The chemist woke up, saw the fire, ran over to his desk, pulled out his CRC (chemistry handbook) and began working out all sorts of fluid dynamics equations. After a couple of minutes, he threw down his pencil, got a graduated cylinder out of his suitcase and measured out a precise amount of water. He threw it on the fire, extinguishing it, with not a drop wasted, and went back to sleep.
The engineer woke up, saw the fire, ran into the bathroom, turned on the faucets full-blast, flooding out the entire apartment, which put out the fire, and went back to sleep.
The mathematician woke up, saw the fire, ran over to his desk, began working through theorems, lemmas, hypotheses, you-name-it and after a few minutes, put down his pencil triumphantly and exclaimed, "I have proven that I can put the fire out!" He then went back to sleep.