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Open problems in the philosophy of information
 Metaphilosophy
"... Abstract: The philosophy of information (PI) is a new area of research with its own field of investigation and methodology. This article, based on the Herbert A. Simon Lecture of Computing and Philosophy I gave at Carnegie Mellon University in 2001, analyses the eighteen principal open problems in P ..."
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Abstract: The philosophy of information (PI) is a new area of research with its own field of investigation and methodology. This article, based on the Herbert A. Simon Lecture of Computing and Philosophy I gave at Carnegie Mellon University in 2001, analyses the eighteen principal open problems in PI. Section 1 introduces the analysis by outlining Herbert Simon’s approach to PI. Section 2 discusses some methodological considerations about what counts as a good philosophical problem. The discussion centers on Hilbert’s famous analysis of the central problems in mathematics. The rest of the article is devoted to the eighteen problems. These are organized into five sections: problems in the analysis of the concept of information, in semantics, in the study of intelligence, in the relation between information and nature, and in the investigation of values.
SINBAD Neurosemantics: A theory of mental representation. Mind
 Brain & Mind
, 2001
"... Abstract: I present an account of mental representation based upon the ‘SINBAD’ theory of the cerebral cortex. If the SINBAD theory is correct, then networks of pyramidal cells in the cerebral cortex are appropriately described as representing, or more specifically, as modelling the world. I propose ..."
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Abstract: I present an account of mental representation based upon the ‘SINBAD’ theory of the cerebral cortex. If the SINBAD theory is correct, then networks of pyramidal cells in the cerebral cortex are appropriately described as representing, or more specifically, as modelling the world. I propose that SINBAD representation reveals the nature of the kind of mental representation found in human and animal minds, since the cortex is heavily implicated in these kinds of minds. Finally, I show how SINBAD neurosemantics can provide accounts of misrepresentation, equivocal representation, twin cases, and Frege cases. 1.
Mathematics and logic as information compression by multiple alignment, unification and search
"... This article introduces the conjecture that mathematics, logic and related disciplines may usefully be understood as information compression by ‘multiple alignment’, ‘unification ’ and ‘search’. As a preparation for the two main sections of the article, concepts of HartleyShannon information theory ..."
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This article introduces the conjecture that mathematics, logic and related disciplines may usefully be understood as information compression by ‘multiple alignment’, ‘unification ’ and ‘search’. As a preparation for the two main sections of the article, concepts of HartleyShannon information theory and Algorithmic Information Theory are briefly reviewed together with a summary of the essentials of the simpler ‘standard ’ methods for information compression. Related areas of research are briefly described: philosophical connections, information compression (IC) in brains and nervous systems, and IC in relation to inductive inference, Minimum Length Encoding and probabilistic reasoning. Then the concepts of information compression by ‘multiple alignment’, ‘unification ’ and ‘search ’ (ICMAUS) are outlined together with a brief description of the SP61 computer model that is a partial realisation of the ICMAUS framework. The first of the two main sections describes how many of the commonlyused forms and structures in mathematics, logic and related disciplines (such as theoretical linguistics and computer programming) may be seen as devices for IC. In some cases, these forms and structures may be interpreted in terms of the ICMAUS framework. The second main section describes a selection of examples where processes of calculation and inference in mathematics, logic and related disciplines may be understood as IC. In many cases, these examples may be understood more specifically in terms of the ICMAUS concepts and illustrated with output from the SP61 model. 1 Associated issues are briefly discussed. Abbreviations used in the article: IC—information compression; ICMAUS—information compression by multiple alignment, unification and search; MA—multiple alignment; ML—mathematics and logic;
Is the Continuum Hypothesis a definite mathematical problem?
"... [t]he analysis of the phrase “how many ” unambiguously leads to a definite meaning for the question [“How many different sets of integers do their exist?”]: the problem is to find out which one of the א’s is the number of points of a straight line … Cantor, after having proved that this number is gr ..."
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[t]he analysis of the phrase “how many ” unambiguously leads to a definite meaning for the question [“How many different sets of integers do their exist?”]: the problem is to find out which one of the א’s is the number of points of a straight line … Cantor, after having proved that this number is greater than א0, conjectured that it is א1. An equivalent proposition is this: any infinite subset of the continuum has the power either of the set of integers or of the whole continuum. This is Cantor’s continuum hypothesis. … But, although Cantor’s set theory has now had a development of more than sixty years and the [continuum] problem is evidently of great importance for it, nothing has been proved so far relative to the question of what the power of the continuum is or whether its subsets satisfy the condition just stated, except that … it is true for a certain infinitesimal fraction of these subsets, [namely] the analytic sets. Not even an upper bound, however high, can be assigned for the power of the continuum. It is undecided whether this number is regular or singular, accessible or inaccessible, and (except for König’s negative result) what its character of cofinality is. Gödel 1947, 516517 [in Gödel 1990, 178]
Everything you always wanted to know about structural realism but were afraid to ask
 EURO JNL PHIL SCI (2011 ) 1:227–276
, 2011
"... ..."
Reflections on Skolem's Paradox
"... In 1922, Thoraf Skolem published a paper titled "Some remarks on Axiomatized Set Theory". The paper presents a new proof of... This dissertation focuses almost exclusively on the first half of this project  i.e., the half which tries to expose an initial tension between Cantor's the ..."
