What is a proof?

"A proof is essentially a chain of reasoning, in which each step can be logically deduced from the ones that preceded it. It’s a way of putting your thought process on display so it can be scrutinized to make sure it holds water. Any step of your reasoning which was unwarranted will be exposed, and perhaps reveal that the conclusion you thought was true isn’t necessarily dependable after all. Here’s an example from everyday life. I’m driving home from work one afternoon, and I believe that my wife and children will be gone when I arrive. I’ll be coming home to an empty house. Now why do I believe this? Well, if I unravel my reasoning, it goes like this. First, today is Wednesday. On Wednesday nights, my wife and children normally go to church for dinner and service. Second, my wife likes to call me ahead of time if this plan changes. My cell phone is in my pocket, and has not rung, and so I conclude that the plan has not changed. I look at my watch, and it reads 5:17pm, which is after the time they normally leave, so I know I’m not going to catch them walking out the door. This is, roughly speaking, my thought process that justifies the conclusion that the house will be empty when I pull into the garage. Notice, however, that this prediction depends precariously on several facts. What if I spaced out the day of the week, and this is actually Thursday? All bets are off. What if my cell phone battery has run out of charge? Then perhaps she did try to call me but couldn’t reach me. What if I set my watch wrong and it’s actually 4:17pm? Etc. Just like a chain is only as strong as its weakest link, a whole proof falls apart if even one step isn’t reliable."