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Natural and Formal Infinities
 Educational Studies in Mathematics 48 (2001
"... Concepts of infinity usually arise by reflecting on finite experiences and imagining them extended to the infinite. This paper will refer to such personal conceptions as natural infinities. Research has shown that individuals’ natural conceptions of infinity are ‘labile and selfcontradictory ’ (Fis ..."
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Concepts of infinity usually arise by reflecting on finite experiences and imagining them extended to the infinite. This paper will refer to such personal conceptions as natural infinities. Research has shown that individuals’ natural conceptions of infinity are ‘labile and selfcontradictory ’ (Fischbein et al., 1979, p. 31). The formal approach to mathematics in the twentieth century attempted to rationalize these inconsistencies by selecting a finite list of specific properties (or axioms) from which the conception of a formal infinity is built by formal deduction. By beginning with different properties of finite numbers, such as counting, ordering or arithmetic, different formal systems may be developed. Counting and ordering lead to cardinal and ordinal number theory and the properties of arithmetic lead to ordered fields that may contain infinite and infinitesimal quantities. Cardinal and ordinal numbers can be added and multiplied but not divided or subtracted. The operations of cardinals are commutative, but the operations of ordinals are not. Meanwhile an ordered field has a full system of arithmetic in which the reciprocals of
Motivation and performance differences in students’ domainspecific epistemological belief profiles
 New Ideas in Psychology
, 2005
"... Cluster analysis and analysis of variance procedures were used to identify students ’ domainspecific epistemological belief profiles and to examine differences in students ’ beliefs, motivation, and task performance. Four hundred eightytwo undergraduates completed measures regarding their belief ..."
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Cluster analysis and analysis of variance procedures were used to identify students ’ domainspecific epistemological belief profiles and to examine differences in students ’ beliefs, motivation, and task performance. Four hundred eightytwo undergraduates completed measures regarding their beliefs about knowledge, competency beliefs, and achievement values relative to history and mathematics and participated in domain learning tasks. Cluster analysis was used to identify epistemological belief profile groups within the domains of history and mathematics. Students with more sophisticated belief profiles had higher levels of motivation and task performance. Although the configuration of profiles differed across domains, crossdomain analyses suggested a tendency for students to be relatively consistent in the sophistication of their beliefs across domains. These findings provide evidence of the dual nature of epistemological beliefs.
ON INTERPRETING CHAITIN’S INCOMPLETENESS THEOREM
, 1998
"... The aim of this paper is to comprehensively question the validity of the standard way of interpreting Chaitin’s famous incompleteness theorem, which says that for every formalized theory of arithmetic there is a finite constant c such that the theory in question cannot prove any particular number ..."
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The aim of this paper is to comprehensively question the validity of the standard way of interpreting Chaitin’s famous incompleteness theorem, which says that for every formalized theory of arithmetic there is a finite constant c such that the theory in question cannot prove any particular number to have Kolmogorov complexity larger than c. The received interpretation of theorem claims that the limiting constant is determined by the complexity of the theory itself, which is assumed to be good measure of the strength of the theory. I exhibit certain strong counterexamples and establish conclusively that the received view is false. Moreover, I show that the limiting constants provided by the theorem do not in any way reflect the power of formalized theories, but that the values of these constants are actually determined by the chosen coding of Turing machines, and are thus quite accidental.
Quantumlike Chaos in Prime Number Distribution and in Turbulent Fluid Flows
 APEIRON
, 2001
"... re applied to derive the following results for the observed association between prime number distribution and quantumlike chaos. (i) Number theoretical concepts are intrinsically related to the quantitative description of dynamical systems. (ii) Continuous periodogram analyses of different set ..."
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re applied to derive the following results for the observed association between prime number distribution and quantumlike chaos. (i) Number theoretical concepts are intrinsically related to the quantitative description of dynamical systems. (ii) Continuous periodogram analyses of different sets of adjacent prime number spacing intervals show that the power spectra follow the model predicted universal inverse powerlaw form of the statistical normal distribution. The prime number distribution therefore exhibits selforganized criticality, which is a signature of quantumlike chaos. (iii) The continuum real number field contains unique structures, namely, prime numbers, which are analogous to the dominant eddies in the eddy continuum in turbulent fluid flows. Keywords: quantumlike chaos in prime numbers, fractal structure of primes, quantification of prime number distribution, prime numbers and fluid flows 1. Introduction he continuum real number field (infinite numbe
Topology and turbulence
, 2001
"... Abstract: Over a given regular domain of independent variables {x,y,z,t}, every covariant vector field of flow can be constructed in terms a differential 1form of Action. The associated Cartan topology permits the definition of four basic topological equivalence classes of flows based on the Pfaff ..."
