High Dimensional Space to Formulate Marriage and Birth Functions

Shuichirou Ike

• Editorial: CRC Press
• Año: 2022
• ISBN:
  9780429594465
• Edición: 1
• Categoría: Mathematics
• Categoría: Differential Equations

With the collapse of Demographic Transition Theory, new theories of population must not just be explanations, but should be falsifiable theories which can compute the number of occurrences of marriages and births. This book reviews computable marriage and birth function using dynamic properties. To do that, the functions are defined in high dimensional space. The reaction-diffusion equation of the number of children in a space is applied to these phenomena, providing solutions to many problems concerning a decline in fertility. The functions are developed as stochastic maps based on the present behaviors of successive behaviors in a geographical space. As we assume that there is an inter-dependence of human behaviors, we use the law of dynamics concerning the function of marriage and birth. The exact mathematical definition of interactions in a space naturally implies a causal relation. For the function concerning the number of children of parents, two geographical-dimensional spaces are required.

The decline in fertility in Belgium due to different languages is explained, and the longer fertility period in Brittany is explained by the Laplacian of the diffusion equation. Depending on the degree of symbolic control over behaviors, we need to add the degree of the dimension of the space. For the marriage function, we add age as a biological dimension to the geographical space. In this higher dimensional space, the mapping from neighboring present marriages to neighboring successive marriages is no less than that of the marriage function. These chain reactions caused the baby boom as an exothermal reaction-diffusion. Birth functions require one to add the marriage-age dimension to two geographical and age dimensions so that it is a five dimensional hypersurface. It can, thus, determine birth probabilities of a female who married at a certain age. The phenomenon of modern fertility decline may only be the result of these chain reactions. These processes are solely dependent upon time-space, and not on socioeconomic conditions. This is the very reason why we are able to predict it mathematically.

The book provides a new thinking in fertility decline for demographic research. Readers need to be aware that the fertility decline experienced throughout the modern era is a spatial pattern formation (as a reaction-diffusion). The author hopes new mathematical applications in human activities are developed through these new models.

III MARRIAGE FUNCTION IN HIGH DIMENSIONAL SPACE

Chapter 7 No Individual Birth Functions Exist

Prop. XXIX. No individual thing, which is entirely different from our own nature, can help or check our power of activity, and absolutely nothing can do us good or harm, unless it has something in common with our nature.

Spinoza, Ethics (Ethica Ordine Geometrico Demonstrata) [16], p. 170.

Are birth behaviors actually products of individualistic decisions? Household economists assume this thesis with simplicity, as do liberals, despite having views on fertility (pronatalism) policies that are contrary to theirs. However, as the marriage function, birth behaviors depend on both time and space neighboring behaviors. This is a scientific fact. Individualism is an illusion that should be recognized as such. “Free will” cannot be defined a deterministic manner; it conceals itself in stochastic processes.

It is easily postulated that parents make decisions individually and depending on their socioeconomic conditions. Is this postulation true? Our intuitive understanding of ourselves is often false; self-recognition methods are not reliable at all.

7.1 Quasi Linearity of the Expected Number of Children Based on Marriage Duration

Birth probabilities specific by duration of marriage have quasi-linearity. This serves as indirect but powerful evidence that human births are stochastic and not deterministic. Whether an event is stochastic or deterministic must be judged based on empirical data rather than introspection. The problem that should be solved is the cause of our birth behaviors, not the manner in which we consciously decide to bear children.

7.1.1 Stochastic Variable

The number of children in a family is approximately proportional to the duration of the parents’ marriage. This regularity was found by the renowned demographer L. Henry.

Figure 7.1: Completed fertility and age at marriage: natural and controlled fertility.

Source: L. Henry, Population Analysis and Models [32], p. 94. Originally Figure 7.5 Completed fertility and age at marriage

Henry calculated “natural fertility” from the data of Hindu villages in Bengal, India, and “controlled fertility” from the completed fertility of women married in 1919 in Great Britain (see Figure 7.1). The development of contraception and the difference in socioeconomic settings do not affect this linearity. Henry described this specific peculiarity as follows:

As in the preceding case, we have a decline in completed fertility with the woman’s age at marriage, but it does not follow a straight line; if we fit a line to the straight part of the curve, the intersection with the horizontal axis will be far beyond age 40 years. The decline is often interpreted without taking into account the influence of psychological and social factors on the age at marriage and on the desired number of children. It is then attributed to:

1. increasing sterility with age: a woman married at age 25 years has a higher probability of not achieving the number of children she wants to have than a woman married at age 20 years.
2. contraceptive failures: A woman married at age 25 years who desires, say, 2 children, is exposed for a shorter time to the risk of having one or several more children than she desires than a woman married at age 20 who also wants 2 children.
   These two factors do not seem to be sufficient to account for the decline in completed fertility, that is actually observed as age at marriage increases. It is likely that the desired number of children is correlated with age at marriage through the influence of psychological and social factors; women who marry earlier also desire or accept more children. There are other reasons as well to believe that the factors enumerated in (1) and (2) are not a sufficient ex...