学习笔记《Introduction to Proofs》

前几天看到一个观点：数学不是科学，Karl Popper 说科学是可以证伪的，而数学无法被证伪，数学只是一系列的推导过程。Karl Popper 最著名的理论，在于对经典的观测-归纳法的批判，提出“从实验中证伪的”的评判标准：区别「科学的」与「非科学的」

Popper is known for his rejection of the classical inductivist views on the scientific method, in favour of empirical falsification: A theory in the empirical sciences can never be proven, but it can be falsified, meaning that it can and should be scrutinized by decisive experiments. Popper is also known for his opposition to the classical justificationist account of knowledge, which he replaced with critical rationalism, namely "the first non-justificational philosophy of criticism in the history of philosophy."

什么是 Mathematical proof 呢？

In mathematics, a proof is an inferential argument for a mathematical statement. In the argument, other previously established statements, such as theorems, can be used. In principle, a proof can be traced back to self-evident or assumed statements, known as axioms, along with accepted rules of inference. Axioms may be treated as conditions that must be met before the statement applies. Proofs are examples of exhaustive deductive reasoning or inductive reasoning and are distinguished from empirical arguments or non-exhaustive inductive reasoning (or "reasonable expectation"). A proof must demonstrate that a statement is always true (occasionally by listing all possible cases and showing that it holds in each), rather than enumerate many confirmatory cases. An unproved proposition that is believed to be true is known as a conjecture.

Proofs employ logic but usually include some amount of natural language which usually admits some ambiguity. In fact, the vast majority of proofs in written mathematics can be considered as applications of rigorous informal logic. Purely formal proofs, written in symbolic language instead of natural language, are considered in proof theory. The distinction between formal and informal proofs has led to much examination of current and historical mathematical practice, quasi-empiricism in mathematics, and so-called folk mathematics (in both senses of that term). The philosophy of mathematics is concerned with the role of language and logic in proofs, and mathematics as a language.

数学的证明也分为几个关键的节点

Axioms（公理）：

Theorems（定理）：

Lemmas（引理）：

我的理解是证明过程中，阶段性的证明出来当定理使用的「小定理」

Corollaries（推论）：

通过定理可以很容易的推导出来的部分

Conjectures（猜想）：

猜想如果被证明了，就可以成为定理