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Hilbert’s Program Then and Now
, 2005
"... Hilbert’s program is, in the first instance, a proposal and a research program in the philosophy and foundations of mathematics. It was formulated in the early 1920s by German mathematician David Hilbert (1862–1943), and was pursued by him and his collaborators at the University of Göttingen and els ..."
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Hilbert’s program is, in the first instance, a proposal and a research program in the philosophy and foundations of mathematics. It was formulated in the early 1920s by German mathematician David Hilbert (1862–1943), and was pursued by him and his collaborators at the University of Göttingen and elsewhere in the 1920s
The Explanatory Power of Phase Spaces
"... David Malament argued that Hartry Field’s nominalisation program is unlikely to be able to deal with nonspacetime theories such as phasespace theories. We give a specific example of such a phasespace theory and argue that this presentation of the theory delivers explanations that are not availabl ..."
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David Malament argued that Hartry Field’s nominalisation program is unlikely to be able to deal with nonspacetime theories such as phasespace theories. We give a specific example of such a phasespace theory and argue that this presentation of the theory delivers explanations that are not available in the classical presentation of the theory. This suggests that even if phasespace theories can be nominalised, the resulting theory will not have the explanatory power of the original. Phasespace theories thus raise problems for nominalists that go beyond Malament’s initial concerns.
Fundamental and derivative truths
, 2007
"... This paper investigates the claim that some truths are fundamentally or really true—and that others are not. Such a distinction can help us reconcile radically minimal metaphysical views with the verities of common sense, and can do essential theoretical work as a unified basis for distinguishing be ..."
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This paper investigates the claim that some truths are fundamentally or really true—and that others are not. Such a distinction can help us reconcile radically minimal metaphysical views with the verities of common sense, and can do essential theoretical work as a unified basis for distinguishing between ‘elite’ and ‘merely abundant’ properties, objects, and the like. I develop an understanding of the distinction whereby fundamentality is not itself a metaphysical distinction, but rather a device that must be presupposed to express metaphysical distinctions. Drawing on recent work by Rayo on antiQuinean theories of ontological commitments, I formulate a rigourous theory of the notion. In the final sections, I show how this package dovetails with ‘interpretationist ’ theories of meaning to give sober content to thought that some things—perhaps sets, or gerrymandered mereological sums—can be ‘postulated into existence’.
Mathematics: Truth and Fiction?
 in Mathematics.’ Philosophia Mathematica
, 1999
"... n advanced against what he argues is the best version of platonism. More specifically, he defends what he calls fullblooded platonism (`FBP'), the view that every mathematical object that could possibly exist does exist. It is important to the conclusions later in the book that FBP is the only viab ..."
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n advanced against what he argues is the best version of platonism. More specifically, he defends what he calls fullblooded platonism (`FBP'), the view that every mathematical object that could possibly exist does exist. It is important to the conclusions later in the book that FBP is the only viable form of platonism, so in this first section Balaguer also attempts to demonstrate that all other platonist positions are indefensible. In the second part of the book, Balaguer tries to show that no good arguments have been advanced against (a broadly Fieldian kind of) fictionalism. Although it is fictionalism that Balaguer defends, he also makes it clear that other antirealist positions, such as deductivism and formalism, are more or less equivalent to fictionalism and so he has no serious quarrel with them. He prefers fictionalism, however, because it "provides a standard semantics for the language of mathematics" (p. 104), whereas other antirealist accounts (such as Chihara [1990], f
ARISTOTELIAN REALISM
"... Aristotelian, or nonPlatonist, realism holds that mathematics is a science of the real world, just as much as biology or sociology are. Where biology studies living things and sociology studies human social relations, mathematics studies the quantitative or structural aspects of things, such as rat ..."
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Aristotelian, or nonPlatonist, realism holds that mathematics is a science of the real world, just as much as biology or sociology are. Where biology studies living things and sociology studies human social relations, mathematics studies the quantitative or structural aspects of things, such as ratios, or patterns, or complexity,
TWODIMENSIONAL BELIEF CHANGE An Advertisement
"... In this paper I compare two different the models of twodimensional belief change, namely ‘revision by comparison ’ (Fermé and Rott, Artificial Intelligence 157, 2004) and ‘bounded revision ’ (Rott, in Hommage à Wlodek, Uppsala 2007). These revision operations are twodimensional in the sense that t ..."
