Edge Cases, or Why Binary Categories Are Broken

Edge Cases, or Why Binary Categories Are Broken

People have a habit of putting everything they encounter into categories. It makes language-based communication possible. If I tell you about the tree at the end of the road you can imagine a scene: you know what a tree is, what a road is, and how the two fit in relation to each other. But you don’t imagine the same scene that I have in mind.

When I write about an older man with a deep, resonant voice, do you think of Orson Welles, James Earl Jones, Richard Burton, Morgan Freeman? They all fit the description, the category. Even though the terms I used are pretty vague: what does “older” mean? Older then you? Than me? Over 50? 60? Showing signs of age such as gray hair or lines on his face?

“Older” must be meaningful in this context or else why would I use it? You see, everybody knows what it means, but you’d not be able to find a consensus, a common definition. Where do you draw the line between older and not-older? It’s a simple binary choice after all.

For argument’s sake, let’s say that older means older than me. That’s easy, right? Oh, you want to know how old I am? Well, I’m not old: I’m middle-aged. So maybe someone who’s close to my age shouldn’t be referred to as older. Let’s change the definition of older to mean an age ten or more years greater than mine. Oh, you still want to know my age?

I guess this is more difficult than it first seemed. Welcome to the territory of the edge case, the point at which we cut from one category to another. The problem arises because our category, older, can’t be defined unambiguously. There is no clear boundary between older and not-older.

That’s not to say that older is useless: it works well as a convenient shorthand, a stereotype. It’s just that the degree to which it applies varies. Morgan Freeman is an older man; George Clooney is less so, Ryan Reynolds probably isn’t, and Daniel Radcliffe is almost certainly not.

And yet older seems such a simple concept. As simple as tree or road. Thinking about trees, you know what a tree is, right? As opposed to a shrub or a bush, or some other plant? And a road: it’s not a freeway, or a track. Right? Or not? There’s some overlap: these terms we use casually, that we understand the meaning of very well, nevertheless are fuzzy around the edges.

We learn the meanings through examples, archetypes. A collection of instances that we are told belong to the category. It’s not about definitions such as you find in a dictionary: those are mere simplified descriptions of what most of us mean when we use a word. When you think about trees you don’t give much, if any, thought to the definition: you just know what a tree is.

It’s the same principle when it comes to gender. You learn about men and women as two mutually-exclusive categories, and you’re taught that everybody fits into one or the other. But just as with any other category, gender has edge cases. There is such a range of variation among humans that there is no characteristic, or set of features, that unambiguously assigns a person to either a male or female gender.

Yes, there are physical characteristics that apply to the majority: these are what are used at birth to decide whether to write M or F on a birth certificate. And for most people that’s fine. But there are edge cases. People who don’t have distinct physical gender characteristics, or people who look like they are male (or female), but are actually female (or male) or neither, or both.

These edge cases, intersex and transgender people, are a minority but not an insignificant one: reliable conservative estimates put the number of transgender people at around 1 in 500, meaning that we account for roughly 15 million people around the world. As a comparison, that’s about the same as the number of Jews worldwide, and more than the population of Greece, Belgium or Sweden.

That’s a lot of people for whom the gender categories don’t properly work. These people exist; gender is an invention. Which do you think might be “wrong”? The sad fact is that there are a lot of people who think of gender, male or female, as absolute binary options: each person has to fit one or the other. Reality doesn’t work that way. It’s not neat or convenient.

Believe me, as a woman myself I’ve tried long and hard to come up with some objective criteria to define womanhood. And for every single characteristic I thought of there are exceptions. It’s not a matter of definitions, it’s a matter of knowing. I have encountered many examples of women in my life and they have shaped my understanding of what a woman is. The same goes for every person who identifies as a woman. It’s an understanding that transcends words. So that is how I know I am a woman.

Saying It Doesn’t Make It So

Saying It Doesn’t Make It So

“When I use a word,” Humpty Dumpty said in rather a scornful tone, “it means just what I choose it to mean — neither more nor less.”

“The question is,” said Alice, “whether you can make words mean so many different things.”

“The question is,” said Humpty Dumpty, “which is to be master — that’s all.”
Lewis Carroll, Through the Looking Glass

There’s an article that I’m seeing a lot on my timeline just lately titled All white people are inherently white supremacists. Now, I have a knee-jerk reaction to generalization in all forms which is to assume that there are holes in the argument.

