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Science Theory Stagnation (6.5)

6.5 Simple Derivation

This section attempts to use mathematical knowledge to explain the changing trend of complexity.

First of all, it is necessary to show that the development of technological complexity is a chaotic system. There are too many factors affecting technological development to accurately predict the direction of future technology. The mathematical derivation in this section is only better than nothing to explain the development process of past technology. There are many factors that affect technology. Among them, human curiosity and desire to explore are variables that are difficult to describe mathematically, and they are often the starting point of technological revolution. Therefore, the following differential equations cannot explain how technology originated, but only describe the trend of change after it appears. Fortunately, human history has repeatedly shown that technological development depends on resource input.

It can be seen from history that in the case of low complexity, a slight technological improvement will bring shocking effects to the society. In today’s high complexity, technology is beginning to encounter negative feedback from society, and the effect of improvement is diminishing. On the other hand, social resources are limited. Those in power always pay attention to social stability, and the share of R&D investment is limited.

The complexity is represented by the variable C. A large C value represents a high level of social technology. K represents the social tolerance or the maximum resource input ratio. Then, the differential equation of complexity evolving with time t can be expressed as follows:

dC/dt = g(C,K,t)+f(C,K,t)

g represents the initial positive feedback effect, but it should be noted that this positive feedback effect will rapidly decrease over time. f represents the proportion of resource input in the middle and late stages, and this resource input decreases as the complexity increases.

For technological development, the best situation is dC/dt=qC (q is a constant of proportionality), and complexity is used to promote complexity, and the curve of C increases exponentially. This is an ideal technological development trajectory, and technology is always in an explosion, but it is impossible in reality.

If dC/dt=q can be achieved, then the complexity will grow linearly from the beginning, and it will never end, which does not conform to human observation facts.

Based on the previous sections, we can see that the changes in complexity have the following characteristics:

  • (1) The development of technology is promoted by resources (in a broad sense). In principle, the larger the proportion of resources invested by a society, the more technological development can be promoted.
  • (2) In the early stage of the emergence of technologies that can be “profitable”, they often have a positive feedback effect. For example, in primitive society, most of the resources were originally used for hunting, but a slight improvement in bow and arrow technology may lead to an increase in hunting efficiency. The tribe will voluntarily invest more resources to improve bow and arrow technology. In other words, the initial increase in complexity can in turn further promote the increase in complexity.
  • (3) Taking into account the law of diminishing marginal utility, the increase in complexity in the middle and late stages means an increase in social maintenance costs, which in turn squeezes other social resources. Continuing the above example, after the bow and arrow technology is improved to a certain extent, on the one hand, the tribe needs to allocate more manpower to produce bows and arrows, on the other hand, all the small animals that are easy to hunt are all caught, and the remaining animals in the animal kingdom are difficult Capture. The ecosystem will give negative feedback.
  • (4) Regardless of the constraints of the global environment, no matter how high the complexity is, society can always squeeze out some resources to continue to increase the complexity, and there will be no eventual decline in complexity.

Taking into account the above characteristics, although the specific expressions of g and f cannot be accurately obtained, if only the trend is observed, it can be simplified as follows:


Among them, a, b, k, d are proportional coefficients, and the value is generally positive. a*C*exp(-b*t) represents the positive feedback term, which rises at the initial stage, but decays quickly; k*(1-C/(C+d)) represents the proportion of social resource input in the middle and late stages (C is very small When it is almost constant k), the larger the C, the smaller the proportion, and k is the maximum investment proportion that the society can tolerate.

Well, take a=5, b=1, d=5, and simulate this equation with Matlab, the approximate curve obtained is as follows:
Technology complexity simulation curve

Again, the derivation of the above model is very rough. The expressions of f and g can be in other forms, and a, b, k, and d can be other values. However, this article only expresses personal opinions and expresses trends, and does not force public approval.

As can be seen from the above curve, the rise of C does not last forever. At the beginning, due to the positive feedback effect, C will rise rapidly, but afterwards, the ascent speed of C will slow down. Finally, the ascent speed of C will tend to 0, and the value of C will approach a certain “ceiling”. The limit is determined by a, b, k, d and other parameters.

If you want to increase the ceiling, the best way is to increase k, which is social tolerance. For example, during the world war, the key to rapid technological advancement is that during the war, various governments faced life and death challenges and went all out. All social resources were used for weapon research and development. When people’s livelihood was put to the end, the value of k was greatly increased.

The achievements of World War II included materials technology, electronic computers, radar, nuclear technology, jet aircraft and rocket technology. To this day, mankind has not broken away from the technological path opened up by World War II.

However, the improvement of k value is limited. In extreme cases, all the resources of the whole society are used for scientific research and development. In peacetime, this is an unimaginable scenario. And in today’s society where nuclear weapons are predominant, it is impossible to imagine a world war breaking out on the earth.


1 Civilization and technology
1.1 Rough talk about paradigm
1.2 The paradigm shift experienced by human civilization
1.3 Science Theory stagnation
1.4 The gap between science and technology
2 The paradigm spring dream advocated by scientific and technological interest groups: the so-called technological explosion
2.1 Rendering and brainwashing
2.2 Papers and patents: the absurdity behind astronomical numbers
2.3-2.4 The bit world and the real world / Part and whole
3 The shadow outside the paradigm spring dream
3.1 The technological dilemma faced by humans
3.2 Numerous technical gimmicks
3.2.5 New gimmicks in recent years
3.3 Frustration of PhD laborers and biotechnology

4 The dilemma of low-entropy body and the technical steps faced
4.1 From the second law of thermodynamics
4.2 The backbone and forks of the technology tree
4.3 Forever 50 years and controlled nuclear fusion
4.4 The future is not always better

5 The Pit Before 5 Steps: The Fate of Human Society
5.1 The Sociological Significance of Dissipative Structure Theory
5.2 The disappearance of the big competitive environment
5.3 Differences erased by globalization and the thermodynamic balance of human society
5.4 Aging self-locking
5.5 How to fill the hole?

6 The essence of 6 steps: complexity devil
6.1 What is complexity
6.2 Two rules behind the complicated world: survival of the fittest and expectation of return on capital
6.3 Technological progress and technological revolution: changes in complexity
6.3.1 Evolution example of transportation / power system
6.3.2 The characteristics and complexity of the technological revolution
6.3.3 The high-complexity science devil facing
6.3.4 Dilemma originating from technical foundation
6.4 Many evil consequences brought by high complexity ( more is different)
6.4.1 I know you have a life and death race
6.4.2 Maintenance costs
6.4.3 Negative feedback from society
6.5 Simple mathematical derivation

7 Silent Star implied by the prospect of terror
7.1 The Great Silence and Fermi Paradox
7.2 Three scenarios for contemplating extreme fear
7.3 The Great Sieve of the Universe
7.4 A small match

8 reflection and summary
8.1 The tragedy of Easter Island
8.2 Calmness does not mean pessimism
8.3 R&D requires a paradigm revolution

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