What are the justifications and historical reasons regarding the choice between the words 'calculus' and 'algebra'?

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?

• Sometimes, names just fall out of the sky and stick. Why "dynamic programming"? (Answer: marketing trick) Why "red-black trees"? – Raphael Feb 28 '14 at 12:08
• @AlexisPetrounias, as far as I can tell, a calculus is supposed to refer to a system of rules/algorithms for symbolic manipulations, whereas an algebra is supposed to refer to a structure imposed on data (unless it is just an algebra). So I don't think we can draw a universal distinction because in a calculus the objects of computation must be represented somehow and in an algebra the manipulation of the data must be subject to rules. So it would just be a question of where the emphasis lies. There's some history on the math.se post. – usul Mar 1 '14 at 15:50
• I have deleted some comments here. Please take discussion regarding this question to the CS chat rooms and/or Meta site to discuss the site's policy and/or the closing of this question. If you can find a way to incorporate feedback on this question such that you feel it becomes appropriate, pleas feel free to flag the question for moderator attention. Thanks for your cooperation and for contributing to the site! – Patrick87 Mar 1 '14 at 19:15
• voting to reopen at raphaels suggestion in chat! its maybe not the greatest question in the world (which ones are?) but the question is clearly on topic about terminology of computer science! but maybe the title is misleading & closers thought its more a math question? some significant amt of terminology of CS is borrowed from math of course.... my analogy/metaphor has been something like "algebra is checkers, calculus is chess"... imho FSM/regular languages are at times something like the "algebra" of CS – vzn Mar 3 '14 at 16:05
• Cheers @vzn, what do you suggest the title be changed to? On my part there is absolutely no uncertainty on the mathematical meaning of these terms. I asked this question because, for the past few years, every time I speak with physicists and theoretical mathematicians about computing they are confused when we call something a calculus as opposed to an algebra. I have not been able to offer a compelling answer for how these terms transfer from mathematics to computing, and how we chose to call one model a calculus and another an algebra. – Alexis Petrounias Mar 3 '14 at 20:53

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.

• Do you happen to know of any languages which use the equivalent of 'analysis' for the English word 'calculus' which would also translate, say, π-calculus into the corresponding π-'analysis' in that language? From the Wikipedia entry for π-calculus all translations currently available use the equivalent of 'calculus' not 'analysis' (e.g. Cálculo pi in French, Pi-Kalkül in Dutch, Пи-исчисление in Russian, and so on). – Alexis Petrounias Feb 28 '14 at 8:49
• In German "Analysis" means "calculus". Note that "analysis" is "Analyse" in German so there is no ambiguity in that regard. – Raphael Feb 28 '14 at 12:11
• Al-Khwarizmi wasn't Arab, and his name doesn't mean the man from Khiva. Rather, he hails from Khwarizm, nowadays known as Khiva, a city in Uzbekistan, and had (apparently) Persian ethnicity. – Yuval Filmus Feb 28 '14 at 16:01
• Italian, for instance, makes the analisi/Calcolo distinction, so it's not just English. Also, Analysis can be used similarly in English. My point is that I don't see this as a very useful point... – Patrick87 Feb 28 '14 at 23:53
• Not to mention that calculus is also used to mean calculation/computation in everyday parlance, as in "that consideration did not enter into my calculus." – Patrick87 Mar 1 '14 at 0:09