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existence





A term that has several different meanings within mathematics. In the broadest sense there is the question of what it means for certain concepts, such as pi, to exist. Was π, for example, invented or discovered? In other words, does π exist only as an intellectual construct or was it somehow already "out there" waiting for people to find it. If it does exist independently of the human mind, when did its existence start? Does π predate the physical universe?

Such ontological questions become even more difficult when applied to more complex or abstract mathematical concepts such as the Mandelbrot set, surreal numbers, or infinity. A narrower and more technical type of "existence" in math is implied by an existence theorem. Such a theorem is used to prove that a number or other object with particular properties definitely exists, but does not necessarily give a specific example. Finally, there is existence in the sense of particular solutions to problems. If at least one solution can be determined for a given problem, a solution to that problem is said to exist. Something of the flavor of all three types of mathematical existence mentioned here are captured in the following anecdote:
An engineer, a chemist, and a mathematician are staying in three adjoining cabins at an old motel. First the engineer's coffee-maker catches fire. He smells the smoke, wakes up, unplugs the coffee maker, throws it out the window, and goes back to sleep. Later that night the chemist smells smoke too. He wakes up and sees that a cigarette butt has set the trash can on fire. He thinks to himself, "How does one put out a fire? One can reduce the temperature of the fuel below the flash point, isolate the burning material from oxygen, or both. This could be accomplished by applying water." So he picks up the trash can, puts it in the shower stall, turns on the water, and, when the fire is out, goes back to sleep. The mathematician has been watching all this out the window. So later, when he finds that his pipe ashes have set the bed sheet on fire, he's not in the least taken aback. "Aha!" he says, "A solution exists!" and goes back to sleep.

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   • LOGIC