Mapping Sex and the Social: The Geometry of Gender
NO-FIGURES VERSION
Moon Duchin
University of Chicago
Males do not represent two discrete populations, heterosexual and homosexual. The
world is not to be divided into sheep and goats. Not all things are black nor all things
white. It is a fundamental of taxonomy that nature rarely deals with discrete categories.
Only the human mind invents categories and tries to force facts into separated pigeonholes. The living world is a continuum in each and every one of its aspects.
–Alfred Kinsey et al, 1948, page 639
Mappings are essential to talk about the world, even casually. Stimuli from the world are
unimaginably complex, and we need simpler structures to render them. Though the domains of
our descriptive maps may be wild and complex, we can choose the target space (the field of
categories by which we structure our descriptions) to be small and tame. Finite target spaces are
simple, and the smaller the better— dichotomies, those neat categorizations of enduring
popularity, can be regarded as maps whose target space is a pair. In the mainstream modern
Western wisdom, sexing maps the big and unruly set of all people to the simple target space
{M, F}. In some narratives of 1950s women’s communities, role-playing maps lesbians to
{butch, femme}. In his ground-breaking study, Kinsey maps the subjects of his study to {white,
Negro} immediately upon meeting them and ultimately to {0,1,2,3,4,5,6} according to his
assessment of their sexuality.
Let us take a closer look at Kinsey’s coding scheme for sexual orientation. He conducted
face-to-face interviews with thousands of subjects, asking hundreds of rapid-fire questions about
the details of their sexual practices and desires.1 From the interview, the researcher would fill a
grid on a single sheet of paper – sometimes two – with an externally inscrutable collection of
numbers and other symbols.
1
Kinsey did the majority of the interviews for his work, set the research program, and is most often
identified with the voice of the books that reported the results. For that reason, I will refer to him as the
author, though he led a larger team.
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Figure 1: "Sample history in code" (Kinsey 1948, 72)
The intricate grid of notations would eventually be distilled into a single numeric value by the
researcher to rank the subject’s sexuality on the famous Kinsey scale from 0 (totally heterosexual)
to 6 (totally homosexual). From a domain unimaginably rich – the field of personal sexual selfdescriptions – and via an array of symbols on paper, Kinsey ultimately maps down to a digit, one
of seven in the target space. He may have a continuum in his mind’s eye, but not in his mapping
scheme practically.
Any system of labeling, indeed, is a mapping, in very nearly the mathematical sense. The
collection of labels – colors, flavors, preferences, names – is the target space, and each input is
associated to one of the label choices. In mathematics, a mapping is a slightly generalized notion
of a function. To the objects in some set or collection, it associates other objects; in other
language, inputs are sent to outputs.2
Like Kinsey, we might soon become impatient with the limited options offered by finite
target spaces. Feeling limited, we might seek to stretch our labeling powers to reflect gradations,
ambiguities, or milder distinctions between alternatives. The standard description of one’s
birthday, for instance, has 366 possible targets in the days of the year. We might object that those
divisions of the calendar are invented categories, or “arbitrarily separated pigeon-holes,” and that
birth moment can conceptually be considered to lie along a continuous timeline—improved
measurement allows for improved accuracy of placement. The time-honored topology for a
labeling field which allows continuous variation is that of the continuum or spectrum. We can
take a standard continuum to be the interval between some two points A and B on the real number
line: it is a smooth, unbroken line segment.
2
I am using an appendix to fix my technical terminology, but I hope that the paper will be readable without
poring over definitions.
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Figure 2: The vaunted spectrum—a line segment from A to B.
Also known as a continuum or an interval.
Binaries have long come up for scathing critique in feminist circles, and there is growing
dissatisfaction with the continuum as well in the feminist record (Sedgwick, Stein, Weston). I
hope to expand the critique of the identity-by-number-line by offering a topological inquiry into
maps of sex and gender, with forays into other realms like geography, politics, and marketing.
This work is based on the main claim that it is productive to look at the taxonomies of
gender theory and identity politics from a mathematical point of view, and in terms of models. In
fact, mathematical language already pervades these fields; as Kinsey’s short quote suggests, we
will frequently encounter notions like discrete, continuous, continuum, and spectrum. The
mathematical lexicon is in play not only because it gives an air of rigor, but because it is useful—
however, the usefulness of this language is immensely boosted when it is used precisely. We are
surrounded by highly vernacularized mappings in the form of descriptive schemes, ways of
talking about birthdays, sexuality, race, and countless other attributes of identity. Failing to see
them as only models, which are subject to evaluation and perhaps replacement by better models,
has political as well as theoretical consequences.
In order to traffic in suggestions as well as critiques, I will introduce a move (termed here
“bending a spectrum”) which provides a family of examples of two-dimensional topologies in the
place of existing continuum models. I also explore some problems and rewards of passing to
many-dimensional mapping. I will put off to the end what might rightly come first: thoughts on
modeling and reality claims, or the ontological status of these mappings. Throughout, I try to
emphasize that models can and should be evaluated in terms of their intended functionality. A
model that provides insight and accurate quantification in one context may easily, when used
beyond the bounds of its applicability, be a source of errors and fallacies. For the converse
reason, a critique leveled here should not be taken to be a blanket condemnation of a model.
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Cartography
One of my main goals is to discuss maps beyond the spectrum; I will begin with a very
familiar collection of examples, though at first glance unrelated to identity politics. Maps in the
cartographic sense are mappings, from physical three-space to a planar rectangle, say.
