# Artificial Intelligence and Society: Philosophy of FallibilityPart 8: Thinking in Terms of Mathematical Formulas

KOBAYASHI Keiichiro
Faculty Fellow, RIETI

In summarizing the discussions from Part 3 to Part 7, I will employ mathematical formulas, rather than merely relying on language, as doing so may better communicate ideas in some cases. The discussions can be summed up through the following three formulas (the meanings of the symbols used in the formulas will be explained later).
(1)$$\:\:\:\:\:q_{ｔ}=αp_{ｔ}$$
(2)$$\:\:\:\:\:p_{ｔ}=βq_{ｔ＋１}$$
(3)$$\:\:\:\:\:q_{ｔ}=α×β×q_{ｔ＋１}＝γq_{ｔ＋１}$$
(Please note that the above formulas are predicated on the following conditions: $$0<α<1; \:0<β<1; \:γ=α×β$$)
The meanings of the above formulas are as follows.

Let us denote individual actions driven by desires and passions (individual virtue) by $$c_{t}$$. The lower-case letter $$c$$ represents individual actions, and the subscript $$t$$ represents the period of time. Therefore, $$c_{t}$$ refers to the individual actions taken in the year $$t$$. Meanwhile, the moral value of $$c_{t}$$ is expressed as $$p_{t}. p_{t}$$ indicates the degree of meaningfulness that the individual finds in his/her goals and actions.

Let us denote the "status of social systems," which indicates the degree of achievement of the goal of human society under the comprehensive doctrine in which the individual believes, by $$k_{t}$$. The goal of human society refers to the "realization of the world spirit" under the Hegelian philosophy, "egalitarian society" under communism, and the "Kingdom of God" under Christianity, for example. $$k_{t}$$ represents the status of the social system as a whole, including political and social institutions intended to realize the goal of human society. The moral value of $$k_{t}$$ (=the status of the system) is expressed as $$q_{t}$$.

The meaning of Formula (1) is that the social system is valuable because it contributes to individual lives. The social system is valuable because it enhances the convenience of the lives of individuals who pursue their respective life goals. For example, if individuals join a religious community, they can easily develop cooperative relationships and engage in mutual help with neighbors within the community, thereby making their own lives easier. When the value of individual life is expressed as $$p_{t}$$, if $$k_{t}$$ (=the status of the system) is to increase $$c_{t}$$ (= individual actions) by the multiplying ratio $$α$$, $$q_{t}$$ (=the value of the system) can be obtained through Formula (1). Here, $$α$$ takes a positive value smaller than 1.

Next, the meaning of Formula (2) is that individual virtue is valuable because it contributes to the system. Hegel's "cunning of reason" idea is represented by this formula. While $$c_{t}$$ (= individual actions) is driven by individual desires and passions, it unwittingly contributes to the progress of reason, or the realization of freedom. That is exactly why individual lives are valuable. Realizing freedom means improving the social system. In terms of mathematical formulas, if the value of the system is expressed as $$q_{t}$$ and if individual actions are to improve $$k_{t}$$ (=the status of the system) by the multiplying ratio $$β$$, $$p_{t}$$ (=the moral value) of $$c_{t}$$ (=individual actions) can be obtained through Formula (2). It should be kept in mind that the time factor is included in this equation. Specifically, as $$k_{t}$$ (=the status of the system) in the current year (year $$t$$) is already fixed, $$c_{t}$$ (=individual actions in the current year) improves $$k_{t+1}$$ (=the status of the system in the next year) by the multiplying ratio $$β$$. In other words, $$p_{t}$$ (=the value of individual actions in the current year) is dependent on $$q_{t+1}$$ (=the value of the system in the next year). Here, $$β$$ takes a positive value smaller than 1.

To summarize the political philosophies discussed from Part 3 to Part 7 based on the individual-system relationships expressed by Formulas (1) and (2), Rawls' philosophy recognizes the relationship expressed by (1), but not the relationship expressed by (2). Under Rawls' philosophy, as $$p_{t}$$ (=the value of individual virtue) is considered to be a given, it does not have to be justified by the social system. The value of the social system ($$q_{t}$$) [the social system that realizes Rawls' "two principles of justice"]) rests in the system's contributions to the pursuit of individual virtue by maximizing individual freedom. Therefore, under Rawls' philosophy, while Formula (1) is valid, Formula (2) is not. Our critique ("Critique 2") presented the following propositions: that it is necessary to provide some justification for $$p_{t}$$,; and that this justification should be provided by the social system—that is, the relationship expressed by Formula (2) is essential.

