One, two, three ... Every human culture can count to three. Ancient Chinese artists knew that if you subdivide larger groups into smaller groups, the viewer's eye/mind system can take it in. Put three geese here, four nearby. The viewer will see three-geese plus slightly more than three: 3 + (3+1) = 7!
(Don't tell me that few ancient Chinese took algebra. Stone age children learned that much algebra just by being what they were and looking around. It's easily established that simple calculations go faster than counting. Besides, counting can miscount past three. Simple calculations are much faster and infinitely more reliable. We all do it. Even the kid who's failing algebra does so only after making a few simple calculations that assure him that he's safe to do so, and perhaps better off.)
The first group of geese in our Chinese scroll forms a triangle no matter how the geese are clumped. Ditto for the second — only there the polygon has four angles and four edges. See the two together: it's another triangle.
A painting with seven geese can't be taken in; a painting where the seven form a triangle of two unequal components can be "seen" by a six year old. Artists don't waste their time with eight. It's too much for the human mind. Maybe God could see it, but few of our representations are ever really for Him.
(We're told of Eight Wonders of the Ancient World. There's an eight. So what am I talking about? Seven is the minimum number greater than the two threes we can see. Eight is the minimum number greater than that. Therefore, each of our primitive minds automatically understands:
It's like raining for forty days and forty nights. Or fasting for forty days and nights in the desert (or wandering or praying). What the "forty" means is a number greater than you can reliably count, calculate, or comprehend. The Romans said a thousand. We say millions. It means the same thing. It's nothing but a linguistic form of inflation.
The experienced backgammon player doesn't have to count his moves. Odd numbers go to an opposite color point; even rolls stay on the same color. Six is a single quadrant away: just go to the same relative point. The novice chess player soon sees how the queen's power radiates over the board; only the experienced chess player sees how the knight can feint here, only to suddenly appear there! The chess player is still sub-dividing his world into manipulable components.
Now: is reality divided into triangles and lines and the little inversible-Ls of the chess knight? Are numbers "true"? Is there an identity between how we see and how the "laws" of the universe actually function?
If not, might there at least be an analogy?
It's my position that we can't know altogether what's true, but that we can know some part of how we perceive and understand things. Monists are forever seeing things as One. In general, the religious impulse is an urge to unify. It's certainly my urge (or has been most of my life) (though not always).
But unification can only go so far. If you make everything the same, then you can't tell anything apart. Information, as Gregory Bateson teaches, is perceived difference. No difference would be the thermodynamic heat death cosmologists have been frightening us with. Total entropy. The end of information.
Monists, like the Scholastic Realists are annoyed by people who see more difference than they, the Monists, like.
One / Many. Continuity / Discontinuity.
St. Thomas Aquinas saw "chair"; Abelard saw this chair and that chair. Aquinas saw "Man"; Abelard saw Alison, Geoffrey, and John. But Aquinas too could address John when he walked into the room. Bertrand Russell was one modern nominalist who annoyed twentieth-century monists by emphasizing the discontinuities of the universe.
Drivers-training vehicles have dual controls. On the road, the public is in the least danger with vehicles having only one set of controls: no matter how drunk a particular driver may be. Even democracies assign one captain to a ship. Even democracies have one executive "at the top." Even modern democracies don't want two hundred and fifty million people all talking at once. You couldn't hear anybody.
I don't doubt that in the earliest days of religion (My piece on Magic argues that) man first invented only one god. But then he found more than one trouble. And the one god wasn't always handling it satisfactorily. By one or two thousand BC who could blame the Jews for wanting a Chief Executive? By then even the Chinese had come to believe that all their gods were merely facets of Hung Ti. From a polynomial standpoint, the Trinity of Christianity is a clear evolutionary advance over the simple monotheism of Judaism. Humans will never be emotionally or perceptually ready for George Gamow's 1, 2, 3 ... Infinity. Lao Tsu was as wise as Homo sapiens can (as a group) get when he wrote:
From Two comes Three
From Three comes Many.
I've long taken the title of Gamow's book to be a translation of those words from the Tao Te Ching. Though, for myself, I Christianize them: From Unity comes Duality. From Duality comes Trinity. From Trinity comes Infinity.
But let's leave unity, trinity, and infinity alone for a moment. I mount this piece today because I suddenly saw a pedagogically graphic use for duality, one custom suited to my current teaching here.
I've been writing here and there about definition, about ambiguity. Take justice for instance. On the one hand, everyone has his own imagined ideal of justice: your enemies under thumb-screws; your daughter married to Prince Somebody; Palestine back in the hands of ... name your group. On the other hand, we have OJ Not Guilty; the mass murder legally insane and not responsible, therefore Not Guilty of Counts 1, 2, & 3; your wife with a new boyfriend so now she gets the Rolls, the Steinway, both the house in Vermont and the condo in Monaco, plus $8,000 per month ...
What you want versus what you get
Your ideal versus what the judge says.
To chart an ambiguity I TRY an example I've already discussed elsewhere:
The physicists' specialized meaning versus The Common Language
force through distance versus Monday to Friday
... what you do in the garden, what you do in your work shop, what you're doing when you push against a wall (even if the wall doesn't move) ...Duality: Binary Semantics
Good & Evil. God & Satan. Life & Death. Flesh & Stone. Ideal & Real. Isn't every culture filled with dualities?
One, two, three ... Every human culture can count to three. Ancient Chinese artists knew that if you subdivide larger groups into smaller groups, the viewer's eye/mind system can take it in. Put three geese here, four nearby. The viewer will see three-geese plus slightly more than three: 3 + (3+1) = 7!