The Earth is roughly a ball, its surface roughly a sphere. A globe can offer a reasonable
representation of the physical features of the Earth’s surface, though of course a particular globe
has some arbitrarily-chosen features like the orientation of the axis of rotation and, indeed, which
way is “up.” However, for many applications, a flat representation is preferable, creating the
problem of rendering a sphere into a rectangle with minimal distortion of the kinds of information
a map is useful for (relative position, relative area, and shape) for the objects of interest (land
masses and bodies of water). The classic rectangular map is called the Mercator projection; it is
achieved by wrapping the sphere in a cylinder, say one that is tangent at the equator, and then
projecting horizontally from the sphere to the cylinder. The cylinder is then unwrapped to
produce a rectangular map. This is close to what Mercator did in his pioneering map of 1569,
except that a translation is then applied to make Europe the literal as well as figurative center of
the world.
Figure 3: Mercator projection (InterCarto). Note that the map is commonly truncated—
gargantuan Antarctica is a clue to the area distortions.
As a result of the direct cylindrical projection method, bodies far from the tangency are
distorted in their areas. On Mercator projections with tangency at the equator, like the one shown
in the figure above, Europe (actually 9.7 million km²) appears larger than South America (17.8
million km²). Scandinavia (1.1 million km²) looks as big as India (3.3 million km²). And,
famously, Greenland (2.1 million km²) seems to equal Africa (30 million km²) and considerably
exceed the size of China (9.5 million km²).
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Maps like Mercator’s blatantly carry values. Some regions are figured as central while
others are peripheral—a literal pushing to the margins, which is of course completely an artifact
of the projection, since the sphere is symmetric and boundaryless. Certain localities are
exaggerated in size and thereby in prominence; others are diminished.
Figure 4: Equal-area projection (InterCarto). This map is corrected to be accurate with respect to
relative area, at the cost of some distortion of shape and position.
The countries and their relative positions and shapes from Mercator’s map are familiar
and comfortable to the average twenty-first-century American; maps made with Peters’
projection, a modern alternative which preserves relative area at the expense of shape, look
strange and distorted. This is an indication that the values carried by the map have likely been
widely internalized as well.
Is all the world a continuum?
Beyond these competing planar mappings, there are also several familiar finite mappings
of the geographic world. For instance, we divide the globe up into seven continents (Europe,
North America, South America, Africa, Asia, Australia, and Antarctica). This suggests a
mapping of the globe whose domain is nearly all of its land mass and whose range is that sevenpoint set. During the Cold War, the nations were commonly aligned another way; as “First
World” (the United States and its Western capitalist allies); “Second World” (the
communist/socialist states); and “Third World” (those unaligned, often undeveloped). This
constitutes a map of the world’s nations to {1,2,3}. With the fall of the Soviet Union, this triad
collapsed to today’s binary of First World and Third.
Several other binaries have some currency for mapping the nations of the world as well;
we speak of East vs. West, and of North vs. South. Would a spectacularly reduced binary map of
the world (East-West; North-South; First World-Third World) be significantly rescued if we
allowed a whole spectrum from A to B? Would such a continuum even make sense?
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It is clear from this example that a spectrum, though it is “infinite” in the sense of having
infinitely many points, should not be regarded as having arrived at a nirvana of sufficient
sophistication. For different purposes, different kinds of maps are appropriate and useful. To
navigate the streets or engage in a nuanced discussion of geopolitics, a spectrum will not do.
An Identity Counting Project: 1, 2, 3, 4, 62
Counting and regulation
In the beginning, either we are all of the same stuff, or we are of two kinds. Totalizing
schemes and binaries are familiar systematizers; they are maps to sets of readily comprehensible
size—one and two. Familiar modern American duos are gay and straight, Republican and
Democrat, Type A and Type B personalities, tops and bottoms. And when a seemingly secure or
natural binary is brought up for criticism, very often it spins off a third category as a sort of
overflow or residual, an Other to absorb its exceptions and inoculate the system against future
nagging critiques of this kind.
Sex and gender (or the more cutting-edge “sex-gender”) come in familiar twos that, when
stressed, admit the logic of a third. A hermaphrodite category for the intermediately sexed is of
ancient vintage, and a locally defined Third Sex can be pointed to in any of a number of cultures.
Kath Weston observes, “The Third has gathered together, under the same rubric, hijras,
transsexuals, guevedoche, butches, eunuchs, xanith, sapphists, kwolu-aatmwol, ‘Balkan virgins,’
and more, not to mention garden-variety gays.”3 With visions of the continuum dancing in their
heads, authors of three-way category schemes usually imagine the Other lying in between the
poles. We might guess that the problems will not disappear as we increment from one to two to
three to four and on; counting identities might have problems in principle.
She continues: “…almost the entire cast of Third Genders performs on a ‘Third World’ stage, sometimes,
it seems, against painted harem and South Sea island backdrops. …where they can be exoticized in ways
fully compatible with the colonialist penchant for new nomenclature and classification” (Weston 42-3).
This observation rings true as well for academic feminist treatment of intersex.
3
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So what’s wrong with enumeration? That there is a count presupposes that the objects or
classes at hand are fixed, consistently distinguishable, and finite in their possibilities. The count
insinuates that the categories are in some way parallel or structurally like to one another; often the
reality that this is not at all so is hidden in the fact that one category holds all the residual as a
broad, untamed Other. Counting and sorting can take the place of describing. And, perhaps most
importantly, counting is very often a means to regulation—regulation that can deepen the
divisions into divides.
Consider the striking example of counting to one, two, and eventually four in Apartheidera South Africa. The ruling white Afrikaaner government sought to prohibit sexual and
romantic pairings between black and white people in the Immorality Act of 1949 and the Mixed
Marriages Act of 1950. To be enforceable, these acts demanded a legal classification scheme.
