**An appropriate topic for February 11th, the "Women in Science Day"**

*Perhaps a more important interpretation of the date: Exactly one year ago, the first gravitational wave detected by LIGO was officially announced*

Kevin Hartnett wrote the article

A Fight to Fix Geometry’s Foundations.He did some good work and it wasn't a waste of time for me to read it. But the bias that penetrates the article, starting from the title, is something I simply cannot accept. Like in so many other cases, the journalist has simply decided who is "right" in a confrontation and we're sort of not surprised that the "heroes" were the side that looked more politically correct to him while the other side were the "villains".

The title already suggests that there is something wrong with the "foundations of geometry" as a subfield of mathematics. Well, first of all, it's not all of "geometry" that's been accused of that lethal disease. It's just symplectic geometry. Second, whether there's something fundamentally wrong – more serious than some minor bugs that can be fixed – was the topic of the fight. Hartnett implicitly decided that those who say that things are basically fine must be wrong even though he seems to believe that they're the majority of researchers in that field. What has led

*him*to that conclusion isn't described but yes, I am 99% confident that it's some dishonesty of the PC writers.

What's going on? Symplectic geometry is the research of some spaces parameterized by some coordinates with the structure that resemble that on the general phase spaces in theoretical physics. In classical mechanics, you have the positions \(x_i\) and the momenta \(p_i\). They have nonzero Poisson brackets \(\{x_i,p_j\}=\delta_{ij}\) which become the commutators in quantum mechanics.

If you unify \(x_i\) and \(p_i\) to "one big column of coordinates" \(q_A\), the matrix appearing in \(\{q_A,q_B\}=\omega_{AB}\) is antisymmetric. You may see that this set of numbers \(\omega_{AB}=-\omega_{BA}\) are functions of all the coordinates \(q_A\) on the phase space (which may be curved and complicated when the particle moves through more complicated manifolds), i.e. functions of both positions and momenta. More generally, a symplectic manifold is one that is equipped with this antisymmetric tensor function \(\omega_{AB}(q_C)\) defined at all of its points.

You can do various things with these objects. One of the things you often need is to find the number of intersections of two submanifolds whose dimensions add up to the total dimension. In most cases, it just works. Sometimes the two objects overlap and the number of intersections is infinite even though the desired, morally correct intersection number should be finite. One may add wiggles to the curves and the problem is solved: the intersection number becomes finite.

Most of us would be satisfied because it works in simple and smooth enough cases. In the most general cases, we would be satisfied with some

*ad hoc*lessons. Mathematicians want these methods to work universally and they have noticed some technical glitches. The wiggles may only be safely added locally so when the intersection numbers are computed, one must combine the information from patches in a manifold-like way.

The Kuranishi structure was invented by Kenji Fukaya and Kaoru Ono when they studied the Gromov-Witten invariants in symplectic geometry (something that the most mathematical or "topological string theorists" know extremely well) to formalize these ideas and prove the so-called V.I. Arnold conjecture relating the number of fixed points of a symplectomorphism to Morse theory. These ideas named "Kuranishi" after the authors' countrymate born in 1924 – with links to category theory and other advanced topics – were visionary in some sense. But the original paper had mistakes and those are being acknowledged by Fukaya and Ono. Well, they were not only acknowledged but pretty much fixed.

The "battle" was about the seriousness of the mistakes. Are they serious? Can they be fixed in a straightforward way? I think that Hartnett's article makes it obvious that most experts believe that they're not serious and they may be fixed so Fukaya has ignited an important project that actually works. For some reason, Hartnett chooses the opposite answer as the one that he wants to promote instead. Relatively young mathematician Katrin Wehrheim teamed up with a senior figure Dusa McDuff in 2009 and they began to publish articles claiming to fix all mistakes in the Kuranishi industry. The Google Scholar link in the previous sentence indicates that these papers haven't been considered too important by the community of mathematicians as of today, at least in comparison with the 1999 Fukaya-Ono book.