In this case, what stands out first and foremost is that it is the old “all men are inherently rapists” trope, recast in terms of race.

But there’s a little twist in this article: the author tells us that when they say supremacist they don’t mean quite what the dictionary tells you the word means, what everybody who hasn’t read the article would assume it means.

No, in this article a supremacist is not someone who advocates for the superiority of a particular group; it is someone who benefits from the advantaged status of that group. In my book (despite the denial of the author) they are talking about privilege.

Sorry, but you don’t get to redefine such a loaded word just because it suits your desire for an attention-grabbing headline. There are already words and phrases in existence that carry the meaning you want, unless you just intend to provoke and incite a reaction.

It’s disingenuous of the author: the redefinition of the word is calculated to undermine the position of anybody asserting that they are not a white supremacist. And because it’s about race, that touchiest of subjects, especially in the US, people are reluctant to call “bullshit” lest they incur the righteous wrath of those who take it all at face value.

The essence of this piece is to imply that all white people are the same: at heart they are indistinguishable from the shaven-headed neo-Nazis and KKK members who proclaim the primacy of the “white race”. I call bullshit.

There’s a particular statement in the article, “All white people […] won’t challenge and disagree with genocide, police brutality, the prison industrial complex, gentrification, the hypersexualization of young Black girls, the criminalization of young Black boys.”

That’s simply not true. In fact I find it hard to credit that more than a handful of white people would agree with those things. But according to the author all white people “are inherently racist” and by implication support the oppression and subjugation of other races.

I must admit I’ve been surprised by the support shown for this deeply flawed article with its sweeping generalizations and unsupported, easily refutable claims. It’s the same sort of polemic as Trump’s attacks on Mexican immigrants in that it caters to the prejudices and insecurities of a particular audience.

But here’s the thing: just because the author believes in what they wrote, just because there are published words on a page does not make it true. I know what a supremacist is, and claiming it means something else just doesn’t hold water. If you want to play games with language, you’ve got to do a damn sight better than that.

Description, Not Prescription

Description, Not Prescription

If the past indicative did not exist, then somebody would have to invent it.

I remember being taught English grammar at school (rather appropriately it was a grammar school). Indeed I received a double dose since I also studied Latin: all those hours spent declining nouns and conjugating verbs. “Amo, amas, amat, amamus, amatis, amant.” Thirty years later and I still remember it.

The number of hours spent learning this huge, unwieldy set of rules that prescribe how to construct sentences was huge, but, certainly at that time, it was believed to be a necessary foundation for the study and academic use of my native language.

However, one small but important fact was omitted: English, like virtually every other spoken language, came into being long before these grammatical rules. I had to deduce for myself that grammar is nothing more than a model, a simplification that attempts to describe the structure of language.

In that respect it is analogous to physical models such as Newton’s Laws of Motion. Those famous statements, at one time seen as perfect representations of the natural world, are now understood to simply be approximations. Very good approximations in the realm of everyday speeds, masses and distances, but increasingly inaccurate as you push their limits.

Nobody with more than a basic understanding of physics these days expects the world to conform to the old Newtonian laws, and yet the equally artificial rules of grammar are often seen as sacrosanct, beyond criticism. To casually break one of those rules is to face opprobrium and censure.

Like the mathematical relationships used to approximate and model the physical world, grammar does have important uses: it is not possible to analyze and describe the structure of language–its syntax–without it. But that is all it is: a specialized vocabulary of a particular domain of study. Language exists and will continue to exist (and evolve) without an awareness of a system of rules.

That is why I take issue with those who slavishly follow the rules of grammar, upbraiding those who transgress. Many criticisms of “bad” or “incorrect” grammar are fair. Missing a preposition changes the meaning of a sentence: “She ate her family” tells you something completely different from “She ate with her family”. Other lapses are less detrimental to comprehension: “He eat an apple” versus “He eats an apple” still conveys the message.

Other commonly quoted rules either have many exceptions, or simply do not apply to colloquial English. Some constructions like the infamous “split infinitive” of Star Trek‘s “to boldly go” are in common use: I slipped one into a sentence earlier and I wonder how many readers even noticed it?

