The politics of math are of newfound concern today, due to the outsize influence of algorithms and code in contemporary life. While only a few years ago, tech authors were still hawking Silicon Valley as the great hope for humanity, today one is more likely to hear how Big Tech increases social inequality, how algorithms are racist, and how math is a weapon. Do algorithms discriminate along gendered lines? Do mathematical systems harbor an essential bias? This essay shows that mathematics has long been defined through an elemental gendering, that within such typing there exists a prohibition on mixing the types, and that the two core types themselves (geometry and arithmetic) are mutually intertwined using notions of hierarchy, foreignness, and priority. The author concludes that whatever incidental biases it may display, mathematics also contains an essential bias.

“Is a woman a thinking unit at all, or a fraction always wanting its integer?”

—Hardy

Math has a gender issue.1 But how and why? The numbers tell a story, a story about the mathematics of gender. How many women study math in school: is it fewer than their male counterparts? How many women pursue careers in math-related fields: is it even fewer than that? Can we count them? What happens when we try to account for them? What about women across the history of mathematics? Do they count? Hermann Weyl summed up the conventional wisdom on women in math with his infamous quip that “there are only two females in the history of math, Sofia Kovalevskaya and Emmy Noether: the former wasn’t a mathematician, the latter wasn’t a woman.”2

Math has a gender issue, and the numbers tell a story, a story about the mathematics of gender. But numbers tell another story as well, a story about the gender of mathematics. The first story is important and has already been explored by others; this essay concerns the second story, a story about the gender of mathematics.3

“What nonsense. Math has no sex or gender. The very question is preposterous. Numbers are unparticular, abstract. Numbers don’t care what sex or gender you are; they don’t care if you are a man or a woman.”

The politics of math are of newfound concern today, due to the outsize influence of algorithms and code in contemporary life.4 Fields like media studies and science and technology studies (sts) have long investigated these types of problems. One need only turn to the important work of scholars like Simone Browne, Wendy Hui Kyong Chun, Lisa Nakamura, or Sadie Plant to understand the complicated machinations of race, class, and gender in software, code, logic, and mathematics. (Or, before them, work on cybernetics and control from the likes of Donna Haraway, James Beniger, or Katherine Hayles.) At the same time, feminist science studies has been in active discussion about feminist epistemology, among other debates, for several decades already, as illustrated by the influential work of Anne Fausto-Sterling, Helen Longino, Sandra Harding, Kristie Dotson, Joan Scott, and many others. The politics of computation has migrated from a scholarly concern to the topic of mainstream discussion. While only a few years ago, tech authors were still hawking Silicon Valley as the great hope for humanity, today one is more likely to see books addressing how Big Tech increases social inequality, how algorithms are racist, or how math is a weapon.5 Even New York congresswoman Alexandria Ocasio-Cortez went on the record claiming that algorithms perpetuate racial bias. And in a New York Times column, legal scholar Michelle Alexander quoted Cathy O’Neil’s argument that “algorithms are nothing more than opinions embedded in mathematics,” suggesting that algorithms constitute the newest system of Jim Crow. Indeed, the gross failures of Big Daddy Mainframe—to borrow the appellation of cyberfeminists vns Matrix—are almost too numerous to mention: a digital camera that thinks Asians have their eyes closed; facial recognition technologies that misgender African American women (or miss them entirely); and search engines that portray young Black men as thugs and threats. A few hours after its launch in 2016, Microsoft’s chatbot “Tay” was already denying the Holocaust.6

“But math is just the pursuit of abstraction and formal relation. Numbers are agnostic to people and their specific qualities. An algorithm has no political or cultural agenda. In math, a correct answer is the only thing that matters.”

Do algorithms discriminate along gendered or racialized lines? Do mathematical systems harbor an essential bias? For many today the answer is an emphatic yes. At the same time, coders and mathematicians, stereotypically liberal (or libertarian) in their political outlook, will frequently maintain that the work they do is politically neutral. Math is pure abstraction, they contest, uncoupled from world-bound details such as race, ethnicity, gender, class, or culture. Perhaps the most common articulation of this position, particularly among mathematicians and computer scientists, is what we might call naive abstraction. For those defending this position, math is entirely insulated from real-world concerns. Math merely expresses the clearest, most rigorous, and most formal relation between abstract entities. (Most mathematicians are still Platonists, after all, even if they are unlikely to admit it publicly; most mathematicians think that mathematical entities exist in a realm of pure, formal abstraction.) Proponents of naive abstraction will thus consider the politicization of math to be a category mistake: that’s just not how math works, they say; politicizing mathematics means subjecting it to an external reality for which it was never intended; to suggest otherwise is to commit an infraction against mathematics. Politics is fine and proper for the public arena. But math is something else. Math wasn’t meant for politics. One would be wise not to mix them.

But what’s to blame for digital technology’s failings if its innards (mostly logic and arithmetic converted into machine operations) are not politically culpable? This dynamic has created a strange paradox, as those activists vehemently decrying algorithmic bias and those coders writing the algorithms are often the same people! Tech communities have created a series of useful mythologies to exit this uncomfortable paradox. One common technique is to suggest that race, class, or gender are “particulars”; as markers of difference, these particulars pertain to particular contexts, to particular bodies, to particular histories. And as particulars, they do not rise to the level of concern for abstract mathematics. In this way, proponents of naive abstraction often think they are operating in good faith even while avoiding or dismissing politics: yes, I care about your plight; but it’s yours alone; I’m simply interested in other things (birdwatching, stamp collecting, prime numbers).

