conceptual and mathematical errors", errors of a magnitude that completely invalidated their claims. Fredrickson wrote a response in which she conceded that the mathematical aspects of the critical positivity ratio were "questionable" and that she had "neither the expertise nor the insight" to defend them, but she maintained that the empirical evidence for the existence of a critical positivity ratio was solid. Brown, Sokal, and Friedman, the rebuttal authors, published their response to Fredrickson's "Update" the next year, maintaining that there was no evidence for a critical positivity ratio. Losada declined to respond to the criticism (indicating to the Chronicle of Higher Education that he was too busy running his consulting business). Hämäläinen and colleagues responded later, passing over the Brown-Sokal-Friedman rebuttal claim of failed criteria for use of differential equations in modeling, instead arguing that there were no fundamental errors in the mathematics itself, only problems related to the model's justification and interpretation. A formal retraction for the mathematical modeling elements of the Losada and Fredrickson (2005) paper was issued by the journal, American Psychologist, concluding that both the specific critical positivity ratio of 2.9013 and its upper limit were invalid. The fact that the problems with the paper went unnoticed for years despite the widespread adulatory publicity surrounding the critical positivity ratio concept contributed to a perception of social psychology as a field lacking scientific soundness and rigorous critical thinking. Sokal later stated, "The main claim made by Fredrickson and Losada is so implausible on its face that some red flags ought to have been raised", as would only happen broadly in graduate student Brown's initiating the collaboration that resulted in the Brown-Sokal-Friedman rebuttal. Background Building on research by Barbara Fredrickson suggesting that individuals with a higher ratio of positive to negative emotions tend to have more successful life outcomes, and on studies by Marcial Losada applying differential equations from fluid dynamics to human emotions, Fredrickson and Losada proposed as informative a ratio of positive to negative affect derived from nonlinear dynamics modelling (based on Lorenz systems), which appeared in 2005 in a paper in American Psychologist. The derived combination of expressions and default parameters led them to conclude that a critical ratio of positive to negative affect of exactly 2.9013 separated flourishing from languishing individuals, and to argue that the ideal positivity/negativity ratio lies between 2.9013 and an upper limit ratio of 11.6346. Hence, they claimed that