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In mathematics, a constructive proof is a method of proof that demonstrates the existence of a mathematical object by creating or providing a method for creating the object. This is in contrast to a non-constructive proof (also known as an existence proof or pure existence theorem ), which proves the existence of a particular kind of object ...
Informally, this means that if there is a constructive proof that an object exists, that constructive proof may be used as an algorithm for generating an example of that object, a principle known as the Curry–Howard correspondence between proofs and algorithms.
In classical real analysis, one way to define a real number is as an equivalence class of Cauchy sequences of rational numbers.. In constructive mathematics, one way to construct a real number is as a function ƒ that takes a positive integer and outputs a rational ƒ(n), together with a function g that takes a positive integer n and outputs a positive integer g(n) such that
Math on Trial: How Numbers Get Used and Abused in the Courtroom is a book on mathematical and statistical reasoning in legal argumentation, for a popular audience. It was written by American mathematician Leila Schneps and her daughter, French mathematics educator Coralie Colmez , and published in 2013 by Basic Books .
proof theory and constructive mathematics (considered as parts of a single area). Additionally, sometimes the field of computational complexity theory is also included together with mathematical logic. [2] [3] Each area has a distinct focus, although many techniques and results are shared among multiple areas. The borderlines amongst these ...
In mathematical logic, realizability is a collection of methods in proof theory used to study constructive proofs and extract additional information from them. [1] Formulas from a formal theory are "realized" by objects, known as "realizers", in a way that knowledge of the realizer gives knowledge about the truth of the formula. There are many ...
The definition of a formal proof is intended to capture the concept of proofs as written in the practice of mathematics. The soundness of this definition amounts to the belief that a published proof can, in principle, be converted into a formal proof. However, outside the field of automated proof assistants, this is rarely done in practice.
The fundamental distinguishing characteristic of intuitionism is its interpretation of what it means for a mathematical statement to be true. In Brouwer's original intuitionism, the truth of a mathematical statement is a subjective claim: a mathematical statement corresponds to a mental construction, and a mathematician can assert the truth of a statement only by verifying the validity of that ...