David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Ezio Di Nucci
Jonathan Jenkins Ichikawa
Jack Alan Reynolds
Learn more about PhilPapers
Context: Consistency of mathematical constructions in numerical analysis and the application of computerized proofs in the light of the occurrence of numerical chaos in simple systems. Purpose: To show that a computer in general and a numerical analysis in particular can add its own peculiarities to the subject under study. Hence the need of thorough theoretical studies on chaos in numerical simulation. Hence, a questioning of what e.g. a numerical disproof of a theorem in physics or a prediction in numerical economics could mean. Method: An algebraic simple model system is subjected to a deeper structure of underlying variables. With an algorithm simulating the steps in taking a limit of second order difference quotients the error terms are studied at the background of their algebraic expression. Results: With the algorithm that was applied to a simple quadratic polynomial system we found unstably amplified round-off errors. The possibility of numerical chaos is already known but not in such a simple system as used in our paper. The amplification of the errors implies that it is not possible with computer means to constructively show that the algebra and numerical analysis will ‘on the long run’ converge to each other and the error term will vanish. The algebraic vanishing of the error term cannot be demonstrated with the use of the computer because the round-off errors are amplified. In philosophical terms, the amplification of the round-off error is equivalent to the continuum hypothesis. This means that the requirement of (numerical) construction of mathematical objects is no safeguard against inference-only conclusions of qualities of (numerical) mathematical objects. Unstably amplified round-off errors are a same type of problem as the ordering in size of transfinite cardinal numbers. The difference is that the former problem is created within the requirements of constructive mathematics. This can be seen as the reward for working numerically constructive.
|Keywords||Numerical mathematics Constructioundations of mathematicsve f|
|Categories||categorize this paper)|
Setup an account with your affiliations in order to access resources via your University's proxy server
Configure custom proxy (use this if your affiliation does not provide a proxy)
|Through your library||
References found in this work BETA
No references found.
Citations of this work BETA
No citations found.
Similar books and articles
Yaroslav Sergeyev (2009). Numerical Computations and Mathematical Modelling with Infinite and Infinitesimal Numbers. Journal of Applied Mathematics and Computing 29:177-195.
Stephen Laurence & Eric Margolis (2007). Linguistic Determinism and the Innate Basis of Number. In Peter Carruthers (ed.), The Innate Mind: Foundations and the Future.
Stanislas Dehaene, Elizabeth Spelke & Lisa Feigenson (2004). Core Systems of Number. Trends in Cognitive Sciences 8 (7):307-314.
Gabriele Gramelsberger (2011). What Do Numerical (Climate) Models Really Represent? Studies in History and Philosophy of Science 42 (2):296-302.
Eric Mandelbaum (2013). Numerical Architecture. Topics in Cognitive Science 5 (1):367-386.
R. Duncan Luce (1978). Dimensionally Invariant Numerical Laws Correspond to Meaningful Qualitative Relations. Philosophy of Science 45 (1):1-16.
Peter H. A. Sneath & Robert R. Sokal (1973). Numerical Taxonomy: The Principles and Practice of Numerical Classification. W. H. Freeman and Co..
Paul Bloom (2002). Enumeration of Collective Entities by 5-Month-Old Infants. Cognition 83 (3):55-62.
H. Barringer, D. M. Gabbay & J. Woods (2012). Temporal, Numerical and Meta-Level Dynamics in Argumentation Networks. Argument and Computation 3 (2-3):143 - 202.
V. Troiani, J. Peelle, R. Clark & M. Grossman (2009). Is It Logical to Count on Quantifiers? Dissociable Neural Networks Underlying Numerical and Logical Quantifiers. Neuropsychologia 47 (1):104--111.
Added to index2012-01-12
Total downloads197 ( #17,491 of 1,934,702 )
Recent downloads (6 months)76 ( #5,552 of 1,934,702 )
How can I increase my downloads?