Not too many people these days argue against the idea that having more women in positions of leadership and authority would be a good thing for most organizations. But turn to the question of why more women leaders is a good goal, or why it’s such an elusive one, and you’re spoiling for a fight.

Maybe, as Sheelah Kolhatkar recently argued in New York magazine, we need more women in senior management because men are generally too emotional to make good decisions (thanks to testosterone that turns them “into a herd of overheated animals”). Or maybe, as a business consultant responding to a recent HBR report on why men get more promotions than women suggested, women have “innate skills” — the writer included among these “authenticity, purpose, empathy, devotion, curiosity, and mindfulness.”

At the root of both arguments is the idea that men and women are different at some fundamental level in their capabilities, and that, contrary to the old-fashioned assumption, the difference makes women especially well-suited (pun intended) to be leaders in business — or in politics, or anywhere else that focus, a cool head, and good social skills are useful. This idea draws on the research of some impressively-credentialed scientists, and on a theory that is so well-accepted that it’s no longer usually presented as a theory, but as a simple fact of human development. Brain organization theory holds that men and women literally have different kinds of grey matter, thanks to different hormones that shape the brain during an early critical period. (To be sure, most scientists describe the difference as an “average” not an absolute one — which makes the theory sound more modern and acceptable than older theories that treated men and women as basically different species.)

There’s only one problem: the theory isn’t supported by the evidence. Naturally, it can’t be tested through direct experimentation on humans (by introducing hormones during development), so that has left us with less direct research methods. But the human studies so far have been marred by such large gaps, outright contradictions, and poor methods (such as 180-degree turns in key definitions) that none of the links between hormone levels and so-called male or female “brain types” is left standing. Meanwhile, existing data from hundreds of studies conducted in the past four decades do not support the hypothesis that human brains are hardwired for masculinity or femininity. When an exciting new study comes out showing a correlation between early hormones and later attributes, people (including scientists) often neglect to take all the prior evidence into account.

In reality, there is far greater variation in skills and personality traits among women, or among men, than there is between the sexes. This is true for all cognitive and personality traits with the single exception of sexual orientation (and, to my knowledge, no one has suggested that being attracted to women versus men is the golden key to being a good manager). So even the finding that there are “average” brain differences between women and men can be misleading. Yes, if you test thousands of men and women, you’ll find that there are slight differences at the group level for some traits, and that’s one meaning of “average differences.” But the common interpretation — that “the average man” or “the average woman” can be accurately described as having a certain brain type — is not true.

Of course, many people are quick to embrace brain organization theory because it seems to account for something they perceive all around them: obvious and often large links between gender and behavior. But these can be explained without implicating the brain. The first cause can be found in the social psychology research on labeling and perceptual biases. (Cordelia Fine’s recent book Delusions of Gender is a treasure trove of such research.) There is overwhelming evidence that men and women are evaluated differently even when they are doing exactly the same thing — in fact, even when they are “paper” people with identical credentials [pdf], and the only feminine or masculine thing about them is their first name. As for the second cause, men and women often are doing different things, because our social norms and obligations remain quite different, even as our education and professional preparation has been getting more similar.

So women do have a difference to offer executive teams, but it can be chalked up to social not biological factors. I talked this over with a woman in top management at a Fortune 75 company. “Until our social fabric changes dramatically,” she said, “men and women will still have different experiences outside the workplace, and that affects the skills and knowledge they bring to business.” She went on to make an interesting observation, however, about today’s organizational hierarchies. “The higher you rise, the less you see gender difference” in leadership style, she told me, because everyone has had to compete on traditional “masculine” terms in order to ascend the ladder.

If we want the attributes that are widely associated with women to infiltrate boardrooms, many interventions are possible. Organizations could, for example, offer greater opportunities for job-sharing, and restructuring professional jobs so that hours of overtime on a weekly or even daily basis is not a routine expectation. That would avoid some of the career derailing that happens when women become mothers — and allow men to have better work-life balance, too. But no one-pronged strategy will do the trick, given how saturated our culture is with the notion that moms are the “real” parents and dads “help out.” Equalizing pay is another important step that companies can and should take. As long as there’s a gender pay gap, it will make more economic sense in heterosexual couples for the woman to fill in on the childcare, and to systematically prioritize his career advancement over hers in the inevitable trade-offs that two-career families involve.

Ultimately, as a society, we should get to the point that, for companies to get the attributes they want in leaders, they will screen for those attributes — and won’t make assumptions about their presence or absence based on gender. But remember those paper people: managers usually aren’t intentionally discriminating when they make hiring and promotion decisions, so we also need to have positive programs in place to neutralize the implicit biases that track employees by gender.

Nothing about women’s brains gives them a monopoly on collegiality, empathy, and so forth. Once we reject the specious theory that women belong — or don’t belong — in leadership positions because of their brain organization, we can let this distraction go and begin to focus on interventions to promote full inclusion.

Rebecca Jordan-Young is an assistant professor of women’s studies at Barnard College and author of the new book Brain Storm: The Flaws in the Science of Sex Differences (Harvard University Press, 2010).