Yet, currently the most robust approach to resolving the paradox of tech bias is what we might call the usage thesis. The usage thesis states that tech bias derives not from any essential trait of the underlying technology but from the social and historical uses of technology by human subjects. According to the usage thesis, code and software are essentially passive substrates that can be “embedded” with values. What values exactly? The values of their creators, which is to say all the biases and assumptions of whoever designs the algorithm. If software contains political values, it is a consequence of how human subjects have designed and used it. Hence, if an algorithm is racist, it is because some racist designer somewhere made it that way. If a piece of software produces a sexist outcome, it is because some coder was negligent. (If this argument sounds familiar, it should; it’s a version of the gun rights argument that “guns don’t kill people, people kill people.” Only now, the argument is that math doesn’t hurt people, negligent mathematicians hurt people.) The usage thesis freely asserts that math is cultural and political, given how math is subject to specific cultural and political uses. For this reason, the usage thesis claims that the best way to understand the culture and politics of math is to look at human subjects, rather than at math. Through a kind of metonymic slippage, technologies become the legible substrates on which the various successes and failings of human beings are inscribed, thereby shifting attention from the device back to the designer or user. For this reason, usage theorists tend either to be Foucauldians (“if you want to understand tech bias, first you have to understand power”), or they tack more toward anthropology and the human sciences (“if you want to understand tech bias, first you have to understand people”).

“Who says math is only an abstraction insensitive to the real world? Math has long paid special attention to real phenomena in disciplines like geometry, topology, or calculus. Math can be applied, even empirical—just ask any working scientist.”

The usage thesis is successful in part because it allows both parties to maintain their political postures without concession: Big Tech is guilty of discrimination at the macro level, satisfying antitech activists; while the workaday act of writing code is exonerated, satisfying computer scientists and mathematicians. Antitech activists are free to find their answers in the failings of designers and technologies, allowing the tech itself to remain neutral, while computer scientists and mathematicians are happy to nestle back into the comfortable familiarity of “naive abstraction” where math is an unsullied foundation for abstract thinking. Either way, math and code fall out of the frame, preserving their status as neutral vessel.

Overall, I see this as a kind of “fear of media.” Whether in denial or acknowledgment, responses to questions of tech bias collude to undermine the notion that math, code, logic, or software are, or could be, a medium at all. If math were a full-fledged medium, one would need to attend to its affordances, its forms and structures, its genres, its modes of signification, its various lapses and slippages, and all the many other qualities and capacities (active or passive) that make up a mode of mediation.

Further, contemporary discourse tends to perpetuate stereotypes rather than remedy them. For instance, the “neutral vessel” is an ancient trope for female sexuality going back at least to Aristotle, if not earlier, as are neutral media substrates more generally (matter as mater/mother, feminine substrates receiving masculine form, and so on). The act of “injecting ethics” or “embedding values” into an otherwise passive, receptive technology resembles a kind of insemination.

Hence the question remains: does mathematical rationality contain an essential bias, and if so, what is it? Not that long ago affirmative answers would have been readily forthcoming. Binary logic is heteronormative (Judith Butler). Rational calculation is an iron cage (Max Weber). The algebraic equation of use-values produces fetishism and alienation (Karl Marx). Still, the notion that mathematical rationality contains an essential bias has slipped away in recent years, replaced by arguments around “usage,” if not outright dismissals. Even today’s most ardent critics of Amazon or Google will frown and backpedal if someone begins to criticize algebra or deductive logic. Math is the third rail of digital studies: don’t touch. Recall, for example, the uproar a few years ago when the enterprising scholar Ari Schlesinger proposed to design a feminist computer language.7 How dare she! The very notion that computer languages might be sexist was anathema to swaths of the Internet public, fomenting a Gamergate-style backlash.

This essay is an attempt to touch that third rail directly. While I’m sympathetic to most all of the work already cited, I’m much more interested today in the question of math’s essential bias than I am in incidental bias, or concerns around use, or “embedded values.” And while I’m compelled by the mathematics of gender—who counts, who gets to count—I’m also intent on exploring a parallel question: whether there is not also a gender of mathematics running apace with the mathematics of gender. We know that humanity is subject to the logic of sex and gender. But is the number 7 gendered? Or the variable x? What about the if . . . then control structure: does it have a sexuality? Or what about an entire computer language, like C++ or Python? Could these kinds of things be gendered or sexuated? And if so, is this a source of inherent bias?

To that end, I argue in what follows that mathematics has been defined since the ancients through an elemental gendering (or typing), that within such typing there exists a pervasive segregation and prohibition on the mixing of types, and that the two core types themselves (geometry and arithmetic) are mutually intertwined using notions of hierarchy, foreignness, and priority. Given the inherently political nature of such conditions—gendering, segregation, hierarchy—I conclude that whatever incidental biases it may display, mathematics also contains an essential bias. In the end, though, the hardest question will be how to evaluate such a bias, and what to do with it.8

Nota bene: I have included some of the attempts to resist such an endeavor, the many voices—so loud, so cocksure—that aim to silence and subdue conversations around the politics of math. These italicized block quotes are paraphrased and condensed from commentary found in various public venues, scholarly or otherwise. Keep these phrases in the front of your mind as we cross over to another kind of inquiry, to a different set of questions, with the intent, by the end, of returning once again to the original position, only then transformed.

Crossing Over to Another Kind

Sex frequently entails counting. One, two, multiple. First, second, many. The Second Sex, Zeros and Ones, This Sex Which Is Not One, “Queer Multitudes,” Third Sex, Third Gender.9 Sex seems to ratify that old aphorism from mathematician Richard Dedekind, ἀεὶ ὁ ἄνθρωπος ἀριθμητίζει, “humanity is always counting.”10 With sex, one comes to count, or not to count. With sex, one furnishes an account. Or as the Lacanians might say, the failure to furnish an account generates the condition called “sex.”

