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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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The notation usually stands for "by definition". – Pedro Tamaroff Feb 17 '12 at 20:46
ALGOL 58 used the ":=" notation to set variables back in the late 1950s, so it clearly pre-dates Mathematica. (I do not claim this is the first major use of the notation.) – arjafi Feb 17 '12 at 20:51
I believe it was originally used in programming languages as a symbol for "is assigned as", but I have no documentation to justify the claim. – JavaMan Feb 17 '12 at 20:56
@Lazar: The sign you describe is symmetric, while := or =: allow to notationally distinguish that which is defined from the expression defining it. – Stefan Feb 17 '12 at 20:57
I read the question and for a split second I thought of posting an answer that says "I think it came from programming languages." So if someone asks about the Riemann hypothesis, and someone else responds with a complete proof in the comments section, I'll post an answer that says "I think that's a conjecture due to Bernhard Riemann." – Michael Hardy Feb 17 '12 at 21:59

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).

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