Pi in the Sky

Differential calculus seems pretty damned-intuitive to me. Where shall we draw the line?

Reminds me of the old joke:

You’re a Mathematician? Wow! Math was always my worst subject. I mean I did OK in grade school and high school and in college. But K-theory completely blew me away!

Belle, keeping track of children doesn’t require counting. Counting is used to map objects to numbers. So, if I have to keep track of four children, and I look around and child C is missing, it need not concern me that there are now three children.

Actually, this whole thing makes me think of the rabbits in watership down, who could count no higher than four. Five and above were “hrair”, which loosely translates to “one thousand.” To mix up literature a bit, Nyarlathotep would be “The Black Goat With Hrair Young.” Anyway, this got me to wondering: let’s say a rabbit goes into a room with five objects. She sees that there are hrair, but wants to know more precisely. So, she moves one out of the room. When she comes back, there are four. So, she knows it’s “four plus one” objects. So now she’s got a procedure for counting to five (which she has to call “four plus one”). Can we build a whole counting system out of this? Yes. The “plus one” thing turns out to be just the successor operator, about which I am too lazy to say more than that it’s sufficient.

And a big “welcome back” to the Sapir- Whorf hypothesis!

That’s:

http://www.theglobeandmail.com/
servlet/story/RTGAM.20040820.
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specialScienceandHealth/

Shub Niggurath is the one with Hrair Young. Nyarlathotep is the one with hrair forms. Wait, is today out-the-geek day?

Anyway, the Globe and Mail story is mindboggling. What an alien people.

in addition to the Watership Down ‘hrair’, there are also the funny aliens in Greg Bear’s Anvil of Stars. they are creatures composed of a number of smaller, simpler, snake-like creatures which, when alone are not fully intelligent (but are very good at chess). when they get together a higher-order consciosness emerges. these aliens are supposed to do math in which it’s all about irrational numbers, in some sense (like, those are the basic units.) yes, this is the geekiest thread ever. all praise el-ahrairah.

There’s more on the Piraha at http://lings.ln.man.ac.uk/Info/staff/DE/cultgram.pdf . The author doesn’t argue for the Sapir-Whorf hypothesis, but rather takes issue with Chomsky etc., from the other side – it’s not that language determines culture, more like culture determines language.

I had the good fortune to speak with a western-educated member of the Amazonian Uru-e-wau-wau tribe at a weekend workshop where other indigenous people from South American were present. He said of the Shuar tribe representative that his people only have four number concepts: one, two, three, and “many.” He said they also essentially only possess three “times” of day: dawn, midday, and sunset. One of his points: these people rarely get psychosomatic illnesses such as cancer, and maybe people in the West (and North) need to reflect on that.

To approach this differently one might ask, “What is it about their existence that makes counting unnecessary?” Maybe precision in counting is a result of economic transactions of a certain type. For survival precise counting isn’t so necessary. I thin about building a fire here. I never count the pieces of wood. I just know when its about enough to produce hot coals for cooking.

I’ve always had a hunch that mathematical thinking was subtended by two very different processes.

One process involved basic visualization of representations of small discreet quantities. This is how we handle manipulations of small sums.

The other process is a complex of concepts, heuristics and literary representations, combined with rote memory. This is how we handle all other mathematical thought, from the multiplication tables we commit to memory to the geometric deductions we learn to do in our heads in the 7th grade, to proofs.

The first is absolutely necessary, it seems to me, for human social life. But it doesn’t require much in the way of precise linguistic representation, since the judgements are usually so easy and obvious, and so universally shared by all participants, that context will almost always painlessly disambiguate.

If I have four guests in the next room, and I send my child to fetch water, but he brings only three glasses, I don’t need a word for ‘four’, ‘three’ or indeed any number to comminicate the problem to him. Nor does he need linguistic concepts like those embodied in the integers to see his mistake and how to remedy it.

The more abstract process(es) have their uses, god knows, but they aren’t necessary for societal existence. At least so it seems to me. . .

“In New Zealand aborigines language you say three on the other foot for 18 ”

In the Maori language 18 is tekau ma waru
ten and eight

Mathematical Cognition, Vol. 1

The Handbook of Mathematical Cognition

A couple good resources. Things are a bit more complex than they appear.