In contemporary usage, sex and gender are commonly distinguished along the lines of biology and culture, or, if you like, material condition versus socio-historical instantiation. At the same time, we might benefit from a slightly different emphasis, relying instead on the roots of the terms, with sex stemming from the Latin secare, meaning “to cut,” and gender from the Greek genos [γένος] (by way of the Latin genus), meaning offspring, family, and hence (those of the same) type, race, or class. Genos itself was formed from the important Greek verb gígnomai [γίγνομαι], meaning “to be born of,” “to come into being,” or “to become.” As technical terms, then, sex and gender both address the same series of events—differentiating and arranging according to kind—albeit from two different perspectives. If sex is a cutting, gender is a grouping. Sex marks the cut of differentiation, while gender collects the differentiated into families. Of course, every cut forms a group (or at the very least, a pair), and every group entails a prior cut (the outline of the group, the membrane between inside and outside). So the story of sex cannot be disengaged from that of gender, and vice versa. Every sexuation is a gendering, and every gendering is a sexuation.

So does mathematics have a cutting? Does it have a grouping? If so, can we claim it is sexuated or gendered?

At the same time, not all counts are the same, and there is a count to counting as well, a first and a second to number, just as there is a first and a second to sex, if not also a third, a fourth, or more. In mathematics, the number of the count is often two, and the name of the count geometry and arithmetic. In other words, mathematics is rooted on an elemental distinction, on the notion that there are two basic ways of doing math, the geometrical way and the arithmetical way.

“We shall do well in general not to overestimate the extent to which arithmetic is akin to geometry,” Gottlob Frege warned his readers in a chapter of the Foundations of Arithmetic unambiguously devoted to the “Distinction between Arithmetic and Geometry” (19).11 Frege’s desire to distinguish neatly between arithmetic and geometry, and, as we will see shortly, to constrain the latter from mixing with the former, is very old, as old as Greek, Egyptian, and Babylonian mathematics. Of course, arithmetic and geometry are intimately intertwined, with the geometric ratio of magnitudes forming the basis of the arithmetical number system, on up to the renaissance of geometry in the twentieth century around the highly abstract (and fully arithmetized) disciplines of topology and algebraic geometry. Yet the indissoluble distinction between arithmetic and geometry was and remains foundational, as Alain Badiou explained in his treatise on Number and Numbers:

The Greeks clearly reserved the concept of number for whole numbers, which was quite in keeping with their conception of the composition of number on the basis of the One, since only natural whole numbers can be represented as collections of units. To treat of the continuum, they used geometrical denominations, such as the relations between sizes or measurements. So their powerful conception was marked through and through by that division of mathematical disciplines on the basis of whether they treat of one or the other of what were held by the Greeks to be the two possible types of object: numbers (from which arithmetic proceeds) and figures (from which, geometry). This division refers, it seems to me, to the two orientations whose unity is dialectically effectuated by effective, or materialist, thought: the algebraic orientation, which works by composing, connecting, combining elements; and the topological orientation, which works by perceiving proximities, contours and approximations, and whose point of departure is not elementary belongings but inclusion, the part, the subset. This division is still well-founded. Within the discipline of mathematics itself, the two major divisions of Bourbaki’s great treatise, once the general ontological framework of set theory is set out, deal with “algebraic structures” and “topological structures.” And the validity of this arrangement subtends all dialectical thought. (10)

In Badiou’s assessment, geometry refers to the continuum and to figures, and more broadly to a topological method “which works by perceiving proximities, contours and approximations.” Geometry is first and foremost a question of measurement and magnitude; the etymology of the word refers to the ancient Egyptian profession of land surveying or “measuring the earth.” For this reason, geometry is associated with space and extension, as inherently enacted or “applied,” as working primarily through diagrams and practical relations rather than symbolic formula, as having a special relationship to intuition and immediacy. “Geometry is valued for its extensive usefulness,” wrote Scottish mathematician Colin MacLaurin in 1742, “but has been most admired for its evidence; mathematical demonstrations being [ . . . ] always supposed to put an end to dispute, leaving no place for doubt or cavil” (95). For his part, mathematician Bernard Bolzano explicitly demarcated “pure” math to be that of “arithmetic, algebra, analysis,” while “merely applied” math was that of “geometry” (“Purely” 228).12

To be sure, many contemporary mathematicians would scoff at the notion that geometry and arithmetic are somehow separate, given how arithmetical and algebraic techniques have so thoroughly infiltrated the domain of geometry, beginning with early modern scientists like Leibniz, Newton, and Descartes, through to the heyday of arithmetization in the nineteenth century with figures like Dedekind, Augustin-Louis Cauchy, and Georg Cantor, on up to David Hilbert’s axiomatization of geometry in 1899. Yet the “nineteenth-century view that all mathematics should be ‘arithmetized’ (based on numbers),” as John Stillwell put it, does not so much dissolve the distinction between geometry and arithmetic as highlight the stubborn persistence of geometry as that which is irreducible to arithmetic and hence must be subdued by it (32). “The XIXth century witnessed a change in the hierarchical order of mathematical domain,” Jacqueline Boniface explains. “[G]eometry, considered since Euclid as a model of rigour [ . . . ], lost precedence to arithmetic” (315).

But why subdued? Arithmetic belongs to a fundamentally different paradigm of knowledge. And these different paradigms come into conflict. If geometry stems from the science of measurement, arithmetic comes from that of counting. If geometry is best suited to answer the question “how much?,” arithmetic is best suited to answer the question “how many?” “The fundamental phenomenon which we should never lose sight of in determining the meaning of arithmos (ἀριθμός) is counting, or more exactly, the counting-off, of some number of things,” explains philosopher Jacob Klein in his important work on Greek Mathematical Thought and the Origin of Algebra. But the key to arithmetic was not just counting; it was the fact that counting implied uniformity of type or kind, as Klein continues,

These things, however different they may be, are taken as uniform when counted; they are, for example, either apples, or apples and pears which are counted as fruit, or apples, pears, and plates which are counted as “objects.” Insofar as these things underlie the counting process they are understood as of the same kind. That word which is pronounced last in counting off or numbering, gives the “counting-number,” the arithmos of the things involved. [ . . . ] Thus the arithmos indicates in each case a definite number of definite things. It proclaims that there are precisely so and so many of these things. It intends the things insofar as they are present in this number, and cannot, at least at first, be separated from the things at all. (46)