To that end, the Race Classification Act and the Population Registration Act (both 1950) were
implemented; they combined to produce the legalized fiction of neatly separable racial categories,
with spurious sorting criteria like the thickness of one’s hair. This classification was the
immediate means to a set of horrifying ends: immediately on the heels of the legal sorting
followed a series of other laws codifying extensive and brutal racial oppression. The Group
Areas Act (1950) divided the land into zones to be inhabited by the members of various races and
people were subsequently evacuated from their homes and relocated by force; the Separate
Amenities Act (1953) mandated the division of public resources, from drinking fountains to
ambulances and even roads, between the races; the Bantu Education Act (1953) declared a
separate school system for Black Africans. In fact, four different educational authorities were
ultimately established, one each for white, Indian, “coloured,” and Bantu or black. The black
schools, needless to say, had extremely low resource levels; they were expressly set up to prepare
workers for white industry.4 This is a clear example of what might be termed “the violence of
4
Apartheid law information comes from McKeever and Oliver. McKeever quotes Minister of Native
Affairs H. Verwoerd saying, in reference to the Bantu Education Act, that “we can see to it that education
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counting”—categorizing people becomes a means to their differential treatment, and rigid
schemes which pretend to partition people into neat subsets make callous regulation possible.5
Problems of this kind are not just “over there,” far away from new millennium America.
The emergent intersex movement in the United States has provided ample grounds for critiquing
the Third Sex for maintaining the divisions of male and female with an all-purpose offload. But
intersex activists are rightly frustrated with academics whose interest in intersex begins and ends
with a problematizing of theoretical categories. Infants are – still – routinely subject to medically
unnecessary surgery; adults are denied access to their hospital records (and with it the ability to
make informed medical decisions) when their history includes infant sex reassignment. The
proliferation of medical categories for intersex becomes a means for forced “normalization.” The
taxonomy, then, has human rights implications as well as theoretical dimensions.
Sixty-two clusters
In a totally different sector, there is ample motivation to disaggregate the nation into
categories: the U.S. marketing industry has elaborate systems set up to typecast money patterns
and movement. Several major firms have competing methods of dividing the population into
quasi-geographic clusters in these terms. Cluster aficionado Michael Weiss has written a book,
The Clustered World, which focuses on the 62-cluster system called PRIZM, arranging
Americans into groups ranked by a “composite score of affluence based on income, home value,
and educational achievement.”6 In terms of SER (socioeconomic rank), they range from Blue
Blood Estates to Southside City. He explains: “Once used interchangeably with neighborhood
type… the term cluster now refers to population segments where, thanks to technological
will be suitable for those who will become the industrial workers in this country… What is the use in
teaching the Bantu mathematics when it cannot use it in practice? that (sic) is quite absurd” (105-6). She
also notes that per capita educational expenditure in 1976 South Africa was still fourteen times higher for a
white child than a black child.
5
“The violence of counting” is Weston’s phrase.
6
Weiss 179. PRIZM, or Potential Rating Index for Zip Markets, is the proprietary cluster system of
Claritas Inc.
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advancements, no physical contact is required for cluster membership. The residents of Pools &
Patios, a cluster of upper-middle-class suburban couples, congregate in La Crescenta, California,
and Rockville, Maryland, but they can also be found on one block in Spring Hill, Tennessee, and
in a few households in Portland, Maine” (Weiss 14).
Why study clusters? More finely targeted mass mailings are a first step, and generally it
is easy to imagine the scheme’s usefulness for those with anything to sell. To Weiss, though, the
clusters have much broader applicability. “As a journalist,” he says, “I saw the dual potential of
clusters: as a clever way to sell soap and an insightful guide to understanding how people live”
(Weiss 3). Indeed, clustering offers a portrait by the numbers of contemporary America, at least
if the adage is true that you are what you buy.
Instead of a geographic unit, a cluster is now an identity type. Weiss’ book literally maps
out our demographics, with images depicting Brie vs. Velveeta purchase density; the
concentration of Wal-Mart shopping, Oprah vs. traditional Book Clubs, Call Waiting, and selfreported Stress; and frequency of the belief that the U.S. government should apologize for
slavery. All this and a great deal more is correlated with cluster membership. If we tell the
marketers, armed with clusters, about your consumption – how much Velveeta and what car and
which telephone features you buy – they will predict with great confidence your racial identity,
how you vote, whether you would like to reduce the stress in your daily life, and even what you
are reading.
These results are obtained by inputting a massive number of measurables into systems
with vast computing power and performing the statistical techniques of cluster analysis or factor
analysis. These techniques find clusters in the data where there is very high correlation among
the variables. In this sense, clustering produces a bottom-up rather than a top-down taxonomy: it
is data-driven in its categorizations.
For me, Weiss’ clustered world calls back to mind what Kath Weston has termed
“number fetishism,” which “continues to lend fluid social relations the appearance of fixity and to
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cloak unjust social arrangements with an aura of the inevitable” (Weston 23). Fixity because the
clusters are a timeless, static description; they include no sense of change or changeability; they
freeze what may be trends into essences.7 Causality disappears, and in its absence, unjust social
arrangements become value-free facts—consider for instance that in the formulation of “SER,”
educational achievement is included as an indicator, rather than a result, of socioeconomic rank.
As with race classification, it is relevant to consider what motivations drive the
development of these clusters, and what social outcomes follow. It matters, to cite one example,
that clusters become a means for predatory lending as banking conglomerates open branches in
different neighborhoods with different lending policies and rates. Clusters may be not just a
freeze-frame of the social configuration they study, but also a mechanism to maintain its
divisions.
Obviously we are not at liberty to jettison the notion of counting in order to avoid
consequences like these. Sorting schemes are unavoidable and are useful, of course, for many
ends which are compatible with social justice. In the presence of a policy prioritizing open
information and patient consent, for instance, the medical classification of intersex conditions can
clearly be of great help. However, we will be justified in approaching neat identity
categorizations with a great deal of skepticism since it will often be the case that their usefulness
for regulatory purposes outweighs their descriptive insight as a raison d’être.
