Robert Sawyer - Calculating God

I thought about this. “Is it something like those word puzzles? You know, the ones in which we write ‘he himself,’ but decode it as the word ‘he’ is adjacent to the word ‘himself,’ and read that as ‘he is beside himself,’ and then take that metaphorically to mean ‘he was in a state of extreme excitement or agitation.”’

“I have not encountered such puzzles, but, yes, I suppose they are vaguely similar,” said Hollus, “but with much more complex thoughts, and much more intricate relationships between the words. Context sensitivity is extremely important to the Wreeds; words mean entirely different things depending on where they are positioned. They also have a language full of synonyms that seem to mean exactly the same things, but only one of which is appropriate at any given time. It took us years to learn to communicate verbally with Wreeds; only a few of my people — and I am not one of them — can do it without a computer’s aid. But even beyond the mere syntactic structures, Wreeds are different from humans and Forhilnors. They fundamentally do not think the same way we do.”

“What’s different about it?” I asked.

“Did you notice their digits?” asked Hollus.

“You mean their fingers? Yes. I counted twenty-three.”

“You counted them, yes,” said the Forhilnor. “That is what I had to do the first time I met a Wreed, too. But a Wreed would not have had to count. It would have simply known there were twenty-three.”

“Well, they are its fingers . . . ,” I said.

“No, no, no. It would not have had to count because it can perceive that level of cardinality at a glance.” He bounced his torso. “It is amusing,” he said, “but I have perhaps studied more human psychology than you have — not that it is my field, but . . .” He paused again. “That is another non-Wreed concept: the idea of having a specialized field of endeavor.”

“You’re making about as much sense as T’kna did,” I said, shaking my head.

“You are correct; sorry. Let me attempt this passage again. I have studied human psychology — as much as one can from monitoring your radio and TV broadcasts. You said you counted twenty-three fingers on T’kna, and doubtless you did. You mentally said to yourself, one, two, three, et cetera, et cetera, all the way up to twenty-three. And, if you are like me, you probably had to redo the counting, just to be sure you had not got it wrong the first time.”

I nodded; I had indeed done that.

“Well, if I showed you one object — one rock, say — you would not have to count it. You would just perceive its cardinality: you would know there was one object. The same thing happens with two objects. You just look at the pair of rocks and in a single glance, without any processing, you perceive that there are two of them present. You can do the same with three, four, or five items, if you are an average human. It is only when confronted with six or more items that you actually start counting them.”

“How do you know this?”

“I watched a program about it on the Discovery Channel.”

“All right. But how was this originally determined?”

“With tests to see how fast humans could count objects. If you are shown one, two, three, four, or five objects, you can answer the question about how many objects are present in roughly the same amount of time. Only for six or more objects does it take more time, and the amount of time it takes to report the tally goes up by an equal increment for every additional item present.”

“I never knew that,” I said.

“Live and learn,” said Hollus. “Members of my species can usually perceive cardinality up to six — a slight improvement over what you can do. But the Wreeds shunt us completely away from the center; the typical Wreed can perceive cardinality up to forty-six, although some individuals can do it as high as sixty-nine.”

“Really? But what happens when there are more items? Do they have to count them all, starting with item one?”

“No. Wreeds cannot count. They literally do not know how. Either they perceive the cardinality, or they do not. They have separate words for the numerals one to forty-six, and then they simply have a word that means ‘many.’ ”

“But you said some of them can perceive higher numbers?”

“Yes, but they cannot articulate the total; they literally do not have the vocabulary for it. Those Wreeds who can perceive larger cardinalities obviously have a competitive advantage. One might offer to swap fifty-two domesticated animals for sixty-eight domesticated animals, and the other, less-gifted Wreed, knowing only that they are both large quantities would have no way to evaluate the fairness of the trade. Wreed priests almost always have a higher-than-normal ability to do this.”

“Real cardinals of the church,” I said.

Hollus got the pun. His eyestalks rippled as he said, “Exactly.”

“Why do you suppose they never developed counting?”

“Our brains have only those abilities that evolution gave them. For the ancestors of your kind and mine, there were real-world, survival-oriented advantages to knowing how to determine quantities greater than five or six: if there are seven angry members of your species blocking your way on the left, and eight on the right, your chances, although slim, are still better with going to the left. If you have ten members of your tribe including yourself, and your job has been to gather fruit for dinner, you better come back with ten pieces, or you will make an enemy. Indeed, fetching just nine pieces will likely mean you yourself will have to forgo your fruit in order to placate the others, resulting in your having expended effort with no personal benefit.

“But Wreeds never form permanent groups larger than twenty or so individuals — a quantity they can perceive as a gestalt. And if there are forty-nine enemies to your left and fifty on your right, the difference is immaterial; you are doomed either way.” He paused. “Indeed, to use a human metaphor, one could say that nature dealt the Wreeds a lousy hand — or, actually, four lousy hands. You have ten fingers, which is a fine number: it lends itself to math, since it is an even number and can be divided into halves, fifths, and tenths; it is also the sum of the first four whole numbers: one plus two plus three plus four equals ten. We Forhilnors did well, too. We count by stomping our feet, and we have six of those — also an even number, and one that suggests halves, thirds, and sixths. And it is the sum of the first three whole numbers: one plus two plus three equals six. Again, a mental basis for mathematics.

“But the Wreeds have twenty-three fingers, and twenty-three is a prime number; it does not suggest any fractions other than twenty-thirds, a divisor too large for most real-world applications. And it is not the sum of any continuous sequence of whole numbers. Twenty-one and twenty-eight are the sums of the first six and first seven whole numbers, respectively; twenty-three has no such significance. With the arrangement of digits they have, they simply never developed counting or the kind of math we perform.”

“Fascinating,” I said.

“It is indeed,” said Hollus. “More: you must have noticed T’kna’s eye.”

That surprised me. “Actually, no. He didn’t seem to have any eyes.”

“He has precisely one — that moist, black strip around the top of his torso. It is one long eye that perceives a complete 360-degree circle. A fascinating structure: the Wreed retina is layered with photoreceptive sheets that rapidly alternate in a staggered sequence between transparency and opacity. These sheets are stacked to a depth of more than a centimeter, providing sharp images at all focal lengths simultaneously.”