All this discussion of the different disciplines reminds me of a study I once read about ...
MIT once studied the differences in approach to problem solving by engineers, physicists and mathematicians, so they invited a sample group up to the lab and ran a simple experiment. The participants were to be admited to the experiment chamber one at a time and told to observe and take whatever action they deemed appropriate.
The Engineer was admitted first and upon entering the room noted a fire blazing in the middle of the carpeted floor, with flames shooting almost to the ceiling. There was a bucket of water a few feet from the fire. The Engineer sprang to the bucket and quickly dumped the whole thing on the fire, which went completely out, filling the room with steam and creating a considerable mess. The Engineer then exited the room and announced, "I've solved the problem."
The study group murmured among themselves, took copious notes, cleaned up the mess and reset the experiement. The Physicist was admitted next.
The Physicist walked in, observed the fire and water, walked quickly but carefully around both, then picked up the bucket and carefully poured about 3/4 of it on the fire, extinguishing all but a single ember, which was then put out with a single extra drop. This produced minimal steam, almost no mess and an incredibly smug grin on the Physicists face while exiting the room and announcing, "I've solved the problem."
The study group repeated the murmur, writing and resetting drill and showed the Mathematician to the door.
The Mathematician walked in, stopped. A slow walk around the fire ensued, followed by a repeat around the water. The Mathematician looked up at the ceiling, seemingly lost in thought for a few seconds, then looked back at the fire, said, "Yes! Of course!" strode briskly out the door and announced, "A solution exists."