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Translation by AB – April 17, 2020

### One, then zero

While the Fields 2018 medal has just been awarded to four exceptional mathematicians^{1}, it should be recalled that Maryam Mirzakhani was, in 2014, the only female recipient of this prize in 80 years: one woman in 60 laureates in total!

The tremendous under-representation of women at the highest level of science and technology is well known, but this “curse” seems to hit mathematics particularly hard. However: women’s associations, model women, testimonies, specialized conferences, all of this exists in mathematics and, if there has been some progress since the end of the 19th century, we are still there in 2018^{2} :

Zero. No girl joined the mathematics department of the École Normale Supérieure (ENS) on rue d’Ulm (Paris) last year. […] After the disappearance, in 1985, of the ENS dedicated to young girls in Sèvres, the predominance of “males” has not ceased to be confirmed.

Going through studies on the subject of women in mathematics, we always come back to more or less the same observations and the same “explanations”^{3} :

- Women represent less than 10% of the editorial boards of specialized journals (these high places of social reproduction).
- They are victims of a representation bias: you would have to be “brilliant” to succeed in mathematics, a quality that they seldom attribute to themselves (whereas it is above all necessary to work and be relentless…)
^{4}. - Women who succeed despite everything are suspected of serving as a “token”, of having been selected as a woman.
- Women produce less work, articles, because they are busier “at home” (but nothing is said about the quality of their publications).
- Women think they are “impostors” and believe they are less capable.
- The Fields medal rewards mathematicians under the age of 40, a period during which women give birth and divert their energy from research (this is obviously completely silly: we can make children and mathematics of very high level, the examples do not lack).
- Etc.

The symptoms are innumerable and draw the state of a situation as appalling as it is inextricable. Why, despite an obvious awareness and despite some very slow progress since the beginning of the 20th century, are women still so little invested in mathematics? We suggest some reflections while recognizing our own bias: whether we are a woman or a man, any “speech” always comes from a body full of experiences and prejudices.

### Mathematics and Power

Studies about the representation of women in mathematics show nothing obvious or clear. Those on the inside, the mathematicians, do not seem to really see the problem (while noticing it) and those on the outside invoke “stereotypes” while losing sight of the essence of mathematics as well as the more general architecture of the question (what is a stereotype the symptom of?). Thus, the under-representation of women, for example on boards of directors, political bodies, digital companies, filmmaking, etc. does it really come from patterns also applicable to mathematics? Perhaps, because we should start simply with the obvious: we still live in a patriarchal society where the places of power and the transmission of symbols are saturated with men. Are we moving away from the subject of the under-representation of women in mathematics? On the contrary! If the under-representation of women is in general the trace of a place of power, then mathematics is (and always has been) a place of power. The first step towards a rebalancing would then be, for mathematicians, to consider their own power, of which here come some effects.

Mathematics is a means of selection and reproduction of elites by selecting individuals who are brilliant, snarling at problems, ingenious, superior, whether or not they subsequently do mathematics. Mathematics also allows us to develop our artefacts, although the relationship between the mathematical cause and the concrete effect is most of the time difficult to perceive (the causal chain is often long, this is partly what “Protects” mathematics in this debate). But mathematics can sometimes deploy a direct power effect, the consequences of which we unfortunately experienced during the 2008 financial crisis (we underline)^{5} :

In reality, mathematical models do not describe professional finance, they create it. For example, credit risk is not modeled by financial mathematics, it is – literally – modeled by them. New scientific tools are creating new professional practices, and we have witnessed a creative drift in mathematics in finance since the 1980s. It is the demiurgic power of financial mathematics, their

shaping of the world, a capacity that the sociology of science calls “performativity”.

Financial mathematics does not give tools or models to finance: they are finance, they are the territory, not the map. More generally, our digital (and therefore social) infrastructures are gradually saturated with mathematical algorithms with very real effects such as, for example, the Gale-Shapley algorithm (supposed to solve a “gendered” problem, that of “stable marriages”! ) which is the pride of the designers of Parcoursup [Parcoursup is an application process designed by both the French Ministry of Education and the French Ministry of Higher Education, Research and Innovation to allocate undergraduate places to high school students and other candidates].

