Why is the word "calculus" used in this context?

The reason I ask is because these usages of "calculus" seem unrelated to the far more popular use of "calculus" in "differential calculus" and "integral calculus" to which something like λ-calculus seems only very distantly related. This word choice seems very misleading yet very popular among many authors. Why?

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    $\begingroup$ Calculi are "rules of calculating", whether it be derivatives or $\beta$ reductions. $\endgroup$ Dec 31 '16 at 9:05
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    $\begingroup$ Perhaps more appropriate for the math history site. $\endgroup$ Dec 31 '16 at 9:06
  • $\begingroup$ @YuvalFilmus Neat, I didn't know about that one, thanks. I guess we'll see how this does here and if it seems like it'll fare better there I'll ask for a migrate. $\endgroup$ Dec 31 '16 at 9:07
  • $\begingroup$ @YuvalFilmus "calculus" connotes more than that and words or phrases without those connotations could have been chosen. Anything could be called a calculus if it only meant "rules of calculating". $\endgroup$ Dec 31 '16 at 9:17

The word calculus comes from the Latin word for limestone, because limestone pebbles were used for counting; you may have a renal calculus: a kidney stone$^1$.

In mathematics, the word is used to describe any system in which symbolic expressions are manipulated by fixed rules$^1$, so we may speak of tensor calculus, vector calculus, predicate calculus, $\lambda$-calculus and integral and differential calculus$^2$. The latter two are related to the former in the sense that both are sets of rules for manipulating expressions. Running a Google Books ngram search on these words, approximately 1890 was the first year in which both "vector calculus" and "tensor calculus" were mentioned, and they have been in frequent use ever since.$^3$

The reason that to you, these systems seem to be using the wrong meaning of calculus is probably that you grew up associating calculus with differentiating and integrating real-valued functions, but this is a bias.


[1] Wiktionary page for Calculus

[2] Wikipedia Disambiguation page for Calculus

[3] Google Books search for integral, differential, vector, tensor calculus

  • $\begingroup$ In the past, would "calculus" in an semi-informal setting have to be qualified to specify which one was being discussed? $\endgroup$ Dec 31 '16 at 19:52
  • $\begingroup$ @Praxeolitic add "calculus" to the ngram search above; it occurs way more often than could be justified by being the sum of the qualified occurrences, but from this data we cannot rule out people thought it should be clear from context. You'll have to talk to a historian. For current-day usage, the Wikipedia disambiguation page can help you. $\endgroup$ Dec 31 '16 at 20:15

I posted the following in answer to the same question on CS Theory a couple of years ago:

A calculus is just a system of reasoning. One particular calculus (well, actually two closely related calculi: the differential calculus and the integral calculus) has become so widespread that it is just known as "calculus", as if it were the only one. But, as you have observed, there are other calculi, such as the lambda calculus, mu calculus, pi calculus, propositional calculus, predicate calculus, sequent calculus and Professor Calculus.


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