### The 1996 Sokal hoax

Readers may be aware of the 1996 “Sokal hoax,” wherein Alan Sokal, a physicist at New York University, wrote a parody of a postmodern science article, entitled Transgressing the Boundaries: Toward a Transformative Hermeneutics of Quantum Gravity, and submitted it to *Social Text*, a prominent journal in the postmodern studies field. The article was filled with page after page of erudite-sounding nonsense, political rhetoric, irrelevant references to arcane scientific concepts and approving quotations from leading postmodern science scholars. In spite of its severe flaws, the article was not only accepted for the journal, but it appeared in a special issue devoted to defending the legitimacy of the postmodern science studies field against its detractors.

As Sokal later noted, “I intentionally wrote the article so that any competent physicist or mathematician (or undergraduate physics or math major) would realize that it is a spoof.” Sokal said he resorted to the hoax out of a deeply felt concern that the postmodern science world has taken a complete about-face from its roots in the Enlightenment, and now instead is embracing obscurantism and pseudoscience. *“Theorizing about ‘the social construction of reality’ won’t help us find an effective treatment for AIDS or devise strategies for preventing global warming. Nor can we combat false ideas in history, sociology, economics, and politics if we reject the notions of truth and falsity.”* See Sokal’s 2008 book *Beyond the Hoax* for additional details and discussion.

### Sokal and colleagues take on the “positivity ratio”

Now in a just-published article in *American Psychologist*, Nicholas Brown, Sokal and Harris Friedman take aim at a 2005 article, *Positive Affect and the Complex Dynamics of Human Flourishing*, which presented a “positivity ratio,” namely a ratio of positive to negative emotions. In their paper Frederickson and Losada asserted that if one’s positivity ratio is greater than 2.9013 (a parameter for a certain Lorenz attractor system), then one will “flourish”; if it is any lower, one is headed downward. The paper has been very widely cited since its publication in 2005: a check today on Google Scholar verifies that it has been cited 964 times while *Scopus* lists 348 citations in nominally serious research articles.

But as Brown, Sokal and Friedman point out, “*The idea that any aspect of human behavior or experience should be universally and reproducibly constant to five significant digits would, if proven, constitute a unique moment in the history of the social sciences.*” In fact, the ratio 2.9013 derives from a Lorenz attractor model — drawn in the paper — that was originally employed by a geophysicist in 1962, with no conceivable connection to psychology. Brown, Sokal and Friedman summarize their detailed analysis as follows:

We find no theoretical or empirical justification for the use of differential equations drawn from fluid dynamics, a subfield of physics, to describe changes in human emotions over time; furthermore, we demonstrate that the purported application of these equations contains numerous fundamental conceptual and mathematical errors. The lack of relevance of these equations and their incorrect application lead us to conclude that Fredrickson and Losada’s claim to have demonstrated the existence of a critical minimum positivity ratio of 2.9013 is entirely unfounded. More generally, we urge future researchers to exercise caution in the use of advanced mathematical tools, such as nonlinear dynamics, and in particular to verify that the elementary conditions for their valid application have been met.

### Conclusion

From all indications, the Fredrickson-Losada article is an exercise in “physics envy” — trying very hard to dress work in the social sciences, which, by definition, are not closely connected to very precise physical laws and processes, in the exalted language of mathematics and mathematical physics.

It is also the case that the whole area of social psychology has been rocked by recent scandals and by a prevalence of sloppy ‘science’. It has been described by Nobel economist Dan Kahneman as a “train wreck waiting to happen.”

But more generally, the lesson for all who would apply mathematics in this or any other arena of modern science and engineering is clear. Mathematics is a powerful tool, but there is no point in attempting to apply it beyond reasonable boundaries, or with a level of numeric precision far beyond what is justified by the original problem in hand. Mathematical excesses can lead to nonsense.

As J. W. Tukey, a co-inventor of the fast Fourier transform, wrote in 1962,

Far better an approximate answer to the right question, which is often vague, than the exact answer to the wrong question, which can always be made precise.