Radical egalitarianism and mathematics

In my article  The Universalism of Mathematics and its Detractors I have tried to sound the alarm  about recent trends of deconstructing mathematics based on what I called radical equality dogma (RED1), a precept which claims that “human beings are roughly identical in terms of talents, intelligence (both cognitive and emotional), interests, motivation, powers of concentration, ability to perform various cognitive  tasks, and that, consequently, every visible disparity between groups of individuals has its origin in some form of bias and discrimination”. 

These, as explained by I. Kendi in his book How to be an antiracist can only be cured by further  discrimination. Thus, he writes: “The only remedy to racist discrimination is antiracist discrimination. The only remedy to past discrimination is present discrimination”. 

This the exact opposite of the traditional notion of fair treatment, once viewed as the gold standard for how Americans are supposed to judge and treat each other, according to which “individuals should only be judged by their intrinsic merits and actions, with no regard for their race, ethnicity, religion, nationality, sex or other personal characteristics and no reference to other individuals, or societal or institutional considerations. Unlike fairness, centered on the individual, that is measurable and dealt with by individual-centered measures, equity understood as equality of outcome requires one to compare individuals based on all possible identity groups to which they belong, a task impossible to accomplish without either setting all against all, in a dystopian state of nature society, or by appealing to a strong authoritarian or totalitarian state to impose its will on how resources are to be divided and how people are to be rewarded or punished”.

A similar dogma, which I will call RED2, holds that all civilizations, cultures or countries have contributed, and continue to contribute, equally to all important cultural developments in science, arts, music, literature, mathematics, philosophy etc., and that any assertion to the contrary is a manifestation of cultural hegemonism, better known as cultural colonization, the ultimate sin which the West is uniquely guilty of. 

According to both versions of RED, mathematics, far from being the universal discipline we used to think it was, is in fact heavily dependent on the group that practices it. It may thus be called ethnocentric, racist and sexist, due to the fact that most of what is taught in schools and universities, was produced, allegedly, by European white male. Here is in the words of Professor Rochelle Gutierrez, holder of many prestigious prizes in education: “Algebra and geometry perpetuate privilege because emphasizing terms like Pythagorean theorem and the symbol ㄫ gives the impression that math was largely developed by Greeks and other Europeans”. Moreover, she writes, “On many levels mathematics itself operates as whiteness. Who gets credit for doing and developing mathematics, who is capable in mathematics, and who is seen as part of the mathematical community is generally viewed as white”. She also claims in the post-modern  style of the times  “Things cannot be known objectively; they must be known subjectively”.

In a peer-reviewed article professor of mathematics education at Vanderbilt University Luis A. Levya, states: “The framework developed and presented here illustrates three dimensions of white institutional space, institutional, labor, and identity, that are intended to support mathematics educators in two ways: (a) systematically documenting how whiteness subjugates historically marginalized students of color and their agency in resisting this oppression, and (b) making visible the ways in which whiteness impacts white students to reproduce racial privilege”.

Professor Levya delivered an invited address at the 2023 Joint Mathematics Meeting, the most prestigious yearly gathering of mathematicians in the US, with the title “Undergraduate mathematics education as a white, cis-heteropatriarchy space and opportunities for structural disruption to advance queer of color justice”.  Laurie Rubel, math education professor of Brookline College, claims (in the same journal) that both meritocracy and color-blindness are ideological precepts that hold back racial minorities from succeeding in math classes.

More recently the Australian National University mathematics professors Rowena Ball and Hongzhang Xu wrote: “Efforts to decolonize mathematics curriculums, professional societies, and the discipline itself are gathering pace in universities, schools and communities. Questions are being asked louder, more often, and more pointedly, such as: What constitutes mathematical knowledge? What is included in mathematics? Who gets to say? How did it happen that British-European mathematics colonized the minds and curriculums of the whole world? Is the runaway process of development of that mathematics starting to saturate? .... Students of under-represented and minority groups and colonized peoples are starting to be more critical about accepting unquestioningly the cultural hegemony of mainstream mathematics”.

