Function: Physical metaphor

Many func­tions can be vi­su­al­ized as phys­i­cal mechanisms of wheels and gears, that take their in­puts on con­veyor belts, ma­nipu­late them (us­ing me­chan­i­cal sen­sors and tools), and pro­duce an out­put which is placed on an out­go­ing con­veyor belt.

A binary function

For ex­am­ple, we can vi­su­al­ize the func­tion \(+\) as a func­tion that takes two stacks of poker chips in as in­put (one on the left con­veyor belt and one on the right con­veyor belt) and pro­duces its out­put by dump­ing both stacks onto the out­put belt. We can vi­su­al­ize the func­tion \(\times\) as a func­tion that also gets a stack of poker chips on each in­put belt, and which puts a copy of the left stack onto the out­put tape for each poker chip in the right stack.

All com­putable func­tions can (in prin­ci­ple) be im­ple­mented by a phys­i­cal sys­tem of wheels and gears (or cir­cuits and wires, etc.), and it is con­jec­tured that the com­putable func­tions are the only func­tions that can be im­ple­mented by phys­i­cal sys­tems. (This con­jec­ture is known as the Church-Tur­ing the­sis.)