INFORMATION, RANDOMNESS & INCOMPLETENESS Papers On
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
21 on Probability and Cosmology: Inference Beyond Data? Martin Sahl´Ena
21 On Probability and Cosmology: Inference Beyond Data? Martin Sahl´ena Slightly expanded version of contribution to the book `The Philosophy of Cos- mology', Khalil Chamcham, Joseph Silk, John D. Barrow, Simon Saunders (eds.), Cambridge University Press, 2017. 21.1 Cosmological Model Inference with Finite Data In physical cosmology we are faced with an empirical context of gradually diminishing returns from new observations. This is true in a fundamental sense, since the amount of information we can expect to collect through as- tronomical observations is finite, owing to the fact that we occupy a particu- lar vantage point in the history and spatial extent of the Universe. Arguably, we may approach the observational limit in the foreseeable future, at least in relation to some scientific hypotheses (Ellis, 2014). There is no guaran- tee that the amount and types of information we are able to collect will be sufficient to statistically test all reasonable hypotheses that may be posed. There is under-determination both in principle and in practice (Zinkernagel, 2011; Butterfield, 2014; Ellis, 2014). These circumstances are not new, in- deed cosmology has had to contend with this problem throughout history. For example, Whitrow (1949) relates the same concerns, and points back to remarks by Blaise Pascal in the 17th century: \But if our view be arrested there let our imagination pass beyond; ... We may enlarge our conceptions beyond all imaginable space; we only produce atoms in comparison with the reality of things." Already with Thales, epistemological principles of unifor- mity and consistency have been used to structure the locally imaginable into something considered globally plausible. -
Can Science Offer an Ultimate Explanation of Reality?
09_FernandoSOLS.qxd:Maqueta.qxd 27/5/14 12:57 Página 685 CAN SCIENCE OFFER AN ULTIMATE EXPLANATION OF REALITY? FERNANDO SOLS Universidad Complutense de Madrid ABSTRACT: It is argued that science cannot offer a complete explanation of reality because of existing fundamental limitations on the knowledge it can provide. Some of those limits are internal in the sense that they refer to concepts within the domain of science but outside the reach of science. The 20th century has revealed two important limitations of scientific knowledge which can be considered internal. On the one hand, the combination of Poincaré’s nonlinear dynamics and Heisenberg’s uncertainty principle leads to a world picture where physical reality is, in many respects, intrinsically undetermined. On the other hand, Gödel’s incompleteness theorems reveal us the existence of mathematical truths that cannot be demonstrated. More recently, Chaitin has proved that, from the incompleteness theorems, it follows that the random character of a given mathematical sequence cannot be proved in general (it is ‘undecidable’). I reflect here on the consequences derived from the indeterminacy of the future and the undecidability of randomness, concluding that the question of the presence or absence of finality in nature is fundamentally outside the scope of the scientific method. KEY WORDS: science, reality, knowledge, scientific method, indeterminacy, Gödel, Chaitin, undecidability, ramdomness, finality. La ciencia, ¿puede ofrecer una explicación última de la realidad? RESUMEN: Se argumenta que la ciencia no puede ofrecer una explicación completa de la realidad debi- do a las limitaciones fundamentales existentes en el conocimiento que puede proporcionar. Algunos de estos límites son internos en el sentido de que se refieren a conceptos dentro del dominio de la cien- cia, pero fuera del alcance de la ciencia. -
Turns in the Evolution of the Problem of Induction*
CARL G. HEMPEL TURNS IN THE EVOLUTION OF THE PROBLEM OF INDUCTION* 1. THE STANDARD CONCEPTION: INDUCTIVE "INFERENCE" Since the days of Hume's skeptical doubt, philosophical conceptions of the problem of induction and of ways in which it might be properly solved or dissolved have undergone a series of striking metamor- phoses. In my paper, I propose to examine some of those turnings, which seem to me to raise particularly important questions about the nature of empirical knowledge and especially scientific knowledge. Many, but by no means all, of the statements asserted by empirical science at a given time are accepted on the basis of previously established evidence sentences. Hume's skeptical doubt reflects the realization that most of those indirectly, or inferentially, accepted assertions rest on evidence that gives them no complete, no logically conclusive, support. This is, of course, the point of Hume's obser- vation that even if we have examined many occurrences of A and have found them all to be accompanied by B, it is quite conceivable, or logically possible, that some future occurrence of A might not be accompanied by B. Nor, we might add, does our evidence guarantee that past or present occurrences of A that we have not observed were- or are- accompanied by B, let alone that all occurrences ever of A are, without exception, accompanied by B. Yet, in our everyday pursuits as well as in scientific research we constantly rely on what I will call the method of inductive ac- ceptance, or MIA for short: we adopt beliefs, or expectations, about empirical matters on logically incomplete evidence, and we even base our actions on such beliefs- to the point of staking our lives on some of them. -
