wrote papers for broader audiences. 1 Gödel drew in his 1951 dramatic philosophical conclusions from the general formulation of his second incompleteness theorem. These conclusions concerned the nature of mathematics and the human mind. The general formulation of the second theorem was explicitly based on Turing’s 1936 reduction of finite procedures to machine computations. Turing gave in his 1954 an understated analysis of finite procedures in terms of Post production systems. This analysis, prima facie quite different from that given in 1936, served as the basis for an exposition of various unsolvable problems. Turing had addressed issues of mentality and intelligence in contemporaneous essays, the best known of which is of course Computing machinery and intelligence. Gödel’s and Turing’s considerations from this period intersect through their attempt, on the one hand, to analyze finite, mechanical procedures and, on the other hand, to approach mental phenomena in a scientific way. Neuroscience or brain science was an important component of the latter for both: Gödel’s remarks in the Gibbs Lecture as well as in his later conversations with Wang and Turing’s Intelligent Machinery can serve as clear evidence for that. 2 Both men were convinced that some mental processes are not mechanical, in the sense that Turing machines cannot mimic them. For Gödel, such processes were to be found in mathematical experience and he was led to the conclusion that mind is separate from matter. Turing simply noted that for a machine or a brain it is not enough to be converted into a universal (Turing) machine in order to become intelligent: “discipline”, the characteristic
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 ..."
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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.
Computing machines can’t be intelligent (...and Turing said so
 In Minds and Machines
, 2002
"... According to the conventional wisdom, Turing (1950) said that computing machines can be intelligent. I don’t believe it. I think that what Turing really said was that computing machines – computers limited to computing – can only fake intelligence. If we want computers to become genuinely intelligen ..."
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According to the conventional wisdom, Turing (1950) said that computing machines can be intelligent. I don’t believe it. I think that what Turing really said was that computing machines – computers limited to computing – can only fake intelligence. If we want computers to become genuinely intelligent, we will have to give them enough “initiative ” (Turing, 1948, p. 21) to do more than compute. In this paper, I want to try to develop this idea. I want to explain how giving computers more “initiative ” can allow them to do more than compute. And I want to say why I believe (and believe that Turing believed) that they will have to go beyond computation before they can become genuinely intelligent. 1. What I Think Turing Said People who try to make computers more intelligent say they are trying to produce “Artificial Intelligence ” (or “AI”). Presumably, they want the word “artificial ” to suggest that the intelligence they are trying to create will – like artificial vanilla – not have developed naturally. But some of their critics are convinced that anything that looks like intelligence in a computer will have to be artificial in another sense – the sense in which an artificial smile is artificial. Which is to say fake. Computers, they believe, cannot be genuinely intelligent because they lack a certain je ne sais quoi that genuine intelligence requires. The more extreme of these critics believe that what computers lack is fundamental. Perhaps they believe that intelligence requires an immortal soul. Perhaps they feel that it can only be implemented in flesh and blood. Perhaps they believe that it requires human experiences or human emotions. Such critics believe that computers cannot be genuinely intelligent, period. Other critics of AI are a bit more generous. They believe that computers cannot be genuinely intelligent until … Perhaps they believe that computers cannot be genuinely intelligent until they have access to better parallel processing or to special neural
The role of heuristics in automated theorem proving – J.A. Robinson’s resolution principle
 Mathware & Soft Computing
, 1996
"... with infinite events and changes, it is impossible one doesn't write, at least one time, Odyssey (J.L. Borges, The Aleph) Abstract The aim of this paper is to show how J.A. Robinson's resolution principle was perceived and discussed in the AI community between the mid sixties and the first seventies ..."
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with infinite events and changes, it is impossible one doesn't write, at least one time, Odyssey (J.L. Borges, The Aleph) Abstract The aim of this paper is to show how J.A. Robinson's resolution principle was perceived and discussed in the AI community between the mid sixties and the first seventies. During this time the so called "heuristic search paradigm " was still influential in the AI community, and both resolution principle and certain resolution based, apparently humanlike, search strategies were matched with those problem solving heuristic procedures which were representative of the AI heuristic search paradigm. 1
Gödel's Path from the Incompleteness Theorems (1931) to Phenomenology (1961)
, 1998
"... this paper I present and discuss the kinds of arguments that led G odel to the work of Husserl. Among other things, this should help to shed additional light on G odel's philosophical and scientific ideas and to show to what extent these ideas can be viewed as part of a unified philosophical outlook ..."
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this paper I present and discuss the kinds of arguments that led G odel to the work of Husserl. Among other things, this should help to shed additional light on G odel's philosophical and scientific ideas and to show to what extent these ideas can be viewed as part of a unified philosophical outlook. Some of the arguments that led G odel to Husserl's work are only hinted at in G odel's 1961 paper, but they are developed in much more detail in G odel's earlier philosophical papers (see especially 1934, *193?, 1944, 1947, *1951, *1953/59). In particular, I focus on arguments concerning Hilbert's program and an early version of Carnap's program.
Gödel's incompleteness theorems and artificial life
, 1997
"... In this paper I discuss whether Gödel's incompleteness theorems have any implications for studies in Artificial Life (AL). Since Gödel's incompleteness theorems have been used to argue against certain mechanistic theories of the mind, it seems natural to attempt to apply the theorems to certain stro ..."
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In this paper I discuss whether Gödel's incompleteness theorems have any implications for studies in Artificial Life (AL). Since Gödel's incompleteness theorems have been used to argue against certain mechanistic theories of the mind, it seems natural to attempt to apply the theorems to certain strong mechanistic arguments postulated by some AL theorists. We find that an argument using the incompleteness theorems can not be constructed that will block the hard AL claim, specifically in the field of robotics. However, we will see that the beginnings of an argument casting doubt on our ability to create living systems entirely resident in a computer environment might be suggested by looking at the incompleteness theorems from the point of view of Gödel's belief in mathematical realism.
Skolem and Gödel
, 1996
"... ... and Gödel (1906–1978) are the two greatest logicians of the century. Yet their styles, philosophies, and careers are strikingly different. Gödel had already published some of his great works and had become world renowned by the time he was 25 years of age. Skolem began to publish his important p ..."
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... and Gödel (1906–1978) are the two greatest logicians of the century. Yet their styles, philosophies, and careers are strikingly different. Gödel had already published some of his great works and had become world renowned by the time he was 25 years of age. Skolem began to publish his important papers only after he was 30, and his impact grew slowly over the years. Gödel was meticulous in writing for publication and published little after he reached 45. Skolem wrote informally, often even casually, continuing to publish into the last days of his life. Gödel was a wellknown absolutist and Platonist who had devoted much effort to studying and writing philosophy. Skolem was inclined to finitism and relativism, and rarely attempted to offer an articulate presentation of his coherent and fruitful philosophical viewpoint about the nature of mathematics and mathematical activity. Apart from mathematical logic, Gödel made contributions to the philosophy of mathematics and to fundamental physics. Skolem divided his work almost equally between logic and other parts of discrete mathematics, particularly algebra and number theory. For many years I have been deeply involved with Gödel’s work and his life. Even though I was for a long time intensely interested in Skolem’s work in logic and made a careful study of it in the sixties, since then I have not followed carefully the important applications and developments of Skolem’s ideas by many logicians. I know very little about his life and his work in fields other than logic. In my opinion, there is much room for interesting and instructive studies of Skolem’s work and his life. One attraction for me in coming here to give the