If you take the "positivity self-test" on the website for Positivity: Top-Notch Research Reveals the 3-to-1 Ratio That Will Change Your Life, a book by psychologist Barbara L. Fredrickson, the results page will tell you that "Dr. Fredrickson’s research indicates that a positivity ratio of 3 to 1 is a tipping point. This ratio divides those who merely get by in life from those who truly flourish." My "positivity ratio" wasn’t even close to 3 to 1. It wasn’t even 1 to 1. I’m a bit negative. I’m merely getting by.

I take comfort in the fact that the main bit of "top-notch research" behind Fredrickson’s theory of the 3–1 positivity "tipping point" has just been resoundingly trashed in a new paper published in American Psychologist by Nicholas Brown, a psychology grad student; Harris Friedman, a psychology professor at the University of Florida; and Alan Sokal, the New York University physicist famous for pranking an academic-literature journal with an elaborately nonsensical paper on "postmodern" physics.

In a 2005 paper published in American Psychologist, Fredrickson, now a professor of social psychology at the University of North Carolina, Chapel Hill, and her co-author, Marcial Losada, a Chilean psychologist and business consultant, used a mathematical technique drawn from a branch of physics called fluid dynamics to establish that positivity obeys the "tipping point" logic of a "nonlinear dynamic system."

Their idea was that one must achieve a very precise takeoff velocity before a positive frame of mind, or positive emotional tone in a relationship or cooperative project, begins to deliver serious benefits. Below the critical threshold of positivity, we remain grounded. But once we get up to speed, we climb upon a chipper wind and flourishingly soar. If you're only twice as positive as negative, you get next to nothing. Three times as positive as negative? The world's your oyster.

Fredrickson and Losada's paper was a huge hit. It became a go-to reference in the literature on positivity and garnered almost 1,000 citations in less than a decade—the academic equivalent of a No. 1 New York Times bestseller. Fredrickson parlayed that success into Positivity, the 2009 mass-market book mentioned above, which makes a big deal about the 3–1 ratio vindicated by Losada's sexy math.

Except Losada's sexy math is totally incompetent.

That’s the upshot of the scathing paper by Brown, Sokal, and Friedman. Losada had recorded the chatter of teams of business professionals collaborating on projects, and researchers later coded the "speech acts" of team members as either positive or negative. They also assessed the performance of those teams along certain metrics. Putting the two together, Losada found that teams with a 2.9013–1 ratio of positive to negative comments performed much better than those with only slightly lower ratios. However, as Brown, Sokal, and Friedman explain, Losada's data are flat out the wrong kind to plug into differential equations, and Losada's attempt to do so produced not a breakthrough about the nonlinear, tipping-point dynamics of positivity, but complete gibberish.

In response to this merciless thrashing, Losada has declared he's all booked up for 2013 and couldn’t possibly carve out a free minute for comment. Fredrickson, in a detailed response published alongside the critique in American Psychologist, bravely confesses that she never actually understood Losada’s debunked math. Even more bravely, she proceeds to argue that subsequent work on positivity ratios has left her ideas more or less intact. "Whether positivity ratios obey one or more critical tipping points" resembling those she and Losada claimed to identify is a question Fredrickson cautiously admits "merits further test."

Fredrickson never admits, or even suggests, that her book or website retail little more than speculation when they say that “a positivity ratio of 3 to 1 is a tipping point” that divides the flourishing from the nonflourishing. Yet the chief evidence that there is a positivity “tipping point”—a critical threshold before which the benefits of positivity are few and after which they really pour in—was Losada’s bad math. The authority and allure of mathematical specificity is great, and it’s doubtful that Fredrickson and Losada’s 2005 paper would have racked up such a huge citation count without it, or that Random House would have been so keen to ink a book deal with Fredrickson, or tout the “top-notch research” about the “3-to-1 ratio” in the subtitle of the paperback edition of Positivity.

Fredrickson tries to salvage what she can, but the attempt only highlights the conceptual shakiness of her larger body of work on positivity and the enterprise of academic social psychology writ large.

It’s not a great time for psychology. Diederik A. Stapel, a Dutch social psychologist, has recently confessed to serial fraud. That he gamed the peer review process of his field’s best journals so often and for so long calls into question the quality-control mechanisms of academic psychology. If garbage can pass peer review, as long as it is well-written and well-formatted garbage, then the authority conferred by appearing in peer-reviewed publications would seem to be slight. That the flaws of Fredrickson and Losada’s study were eventually exposed may seem like a sort of success, but they probably would have gone unnoticed if it hadn’t been such a blockbuster paper. If it was cited 1,000 times, how many other academic research psychologists actually read it? Twenty? Two hundred? It is not heartening that it took eight years for a grad student to catch the bogus math.

But matters in psychology are even worse than that. For lack of a less expensive and convenient alternative, academic psychologists mainly study the inner workings of American college students, observing them in labs and paying them to fill out surveys. While this method is sure to reveal interesting truths about the distinctive cultural milieu of the American college student, it is unlikely to uncover solid, universal truths about the psychology of Homo sapiens. A number of recent studies suggest that the rich, well-educated youths of industrialized Western democracies who make up the subject pool of most psychological research are not actually the generic, typical humans that the rich, well-educated elders of industrialized Western democracies take them to be. The minds of our young Aggies and Buckeyes operate a bit differently than those of people laboring under different conditions and in sometimes surprising and important ways. Academic psychology has become aware of this problem, but has yet to absorb how serious it really is.