 “Mathematics has been gate-kept by the West and defined to exclude entire cultures. Almost all mathematics that students have ever come across is European based”, Professor Ball explains: “We would like to enrich the discipline through the inclusion of cross-cultural mathematics”.

According to her argument schools should teach less algebra or calculus to leave more time for teaching about indigenous “lived  mathematics” which, she writes, is less interested in “numbers and arithmetic and accounting'' and more in “recognizing and classifying patterns”, that is, “the science of patterns and periodicities and symmetries”.   

Leaving aside some of her more incredible claims, such as that of the ability of the Aboriginal Australians to predict eclipses, from a purely anthropological point of view Professor Ball is not wrong. Yes, it is important to know more, as part of the cultural history of the world, about how different societies have developed the rudimentary, or less rudimentary, mathematics needed in their day-to-day “lived experience”. But such considerations have little to do with the subject of Mathematics as such- a uniquely structured body of knowledge which starts with numbers and arithmetic, and yes accounting, and builds on it, layer by layer, the branches of mathematics we learn in schools and universities: geometry, algebra, calculus, differential equations, topology, group theory, etc. To understand and make use of this magnificent, uniquely coherent and “unreasonably effective” structure it is entirely irrelevant how different layers have developed. The entire body of mathematics can be studied and applied by any country, culture or any group of people, anywhere in the world with no reference whatsoever to where and how its content has developed. This is precisely what the great mathematician David Hilbert meant when he wrote “Mathematics knows no races. For mathematics, the whole cultural world is a single country”- a unique cultural stage where individuals all over the world are free to participate, according to their personal interests and talents, to the greatest and most enduring human achievement – what Hilbert called “the highest pinnacle and highest height of the culture of rigorous knowledge”.

This is not to say that the history of mathematics is not important, on the contrary, it is a fascinating and inspiring subject, with a lot to add to the cultural history of mankind, but not in the least needed to learn and understand any specific mathematical content. It is fascinating to wonder about the mysterious personality of Pythagoras or learn about the idiosyncratic one of Newton, but neither knowledge is needed to understand either the Pythagoras' theorem or the Calculus. More importantly we don't need to delve into Newton's confusing method of fluxions, with its awkward notations; the modern exposition of the subject in a decent high school manual is greatly superior. Similarly, it would make no sense to learn arithmetic using any other numbering system than the one introduced by Indian mathematicians in the 1st to the 4th centuries, and later adopted by the Arabs in the 9th century. The Indian-Arab system in use today is vastly superior to the ones previously used by the Egyptians, the Greeks, the Romans, Chinese etc. It would be equally foolish to try to learn algebra using the 9th century text of Al-khwarizmi, or even the more advanced 16th century one of Viete; the modern texts, at least the ones before the advent of ethno-mathematics, have far better notation and content and are thus much easier to understand and apply. 

A lot of the confusion surrounding the pronouncements made by professors Gutierrez, Ball and Levya, and many others, stem from a fundamental misunderstanding of the nature of mathematical concepts. Are they invented or discovered? If invented then, like all other cultural artifacts, they reflect the unique idiosyncrasies of the societies or the individuals that have created them. It would then be justified to entertain the idea that the discipline of mathematics might be enriched “through the inclusion of cross-cultural mathematics”.  But mathematics deals with its unique, abstract, form of objective reality. The theorem of Pythagoras is no less valid today than the day it was first discovered, probably in ancient Babylon 4000 years ago. Euclidean geometry is still valid within its precise axiomatic 3rd century BC framework, even though mathematicians have later discovered that there are other types of geometries, equally valid within a broader axiomatic system. Like all other scientists, mathematicians discover objective reality, they do not create it. If mathematics was in fact a cultural artifact, like music, literature or the arts, it would be impossible to explain its extraordinary effectiveness in the physical sciences, weather prediction, engineering or artificial intelligence.    