Uniform, of the same kind, a definite number—such is the essence of counting, not to ignore difference, but to stress that in the count things are taken to be of the same kind. (Such a move aggravates some—particularly skeptics, empiricists, pragmatists, and nominalists—who object to type and kind on the grounds that they are fictions, and pernicious fictions at that; my preference is not to indict arithmetic and invalidate it, but to acknowledge that a technology of uniform type exists, a technology that comes under the name arithmetic.) Arithmetic entails the generalization of a single type; this is true in Klein’s example of the apples and pears but also with the simple counting numbers 1, 2, 3, and so on. The counting numbers are valid arithmetical elements, if not also the very paradigm of arithmetical thinking, because they all conform to one or more well-defined types, most importantly the monad or unit that sits at the basis of all whole numbers and the rational numbers more generally. Bound together by type, arithmetical numbers are at the same time distinguished one from the other by strict discretization. A gap always remains between arithmetical elements. No continuous gradient exists between the integers 0 and 1. And while the rational numbers are described by mathematicians as “dense,” a gap always exists between any two rational numbers no matter how close in value.

The question of number is thus indistinguishable from the question of boundary, at least when it comes to arithmetical number. Do well-defined boundaries exist at all? Many will want to say no, instead transgressing boundaries via contradiction, ambiguity, or irony. Yet arithmetic is one place where it is possible, necessary even, to answer in the affirmative. Well-defined boundaries do exist in arithmetic (since arithmetic is little more than the science of the monad). Or one might formulate the same idea in reverse: assign a name to the science of well-defined boundaries; that name is arithmetic.

Edmund Husserl, in the “Prolegomena to Pure Logic” that begins his large treatise Logical Investigations, wrote in praise of “questions of principle” and advocated well-defined boundaries for fields.

There is another, much more dangerous fault in field-delimitation: the confusion of fields, the mixture of heterogeneous things in a putative field-unity, especially when this rests on a complete misreading of the objects whose investigation is to be the essential aim of the proposed science. Such an unnoticed μετάβασις εἰς ἄλλο γένος can have the most damaging consequences: the setting up of invalid aims, the employment of methods wrong in principle, not commensurate with the discipline’s true objects, the confounding of logical levels so that the genuinely basic propositions and theories are shoved, often in extraordinary disguises, among wholly alien lines of thought, and appear as side-issues or incidental consequences etc. These dangers are considerable in the philosophical sciences. Questions as to range and boundaries have, therefore, much more importance in the fruitful building up of these sciences than in the much favoured sciences of external nature, where the course of our experiences forces territorial separations upon us, within which successful research can at least be provisionally established. It was Kant who uttered the famous special words on logic which we here make our own: “We do not augment, but rather subvert the sciences, if we allow their boundaries to run together.” (13)13

Such a full-throated rejection of interdisciplinarity clashes with today’s academic mores, where “allowing boundaries to run together” is canon and creed for provost and freshman alike.14 Husserl’s advice also clashes with many forms of feminist theory, which frequently celebrate “allowing boundaries to run together” through an attention to ambiguity, irony, hybridity, heterogeneity, intermixing, and the slippage between terms.

But what about Husserl’s enigmatic Greek phrase “μετάβασις εἰς ἄλλο γένος,” or “crossing over to another kind”? What does this mean, and where does the expression come from? Why was Husserl so keen to avoid crossing over to another kind, to avoid boundaries running together? The expression “crossing over to another kind” is found in Aristotle, who considered it a logical fallacy to prove results in one domain via techniques from another domain. For example, if, during an arithmetical proof, one deploys techniques or concepts borrowed from geometry, one would be “crossing over” from the domain of arithmetic to the domain of geometry. Aristotle considered this a flaw and stipulated that crossing over to another kind be marked as an error, and thus prohibited from proper logical proof.15

“[I]t is an intolerable offence against correct method to derive truths of pure (or general) mathematics (i.e. arithmetic, algebra, analysis) from considerations which belong to a merely applied (or special) part, namely geometry,” wrote Bolzano, in an echo of Aristotle’s skepticism toward arithmetical proofs that “cross” into geometry. “Indeed, have we not felt and recognized for a long time the incongruity of such μετάβασις εἰς ἄλλο γένος? Have we not already avoided this whenever possible in hundreds of other cases, and regarded this avoidance as a merit?” (“Purely” 228).

There’s that Greek expression again—crossing over to another kind—and here, as it was in Aristotle, the expression stipulates prohibition on transit from arithmetic to geometry. Elsewhere, Bolzano maintained he “could not be satisfied with a completely strict proof” if it “made use of some fortuitous alien, intermediate concept [Mittelbegriff ], which is always an erroneous μετάβασις εἰς ἄλλο γένος” (“Preface” 173). The “Mittel” of Mittelbegriff and the “meta” of μετάβασις both refer to a “medium” condition, the state of being between, through, or across. Being both “erroneous” and “alien,” this metabasis, this middle transit, was too risky for Bolzano. (“Alien” was Husserl’s word too, along with “damaging,” “invalid,” and “confusion.”) So it wasn’t geometry per se that riled Bolzano, but something about the transit from arithmetic to geometry, the fact of transiting, of moving through, of crossing through the middle, and hence being forced to acknowledge the alternate term (geometry) while also validating the existence of the middle along the way.16

So μετάβασις was a problem for mathematicians. But a bigger problem was the ἄλλο γένος, the “other kind,” or more literally the other genus. Why was arithmetic a “genus”? And why was geometry its “other”? The arithmetic-genus bond is very old, going back at least to Euclid if not also to the Pythagoreans. In the important book 5 of the Elements, Euclid defined arithmetical numbers as a logos or ratio. Further, he specified that this ratio was formed from “two magnitudes of the same genus” (δύο μεγεθῶν ὁμογενῶν).17 For Euclid, arithmetical numbers were thus, by definition, “homogenous.” (Against this backdrop, Aristotle’s distaste for the “other genus” begins to take on a new sense.) And Euclid had a precise explanation for what it meant to be of the “same genus,” which for the sake of brevity I will whittle down to mean derived from the same unit.18 Arithmetic is thus in a very literal sense a “genetic” science, given that arithmetical numbers are those numbers constructable from a common gene or monadic unit.19

