11
$\begingroup$

The symbol $\triangleq$ is sometimes used in mathematics (and physics) for a definition. It is instantiated for instance in the Unicode Character 'DELTA EQUAL TO' (U+225C). The notation $t \triangleq m$ (often) means: "$t$ is defined to be $m$" or "$t$ is equal by definition to $m$" (often under certain conditions). In a similar sense, some uses $:=$ or $=:$ (see for instance Symbols based on equality). The SE. Maths post What is meant by the delta equivalent sign? proposes a slight distinction (not crystal-clear to me) between the above similar senses:

Sometimes it is used with the slightly different meaning of "equal by definition", to underline the difference w.r.t. "$:=$ " which is the definition itself. i.e.

$$ a:=3;\\ 5+a \triangleq 5 + 3 = 8 $$

I always took for granted that the $\Delta$ stood for "D", i.e. for the initial of "definition". Indeed, one sometimes finds: $\overset{\mathrm{def}}{=}$ too. Now I am in doubt.

  • Who introduced this dual symbol first, and where?
  • What motivated the Greek $\Delta$ notation? The abbreviation of some word, a symbol?
  • Why not merge the lower bar of the Delta with the upper bar of the equal sign, to save some ink, and create a lighter symbol?

References: the symbol itself was already discussed in StackExchange:

$\endgroup$
  • 5
    $\begingroup$ This notation seems very strange, as it suggests symmetry between the definiens and the definiendum. I am more used to := or =:, with the colon on the side of the definiendum. $\endgroup$ – Margaret Friedland Sep 5 '16 at 1:45
  • 2
    $\begingroup$ Mathsym jeff560.tripod.com/mathsym.html has only $=_{\mathrm{Def}}$ (from 1894) but not the Delta version. $\endgroup$ – Gerald Edgar Sep 5 '16 at 13:44
  • 1
    $\begingroup$ @MargaretFriedland The asymmetry is given by what is before the symbol and what is after the symbol. Thus $A \triangleq B$ always means "A is defined to be B". $\endgroup$ – juanrga Mar 19 '18 at 19:42
  • 2
    $\begingroup$ @MargaretFriedland, I am sympathetic to your objection, and, in fact, would go so far as to object somewhat to the $:=$ usage, since it lends itself to typos so easily. Rather, I'd strongly prefer that the context clarify any asymmetry (such as a definition or notational abbreviation) in a mathematical equality. (Mathematicians seem unlikely to take these asymmetries as seriously as programmers may, so I don't necessarily trust them!) $\endgroup$ – paul garrett Jun 28 '18 at 23:43

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.