(Don't tell me that few ancient Chinese took algebra. Stone age children learned that much algebra just by being what they were and looking around. It's easily established that simple calculations go faster than counting. Besides, counting can miscount past three. Simple calculations are much faster and infinitely more reliable. We all do it. Even the kid who's failing algebra does so only after making a few simple calculations that assure him that he's safe to do so, and perhaps better off.)
The first group of geese in our Chinese scroll forms a triangle no matter how the geese are clumped. Ditto for the second — only there the polygon has four angles and four edges. See the two together: it's another triangle.
A painting with seven geese can't be taken in; a painting where the seven form a triangle of two unequal components can be "seen" by a six year old. Artists don't waste their time with eight. It's too much for the human mind. Maybe God could see it, but few of our representations are ever really for Him.
(We're told of Eight Wonders of the Ancient World. There's an eight. So what am I talking about? Seven is the minimum number greater than the two threes we can see. Eight is the minimum number greater than that. Therefore, each of our primitive minds automatically understands:
It's like raining for forty days and forty nights. Or fasting for forty days and nights in the desert (or wandering or praying). What the "forty" means is a number greater than you can reliably count, calculate, or comprehend. The Romans said a thousand. We say millions. It means the same thing. It's nothing but a linguistic form of inflation.
The experienced backgammon player doesn't have to count his moves. Odd numbers go to an opposite color point; even rolls stay on the same color. Six is a single quadrant away: just go to the same relative point. The novice chess player soon sees how the queen's power radiates over the board; only the experienced chess player sees how the knight can feint here, only to suddenly appear there! The chess player is still sub-dividing his world into manipulable components.
Now: is reality divided into triangles and lines and the little inverted-Ls of the chess knight? Are numbers "true"? Is there an identity between how we see and how the "laws" of the universe actually function?
If not, might there at least be an analogy?
It's my position that we can't know altogether what's true, but that we can know some part of how we perceive and understand things. Monists are forever seeing things as One. In general, the religious impulse is an urge to unify. It's certainly my urge (or has been most of my life) (though not always).
But unification can only go so far. If you make everything the same, then you can't tell anything apart. Information, as Gregory Bateson teaches, is perceived difference. No difference would be the thermodynamic heat death cosmologists have been frightening us with. Total entropy. The end of information.
Monists, like the Scholastic Realists are annoyed by people who see more difference than they, the Monists, like.
One / Many. Continuity / Discontinuity.
St. Thomas Aquinas saw "chair"; Abelard saw this chair and that chair. Aquinas saw "Man"; Abelard saw Alison, Geoffrey, and John. But Aquinas too could address John when he walked into the room. Bertrand Russell was one modern nominalist who annoyed twentieth-century monists by emphasizing the discontinuities of the universe.
Drivers-training vehicles have dual controls. On the road, the public is in the least danger with vehicles having only one set of controls: no matter how drunk a particular driver may be. Even democracies assign one captain to a ship. Even democracies have one executive "at the top." Even modern democracies don't want two hundred and fifty million people all talking at once. You couldn't hear anybody.
I don't doubt that in the earliest days of religion (My piece on Magic argues that) man first invented only one god. But then he found more than one trouble. And the one god wasn't always handling it satisfactorily. By one or two thousand BC who could blame the Jews for wanting a Chief Executive? By then even the Chinese had come to believe that all their gods were merely facets of Hung Ti. From a polynomial standpoint, the Trinity of Christianity is a clear evolutionary advance over the simple monotheism of Judaism. Humans will never be emotionally or perceptually ready for George Gamow's 1, 2, 3 ... Infinity. Lao Tsu was as wise as Homo sapiens can (as a group) get when he wrote:
From Two comes Three
From Three comes Many.
I've long taken the title of Gamow's book to be a translation of those words from the Tao Te Ching. Though, for myself, I Christianize them: From Unity comes Duality. From Duality comes Trinity. From Trinity comes Infinity.
But let's leave unity, trinity, and infinity alone for a moment. I mount this piece today because I suddenly saw a pedagogically graphic use for duality, one custom suited to my current teaching here.
I've been writing here and there about definition, about ambiguity. Take justice for instance. On the one hand, everyone has his own imagined ideal of justice: your enemies under thumb-screws; your daughter married to Prince Somebody; Palestine back in the hands of ... name your group. On the other hand, we have OJ Not Guilty; the mass murder legally insane and not responsible, therefore Not Guilty of Counts 1, 2, & 3; your wife with a new boyfriend so now she gets the Rolls, the Steinway, both the house in Vermont and the condo in Monaco, plus $8,000 per month ...
What you want versus what you get
Your ideal versus what the judge says.
I've already discussed the example of Work elsewhere:
The physicists' specialized meaning of force through distance versus the common language's multiple meanings of Monday to Friday ... what you do in the garden ... what you do in your work shop ... what you're doing when you push against a wall (even if the wall doesn't move) ... Do you think you can use that tool inside your own head when you notice an ambiguity? So what? you may be thinking. Take it a step further. Use it to distinguish useful ambiguities from deleterious ones.
In his Point Counterpoint, Aldous Huxley offers a masterful contrast between what the politician is thinking when he offers speeches about liberty and what the audience hears as meaning. Talking about Liberty the politician is thinking
Limit liability for medical malpractice.
For God's sake, don't elect Labour.
while the audience is thinking
Stand around, in your undershirt.
Feel up who you want on the street.
If I remember correctly, Huxley made his reader aware that the politician knew exactly how his audience was misunderstanding his words. Whether or not that's so I hope you see it.
from 1998 10 27