Interpretations of Probability First Published Mon Oct 21, 2002; Substantive Revision Mon Dec 19, 2011
Open access to the Encyclopedia has been made possible, in part, with a financial contribution from the Australian National University Library. We gratefully acknowledge this support. Interpretations of Probability First published Mon Oct 21, 2002; substantive revision Mon Dec 19, 2011 ‘Interpreting probability’ is a commonly used but misleading characterization of a worthy enterprise. The so-called ‘interpretations of probability’ would be better called ‘analyses of various concepts of probability’, and ‘interpreting probability’ is the task of providing such analyses. Or perhaps better still, if our goal is to transform inexact concepts of probability familiar to ordinary folk into exact ones suitable for philosophical and scientific theorizing, then the task may be one of ‘explication’ in the sense of Carnap (1950). Normally, we speak of interpreting a formal system , that is, attaching familiar meanings to the primitive terms in its axioms and theorems, usually with an eye to turning them into true statements about some subject of interest. However, there is no single formal system that is ‘probability’, but rather a host of such systems. To be sure, Kolmogorov's axiomatization, which we will present shortly, has achieved the status of orthodoxy, and it is typically what philosophers have in mind when they think of ‘probability theory’. Nevertheless, several of the leading ‘interpretations of probability’ fail to satisfy all of Kolmogorov's axioms, yet they have not lost their title for that. Moreover, various other quantities that have nothing to do with probability do satisfy Kolmogorov's axioms, and thus are interpretations of it in a strict sense: normalized mass, length, area, volume, and other quantities that fall under the scope of measure theory, the abstract mathematical theory that generalizes such quantities. -
Probability Disassembled John D
Brit. J. Phil. Sci. 58 (2007), 141–171 Probability Disassembled John D. Norton ABSTRACT While there is no universal logic of induction, the probability calculus succeeds as a logic of induction in many contexts through its use of several notions concerning inductive inference. They include Addition, through which low probabilities represent disbelief as opposed to ignorance; and Bayes property, which commits the calculus to a ‘refute and rescale’ dynamics for incorporating new evidence. These notions are independent and it is urged that they be employed selectively according to needs of the problem at hand. It is shown that neither is adapted to inductive inference concerning some indeterministic systems. 1 Introduction 2 Failure of demonstrations of universality 2.1 Working backwards 2.2 The surface logic 3 Framework 3.1 The properties 3.2 Boundaries 3.2.1 Universal comparability 3.2.2 Transitivity 3.2.3 Monotonicity 4 Addition 4.1 The property: disbelief versus ignorance 4.2 Boundaries 5 Bayes property 5.1 The property 5.2 Bayes’ theorem 5.3 Boundaries 5.3.1 Dogmatism of the priors 5.3.2 Impossibility of prior ignorance 5.3.3 Accommodation of virtues 6 Real values 7 Sufficiency and independence The Author (2007). Published by Oxford University Press on behalf of British Society for the Philosophy of Science. All rights reserved. doi:10.1093/bjps/axm009 For Permissions, please email: [email protected] Advance Access published on May 23, 2007 142 John D. Norton 8 Illustrations 8.1 All properties retained 8.2 Bayes property only retained 8.3 Induction without additivity and Bayes property 9 Conclusion 1 Introduction No single idea about induction1 has been more fertile than the idea that inductive inferences may conform to the probability calculus. -
The Limits of Reason
Ideas on complexity and randomness originally suggested by Gottfried W. Leibniz in 1686, combined with modern information theory, imply that there can never be a “theory of everything” for all of mathematics By Gregory Chaitin 74 SCIENTIFIC AMERICAN M A RCH 2006 COPYRIGHT 2006 SCIENTIFIC AMERICAN, INC. The Limits of Reason n 1956 Scientific American published an article by Ernest Nagel and James R. Newman entitled “Gödel’s Proof.” Two years later the writers published a book with the same title—a wonderful Iwork that is still in print. I was a child, not even a teenager, and I was obsessed by this little book. I remember the thrill of discovering it in the New York Public Library. I used to carry it around with me and try to explain it to other children. It fascinated me because Kurt Gödel used mathematics to show that mathematics itself has limitations. Gödel refuted the position of David Hilbert, who about a century ago declared that there was a theory of everything for math, a finite set of principles from which one could mindlessly deduce all mathematical truths by tediously following the rules of symbolic logic. But Gödel demonstrated that mathematics contains true statements that cannot be proved that way. His result is based on two self- referential paradoxes: “This statement is false” and “This statement is un- provable.” (For more on Gödel’s incompleteness theorem, see www.sciam. com/ontheweb) My attempt to understand Gödel’s proof took over my life, and now half a century later I have published a little book of my own. -