When Fredrickson sets out in her rebuttal to defend the plausibility of the 3–1 ratio as a “tipping point,” she refers to a study she conducted with a colleague that looked at the self-reported emotions of, yes, American college students, and examined how the “positivity” or their feelings related to how well they got along with their American college-student roommates in a dormitory of the University of Michigan, an elite American school. “Strikingly,” Fredrickson writes, “for students with ratios below 2.9:1, absolutely no evidence emerged to suggest growth in relational resources ... By sharp contrast, among those with ratios above this same threshold, growth in relational resources was both evident and statistically significant.” Fredrickson then suggests that this counts as general evidence of nonlinearity in the effects of positivity. Yet it’s hard to take this sort of thing, in isolation, as evidence of much more than the character of the undergrad experience in Ann Arbor, and the attempt to use it to fill in the gap left by Losada’s annihilated equations amounts to hand waving.

Finally, most work in the psychological and social sciences suffers from a lack of conceptual rigor. It’s a bit sloppy around the edges, and in the middle, too. For example, “happiness research” is a booming field, but the titans of the subdiscipline disagree sharply about what happiness actually is. No experiment or regression will settle it. It’s a philosophical question. Nevertheless, they work like the dickens to measure it, whatever it is—life satisfaction, “flourishing,” pleasure minus pain—and to correlate it to other, more easily quantified things with as much statistical rigor as deemed necessary to appear authoritative. It’s as if the precision of the statistical analysis is supposed somehow to compensate for, or help us forget, the imprecision of thought at the foundation of the enterprise.

It’s interesting that Fredrickson in Positivity avoids the term "happiness," because she feels "it's murky and overused." One may say the same of "positivity." There is definitely murk. According to Fredrickson, the constituents of positivity are joy, gratitude, serenity, interest, hope, pride, amusement, inspiration, awe, and love. But why these emotions? Why not others? As an inventory of positivity, this seems arbitrary.

Fredrickson declines to include “bodily pleasures” as an aspect of positivity, though they are obviously desirable, sought-after, positive psychological states that contribute to psychological flourishing. Why? Because, it seems, she defines positivity as a psychological state that "broadens the mind" and brings us closer to others. Sensuous satisfactions "narrow your focus ... and help you meet a current survival need," making them "more akin to negativity than to positivity."

A less question-begging way to put this is simply to say that certain positive emotional states broaden our outlook while others narrow it. In Positivity, Fredrickson assembles a list of emotions that help us to take a wide perspective and see our continuity with those around us, and she develops a number of interesting ideas about them. That’s worth doing. Calling an account of these broadening, ego-effacing emotions an account of "positivity" may be a savvy rhetorical decision, but it cuts no theoretical ice.

How about erotic charge, social dominance, and physical bravery? These elements of “swagger” would seem to be positive attitudes predictive of a certain sort of success. Are the elements of swagger omitted in Fredrickson’s account of positivity because they do not broaden our outlook in the right way? Because Fredrickson has limited herself to nice emotions that leads to a kumbaya fusion of egos?

In any case, it’s not at all clear that Fredrickson’s list of nice emotions sensibly aggregates into a single sum of “positivity” that can be sensibly compared to a sum of aggregated negativity. Maybe there’s a sense in which “interest” belongs in the same bin as “love,” but what’s the exchange rate? How many pounds of interest does it take to buy an ounce love? When it comes to offsetting the negativity of disgust, does pride really work just as well as serenity? Does awe weigh against fear or amplify it? Are fear and embarrassment equal in their antagonism to the positivity of amusement?

This is what’s so irksome about the positivity self-test and the “science” of the positivity ratio more generally. It’s not just that that there’s no credible scientific finding to the effect that it takes three episodes of awe to pay back one episode of embarrassment; it’s that no one has ever looked. It’s arbitrary to give all the elements of positivity and negativity equal weight. But then nobody actually knows precisely, or imprecisely for that matter, how each stacks up against the others. It’s possible that this can’t be known, as a general matter, because the relative value of the various positive and negative emotions in terms of psychological well-being must be to some degree culturally and individually variable.

If Fredrickson would let us throw all good feelings, whatever they are, in a single positivity bin, and all bad feelings, whatever they are, in the negativity bin, there might be some sense in talking about ratios of positive to negative. However, that’s not where she leaves us. She leaves us with the ratio of an arbitrarily edited and arbitrarily weighted list of positive emotions to an arbitrarily edited and arbitrarily weighted list of negative emotions. This is not a quantity of scientific interest. This is specious quantification, whether or not differential equations have been incompetently applied.

The problem is not that Fredrickson is a bad research psychologist. The problem is that she’s one of the best. This is how it’s done by chaired professors at major universities at the top of the game. Filigrees of rhetorical precision atop unsteady pillars of conceptual bluff. Now and again, someone sees through the decorative math, but the semipro philosophizing just goes on and on.

As the great economic historian Deirdre McCloskey once wrote, "The scientist is always trying to persuade people that her evidence is just like the proof of the irrationality of the square root of two. If she can get people to agree that she has demonstrated the impossibility of monetary policy under rational expectations, she can knock of work early." Or, one might add, land a mass-market book contract, give a TED talk, and make more than few bucks consulting.