What is a secret, neurobiologically? Imagine constructing an artificial neural network of millions of interconnected neurons---what would a secret look like here? Could a toaster, with its interconnected parts, harbor a secret? We have useful scientific frameworks for understanding Parkinson's disease, color perception, and temperature sensation---but none for understanding what it means for the brain to have and to hold a secret.
Within the team-of-rivals framework, a secret is easily understood: it is the result of struggle between competing parties in the brain. One part of the brain wants to reveal something, and another part does not want to. When there are competing votes in the brain---one for telling, and one for withholding---that defines a secret. If no party cares to tell, that's merely a boring fact; if both parties want to tell, that's just a good story. Without the framework of rivalry, we would have no way to understand a secret.* The reason a secret is experienced consciously is because it results from a rivalry. It is not business as usual, and therefore the CEO is called upon to deal with it.
The main reason not to reveal a secret is aversion to the long-term consequences. A friend might think ill of you, or a lover might be hurt, or a community might ostracize you. This concern about the outcome is evidenced by the fact that people are more likely to tell their secrets to total strangers; with someone you don't know, the neural conflict can be dissipated with none of the costs. This is why strangers can be so forthcoming on airplanes, telling all the details of their marital troubles, and why confessional booths have remained a staple in one of the world's largest religions. It may similarly explain the appeal of prayer, especially in those religions that have very personal gods, deities who lend their ears with undivided attention and infinite love.
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As you have doubtless noticed, venting a secret is usually done for its own sake, not as an invitation for advice. If the listener spots an obvious solution to some problem revealed by the secret and makes the mistake of suggesting it, this will frustrate the teller---all she really wanted was to tell. The act of telling a secret can itself be the solution. An open question is why the receiver of the secrets has to be himan---or human-like, in the case of deities. Telling a wall, a lizard, or a goat your secrets is much less satisfying.
* Some people are constitutionally incapable of keeping a secret, and this balance may tell us something about the battles going on inside them and which way they tip. Good spies and secret agents are those people whose battle always tips toward long-term decision making rather than the thrill of telling.