Mind in Leibnizian Metaphysics
1 A Computational Theory of World: Mind in Leibnizian Metaphysics Natalie Hastie Student number: 30454393 Bachelor of Arts in Philosophy with Honours 2 STATEMENT OF PRESENTATION The thesis is presented for the Honours degree of Bachelor of Arts in Philosophy at Murdoch University. 2014 I declare that this thesis is my own account of my research and contains, as its main content, work that has not previously been submitted for a degree at any tertiary educational institutions, including Murdoch. Signed: ______________________________________________ Full Name: ___________________________________________ Student Number: ______________________________________ Date: ________________________________________________ 3 COPYRIGHT ACKNOWLEDGEMENT I acknowledge that a copy of this thesis will be held at the Murdoch University Library. I understand that, under the provisions of s51.2 of the Copyright Act 1968, all or part of this thesis may be copied without infringement of copyright where such a reproduction is for the purposes of study and research. This statement does not signal any transfer of copyright away from the author. Signed: …………………………………………………………... Full Name of Degree: …………………………………………………………………... e.g. Bachelor of Science with Honours in Chemistry. Thesis Title: …………………………………………………………………... …………………………………………………………………... …………………………………………………………………... …………………………………………………………………... Author: …………………………………………………………………... Year: ………………………………………………………………....... 4 ABSTRACT Computational theory of mind (CTM) is a dominant model found in much of -
Critical Thinking
Critical Thinking Mark Storey Bellevue College Copyright (c) 2013 Mark Storey Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.3 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is found at http://www.gnu.org/copyleft/fdl.txt. 1 Contents Part 1 Chapter 1: Thinking Critically about the Logic of Arguments .. 3 Chapter 2: Deduction and Induction ………… ………………. 10 Chapter 3: Evaluating Deductive Arguments ……………...…. 16 Chapter 4: Evaluating Inductive Arguments …………..……… 24 Chapter 5: Deductive Soundness and Inductive Cogency ….…. 29 Chapter 6: The Counterexample Method ……………………... 33 Part 2 Chapter 7: Fallacies ………………….………….……………. 43 Chapter 8: Arguments from Analogy ………………………… 75 Part 3 Chapter 9: Categorical Patterns….…….………….…………… 86 Chapter 10: Propositional Patterns……..….…………...……… 116 Part 4 Chapter 11: Causal Arguments....……..………….………....…. 143 Chapter 12: Hypotheses.….………………………………….… 159 Chapter 13: Definitions and Analyses...…………………...…... 179 Chapter 14: Probability………………………………….………199 2 Chapter 1: Thinking Critically about the Logic of Arguments Logic and critical thinking together make up the systematic study of reasoning, and reasoning is what we do when we draw a conclusion on the basis of other claims. In other words, reasoning is used when you infer one claim on the basis of another. For example, if you see a great deal of snow falling from the sky outside your bedroom window one morning, you can reasonably conclude that it’s probably cold outside. Or, if you see a man smiling broadly, you can reasonably conclude that he is at least somewhat happy. -
Explication of Inductive Probability
J Philos Logic (2010) 39:593–616 DOI 10.1007/s10992-010-9144-4 Explication of Inductive Probability Patrick Maher Received: 1 September 2009 / Accepted: 15 March 2010 / Published online: 20 August 2010 © Springer Science+Business Media B.V. 2010 Abstract Inductive probability is the logical concept of probability in ordinary language. It is vague but it can be explicated by defining a clear and precise concept that can serve some of the same purposes. This paper presents a general method for doing such an explication and then a particular explication due to Carnap. Common criticisms of Carnap’s inductive logic are examined; it is shown that most of them are spurious and the others are not fundamental. Keywords Inductive probability · Explication · Carnap 1 Introduction The word “probability” in ordinary English has two senses, which I call induc- tive probability and physical probability. I will begin by briefly reviewing the main features of these concepts; for a more extensive discussion see [27, 28]. The arguments of inductive probability are two propositions and we speak of the probability of proposition Hgivenproposition E; I will abbreviate this as ip(H|E). For example, if E is that a coin is either two-headed or two- tailed and is about to be tossed, while H is that the coin will land heads, then plausibly ip(H|E) = 1/2. By contrast, the arguments of physical probability are an experiment type and an outcome type, and we speak of the probability of an experiment of type X having an outcome of type O; I will abbreviate this as ppX (O). -
The Algorithmic Revolution in the Social Sciences: Mathematical Economics, Game Theory and Statistics