It is also true that mathematicians, just like chemists or biologists, rely on their creativity when making specific choices in their probing of objective reality. These choices may reflect the cultural environment in which the discoverer operates, but do not affect the reality of what is discovered. America was there before Columbus, or whoever other, non-indigenous, explorers first discovered it. The history of its discovery does indeed reflect the European cultural environment in which Columbus lived, but that does not call into question the reality of the continents we refer to as the Americas.

If Red2 is dangerous for how Mathematics and Science are taught, by putting at risk¹ the education of the very people it purports to help, Red1 is having a measurable adverse impact on how individual contributions to research and scholarship are being assessed and rewarded by academic institutions and various government and non-government funding agencies. A recent, highly concerning,   peer reviewed article,  documents how federal funding agencies in the US are ``changing the criteria by which they distribute taxpayer money intended for scientific research'' based on Diversity, Equity and Inclusion (DEI) considerations rather than scientific merit. These heavy-handed bureaucratic efforts are all justified in the name of combatting biases and discrimination. Insofar as these biases are real such efforts may be justified, but are they?  

The recent book ''Matheuse” authored by three prominent women researchers in France, strongly promoted by the prestigious Centre National des Recherches Scientifique (CNRS), claims that the present academic system is heavily stacked against women everywhere, and calls for a radical transformation of Mathematics. In translation from the original French: “Mathematics needs an internal and collective transformation of practices, based on the refusal to build the discipline on the personal success of a few individuals considered exceptional, and on the systematic rejection of all naturalizing approaches for women and men, but also of questions of taste, talent and merit”.

The book starts from the obvious reality that women are far less represented in the mathematical sciences and the occupations based on them, and attributes this fact entirely to societal biases (that is self-fulfilling societal expectations that women are naturally less talented or simply less interested in mathematics), as well as to overt discrimination, harassment and even sexual violence. The authors refuse to consider the possibility that sex differences may have any role in personal preferences or even abilities, and vehemently reject that any such differences could have biological origins. They go so far as to claim that any dearth of women in informatics is due to “brutal eviction based on sexual violence”. According to the authors “Les filles sont repoussées de façon humiliante, rejetées parce que jugées sans valeur et dégoutées et découragées (Young women are pushed away in a humiliating fashion, rejected because they are considered without value, they feel disgusted and discouraged [by this treatment].)”  

Even the fact that women are the majority in medicine, biology, pharmacy or veterinary sciences is presented in a negative light, as if somehow women are pushed by society to these disciplines which the authors claim to be inferior to the more “noble” mathematically based ones, in which men are “over-represented”.

To support its radical claim of ongoing, unrelenting, discrimination the authors deny the existence of natural differences in cognitive aptitudes. Nobody has more “natural talent” in math than anyone else, and intelligence is not an intrinsic quality of a person but the result of personal interactions and training. This RED view of human nature leads the authors to state  “L’intelligence des femmes s’arrête là où la domination commence (women's intelligence stops where domination starts)”. The question, according to the authors, “is not to know who is intelligent but to understand who decides the attribution”. Thus, in a postmodernist fashion, intelligence is “what is measured by tests of intelligence, or by the school system” in which the child is educated. The school is nothing more than a social “alchemy which transforms social inequalities into school inequalities”. Or, in a Marxist doctrinaire fashion, it claims that school “transforms social differences into natural differences, and then uses the pretended natural differences to justify social differences”. The book also states, with just a minimum of equivocation, that “the consensus in life and social sciences is that intelligence or the so-called `mathematical talent’ are principally, if not totally, acquired”. Really? Is it also the consensus that musical or literary talent are also socially constructed? What about sports? 

The book contains (page 166) these astonishing accusations: 

“Les savoirs mathématiques légitiment les hiérarchies raciales (Mathematical knowledge legitimizes racial hierarchies)”.  

