Seldon was uncertain whether to laugh or cage. After a pause, he did neither, but managed to speak calmly. "Progress? In less than two months? Hummin, this is something that might easily take me my whole life and the lives of the next dozen who follow me.-And even then end in failure."
"I'm not talking about anything as final as a solution or even as hopeful as the beginning of a solution. You've said flatly a number of times that a useful psychohistory is possible but impractical. All I am asking is whether there now seems any hope that it can be made practical."
"Frankly, no."
Dors said, "Please excuse me. I am not a mathematician, so I hope this is not a foolish question. How can you know something is both possible and impractical? I've heard you say that, in theory, you might personally meet and greet all the people in the Empire, but that it is not a practical feat because you couldn't live long enough to do it. But how can you tell that psychohistory is something of this sort?"
Seldon looked at Dors with some incredulity. "Do you want that explained."
"Yes," she said, nodding her head vigorously so that her curled hair vibrated.
"As a matter of fact," said Hummin, "so would I."
"Without mathematics?" said Seldon with just a trace of a smile.
"Please," said Hummin.
"Well-" He retired into himself to choose a method of presentation. Then he said, "-If you want to understand some aspect of the Universe, it helps if you simplify it as much as possible and include only those properties and characteristics that are essential to understanding. If you want to determine how an object drops, you don't concern yourself with whether it is new or old, is red or green, or has an odor or not. You eliminate those things and thus do not needlessly complicate matters. The simplification you can call a model or a simulation and you can present it either as an actual representation on a computer screen or as a mathematical relationship. If you consider the primitive theory of nonrelativistic gravitation-"
Dors said at once, "You promised there would be no mathematics. Don't try to slip it in by calling it 'primitive.' "
"No, no. I mean 'primitive' only in that it has been known as long as our records go back, that its discovery is shrouded in the mists of antiquity as is that of fire or the wheel. In any case, the equations for such gravitational theory contain within themselves a description of the motions of a planetary system, of a double star, of tides, and of many other things. Making use of such equations, we can even set up a pictorial simulation and have a planet circling a star or two stars circling each other on a two-dimensional screen or set up more complicated systems in a three-dimensional holograph. Such simplified simulations make it far easier to grasp a phenomenon than it would be if we had to study the phenomenon itself. In fact, without the gravitational equations, our knowledge of planetary motions and of celestial mechanics generally would be sparse indeed.
"Now, as you wish to know more and more about any phenomenon or as a phenomenon becomes more complex, you need more and more elaborate equations, more and more detailed programming, and you end with a computerized simulation that is harder and harder to grasp."
"Can't you form a simulation of the simulation?" asked Hummin. "You would go down another degree."
"In that case, you would have to eliminate some characteristic of the phenomenon which you want to include and your simulation becomes useless. The LPS-that is, 'the least possible simulation' gains in complexity faster than the object being simulated does and eventually the simulation catches up with the phenomenon. Thus, it was established thousands of years ago that the Universe as a whole, in its full complexity, cannot be represented by any simulation smaller than itself.
"In other words, you can't get any picture of the Universe as a whole except by studying the entire Universe. It has been shown also that if one attempts to substitute simulations of a small part of the Universe, then another small part, then another small part, and so on, intending to put them all together to form a total picture of the Universe, one would find that there are an infinite number of such part simulations. It would therefore take an infinite time to understand the Universe in full and that is just another way of saying that it is impossible to gain all the knowledge there is."
"I understand you so far," said Dors, sounding a little surprised.
"Well then, we know that some comparatively simple things are easy to simulate and as things grow more and more complex they become harder to simulate until finally they become impossible to simulate. But at what level of complexity does simulation cease to be possible? Well, what I have shown, making use of a mathematical technique first invented in this past century and barely usable even if one employs a large and very fast computer, our Galactic society falls short of that mark. It can be represented by a simulation simpler than itself. And I went on to show that this would result in the ability to predict future events in a statistical fashion-that is, by stating the probability for alternate sets of events, rather than flatly predicting that one set will take place."
"In that case," said Hummin, "since you can profitably simulate Galactic society, it's only a matter of doing so. Why is it impractical?"
"All I have proved is that it will not take an infinite time to understand Galactic society, but if it takes a billion years it will still be impractical. That will be essentially the same as infinite time to us."
"Is that how long it would take? A billion years?"
"I haven't been able to work out how long it would take, but I strongly suspect that it will take at least a billion years, which is why I suggested that number."