769 ⁄ 0022-2496/02 $35.00 © 2002 Elsevier Science (USA) All rights reserved. Journal of Mathematical Psychology 46, 769–788 (2002) doi:10.1006/jmps.2002.1429 The Irony of Measurement by Subjective Estimations Louis Narens Address correspondence and reprint requests to Louis Narens, Department of Cognitive Sciences, University of California, 3151 Social Science Plaza, Irvine, CA 92697-5100. University of California, Irvine In 1948 S. S. Stevens, in his famous Science article, proposed a theory of measurement that radically differed from the dominate theory of the time. The dominate theory held that all strong forms of scientific measurement— for example, those that yielded ratio scales—had to be based on an observ- able ordering and an observable commutative and associative operation. Stevens proposed different criteria and introduced his method of magnitude estimation. Stevens as well as measurement theorists considered his method to be radically different from those based on commutative and associative operations. Although his method was controversial, it became a standard tool in the behavioral sciences. This article argues that Stevens’ method, together with implicit assumptions he made about the scales of measurement it gener- ated, is from a mathematical perspective the same as the measurement process based on commutative and associative operations. The article also provides a theory of qualitative numbers and shows an interesting relation- ship between qualitative numbers and Stevens’ method. © 2002 Elsevier Science (USA) 1. INTRODUCTION In 1946, the psychologist S. S. Stevens published an article in the journal Science that has had an enormous impact on how behavioral scientists thought about and used measurement. The article was Stevens’ response to a widely held view of the time that justifiable forms of measurement that were stronger than counting or numerical ordering necessarily relied on the existence of an observable, empirical form of addition. Because psychological phenomena generally lacked such forms of addition, many scientists believed that psychology was not—and could never be—a quantitative science founded on philosophically sound principles. Stevens and other psychologists thought otherwise. This led the British Association for the Advan- cement of Science to appoint a committee to look into the matter. Stevens (1946) comments,