# Why is it called lambda "calculus"?

Why is lambda calculus a calculus when it has nothing to do with infinitesimal change? Same question arises with relational calculus. These aren't calculi, are they?

• Calculus is a system of reasoning, not necessarily differential & integral calculus. Think 'calculate'. Mar 22, 2018 at 20:04
• Calculus from Latin: computation. Thus, in the past there were already "calculus" (plural: calculi) before "the" calculus. Mar 23, 2018 at 7:00

## 3 Answers

This is a good opportunity for using a dictionary. The Oxford dictionary (used in Apple's Dictionary app, which here gives the same result as "calculus" at Oxford Dictionaries online) indicates "calculus" comes from the latin calculus, meaning "small pebble (as used on an abacus)." Presumably that's what was in mind when the term was used for what we most commonly call "calculus," which Oxford also notes is more specifically called infinitesimal calculus.

However, Oxford also notes that the term is used more widely in mathematics and logic, as "a particular method or system of calculation or reasoning." It's in this latter sense that you see it used in "lambda calculus," "propositional calculus," "relational calculus," "probability calculus," "umbral calculus," etc..

See mathematics - Who first used the word "calculus", and what did it describe? - History of Science and Mathematics Stack Exchange for more on the history of usage of "calculus."

Your use of "calculus" is actually an abbreviation for "infinitesimal calculus" — the art of doing 'calculation' with infinitesimals.

The term "calculus" is used for a number of different frameworks for doing 'calculation'; it just so happens that infinitesimal calculus is the most widely known example.

• Widely known? It was also the original was it not? Mar 22, 2018 at 20:14

I would rather ask why we say calculus to refer almost exclusevely to infinitesimal calculus.

Around the first years of the 20th century, many mathematicians were interested in answering some questions regarding mechanisms of calculation. They wonder if it was possible to have a mechanism capable of proving all theorems to be either right or wrong, avoiding the inescapable likelihood of human mistakes. This is known as the Entscheidungsproblem.

Before answering to that question, it was mandatory to define a formal definition of mechanism of calculation, a definition of algorithm.

That's when the Lambda Calculus comes into play. Both Lambda Calculus and Turing Machines were accepted to be the model of all that is effectively calculable. So I can't think of a better place to use that word.