Argument
Commutative property
Overview
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Perhaps most familiar as a property of arithmetic, e.g. "3 + 4 = 4 + 3" or "2 × 5 = 5 × 2", the property can also be used in more advanced settings. The name is needed because there are operations, such as division and subtraction, that do not have it (for example, "3 − 5 ≠ 5 − 3"); such operations are not commutative, and so are referred to as noncommutative operations.
The idea that simple operations, such as the multiplication and addition of numbers, are commutative was for many centuries implicitly assumed. Thus, this property was not named until the 19th century, when new algebraic structures started to be studied.
Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).
Possible passages
Found by search and not yet checked by a person; they may only mention the idea in passing.
It is obvious, however, that any hereditary property possessed by Adam must belong to all men; and similarly any hereditary property possessed by 0 must belong to all finite numbers. This is the principle of what is called "mathematical induction." It frequently happens, when we wish to prove that all finite numbers have some property, that we have first to prove that 0 has the property, and then that the property…
Bertrand Russell, Our Knowledge of the External World as a Field for Scientific Method in Philosophy
States what Commutative property is.
In respect to any complex eternal object, _S(D1, D2, ... Dn)_, the eternal objects _D1, ... Dn_, whose individual essences are constitutive of the individual essence of _S(D1, ... Dn)_, are called the components of _S(D1, ... Dn)_. It is obvious that the grade of complexity to be ascribed to _S(D1, ... Dn)_ is to be taken as one above the highest grade of complexity to be found among its components.
Alfred North Whitehead, Science and the modern world
States what Commutative property is.
But most frequently we consider only the cases in which the functions are such as are called _algebraic_, and to which the idea of _degree_ is applicable. In this case we can give more precision to the general proposition by determining the analytical character which must be necessarily presented by the equation, in order that this property may be verified. It is easy to see, then, that, by the modification just…
Auguste Comte, The philosophy of mathematics
States what Commutative property is.
Related ideas
- Integer
- Logical conjunction
- Mathematics
- Group (mathematics)
- Rational number
- Real number
- Natural number
- Intersection (set theory)
- Euclid's Elements
- Truth table
- Set (mathematics)
- Union (set theory)
- Complex number
- Royal Society of Edinburgh
- Truth function
- Egypt
- Mathematical proof
- Euclid
- Function composition
Place in the tradition
Part of Propositional logic.