Probably the most famous demographic riddle of all time is the one that the Sphinx was said to have posed to travellers outside the Greek city of Thebes: ‘Which creature walks on four legs in the morning, two at noon, and three in the evening?’ Unfortunate travellers who could not answer the riddle correctly were immediately devoured. Oedipus, fresh from killing his father, was the first to have got the answer right. The correct answer was ‘humans’. People crawl on their hands and knees as infants, walk on two feet in adulthood, and walk with a cane in old age. We easily recognize the three ages of humans. Humans are born dependent on the care of others. As they grow, their capacities and productivities generally increase, but eventually these reach a peak. After a while, capacities and productivities decline and, eventually, if they are lucky enough to survive, people become elderly, often again requiring transfers and care from others. The human life cycle is the basis of all studies of population ageing, and so we cannot begin to study population ageing without first answering the Sphinx’s riddle. However, answering the Sphinx’s riddle is not enough to get us started on a study of population ageing. We must take two more steps before we begin. First, we must recognize that not all people age at the same rate. As seen in Chapter 5, nowadays more educated people tend to have longer life expectancies than less educated people. Second, we must realize that there is no natural generalization of the Sphinx’s riddle to whole populations. Populations cannot be categorized into the stages of infancy, adulthood, and old age. Indeed, if the Sphinx was reborn today, we might find her sitting near another city and posing an equally perplexing riddle, one especially relevant for our times: ‘What can grow younger as it grows older?’ Answering this riddle correctly is the central challenge of this chapter and the key to understanding population ageing in the twenty-first century.
Enveloping semigroups and quasi-discrete spectrum