The Algorithmic Revolution in the Social Sciences: Mathematical Economics, Game Theory and Statistics K. Vela Velupillai Department of Economics University of Trento Via Inama 5 381 00 Trento Italy August 9, 2009 Prepared for an Invited Lecture to be given at the Workshop on Information Theoretic Methods in Science and Engineering (WITMSE), August 17-19, 2009, Tampere, Finland. I am most grateful to Jorma Rissanen and Ioan Tabus for inviting me, an economist, to give a talk at an event with experts and specialists who are pioneers in information theory. 1 Abstract The digital and information technology revolutions are based on algorith- mic mathematics in many of their alternative forms. Algorithmic mathemat- ics per se is not necessarily underpinned by the digital or the discrete only; analogue traditions of algorithmic mathematics have a noble pedigree, even in economics. Constructive mathematics of any variety, computability theory and non-standard analysis are intrinsically algorithmic at their foundations. Eco- nomic theory, game theory and mathematical …nance theory, at many of their frontiers, appear to have embraced the digital and information technology revo- lutions via strong adherences to experimental, behavioural and so-called compu- tational aspects of their domains –without, however, adapting the mathemati- cal formalisms of their theoretical structures. Recent advances in mathematical economics, game theory, probability theory and statistics suggest that an al- gorithmic revolution in the social sciences is in the making. In this paper I try to trace the origins of the emergence of this ‘revolution’ and suggest, via examples in mathematical economics, game theory and the foundations of sta- tistics, where the common elements are and how they may de…ne new frontiers of research and visions. -
APA Newsletters NEWSLETTER on PHILOSOPHY and COMPUTERS
APA Newsletters NEWSLETTER ON PHILOSOPHY AND COMPUTERS Volume 10, Number 1 Fall 2010 FROM THE EDITOR, PETER BOLTUC FROM THE CHAIR, DAN KOLAK ARTICLES SUSAN A. J. STUART “Enkinaesthesia: Reciprocal Affective Felt Enfolding, a Further Challenge for Machine Consciousness” GREGORY CHAITIN “Metaphysics, Metamathematics, and Metabiology” RICARDO SANZ “Machines among Us: Minds and the Engineering of Control Systems” THOMAS W. POLGER “Toward a Distributed Computation Model of Extended Cognition” COLIN ALLEN, JAIMIE MURDOCK, CAMERON BUCKNER, SCOTT WEINGART “An API for Philosophy” © 2010 by The American Philosophical Association ISSN 2155-9708 APA NEWSLETTER ON Philosophy and Computers Piotr Bołtuć, Editor Fall 2010 Volume 10, Number 1 author argues that we could do a much better job if we could ROM THE DITOR design a human being out of scratch! Incidentally (talking about F E controversial claims!), both Chaitin and Stuart seem to belong to an exceedingly small group of serious authors who seem to question the standard interpretations, or applications, of the Piotr Boltuc Church-Turing thesis. University of Illinois at Springfield The second block of articles comes from the invited session of this Committee at the Central Division APA meeting On the first pages of his recent book Steven Hawkins1 argues that in Chicago in 2009. Ricardo Sanz, one of the top EU experts philosophers, entangled in their narrow theoretical framework in robotics, discusses the issue of minds in the engineering of methodology and philosophy of language, are no longer of control systems. To put it somewhat crudely, Sanz argues relevant in explaining the traditionally philosophical issues that controlled systems require a mind since they need to essential to understanding humans’ role in the universe. -
Chaitin Chats with Kurt Gödel
The Chaitin Interview I: Chaitin Chats with Kurt Gödel https://mindmatters.ai/podcast/ep124 Robert J. Marks: We talk today with the great mathematician and computer scientist, Gregory Chaitin, on Mind Matters News. Announcer: Welcome to Mind Matters News, where artificial and natural intelligence meet head on. Here's your host, Robert J. Marks. Robert J. Marks: There are few people who can be credited without any controversy with the founding of a game changing field of mathematics. We are really fortunate today to talk to Gregory Chaitin who has that distinction. Professor Chaitin is a co-founder of the Field of Algorithmic Information Theory that explores the properties of computer programs. Professor Chaitin is the recipient of a handful of honorary doctorates for his landmark work and is a recipient of the prestigious Leibniz Medal. Professor Chaitin, welcome. Gregory Chaitin: Great to be with you today. Robert J. Marks: Thank you. Algorithmic Information Theory, when people first hear the term, at least at the lay level, their reaction is the field might be pretty dry and boring. But I tell them that Algorithmic Information Theory, if you understand it, is more mind-blowing than any science fiction I've ever read or watched. So I hope in our chats that we can convey some of the mind-blowing results springing out of Algorithmic Information Theory. Robert J. Marks: So before diving in, I hope you don't mind if I get a little personal. Your life has changed recently. You have two new children in your life. Juan, and how did I do in the pronunciation? That's a rough name to pronounce.