Biological Sex: Continuity and Discreteness
If counting schemes, and especially small finite ones, are vulnerable to critique as
arbitrary and potentially unjust, maybe we can revive our faith in binaries with the one that seems
the most secure of all: the biological sex programmed in the code of our genes and written on our
bodies. The division of the species into male and female seems to many to be the most basic and
7
The clusters need to be completely recalculated with each new census since there is no room for dynamic
change in the model. Most of the other models described here are static as well, and how much this
undermines the usefulness of the model varies.
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inalienable instance of twoness underlying all human experience, challenged only perhaps by the
distinction of Self and Other.8 Ironically, even though Kinsey’s epigram exalted the continuum
above all things, he divided his oeuvre into Male and Female volumes and seemed perfectly
comfortable with that essential binary.
The Five Sexes and the Intersex Spectrum
Feminist biologist Anne Fausto-Sterling published an enormously influential article, “The
Five Sexes,” in 1993. In it, she argued for the recognition, besides male and female, of so-called
merms (male pseudo-hermaphrodites), ferms (female pseudo-hermaphrodites), and herms (the
rare but theoretically possible true hermaphrodites) rounding the number of sexes out to five.9
Upon inspection, her scheme really amounts to a scientifically dubious three-way splitting of the
Other, and in fact Fausto-Sterling meant the scheme somewhat humorously, then disavowed it a
few years later in her follow-up article “The Five Sexes, Revisited” (2000).
Part of the explanation for the amount of attention given to her “Five Sexes” is
explainable by its fiveness—more than three, but not too much more—which frightened some
and comforted others. In one camp, the five was a worrisome move, a sign that the time-honored
overflow category was expanding and demanding more psychic space. In another camp, at the
same time, the five had the soothing effect that science had caught up to sex and order was
restored. The state of the art had progressed, and now we had a neat categorization; the sexes
even had their own bite-sized names, and everything was under control. The regulatory
possibilities are immediate: a few new sets of legal codes might be called for, or perhaps new
surgical protocol, but nothing more.
8
In fact, biological sex is not the only discrete designation which is arbitrarily fetishized as a means to
partner choice. In Western astrology, there are twelve signs of the zodiac; in Japan, wisdom on
compatibility is organized by blood type.
9
The distinction was supposed to be made on the basis of internal morphological sex—that is, on sexdifferentiated organs and structures inside the body.
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The topological possibilities are also immediate. The three intersex categories lend
themselves to an arrangement between the poles of male and female: from female to ferm to
herm to merm to male. And indeed even in Fausto-Sterling’s update of her logic, the new ideal
she is promoting is precisely the blurring of categories whose consequence is the linear spectrum
suggested by that arrangement. This is clear in passages from her most recent book, Sexing the
Body: “While male and female stand on the extreme ends of a biological continuum, there are
many other bodies… that evidently mix together anatomical components conventionally
attributed to both males and females” (Fausto-Sterling 2000, 31). It is stunning that FaustoSterling would use such a description, being intimately acquainted with an array of intersex
configurations of sex indicators that absolutely defy rankability on the linear M-F scale that she
suggests in passages like these. A partial list of conditions includes Congenital Adrenal
Hyperplasia, Testosterone Biosynthetic Defects, Gonadal Dysgenesis, 5-alpha Reductase
Deficiency, Micropenis, Klinefelter Syndrome, Turner Syndrome, and Timing Defect10—these
range tremendously in kind, with some located in chromosomal anomalies, others in atypical
hormonal levels, and one (“micropenis”) simply describing an unusual cosmetic outcome.
I
consider their pairwise comparability with respect to male and female poles to be a theoretical
impossibility. (I also consider their exoticization, stigmatization, and authoritarian medical
regulation to be an outrage.)
Discrete or continuous?
Even if counting is problematic and the spectrum is inadequate, there is still hope for
identity mappings. I want to make a forceful intervention into the discourse around taxonomies
on the issue of discrete and continuous. There is a frequent hidden assumption, and sometimes
explicit claim, that categories only come in these two types. In combination with a confusion of
This is the list compiled by Carl Gold for NOVA’s “Sex: Unknown” site (NOVA). The list, with
descriptions of the extremely dissimilar causes and effects of the “intersex conditions” it tabulates and an
explicit admission of its own incompleteness, is inexplicably titled “The Intersex Spectrum.”
10
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continuous with continuum, this impoverishes models of identity in just the way described
above—whatever defies simple enumeration is forced onto a spectrum between two extremes. In
the following paragraphs, I want to argue that there are more types of models available than
counting and the continuum, and even than discrete and continuous.
Ed Stein, in The Mismeasure of Desire, spent considerable energy on the question of the
mapping geometry of sex and sexuality. He says,
Categories come in two different types—continuous or discrete—depending on the nature
of the entities to which they apply. A category is discrete if any entity that falls into that
category is as much a member of that category as every other member and if there is a
clear-cut line between the category and rival categories. A category is continuous if there
are various degrees of being a member of that category and if the category shades into
rival categories (Stein 29).
Mathematically, this is not sound. There are many possible topologies, and discrete is only one—
namely, it is the topology where every point is isolated. Using “continuous” for everything
indiscrete is misleading. As points on the number line, the set
{1, 1/2, 1/3, 1/4, 1/5, 1/6, … , 1/n, …}
is not discrete – because as the denominators grow, the points accumulate at zero – but neither
should we be tempted to call it continuous, since there are jumps between successive points.
It is not immediately clear how to formalize the vernacular idea of a “continuous”
category, but we need a formalization if we want the benefits of rigor and not just the aura of
mathematical language. A promising possibility: a space is path-connected if an unbroken path
exists connecting every two points. Path-connectedness is certainly false for discrete spaces and
true for a continuum.11 Perhaps this notion – the path-connectedness of the target space –
succeeds in capturing what is intended by calling a categorization continuous, since it means that
11
In math, we have continuous maps, but not spaces, so we should seek a concept which captures the
intuition of a “continuous category” and can be applied to a target space. A map is continuous if nearby
points in the domain are mapped to nearby points in the range. This is only well-defined if the domain is
topologized, which we would not want to assume for these identity maps where the domain is an a priori
unstructured collection of inputs. Formally, a space X is path-connected if for any pair x and y of elements
of X, there is a continuous map f: [0,1]  X such that f(0)=x and f(1)=y (the image of this map is the path
in X from x to y). In this way, continuity figures into the definition.
