2.4: What is a Proof?

People often argue in order to prove something. But that word “proof” is a tricky word. There are different standards of proof in different situations. You have to meet a higher standard if you are proving a new theorem in mathematics than if you are proving to your neighbor that you saw the same film he did last week. Basically, though, a proof is a convincing argument, an argument that should convince your audience, not simply an argument that does convince them.

Exercise \(\PageIndex{1}\)

AdSuppose you cannot locate that favorite blue shirt you want to wear. You’ve looked in the closet where you usually keep your shirts. You remember washing it at the Laundromat in your apartment building last week. Maybe you hung it back in the closet after that, or maybe you didn’t. You can’t remember. You don’t remember any other time it has been out of the apartment recently. Could you be having a memory problem? You do remember your worst case of bad memory; last year you were sure your apartment key was on the kitchen table, but then you found it an hour later on a shelf in your refrigerator. But after thinking about this you decide that is very unlikely the shirt loss is because of memory failure. You decide to do a more careful search. You look through each item of clothing in your closet, on the closet floor, and in the drawers in your dresser where you place other clothes. You look a few more places in your apartment. Then you remember that occasionally you hang clothes in the closet on top of other clothes hanging there because you don’t have enough coat hangers. So, you search your closet one more time looking under everything hanging there. Still no shirt. So you conclude, “This proves the shirt was stolen.” You start thinking about your three friends who have been in your apartment since the last time you saw that blue shirt. David was there when you went out for an hour to get party supplies. The shirt would fit him. That proves the shirt was stolen.

A logical reasoner hearing this story might say, “That’s not really a proof,” and this judgment would be correct. What else would it take for you to have a real proof the shirt was stolen by David?.

Answer

It’s more likely you lost your shirt in the Laundromat than to a thief in your apartment. You can’t have a proof without being sure that the shirt wasn’t lost at the Laundromat or on your travels back from there. If you could rule this out, then you’d have a stronger case that it was stolen. Even so, that evidence about the Laundromat is not going to be available to you. Also, for a decent proof you’d need some more direct evidence of a thief, such as a friend telling you he saw David wearing it yesterday, or a neighbor telling you she noticed someone leaving your apartment yesterday carrying a blue shirt. Most probably you’ll never get a proof your shirt was stolen even if it was, because having a proof requires having a totally convincing case.