My preference is to always try to find a construction that feels natural, that mirrors the way I speak. I’m reminded of a phrase commonly attributed (in various forms though without evidence) to Winston Churchill, “This is the kind of arrant pedantry up with which I will not put.” It’s a great illustration of the contortions necessary to satisfy an arbitrary rule of grammar (don’t end a sentence with a preposition) that has no place in a description of English syntax.

Because that’s the point: grammar rules are descriptive. For all that a complete record of them would (and does) fill many volumes, it remains an indispitable fact that the language was there first and grammar is something invented as a tool to model it. The English language in particular is immensely flexible, liquid in its ability to assume different forms. Its complexity and fluid nature defy complete documentation.



Reading a play in Spanish earlier–a language in which I am far from fluent–it struck me that as I read I don’t think about the English equivalent of the words and sentences. They have meaning without any conscious translation to my native written (and spoken) language.

I’ve found I have a certain facility for learning to read some languages: basically those that share some common roots with English. This includes Latin, Middle English, French, Dutch, German and Spanish. (I’m keen to add Celtic languages such as Welsh and Irish to my list as well, but my available time is finite.) There’s no trick to it: it appears to be a consequence of how my mind works.

My lifelong immersion in English had hidden from me the non-verbal nature of my thoughts, because whenever I come to communicate them I use that language. But, as I have described before, inside my mind there are no words, only impressions: feelings and visions. Meanings in elemental form.

When I read a text the parts I recognize cause these impressions of meaning directly. The parts I don’t recognize I must consciously, mechanically translate to English first. I quickly become familiar with new words, at which point this intermediate English stage no longer happens. At that point I’m reading fairly fluently. I generally find it takes several hours, maybe a few days, to reach that stage.

There’s a curious effect of understanding the meaning directly (i.e. without translating via English): if I express that meaning in English I use my own phrasing and structure. I often find it more difficult to render a literal translation than simply to convey the essence of the original. This is something I have also noticed when I’m reporting information (text or conversation–it doesn’t make a difference) that I encountered in English.

If somebody asks me what another person said to me in conversation I am usually unable to recall their words, instead simply retaining the underlying meaning. The implication is that I do not require English (or any equivalent language) for thought. I describe my thinking as visual but the reality is much richer and involves more senses than just sight: touch, smell, taste, hearing and emotions are also involved. Perhaps it would be more accurate to describe it as sensory rather than visual?

Most of the time it is effortless to translate my thoughts into English, expressing myself through writing or speech. But sometimes I will hit a block: I know what I want to express but can’t immediately find the words to convey that meaning. My choice of words is also very important to me. I like to find le mot juste, the exact word or phrase that matches what I mean. Failing to do so causes much frustration.

Exactly the same mechanism applies to programming. I “see” the structure of the code, how it operates, how the parts interact. The actual computer language used to express this is mostly irrelevant: I’m fluent in several and have no difficulty expressing equivalent meaning in any of them. To me there is no essential difference between writing in English and writing in C++ or Perl. It is only that computer languages operate in a restricted domain, defining sequences of operations. Natural languages are much more extensive in scope.

The common link is that they are all forms of communication, ways to express or record what is inside one’s mind so that it may be transferred to another mind. This also applies to computer languages: sure, programs are converted into machine instructions, but the reason that programming languages exist–as opposed to using those low-level instructions directly–is that they provide a means for programmers to express what is in their minds in a more natural way, one that is closer to the mental model than the binary codes that are the end result.

Light Show

Light Show

I fall into the diamond sparkle of light refracted by my cut-glass tumbler. Shards of shattered rainbows glint and swirl kaleidoscopically as I spin round and round, drawn deeper into this well of visual stimulation. My world is reduced to the immediacy of this pyrotechnic extravaganza: colors explode, burst, shine briefly and fall to earth while still more rise on trails of glitter to take their place. As I am drawn deeper a cloud of iridescent bubbles floats on the air all around; I am lost in a cloud of polychromatic ephemera. The bubbles burst to reveal a swarm of butterflies, nacreous wings flashing in the light. I float among them, borne by their fluttering, reveling in my weightlessness; they carry me up, higher and higher, until they metamorphose and dissipate into a whirlwind of flashing color, fragments resolving into spectra until I am returned to where I started, captivated by the beauty of the prismatic play of light on my glass.