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gradations between any two positions would make sense (and thus matches Stein’s informal
description given above).
Frequently, authors like those cited here conflate the continuous with the continuum,
rather than regarding the latter as just an example. Consider Ed Stein’s statement: “None of the
evidence that supports the view that there are more than two sexes counts against the view that
sex is a continuous category (that is, with male and female at either end of the continuum and
with various intermediary forms existing between them)” (Stein 31). Like Fausto-Sterling, he can
go no further than the spectrum. Later, he says, “The relevant difference between continuous and
discrete categories is that a continuous category involves, at least implicitly, an ordering among
the entities that it characterizes” (Stein 52). Not so, although the last claim – that there is an
implicit order – is true for a continuum. A standard circle, for example, is a topological space
which should clearly be considered continuous, but which has no intrinsic poles, or starting point,
or end. We could for instance model the stages of the menstrual cycle by a circle, and that model
carries no implicit order (it is neutral as to whether ovulation should be considered to precede or
to follow the onset of menstruation, say).
Discrete and continuous are valuable paradigms, as they capture noteworthy features of
models—that they are comprised of singular, separate categories on one hand, or that they allow
differences of degree on the other. A few observations have been made here: not every set of
points is discrete; not everything continuous is a continuum; and there are spaces which are
neither discrete nor continuous.
The assignment of sex
We have seen that mathematics offers more possibilities for target spaces than just
discrete and continuous. This should be very welcome news, because the world also offers more
than just discrete and continuous phenomena to study. I will offer sex as an important example
here. I have already argued above that the idea of arraying all possible sexual configurations on a
spectrum is not workable. Perhaps, though, each single sex-related attribute is either discrete or
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continuous. Hormone level might be considered to vary continuously, for example, while
chromosomes may seem to come in discrete possibilities.
Kinsey held that discrete categories are not found in nature and Stein told us that no
evidence contradicts the idea that sex is a continuous category. Let me do Kinsey one better and
give a lie to Stein: I claim that not only the discrete but even the continuous modeling paradigm,
however useful, is virtually never neatly actualized by nature, and that biological sex furnishes an
excellent example of this principle. Even karyotype (chromosomal genotype) – the last vestige of
human insistence in twoness of sex, XX and XY – is not only not two, but not really discrete in
any meaningful way. Chromosomal anomalies are very frequently trotted out in discussions like
these: Turner’s syndrome (annotated in medical texts as XO), Klinefelter’s syndrome (XXY), and
a constellation of others are recorded. This seems to shake up the size of the target set but not its
finiteness. What this scheme simplifies for the sake of intelligibility is the prevalence of
mosaicism, the presence of different genotypes in the various cells of the same organism. For
instance, no fetus is viable (can survive to term) if all of its cells have the XO karyotype, so every
person with Turner’s syndrome has in fact a mix – most often of XX and XO – cells, some in
different regions of the body. Mosaicism is not an anomaly. Tortoiseshell cats display splotches
of several different coat types for this reason—the coat pattern is encoded on the X chromosome,
so the fur actually displays a map of the cat’s genetic mosaic. And in fact the work of Mary Lyon
indicates that among humans, all women are to some degree mosaics. This is because every cell
will express only one X chromosome: in XX individuals, one of the X chromosomes must be
deactivated in every cell, so that the other can be unambiguously expressed. Lyon hypothesized
that each cell decides independently which X chromosome to deactivate. This process, known as
Lyonization, has now been scientifically corroborated.
In Turner individuals, the degree of severity of symptoms is roughly proportional to the
degree of XO present in the mosaic, a matter of quantity which is impossible to capture in an
actual count, since counting the number of cells in the body is a theoretical impossibility. XO is
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usually considered phenotypically female; however, there are cases of XO/XY mosaic on
record.12 How then to argue that even karyotype is truly discrete, let alone the other indicators of
sex? At the same time, it is clearly equally absurd to try to array the options along any sort of
continuum. Even path-connectedness is out of the question, as there is no sensible notion of a
smooth variation between A and B. To see this, consider (A) an XX individual with adrenal
gland overproduction of testosterone, resulting in sex-ambiguous genitalia (Congenital Adrenal
Hyperplasia) and (B) an XY individual with no receptors for androgens, who manifests
exaggerated female sex characteristics, has a short vagina, and no ovaries (Androgen Insensitivity
Syndrome). It is absolutely not the case that every possible permutation of sex characteristics is
biologically viable, and it is nonsensical to argue that a full spectrum of intermediate positions
from A to B is possible.
In short, biological sex, or even its single attribute of chromosomal configuration, is a
clear example of a phenomenon whose description requires both matters of kind and matters of
degree.
Gender, Sexuality, Presentation: Taming Human Nature
If new insights are available from complicating the mapping geometry of biological sex,
then we might be hopeful about turning our attention to matters more obviously amenable to
social construction, because it is more clear that the parameters of description are not naturally
mandated. I will focus on two familiar continua: those used to interpolate between straight and
gay and between masculine and feminine.
A continuum in seven pieces
Alfred Kinsey was a devoted taxonomist, and taxonomies belong to a counting tradition.
As Kinsey observed, “Taxonomy is a development of systematic botany and systematic zoology”
12
See for instance Max Beck’s “My Life as an Intersexual” on Nova Online’s “Sex: Unknown” site.
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(Kinsey 1948, 17). And indeed his background is in precisely these fields: though his name is
strongly associated with his two massive tomes on human sexuality, his other monographs are
“The origins of higher categories in Cynips” (being a kind of wasp) and “Edible wild plants of
Eastern North America.”
