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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]
Ethics and Finance: the role of mathematics ∗
, 2012
"... This paper presents the contemporary Fundamental Theorem of Asset Pricing as being equivalent to approaches to pricing that emerged before 1700 in the context of Virtue Ethics. This is done by considering the history of science and mathematics in the thirteenthandseventeenth century. Anexplanationas ..."
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This paper presents the contemporary Fundamental Theorem of Asset Pricing as being equivalent to approaches to pricing that emerged before 1700 in the context of Virtue Ethics. This is done by considering the history of science and mathematics in the thirteenthandseventeenth century. Anexplanationastowhytheseapproachestopricing were forgotten between 1700 and 2000 is given, along with some of the implications on economics of viewing the Fundamental Theorem as a product of Virtue Ethics. TheFundamental Theorem was developed in mathematics to establish a ‘theory ’ that underpinnedtheBlackScholesMerton approach topricingderivatives. Indoingthis, the Fundamental Theorem unified a number of different approaches in financial economics, this strengthened the status of neoclassical economics based on Consequentialist Ethics. We present an alternative to this narrative. 1
1 Clifford Algebraic Computational Fluid Dynamics: A New Class of Experiments.
, 2010
"... Though some influentially critical objections have been raised during the ‘classical ’ precomputational simulation philosophy of science (PCSPS) tradition, suggesting a more nuanced methodological category for experiments2, it safe to say such critical objections have greatly proliferated in philos ..."
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Though some influentially critical objections have been raised during the ‘classical ’ precomputational simulation philosophy of science (PCSPS) tradition, suggesting a more nuanced methodological category for experiments2, it safe to say such critical objections have greatly proliferated in philosophical studies dedicated to the role played by computational simulations in science. For instance, Eric Winsberg (19992003) suggests that computer simulations are methodologically unique in the development of a theory’s models3 suggesting new epistemic notions of application. This is also echoed in Jeffrey Ramsey’s (1995) notions of “transformation reduction,”—i.e., a notion of reduction of a more highly constructive variety.4 Computer simulations create a broadly continuous arena spanned by normative and descriptive aspects of theoryarticulation, as entailed by the notion of transformation reductions occupying a continuous region demarcated by Ernest Nagel’s (1974) logicalexplanatory “domaincombining reduction ” on the one hand, and Thomas Nickels ’ (1973) heuristic “domainpreserving reduction, ” on the other. I extend Winsberg’s and Ramsey’s points here, by arguing that in the field of computational fluid dynamics (CFD) as well as in other branches of applied physics, the computer plays a constitutively experimental role—supplanting in many cases the more traditional experimental methods such as flowvisualization, etc. In this case, however CFD algorithms act as substitutes, not supplements (as the notions “simulation ” suggests) when it comes to experimental practices. I bring up the constructive example involving the CliffordAlgebraic algorithms for modeling singular phenomena (i.e., vortex
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