200 Are the Androids Dreaming Yet? The Game of Mathematics When I was a child, our living room carpet had a square pattern. You could use boiled sweets to play checkers on it. Even though there was no board and no pieces, it was clearly a game of checkers because we followed the right rules (with the one exception that if you jumped over a sweet you got to eat it). Mathematics is like a game with a set of rules. If you follow the rules, you are doing mathematics. Consider the simple mathematical theory that if A equals B, then B equals A. This seems clear-cut, but you can get into trouble if you're not careful when defining the word ‘equals. ‘My dog equals naughty’ does not imply ‘naughty equals my dog. Here I have used ‘equals’ to mean ‘has the property of? My dog has the property of being naughty. This is an attribute, not equivalence. You must be careful with mathematics. A equals B implying B equals A is a property of numbers when the equals sign is used to mean equivalence. Here are the rules of the game that provide a proof for this theory. Let us start with the position in which we don’t know whether A equals B implies B equals A. We have these three axioms, call them rules for now since we are using the game analogy. Rule 1: If] have no minus sign in front of a letter I can assume there is an invisible + sign there. Rule 2: If I have a positive letter (or a letter with no symbol in front of it) I can put a minus in front of it and put it on the other side of the equals sign. Rule 3: I can swap the plus and minus signs of all the letters in my equation if I do it to all of them. Now I am ready to prove my theorem. A = Bis the same as +A = + B. (rule 1) +A = + Bis the same as -B = - A (rule 2 done twice) -B = -A is the same as B = A (rule 3) Success. So A = Bis the same as B= A. I have my proof. It might be glaringly obvious, but that’s not the point. The point is you can apply rules to symbols and derive new rules. It does not matter what the sy