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Many mathematical definitions have the following form using real numbers as an example:

A real number is called X iff it is integer and greater than $2$.

As we know definitions serve only as abbrevations. Could we define "X" in the following form:

For a real number X $ \stackrel{\text{def}}{=} integer \, \, and \, \, greater \, \, than \, \, 2 $.

This seems like a "dictionary" definition. Does this notation introduce any problem/confusion?

For example we can say that "$5$ is X" and if we substitute "X" we get the statement "$5$ is integer and greater than $2$" which is a true statement. Why it is the first form that is used if a definition serves only as an abbrevation?

Mauro ALLEGRANZA
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user599310
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  • The example is kind of poorly chosen. Usually you define a set first and then just call upon elements. That way you can differentiate between them. For example the set of solutions to a function. If you would call all the solutions by X the expression f(X)=f(X)=0 is unclear when compared to :for x,y in the set of solutions f(x)=f(y)=0. – MathR May 14 '21 at 23:50
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    The former is easier to read - remember, for the most part math is written by humans for humans. Adding more symbols doesn't always make things clearer; often, sticking to precise natural language is better. – Noah Schweber May 15 '21 at 00:03
  • Could we define "X" in the following form: [displayed sentence] Does this notation introduce any problem/confusion? --- A good way to see what some of the concerns are would be to try to diagram the displayed sentence. – Dave L. Renfro May 15 '21 at 05:25

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Well, you could say

Definition. A natural number is even if and only if it is a multiple of $2$.

You could also first introduce the set $E = \{2n \mid n \in {\Bbb N}\}$ and then say

Definition. An element of $E$ is called an even number.

I prefer the first version, but the second one might be used if you need to introduce the notation $E$ for other purpose.

That said, writing

Definition. For a natural number, even $\stackrel{\text{def}}{=}$ belongs to $E$.

should just be banished. It makes use of an unnecessary (and disputed) symbol, and it mixes mathematics and English in some awful way. If you really want to insist on the abbreviation aspect, you could use the first definition and add as a comment: "in other words, the term even can be seen as a shortcut for belongs to $E$".

J.-E. Pin
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