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I've noticed that sometimes people use ":=" to set variables, like "With $f(x):=x^{2}$, we have $f(1) = 1$." This is also the variable definition operation in Mathematica. My question is, did Mathematica come first and then people started using ":=" or vice versa? And when was the first documented use of such notation?

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    @Stefan: Agreed, but I'd say that by convention we read from left to right, so unless stated otherwise, assigned object appears on the right, while the label on the left. But I agree that $:=$ is better than $=$, since $=$ may lead to confusion - "is it something that follows obviously, or is it a definition?." Even when one precedes the definition by, say, "let us define...", the equation with $=$ appearing on the next line leads to confusion if it is referenced later in the text (and the reader doesn't immediately refer to the line above it, but only to the equation).2012-02-17

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Edsger Dijkstra, who probably thought more about notation than most of us ever will, and was old enough to maybe know for sure, attributes the mathematical community's use of it as a borrowing from ALGOL (see this link).

For what it's worth, Dijkstra preferred to use $:=$ for assignment (which is actually its use in ALGOL), and if I am reading that opinion right, did not see the need for a "definitional" equality symbol (probably because if you write proofs in his favored style, "definitional" equality is explicitly signaled in words between formulas).