So arithmetic revolved around the “genus” and the rote repetitions of the genus to generate the “same genus” (the “homogenous”). And, as we have seen, arithmetic is the science of well-defined boundaries. A potent ideological posture emerged: it wasn’t just that arithmetic and geometry should not be mixed but that geometry was a kind of alien domain that must be subdued in order that arithmetic may assume its rightful position of privilege over geometry. It sounds like I’m exaggerating for effect. But this was precisely the position of mathematical luminaries like Dedekind, Frege, and many others. “[I]t is possible to avoid all importation of external things and geometrical intuitions into arithmetic,” Frege put it, evoking that age-old affiliation of geometry with intuition. “If we turn for assistance to intuition,” he warned, “we import something foreign into arithmetic” (Foundations 119, 114).20 Likewise for Dedekind, “geometric ideas” were “foreign ideas” that had to be replaced by “purely arithmetical” foundations (“Continuity” 768). And of course, Dedekind’s great masterstroke, his most important contribution to the history of mathematics, was to provide an arithmetically rigorous definition of irrational numbers (and hence for real numbers more generally)—thanks to his famous Schnitt or cut—thereby excluding the foreign admixture of geometry and replacing it with strictly arithmetical techniques. “I wholly reject the introduction of measurable quantities,” Dedekind stated in defiance, “measurable quantities” being a codeword for geometry; “I demand that arithmetic shall develop out of itself” (“Was sind?” 793 and “Continuity” 771).21 Translation: don’t cross over to the other kind.

From this we may conclude that the geometry-arithmetic distinction was not simply a segregation of kinds; it was also a hierarchy of kinds, with arithmetic in the superior position, geometry the inferior. It is true that geometry experienced something of a renaissance in the wake of figures like Bernhard Riemann and Benoit Mandelbrot that continues in today’s differential topology and symplectic geometry, among other exotic variants. Yet, the resurgence of geometry was only possible because it had come under the yoke of arithmetic, algebra, and logic, that is, because it had already been arithmetized.22

Is this segregated hierarchy of two kinds, arithmetic and geometry, enough to demonstrate the gendering (or sexuation) of math? Perhaps not. Not every binary distinction is a gendering, and I acknowledge that formal similarities between things do not necessarily prove connection.23 However, in this case geometry is explicitly feminized, almost to the point of cliché, particularly around the theme of intuition. If arithmetic extracts true theorems using axioms and proofs, geometry finds its truth in the pure intuition of material experience. If algebra displays the masculine virtues of rigor or abstraction, geometry is, classically speaking, embodied and materially extended. If arithmetic is paradigmatically “whole” and “natural,” geometry is the place of the “real” and even the “irrational.” So is arithmetic male and geometry female? Me Tarzan, you Jane?

Here I’m reminded of Naomi Schor’s approach to the thorny question of whether a specific kind of techne is essentially “feminine.” For Schor, the topic wasn’t the intuitive, embodied, experiential nature of geometry, but the particular itself, the specific, the detail. Or as Schor famously asked, “Is the detail feminine?” (4). In fact, Schor sought not to answer the question, if answering might bring closure. The question “remains open” (116), she wrote, even as people remain “prisoners of the paradigms” during any attempt to deconstruct them: “[W]e remain, of course, prisoners of the paradigms, only just barely able to dream a universe where the categories of general and particular, mass and detail, and masculine and feminine would no longer order our thinking and our seeing” (xlii–xliii).

Ellen Rooney has described Schor’s strategy as “press[ing] the claims of the detail as we dismantle its enabling conditions” (xxxii). In other words, however hackneyed these clichés may be—arithmetic male, geometry female—they generate real consequences. Math has a gender issue, as I mentioned at the outset, tallied through the mathematics of gender, which is to say the deeply asymmetrical participation rates between men and women in the fields of mathematics and computer science, not to mention engineering, philosophy, and other fields. It is hard not to read the mathematics of gender through the lens of the gender of mathematics. Mathematicians typically do not openly lobby for the exclusion of women—although there are those who do—even as some of them openly lobby for the excision of feminized methods like geometry or feminized epistemological structures like intuition. “It is very much a present-day concern of mathematical thought to eliminate the intuitive elements as thoroughly as possible,” explained Jacques Lacan in a moment of candor. “The intuitive element is considered to be an impurity in the development of the mathematical symbolic” (Ego 316).24 Or, as Dedekind himself admitted, “geometrical intuition” might be a useful shorthand, but it “can make no claim to being scientific” (“Continuity” 767). Or as Frege explained in the preface to his “Begriffsschrift,” his intention in the treatise was “that nothing intuitive could intrude here unnoticed” (48). For many modern mathematicians, granting admittance to geometric intuition was “an intolerable offence,” to reiterate Bolzano’s injunction from above. Considering all of this, my intent here has been not so much to bring sex to math, as it were, but rather to reveal the logic of sexuation already inherent in the structure of math.