I recommend taking a look at the original research article that tim f. mentions, which, as always, make more interesting reading than the journalists’ versions. Everett doesn’t simply _claim_ that the Pirahã can’t count: he argues with ethnographic evidence. As to “how anyone could get through life” — well, that’s kind of tough anywhere, and it sounds like the Pirahã haven’t been spectacular successes by most measures, but Everett indicates that they deal with their cognitive blindspots the same way we deal with our own: culturally. With communal (and transient) ownership, valorization of hardships, and a very simple kin system, they can scrape by amongst themselves.

I agree that perhaps the original research has been misrepresented. I shall look into it as soon as I have some time.

But even in the way you try to explain what is going on, the idea that a certain group of humans can measure magnitude and not enumerate put in rough terms is an ethnocentric absurdity.

I give you one example of how the things could be conversed: Eskimo has many words for snow. This does not and cannot mean that English Canadians cannot grasp or distinguish the several kinds of snow accordingly to Eskimo classification. But, of course, an ethnocentric Eskimo could argue that if Eskimos decided to make anthropological studies of us Whites.

And mind you that in the example above we think of taxonomy, something that really depends on conventions. The capacity of enumerate is too radical of human cognition to be impaired in the manner described. It does not depend on language or culture. To think the contrary would be the same as to claim that another People could not memorise things because their language and their cultural conventions are different from others. That is not a possibility. Not at all.

There are precedents. Once years ago someone just stated that a certain language lacked temporal expressions, an error which later was fully exposed. But ok, I shall look into the research before expressing any final verdict. Perhaps I change my point of view.

The capacity of enumerate is too radical of human cognition to be impaired in the manner described.

Whoever said that the Piraha have no capacity to enumerate? Did the human race suddenly develop the cognitive capacity to browse the Web in the mid-90’s (to use my fanciful example from above)? Were North Americans incapable of conceiving of a single lineup at the bank until some time in the seventies? And are they somehow incapable of conceiving of a single lineup at the supermarket today?

Incredibly simple, retrospectively obvious, magnificently life-improving conceptual innovations are continually being introduced into all societies, though clearly at different rates. Sometimes, looking back on them, one wonders what on earth was wrong with the people who hadn’t thought of them yet. One is even tempted, for an instant, to imagine some kind of cognitive impairment that caused them not to see the foolishness of the old way of doing things–but of course there is no need for such a ludicrous explanation, when “they simply hadn’t thought of the alternative” suffices.

I find it perfectly plausible that just as nobody had thought of “zero” anywhere before about 1500 years ago, nobody among the Piraha has thought of “one” yet. (Or perhaps several have, but have been unable to persuade the others of its utility.) Many things that we think are obvious are actually wonderfully subtle ideas that familiarity has prevented us from appreciating. I see it as no criticism of the Piraha that they have given me the opportunity to appreciate another one.

Do people *still* believe that Eskimos have many words for snow?

Do people still use insult-words like “Eskimo” to describe circumpolar people?
A diverse group of folk who, in a neat way, circle the earth.
There may not have been 300 different words for types of snow, but the ability to describe exactly the conditions of an unforgiving environment were crucial to them, and the means of that expression were there, obviously, regardless of the linguists documentation.
As Ray Davis points out, it’s contact and taking the “test” that are problematic for the Pirahã. These natives will make soft meat for the merchants they encounter but it needs emphasizing, they were living successfully, without numbers, for countless thousands of years.
The assumption is that “our” way of living is more adaptable – even through incipient, and catastrophic, ecological disruption – and likely to be still more successful in future.
That assumption is baseless and void of proof. A teenager driving his father’s car at 100mph knows he’s immortal just as certainly, and just as validly.
–
As far as people spitting up over someone else describing cancer as “psychosomatic”…
That the psyche impinges on somatic function is beyond debate, the question being only degree and mechanism.
Less than two hundred years ago the majority view of medical professionals as to the cause of disease was “miasma”.
In a more local context having seriously propounded that absurdity would be enough to delegitimize any further expression.

The use of machines, such as computers or airplanes, or of simple implements, such as pencils or knives, involve different kinds of knowledge, which are acquired through human experience. And of course, such examples, as the examples given by Dan Simon, depend on technological development and social-economic organisation.

But some cognitive capacities are radical or fundamental and have not been invented. There is no term of comparison between the capacity to enumerate and that to browse the internet. If there is no internet in the society where one lives, he or she will probably not know how to browse the internet. And, this is the same for other non-elementary instances of mathematical knowledge, such as the use of zero and negative and complex numbers, or the method of solving second degree equations.

