Beauty Is Truth In Mathematical Intuition: First Empirical Evidence
Rolf Reber, together with mathematician Morten Brun and psychologist Karoline Mitterndorfer, all from the University of Bergen, Norway, have reported first empirical evidence for the use of beauty as truth and they have provided an explanation for this phenomenon, based on the processing fluency theory of beauty.
Mathematicians and scientists reportedly used beauty as a cue for truth in mathematical judgment. French mathematician Jacques Hadamard, for example, wrote in 1954 in his famous book, “The Psychology of Invention in the Mathematical Field,” that a sense of beauty seems to be almost the only useful “drive” for discovery in mathematics. However, evidence has been anecdotal, and the nature of the beauty-truth relationship remained a mystery.
In 2004, Rolf Reber (University of Bergen), Norbert Schwarz (University of Michigan), and Piotr Winkielman (University of California at San Diego) suggested – based on evidence they reviewed – that the common experience underlying both perceived beauty and judged truth is processing fluency, which is the experienced ease with which mental content is processed. Indeed, stimuli processed with greater ease elicit more positive affect and statements that participants can read more easily are more likely to be judged as being true. Researchers invoked processing fluency to help explain a wide range of phenomena, including variations in stock prices, brand preferences, or the lack of reception of mathematical theories that are difficult to understand.
Applied to mathematical reasoning, processing fluency, stemming either from familiarity with problems or from attributes of a task, is predicted to increase intuitively judged truth. As a first step towards testing this assumption, the authors of the study demonstrated in two experiments that symmetry, a feature known to facilitate mental processing and to underlie perceived beauty, is used as heuristic cue to correctness in arithmetic problems.
The researchers constructed additions, consisting of dots. For example, 12 dots plus 21 dots equaled 33 dots. Half of the additions were correct; the others were wrong, such as 12 dots plus 21 dots equaled 27 dots. Half of the additions had symmetric dot patterns (symmetric additions), the other half asymmetric patterns (asymmetric additions). These additions were presented briefly, e.g., in one experiment 1800 milliseconds, and student participants without training in mathematics had to decide immediately after the addition disappeared whether it was correct or incorrect.
Participants were more likely to judge symmetric additions than asymmetric additions to be correct. As this was also the case when additions in fact were incorrect, the finding cannot be explained by the fact that symmetric additions were easier to count or to estimate: In this case, symmetric additions that were incorrect would have been less likely to be judged correct. The results clearly show that participants used symmetry as an indication to correctness, or beauty as truth.
The authors have shown that people who do not have enough time to analyze the problem use heuristic cues in order to assess the correctness of a proposed solution. This simple setup serves as a model for the more complicated situation where a mathematician has discovered a plausible solution to a problem and now wants a quick assessment of whether this solution “feels” right. These findings suggest a solution to the mystery why beauty serves as a cue for truth in the context of mathematical discovery.
Psychonomic Bulletin & Review, 15, 1174-1178. doi: 10.3758/PBR.15.6.1174
The use of heuristics in intuitive mathematical judgment
Rolf Reber, Morten Brun, and Karoline Mitterndorfer
Anecdotal evidence points to the use of beauty as an indication of truth in mathematical problem solving. In the two experiments of the present study, we examined the use of heuristics and tested the assumption that participants use symmetry as a cue for correctness in an arithmetic verification task. We manipulated the symmetry of sets of dot pattern addition equations. Speeded decisions about the correctness of these equations led to higher endorsements for both correct and incorrect equations when the addend and sum dot patterns were symmetrical. Therefore, this effect is not due to the fact that symmetry facilitates calculation or estimation. We found systematic evidence for the use of heuristics in solving mathematical tasks, and we discuss how these findings relate to a processing-fluency account of intuition in mathematical judgment.