Posts tagged with binary

Transgressing boundaries, smashing binaries, and queering categories are important goals within certain schools of thought.

Reading such stuff the other week-end I noticed (a) a heap of geometrical metaphors and (b) limited geometrical vocabulary.

In my opinion functional analysis (as in, precision about mathematical functions—not practical deconstruction) points toward more appropriate geometries than just the `[0,1]` of fuzzy logic. If your goal is to escape “either/or” then I don’t think you’ve escaped very much if you just make room for an “in between”.

By contrast `ℝ→ℝ` functions (even continuous ones; even smooth ones!) can wiggle out of definitions you might naïvely try to impose on them. The space of functions naturally lends itself to different metrics that are appropriate for different purposes, rather than “one right answer”. And even trying to define a rational means of categorising things requires a lot—like, Terence Tao level—of hard thinking.

I’ll illustrate my point with the arbitrary function ƒ pictured at the top of this post. Suppose that ƒ∈𝒞². So it does make sense to talk about whether ƒ′′≷0.

But in the case I drew above, ƒ′′≹0. In fact “most” 𝒞² functions on that same interval wouldn’t fully fit into either “concave" or "convex”.

So “fits the binary” is rarer than “doesn’t fit the binary”. The “borderlands” are bigger than the staked-out lands. And it would be very strange to even think about trying to shoehorn generic 𝒞² functions into

• one type,
• the other,
• or “something in between”.

Beyond “false dichotomy”, ≶ in this space doesn’t even pass the scoff test. I wouldn’t want to call the ƒ I drew a “queer function”, but I wonder if a geometry like this isn’t more what queer theorists want than something as evanescent as “liminal”, something as thin as "boundary".

hi-res

Railing against “grey areas” has become a favourite rant topic. People think that they’ve covered their bases and are being really open-minded when they switch from {0,1} to [0,1]—but no false dichotomies are avoided in this transition from discrete to continuous.

Let’s take the example of sex & gender. Most of the tick-boxes and bathrooms we face in life are labelled “M” or “F”, which covers most of us but not all.

(And I want to apply a kernel weighted to extra-count the forgotten individuals, since as minorities they’re more vulnerable. This can be seen in data such as e.g. higher suicide rates and higher murder rates.) The University of Hawai’i’s guidelines for dealing with individuals possessing ambiguous genitalia (Archives of Pediatrics and Adolescent Medicine) use words like

• chromosomes—XX, XY, or other
• micropenis, labia-scrotum fusion, gonadal dysgenesis
• androgen insensitivity syndrome, hypospadias, kiinefelter syndrome, congenital adrenal hyperplasia, Turner’s syndrome
• true hermaphroditia

which raises the question of where the “grey area” between [M,F] ~= [0,1] could come from. Chromosomes either come in whole units — for example people with Klinefelter’s syndrome have 47 chromosomes “XXY” — or have a much more complicated structure if you want to dig into the DNA string. Other aneuploidies include XYY, monosomy or partial monosomy, trisomy 21 (which I don’t think affects genitals or sex assignment), distal 18q−, mosaicism, the list goes on. How are we going to assign a total order there in order to define a continuous variable? I don’t see any way to—just more possibilities to add to the domain of a categorical variable (and making it much more confusing than the usual gender dummy!).

The paper above, to give another example of non-orderability, notes that various chemicals usually squirt at you in fœtal development but they vary in their squirtular timing. So androgen, progesterone, and so on aren’t mutually fungible (as the different “coloured edges” in Ramsey theory), and además we’re dealing with time series like Ed Küpfer’s pictures of sports scores:

Those kinds of pitures, but with different coloured spiketrains representing the incommensurability of androgen vs testosterone and so on.

So how do you get total orderability (necessary for a “grey area”) from a time series of incommensurable chemtrains? I don’t see it. The geometry is more interesting than just a line segment.

Further reading: transgender mathematician (Leigh Noble), transgender computer programmer (Tim Chevalier @eassumption), transgender economist (Deirdre McCloskey @deirdremcclosk), transgender electrical engineer (Lynn Conway). Jeff Eugenides’ Middlesex.

## 2¹⁰⁰

Readers of isomorphismes, you might enjoy powers of two tumblr.

2100 = 1,267,650,600,228,229,401,496,703,205,376 — one nonillion, two hundred sixty-seven octillion, six hundred fifty septillion, six hundred sextillion, two hundred twenty-eight quintillion, two hundred twenty-nine quadrillion, four hundred one trillion, four hundred ninety-six billion, seven hundred three million, two hundred five thousand, three hundred seventy-six (31 digits, 320 characters)

I think I’ve been subscribed since the 30’s. Never a letdown. And of course it’s only going to get more exciting.

