The path of great female mathematicians has led the way for future women wanting to enter mathematics, sometimes giving them opportunities that they wouldn’t have normally had. This can be seen with Maria Pastori, she attended the Maria Agnesi School. The Pastori family were not wealthy like the Agnesi’s, her early education was down to her own motivation and study. The Maria Agnesi school gave her the opportunity to learn the relevant subjects and therefore to nurture her career. Pastori went on to become the professor at the Istituto Matematico of the University of Milan. She studied in the field of tensor analysis and relativity. Tensor analysis was essential for the pure mathematical investigation of generalised spaces during the nineteenth and twentieth century when algebraic and differential geometry greatly developed. Pastori’s work helped to develop the tools necessary for such investigations. Pastori has been called a twentieth-century disciple of Agnesi, if Agnesi had not been such a prominent female maybe Pastori wouldn’t have received the education she needed and wouldn’t have shone quite as brightly as she did!
Sonya Kovalevskaya’s successor was Maria Cibrario. Her work helped to achieve the classification of linear partial differential equations of the second order of mixed type. She has also done research into non-linear hyperbolic equations and systems of such equations, and solved the Goursat problem for the hyperbolic non-linear equation of the second order. Sonya’s work would have been crucial background knowledge, what would have happened if Sonya had tumbled at the first hurdle and not continued her career against the odds and managed to make such advances in the topic? Maria Cibrario filled the chairs of mathematical analysis at both the Universities at Modena and Pavia.
Another twentieth century great woman mathematician was Jacqueline Lelong-Ferraud. Like Emilie du Châtelet, Jacqueline was among the first women to sit the examination for entry into the famous Paris Ecole Normale Supérieure. She was a professor of mathematics at the University of Paris. Her research topics included the behaviour of conformal transformations and representations, Riemann manifolds and harmonic forms and potential theory. Jacqueline also originated the concept of preholomorphic functions, consequently utilising them to produce new methodology for proofs.
Paulette Liberman was prolific in the field of algebraic topology, working on differentiable fiber spaces, and almost complex manifolds and their generalisations at the Charles Ehresmann’s research school. Like Emmy and Olga, World War II also disrupted Paulette’s work. She became a professor at the University of Rennes.
Sophie Picard occupied the chair of higher geometry and probability theory at the University of Neuchâtel. Current day students of probability and statistics would encounter her name when studying group theory, function theory, the theory of relations etc.
There have been numerous other twentieth century female mathematicians such as Emma Lermer who researched special cases of Fermats last theorem, Julia Robinson who contributed to Hilbert’s tenth problem, Elizabeth Scott, Grace Hopper and Dorothy Maharam Stone are all notable names of the current mathematical world.
Female mathematicians do appear to be more numerous in the later centuries, but is it because their access to academia has been easier, or is it just because more women are ready to take up the challenge? Looking at the women studied here there does seem to be a pattern.
Hypatia of Alexandria had access to education as a result of her father’s encouragement. She had a prolific career until a misogynist male Christian leader brought her and her career to a sticky end.
Maria Gaetana Agnesi again only received an education because her father encouraged it, he was proud to have such an intelligent daughter. She produced a two-volume book that had widespread popularity. Even with such an achievement the French academy of sciences would not allow her to become a member of the institute as it prohibited females. Maria dedicated her time to her only love - caring, her career was not as driven as some. Consequently she did not even attempt a position in academia.
Sonya Kovalevskaya had to fight for her education, employment, recognition and pay during the whole of her career. Her main obstacle was male prejudices against women. She was forced into a loveless marriage so that she could pursue higher education, but then was not permitted to enrol in any universities because she was a woman. Finally after much hard work, against all odds, she managed to attain her doctorate, to then discover that absolutely no employment was available to women in academia. When she eventually did secure an unofficial academic position years later, she got paid very little, much less than her male counter-parts. She won the famous ‘Prix Bordin’ prize from the French Academy of Sciences and produced many papers that developed the mathematical world. After many years of hard work she was finally recognised for her talent and was granted membership at the Imperial Academy. She was the first woman at the Russian Academy of Sciences but still no teaching/academic position was offered in Russia. Sonya was apparently a very beautiful lady, unfortunately this disadvantaged her in her professional career, and even in historical documentation of her life, people believed she used her beauty and persuasive powers to her advantage to get what she wanted. Could researchers not recognise that in most cases it was her intelligence that got her over such obstacles, as her achievements demonstrated that she was obviously gifted?
