## Reasoning and Proof Standards

1. Read through the text below on the difference between reasoning and proof.
2. See the directions that follow for the assignment (If you are a math minor and want an assignment that is less related to teaching, then come see me in office hours.)

### The Difference Between Reasoning and Proof

Convincing Arguments There are many ways to convince someone that a statement is true, for example:
• Experimental evidence , e.g., physics experiments to validate theories of General Relativity; medical experiments to validate that rats that eat sugar get cancer.
• Statistical sampling e.g., Gallup polls to convince people that someone will be elected President.
• Citing a reputable authority. "The professor/boss said the answer to the problem was...".
• Acting confident, talking loudly. "I'm sure my program works now." I don't see why not.
• Shifting the burden of proof to someone who disagrees with you.
• Legal system. Uses "proof beyond a reasonable doubt"-convince a judge and jury.
But these are not proofs, at least, not in the mathematical sense, as these methods don't convince everyone and in fact can lead to false conclusions. For example:
• Experiments can be contaminated or otherwise messed up.
• Statistics can be misleading. Consider the Florida election!
• Authorities can make mistakes, e.g., Intel said the original Pentium chip worked fine.
• Juries can be fooled.
Mathematical Proofs Mathematics uses a particularly convincing way to argue that something is true:
A mathematical proof is a formal verification of a proposition by a chain of logical deductions starting from a base set of axioms.
Formal language: Need a precisely defined language for expressing everything.
Proposition: Statement written in the language, e.g., what you want to prove.
Axioms: Statements that you are assuming in your proof.
Logical deductions: Getting a new truth from old ones, according to agreed-upon, formal rules of deduction.

The main idea is to specify everything so precisely that everyone is convinced. If people accept the assumptions and deduction rules, they must also accept the conclusions. Even if we can't understand the entire argument at once, if we believe each deductive step, then we can be comfortable accepting the conclusions. This is a philosophical position, and thus debatable, but most of modern civilization accepts it. By learning to read and do proofs, you should be able to
• Convince others of your arguments.
• Find flaws in others' attempts to convince you.
• Find flaws in your own proofs.
• Develop the ability to reason carefully about, attack, and defend ideas.

### Assignment

Read through the National Council of Teachers of Mathematics (NCTM) web page on Reasoning and Proof Standard for Grades 9-12.