Introduction to mathematical arguments（介绍数学参数）.pdf

Introduction to mathematical arguments (background handout for courses requiring proofs) by Michael Hutchings A mathematical proof is an argument which convinces other people that something is true. Math isn’t a court of law, so a “preponderance of the evidence” or “beyond any reasonable doubt” isn’t good enough. In principle we try to prove things beyond any doubt at all — although in real life people make mistakes, and total rigor can be impractical for large projects. (There are also some subtleties in the foundations of mathematics, such as G¨odel’s theorem, but never mind.) Anyway, there is a certain vocabulary and grammar that underlies all mathematical proofs. The vocabulary includes logical words such as ‘or’, ‘if’, etc. These words have very precise meanings in mathematics which can differ slightly from everyday usage. By “grammar”, I mean that there are certain common-sense principles of logic, or proof techniques, which you can use to start with statements which you know and deduce statements which you didn’t know before. These notes give a very basic introduction to the above. One could easily write a whole book on this topic; see for example How to read and do proofs: an introduction to mathematical thought process by D. Solow). There are many more beautiful examples of proofs that I would like to show you; but this might then turn into an introduction to all the math I know. So I have tried to keep this introduction brief and I hope it will be a useful guide. In §1 we introduce the basic vocabulary for mathematical statements. In §2 and §3 we introduce the basic principles for proving statements. We provide a handy chart which summarizes the meaning and basic ways to prove any type of statement. This chart does not include uniqueness proofs and proof by induction, whi