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{{Short description|Concept in logic and mathematics}}
{{other uses|Proof (disambiguation)}}

A '''proof''' is a logical argument that demonstrates the truth of a given statement or proposition. Proofs are essential in [[mathematics]], [[logic]], and the [[sciences]], where they are used to establish the validity of [[theorems]], [[hypotheses]], and [[scientific theories]].

== Types of Proof ==
Proofs can be categorized into several types, each with its own methodology and application:

=== Direct Proof ===
A [[direct proof]] involves a straightforward application of logical steps to arrive at a conclusion. It starts with known facts and uses logical operations to derive the statement to be proven.

=== Indirect Proof ===
An [[indirect proof]], also known as a [[proof by contradiction]], assumes the negation of the statement to be proven and shows that this assumption leads to a contradiction, thereby proving the original statement.

=== Constructive Proof ===
A [[constructive proof]] not only demonstrates that a statement is true but also provides a method for constructing an example of the statement.

=== Non-constructive Proof ===
A [[non-constructive proof]] shows that a statement is true without necessarily providing a concrete example.

=== Proof by Induction ===
[[Mathematical induction]] is a method of proof used primarily in [[number theory]] and [[combinatorics]]. It involves proving a base case and then showing that if the statement holds for an arbitrary case, it holds for the next case.

=== Proof by Exhaustion ===
[[Proof by exhaustion]] involves checking all possible cases to ensure that a statement holds in each one.

== Importance of Proof ==
Proofs are fundamental to the advancement of knowledge in various fields. In mathematics, they provide a rigorous foundation for [[theorems]] and [[lemmas]]. In the sciences, proofs underpin the development of [[scientific theories]] and [[laws of nature]].

== History of Proof ==
The concept of proof has evolved over centuries, with significant contributions from ancient Greek mathematicians such as [[Euclid]] and [[Archimedes]]. The development of formal logic in the 19th and 20th centuries further refined the methods and standards of proof.

== Related Concepts ==
* [[Axiom]]
* [[Theorem]]
* [[Lemma]]
* [[Corollary]]
* [[Hypothesis]]
* [[Scientific method]]
* [[Formal system]]
* [[Logical argument]]

== See Also ==
* [[Mathematics]]
* [[Logic]]
* [[Scientific theory]]
* [[Number theory]]
* [[Combinatorics]]
* [[Euclid]]
* [[Archimedes]]

== References ==
{{Reflist}}

[[Category:Mathematics]]
[[Category:Logic]]
[[Category:Scientific method]]
[[Category:Philosophy of science]]

{{Mathematics-stub}}

Proof - Revision history

Prab at 13:10, 18 March 2025

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