Figure 5: The famous diagram of the seven-point Kinsey scale (Kinsey 1948, 638)
At first glance, Kinsey is making a break with his pedigree in counting and sorting by
conceiving of sexuality as lying as along a continuum. However, the continuum for sexuality to
which Kinsey was so theoretically attached remains there – in theory – because he was prepared
to produce only a seven-way classification. He would simply have contended that detailed
enough measurement (longer interviews, more precise questions, bigger grids) would allow a
research team more and more precision, until the image dots falling along the line segment
between his poles – absolute positions marked 0 and 6 – would grow so dense as to approach the
full continuum. But in fact, it is hard to make the case that high standards of precision are even
possible—Kinsey spoke of the measurement abstractly as a “ratio” of male to female attraction,
but gave no indication in his methodology section of how such a calculation was to be made.
Kinsey had a potential powerful critique of his own exaltation of the continuum right in
front of him in the form of his “category X.” In his volume on male sexuality, a small residue of
the boys and men he classifies are denoted not 0 through 6, but X. It is not defined along with the
other designations, but about twenty pages later we may discover in the small print that “Percent
shown as ‘X’ have no socio-sexual contacts or reactions” (Kinsey 1948, 656). X men get little
attention because their numbers dwindle by late puberty, but in the second volume, where females
are the subject, the X comes to the fore because “a goodly number of females belong in this
category at every age group”—indeed, among single women, for no age are there fewer than 14%
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whose attractions and experiences do not register.13 His charts show that in the aggregate of all of
his subjects, a small majority is in category 0 (totally heterosexual) and the women of category X
total about the same number as those in 1-6 together—that is, the erotically unresponsive total as
many as those who register any attraction to women (Kinsey 1954, 472). Needless to say, in both
the male and female volumes, people in category X are simply excluded from the seven-way
graded continuum and the exposition that accompanies it. This is only given attention in the
female case, since the residual in his category scheme is embarrassingly large—twenty percent of
the total.
Masculine and feminine
The systematic measurement of masculinity and femininity is practiced in the field called
psychometrics, which has its roots in the numerical reckoning of intelligence. Among intangible
human attributes, intelligence has a rich history of being the subject of obsessive study,
quantification, and comparison. Modern intelligence measurement began with the theories of
Francis Galton (also the father of eugenics) and the test design of Alfred Binet, who introduced
the notion of I.Q. or intelligence quotient to capture the ratio of mental age to physical age. The
field was brought to the United States in the early twentieth century, in particular by Lewis
Terman, a Stanford psychologist who reformulated Binet’s test in 1917 and popularized it
stateside. He enjoyed a long-lived surge of attention and respect at the vanguard of this
burgeoning psychometric movement. By the 1930s, Terman needed a new human wilderness to
colonize and tame with testing regimes, measurement, and appraisal. He chose to formulate a test
of “mental masculinity and femininity.” With Catherine Cox Miles, Terman designed a test to
validate the prior conviction that “the sexes differ fundamentally in their instinctive and
emotional equipment and in the sentiments, interests, attitudes, and modes of behavior which are
13
The Kinsey X resonates remarkably well with the notion of a Zero figure for the unsexed recently
advanced by Kath Weston in her text Gender in Real Time. That is, it confounds the number line and calls
the poles into question.
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the derivatives of such equipment” (Terman 2). The test would be a series of binary questions;
out of a large initial pool of questions, only those 455 questions were retained for which “the
pretest males and females did, in fact, answer differently.” As a small illustration of the nature of
the resulting test, consider that a femininity point was awarded for liking “nursing, Rebecca of
Sunnybrook Farm, babies, and charades” in addition to a masculinity point for disliking these
same; conversely for “soldiering, Robinson Crusoe, people with loud voices, and hunting” (Bem
103). After the suite of questions, the M-F score was tabulated by subtracting the femininity
points from the masculinity points—for those keeping track, this effectively doubles the spacing
(a change in one answer changes the score by two points) so that scores seem farther apart. The
Terman M-F test, then, quantifies masculinity versus femininity—strictly, by construction,
opposed.
Feminist psychologist Sandra Bem, working in the 1960s, recognized that the TermanMiles test, and all others like it, “force[d] masculinity and femininity to be bipolar ends of a
single dimension” (Bem 103). Her contribution was a new test called the Bem Sex Role
Inventory (BSRI) whose design disputes the necessary bipolarity of gender roles. Instead of
choosing questions for which males and females were already known to give different answers,
Bem pre-tested attributes to find those which were deemed “more desirable in American society”
for men or for women by a sample of 100 undergraduates.14 Men and women cumulatively
agreed that characteristics like aggressive, independent, and analytical were desirable for men
significantly more than they were desirable for women, while childlike, flatterable, and yielding
went the other way.15 A second sample of students took the test that asked its subjects to selfscore their conformity with sixty attributes—twenty feminine, twenty masculine, and twenty
neutral that would not figure in the scoring.
14
15
Coincidentally or not, Bem, like Terman, relied heavily on testing the Stanford undergraduate population.
Of course, it is remarkable that some of these characteristics were deemed desirable at all.
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Her results, vindicating her test design, showed that the femininity score was uncorrelated
with the masculinity score.16 Furthermore, 36% of the men in her study and 29% of the women
had nearly identical scores in Masculinity and Femininity; she called these subjects androgynous.
Bending a Spectrum
Let me propose a move to apply to continuum models. Using it as a tool, I will show that
even slightly complicating the topology of identity mappings often has descriptive advantages
and suggests insights that are hidden by the spectrum. The move is inspired by Bem’s work, but
seems to have quite broad applicability. Very often—in fact, whenever the poles of a spectrum
each have their own identity (straight/gay; masculine/feminine) rather than measuring quantity of
one attribute (amount of Velveeta consumption; aggregate test scores)—there is room to
reconsider the opposedness of the poles. A subject in a test to measure masculinity and
femininity might be able to make sense of the question if asked to rank herself on a scale from
aggressive to conciliatory, say, without questioning the validity of the opposition. If instead she
were asked to rank her level of aggressiveness and her level of conciliation, though, she might
well consider herself to merit high scores in both. The move I will call “bending a spectrum”
consists in pulling apart the poles and measuring each independently.
