Is it possible and useful to mathematically formalize human behavior on large scales sufficient to probabilistically predict the future?
Answer by Joshua Engel, Polymath.
Probably not. Asimov spent most of the Foundation novels explaining why it doesn't work. It's parallel to the way he designed laws of robotics, and then wrote stories that showed how things went wrong despite and even because of those laws.
The novels claimed only to be able to predict the vastest aggregation: trillions of people over the course of hundreds or thousands of years. That kind of scale is so far beyond anything we can manage right now that the only question we can ask is "could psychohistory exist?"; actually speculating on what it might look like is impossible and making the question itself so speculative as to be irrelevant.
One of the things that Asimov noted would cause psychohistory to fumble is technology. Right now, and for the forseeable future, human history is limited to the carrying capacity of the earth, too small for psychohistory to apply. The kind of technology that would change that is completely unpredictable; it could happen 10 years from now or literally never. Most responsible scientists would veer towards "never," in fact, unless you catch them in a dreamy mood.
The form such a miraculous discovery would take is so variable that predictions are impossible: a Star Trek transporter or hyperdrive or something science fiction writers haven't yet patted themselves on the back for foreseeing. It's at least as believable as the ability to manipulate human minds, which as Asimov noted makes psychohistory moot.
Technological changes complete revise the way humans interact with each other. Hari Seldon would have to spit out things like "You've probably invented X by now," which is an odd thing to say. If you can predict how long it will take to invent X, you can invent X right now. Even if you could devise an equation for how an invention would affect large-scale patterns of human beings (an odd insight to have), predicting when you'd get it or the order they'd come in doesn't seem realistic.
Asimov also speculated that psychohistory would be made moot if people were aware of it. Predict the future and people will perversely alter it. It would be impossible to keep a discovery like psychohistory secret: anything that can be discovered once will be discovered again, especially if you throw trillions of people at the problem. Asimov had to deliberately concoct a Middle Ages-style interregnum to keep people from rediscovering psychohistory.
There is economics, and economics does already make some kinds of very limited predictions about large-scale human behavior. It will get better at that over time, as it grows more sophisticated. But sophistication brings complexity, and complexity brings chaos. Chaotic systems can be predicted only for extremely broad, general numbers like the temperature of a system.
Economics predicts a steady rising of GDP and population, a nice single number on the order of temperature. But that's hardly a psychohistorical kind of prediction, and economists are well aware that this prediction won't apply forever due to resource constraints on the earth. They can even predict a leveling off, and with more sophisticated models can even take a guess as to where. But the numbers will be vague and with large error bars, hardly the kind of power we attribute to psychohistory, and adding more math won't really improve the power of its predictions.More questions on psychohistory: