As if there wasn't enough real science for us to learn, now science fiction writers have to go and create their own. But to give credit where it's due, psychohistory does serve an important role as a symbol for science's purpose in society. Psychohistory represents two aspects of science.
…how awful, terrifying, and dreadful scientific facts can sometimes be. We see this particularly in "The Psychohistorians."
At one point, Seldon is put on trial for predicting the downfall of the Galactic Empire. But it's not Seldon's fault. It's just the data. But the Commission of Public Safety is afraid of Seldon's predictions and tries to silence him to mask their fears, even though they ultimately can't change the facts.
That's because, as Seldon says, his facts are a "prediction which is made by mathematics." He passes "no moral judgment" and personally "regret[s] the prospect" of the end of the Galactic Empire (I.6.75). Just because the facts are disagreeable and scary, doesn't mean they aren't the facts. Instead of getting mad at the facts—like the Commission does at psychohistory—people should find ways to improve and work with those facts for the betterment of mankind.
Psychohistory represents the need to let science help human development. This one is Asimov through and through, as the man was a big supporter of science, not only as a field of study but also as a leading force in social and political decision-making.
In every story of Foundation, the people who attempt to change the facts or move against them are ultimately defeated—like Chen and the Galactic Empire Barons when they try Seldon for treason. In contrast, the characters who allow the facts of psychohistory to guide them and their decision-making are the ones who ultimately succeed—just like Hari Seldon.
And this is what psychohistory ultimately represents: science (and rational thinking) as the ultimate means for people to succeed over the obstacles in their way.