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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]
There are no abstract objects
 In
, 2008
"... Suppose you start out inclined towards the hardheaded view that the world of material objects is the whole of reality. You elaborate: ‘Everything there is is a material object: the sort of thing you could bump into; the sort of thing for which it would be sensible to ask how much it weighs, what sh ..."
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Suppose you start out inclined towards the hardheaded view that the world of material objects is the whole of reality. You elaborate: ‘Everything there is is a material object: the sort of thing you could bump into; the sort of thing for which it would be sensible to ask how much it weighs, what shape it is, how fast it is moving, and how far it is from other material objects. There is nothing else. ’ You develop some practice defending your thesis from the expected objections, from believers in ghosts, God, immaterial souls, Absolute Space, and so on. None of this practice will do you much good the first time you are confronted with the following objection: What about numbers and properties? These are obviously not material objects. It would be crazy to think that you might bump into the number two, or the property of having many legs. One would have to be confused to wonder how much these items weigh, or how far away they are. But obviously there are numbers and properties. Surely even you don’t deny that there are four prime numbers between one and ten, or that spiders and insects share many important anatomical properties. 1 But these wellknown truths evidently imply that there are numbers, and that there are properties. So
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...
Challenges to Predicative Foundations of Arithmetic
 in Between Logic and Intuition Essays in Honor of Charles Parsons
, 1996
"... This paper was written while the first author was a Fellow at the Center for Advanced Study in the Behavioral Sciences (Stanford, CA) whose facilities and support, under grants from the Andrew W. Mellon Foundation and the National Science Foundation, have been greatly appreciated. ..."
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This paper was written while the first author was a Fellow at the Center for Advanced Study in the Behavioral Sciences (Stanford, CA) whose facilities and support, under grants from the Andrew W. Mellon Foundation and the National Science Foundation, have been greatly appreciated.
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
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
Of numbers and electrons
 In Proceedings of the Aristotelian Society
, 2010
"... The sciences are full of theories which, in the course of making detailed claims about the physical world, say things which entail that there are mathematical entities like numbers and sets. According to an influential tradition stemming from Quine (1948) and Putnam (1972), good scientific reasoning ..."
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The sciences are full of theories which, in the course of making detailed claims about the physical world, say things which entail that there are mathematical entities like numbers and sets. According to an influential tradition stemming from Quine (1948) and Putnam (1972), good scientific reasoning—induction, broadly construed—requires us to believe some such theory, or some disjunction of such theories. And it is because of this that we ought to believe that there are mathematical entities. The belief that there are numbers is, according to this tradition, on a similar epistemological footing to the belief that there are electrons, viruses, quasars, etc. 1 Some will regard this analogy as unhelpful because they think that we can know that there are mathematical entities in the same way—whatever it is—that we know that all dogs are dogs, or that all bachelors are unmarried. 2 Others may regard this analogy as unhelpful because they think that we can directly perceive that there are mathematical entities—such as sets of
Empirical
 of International Economics
, 1990
"... promoting access to White Rose research papers ..."
Putnam, Context, and Ontology 1
"... When a debate seems intractable, with little agreement as to how one might proceed towards a resolution, it is understandable that philosophers should consider whether something might be amiss with the debate itself. Famously in the last century, philosophers of various ..."
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When a debate seems intractable, with little agreement as to how one might proceed towards a resolution, it is understandable that philosophers should consider whether something might be amiss with the debate itself. Famously in the last century, philosophers of various