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In this paper I compare two different the models of twodimensional belief change, namely ‘revision by comparison ’ (Fermé and Rott, Artificial Intelligence 157, 2004) and ‘bounded revision ’ (Rott, in Hommage à Wlodek, Uppsala 2007). These revision operations are twodimensional in the sense that they take as arguments pairs consisting of an input sentence and a reference sentence. Twodimensional revision operations add a lot to the expressive power of traditional qualitative approaches to belief revision and refrain from assuming numbers as measures of degrees of belief. 1.
AntiRealist Classical Logic and Realist Mathematics
, 2009
"... Abstract. I sketch an application of a semantically antirealist understanding of the classical sequent calculus to the topic of mathematics. The result is a semantically antirealist defence of a kind of mathematical realism. In the paper, I begin the development of the view and compare it to ortho ..."
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Abstract. I sketch an application of a semantically antirealist understanding of the classical sequent calculus to the topic of mathematics. The result is a semantically antirealist defence of a kind of mathematical realism. In the paper, I begin the development of the view and compare it to orthodox positions in the philosophy of mathematics.
On the imaginative constructivist nature of design: a theoretical approach
"... Abstract: Most empirical accounts of design suggest that designing is an activity where objects and representations are progressively constructed. Despite this fact, whether design is a constructive process or not is not a question directly addressed in current design research. By contrast, in other ..."
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Abstract: Most empirical accounts of design suggest that designing is an activity where objects and representations are progressively constructed. Despite this fact, whether design is a constructive process or not is not a question directly addressed in current design research. By contrast, in other fields such as Mathematics or Psychology, the notion of constructivism is seen as a foundational issue. The present paper defends the point of view that forms of constructivism in design need to be identified and integrated as a foundational element in design research as well. In fact, a look at the literature reveals at least two types of constructive processes that are well embedded in design research. First, an interactive constructivism, where a designer engages a conversation with media, that allows changing the course of the activity as a result of this interaction. Second, a social constructivism, where designers need to handle communication and negotiation aspects, that allows integrating individuals ’ expertise into the global design process. A key feature lacking to these wellestablished paradigms is the explicit consideration of creativity as a central issue of design. To explore how creative and constructivist aspects of design can be taken into account conjointly, the present paper pursues a theoretical approach. We consider the roots of
Derivative Properties in Fundamental Laws
"... Orthodoxy has it that only fundamental properties can appear in fundamental laws. We examine Newton’s second law F=ma—a paradigm historical candidate for a fundamental law—and find properties that are evidently derivative by the lights of Newtonian mechanics: Newtonian acceleration is a defined prop ..."
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Orthodoxy has it that only fundamental properties can appear in fundamental laws. We examine Newton’s second law F=ma—a paradigm historical candidate for a fundamental law—and find properties that are evidently derivative by the lights of Newtonian mechanics: Newtonian acceleration is a defined property (the second derivative of position), and Newtonian resultant force is a defined property as well (the vector sum of the component forces). We conclude that derivative properties can appear in fundamental laws when their inclusion simplifies the equations or renders the resulting system of equations more modular, and we draw a deflationary moral about laws themselves: laws are merely convenient summaries. Overview: In §1 we introduce the orthodox view that only fundamental properties can appear in fundamental laws. In §2 we examine F=ma and argue that Newtonian acceleration and Newtonian resultant force are derivative properties by the lights of Newtonian mechanics, and in §3 we discuss three possible replies. In §4 we conclude by recommending a deflationary view of laws. 1. Orthodoxy Armstrong and Lewis each posit fundamental properties: metaphysically elite perfectly natural properties that “carve at the joints. ” One role this posit plays is to restrict the candidate fundamental laws via:
Inferentialism, Logicism, Harmony, and a Counterpoint by
, 2007
"... Inferentialism is explained as an attempt to provide an account of meaning that is more sensitive (than the tradition of truthconditional theorizing deriving from Tarski and Davidson) to what is learned when one masters meanings. The logically reformist inferentialism of Dummett and Prawitz is cont ..."
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Inferentialism is explained as an attempt to provide an account of meaning that is more sensitive (than the tradition of truthconditional theorizing deriving from Tarski and Davidson) to what is learned when one masters meanings. The logically reformist inferentialism of Dummett and Prawitz is contrasted with the more recent quietist inferentialism of Brandom. Various other issues are highlighted for inferentialism in general, by reference to which different kinds of inferentialism can be characterized. Inferentialism for the logical operators is explained, with special reference to the Principle of Harmony. The statement of that principle in the author’s book Natural Logic is finetuned here in the way obviously required in order to bar an interesting wouldbe counterexample furnished by Crispin Wright, and to stave off any more of the same.