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Cited by 7 (3 self)
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Abstract: Over a given regular domain of independent variables {x,y,z,t}, every covariant vector field of flow can be constructed in terms a differential 1form of Action. The associated Cartan topology permits the definition of four basic topological equivalence classes of flows based on the Pfaff dimension of the 1form of Action. Potential flows or streamline processes are generated by an Action 1form of Pfaff dimension 1 and 2, respectively. Chaotic flows must be associated with domains of Pfaff dimension 3 or more. Turbulent flows are associated with domains of Pfaff dimension 4. It will be demonstrated that the NavierStokes equations are related to Action 1forms of Pfaff dimension 4. The Cartan Topology is a disconnected topology if the Pfaff dimension is greater than 2. This fact implies that the creation of turbulence (a state of Pfaff dimension 4 and a disconnected Cartan topology) from a streamline flow (a state of Pfaff dimension 2 and a connected topology) can take place only by discontinuous processes which induce shocks and tangential discontinuities. On the otherhand, the decay of turbulence can be described by continuous, but irreversible, processes. Numerical procedures that force continuity of slope and value cannot in principle describe the creation of turbulence, but such techniques of forced continuity can be used to describe the decay of turbulence. 1
University students’ embodiment of quantifier
 In Anne D. Cockburn & Elena Nardi (Eds), Proceedings of the 26th Conference of the International Group for the Psychology of Mathematics Education
, 2002
"... This paper investigates the novice university students ’ understanding of the formal definition of “equivalence relations”, especially their understanding of the quantifiers in the definition. Even though the definition is relatively simple and only involves the universal quantifier, we find that ha ..."
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This paper investigates the novice university students ’ understanding of the formal definition of “equivalence relations”, especially their understanding of the quantifiers in the definition. Even though the definition is relatively simple and only involves the universal quantifier, we find that half of a class of highly qualified university students are unable to test whether an explicit relation on a set with three elements is an equivalence relation. Analysis of the data, from a questionnaire answered by 277 students and interviews with 36, reveals subtle influences of language and of conceptual embodiments. In particular, the transitive law, which is shared with the notion of order relation, may evoke an embodied image of order that is highly misleading.
Averaging for ordinary differential equation and functional differential equations
 The Strength of Nonstandard Analysis
, 2007
"... A nonstandard approach to averaging theory for ordinary differential equations and functional differential equations is developed. We define a notion of perturbation and we obtain averaging results under weaker conditions than the results in the literature. The classical averaging theorems approxim ..."
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A nonstandard approach to averaging theory for ordinary differential equations and functional differential equations is developed. We define a notion of perturbation and we obtain averaging results under weaker conditions than the results in the literature. The classical averaging theorems approximate the solutions of the system by the solutions of the averaged system, for Lipschitz continuous vector fields, and when the solutions exist on the same interval as the solutions of the averaged system. We extend these results to perturbations of vector fields which are uniformly continuous in the spatial variable with respect to the time variable and without any restriction on the interval of existence of the solution.
JAMES J KAPUT (19422005) IMAGINEER AND FUTUROLOGIST OF MATHEMATICS EDUCATION
"... Jim Kaput lived a full life in mathematics education and we have many reasons to be grateful to him, not only for his vision of the use of technology in mathematics, but also for his fundamental humanity. This paper considers the origins of his ‘big ideas’ as he lived through the most amazing innova ..."
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Jim Kaput lived a full life in mathematics education and we have many reasons to be grateful to him, not only for his vision of the use of technology in mathematics, but also for his fundamental humanity. This paper considers the origins of his ‘big ideas’ as he lived through the most amazing innovations in technology that have changed our lives more in a generation than in many centuries before. His vision continues as is exemplified by the collected papers in this tribute to his life and work.