It is possible to describe psychological phenomena with mathematics. As is stated in the opening to one of my favourite TV shows, Numb3rs (a CBS 'solve-crimes-with-math series', 2005-2010, starring Rob Morrow and David Krumholtz), math is about logic and rationality. If you can follow good mathematical structure, then the logic is built into your conclusions. In this two-post series, I use mathematical structure to decipher what might be behind one of the most sought-after human emotions and habits of mind: the experience of joy, and the attitude of hope.
Leaving the concept of hope for right now, let's start at the ‘other end' by looking at how experts predict disaster. The insurance industry uses a mathematical formula called the hazard function. This function yields, through frequency counts of disastrous events and the use of basic calculus and probability theory, the likelihood that a specified ‘hazard' of interest will happen in the very next instant. This could mean a car accident, a house fire, or a fatality. These are events for which people are often insured. If someone is a member of a given age group, sex, region, occupational category, or other grouping that predict a higher ‘hazard function' value (higher moment-by-moment likelihood of a given disaster), then their insurance rates for that type of hazard will be higher. An example of this is higher automobile insurance premiums for young males. They are one of the higher risk groups for an insurance company payout.
Something called the reverse hazard function (see Chechile, 2011) looks at a similar quantity, but in reverse. The reverse hazard function describes the likelihood that something (i.e., a disaster or ‘hazard') has just happened in the preceding moment. In keeping with the semantic field of the wording for the ‘hazard' function, the reverse hazard function might be termed the ‘shock function'. By calling it the ‘shock function' the intent is to capture the emotion just after the ‘hazard' has occurred. The shock function speaks to the likelihood in any given instant in a given place and time that people and systems will be in a state of shock. The 'shock function' specifically describes the likelihood that a disaster has just occurred in the preceding moment.
Good mathematical structure can preserve logic while turning it to new ends. If ‘hazard' is viewed neutrally as the likelihood of an event in the next instant, then replacing the ‘disaster' component with ‘miracle' or simply 'a good event' changes the emotional interpretation of the ‘hazard' function. A measure for how much something good can be expected to happen in the next instant could be called: an index of hope. Proceeding from the reverse hazard function through a positive emotional logic, the chance that something good has just happened -- the opposite of a ‘shock function' as defined above -- would be a measure of joy. In my second post, I will explore potential applications of this concept.
Chechile, R.A. (2011). Properties of reverse hazard functions. Journal of Mathematical Psychology, 55, 203-222.