Difference between a 'calculus' and an 'algebra'

What is really the conceptual difference between a calculus and an algebra. Eg. Is SKI combinator calculus really a calculus?

A friend claims that free variables are fundamental for a calculus, and as such that SKI is not a calculus, but an algebra.

• This is an interesting question as I doubt many 'research mathematicians' really could write down a definition of what is a 'calculus' and what is an 'algebra' -- they just "know" what they are. And so these definitions are rarely given in (undergraduate) classes by same mathematicians, and so the next generation also does not know a proper definition, although they too pick up a general feeling as to what the topic is. Aug 26, 2010 at 13:41
• In some logics, a sentence is only considered to be well-formed if it is closed. Those logics however have the feel of a calculus still, not an algebra. I wonder if another relevant aspect is that in a calculus you have the notions of model theory ("semantics") versus proof theory. An algebra might be concrete or abstract. [These might just be further terms than need to be pinned down.] Although you might draw an analogy between concrete instances of abstract algebras, that still doesn't feel like semantics. Aug 26, 2010 at 14:58
• I'm pretty sure the words "calculus" and "algebra" have many different meanings within mathematics. First, there are the words that refer to standard undergraduate curriculum, and my first thought upon reading the question is that OP is asking about that distinction, whence the difference is mostly determined by historical accident. But words like "calculus" and "algebra" also have technical meanings within parts of logic. And researchers use even the nontechnical words subtly differently from undergrad curriculum. I recommend you revise the question to clarify which sense you mean. Aug 26, 2010 at 15:05
• you should change the title to something that reflects the question a bit better, like "difference between a calculus and an algebra" Aug 27, 2010 at 0:09

In logic, the terminology seems to have been influenced by two factors. The very early development of various deductive systems was done by people who were more philosophers than mathematicians and who seem to have used "calculus" to refer to anything that looked mathematical. Also, that development took place before "algebra" had acquired all of its current meanings.

My impression is that the use of "calculus" in logic is restricted to the meaning of "formal deductive system" --- and usually rather old systems. As for the SKI system of combinators, I would call it a calculus if you're talking about rules of inference. But if you mean the system of all combinators, with the operation of application, generated by S and K (I is redundant), then this is an algebra.

• This answers the question and yet somehow misses the point entirely. The term 'calculus' was coined when Latin was the language of all academics in Europe and was used in a very general sense. Sep 14, 2010 at 21:37

Webster's defines the primary definition of a calculus as follows:

a method of computation or calculation in a special notation (as of logic or symbolic logic)

Wiktionary gives a similar definition:

Any formal system in which symbolic expressions are manipulated according to fixed rules.

This agrees very much with the definitions I have encountered within mathematics. Free variables are not a requirement; indeed, even variables are not strictly required as objects within a calculus. (I am aware that there exist proof calculi that don't deal with the concept of variables; perhaps someone could give the name of such a one.)

• +1 Noldorin; I quite agree with all of this. I would consider the method of string diagrams (for calculations in monoidal categories) as a calculus, and also C.S. Peirce's Existential Graphs as calculi. These indeed circumvent explicit mention of variables and related apparatus like alpha conversion, even though it is straightforward to translate each of these calculi into a variable-based syntax. Of course, Tarski and Givant wrote a famous monograph in logic whereby variables are eliminated. Mar 28, 2011 at 12:43
• @Todd: Absolutely. And thank you for reminding me, I believe I was thinking of Tarski and Givant's formulation. Apr 30, 2011 at 22:19
• I always remind my students in Calculus I that the algorithm they learned in elementary school for long multiplication of natural numbers is also a calculus. Dec 9, 2011 at 0:25

Mathematics is an activity of investigation and exploration. Informally, both calculi and an algebras are tools which consist of sets of symbols and systems of rules (usually called axioms) for manipulating those symbols.

Calculi tend to be specified/defined/explored/used to answer questions of "calculation" or reckoning, in some very general sense. Calculi tend to be used to investigate properties of objects (i.e "What is the area under the curve?")

Algebras tend to be specified/defined/explored/used to answer questions about how different "things" are related, in some very general sense. Algebras tend to be used to study the relationship between objects. (i.e. "Is this equation 'the same' as that equation?")

I think it is safe to say that the term "algebra" today, carries a bit more meaning to most mathematicians than the general teram "calculus".

As examples:

The Calculus (as taught in high-school or undergraduate university), also known as "infinitesimal calculus", is a calculus focused on limits, functions, derivatives, integrals, and infinite series. It is chiefly concerned with calculations or answering questions about change. The Calculus uses the complex numbers (chiefly) as a foundation for this investigation.