Of course, specialists in algebraic geometry, logicians, categoricians … all those who deal with abstract mathematics (in the first place) probably do not conceive this kind of performativity for their discipline. But mathematics forms a coherent “body” to which one apply indifferently, according to their relationship with the political, social, education, etc. the strategies and effects mentioned above. Each mathematician is, in fact, committed to the name of mathematics as a whole. But it is probably a lot to ask a specialist, say from Landau’s theory of damping, to report on the 2008 financial crisis … or to ask him to report on the under-representation of women in his discipline. But he can at least consider the nature of his power.

The question of parity often seems to arise from the same dilemma: either we are too much “inside” which, in the technical world, means in its extreme specialty and we do not perceive its own effects of power, or we are too much “outside” and we come up against this impossible challenge: to change the “structure” of power itself.

### Mathematics and History

The real problem could be better perceived (less concealed) if there was already a counterintuitive change: stop making icons of the few women spotted in the history of mathematics. They are very bad examples for young girls! The main reason is that they seem to attest to the possibility of “succeeding” in mathematics … in a society of men (but at what cost!).

As far as antiquity is concerned, we cling to the name of Hypatia, better known for her tragic death than for her work. Elsewhere, we dwell on the names of Sophie Germain, Sofia Kovalevskaïa, Émilie du Châtelet, Maria Agnesi, Ada Lovelace … But, if they were all valuable “mathematicians” (if we take into account the conditions under which they were to exercise) , their life is not enviable as the biases and tactics for accomplishing their work are sometimes appalling. They were exceptional beings rather than “models”.

We can guess the intention: to show young girls, teachers, parents … that women are capable of doing mathematics which, by the way, should not be a question. We therefore summon these mathematicians and summon them to set an example. But we cannot ask for anything like that, say to Sofia Kovalevskaïa who in the society of the time inspired this^{6} :

The sad sight of this woman whom nature had showered with her favors and who perhaps a clumsy work, certainly excessive, made irritable and unsightly, of this woman who, at thirty, finds life too long and fade away exhausted at thirty-seven, can and must serve as a salutary warning for inexperienced young girls who, following the suggestions of a real or apparent vocation, propose to adopt mathematics as a professional and scientific occupation; before making such an important decision, they will have to weigh well if they have sufficient strength to bear throughout their life the heavy burden that awaits those who aspire to follow in the glorious footsteps of Euclid. The example of the deeply troubled life of Sofia Kovalevskaïa raises the question whether the ruler, the compass and the logarithms are not instruments too heavy for female arms.

This author, Gino Loria, quotes a certain Moebius, “*whose ancestor was a known mathematician*” (the author of the eponymous strip) and who retains nothing of his contempt:

So, we can say that a female mathematician is unnatural; it is in a sense a hermaphrodite. Learned women and artists are products of degeneration. It should be noted that these women take the masculine type: Sophie Germain has the appearance of a man, “The Kovalevsky” proves that women can hardly possess genius and health; she was extremely nervous and the ailments from which she suffered made her age early. “The Germain” was a good nature, but “The Châtelet” represents the brutal type of the degenerate […]

Closer to us, Emmy Noether was a very great mathematician of the 20th century which was attested by … by her rise to the rank of “man”^{7} :

She was nicknamed “der” Noether by some contemporaries and it is customary to emphasize her “ugliness”, her lack of femininity, all indicating that she was “in fact” a man, or at least a hybrid being. At the same time, a large number of feminine traits, derogatory, are often called to describe it, in a pleasant way (mother hen, talkative, etc.).

When we look for models, we find them again and again, like this wave initiated by the film “*Hidden Figures*” in 2016 which recounts the role of “*black mathematical women*” in the American space conquest^{8} :

They broke down prejudices and social barriers at the time of social segregation. Awesome black mathematician women played a very important role in the great successes of the American aerospace program.

