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The Berry Paradox
, 1994
"... was Godel's secretary. She said that Godel was very careful about his health and because of the snow he wasn't coming to the Institute that day and therefore my appointment was canceled. And that's how I had two phone conversations with Godel but never met him. I never tried again. I'd like to tell ..."
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Cited by 17 (1 self)
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was Godel's secretary. She said that Godel was very careful about his health and because of the snow he wasn't coming to the Institute that day and therefore my appointment was canceled. And that's how I had two phone conversations with Godel but never met him. I never tried again. I'd like to tell you what I would have told Godel. What I wanted to tell Godel is the difference between what you get when you study the limits of mathematics the way Godel did using the paradox of the liar, and what I get using the Berry paradox instead. What is the paradox of the liar? Well, the paradox of the liar is "This statement is false!" Why is this a paradox? What does "false" mean? Well, "false" means "does not correspond to reality." This statement says that it is false. If that doesn't correspond to reality, it must mean that the statement is true, right? On the other hand, if the statement is true it means that what it says corresponds to reality. But it says that it is false. Therefore the sta
A New Version Of Algorithmic Information Theory
, 1995
"... Algorithmic information theory may be viewed as the result of adding the idea of programsize complexity to recursive function theory. The main application of algorithmic information theory is its informationtheoretic incompleteness theorems. This theory is concerned with the size of programs, but ..."
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Algorithmic information theory may be viewed as the result of adding the idea of programsize complexity to recursive function theory. The main application of algorithmic information theory is its informationtheoretic incompleteness theorems. This theory is concerned with the size of programs, but up to now these have never been programs that one could actually program out and run on interesting examples. I have now figured out how to actually program the algorithms in the proofs of the all the key informationtheoretic incompleteness theorems in algorithmic information theory. I have published this material electronically in a series of detailed reports [1,2,3,4]
Pocket Mathematics
, 1995
"... Mathematics is in a dramatic and massive process of changing, mainly due to the advent of computers and computer science. Our aim is to present a pocket image of this phenomenon; a "case study" will give us the opportunity to describe some of these new ideas, problems, and techniques. Particularly, ..."
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Mathematics is in a dramatic and massive process of changing, mainly due to the advent of computers and computer science. Our aim is to present a pocket image of this phenomenon; a "case study" will give us the opportunity to describe some of these new ideas, problems, and techniques. Particularly, we will be concerned with foreseeable mutations in the interaction between deductive and experimental trends.
What’s experimental about experimental mathematics? ∗
, 2008
"... From a philosophical viewpoint, mathematics has often and traditionally been distinguished from the natural sciences by its formal nature and emphasis on deductive reasoning. Experiments — one of the corner stones of most modern natural science — have had no role to play in mathematics. However, dur ..."
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From a philosophical viewpoint, mathematics has often and traditionally been distinguished from the natural sciences by its formal nature and emphasis on deductive reasoning. Experiments — one of the corner stones of most modern natural science — have had no role to play in mathematics. However, during the last three decades, high speed computers and sophisticated software packages such as Maple and Mathematica have entered into the domain of pure mathematics, bringing with them a new experimental flavor. They have opened up a new approach in which computerbased tools are used to experiment with the mathematical objects in a dialogue with more traditional methods of formal rigorous proof. At present, a subdiscipline of experimental mathematics is forming with its own research problems, methodology, conferences, and journals. In this paper, I first outline the role of the computer in the mathematical experiment and briefly describe the impact of high speed computing on mathematical research within the emerging subdiscipline of experimental mathematics. I then consider in more detail the epistemological claims put forward within experimental mathematics and comment on some of the discussions that experimental mathematics has provoked within the mathematical community in recent years. In the second part of the paper, I suggest the notion of exploratory experimentation as a possible framework for understanding experimental mathematics. This is illustrated by discussing the socalled PSLQ algorithm.
On the ChurchTuring Thesis
, 2008
"... After a brief description of the ChurchTuring Thesis, we suggest that, according to the latest results on classical recursive probabilistic solution of the Halting Problem, such thesis is asymptotically false. 1 ..."
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After a brief description of the ChurchTuring Thesis, we suggest that, according to the latest results on classical recursive probabilistic solution of the Halting Problem, such thesis is asymptotically false. 1
Asymptotic behavior and halting probability of Turing Machines ∗
, 2006
"... Through a straightforward Bayesian approach we show that under some general conditions a maximum running time, namely the number of discrete steps performed by a computer program during its execution, can be defined such that the probability that such a program will halt after that time is smaller t ..."
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Through a straightforward Bayesian approach we show that under some general conditions a maximum running time, namely the number of discrete steps performed by a computer program during its execution, can be defined such that the probability that such a program will halt after that time is smaller than any arbitrary fixed value. Consistency with known results and consequences are also discussed. 1 Introductory remarks As it has been proved by Turing in 1936 [1], if we have a program p running on an Universal Turing Machine (UTM), then we have no general, finite and deterministic algorithm which allows us to know whether and when it will halt (this is the well known halting problem). That is to say that the halting behavior of a program, with the trivial exception of the simplest ones, is not computable and predictable by a unique, general procedure. In this paper we show that, for what concerns the probability of its halt, every program running on an UTM is characterized by a peculiar asymptotic behavior in time. Similar results have been obtained by Calude et al. [2] and by Adamyan et al. [3] through a different approach, which makes use of quantum computation.
HOW TO RUN ALGORITHMIC INFORMATION THEORY
, 1995
"... Hi everybody! It’s a great pleasure for me to be back here at the new, improved Santa Fe Institute in this spectacular location. I guess this is my fourth visit and it’s always very stimulating, so I’m always very happy to visit you guys. I’d like to tell you what I’ve been up to lately. First of al ..."
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Hi everybody! It’s a great pleasure for me to be back here at the new, improved Santa Fe Institute in this spectacular location. I guess this is my fourth visit and it’s always very stimulating, so I’m always very happy to visit you guys. I’d like to tell you what I’ve been up to lately. First of all, let me say what algorithmic information theory is good for, before telling you about the new version of it I’ve got. 1
A NEW VERSION OF ALGORITHMIC
, 1995
"... Algorithmic information theory may be viewed as the result of adding the idea of programsize complexity to recursive function theory. The main application of algorithmic information theory is its informationtheoretic incompleteness theorems. This theory is concerned with the ..."
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Algorithmic information theory may be viewed as the result of adding the idea of programsize complexity to recursive function theory. The main application of algorithmic information theory is its informationtheoretic incompleteness theorems. This theory is concerned with the
THE BERRY PARADOX
, 1994
"... videotaped; this is an edited transcript. It also incorporates remarks made at the Limits to Scientific Knowledge meeting held at the Santa ..."
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videotaped; this is an edited transcript. It also incorporates remarks made at the Limits to Scientific Knowledge meeting held at the Santa
1.4. On Gödel’s theorem and Algorithmic Complexity........... 20
, 704
"... Information, complexity, brains and reality (Kolmogorov Manifesto) ..."