With such vociferous railing against intuition, is it any wonder that a backlash ensued, the trend inverting suddenly, with a new school of thought emerging at the start of the twentieth century, developed first by L. E. J. Brouwer and ultimately dubbed Intuitionism, intent on defending mathematical intuition against the most esoteric abstractions? Brouwer and his ilk were inspired in part by Husserl and the burgeoning school of phenomenology, although these connections often remained underdeveloped. Brouwer and phenomenology were linked via Heidegger’s student Oskar Becker who was proficient in mathematics (Mancosu and Ryckman). At the same time, Hermann Weyl—our opening joker—was a onetime liaison between Brouwer and phenomenology, before Weyl changed his intellectual course. As for feminism, that hoary notion of “woman’s intuition” was, in a certain sense, far removed from what Brouwer meant by Intuitionism, which he considered to be more strictly grounded in classical rationality rather than less (Brouwer). In his rejection of Frege, Husserl had sought to ground arithmetic in intuition and psychology, while Frege adamantly refused to do so, opting instead for a “purely” logical ground.25 Brouwer hoped that his new approach, modest but also mystical, which admitted only mathematical objects and concepts directly constructable from human intuition, would avoid the problems of more formalist approaches. Indeed, every mathematician dreams of solid foundations, axioms so intuitive and obvious that they are self-evident and theorems deduced from the axioms using clearly demonstrated proofs. Yet “intuitive” and “self-evident” are the qualities of geometry, further confounding the foundationalists’ original dilemma.26 The difficulty thus remains in this bulwark of intuition—woman as human shield—and, further, how such a bulwark divides geometry from arithmetic, the latter defended from the former.

“We will concede that mathematical abstraction may be understood culturally and politically. But to define math in this way means assigning it an essence, and essentialism is the worst form of cultural and political misuse. Technology has no essence; to offer a rigid definition is to be guilty of essentialism.”

Crossing Back Again

I have identified three basic postures. The first, dubbed “naive abstraction,” claims that mathematics is unconnected from culture and politics and hence is more or less neutral on the question of gender or sexual difference. Second, “usage” theorists take a more social-constructivist position, arguing that, while math is ultimately a neutral vessel, math can and will be influenced, as it were, “from the outside,” that is, influenced socially or anthropologically. What has occupied me most here, though, is a third group, which I will call the “arithmetical chauvinists,” those who turned math into a contest of internal otherness, isolating and subduing geometry as the “other kind,” the “alien,” and the “intuitive.”

While the first group (naive abstraction) has its own obvious shortcomings, the gendering of math really starts with the second and third groups. The social constructivists talk of “embedding” social values in an otherwise neutral vessel. For them, society acts as the patriarch, with math a kind of breeder mother tasked to reproduce his materiality (and his ideology). What might not be evident, though, is how arithmetical chauvinists are also voicing a form of social construction, namely that geometry is constructed as intuitive or feminine and arithmetic constructed as rigorous or masculine. The arithmetical chauvinists, in other words, are also enacting a version of the usage thesis, albeit locating the site of social construction at a deeper level.27

One difficulty in pointing this out is that even the critique of the gender of math begins to mimic some of the very things it indicts. For example, in demeaning usage one is also unwittingly demeaning the feminized term, while ironically also renaturalizing it, the assumption being that if “masculine methods” and “feminine methods” were somehow culled from the equation, a newly uncoded underbelly of pure geometry and pure arithmetic would spring forth, like Aphrodite from the sea foam.

So while the present argument has relied heavily on the usage of “genus” in mathematical discourse, and the gendering implied by such a term, I am ultimately more persuaded by Joan Copjec’s preference for sex over gender. In Copjec’s view, recent feminist theory has tended to replace sex with gender, or, as she puts it, to remove the category of sexual difference “in favor of the neutered category of gender” (“Sexual” 193).28 What if something similar has taken place in math? What if math is not so much gendered (and neutered) as it is sexuated?

Back, then, to cutting and grouping. If math has a grouping, does it not also have a cutting? If math is gendered, is it not also sexuated? For Copjec, symbolic languages (including mathematics) are marked by a splitting or cutting. Or she would likely say it in reverse: take the difference at the heart of math; such a distinction is “coincident” with math, meaning it is math.29 For Copjec, language fails and “[s]ex coincides with this failure, this inevitable contradiction. Sex is, then, the impossibility of completing meaning [ . . . ]. [S]ex is the structural incompleteness of language” (“Sex” 206). Indeed, the sexuation of language was itself already defined in mathematical terms by Lacan, namely in the simple algebraic mathemes called the “formulas of sexuation” that appeared at the end of his Seminar XIX and again at the start of Seminar XX. Lacan’s mathemes are controversial, to be sure, and Lacan’s formulas of sexuation have been rejected by many feminists, not to mention most mathematicians. Yet, in my perhaps unorthodox reading, Lacan was not so much showing that sex has a mathematics as he was showing that mathematics has a sex. This, then, is the sexuation of math, after Copjec (after Lacan): mathematics might have a type or genus, but more importantly, mathematics is also the coincident consequence of a contradiction.

In other words, the villains of this story—all those like Dedekind or Frege who were calling for the elevation of arithmetic at the expense of geometry—were nevertheless correct on one point, a crucial point. Not to their cartoonishly bad gender politics (intuition bad, rigor good), I refer instead to the split itself, the elemental assumption that mathematics arrives already riven, already cleaved in two. Yet, this is not two as a disciplinary technology, enforcing binary identity or two genders or what have you, but quite the opposite, as the proper fabric of dialectical contradiction, as a way “to unground any ground that might be attributed to sexuality as such” (Copjec, “Sexual” 205).

Does mathematics contain an essential bias? Are algorithms discriminatory? The answer will lie in the mathematics of gender—who counts, who gets to count. But at the same time, as I have tried to argue here, the answer also lies in the gendering of math, and, ultimately, its sexuation. The gendering of math entails both the segregation of two types of knowledge, geometry and arithmetic, and the prohibition on mixing or transitioning between the types; so in one sense, human values are most certainly embedded in technologies, even if the metaphor of the “neutral vessel” remains rather dubious. But from another view, technology itself, in its very matter and form, harbors essential contradictions or cuts. I have erstwhile called one of these contradictions gender, because of its discursive grounding in “genus” and because of the feminized nature of the bias. Although, as we have seen, when asking after the gender of math, we might do better, in the end, by switching to sex. Any analysis of the culture and politics of math will need to address these structuring contradictions directly, if not now then soon.

This essay is dedicated to Ellen Rooney, Phil Rosen, and Elizabeth Weed.