To describe these women as “*awesome mathematicians*”, one must not exaggerate (and moreover the social barriers and prejudices have they really broken down?). Pay homage to them, of course, but on the condition of not falling into a kind of dishonest “women washing”. The reality is that most of them were more like auxiliaries of calculation, which is reminiscent of this prejudice allowing men to retain girls from doing real mathematics while seeming to open the field to them, as Evelyne Barbin relates, evoking the situation in the 19th century^{9} :

The feminization of mathematics education also involves choosing the type of mathematics to teach girls, but abstract education does not suit them. On the other hand, as Camille Sée writes, women excel in calculation: “everything in mathematics that prepares for special schools must disappear [from the education given to girls]; but women need to know how to count very well, and with good preparation, they excel at it”.

These stories, these prejudices, these bad examples … force us to hear the reluctance of today’s mathematicians to be summoned as women in their turn, like the great Maryam Mirzakhani herself^{10} :

[…] while she feels greatly honored to have been awarded a Fields Medal, she has no desire to be the face of women in mathematics, she said. Her ambitious teenage self would have been overjoyed by the award, she said, but today, she is eager to deflect attention from her achievements so she can focus on research.

The second shift to make would be to recognize not the talent of these rare women offered as an example, but above all the difficulty of their journey and why it was so: it is an integral part of the history of mathematics.

We can still be indignant today but, as we have seen, we still come from very far!

### Mathematics and Ethics

We have to mention this about mathematics itself: absolutely nothing in the mathematical matter, in its representations, concepts and symbols seems “gendered”. It is impossible to distinguish, except perhaps having an unimaginable hindsight or a bad spirit, the slightest trace of masculinity in this building, however, almost exclusively constructed by men.This absence of gender, and more generally of any cultural trace, in a way exempts mathematical practice from all ethical questions. It is therefore very unnatural to consider, when one is a mathematician, anything that should be corrected or made more “ethical”, or even to consider one’s own biases. So, as Jane C. Hu writes:

Those committed to scholarly objectivity may pass off their personal beliefs as ultimate truths without recognizing their own biases may have crept in.

A professional distortion in a way, quite marked in a discipline where Platonism always outcrops^{11}, where it remains possible that mathematical entities (numbers, geometric figures, etc.) have an independent objective existence, and are not phantasms of the human mind. In this case, there can be no trace, in mathematics, of human indignities.

### Mathematics and Philosophy

Philosophy shares with mathematics the historically masculine privilege of producing concepts for which no immediate performativity is required^{12}. As a result, the situation of women in philosophy is almost the same as that of women in mathematics. It’s even quite striking^{13} :

Studies confirm that the higher you go up the academic ladder, the fewer women there are in philosophy. The proportion of women shrinks when graduating from baccalaureate to master’s, from master’s to doctorate and from doctorate to professorship. Ultimately, women make up only 20% to 30% of the faculty in philosophy departments.

We find the same observations about an elusive situation:

The lack of models to identify with among its professors, among the philosophers that we read or hear in lectures promotes the feeling of impostor: we have the constant impression of not being in our place as a woman in philosophy.

The same “history”:

If I asked you to name five female philosophers, chances are you’d be hard pressed to quickly find names other than Hannah Arendt and Simone de Beauvoir. We will therefore not surprise anyone by saying that the philosophical tradition was until the twentieth century (and still today), especially a case of white men.

The same platonist “innocence”:

This claim to the universal and to the objectivity, so dear to philosophers, also fuels their resistance to changes in the discipline. Considering having objective and rational judgment helps to mask the equity issues arising from their practices.

And thereafter, the same conclusion:

The real issue for women in philosophy is not just achieving digital equality, it is about achieving real power sharing.

Women generally remain excluded from the real places of power, among which these sources of production of symbols and of ordering of the world which constitute mathematics and philosophy. Pierre Bourdieu theorized this by speaking of “*symbolic violence*“. His analysis had been received to say the least in a mixed way at the time, notably by certain feminist movements, accused that it was to downplay or forget other more immediate and concrete means of domination, economic or physical for example. Anyway, it would be necessary today to re-read and probably update its contributions: our symbols obviously play a fundamental role in social organization and certain activities, including mathematics, produce more of them than others.

