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[Solved]: What are the justifications and historical reasons regarding the choice between the words 'calculus' and 'algebra'?

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Problem Detail: 

The principles of calculus, historically, are differentials and integrals [1], while those of algebra are operators and equation solving [2]. Contemporary principles are analysis and abstract objects, respectively.

As an example case, why is relational algebra not called a calculus, and why is the π-calculus not called an algebra?

Asked By : Alexis Petrounias

Answered By : babou

What is in a name? Calculus is called analysis in some languages other than English, while the word calculus itself means computation.

The name π-calculus was most likely chosen by Milner because it has to do with computation, and is intended to be for parallel computation similar to what $\lambda$-calculus is for classical sequential computation. Computation is actually the original meaning of calculus, referring to pebbles used as counters in ancient time.

The word algorithm, as everyone knows, comes from the name al-Khwarizmi (the man from Khiva, a city formerly called Khwarezm), given to the 9th century Persian mathematician Abū Ja'far Muhammad ibn Mūsa. Algebra comes from the name of the treatise Hisab al-jabr wa'l muqabala, that he wrote about the resolution of equations.

Al-Khwarizmi systematized the study of equations (algebra), and gave procedural techniques to solve them (algorithms), which implies some form of computation (calculus). Could this be seen as a kind of Curry-Howard situation, where the mathematics for proving go in hand with the corresponding algorithms to actually compute ?

Names may be chosen for strange reasons, and their meaning evolves along with the objects they initially denote. The internationalization of sciences also leads to different interpretations of words, as the same word (or its local variant) may have a different meaning in different country (and that is also true is other areas of language, possibly creating some awkward situations).

This issue has already been discussed on math.SE.

Further remarks

Actually, according to wikipedia, Analysis is used in all languages, English included, with the same meaning. English appears to use the word Calculus which refers only to elementary concepts of Mathematical Analysis, differential and integral calculus. If you look for the English Calculus in wikipedia, you find out that translation is missing for many languages (no equivalent entry in German or French for example), and when an entry is proposed for another language it may corresponds to a different meaning (Cálculo in Spanish is for calculation or computation, as it is in some other languages).

However, Cálculo in Portuguese has the same meaning as in English and covers the same topics. They actually explain, for Portuguese, the origin of this use of the word, and the explanation is likely to be the same in English. Calculus is very simply an abbreviation for "differential and integral calculus". And indeed, it corresponds to computations expressed with algorithms, and is, in this sense, close to Algebra.

I was mislead, in understanding the question and writing the first part of my answer, because I took Calculus to mean the whole of Mathematical Analysis, as there is no single word in my own language to cover specifically differential and integral calculus. Comments show that I am not the only non-american user to be thus mislead.

Hence there is no linguistic inconsistency. The name λ-calculus denotes a formalisation of algorithms and computation, and the use of the word calculus is adequate. From it was derived the name π-calculus for parallel computations.

Note that initially, calculus just means computation (calculation has a connotation as pertaining to numbers). When applied to a specific domain, it is qualified so that the domain is explicit (integral calculus, π-calculus, ...). The problem comes from the fact that it has a common use in some languages, including English and Portuguese, where it is domain specific without making the domain explicit, while also retaining its more general use.

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