[x.com](https://twitter.com/burny_tech/status/1772484528545689876) Mathematics of reality I will provide a list of mathematical fields, subfields, concepts, and equation names used in explaining each of the given objects, followed by explanations and concrete examples for each concept mentioned. Please note that the explanations and examples will be brief and not exhaustive due to the scope of the question. 1. Fundamental particle: - Quantum mechanics, quantum field theory, particle physics, Schrödinger equation, Dirac equation, Klein-Gordon equation, Standard Model. Example: The Schrödinger equation, iℏ∂Ψ/∂t = ĤΨ, is used to describe the quantum state and time evolution of a fundamental particle, where Ψ is the wave function, ℏ is the reduced Planck's constant, and Ĥ is the Hamiltonian operator representing the total energy of the system. 2. Atom: - Quantum mechanics, atomic physics, Schrödinger equation, Bohr model, Rydberg formula, Zeeman effect, Stark effect. Example: The Rydberg formula, 1/λ = R(1/n₁² - 1/n₂²), is used to calculate the wavelengths (λ) of photons emitted or absorbed during electron transitions in an atom, where R is the Rydberg constant, and n₁ and n₂ are the principal quantum numbers of the initial and final states, respectively. 3. Molecule: - Quantum chemistry, molecular physics, Schrödinger equation, Born-Oppenheimer approximation, molecular orbital theory, valence bond theory, Hückel method. Example: The Born-Oppenheimer approximation, Ĥ = T̂ₙ + Ĥₑ(r;R), separates the Hamiltonian (Ĥ) of a molecule into the nuclear kinetic energy (T̂ₙ) and the electronic Hamiltonian (Ĥₑ), which depends on the electronic coordinates (r) and parametrically on the nuclear coordinates (R), allowing for the separate treatment of nuclear and electronic motions. 4. Protein: - Biophysics, computational biology, molecular dynamics, force fields, potential energy functions, Ramachandran plot, protein folding algorithms. Example: Molecular dynamics simulations use Newton's equations of motion, F₁ = m₁a₁, to calculate the forces (F₁) acting on each atom (with mass m₁ and acceleration a₁) in a protein, based on the potential energy functions describing the interactions between atoms, to predict the protein's conformational changes over time. 5. Cell: - Mathematical biology, systems biology, differential equations, reaction-diffusion equations, compartmental models, Michaelis-Menten kinetics, Hodgkin-Huxley model. Example: The Michaelis-Menten equation, v = V_max[S] / (K_m + [S]), describes the rate (v) of an enzyme-catalyzed reaction in a cell, where V_max is the maximum reaction rate, [S] is the substrate concentration, and K_m is the Michaelis constant, which represents the substrate concentration at which the reaction rate is half of V_max. 6. Neuron: - Computational neuroscience, neural networks, differential equations, cable theory, Hodgkin-Huxley model, FitzHugh-Nagumo model, integrate-and-fire model. Example: The Hodgkin-Huxley model uses a set of coupled differential equations to describe the electrical properties of a neuron, such as the membrane potential (V) and the gating variables (m, n, h) for ion channels: C(dV/dt) = I - g_Na m³h(V-E_Na) - g_K n⁴(V-E_K) - g_L(V-E_L), where C is the membrane capacitance, I is the applied current, g_Na, g_K, and g_L are the conductances, and E_Na, E_K, and E_L are the reversal potentials for sodium, potassium, and leakage channels, respectively. 7. Organism: - Population dynamics, ecology, differential equations, Lotka-Volterra equations, logistic equation, allometric scaling, metabolic theory of ecology. Example: The logistic equation, dN/dt = rN(1 - N/K), models the growth of a population (N) over time (t), where r is the intrinsic growth rate, and K is the carrying capacity of the environment, taking into account the limiting factors that constrain population growth. 8. Human: - Anthropometry, biomechanics, allometry, body mass index, basal metabolic rate, Kleiber's law, Dubois and Dubois formula. Example: The basal metabolic rate (BMR) of a human can be estimated using Kleiber's law, BMR = a(M^b), where a is a proportionality constant, M is the body mass, and b is the scaling exponent (approximately 3/4). This equation relates the metabolic rate to body size across different species. 