Art Experience

Art Experience

There’s something about a good piece of art that speaks to me. No, it’s more than that, much more: there’s something about it that resonates, a positive feedback loop. My initial visceral gestalt stimulates my mind and senses, filtering them through that raw emotion so that the detail becomes imbued with context.

The tone of the piece in its entirety informs each part, marking and tying it into the coherent assemblage. Reductionism is destined to fall at the first hurdle: just as an individual instrument cannot hope to convey the sense of the entire orchestra, so one component of an artwork can only offer a faint hint of the scope of the whole.

As I ingest the artwork through my senses — seeing, feeling, hearing, tasting — it becomes part of me while simultaneously I see myself reflected in it, part of it. There can be no illusions here, no falsehood. We are both laid bare, opened to deep scrutiny, and the further I stare into the piece the further I see into the very core of my own being.

This is communication of the highest order, as far beyond language and words as the works of Shakespeare are beyond the first scribblings of a child. And yet there is a simplicity to this dialog verging on the magical. A conjunction of my mind and that of the artist, synchronized through the medium of the art.

The experience of art is one that transcends language to wash through the mind, and when it is over the signs of its passage remain. The person who stepped up to the piece is not the same one who later walks away.

Thought Transference

Thought Transference

I’ve written before about being a visual thinker, but a presentation by a colleague of mine this morning at work set me thinking about it again. I won’t go into the detail of his talk which was based on this talk at ACCU 2013. Suffice to say that he has a strong interest in understanding thought processes as they relate to software development in particular.

There are different styles of thinking. Some are visual; some are language-based. Some are grounded in rationality, others take soaring flights of fancy. A person who thinks in a particular way will find it difficult if not impossible to imagine how somebody with a different cognitive style thinks: I am completely unable to imagine thinking in words.

Most people appear to use a combination of verbal and non-verbal thinking. Purely non-verbal thinkers are a minority, although they are reportedly more common among the autistic population. Indeed one of the best known autistic women, Temple Grandin, is a purely visual thinker. She wrote this informative article about her own experience.

When considering cognitive styles I encounter this conundrum: how can a purely verbal thinker imagine concepts that transcend language? How can they hold something in their mind that they lack the vocabulary to describe? I do not know the answer to this.

Language is a tool for communication. It allows something to be transferred from one mind to another, but the process is imperfect, incomplete. It’s like emailing a photo of a scene, reducing all the sensory impressions and feelings to a collection of colored pixels. So much context is lost.

How can I describe a walk through woodland? I could take a picture, freeze one instant. I could describe the feel of the ground underfoot; the earthy, damp smell; the sound of the wind through the leaves overhead overlaid with the songs of birds and the zip and hum of insects; the play of the dappled sunlight through the canopy onto the undergrowth. I can see that walk in my mind even though I never experienced it directly in that exact form. But can I conjure those same thoughts in your mind?

Much of what I hear or read has an emotional impact that derives from my own experiences, my own personal set of likes and dislikes, my own moral sense. Shared culture means that there will be overlap between people depending on how much they have in common. But ultimately what I experience inside my own head is unique to me.

Which leads me, in a round-about way, to another attempt to use words to build a mental model in your mind of what it means to “think in pictures”. Even that phrase is misleading: perhaps I ought to call it non-verbal thinking. Because what I “see” is not just like an array of photographs.

Consider a simple mechanism like a door hinge. Suddenly in my mind I am holding a 3″ steel hinge. I am feeling the weight of the cold, hard metal; I am opening and closing the two halves, feeling the friction in the joint; I am seeing how the screw holes — three per side — are arranged and their edges countersunk. I am fitting such a hinge to a door, seeing the process of first removing the screws, removing the old hinge, aligning the new one in its place and then driving the screws in to hold it there.

Visual thinking, for me, is also spatial and temporal. I see objects in relation to each other, and I see how the objects and their relationships change over time. The models in my mind are not static but dynamic. Communicating them to others is difficult: it requires the use of visual metaphors and analogies to “real” examples. My visual representations must be translated into language. When speaking or writing that is typically English, when programming it would usually be C++. The process is the same, and over the years has become largely intuitive.