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Where Do Functions Come From?

The study of calculus commonly begins with the question, "What is the instantaneous velocity of a falling body?" It's seldom noted that 2,000 years elapsed between the time Aristotle began to study velocities of falling objects and Newton's discovery that the derivative of position yields velocity. This paper examines the reasons for this long delay. Part of the explanation is, of course, that functions come before derivatives, so we also consider the question asked in the title. This study is an interesting thread in the history of mathematics, which throws into relief the importance of functions in the development of modem mathematics, something calculus students often fail to appreciate. It is conventional, but in the present case wholly accurate, to say that this brief exposition will merely scratch the surface of the subject. Suggestions for further reading, themselves containing many more suggestions, can be found in the references.