Figure 6: Bending a spectrum. If the poles (A and B) represent different attributes
that are a priori independent, then they can be measured separately;
the compound attribute is mapped to a square target space with four extreme points.
This new model made by a bent spectrum provides only modest theoretical gains: it
creates two spectra (zero to A; zero to B) in the place of one (A to B). This two-dimensionality,
16
in the four subpopulations of subjects, the correlation scores were r =.11, r = –.14, r =.02, r = –.07.
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though, creates a planar grid instead of linear field of measurement and undermines the
hierarchizabilty of the measured attribute.17
Of course, there is a danger here of confusing the model for the thing. Looking at the
resulting square plot as itself a faithful picture of the identity attribute it models is like concluding
from (unimpeachably useful) rectangular geographic maps that, indeed, the world is flat. With
this caveat, we can proceed to try to get insight about identity from bent spectra.
Examples: Bending familiar spectra
With a moment’s reflection, some continuum models for identity attributes can be readily
seen to be deeply conceptually flawed. One glaring example is the Left-Right spectrum for
political identification, an artifact of the French parliamentary seating arrangement and a curious
mapping indeed. Because of the many issue-attitudes that constitute political position, there are
clear examples of people who confound the spectrum; in particular, it can be extremely
befuddling when considering some two politicians to try to put one to the Left of the other, which
would necessarily be possible for orthodox adherents to the validity of the spectrum. Libertarians
have long bemoaned the consequences of this model, arguing that the Left-Right spectrum leaves
them in the lurch, and the libertarian organization Advocates for Self-Government developed a
teaching tool called “The World’s Smallest Political Quiz” to drive home the point.
Figure 7: Chart for "The World's Smallest Political Quiz"
(Advocates for Self-Government)
The quiz and accompanying schematic can be read to be making a claim that is very interesting in
its own right: that the Left-Right political spectrum is premised on the false opposition of
political self-governance with economic self-governance. Their chart, then, is a bending of this
spectrum. If their claim is right about what makes a leftist (preference for political self-
17
Mathematically, the new model is a map to R2, the Cartesian plane, instead of to the number line R.
There is no “natural” ordering on the plane.
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governance and central economic control) and what makes a rightist (the reverse), then an
excellent point is made about the consequences of the standard continuum geometry for political
affiliation. In particular, a libertarian and an “authoritarian” (someone who believes in total state
control) both uncomfortably sit at the center of the Left-Right spectrum, when in fact in any
reasonable political metric they are very far apart.
In this way, new and valuable positions can be revealed at the two corners of the grid
which were previously collapsed to the center of the spectrum. For instance, in Bem’s two-axis
scheme for sex role, people who do not display attributes of either gender lie at one corner, and
people who register as both highly masculine and highly feminine lie at the opposite corner (see
Figure 6), prompting Eve Kosofsky Sedgwick to remark that “some people are just plain more
gender-y than others” (Sedgwick 16). Bem herself, whose original aim was to quantify
androgyny, had early on decided to consider people androgynous whose masculinity and
femininity scores were nearly the same. “Somewhat later,” she says, “the decision was made to
reserve the term androgynous for those individuals who earned their small difference scores by
scoring high on both masculinity and femininity and to label as undifferentiated those who earned
their small difference scores by scoring low in both masculinity and femininity” (Bem 120). This
distinction makes use of the topology of the plane, seeing a difference where the spectrum finds
none.
Likewise, the Kinsey scale admits bending. For instance, psychologist Michael Storms
has proposed various two-dimensional sexuality models, on the grounds that attraction to men is
a priori independent of attraction to women for a particular subject (Stein 55). This bent scale
locates at one corner of its grid the pansexual subject, highly attracted to women and men alike;
the nagging X of Kinsey’s “erotically unresponsive,” recalled from its banishment, lies at the
other corner.
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Modeling and Reality Claims
I have already alluded to the danger of “mistaking the model for the thing.” Indeed, the
literature on the subject of identity mappings contains many conflations between statements of
the type “sex is…” with those of the type “sex is well modeled by…”. Sex surely is neither
discrete nor continuous, but models of each of those types may be useful and perfectly adequate
to a particular task at hand. It is incumbent upon feminist thinkers, though, to turn a critical eye
to the tasks that demand sexual taxonomies, just as the motivations for Afrikaaner race
classification and for marketing industry development of consumer clusters bear scrutiny.
Several questions are still pressing for this work. What can the model say, if anything,
about the thing? What is the ontological status of these various kinds of mappings and what is the
nature of their truth claims? Finally, are there new ways to imagine mapping identity that are
suggested by the ideas elaborated here?
Axes, orthogonality, and independence
Inspired by Bem’s BSRI results, Eve Kosofsky Sedgwick concluded that “Masculinity
and femininity are in many respects orthogonal to each other… that is, instead of being at
opposite poles of the same axis, they are actually in different, perpendicular dimensions and
therefore are independently variable” (Sedgwick 15-16).
When you represent measurables by orthogonal axes, you are making a model. What you
can not be doing, or not without further justification at any rate, is claiming true, global
independence of the attributes being measured. That is, Sedgwick has it backwards if her
“therefore” is taken literally. Measurables should be a priori independent in order to merit
representation on orthogonal axes; then, the model given by the axes can be called on to elucidate
how the attributes in fact depend on each other.
