Mathematical Theories of Everything
This bullet-point introduction sets the stage for discussing influential theories and approaches over the decades. Each theory will presented mostly with an equation, a plot and some lines of explanation:
1960s: Cybernetics: Control Systems and Feedback Loops
Output = (G(s) / (1 + G(s)H(s))) * Input
Norbert Wiener pioneered cybernetics, which became a dominant framework in the 1960s. Cybernetics studies control and communication in animals, machines, and organizations. It introduced concepts such as feedback loops, self-regulation, and homeostasis. Cybernetics has influenced several fields, including engineering, biology, and the social sciences. The emphasis is on feedback mechanisms in maintaining system stability.
1970s: System Dynamics - Modeling and Simulation of Complex Systems
dS/dt = Inflow - Outflow
Here, S (Stock) represents the quantity of interest that accumulates or depletes over time. It can be capital in a bank account of stock, or water level in a tank, but also the population of a species so that the inflow can be the birth rate in a population model and outflow the death rate.
Developed by Jay W. Forrester and others, system dynamics emerged as a method for understanding the behavior of complex systems over time. It uses differential equations (mainly ODEs) to model the interactions between different system components. System dynamics became useful for policy analysis, organizational development, and understanding ecological and economic systems.
Interestingly, Forrester stated that our mental models are fuzzy in his "Counterintuitive Behavior of Social Systems" and this gives the opportunity to introduce the next chapter...
1980s: Fuzzy Sets and Fuzzy Logic - Handling Uncertainty and Vagueness
Lotfi Zadeh introduced fuzzy sets and fuzzy logic in the 1960s, but widespread application and development surged much later in the 1980s. Fuzzy logic provides a way to handle imprecise/ambiguous information, allowing for degrees of membership rather than binary classification. This approach has been applied in control systems, decision-making processes, and artificial intelligence, enabling more flexible and human-like reasoning in systems.
More on Fuzzy logic and membership functions here:
If you think that fuzzy systems are chaotic, you are on the wrong track .
Chaos is chaotic!
1990s: Chaos Theory - Nonlinear Dynamics and Sensitivity to Initial Conditions
x_{n+1} = r x_n (1 - x_n) (r=growth rate parameter)
Figures like Edward Lorenz popularized chaos theory and studied how small changes in initial conditions (the butterfly flapping its wings) can lead to very different outcomes in nonlinear systems (a tornado, for instance).
Chaos theory focuses on the inherent unpredictability of complex systems. It has influenced fields such as meteorology, engineering, economics, and even philosophy, highlighting the limits of predictability in complex systems.
Not to mention its role in "Jurassic Park"!
领英推荐
Now let us discuss complex systems from another point of view.
2000s: Agent-Based Modeling (ABM) - Individual-Based Simulation of Complex Systems
Agent_State_{t+1} = f(Agent_State_t, Environment_State_t)
Agent-based modeling became prominent in the 2000s as a method for simulating the actions and interactions of autonomous agents to assess their effects on the system as a whole. It is particularly useful for modeling social systems, ecological interactions, and market dynamics. ABM explores emergent phenomena and helps understand complex adaptive systems, providing insights into collective behavior and system-level outcomes from individual actions.
Talking about individuals, let us see them as dots A_{ij}.
2010s: Network Theory - Analyzing and Modeling Networked Systems
A_{ij} = 1 if there is an edge from node i to node j
A_{ij} = 0 otherwise
Network theory, or network science, gained prominence in the 2010s. It focuses on studying complex social, biological, and technological networks. It analyzes the structure, dynamics, and function of interconnected systems. Network theory has enhanced our understanding of phenomena like the spread of information, disease, and systemic risk in financial networks, leading to applications in epidemiology, sociology, and infrastructure resilience.
What have we learned?
When the discussion is about learning, we cannot avoid Deep Learning!
Deep learning probably started in the 2010s or earlier but it exploded in the 2020s.
2020s: Machine Learning and Data-Driven Modeling
θ_{t+1} = θ_t - η * ? C(θ_t)
The rise of data-driven approaches to system modeling marks the current decade. Techniques like deep learning and reinforcement learning use patterns from large datasets to model complex systems. This approach is revolutionizing fields such as predictive analytics in engineering, or medicine. Deep Learning is data-driven and perception-based and can create accurate models without explicit programming and physics-driven knowledge.
More here:
2030s
Up to you, next generations! Good luck and all the best!!
Specialista IT | Analista CAE
4 个月In theoretical physics, "theories of everything" are usually wrong or useless ??. Applied maths is focused on modelling rather than reductionism, so it has more chances to be relevant.