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Where does the choice of the Greek letter $\lambda$ in the name of “lambda calculus” come from? Why isn't it, for example, “rho calculus”?

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    $\begingroup$ So that in Haskell, the symbol could easily be replaced with `\`! $\endgroup$ – leftaroundabout Sep 14 '11 at 18:21
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    $\begingroup$ Not really relevant to the question, but: Anthony Morse's book "A Theory of Sets" is written in an extremely formalistic style, so I doubt that many people read enough of it to encounter the joke (embedded as part of the formalism) suggesting that the $\lambda$-abstraction operator be read as "lonzo". $\endgroup$ – Andreas Blass Dec 19 '13 at 15:13
  • $\begingroup$ Not for nothing, but $\lambda$ is also the most fun Greek lower-case character to write. $\endgroup$ – Emily Feb 22 '15 at 4:39
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    $\begingroup$ Another source claims that $\lambda$ was a "random choice" $\endgroup$ – Omnomnomnom May 16 '16 at 17:23
  • $\begingroup$ I wish I could upvote this question at $≥ 10$ times. I was drawn here after pondering the name "lambda function" in programming. $\endgroup$ – Jonathan Komar May 8 at 11:41
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The symbol “λ” is used for one of two basic constructions in the system introduced by Alonzo Church, specifically abstraction. The notation did not just happen to be chosen but was to distinguish it from another construction by Whitehead and Russell represented as “xˆ.” For his new system, Church initially used “∧x,” then replaced it to “λx” to ease printing, obviously, interpreting the former logical symbol as the capital Greek letter “Λ.”

See “History of λ-calculus and Combinatory Logic” by J. R. Hindley, F. Cardone (Handbook of the History of Logic, 5: 723–817, Elsevier, 2009).

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    $\begingroup$ Here's a working link version, curtesy of Hindley himself $\endgroup$ – Sarah Apr 11 '15 at 16:46
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    $\begingroup$ Dana Scott denies this - see Hisham H M's answer and this question on CS. $\endgroup$ – reinierpost Nov 6 '17 at 13:42
  • $\begingroup$ By why is it a calculus? It has nothing to do with infinitesimal change. $\endgroup$ – Jordan Mar 22 '18 at 15:23
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    $\begingroup$ A calculus is simply a formal, mechanical system for calculation. The word "calculus" literally means "stone" — imagine someone using stones to calculate, abacus-style. So differential calculus is a formal system for calculating derivatives, and integral calculus is a system for calculating integrals. $\endgroup$ – andru Dec 10 '18 at 21:57
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I heard that Church originally used the hat symbol above the bounded variable, like $\hat x.x$, in handwritten papers. Then, the notation became ^x.x because of old-fashioned typewriters. I think it first became $\Lambda x.x$ when more recent text processors appeared, and $\lambda x.x$ after that (for aesthetics reasons probably).

Thanks to the silent downvoter, I did some googling and found Barendregt version in “The Impact of the Lambda Calculus in Logic and Computer Science”:

We end this introduction by telling what seems to be the story how the letter 'λ' was chosen to denote function abstraction. In [100] Principia Mathematica the notation for the function $f$ with $f(x) = 2x + 1$ is $2 \hat x + 1$. Church originally intended to use the notation $\hat x.2x + 1$. The typesetter could not position the hat on top of the $x$ and placed it in front of it, resulting in $\wedge x.2x + 1$. Then another typesetter changed it into $\lambda x.2x + 1$.

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    $\begingroup$ The reference [100] belongs to Whitehead and Russell, not Church. The “History of λ-calculus and Combinatory Logic” paper distinguishes these two constructions, saying explicitly that Church had to introduce another notation. $\endgroup$ – user13300 Sep 16 '11 at 5:14
  • $\begingroup$ Indeed, meta-lambda abstraction is denoted another way, and is not the same as that of the system by Church. $\endgroup$ – user13300 Sep 16 '11 at 5:16
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    $\begingroup$ That reminds me of a section in Littlewood's "Mathematician's Miscellany" (now available as "Littlewood's Miscellany" and highly recommended) where he wrote of a paper he wrote with Hardy which had this as its last phrase: "where $\sigma$ must be made as small as possible." The phrase did not appear in the paper, but, upon examination with a magnifying glass, there was an extremely small "$\sigma$" there. Littlewood wrote that the printer must have scoured London to find such a small $\sigma$. $\endgroup$ – marty cohen Feb 22 '15 at 6:06
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    $\begingroup$ Dana Scott, Church's student, recently specifically debunked this version and asked people to stop spreading it. $\endgroup$ – Hisham H M Jan 13 '17 at 4:11
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Dana Scott, who was a PhD student of Church, addressed this question. He said that, in Church's words, the reasoning was "eeny, meeny, miny, moe" — in other words, an arbitrary choice for no reason. He specifically debunked Barendregt's version in a recent talk at the University of Birmingham.

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$\lambda$ symbol has (at least) 2 (related) meanings in mathematics (and logic), stemming directly from the ancient greek texts.

Its meaning stems form the fact that is the initial letter of the word ratio (greek: λόγος), meaning both analogy (i.e ratio) and logic (i.e rationality/reasoning)

Presumably this could be the reason Church used that symbol.

Furthermore, there is indeed a $\rho$-calculus which combines (or generalises) $\lambda$-calculus with pattern matching rewritting systems

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