“La communauté mathématique entretient l’exclusion et l’oppression (The mathematical community sustains exclusion and oppression)” 

and ends with this Leninist-style slogan:

“Les filles sont l'avenir des mathématiques féministes, antiracistes et égalitaires dont le pouvoir ne serra plus oppresseur, mais  émancipateur (Women are the future of feminist, anti-racist, and egalitarian mathematics, whose power will no longer be oppressive but emancipating).” 

The book, highly advertised by CNRS, is only the most extreme version of a RED-informed  ideology, now widely accepted throughout academia, willing to accept nothing other than full numerical parity in all areas of the sciences and engineering in which men are still “over-represented”. Universities, research institutes and professional societies have put in place various bureaucratic measures by which they hope, in time, to achieve this goal. Here is, for example, how Susan Wexler, the former home secretary of the US  National Academy of Sciences, describes the selection process² of new members of the academy: “We assign slots based on the diversity of the lists of nominees that they have forwarded... Classes presenting a more diverse list get extra slots." The next year the council reviews how those slots have been filled and adjusts the distribution based on performance. “If [the selectors] used them to pick a bunch of white guys from Harvard, they get penalized.”  The home secretary is also quoted as saying that the process has been “spectacularly” successful in recent years with regard to women. “I was one of nine women in my class [of 60 in 1998]. Last year (2020), we had 44 [out of 120], and this year (2021) it's essentially even”.

Once more, such efforts would be understandable, even if heavy handed, if the claims made in the “Matheuse” book were correct. But are they? The best recent refutation comes from an evolutionary biologist. In her excellent essay  “Why do Men dominate chess” Carole Hooven elegantly demolishes the RED1 myth that the under- performance of women in chess is due to nothing more than overt societal discrimination. Chess is, of course, in no way equivalent to mathematics, it is  just a game after all, but it is well known that interest and ability for math correlate well with those for chess. The level of participation of women in chess is remarkably similar to that of their participation in higher-level math or the STEM fields that require it. Moreover the success of girls in chess competitions mirrors their level of success in math competitions, such as the International Mathematics Olympiad (IMO).  

To understand why men dominate chess Dr. Hooven, Harvard PhD and author of a popular  book on the importance of testosterone on human behavior, analyzes various hypotheses. One of them, the greater male variability (GMV) hypothesis, is based on the observation that, though men and women exhibit equal average level if intelligence (with maybe a slight advantage for women) males exhibit greater variability than females in a range of traits, such as height, intelligence, preferences for leisure activities, risky behavior, and competitive drive.  Dr. Hooven points out that the GMV hypothesis can explain sex differences in the STEM disciplines. She writes: “The idea is that even if there’s no male-female difference in average math or physics ability, there would still be more men at the very high (and low) end of the ability spectrum. These are the extreme outliers who are most likely to earn prestigious faculty positions, file many patent applications, and win career achievement awards. And there is, in fact, strong evidence supporting the hypothesis; many traits do tend to be more variable in men than in women”.

A second explanation is the Participation Rate Hypothesis (PRH).  The idea here is that even though schoolboys and girls may have similar aptitudes for chess, the higher number of boys who become interested in playing competitively results in more of them eventually reaching the highest echelons of performance. This phenomenon could also explain the higher achievement rates of high school boys in mathematical competitions, such as IMO.    

She then goes on to undermine the explanatory power of PRH by pointing out that in Scrabble, a game completely dominated by women in terms of recreational participation rates, men still outperform women at a competitive level.  She points out that the same goes for bridge, a game also heavily dominated by women at the recreational level.

She concludes with what seems to be the most realistic explanation, one maybe also consistent with the GMV hypothesis,  “...that men are more likely than women to exhibit a `rigid persistence’ in an activity, by which the passion controls the individual (`obsessive passion’ in the literature).  In anecdotal terms, we are talking here about the man who drops everything to become, say, a 16-hour-per-day video-gamer, or a day-trader, or chess addict. Yes, some women take on these kinds of fixations. But men do it more often, and with greater intensity”. 