Notes

1

The passage cited in the epigraph—Jude speaking to Sue—comes relatively late in the novel (351), after Sue, formerly sensible and level headed, has been broken down emotionally and turned to religion for solace. George Eliot also satirized the notion of woman as a mathematical complication in Middlemarch: “[W]oman was a problem which, since Mr. Brooke’s mind felt blank before it, could be hardly less complicated than the revolutions of an irregular solid” (41). For more on the relation between Victorian literature and mathematics, see in particular Henderson; and Kornbluh.

2

Weyl’s joke is recounted in Hottinger (15). In fact, Weyl knew Noether and admired her a great deal; his eulogy for her was published in Levels of Infinity (49–66). For more on women in mathematics, see Osen; and Williams.

3

On the mathematics of gender in the fields of math and computer science, see Hicks; Margolis and Fisher.

4

An important caveat before starting: math and computation are not the same thing, and given more time it would be necessary to define these terms more clearly and show how they are related. I acknowledge that others might be uncomfortable with reductive claims like “software is math.” Yet I consider math, logic, and computation to be intimately connected, often so intimately connected as to reduce one to another. Similar to algebra, software combines variables and operations together to manipulate symbols and the values they represent. Most of these operations are drawn from logic (e.g., comparing the truth value of two expressions) or from arithmetic in the form of addition, subtraction, multiplication, and division. In code, these variables and operations are typically grouped into larger, more complex data types and structures. Yet, while math may remain as a pure figment of abstract imagination, computer software is a special kind of symbolic abstraction in that it exists in physical form as a set of executable operations in electricity, metal, plastic, glass, and silicon. Always bound to physical hardware, computer science thus tends to be more pragmatic than its mathematical cousin, computer science being indelibly marked by questions of finitude, physical affordance, and practical operation. This realization once led media theorist Friedrich Kittler to pen a now notorious essay titled “There Is No Software,” arguing just that: software has no autonomous existence per se, since algorithms are always already embedded in physical devices. Kittler was wrong empirically, of course—there is software—and his short piece was clearly tailored to be more provocative than accurate. But I suspect he was also wrong philosophically; in order to understand computers, we must directly engage (not try to eliminate) the realm of logic and mathematics, which resides within what Marxism calls the superstructure and what psychoanalysis calls the symbolic order. In sum, I prefer not to reduce software “down” into the material base but to keep it “up” within the domain of mathematics, logic, formal abstraction, language, and the symbolic. Here I mimic the cultural turn in twentieth-century Marxism, where sometimes being a good materialist means updating one’s definition of materiality.

5

These being the arguments of three widely read books: see Eubanks; Noble; and O’Neil.

6

For more on Tay, see Handelman.

7

See Schlesinger. I am inspired by these kinds for proposals for reimagining the infrastructure of computation, including computer languages—although, as my colleague Erica Robles-Anderson recently reminded me, it may not be so easy to take a computer language and, as it were, “make it feminist.” Historically the computer has been so intimately tied to the logic of material production, while feminist concerns have tended to be more aligned with social reproduction, that a feminist computer language, were one even possible, would most likely not resemble existing computer languages at all. And a similar rebuttal would likely be forthcoming from Black studies: what is the machine language of social death (when leveraged as social reproduction)?

8

Bias is also a technology in its own right. A whole separate study would have to be made “on the diagonal”—or “on the bias,” as they say in sewing—from the diagonal of the unit square (which nearly destroyed Pythagoreanism and, later, played an important role in Plato’s Meno) and the clinamen, or oblique swerve, in Epicurus and Lucretius, to the modern intervention of Georg Cantor’s “diagonal argument” (where in 1891 he demonstrated that the real numbers are uncountable), to Kurt Gödel’s “diagonalization lemma” and Alan Turing’s own use of the diagonal argument, to Gilles Deleuze and Félix Guattari’s postmodern “machine,” defined as a diagonal that cuts through an assemblage.

9

These are but a small fraction of the many titles that frame sexuality as count. See Beauvoir; Herdt; Irigaray; Plant; and Preciado.

10

Or, if translated literally, “mankind is always doing arithmetic.” The Greek expression, Dedekind’s own, is a tidbit of mathematical wit: the expression ἀεὶ ὁ θεὸς γεωμετρεῖ (“God is always doing geometry”) had long been associated with Plato; Dedekind updated the sentiment for the modern era, replacing God with mankind, and geometry with arithmetic. The aphorism appears at the outset of Dedekind’s influential 1888 essay “Was sind und was sollen die Zahlen?” (796), usually assigned the rather limp English title “The Nature and Meaning of Numbers.”

11

Frege’s goal was to ground arithmetic in pure logic rather than psychology, intuition, or empirical experience. To achieve this, he had to dismiss Mill’s empirical definition of number, but also Kant’s definition of mathematics as “synthetic a priori.” Frege sought to “put an end to the widespread contempt for analytic judgements and to the legend of the sterility of pure logic” (Foundations 24) and ultimately to prove (contra Kant) that “the laws of arithmetic are analytic judgements and consequently [analytic] a priori” rather than synthetic a priori (99).

12

For more on the geometry-arithmetic distinction, particularly as it pertains to mathematical diagrams and drawings, see Kennedy.

13

While Husserl is best known for phenomenology, he also made major contributions to the study of arithmetic and geometry, namely in Philosophy of Arithmetic and “The Origin of Geometry,” although the latter is less about geometry in the parochial sense than it is about how something becomes an objective idea at all if it starts, as it must according to Husserl, from a “conscious space” that is “intrasubjective” (“Origin” 358). Husserl’s geometry essay was made famous retroactively, after Jacques Derrida glossed the text in his book Edmund Husserl’s Origin of Geometry: An Introduction.