the Good People and the misguided

HT @jaredwoodard (supervenes)

hi-res

In the beginning, at the birth of computing, there were no programming languages. Programs looked something like this:
```00110001 00000000 00000000
00110001 00000001 00000001
00110011 00000001 00000010
01010001 00001011 00000010
00100010 00000010 00001000
01000011 00000001 00000000
01000001 00000001 00000001
00010000 00000010 00000000
01100010 00000000 00000000```

That is a program to add the numbers from one to ten together, and print out the result `(1 + 2 + ... + 10 = 55)`. It could run on a very simple kind of computer.
Marijn Haverbeke

(Source: eloquentjavascript.net)

Early in his academic career, [Paul] Schervish was a committed Democratic Socialist. But around 1990, he began interviewing wealthy people and decided that his Marxist instinct to criticize the rich was misguided.

“I realized good and evil are equally distributed across the economic spectrum and not particular to the wealthy or the poor,” he says. “A lot of wealth holders were very sincerely concerned about others and were doing something about it.”

Graeme Wood

(Source: The Atlantic)

## Beyond Between Good and Evil

• "Adults have to deal with moral grey areas"
• "I’m not liberal or conservative, I guess I’m somewhere in the middle"
• "It may be helpful to think of data science and business intelligence as being on two ends of the same spectrum” (source)
• "On a sliding scale from 1 to 10, how happy are you with life?"
• "[S]cientific bias…is a model for separating plausible hypotheses from their opposite.” (source)
• Please rate your attitude toward the following statements from “strongly agree” to “strongly disagree”.
• How did you like that book, movie, play, album? Please answer anywhere between ★ and ★★★★★.
• "The truth lies somewhere in between"

People talk about “grey areas” as if [0,1] is so much more sophisticated than {0,1}. I find such rhetoric limiting. After all, the convex combinations of black and white are totally ordered, completely linear, and only one-dimensional! A painting in B&W couldn’t display much variation. (Not that it couldn’t be interesting.) We deal everyday with things more complicated than “a grey area” because the world is 3-D and colour is Lab (3-D nonlinear). Add in texture and smell and you’ve increased the psychological dimensionality manyfold.

The metaphor is insufficiently rich. Adult situations don’t fall on a straight line. Political viewpoints don’t sit neatly next to each other in 1-D. Moral ambiguity is certainly more colourful and convoluted than the path from `#000000` to `#FFFFFF`.

Me, I’m more interested in 2.7-dimensional hornspheres, quartz crystal spires, hot-air balloons with a row of golden rings piercing the spine, and quasi-polar negatively bent inside-out torii-cum-logcabins. Or even just something as “pedestrian” as a mountaintop pine forest, which is already much more intricate than, cough cough, the unit interval [0,1].

So—back to my original point—I think moral ambiguity resembles a cell complex more than a line segment. Real situations—the layered tragedies, ironies, comedies, and lengthy mediocrities that desirous, egocentric humans instinctively generate—have a much more interesting shape than “the span between 0 and 1.”

I guess I shouldn’t be so critical. The people using the grey-area metaphor probably don’t avail themselves of the whimsical thought-gardens in which more exciting shapes live. Sorry there, I was just feeling constricted.

I hope you’ve enjoyed these drawings by Robert Ghrist from his (free) notes on homotopy.

## Gender Wars

"There is more difference within the sexes than between them."
‒Ivy Compton-Bennett, Mother and Son

"In all of human biology, there is no greater difference than of that between men and women."
—Some biology notes I found online

These two statements sound like rhetorical opposites, but in fact both are true.

(Says me. I can’t prove this, but I bet that taking everything into consideration, divisions between men & women are greater than those between liberals & conservatives, blacks & non-blacks, tall & short, sick & well, D&D players and people who get laid, etc.)

Let me show how both statements can logically live together harmoniously.

Just like how most men are slower than female Olympians, but at the same time the average man is faster than the average woman.

NB: Not real data.

Thanks to Stats in the Wild, here’s some real data on the ages of Olympians that makes the point:

### Measurement

Even when differences are statistically significant enough to draw conclusions (such as: “boys sprint faster than girls”), the magnitude may be really small so that the difference, while indisputable, is also unimportant. (“Statistical significance” is a confusing term in this respect.)

Consider that there are many ways you could measure differences among people. Here are some that come up frequently in the gender wars:

• height, weight, curvature, angle of femur
• IQ, SAT scores, reading tests
• speed, throwing distance, fine motor skills
• communication skills, emotional intelligence
• went to college, profession is engineer
• finding things in the refrigerator, ability to focus, ability to multitask

There are many ways to measure each of these. For example, does "speed" mean in the 100m dash, 200m dash, marathon, trail running, bike race, or triathlon? While the answers wouldn’t be independent, they wouldn’t be one-to-one either.

### A billion points in a million-dimensional space

Now you are faced with 6.7 billion points in an N-dimensional space, where N is the number of things you could measure. Let’s say like a billion points in a million-dimensional space. (Some dimensions may be collinear.)

On the one hand, there are always lots of pink and blue dots mixing in with each other (e.g. men who sew better than most women)‒and directly from Ivy’s point, the distance among pinks (variation among men) is greater than the distance from the pink centroid to the blue centroid (variation between men and women).

At the same time, though, if you had to choose just one factor by which to color these dots and get maximal classification power, it would have to be gender.

In other words, gender differences may generate a maximally separating hyperplane, but Euclidean distances between differently-gendered points are often small, and Euclidean distances between same-gendered points are often large.

## sqrt10

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