Finally, Dr. Hooven states, in stark contradiction to the main premise of the “Matheuse” book:  “Ultimately, sex differences in complex behaviors and skills are always a product of interactions between biology on the one hand (that is, our genes and their relatively fixed effects, such as hormone levels and body size) and our environment on the other (that is, factors such as our family circumstances, social dynamics, and cultural norms). Interactions between the two shape not only our skills and abilities, but also any emerging group differences. But none such complicating factors change the fact that the sex gap in chess [or mathematics]  is real and persistent”. 

These differences have only statistical relevance; individual women can, and often do, exhibit the same obsessive passion for chess or mathematics as men. I know that to be true by my direct professional experience with extremely talented women mathematicians. Yet differences between human beings in mathematical abilities and interests are “real and persistent”. There is no better illustration of this fact than the educational gender equality paradox (GEP), identified by G. Stoet  and D.C  Gerry . They found that countries with high levels of gender equality have some of the largest STEM gaps in secondary and tertiary education: “For example, Finland excels in gender equality (World Economic Forum, 2015), its adolescent girls outperform boys in science literacy, and it ranks second in European educational performance (OECD, 2016b). With these high levels of educational performance and overall gender equality, Finland is poised to close the STEM gender gap. Yet, paradoxically, Finland has one of the world’s largest gender gaps in college degrees in STEM fields, and Norway and Sweden, also leading in gender-equality rankings, are not far behind. We will show that this pattern extends throughout the world, whereby the graduation gap in STEM increases with increasing levels of gender equality’’.

Even though statistical disparities are to be expected one can still ask the question whether mathematically gifted girls continue to be discriminated against and discouraged from pursuing STEM careers. There is no doubt that societal norms and expectations, and sometimes ugly prejudice and predatory behavior, did in the past prevent full participation of women in science. But is that still there today? Recent comprehensive meta-analysis study by Stephen J. Ceci  and co-authors explore gender bias in six key domains of academic science: tenure-track hiring, journal peer reviewing, grant funding, letters of recommendation for faculty applicants, salary, and teaching evaluations. While they still find a wage gap of about 3.6%, though not the 18% which is often reported, they find no bias in all other metrics. “In peer-review success and letters of recommendation, we found no systematic gender bias. Manuscripts by women were accepted for publication at equivalent rates to men’s, and women applicants for tenure-track jobs received comparably strong letters of recommendation, despite widespread claims to the contrary”. Moreover in the tenure-track hiring, arguably the most important metric of all, their analyses revealed “that women applicants have a substantial advantage over comparable male applicants”.

Though numerical disparities are bound to persist, no matter what we do, we can alleviate their negative impact on society by rewarding people based on the interests and talents they have, rather than those they do not, and treat the interests themselves as worthy of equal respect, rather than insist that all individuals are equally qualified to pursue them. The notion that Mathematics or Physics are nobler occupations, as the “Matheuse” book claims, nobler than raising and educating children, taking care of the sick, or gardening, building, plumbing, or any other occupation without which civilized life would not be possible, is patently false and should be widely rejected. And yes, we should continue to make all efforts, short of bureaucratic dictates, to make it possible for talented women, indeed all talented individuals, to thrive in STEM and the hard sciences.  

1. It also puts at risk the present supremacy of the U.S. in the sciences and technology.

2. A few years ago NAS has introduced, in parallel to the traditional selection process  centered on the scientific contributions of a nominated person, the so-called “temporary nominating groups” (TNGs) with the understanding that the academy needed a “one time shot in the arm” to correct the large imbalance regarding women, underrepresented minorities, young scientists and geographically unrepresented research institutions. These TNGs have now become a permanent feature of the selection process.