But to count, to enumerate, or to use natural numbers is a natural capacity, something that does not depend on the cultural environment. The same applies to the cognitive capacities of memorising things or of commanding your legs to walk.

To belief the contrary is the same as to claim that culture or language can have a more devastating impact on the lives of people than a stroke or a any form of cerebral damage.

to count, to enumerate, or to use natural numbers is a natural capacity, something that does not depend on the cultural environment.

Really? What about counting by twos? By fives? By seventeens? Addition? Multiplication?

My understanding is that even in our mathematically advanced society, dealing with numbers beyond five or six is a completely mechanical, non-intuitive process. For the untutored Piraha, the threshold seems to be three. That doesn’t sound to me like the kind of fundamental difference in cognitive capacities that couldn’t be put down to lack of familiarity and practice.

they were living successfully, without numbers, for countless thousands of years.
The assumption is that “our” way of living is more adaptable – even through incipient, and catastrophic, ecological disruption – and likely to be still more successful in future.
That assumption is baseless and void of proof.

Well, given that “our” way has resulted in billions of healthy, long-lived people dominating the habitable parts of the planet, while their way has resulted in a tiny tribe of low-lifespan, resource-poor tribespeople in a remote patch of South America, I’m inclined to place my bets with “our” way and take my chances.

In fact, I’d go further and surmise that the invention of the natural numbers would find some application–indeed, likely a whole bunch of applications–in the world of the Piraha, even in their state. Arithmetic is, after all, an extraordinarily versatile conceptual tool.

Psychological stress exacerbates cancers. It also exacerbates heart disease, diabetes, MS, and every other ailment known to man. This does not mean that all diseases are “psychosomatic.” (I could go into detail about the fact that ‘cancer’ is a name for a class comprising hundreds of different diseases with different causes and symptoms, and that it seems to always have a genetic aspect *and* a triggering aspect, as well as a basic explanation of why emotional strain has objective physical consequences, but I’ll just join Doctor Memory in finishing wiping off my screen here…)

dan simon-
“given that “our” way has resulted in billions of healthy, long-lived people dominating the habitable parts of the planet, while their way has resulted in a tiny tribe of low-lifespan, resource-poor tribespeople in a remote patch of South America, I’m inclined to place my bets with “our” way and take my chances.”

Since the teenager in his dad’s car didn’t seem to have much effect, how about this:
Crack cocaine is probably about as good as pharmacological euphoria’s going to get. It’s pretty seductive, I’ve been near it, but never on it. I’ve seen it “work”, as a drug and as a lifestyle. The parallel with “our” way, as opposed to the primitive way of the indigenes in question, is pretty symmetrical.
The “way” you defend has been around for less than a century, or if you want to stretch it, less than two centuries. In that time it’s brought its own adherents to the brink of extinction, though most of them haven’t confronted that yet. It has spent the capital of its, and the world’s, natural resources beyond replenishing in “our” lifetimes. Children are kept indoors now, moving from one controlled environment to another; the fields that surround “our” cities are hostile landscapes of poison and industrial geometries, birdless monoculture deserts. The US alone burns 380 million gallons of gas every day.
The automobile is the single largest killer of people under 30, and yet it’s the first technology most of the chauvinists of “our” way would point to as indispensable.
Billions of healthy people for how much longer. Ten years? A century?
And how much of a factor is happiness in your definition of healthy? With enough medical technology a brain-dead body in a coma can be kept “healthy” for years. Happiness doesn’t mean anything in that context though, does it?
My main point is the timeline. The aborigines – thousands of years, tens of thousands. You, “us”, this – a few decades.
Decades in which the conditions have been set for massive climate and ecological disruption to be made ineluctable, and inadvertently, blindly.
But like the crack-head’s minutes of rushing bliss those few years have been so exhilarating, and so saturated in power, there’s no way you’ll go back. No way.

Well, Serial Catowner, I was talking about what is natural for humans. I thought this was understood.

But, ok, if you read that Ducks can count up to three, so it means that even Ducks can count. One extra example of how absurd is to assume that the Pirahãs would lack a capability that, according to your post, even some birds possess.

They haven’t been living successfully, without numbers, for countless thousands of years, they have been living successfully, without numbers, for countless manies of years.

Similarly, we have many years of oil left, etc.