Emmy Noether was influenced and encouraged by her father to develop her intelligence. After her basic education she was lucky enough to be able to sit-in on lectures given at the University of Erlangen, but was not an official student. She had to wait five years before she could gain this title and be granted a doctorate. Once again it was hard to obtain employment, so taking on her father’s duties at the university was the closest she got to an academic post after graduation. Although her reputation was growing following the fine research she partook in, she received no recognition or pay from the university. On request she then went to work with Einstein, Hilbert and Klein aiding with breakthrough mathematics on the relativistic theory of gravity at the University of Göttingen, still with no pay or official position, having to lecture under the name of a male professor. Seven years later she was finally given the pathetic title ‘Unofficial Associate Professor’, but for the eighteen years she worked there she received no pay or benefits. Only after a World war and migration to America did she get at least a little of the respect she deserved (her title became ‘Visiting Professor’). Emmy’s accomplishments in algebra were vast, she published many papers, and has left a great mark on the development of mathematics.
Olga Taussky Todd also was encouraged by her father and family to seek further education. At this point in history females began enrolling officially at more liberal universities. Olga earned her doctorate in mathematics and was actually appointed an official position of ‘assistant’ at the University of Göttingen where she was helping to edit Hilbert’s works on number theory. She then went on to secure a position at Bryn Mawr College in America, where she had to be classed once again as a graduate student. She then moved to Girton College where she became a fellow but was not allowed to supervise theses, as she was a woman. During the war effort she accomplished great achievements on the subject of ‘flutter’. Then at the National Bureau of Standards she was given the appointment of mathematical consultant. She went on to secure a teaching/research position at Caltech, where only her husband could be appointed professor, and as his wife she had to be the research associate, this was as late as 1957! Then finally in 1971 she was granted the title of Professor. From only the 1960’s Olga began to be accepted into the prestigious academies and institutes.
As is demonstrated here, women throughout mathematical history have had a tough time being taken seriously and respected for their equally ambitious and prolific works in mathematics. They have had to work twice as hard as their male counterparts to receive any recognition just because of their sex. As once proclaimed about the acknowledgement (or lack of it) for such women:
“…for if Ginger Rogers had to do everything Fred Astaire did but backwards and in high heels, these women had to do everything their male colleagues did but they may as well have been doing it backwards, in high heels, blindfolded and up a steep slope…” 
Although women found it hard to break into academic careers due to male prejudices, it must be noted that if it weren’t for a minority of liberal men (mainly fathers) that were sympathetic to their cause they may not have even achieved what they have. For example if Hypatia, Maria, Emmy and Olga did not have such encouraging fathers that were in the position to aid their education and encourage them they may not have achieved so much. Although Sonya’s father was dubious, if it was not for Vladimir being socially aware of the problem and willing to commit to a fictitious marriage she would never have had a chance of continuing her education. Also if Weierstrass had not been kind enough to teach her separately, as she could not enter the universities she may not have earned her doctorate. Emmy also may have never begun her career, or achieved so much if Hilbert and Klein had not recognised her talent beyond her sex. Although unsuccessful, Hilbert and Klein also actively demonstrated for Emmy, for her to be recognised and rewarded with at least a title and maybe a salary for all her hard work.
Therefore in comparison of these five ladies (Hypatia, Maria, Sonya, Emmy and Olga) it does seem that as the centuries have passed things have been getting slightly easier for women in academia. At least by the twentieth century women were beginning to be officially accepted into the more liberal universities, and were being more readily employed, although they still weren’t receiving the same status or recognition as their male counterparts were. The publishers of ‘American Men of Science’ changed the title of their book to ‘American Men and Women of Science’ to highlight such a change of social views towards female scientists in this century. This was also reflected in their listings – in 1921 only forty-two women mathematicians were noted, but by 1938 one hundred and fifty-one women mathematicians were noted. Quite a vast increase (approximately a three hundred and sixty percent increase) over only seventeen years.