### Mathematics

These few reflections do not lead to clear balancing tracks but show, it seems to us, the limits of a fight that would be too directly gendered (associations of women, exemplary women, etc.) or classified “Fight against stereotypes”. These approaches always present the risk of contradictory results, or even of a status quo through the reproduction of gender opposition: because she is a woman … rather than because she is an excellent mathematician …

Here are some observations, however.

First, we can at least recognize the responsibility of mathematics in economic, political and social production: it is an eminently performative field, for which there is a form of global responsibility to be better considered, especially from within mathematics and also with non-mathematics.

Second, we must modify the way of entering mathematics which is terribly discouraging for children, at least in France: it is everyone’s taste for mathematics that we must develop as well as confidence in oneself by an appropriate pedagogy. This point somewhat resembles the previous one: the very concrete role of mathematics in the mediation of humans with the world are to be demonstrated to children, where we still brilliantly theorize and intimidate the weakest.

Third, we must assist the parents, the main propagators of stereotypes, to support their daughters in mathematics. History shows at least this: a mathematician woman has always been supported by one of her parents.

Four, it is through an inclusive approach of this problem, also conducted with men, with mathematicians and non-mathematicians, that we can move forward. In this context, we can then consider acceptable and useful forms of positive discrimination for the benefit of women but especially mathematics. In other areas, this radical method has proven itself. The problem is that this seems unthinkable for the very reason that mathematics pretends to be objective: we are selected by our suitability for concepts, not by our gender. Well, precisely it’s too simple and it shouldn’t be an argument.

Finally, women will not accept being pushed in the back simply to balance a ratio: they must simply have an interest in it. Women in search of empowerment are here at the forefront^{14}. It is not entirely surprising that Maryam Mirzakhani was Iranian and attended secondary school in her native country.

1. ↑ Thomas Messias / Slate – August 2, 2018 – *La médaille Fields et le club très fermé de l’élite des maths*

2. ↑ Philippe Douroux and Magalie Danican / Libération – June 21, 2018 – *En mathématiques, les filles restent des inconnues*

3. ↑ Jane C. Hu / The Atlantic – November 4, 2016 – *Why Are There So Few Women Mathematicians?*

4. ↑ Pierre Bourdieu, interviewed in 1998 by Catherine Poitevin: « *As in Kabylia, for the olive harvest the man arrives with a large sapling, masculine symbol, okay, but above all he hits the branches, brief male activity, it lasts ten minutes, and then the woman and the children pick up olives under the sun for whole days* ». Men show a brief and efficient power, women persevere.

5. ↑ Christian Walter / Libération – May 21, 2012 – *Le pouvoir démiurgique des mathématiques financières*

6. ↑ Gino Loria, Italian mathematician / Revue Scientifique – September 26, 1903 – *Les femmes mathématiciennes*

7. ↑ Catherine Goldstein / Sextant 2 – 1994 – *Une créativité spécifique des femmes en mathématiques ?*

8. ↑ 20 minutes – September 20, 2016 – *Nasa : Des mathématiciennes noires géniales au cœur de la conquête spatiale américaine*

9. ↑ Evelyne Barbin / Repères – IREM N° 97 – October, 2014 – *L’enseignement des mathématiques aux jeunes filles et les stéréotypes de genre* (broken link)

10. ↑ Erica Klarreich / Quanta Magazine – August 13, 2014 – *A Tenacious Explorer of Abstract Surfaces*

11. ↑ Wikipedia – *Philosophy of mathematics*

12. ↑ Women themselves, of course, conformed or supported the stereotype of manly conceptual creation. For men the “official” science, that which makes and produces, for women the “unofficial” science , that is to say magic, hypnosis or magnetism … this science preceded the science of men from so far that it never overshadowed it, as this author suggested in 1904: “*Women therefore have an important role in unofficial science, and often hidden behind the personalities of their husbands and brothers, they can even take place in the debates of official science*“. What a privilege!

13. ↑ Marianne Di Croce / Raisons Sociales – November 19, 2015 – *Place des femmes en philosophie : un panorama de la question* (broken link)

14. ↑ Claire Levenson / Slate – February 20, 2018 – *Plus un pays est égalitaire, moins les femmes s’orientent vers des études scientifiques*