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Formalized mathematics
 TURKU CENTRE FOR COMPUTER SCIENCE
, 1996
"... It is generally accepted that in principle it’s possible to formalize completely almost all of presentday mathematics. The practicability of actually doing so is widely doubted, as is the value of the result. But in the computer age we believe that such formalization is possible and desirable. In c ..."
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It is generally accepted that in principle it’s possible to formalize completely almost all of presentday mathematics. The practicability of actually doing so is widely doubted, as is the value of the result. But in the computer age we believe that such formalization is possible and desirable. In contrast to the QED Manifesto however, we do not offer polemics in support of such a project. We merely try to place the formalization of mathematics in its historical perspective, as well as looking at existing praxis and identifying what we regard as the most interesting issues, theoretical and practical.
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.
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 theorem and the Lö ..."
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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...
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]
particles and structural realism
"... Even if we are able to decide on a canonical formulation of our theory, there is the further problem of metaphysical underdetermination with respect to, for example, whether the entities postulated by a theory are individuals or not... We need to recognise the failure of our best theories to determi ..."
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Even if we are able to decide on a canonical formulation of our theory, there is the further problem of metaphysical underdetermination with respect to, for example, whether the entities postulated by a theory are individuals or not... We need to recognise the failure of our best theories to determine even the most fundamental ontological characteristic of the purported entities they feature... What is required is a shift to a different ontological basis altogether, one for which questions of individuality simply do not arise. Perhaps we should view the individuals and nonindividuals packages, like particle and field pictures, as different representations of the same structure. There is an analogy here with the debate about substantivalism in general relativity. (Ladyman, 1998) In his paper “What is Structural Realism? ” (1998) James Ladyman drew a distinction between epistemological structural realism (ESR) and metaphysical (or ontic) structural realism (OSR). In recent years this distinction has set much of the agenda for philosophers of science interested in scientific realism. It has also led to the emergence of a related discussion in the philosophy of physics that concerns the alleged difficulties of interpreting general relativity that revolve around the question of the ontological status of spacetime points. Ladyman drew a suggestive analogy between the perennial debate between substantivalist and relationalist interpretations of spacetime on the one hand, and the debate about whether quantum mechanics treats identical particles as individuals or as ‘nonindividuals ’ on the other. In both cases, Ladyman’s suggestion is that a structural realist interpretation of the physics—in particular, an ontic structural realism—might
1 Induction and Comparison
"... Frege proved an important result, concerning the relation of arithmetic to secondorder logic, that bears on several issues in linguistics. Frege’s Theorem illustrates the logic of relations like PRECEDES(x, y) and TALLER(x, y), while raising doubts about the idea that we understand sentences like ‘ ..."
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Frege proved an important result, concerning the relation of arithmetic to secondorder logic, that bears on several issues in linguistics. Frege’s Theorem illustrates the logic of relations like PRECEDES(x, y) and TALLER(x, y), while raising doubts about the idea that we understand sentences like ‘Carl is taller than Al ’ in terms of abstracta like heights and numbers. Abstract paraphrase can be useful—as when we say that Carl’s height exceeds Al’s—without reflecting semantic structure. Related points apply to causal relations, and even grammatical relations like DOMINATES(x, y). Perhaps surprisingly, Frege provides the resources needed to recursively characterize labelled expressions without characterizing them as sets. His theorem may also bear on questions about the meaning and acquisition of number words.
Macroscopic Ontology in Everettian Quantum Mechanics’. Philosophical Quarterly 61:243
 Objective Probability in Everettian Quantum Mechanics’. To appear in British Journal for Philosophy of Science
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Ontological Commitment and Reconstructivism
, 2000
"... . Some forms of analytic reconstructivism take natural language (and common sense at large) to be ontologically opaque: ordinary sentences must be suitably rewritten or paraphrased before questions of ontological commitment may be raised. Other forms of reconstructivism take the commitment of ordina ..."
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. Some forms of analytic reconstructivism take natural language (and common sense at large) to be ontologically opaque: ordinary sentences must be suitably rewritten or paraphrased before questions of ontological commitment may be raised. Other forms of reconstructivism take the commitment of ordinary language at face value, but regard it as metaphysically misleading: commonsense objects exist, but they are not what we normally think they are. This paper is an attempt to clarify and critically assess some common limits of these two reconstructivist strategies. 1. Introduction Ordinary language describes a world inhabited by entities of different sorts people, tables, trees, smiles, heaps of sand, penalty kicks. We utter sentences such as (1) There is a table in the kitchen (2) There are deeds, such as a smile, that are worth a thousand words, which contain explicit existential idioms and which therefore seem to commit us to the existence of corresponding entities (here: tables...
Objective Subjectivity: Allocentric and Egocentric Representation in Thought and Experience
, 2000
"... this dissertation I address both questions of application and questions of constitution. The primary aim of this chapter is to begin by addressing the question of what it means for judgments to be objective or subjective. One common use of the notions of objectivity and subjectivity is to demarcate ..."
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this dissertation I address both questions of application and questions of constitution. The primary aim of this chapter is to begin by addressing the question of what it means for judgments to be objective or subjective. One common use of the notions of objectivity and subjectivity is to demarcate kinds of judgment (or thought or belief). On such a usage, prototypically objective judgments concern matters of empirical and mathematical fact such as the moon has no atmosphere and two and two are four. In contrast, prototypically subjective judgments concern matters of value and preference such as Mozart is better than Bach and vanilla ice cream with ketchup is disgusting. I offer these examples not to take sides on whether such judgments actually are objective or subjective, but only to call attention to a typical way of using "objective" and "subjective". The question arises as to what it means in this context to call these respective judgments "objective" and "subjective". Some have proposed that the difference hinges on truth. Objective judgments are absolutely true, whereas the truth of subjective judgments is relative to the person making the judgment: my judgments are true for me, your judgments are true for you. You and I can each utter "vanilla tastes great" but in your mouth this may constitute a truth and in my mouth it may constitute a falsehood. Subjective judgments are subject relative. Some philosophers have noted an analogy between this kind of subject relativity and a kind that obtains for indexical expressions. You and I can both utter "I am here" and thereby express different propositions. Some philosophers have construed indexicality as an instance of subjectivity and some others have even gone so far as to argue that subjectivity just is indexicality....