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In 1922, Thoraf Skolem published a paper titled "Some remarks on Axiomatized Set Theory". The paper presents a new proof of... This dissertation focuses almost exclusively on the first half of this project  i.e., the half which tries to expose an initial tension between Cantor's theorem and the LöwenheimSkolem theorem. I argue that, even on quite naive understandings of set theory and model theory, there is no such tension. Hence, Skolem's Paradox is not a genuine paradox, and there is very little reason to worry about (or even to investigate) the more extreme consequences that are supposed to follow from this paradox. The heart of my...
Categories, structures, and the fregehilbert controversy: The status of metamathematics
 Philosophia Mathematica, 13:61–77. Pagenumbers in
, 2005
"... There is a parallel between the debate between Gottlob Frege and David Hilbert at the turn of the twentieth century and at least some aspects of the current controversy over whether category theory provides the proper framework for structuralism in the philosophy of mathematics. The main issue, I th ..."
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There is a parallel between the debate between Gottlob Frege and David Hilbert at the turn of the twentieth century and at least some aspects of the current controversy over whether category theory provides the proper framework for structuralism in the philosophy of mathematics. The main issue, I think, concerns the place and interpretation of metamathematics in an algebraic or structuralist approach to mathematics. Can metamathematics itself be understood in algebraic or structural terms? Or is it an exception to the slogan that mathematics is the science of structure? The slogan of structuralism is that mathematics is the science of structure. Rather than focusing on the nature of individual mathematical objects, such as natural numbers, the structuralist contends that the subject matter of arithmetic, for example, is the structure of any collection of objects that has a designated, initial object and a successor relation that satisfies the induction principle. In the contemporary scene, Paul Benacerraf’s classic
Conceptions of the Continuum
"... Abstract: A number of conceptions of the continuum are examined from the perspective of conceptual structuralism, a view of the nature of mathematics according to which mathematics emerges from humanly constructed, intersubjectively established, basic structural conceptions. This puts into question ..."
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Abstract: A number of conceptions of the continuum are examined from the perspective of conceptual structuralism, a view of the nature of mathematics according to which mathematics emerges from humanly constructed, intersubjectively established, basic structural conceptions. This puts into question the idea from current set theory that the continuum is somehow a uniquely determined concept. Key words: the continuum, structuralism, conceptual structuralism, basic structural conceptions, Euclidean geometry, Hilbertian geometry, the real number system, settheoretical conceptions, phenomenological conceptions, foundational conceptions, physical conceptions. 1. What is the continuum? On the face of it, there are several distinct forms of the continuum as a mathematical concept: in geometry, as a straight line, in analysis as the real number system (characterized in one of several ways), and in set theory as the power set of the natural numbers and, alternatively, as the set of all infinite sequences of zeros and ones. Since it is common to refer to the continuum, in what sense are these all instances of the same concept? When one speaks of the continuum in current settheoretical
How applied mathematics became pure
 Review of Symbolic Logic
"... Abstract. This paper traces the evolution of thinking on how mathematics relates to the world— from the ancients, through the beginnings of mathematized science in Galileo and Newton, to the rise of pure mathematics in the nineteenth century. The goal is to better understand the role of mathematics ..."
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Abstract. This paper traces the evolution of thinking on how mathematics relates to the world— from the ancients, through the beginnings of mathematized science in Galileo and Newton, to the rise of pure mathematics in the nineteenth century. The goal is to better understand the role of mathematics in contemporary science. My goal here is to explore the relationship between pure and applied mathematics and then, eventually, to draw a few morals for both. In particular, I hope to show that this relationship has not been static, that the historical rise of pure mathematics has coincided with a gradual shift in our understanding of how mathematics works in application to the world. In some circles today, it is held that historical developments of this sort simply represent changes in fashion, or in social arrangements, governments, power structures, or some such thing, but I resist the full force of this way of thinking, clinging to the old school notion that we have gradually learned more about the world over time, that our opinions on these matters have improved, and that seeing how we reached the point we now occupy may help us avoid falling back into old philosophies that are now no longer viable. In that spirit, it seems to me that once we focus on the general question of how mathematics relates to science, one
NeoFregean foundations for real analysis: Some reflections on Frege’s constraint
 Notre Dame Journal of Formal Logic
, 2000
"... We now know of a number of ways of developing Real Analysis on a basis of abstraction principles and secondorder logic. One, outlined by Shapiro in his contribution to this volume, mimics Dedekind in identifying the reals with cuts in the series of rationals under their natural order. The result is ..."
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We now know of a number of ways of developing Real Analysis on a basis of abstraction principles and secondorder logic. One, outlined by Shapiro in his contribution to this volume, mimics Dedekind in identifying the reals with cuts in the series of rationals under their natural order. The result is an essentially structuralist conception of the reals. An earlier approach, developed by Hale in his "Reals by Abstraction", differs by placing additional emphasis upon what I here term Frege's Constraint, that a satisfactory foundation for any branch of mathematics should somehow so explain its basic concepts that their applications are immediate. This paper is concerned with the meaning of and motivation for this constraint. Structuralism has to represent the application of a mathematical theory as always posterior to the understanding of it, turning upon the appreciation of structural affinities between the structure it concerns and a domain to which it is to be applied. There is therefore a case that Frege's Constraint has bite whenever there is a standing body of informal mathematical knowledge grounded in direct reflection upon sample, or schematic, applications of the concepts of the theory in question. It is argued that this condition is satisfied by simple arithmetic and geometry, but that in view of the gap between its basic concepts (of continuity and of the nature