Acceptance into education has dramatically improved, in 1991 women earned forty seven percent of the mathematics and statistics bachelor degrees granted in America. Which demonstrates that at undergraduate level the number of males and females appears reasonably even, implying that female students are equally likely to get places in the universities and succeed as the male students. The equality all women mathematicians have been striving for. When looking at postgraduate education the statistics slightly change, only 19% of those earning Ph.D’s in mathematics were women. Which is a dramatic decrease compared to degree level. Comparing this with social sciences where women hold forty-seven percent of the Ph.D’s, it demonstrates the lack of popularity for mathematics, but why? Maybe the education system is not thoroughly as equal as it should be. Even though the universities may not be prejudice towards the sex of a candidate it takes time to eradicate beliefs such as female intelligence inferiority in the current generations of women and men, especially in the more classically academic subjects such as mathematics. They may still be affected psychologically by the prejudices of the past. 
Even though the number of women in mathematics is increasing, the number of men is also increasing at a faster rate, therefore the proportion of women mathematicians is actually supposed to be declining compared to men. This is supported by the fact that the proportion of women being employed in academic positions is much smaller than the proportion receiving higher education in it. Salary comparisons between males and females also support this theory that the position of women in mathematics has been actually declining since the early part of the twentieth century. 
A difference between men and women mentally was highlighted from a very early stage in mathematical history, embedding an idea of inequality into social culture. As mentioned in chapter one Pythagoras, although it is believed not intentionally, began such thoughts by emphasising females association with all that is earthly and material, and males with all that is heavenly and intellectual. This then led to intelligence being linked to the divine, and therefore to males. Eventually only male priests could participate in any learning. This was emphasised by Aristotle who declared women’s bodies were inferior to males and therefore their minds were also inferior. Hence males were the only ones who could transcend to the divine and possess such intelligence. In comparison the material, the physical, the personal and the domestic would always ground women. As a result women were classed as inferior to men for many centuries. Therefore it is not surprising that women became more closely linked to the ‘earthly’ life sciences, and males to the more ‘heavenly’ mathematics and mathematically based sciences when women were introduced to education. Which explains maybe why the Ph.D’s for social sciences are approximately equal between the sexes.
From a young age girls are trained to become more acquainted with the ‘material’ or domestic matters of life, unlike boys. It must be noted that if women had not been trained in such domestic matters, enabling them to support their husbands, a lot of great male mathematicians would not have been able to dedicate quite as much time to their studies and maybe not have been quite as successful. For example, Einstein’s second wife, Elsa, cooked, cleaned etc. so that he only had to care for his work. Where would he have been without her? Most probably suffering from malnutrition and unable to do work through hunger! When great male mathematicians achieve great things, they receive sole recognition, their wives who have supplied domestic bliss enabling them to fully concentrate and produce results are dismissed. But where do female mathematicians find their domestic help, so they can concentrate on their work? The majority of the time they don’t, once again they are at a disadvantage.
Therefore it is not surprising that females feel they have to shed their ‘womanliness’ to be considered seriously in the ‘heavenly’ world of male intelligence. For example the current social perceptions of a ‘bimbo’ or ‘airhead’ is a woman whose appearance is envied and admired, but is lacking in intelligence. This stereotype of linking well-groomed females with little grey matter can suggest to women that appearance may inhibit fair judgement of their intelligence. It seems unfair and unnecessary that women must rid themselves of all female conventions so that their intelligence may be taken seriously.
As mentioned earlier in chapter one and four, brains and skulls have actually been measured in the past (including Sonya Kovalevskaya’s brain) to determine the largest/most dominant brain between the two sexes. More recently the male and female brains have been studied by psychologist Doreen Kimura to determine if either one are more naturally substantial in any one domain. It has been concluded that intelligence levels are the same, but it has been suggested that women’s brains are more naturally accustomed to linguistic skills, and men’s are more naturally accustomed to mathematical skills. Studies have been done to specifically research mathematical abilities in both males and females. Unlike with skull size experiments in the past the researchers of these studies are of both sexes, so hopefully no prejudices would be seen in the results. The studies demonstrate that males perform better in the mathematical reasoning tests and the tests requiring analysis of spatial relationships, consequently males are better equipped for conducting mathematics. In comparison women are better at perceptual and verbal skills  . These studies suggest that mathematics does naturally fall into the male domain. Maybe Pythagoras’ belief that male minds were more suited to practising mathematics was actually very accurate all those centuries earlier, when the number one dictated it!