9. Intelligence: - Cognitive science, artificial intelligence, machine learning, neural networks, Bayesian inference, reinforcement learning, Hebbian learning rule. Example: The Hebbian learning rule, Δw_ij = ηx_i x_j, describes the change in the synaptic weight (Δw_ij) between two neurons (i and j) based on the product of their activations (x_i and x_j) and a learning rate (η), which is a key concept in understanding the development of neural connections and learning in both biological and artificial neural networks. 10. Electricity: - Electromagnetism, circuit theory, Ohm's law, Kirchhoff's laws, Maxwell's equations, Faraday's law, Ampère's law. Example: Ohm's law, V = IR, relates the voltage (V) across a conductor to the current (I) flowing through it and the resistance (R) of the conductor, which is a fundamental equation in analyzing electrical circuits and understanding the behavior of electrical components. 11. Computer: - Computer science, information theory, Boolean algebra, binary arithmetic, logic gates, algorithms, computational complexity, Turing machines. Example: Boolean algebra uses logical operators such as AND (∧), OR (∨), and NOT (¬) to perform operations on binary variables (0 and 1), which form the basis of digital logic circuits in computers. For instance, the AND operation can be represented as: x ∧ y = 1 if and only if both x = 1 and y = 1; otherwise, x ∧ y = 0. 12. ChatGPT: - Natural language processing, deep learning, transformer architecture, attention mechanism, unsupervised learning, transfer learning, perplexity. Example: The attention mechanism in the transformer architecture used by ChatGPT can be represented as: Attention(Q, K, V) = softmax(QK^T / √d_k)V, where Q, K, and V are the query, key, and value matrices, respectively, and d_k is the dimension of the key vectors. This equation computes the weighted sum of the value vectors based on the similarity between the query and key vectors, allowing the model to focus on relevant information when generating responses. 13. Society: - Social network analysis, graph theory, centrality measures, power law, preferential attachment, homophily, game theory. Example: The degree centrality of a node (C_D(v)) in a social network can be calculated as: C_D(v) = deg(v) / (n-1), where deg(v) is the number of edges connected to node v, and n is the total number of nodes in the network. This measure quantifies the importance or influence of an individual in a social network based on their direct connections. 14. Planet: - Celestial mechanics, orbital mechanics, Kepler's laws, Newton's law of universal gravitation, tidal forces, planetary equations, Milankovitch cycles. Example: Kepler's third law, T^2 = 4π^2a^3 / (GM), relates the orbital period (T) of a planet to its semi-major axis (a), where G is the gravitational constant, and M is the mass of the central body (e.g., the Sun). This equation helps in understanding the motion and characteristics of planets in the solar system. 15. Galaxy: - Astrophysics, cosmology, Hubble's law, Friedmann equations, dark matter, dark energy, Tully-Fisher relation, Faber-Jackson relation. Example: Hubble's law, v = H_0 × d, describes the relationship between the recessional velocity (v) of a galaxy and its distance (d) from the observer, where H_0 is the Hubble constant. This law is crucial in understanding the expansion of the universe and estimating the distances to galaxies. 16. Universe: - Cosmology, general relativity, Friedmann equations, cosmic microwave background, inflation theory, dark matter, dark energy, cosmological constant. Example: The Friedmann equations, (da/dt)^2 = (8πG/3)ρa^2 and d^2a/dt^2 = -(4πG/3)(ρ + 3p)a, describe the evolution of the scale factor (a) of the universe over time (t), where G is the gravitational constant, ρ is the energy density, and p is the pressure. These equations form the basis of the standard model of cosmology, known as the Big Bang theory, and help in understanding the expansion and fate of the universe.