A "comprehensive doctrine" as referred to by Rawls means a political philosophy encompassing both Formulas (1) and (2). Meanwhile, a "political conception" as referred to by him focuses exclusively on developing the relationship expressed by Formula (1) while refraining from developing the relationship expressed by Formula (2). Hegel's philosophy is a comprehensive doctrine that recognizes the relationship expressed by Formula (2) as it maintains that the value of individual lives is justified by the progress of the world spirit. Conventional religions and the totalitarian ideologies of the 20th century (e.g., communism and Nazism) are also comprehensive doctrines.

In the modern world, the Rawlsian liberal political philosophy alone cannot provide value to individual lives ($$p_{t}$$). Until now, we have held on to comprehensive doctrines as implicit premises, and that is the belief in the grand narrative that was mentioned by Lyotard, namely the worship of reason, which is strongly related to the Hegelian philosophy. The worship of reason, which implies a belief in the unlimited progress of reason, is also a form of humanism. In the worship of reason, or under humanistic thoughts, reason is assumed to continue to progress forever. When the view spread that reason has its limitations, humanism faced a crisis. Why is it?

The answer to this question is provided by Formula (3), which is obtained by substituting the symbol $$p_{t}$$ in Formula (1) with Formula (2). Formula (3) expresses the following relationship. $$q_{t}$$ (=the status of the system [$$k_{t}$$] in the current year [the year $$t$$]) is obtained by discounting $$q_{t+1}$$ (=the value of the system in the next year [the year $$t＋1$$]) by the multiplying ratio $$γ$$.

In other words, according to Formula (1), $$q_{t}$$ (=the value of the system in the current year) is obtained through the equation $$q_{t} =αp_{t}$$, namely by multiplying $$p_{t}$$ (= the value of individual virtue) by $$α$$. According to Formula (2), $$p_{t}$$ (= the value of individual virtue) is obtained through the equation $$p_{t} =βq_{t+1}$$, that is, by multiplying the value of the system in the next year ($$=q_{t+1}$$) by $$β$$. According to Formula (3), $$q_{t}$$ in the current year is obtained through Formula (3), which combines those two equations, and this means that $$q_{t}$$ is dependent on $$q_{t+1}$$ (=the value of the system in the next year).

Formula (3) continues to be valid beyond the next year. If Formula 3 is extended to future years, the following formula is obtained:

(4）$$\:\:q_{t} =γq_{t+1}=γ^{2}q_{t+2}=γ^{3}q_{t+3}= … ＝γ^{N}×q_{t+N}$$
The year $$t+N$$ represents an infinite future. In other words, $$q_{t}$$ (=the value of the system in the current year) depends on $$q_{t+1}$$ (=the value of the system in the next year), which in turn depends on $$q_{t+2}$$ (the value of the system two years later), and this chain of relationship continues forever. If so, $$q_{t}$$ (=the value of the system in the current year) ultimately depends on $$q_{∞}$$ (=the value of the system in the future). If $$γ$$ is larger than 0 and smaller than 1, that means $$q_{t}$$ is smaller than $$q_{t+1}$$ in Formula 3. This indicates that the value of the system must continue to increase with the passage of time (Formula (5)).

(5）$$\:\:q_{t}\lt q_{t+1}\lt q_{t+2}\lt q_{t+3}\lt … \lt q_{t+N}\lt q_{t+N+1}\lt …$$
As indicated by Formulas (4) and (5), if the social system (including ideals such as the progress of reason, and institutions based thereon) is to have a positive present value ($$q_{t}$$), the value of the system must continue to be positive and growing forever. Therefore, the value of an ideal advocated by a comprehensive doctrine needs not only to be maintained forever but also to continue to increase with the passage of time.

In the context of Lyotard's "grand narrative," the reason why we find value $$q_{t}$$ in the "progress of reason" is that we believe that the progress of reason will maintain its value forever and that the value $$q_{t}$$ will continue to increase with the passage of time. If the progress of reason comes to a halt midway or if the value of the progress of reason is reduced to zero at some point in the future ($$q_{t+N}= 0$$), Formulas (4) and (5) cease to be valid. This means that the present value of the progress of reason is also reduced to zero ($$q_{t}=0$$).

That is the situation meant by the "collapse of the grand narrative." Speaking more generally, if we stop believing that the value of a comprehensive social ideal (ideology) will continue to exist and grow forever, the ideal will instantly lose its value. That is why no sooner did we face the limitations of reason and stopped believing that the "progress of reason" would continue forever than the value of humanism came under critical threat.

June 24, 2022

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