A theorem It is a word of Greek origin that is a proposition that indicates a truth for a certain field of science, which has the particularity of being demonstrable by resorting to other previously demonstrated propositions, called axioms. Typically the theorems hold the sciences called ‘exact‘, especially the’ formal ‘(mathematics, logic), which are those that use ideal elements to draw general conclusions. For instance: Pythagorean theorem, binomial theorem, Euler’s theorem.
The thought that underlies the concept of theorem is that, as long as they are founded on true propositions articulated logically and correctly, what the theorem expresses is a truth of absolute validity. This is precisely what allows them to serve as a support for the development of any scientific theory, without the need to prove it again.
The central quality of theorems is their character of logical. In general, and again in comparison with other kinds of scientific knowledge (such as those that are produced through inference or observation), its origin is from the performance of a logical procedure that can be easily ordered. In this sense, the theorems start from a fundamental hypothesis, which is what one wants to prove; a thesis, which is precisely the demonstration, and a corollary, which is the conclusion that is reached once the demonstration is completed.
As said, the main idea of the theorems is the question of constant feasibility and the possibility of being countersigned and accepted again at all times. However, if a single situation arises in which the theorem loses its universality, the theorem immediately becomes invalid.
The concept of theorem has been taken by other sciences (economics, psychology or political science, among others) to designate certain important or foundational concepts that govern those fields, even when these do not arise through the procedure explained. In those cases, axioms are not used but rather inferences made by procedures such as observation or even statistical sampling.
Examples of theorems
The following list gathers examples of theorems and a brief description of what it postulates:
- Pythagoras theorem. Relationship between the measure of the hypotenuse and that of the legs, in the case of right triangles.
- Prime number theorem. As the number line grows, there will be fewer and fewer numbers from that group.
- Binomial theorem. Formula for solving powers of binomials (additions or subtractions of elements).
- Frobenius theorem. Solving formula for systems of linear equations.
- Thales’ theorem. Characteristics in terms of angles and sides of similar triangles, and other properties of them.
- Euler’s theorem. The number of vertices plus the number of faces equals the number of edges plus 2.
- Ptolemy’s theorem. The sum of the products of the diagonals is equal to the sum of the products of opposite sides.
- Cauchy-Hadamard theorem. Establishment of the radius of convergence of a series of powers that approximates a function around a point.
- Rolle’s theorem. In an interval whose evaluated ends in a differentiable function are equal, there will always be a point where the derivative vanishes.
- Mean value theorem. If a function is continuous and differentiable over an interval, there will be a point in that interval where the tangent will be parallel to the secant.
- Cauchy Dini’s theorem. Conditions for the calculation of derivatives in the case of implicit functions.
- Calculus theorem. The derivation and integration of a function are inverse operations.
- Arithmetic theorem. Every positive integer can be represented as a product of prime factors.
- Bayes theorem (statistics). Method to obtain conditional probabilities.
- Cobweb theorem (economics). Theorem to explain the formation of products that are made based on the previous price.
- Marshall Lerner’s theorem (economics). Analysis of the impact of a currency devaluation in terms of quantities and prices.
- Coase theorem (economics). Solution for cases of externalities, tending towards deregulation.
- Median voter theorem (political science). The majority election system tends to favor the median vote.
- Baglini’s theorem (political science, Argentina). The politician tends to bring his proposals closer to the center when he approaches positions of power.
- Thomas’s theorem (sociology). If people define situations as real, they become real in their consequences.