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| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Mathematics | 7/10 | https://en.wikipedia.org/wiki/Mathematics | reference | science, encyclopedia | 2026-05-05T03:56:46.299650+00:00 | kb-cron |
=== Unreasonable effectiveness === The unreasonable effectiveness of mathematics is a phenomenon that was named and first made explicit by physicist Eugene Wigner. It is the fact that many mathematical theories (even the "purest") have applications outside their initial object. These applications may be completely outside their initial area of mathematics, and may concern physical phenomena that were completely unknown when the mathematical theory was introduced. Examples of unexpected applications of mathematical theories can be found in many areas of mathematics. A notable example is the prime factorization of natural numbers that was discovered more than 2,000 years before its common use for secure internet communications through the RSA cryptosystem. A second historical example is the theory of ellipses. They were studied by the ancient Greek mathematicians as conic sections (that is, intersections of cones with planes). It was almost 2,000 years later that Johannes Kepler discovered that the trajectories of the planets are ellipses. In the 19th century, the internal development of geometry (pure mathematics) led to definition and study of non-Euclidean geometries, spaces of dimension higher than three and manifolds. At this time, these concepts seemed totally disconnected from the physical reality, but at the beginning of the 20th century, Albert Einstein developed the theory of relativity that uses fundamentally these concepts. In particular, spacetime of special relativity is a non-Euclidean space of dimension four, and spacetime of general relativity is a (curved) manifold of dimension four. A striking aspect of the interaction between mathematics and physics is when mathematics drives research in physics. This is illustrated by the discoveries of the positron and the baryon
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In both cases, the equations of the theories had unexplained solutions, which led to conjecture of the existence of an unknown particle, and the search for these particles. In both cases, these particles were discovered a few years later by specific experiments.
=== Specific sciences ===
==== Physics ====
Mathematics and physics have influenced each other over their modern history. Modern physics uses mathematics abundantly, and is also considered to be the motivation of major mathematical developments.
==== Computing ====
Computing is closely related to mathematics in several ways. Theoretical computer science is considered to be mathematical in nature. Communication technologies apply branches of mathematics that may be very old (e.g., arithmetic), especially with respect to transmission security, in cryptography and coding theory. Discrete mathematics is useful in many areas of computer science, such as complexity theory, information theory, and graph theory. In 1998, the Kepler conjecture on sphere packing seemed to also be partially proven by computer.
==== Statistics and other decision sciences ====
The field of statistics is a mathematical application that is employed for the collection and processing of data samples, using procedures based on mathematical methods such as, and especially, probability theory. Statisticians generate data with random sampling or randomized experiments. Statistical theory studies decision problems such as minimizing the risk (expected loss) of a statistical action, such as using a procedure in, for example, parameter estimation, hypothesis testing, and selecting the best. In these traditional areas of mathematical statistics, a statistical-decision problem is formulated by minimizing an objective function, like expected loss or cost, under specific constraints. For example, designing a survey often involves minimizing the cost of estimating a population mean with a given level of confidence. Because of its use of optimization, the mathematical theory of statistics overlaps with other decision sciences, such as operations research, control theory, and mathematical economics.
==== Biology and chemistry ====
Biology uses probability extensively in fields such as ecology or neurobiology. Most discussion of probability centers on the concept of evolutionary fitness. Ecology heavily uses modeling to simulate population dynamics, study ecosystems such as the predator-prey model, measure pollution diffusion, or to assess climate change. The dynamics of a population can be modeled by coupled differential equations, such as the Lotka–Volterra equations. Statistical hypothesis testing is run on data from clinical trials to determine whether a new treatment works. Since the start of the 20th century, chemistry has used computing to model molecules in three dimensions.
==== Earth sciences ====
Structural geology and climatology use probabilistic models to predict the risk of natural catastrophes. Similarly, meteorology, oceanography, and planetology also use mathematics due to their heavy use of models.
==== Social sciences ====
Areas of mathematics used in the social sciences include probability/statistics and differential equations. These are used in linguistics, economics, sociology, and psychology.