All such papers are complicated enough and are being consciously read by a very small number of people on Earth. It cannot be surprising that there are mistakes in them. But "a mistake" doesn't imply that "everything is trash and should be thrown away". A mistake may often be fixed and even the fix just isn't always the #1 priority of top mathematicians' work. Wehrheim was systematically overselling these mistakes and she basically wanted to grab the credit for Fukaya's work just because she emphasized the errors and fixed some. (There's obviously no rigorous proof that there aren't other errors and I think that there probably are.)

In reality, it's questionable whether she and McDuff should even get the credit for correcting the mistakes. Google Scholar indicates that even when it comes to these clarifications, Fukaya-Oh-Ohta-Ono were far more important.

Katrin Wehrheim is a skillful young mathematician who may be assertive and perhaps annoying. But it's OK and Fukaya has unbelievably patiently answered all the questions that were asked – and wrote hundreds of extra pages of explanations in papers and a special newsgroup. But what I dislike is especially the attitude by the journalists, in this case Hartnett. Wehrheim refused to be interviewed because "she didn't want to politicize things further" – which is reasonable – but Hartnett politicized it, anyway, and decided to find Wehrheim's views otherwise. What did he do? Well, he read her

profile in MIT's "Women In Mathematics"Like other pages of this kind, "women in mathematics" isn't just about some mathematicians who happen to be women. The very reason why these publications exist is to

*selectively strengthen the voice of and oversell*women in mathematics. The whole publication is obviously an arm of the feminist movement that attempts to contaminate mathematics (and science) by similar ideological biases. It's wrong, wrong, wrong for the Quanta Magazine to use such dubious sources.

In the first part of the profile, she complains that mathematicians don't write good papers anymore. It could be right or wrong. But her climbing metaphor makes it clear that I just couldn't possibly agree with the reasons behind her criticism:

It's like if some really well trained climber made it to the mountain top. But they didn't leave any hooks along the way, so someone with less training will have no way of following it without having to find the route for themselves.I am sorry but it is totally OK for skillful climbers to reach the top without hooks. And such achievements – in mathematics, science, and adrenaline sports – are an important source of the wow factor which is what makes some practitioners special. The key work done by the best researches, especially the geniuses, is something else than teaching. If they can make it to the peaks more effectively, they should do so and they must have the freedom to do so. It would be a waste of their precious time to demand that these people spend most of their time by detailed explanations for less trained – and less talented – followers or by adding the hooks everywhere.

*This cyclist is using no hooks to move in the mountains. Maybe he's violating laws and some of us are close to vomiting while watching the scenes but who doesn't admire – or who wants to criminalize – things like that and their counterparts in other fields is really working hard to slow down the genuine progress because most of the biggest steps often need exceptional guys like that who can't be followed by the generic colleagues. Wehrheim has impressively strong muscles to have nearly participated at the Olympic rowing tournament but this guy has an extra X factor, doesn't he?*

At this point, I should add an extensive discussion about the need for absolute rigor (more precisely, the absence of such a need) and the role for heuristic methods or even speculations in mathematics. If you haven't read them, I recommend you the 1993 Jaffe-Quinn paper and especially the 1994 responses by top mathematicians that made it very clear that most of the really best ones think that the methods and levels of rigor favored by theoretical physicists have their place and play an important role in mathematics, too. If Wehrheim wanted to delegitimize all similar efforts that aren't just full-fledged paths with all the hooks ready to be used by the untrained folks, she would throw the baby out with the bath water. The purpose of mathematics isn't and cannot be the easiest possible accessibility of the body of results by everyone. Indeed, progress and new advances, including conceptually new ones, are what mathematicians try to get, and whenever they do, the body of mathematics unavoidably becomes less accessible in average.

The later part of the profile, basically 50%, is all about the feminist cause. For example, we read:

In Katrin's experience, women bring a different culture of thinking into mathematics that helps deal with these issues.This is just garbage and the unproductive mess that Wehrheim helped to cause is one of the more extreme examples showing why the proposition above is totally wrong. Mathematics just doesn't depend on sexual organs. It's mainly a men's game and if a woman becomes a good mathematician, it's because she learns the game basically just like anyone else who is good – she learns to think in a way that has been associated almost entirely with men and she does it well. She becomes an "honorary man" in this intellectual sense. The idea that her womanhood, characteristics by which she differs from males, is very useful for mathematics is just rubbish.

