to this issue. My approach is that of a demographer and statistician who attempts to define what limits might look like in terms of aggregate mortality data and then reviews the existing empirical evidence within that framework.

If limits to human longevity exist, they must have both a mechanism and a manifestation. As noted already, I do not address the issue of mechanisms in this chapter but focus instead on the question of how limits might manifest themselves in aggregate mortality patterns. Based on a review of the literature and my own understanding of this topic, there appear to be three hypotheses about the age distribution of human mortality—and, in particular, about changes in the shape of that distribution over time—which may have a bearing on the issue of limits. In this section, I seek to explain these hypotheses clearly, to review some of the relevant literature, and to examine the most important empirical evidence in light of this theoretical framework. I also discuss briefly the logical connections between the three hypotheses. I conclude by offering a summary view of what a demographic perspective suggests at present about the limits of human longevity.

The three hypotheses can be stated using the notation of life tables or survival analysis. The former notation is most familiar to demographers and actuaries, while the latter is often used by statisticians, reliability engineers, and many biomedical researchers. To facilitate understanding by a diverse crowd, I will mention both sets of notation whenever convenient. I will favor the notation of survival analysis, however, because it is more general and probably more widely accessible to an interdisciplinary audience.

Mortality distributions can be effectively summarized by any one of several complementary functions. Three functions are particularly useful: the density function, the survival function, and the hazard function. Let X be a (positive) random variable representing the life span of an individual drawn at random from a population of newborns. Following standard statistical practice, let f(x) be the probability density function describing the distribution of life spans in this population. The cumulative density function, F(x), gives the probability (Pr) that an individual dies before surpassing age x:

The survival function, S(x), gives the complementary probability that an individual is still alive at age x: