Abstract
Weber-Fechner Law states that the perceived intensity is proportional to the logarithm of the stimulus. Recent experiments suggest that this law also holds true for perception of numerosity. Therefore, the use of a logarithmic scale for the quantification of the perceived intensity may also depend on how the cognitive apparatus processes information. If Weber-Fechner law is the result of natural selection, then the logarithmic scale should be better, in some sense, than other biologically feasible scales. We consider the minimization of the relative error as the target of natural selection and we provide a formal proof that the logarithmic scale minimizes the maximal relative error.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Batschelet, E. (1979). Introduction to mathematics for life scientists. New York: Springer.
Burden, R. L., Faires, J. D., & Reynolds, A. C. (1978). Numerical analysis. Boston, Mass: Prindle, Weber & Schmidt.
Dehaene, S. (2003). The neural basis of the Weber-Fechner law: a logarithmic mental number line. Trends in Cognitive Sciences, 7(4), 145–147.
Fechner, G. T. (1860). Elemente der Psychophysik, Vol. 2. Leipzig: Breitkopf und Haertel.
Higham, N. J. (1996). Accuracy and stability of numerical algorithms. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM).
Laski, E. V., & Siegler, R. S. (2007). Is 27 a big number? Correlational and causal connections among numerical categorization, number line estimation, and numerical magnitude comparison. Child Development, 78(6), 1723–1743.
Miller, G. A. (1956). The magical number seven plus or minus two: Some limits on our capacity for processing information. Psychological Review 63(2), 81–97.
Nieder, A., & Miller, E. K. (2003). Coding of cognitive magnitude: Compressed scaling of numerical information in the primate prefrontal cortex. Neuron, 37(1), 149–157.
Ortega, J. M. (1972). Numerical analysis. A second course. New York: Academic Press. Computer Science and Applied Mathematics.
Stevens, S. S. (1961). To honor Fechner and repeal his law: A power function, not a log function, describes the operating characteristic of a sensory system. Science, 133(3446), 80–86.
Stewart, G. W. (1996). Afternotes on numerical analysis. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM).
Weber, E. H. (1850). Der Tastsinn und das Gemeingefuhl. In Handwörterbuch der Physiologie, Vol. 3. Braunschweig.
Acknowledgments
Partially supported by CNPq grants 303583/2008-8, 480101/2008-6 , FAPERJ grant E-26 102.821/2008 and by PRONEX-Optimization.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Portugal, R.D., Svaiter, B.F. Weber-Fechner Law and the Optimality of the Logarithmic Scale. Minds & Machines 21, 73–81 (2011). https://doi.org/10.1007/s11023-010-9221-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11023-010-9221-z