But it's pretty much every following sentence that is offensively wrong.

It's not that all men and women have distinct ways of doing math, but she notices that many women tend to focus on what they do not understand, while their male colleagues often rush to push together pieces they do understand and just take certain things for granted along the way.I am sorry but men are rather typical for their focus on details – and their occasional desire to question the details. If this "maximum amount of rigor one may demand and achieve" is the real contest, be sure that men would almost always win this contest, too. Women are actually better in multitasking. These differences are similar to many biological differences outside the brain. For example, the sexual arousal of men is much more linked to some more or less small ;-) fractions of their body while women may get excited at many more places on their skin.

If something similar to the statement above is true, it's that women usually realize that they misunderstand something – much like men realize it – but because the bulk of girls in a class often misunderstands something about lectures in mathematics, it's enough to force the teacher to do things again and very slowly. But all of these things are

*negative*for the pace of progress in mathematics or science. These characteristics are not advantages which is how they're ludicrously painted.

You know, after I read Hartnett's article and thought why he would pick Wehrheim's perspective so uncritically, I decided that Wehrheim cannot be "just a female mathematician". There had to be something more "intense" about her. My prediction was confirmed at some point:

Katrin came out as a gay woman during a panel discussion at an MIT women’s math conference, and to loud applause.I see. Sorry but there's no justifiable reason to loudly applause when someone points out that she is lesbian. Someone is straight, someone is lesbian, and those things have nothing to do with mathematics. It's exactly as wrong for participants of a mathematics conference to loudly prefer one of these sexual orientations as it is wrong for a conference in Germany of the 1930s to loudly prefer Aryan scholars. The loud applause at a women's mathematical conference strongly suggests that a big fraction of women in mathematics are there for reasons that are more political than mathematical and that's very bad.

Since then, she's hoped that queer women would become more visible in academia and become role models for others.Women represent a small minority of mathematicians and lesbians are a small minority of this minority. There is no reason to expect that they should be "visible" and if they're visible, it almost certainly means that something else than meritocracy has taken over their environment.

Her "PC credentials" go on and on and on:

She's been helping with an intense and successful effort by MIT to recruit students and post docs from nonstandard backgrounds.But I had to laugh out loud when I read the last sentences about her own hiring:

When looking at applications, she says the committee looks for evidence of a strong trajectory rather than how the students currently compare to peers or if they took AP calculus in high school. Oftentimes, those people are women. "When we find those students, we don't just email an offer," Katrin says, "We tell them in person how much we will do to help them succeed at MIT." "Actually that's why I'm here," she points out. "I had tons of offers, but MIT was one place where I was sure they didn't just need to fill a quota, they actually wanted me. They went out of their way to sweeten the deal. They even would have found me a place to store my boat."I am sorry lady but you have proven that you haven't mastered basics of rational reasoning. First, you tell us that you are aware that MIT does everything it can to enforce affirmative action and gives all sorts of special advantages, encouragement, and assistance to members of the privileged groups and to their boats. You have seen it, you have done it yourself. And you correctly say that this is why you were hired, too. But seconds afterwards, you tell us that that you weren't hired just because they needed to fill a quota. Can't you see the contradiction? If the new women and lesbians are being hired because MIT needs to do miracles to increase their percentage at MIT, chances are high that it's exactly the reason why you were hired, too.

It also looks weird that someone brags that she has picked the best offer from the viewpoint of her finances and her boat. Most people with a plethora of job offers probably choose the "sweetest" offer (let me admit that I would say that this is what I did e.g. in 2001, too) but they're not bragging about it. It's common sense and there's nothing to brag about.

At any rate, it's unethical for journalists claiming to be science journalists – like Hartnett – to make someone and her viewpoint more visible just because she is female and lesbian. When I compare the positives and negatives that articles such as Hartnett's may bring to mathematics, I think that the negatives prevail.

When Thunderfoot said that feminism poisons everything, he really meant everything. And yes, that includes symplectic geometry. Shame on all the people who are polluting even most esoteric fields of serious scholarship by this toxic garbage.

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