Opening a book on computer science, you might find a "calculus of computation" which might involve symbols and rules which let one "calculate" or "discover" behavioral properties of a computer program. As a foundation, such a calculus might use "states" and "transitions", instead of the complex numbers, to ground the investigation.

Elementary Algebra (ie. high-school algebra) is, informally, the study of relationships of variables and structures (e.g. equations) arising from combining variables according to certain rules (i.e. performing "operations"). It uses the complex numbers as the basic foundation in which one could "check" or "verify" statements, but quickly one finds that "calculating with numbers" is not that useful (or practical) in investigating relationships between equations.

"The general theory of arithmetic operations is algebra: so we can also develop an algebra of set theory." - Concepts of Modern Mathematics, Ian Stewart

In that sense, Elementary Algebra is more "abstract" than arithmetic, and is often the subject where schools (specifically bad teachers) lose a student's interest and attention in mathematics. It is a tragedy, since it is exactly at Elementary Algebra that things get interesting.

In computer science or other engineering disciplines, you might find a "process algebra" when reasoning about how various states of a computer program relate to each other. We can ask questions like "is a specification of a collection of processes 'functionally equivalent' to another specification (i.e do they do the same thing? as in the case of a particular hardware design versus a software program)? The same "process algebra" could possibly be used to reason about how the various "states" of a garage door opener relate to each. Such an algebra might use states, transitions, and time as a foundation.

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• Relational calculus and Relational algebra are all involved querying databases. But calculus seems to have a very declarative standard library whereas the algebra is more focused on symbolic manipulation. The relational calculus reads like a natural language for end users, the relational algebra reads like a programming language for developers. What do you think? Mar 30, 2015 at 2:35
• I'm enjoying this post + answers enormously, arriving from the computing camp myself. I expected you lot would have a deep formal distinction between algebra and calculus and imagined: "a calculus is an algebra equipped with a reduction function leading to a fixed point" or some such :-) But no! It appears its all a bit historic and fairly arbitrary, which is an equally good answer. FWIW – Wikipedia page's title on the subject you mention is "Process Calculus" and begins "In computer science, the process calculi (or process algebras) are…"'. Jul 10, 2017 at 20:37
• @CMCDragonkai - Here's a quote from C. J. Date's Introduction to Database Systems: "Whereas the algebra provides a collection of explicit operators—join, union, project, etc.—that can be used to tell the system how to construct some desired relation from given relations, the calculus merely provides a notation for stating the definition of that desired relation in terms of those given relations." If we think of a calculus as being a formal system that defines the properties of an object, rather than its relationships to other objects, then the definition given here still applies. Jan 23, 2022 at 17:29

As I vaguely feel it, and trying to follow the ethimology, calculus is linked to computing, while algebra is linked to solving.

• The "propositional calculus" and "predicate calculus" might be considered to be exceptions to that, as areas of meta-mathematics. Aug 26, 2010 at 15:04
• Funnily, this makes quite unclear what numerical analysis is, since it is essentially "computing solutions". :) May 12, 2011 at 19:23

a calculus is a symbolic system for computation where computation can most generally be seen to be a spatial reorganization of symbols. any kind of numerical computation can be described in terms of recursion which can fundamentally be seen as a symbolic manipulation process. logic calculi and the calculus that set theory uses is also describable as a symbolic computation system.

an algebra is a mathematical structure in the informal sense which turns out to be a vector space with the added ability to multiply the actual vectors together. the complex numbers with addition and multiplication is an algebra over the complex numbers; http://en.wikipedia.org/wiki/Algebra_over_a_field

• You started off pretty well in describing a "calculus", but that is too restricted a reading of the term "algebra" for the purpose of this question. The word is used in other ways, e.g., combinatory algebra (as vs. $\lambda$-calculus) or relational algebra (as vs. calculus of relations). OP wants to know what guides the choice of terminology in such closely related situations as these. Mar 24, 2013 at 12:05
• Here you mixed two different meanings of "algebra" - a part of mathematics and a mathematical structure. Oct 12, 2016 at 0:33

In an algebra we would work within a known system and the results tend to be within those domains of interest and nothing new is found. In calculus the situation is different and we would encounter new things other than those we have been dealing. For example if we are working with one kind of curves and think about an operation like an integration or a differentiation we would be taken up/down another system. There is always a scope to see a new thing in a calculus and never a new thing in algebra. Whether it is a conventional mathematical system or relational algebra/calculus or logic (first order and more) we can observe this tendency.