Anne Fausto-Sterling, professor of biology and medicine at Brown University has analysed the results of these studies and she believes it’s not quite so black and white though. She suggests that the results are not wrong, but that the statistical methods used were often open to interpretation, and so do not stand up under rigorous scrutiny. As if one wants to find a difference between two groups of people, and one tries enough methods of comparisons, then a difference can usually be found. The results that were genuinely different between the sexes were only negligibly different with just a few percentage points in it. Therefore Fausto-Sterling concludes (along with a number of other researchers looking into this topic) that so far there is no evidence of a difference between male and female brains. She suggests the negligible difference that sometimes was seen, could be due to socialisation differences rather than biological differences. She believes that the mathematical sphere of a girl’s brain is sometimes not always developed to its potential, unlike boys. For example, stereotypically boys are more likely to play with lego when young, which aids the development of spatial relationships. The majority of the time they are also exposed to such activities as woodwork, metalwork, baseball and basketball while they are growing, all of these activities help with mathematical development. In contrast, girls activities typically do not provide such informal mathematical training (you cannot learn spatial relationships from a dolls tea party!). Maybe Pythagoras wasn’t right after all.  
One differing aspect between males and females though can be their perspective. It has been noted that women tend to ask different kinds of questions to men. This phenomenon has been noted in the biological sciences since the 1970’s, the increased introduction of women’s perspectives has proved to be a catalyst for the introduction of new methods and new viewpoints. Women in all communities and relationships naturally provide a balancing influence to the male’s persona. Therefore the increased presence of female mathematicians would hopefully bring a fresh insight into the current and future research fields, should male mathematicians not welcome this?
Unfortunately the perspective that very academic subjects, like mathematics, lie within the male domain is heightened by social and cultural views at all levels of one’s life.
During a child’s education such a viewpoint can be prominently placed, while they are naïve to the gender inequalities. A study of American schools was done by Myra and David Sadker, which highlighted this failure within the system. They found that teachers tended to encourage boys over girls, giving boys more attention and time, in mathematics and sciences classes especially, independent of the sex of the teacher (this was also found in Australian and European studies). This was demonstrated as boys questions were answered more readily, they were given longer to speak and they received more feedback concerning their answers. For example, in a particular elementary school a teacher ordered a group of girls away from the mathematics department to allow the boys to get on with their work!   This problem was also highlighted in a popular ladies magazine survey in 1992. Seventy-four percent of the respondents claimed that at school the boys received more attention in preference to girls, or a teacher demonstrated favouritism towards males. The survey also demonstrated that the most inequality occurred in mathematics lessons!
Although the Sadker’s research did demonstrate that sometimes it wasn’t always the teachers fault that the girls were not successful in maths and science, a lot of the time it was actually the girl’s prejudices against such subjects that decreased their interest in them. It has been demonstrated that when girls reach puberty they increasingly believe that being intelligent in such subjects, i.e. being a ‘swot’, is not very socially acceptable and unfeminine. Hence they hide their intelligence, and as a result do badly. Once again this raises the issue that mathematics is perceived as a very masculine subject.
This inhibiting social viewpoint is reflected in the results of a survey done in 1990 by the United States government funded National Assessment of Education Progress (NAEP). The survey recorded and compared the exam results of nine, thirteen and seventeen year old students. At nine years of age the mean score for maths out of five hundred, for girls was 230.2, and for boys was 229.1. Therefore initially the difference between the scores was negligible, i.e. their abilities were very similar. This was also true at thirteen years of age. When they reached seventeen years of age though (the onset of puberty and need for social acceptance) the girls began to fall behind the boys by one percent. This change is highlighted in the NAEP maths assessment taken by students at the end of their secondary education, far fewer girls rated in the top percentile bracket. Also fewer girls took up the opportunity to enrol in advanced courses in mathematics. Even in the Scholastic Assessment Test (SAT), that is necessary for entry to college, the males scored approximately fifty more points in the mathematics section than the girls. Therefore from seventeen onwards the boys appear to do better in mathematics. Although (as noted earlier in this chapter) the amount of women earning mathematics and statistics degrees in America was equal to men, demonstrating that girls are just as capable in the subject at advanced levels. Looking at Ph.D’s in mathematics once again the boys dominate the girls. Therefore the girls are just as capable as the boys are obviously, but maybe social/cultural views or lack of proper development may hinder their achievements in mathematics.
These ideals of what is socially acceptable for young girls are not only enforced by friends or school but can also be found in the media. It is interesting to note that in films, on television, in music videos, and comic books, all of the scientific/mathematical ‘brains’ are male. Even as recently as 1997 a film was released entitled ‘Good Will Hunting’, which documented a mathematical boy genius. There have been numerous films like this, but there appears to be none documenting female geniuses. This type of stereotyping highly influences the younger generation, especially as television, films etc. play an important role in their relaxation.
