# Why is the word “the” used before different categories of calculus?

I've noticed that, when referring to certain branches of calculus, mathematicians sometimes precede the name of those branches with the word "the". For example, "the lambda calculus" or "the predicate calculus".

To be fair, a Google search turns up plenty of references to "lambda calculus" (without "the" as a prefix). But it also turns up an equally large number of references to the phrase with the prefix included. Further, I haven't noticed this with other branches of math. For example, I haven't heard anyone use the phrase "the linear algebra" or "the Euclidean geometry". They just say "linear algebra" or "Euclidean geometry".

My question is, why the difference?

EDIT:

As requested by @Lawrence in the comments, here are some examples of full sentences where "the" is used before "calculus":

1) "The λ-calculus can be called the smallest universal programming language in the world. The λ-calculus consists of a single transformation rule (variable substitution, also called β-conversion) and a single function definition scheme. It was introduced in the 1930s by Alonzo Church as a way of formalizing the concept of effective computability." Source- https://arxiv.org/pdf/1503.09060.pdf

2) "Church (1936) invented a formal system called the lambda calculus and defined the notion of computable function via this system." Source- http://www.cse.chalmers.se/research/group/logic/TypesSS05/Extra/geuvers.pdf

3) "The Lambda calculus is an abstract mathematical theory of computation, involving functions. The lambda calculus can be thought of as the theoretical foundation of functional programming." Source- https://brilliant.org/wiki/lambda-calculus/

In response to @Trevor's comment, we can replace "Euclidean geometry" with "boolean algebra", meaning we're now comparing two types of algebra (linear and boolean). I haven't heard either of these genres of algebra used with "the" as a prefix, yet they're both 2 sub-branches of the same branch of mathematics.

I'm able to find the phrase "the linear algebra" as part of a broader phrase (i.e. "The Linear Algebra Survival Guide" or "The Linear Algebra Behind Search Engines"), but the usage of the phrase in these examples is different from the usage in the examples I mentioned in response to Lawrence's comment.

• It would help for you to quote one or two full-sentence examples in your question so that we can consider the context that influenced the choice (of whether to add or omit the) in each case. – Lawrence Oct 25 '18 at 4:27
• Your first examples are both variants of 'calculus', so they might be comparing usages. Your later examples are from different fields, algebra and geometry. Consider we have some books. When we talk about books, we need no articles. But once inside our book collection we might switch to referring to them as 'use the red book' and 'use the blue book'. So the might be indicating the same: 'blah blah and we solve this using the Lambda calculus blah blah'. – Trevor Christopher Butcher Oct 25 '18 at 7:10
• The thing is, the usage of "the" before "lambda calculus" is not limited to one author. I've encountered it repeatedly in my studies of functional programming and it's always bugged me. I never had the motivation to delve into it until now though lol. – Richie Thomas Oct 26 '18 at 18:03
• I don't know enough to expand this into a full answer, hence just a comment: "calculus" was first used as a generic term for "method of calculation". Only later (and nowadays) is the word used specifically to refer to integral and differential calculus, as taught in schools. So, I believe the use of "the" is a reminder of the earlier, more general usage of the word. Writing "the lambda calculus" reminds the reader that we're referring to a specific set of concepts, a specific method. Of course, writing "lambda calculus" would not necessarily imply anything different, but this is my guess. – Brendan W. Sullivan Oct 30 '18 at 20:36
• I have no direct information on the naming here, but whether to use "the" is arbitrary in other areas: countries and cities (the Ukraine, the Hague, but Russia, Belgium); diseases (he had the flu; he had dysentery). – Maverick Nov 8 '18 at 14:39