14

To be sure, crossing over into other disciplines or other methods is not always so easy. As Louis Althusser reminds us in Reading Capital and various other writings, every way of thinking will have blind spots that render it impossible (regardless of its desirability) to cross “over” or “through” itself, at least not completely. For instance, bourgeois economics entails a number of constitutive blind spots—most famously around the labor theory of value—that make it difficult for the bourgeoisie to grasp the conditions of its own existence. Or as the old line goes: the bourgeoisie is the only class that, if it were to understand itself, would cease to exist.

15

“Hence it is not possible to prove a fact by passing from one genus to another [ἐξ ἄλλου γένους μεταβάντα]—e.g., to prove a geometrical proposition by arithmetic,” Aristotle argued in Posterior Analytics (75a, lines 39–40). A similar expression also appears in Aristotle, On the Heavens (268b, lines 1–2). By the mid-twentieth century, following the work of Gilbert Ryle (Concept of Mind), Aristotle’s fussiness over incompatible types came to be subsumed under the larger notion of a “category mistake,” that is, mistakenly using a term in a context in which it does not belong.

16

This is not the place to comment at length on such “fear of mediation.” Suffice it to say that Western philosophy and culture is overflowing with anxieties toward middles, matter, mater, and media of various kinds. Perhaps the most unambiguous illustration is found in the “law of excluded middle” (lem), a foundation of logic stating that something either is the case or is not the case, but that no middle option exists. The lem is sometimes expressed using the Latin phrase tertium non datur, literally, “no third (way) is given.” So the fear of the middle is also, in a sense, the fear of the third way, an unfortunate predicament already chronicled for decades by feminist and postcolonial theorists.

17

Definition 3 in book 5 states that “[a] ratio [Λόγος] is a sort of relation in respect of size between two magnitudes of the same kind [δύο μεγεθῶν ὁμογενῶν]” (Euclid 116).

18

Euclid’s definition of arithmetical number was based on the concept of a monad or unity. Given a monad, the counting numbers were constructed as repetitions of the monad, hence repeat the monad once to construct 2, again to construct 3, and so on. Likewise the fractions or rational numbers were, as it were, “internal” repetitions of the monad: divide once for 1/2, again for 1/4, and iterate onward accordingly, combining fractions to make more complex values like 3/4. This system of external and internal ratios—like the standing waves and harmonics in a vibrating string—had two key consequences. First, it axiomatized the monad; we don’t know what this monad is or where it comes from, yet after its appearance the monad generates the rest of the system in its image. So while the monad is the absolute precondition for arithmetic, the monad is, in some elemental sense, pre- or nonarithmetical. (The monad thus acts as the general equivalent within its own domain, similar to the phallus in psychoanalysis, the sign in semiotics, or money in economics.) And second, this set of “ratio numbers,” or in proper parlance “rational numbers,” reveals, through the silhouette of its construction, a whole secondary set of numbers irreconcilable with either the internal or external ratio. Given the absence of ratio, or the failure of the logos, Euclid called these other numbers nonratio or “irrational” numbers, as the Pythagoreans had before him. No mere curiosity, irrational numbers are ubiquitous, including common and significant values such as √2 and π. So Euclid used geometry to derive arithmetic (the monad as geometric cypher, followed by the ratio as geometric proportion), and he also showed the outer boundary of arithmetic, irrational numbers, which remain unconstructable in arithmetic but indeed pose no great threat to geometry—unconstructable, that is, until Dedekind’s “Continuity and Irrational Numbers” of 1872, based on work first developed in the 1850s.

19

David Hilbert also described arithmetic as a “genetic” method of “engendering”: “Starting from the concept of the number 1, one usually imagines the further rational positive integers 2, 3, 4 . . . We can call this method of introducing the concept of number the genetic method, because the most general concept of real number is engendered [erzeugt] by the successive extension of the simple concept of number” (“On the Concept” 1092).

20

On the question of excluding “foreign elements” (Foundations 119), it is worth remembering that Frege was an avowed anti-Semite.

21

See Pourciau for more on Dede-kind’s “quintessentially German Idealist” posture (630).

22

A signal event in this history was Hilbert’s 1899 axiomatization of geometry, published as The Foundations of Geometry.

23

Ironically, the old logical fallacy that “correlation does not imply causation” has been eroded in recent years by computer science itself. Contemporary data science, from Google all the way on down, is very much invested in the notion that formal similarity is sufficient for determining meaning and value. Literary critics have long “read for form” and will sometimes base their arguments on how something “looks like” something else; today, science is trending in a similar direction, away from absolute causal laws and toward the more contingent methods of inductive empiricism.

24

I thank Sam Kellogg for pointing out to me that resistance to geometric intuition is also an important subtheme in Gaston Bachelard’s The Poetics of Space.

25

See Frege, Foundations; and Husserl, Philosophy. Note that Frege offered no other evidence for the viability of the principle of identity (the basis of his logical theory of number) than that it was intuitively true.

26

In his essay “The Origin of Geometry,” Husserl wrote of geometry as “an implicit knowledge [ . . . ] a knowledge of unassailable self-evidence” (355). Bolzano made a distinction between evidence and justification, the former being a quality of geometry, the latter associated with analysis, logic, and arithmetic. “For while the geometrical truth to which we refer is (as we have already said), extremely evident, and therefore needs no proof in the sense of confirmation, it none the less needs justification” (“Purely” 228). Hence geometry is evident and unproven (or at least requires no proof), while Bolzano’s “purely analytical” method sought to justify itself properly, which is to say, without the weakening and diluting admixture of geometry.

27

I thank both my colleague Salwa Hoque and an anonymous peer reviewer for pointing out this important detail.

28

Copjec made a similar argument already in the early 1990s directed at Judith Butler’s preference for gender (“Sex and the Euthanasia of Reason”).

29

Elsewhere, in an essay titled “Mathification,” I have also labeled this “Badiou’s Principle,” or the notion (derived from the work of Alain Badiou) that math itself is properly defined as the difference between geometry and arithmetic. Here we may assign the same impulse to Copjec and Lacan, the latter of course also having been a key influence on Badiou.

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