“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.”

“Eyes have evolved dozens of times in Earth’s history,” I said. “Insects and cephalopods and oysters and vertebrates and many others all developed eyes independently of each other. But I’ve never heard of an arrangement like that.”

“Nor had we until we met the Wreeds,” said Hollus. “But the structure of their eye also has an impact on the way they think. To stick with mathematics a moment longer, consider the basic model for all digital computers, whether made by humans or Forhilnors; it is the model, according to a documentary I saw on PBS, that you call the Turing machine.”

The Turing machine is simply an infinitely long strip of paper tape divided into squares, coupled with a print/erase head that can move left, right, or remain motionless and can either print a symbol in a square or erase the symbol already there. By programming movements and actions for the print/erase head, any computable problem can be solved. I nodded for Hollus to go on.

“The Wreed eye sees a complete, all-around panorama, and it requires no focusing — all objects are perceived with equal clarity at all times. You humans and we Forhilnors use the words concentrate and focus to describe both setting one’s attention and the act of thinking; you concentrate on an issue, you focus on a problem. Wreeds do neither; they perceive the world holistically, for they are physiologically incapable of focusing on one thing. Oh, they can prioritize in an intuitive sense: the predator up close is more important than the blade of grass far away. But the Turing machine is based on a kind of thought that is foreign to them: the print head is where all attention is concentrated; it is the focus of the operation. Wreeds never developed digital computers. They do, however, have analog computers and are adept at empirically modeling phenomenons, as well as understanding what factors go into producing them — but they cannot put forward a mathematical model. To put it another way, they can predict without explaining — their logic is intuitive, not deductive.”

“Amazing,” I said. “I’d have been inclined to think that mathematics would be the one thing we’d share with any other intelligent lifeform.”

“That was our assumption, too. And, of course, the Wreeds have been disadvantaged in some ways by their lack of math. Radio eluded them — which is why despite all the listening your SETI projects have done to Delta Pavonis, they were never detected. My race was monumentally surprised to find a technological civilization when our first starship arrived there.”

“Well, maybe Wreeds aren’t really intelligent,” I said.

“They are. They build the most beautiful cities out of the clay that covers most of their world. Urban planning is an art form for them; they see the whole metropolis as one cohesive entity. In fact, in many ways, they are more intelligent than we are. Well, perhaps that is an overstatement; let us say they are differently intelligent. The closest we come to having a common ground is in our use of aesthetics to evaluate scientific theories. You and I agree that the most beautiful theory is probably the correct one; we look for elegance in the way nature works. Wreeds share that, but understanding what constitutes beauty is much more innate in them; it lets them discern which of several theories is correct without testing them mathematically. Their sense of beauty also seems to have something to do with why they are so good at matters that perplex us.