Even later on in life during employment female’s can still be hindered by male domination, especially in mathematical/scientific roles. Men’s social clubs play a major role in the dismissal of women, this is where work is discussed, networking is done, and deals are made, all without the presence of females. In some countries, such as Japan, it is not socially acceptable for women to attend dinner with their male colleagues, and such informal situations are precisely the place where new ideas are discussed, Japanese women miss out on these.
A study was conducted to see if employers were bias towards male candidates. Two identical CV’s were dispatched to universities to make a recommendation for an associate professorship position, the only difference being one was under the name Joan, and the other John. The majority of the universities recommended that John should have the associate professorship and Joan should have the assistant professorship which smacks of the situation Olga Taussky Todd found herself in when applying to Caltech with her husband Jack. She was only allowed to undertake the position of associate researcher whilst her husband got the title of professor.
Unfortunately such prejudices can still be reflected in the salary of women employees. In 1969 the Women’s Bureau of the Labour Department compiled statistics that stated the annual salary for female full-time civilian scientists was $9,400, but males in the corresponding role received $13,000. Considering this was approximately only three decades ago it is quite startling.
Therefore as demonstrated here equality does not seem to be achieved as yet, and more readily mathematical literacy is needed in other disciplines consequently a lack of basic knowledge blocks numerous paths in education. Hence mathematics has become an important part of any career. To improve this situation more women need to be encouraged into the field of mathematics. It is not just as simple as changing women’s attitudes so that they show an interest in this field, the whole culture of mathematics must be reassessed to be more welcoming to them i.e. it is not just a masculine subject. The best way to tackle this situation is to start with the younger females, so that the next generation is already established in their interest. Therefore the education system must be looked at. Teachers must be trained to be aware of such bias against girls, be able to allocate equal time with both sexes, and maybe to develop different methods of learning that appeals to girls, for example group activities (girls have been shown to adapt to these more readily). Young girls also need to be exposed to more female role models, such as teachers and mathematical/scientific brains in the media. Even just the addition of feminine content in textbook problems – as a lack of it has proved to have a negative effect also. (In Sweden and Denmark they have actually done this already). In higher education, course scheduling, fellowships, assistantships, loans etc. must all be improved so that they are more acceptable to females. As it has been demonstrated that the low admission of females maybe due to such practices and polices being ideally designed for male students and teachers only.
An important factor to keeping and welcoming women into the profession of mathematics is knowing what is currently wrong, and what needs to be improved within academia. The roles of women’s associations are essential in this task. In the 1970’s the ‘Association for Women in Mathematics’ was set up with such a purpose. They publicise the roles of successful women mathematicians giving younger females mentors to associate with as well as dealing with any problems. They also provide communication to other women in the same position.
Therefore the current position of females in the mathematical world seems to be improving, but there is much still to do and maintain. Funnily enough this struggle parallels the one that women face trying to break into the clergy. As women are beginning to become ministers in many of the denominations of the Christian Church, they are also beginning to become leaders in all the denominations of mathematics. As demonstrated in chapter one, this parallel between religion and mathematics is not fluke, there is a deep connection dating right back to Pythagoras. The cultural academic male Christian priesthood is gradually changing.
The nuclear physicist, Fay Ajzenberg-Selove, once said:
“I will believe that discrimination against women has stopped when I observe that second-rate women are given tenure” 
As unfortunately the majority of the time it is still only the first-rate women mathematicians that achieve the positions and success in the discipline.
Equality will hopefully one day be achieved in mathematics, enough great women mathematicians have invested time, motivation, financial hardship and social in-acceptance to the cause, hopefully their efforts will be repaid.
 Margaret Wertheim, “Pythagoras’ Trousers. God, Physics, and the Gender Wars”, 1997, Fourth Estate Ltd, London.
 Lynn M. Osen, “Women in Mathematics”, 1974, The MIT Press, Cambridge, Massachusetts and London, England.
 E. T. Bell, “Men of Mathematics”, 1937, Simon & Schuster, New York.
 M. Dzielska, “Hypatia of Alexandria”, 1995, Harvard.
 T. L. Heath, “A History of Greek Mathematics” (two volumes), 1921, Oxford.
 B. L. Van der, “Science Awakening”, 1954, New York.