#proof A mathematical proof is a rigorous and logical demonstration that establishes the truth or validity of a mathematical statement or theorem. It is a way to convince others, and most importantly, oneself, that a particular statement is true under certain assumptions or conditions. In mathematics, [[Mathematical proof|proofs]] follow a set of rules and conventions to ensure clarity, precision, and rigor. They are based on axioms and previously established results, building upon existing knowledge. Proofs can be presented in various forms such as direct proofs, indirect proofs (e.g., proof by contradiction), or constructive proofs. To construct a mathematical proof, one typically starts with stating the theorem or proposition that needs to be proven. Then, using logical reasoning and deductive steps, the proof proceeds by establishing a chain of arguments leading from known facts or assumptions to the desired conclusion. Each step must be justified using previously established results or axioms. The key elements of a mathematical proof include: 1. Assumptions: Clearly stating any initial assumptions or premises from which the proof begins. 2. Definitions: Providing precise definitions for any relevant terms used in the statement being proved. 3. Logical reasoning: Applying logical rules such as modus ponens (if A implies B and A is true, then B must be true) or modus tollens (if A implies B and B is false, then A must be false) to draw valid conclusions. 4. Theorems/lemmas: Utilizing previously proven theorems or lemmas as building blocks in the proof. 5. Justification: Explaining each step in the proof by explicitly stating the reason behind it. 6. Clarity: Ensuring that all statements are clear and unambiguous by using precise mathematical language. Mathematical proofs serve multiple purposes beyond just verifying correctness. They provide insight into why something is true and often lead to deeper understanding of underlying principles and concepts. Additionally, proofs allow mathematicians to communicate and share their discoveries, enabling others to build upon existing knowledge and contribute to the advancement of mathematics as a whole. ## References What is Curry-Howard-Lambek isomorphism? The [[Curry-Howard-Lambek isomorphism]] is a fundamental concept in computer science and mathematical logic that establishes a connection between logic, type theory, and category theory. Named after logicians Haskell Curry, William Howard, and Joachim Lambek, the isomorphism states that there is a correspondence between proofs in intuitionistic logic (or constructive logic), typed lambda calculus, and Cartesian closed categories. In this correspondence, propositions in logic are treated as types in type theory, and proofs of these propositions are identified with terms of these types. This means that logical statements can be seen as types that have values (proofs) associated with them. Similarly, functions in lambda calculus correspond to logical implications. The function's input type represents the assumption or premise of the implication, while the output type represents the conclusion. Category theory comes into play by providing a framework for expressing the connections between types and functions. Cartesian closed categories are used to model the structure of typed lambda calculi and intuitionistic logic. Overall, the Curry-Howard-Lambek isomorphism provides a deep insight into the relationships between logic, computation, and category theory. It has had significant impacts on programming language design, proof theory, and formal methods in computer science.