Sandra Bem makes this point when she distinguishes between two kinds of independence
for pairs of attributes. She says, “the Masculinity and Femininity scores of the BSRI are logically
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independent. That is, the structure of the test does not constrain them in any way, and they are
free to vary independently” (Bem 159). Her logical independence is what I have called a priori
independence. Empirical data can then confirm independence-in-fact, which she calls “empirical
independence”: scores in either factor do not permit better than random predictions of the score
in the other.
Onward to n dimensions
Why stop at two measurables for describing complex social phenomena? Multiparameter
models, where axes proliferate in the hopes of ever sharper descriptive powers, are a short
conceptual leap away. Ed Stein, for instance, points out that “sexual object choice” could be
regarded as much more complicated than the sex-gender of the target of one’s attraction; for
instance, one could be preferentially attracted to gay men, or to people of certain age ranges, body
types, races, hair colors, personality types, professions, and so on without end. “There is a
serious danger here of an explosion of dimensions,” he concludes (Stein 64). For him, the
complexity of such models and the difficulty of visualizing them is a deal-breaker, even at the
level of just three or four descriptors. “Although they would be easy to articulate, three- and
four-dimensional views of sexual orientation seem so complex that any reasonable way to avoid
having to embrace them is worth considering” (Stein 61).
Not everyone agrees that an explosion of measurables is scary and should be shunned,
however. Sedgwick finds the possibility exhilarating.
“If we may be forgiven a leap from two-dimensional into n-dimensional space, I think it
would be interesting, by the way, to hypothesize that not only masculinity and femininity,
but in addition effeminacy, butchness, femmeness, and probably some other superficially
related terms, might equally turn out instead to represent independent variables—or at
least, unpredictably dependent ones. I would just ask you to call to mind all the men you
know who may be both highly masculine and highly effeminate—but at the same time,
not a bit feminine. Or women whom you might consider very butch and at the same time
feminine, but not femme. Why not throw in some other terms, too, such as top and
bottom? And an even more potent extension into n-dimensional space could, ideally,
make representable a factor such as race as well” (Sedgwick 16).
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My own stance is to urge a return to the principle of usefulness for models, remembering that a
model is appropriately assessed in terms of the questions for which it was developed. I imagine
identity modeling being very illuminating if it follows a program that makes use of the empirical
and looks to the theoretical. We can select a large collection of a priori independent measurables
of identity, such as the ones cited by Stein or by Sedgwick, and can choose a certain number of
them to study at one time – if we are not pretending to draw an absolute identity picture, then we
are under no compulsion to represent all relevant factors in the same model. Then, with regard to
data about actual people describing their actual lives, we may finally emerge with a conclusion
about the relatedness of the measurables.
In closing and to illustrate this ambitious program, I want to return to the example of flat
map vs. globe. Suppose instead of thinking of the problem as making a flat map from the globe
(in whose accuracy we are fairly confident), we imagine the inverse problem. We have a notion
of N, S, E, W and with it we can make flat maps, but our aim is to learn the real physical shape of
the world. We might notice that the sun’s passage from East to West takes different time at
different values on the North-South axis; this would be a clue from which the global relationship
of those factors could be deduced, leading to a new theory, namely the inference of a sphere as a
better model for the Earth’s surface than the flat plane.
This is something of the situation we will be in when we posit our several factors for
mapping sex, gender, and social identity: we can make a flat map, but this should not be confused
with a true or natural topology of the structure of study. Large-scale interactions of the measured
attributes may combine to suggest a manifold whose shape we had not guessed in advance—the
world is round, in a meaningful way, and the geometry of gender is an intriguing open question.
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Appendix: Terminology and Definitions
Let me fix mathematical terminology. For an arbitrary map, let the space of inputs be the source
and the space of possible outputs be the target. Those inputs which do get assigned to some
outputs comprise the domain and the collection of actual outputs achieved is the image (or the
range). Functions are usually taken to assign a singleton output (a unique element of the range)
to a fixed input, but I allow a mapping to be one-to-many, though I will not discuss such
mappings here. Note that map and mapping are synonymous, and that many authors take them to
be also synonymous with function, disallowing the one-to-many associations that I am willing to
accept as mappings.
Any descriptive scheme, such as the mappings I consider here, can also be regarded as a model. I
will use this language when trying to emphasize that choices are made when setting up the
scheme, that rival schemes are possible, and that it should be evaluated in terms of how
successfully it aids understanding of the objects of study.
Topology is the mathematical study of shapes (that is, spaces with a notion of nearness of points),
and the shapes are permitted to stretch and bend (though not to tear or to fold back on themselves)
while retaining their topological identity. The objects of topology are topological spaces, and
especially certain topological spaces with extra structure called manifolds—curves are onedimensional manifolds and surfaces are two-dimensional manifolds, for instance. Geometry is
topology plus a notion of distance, and its objects are metric spaces. One topological space may
admit many different metrics; for example, a unit circle and a unit square are different
geometrizations of the same topological object, a simple closed curve.
I am using the term field (as in “field of categories” or “field of personal sexual selfdescriptions”) nontechnically. Here, field means available collection and does not denote the
algebraic structure with the same name.
The subsection “Discrete or continuous?” includes descriptions of the terms continuous, discrete,
and path-connected. The terms interval, spectrum, and continuum are all used interchangeably to
refer to a segment of the real number line. (In many of the examples cited, authors use continuum
to mean interval with the endpoints included, but this is not definitionally required.)
As to the terms associated with modeling, they present a challenge. Several fields (pure
mathematics, mathematical modeling, statistics and factor analysis, and others) each have
lexicons in which key terms are defined subtly differently. For instance, in modeling contexts, a
parameter is usually thought of as an arbitrary constant whose value sets up the model; in
mathematics, a parameter is an independent variable which sweeps out an object of interest as it
varies over its domain. I have used the term measurables – where I might have used parameters,
variables, or factors – to avoid this confusion as much as possible. In my terminology, human
features under study are called attributes, and once they are quantified (or considered
quantifiable) they are measurables.
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