Is there a deep reason why math books use the phrase "is called" in definitions?

Question

I am trying to struggle through some basic math textbooks and I noticed that in definitions they usually use the following phrasing: "A function $f:X \to Y$ is called one-to-one if..." I was wondering if there was a deep reason why they don't say "A function $f:X \to Y$ is one-to-one if..." or if it could be said either way.

Answer 1

It's to distinguish the act of introducing a name from stating a claim.

If I write

A foo is blah iff it gleens,

it's unclear whether I'm defining the notion of a "blah foo" or if I'm claiming that I can prove that the blah foos are exactly the ones which gleen. By contrast, if I write

A foo is called blah iff it gleens,

it's clear that what I mean is that I'm introducing the term "blah" here. Basically, this is letting the reader know that they haven't missed the definition of "$blah$" earlier in the text, and that I'm not tacitly assuming that they already know what "blah" means. Note that in mathematical writing, we often do state "immediate" observations without proof (e.g. "A function is injective iff it has a left inverse"), so confusion is definitely possible here.

(Unfortunately this isn't universally used.)

Answer 2

"is called" is a statement about phrases and meaning. "is" is a statement about actual existence.

To state: "A function, $f:X\to Y$ is one-to-one if for any $x\in X; y\in Y$ so that $f(x)=y$ then there is no other $z \in X; z \ne x$ so that $f(z) = f(x)$" when "one-to-one" has not been defined is meaningless because "one-to-one" is meaningless (at this time).

To state: "A function, $f:X\to Y$ is one-to-one if for any $x\in X; y\in Y$ so that $f(x)=y$ then there is no other $z \in X; z \ne x$ so that $f(z) = f(x)$" when "one-to-one" has been defined is redundant because that is what one-to-one means so this statement is pointless and repetitive.

To define what "one-to-one" means the first time we must make a statement that says: "We will use a phrase $X$ to mean this particular condition $C$". But this is a statement about the phrase "one-to-one" and declaring its meaning. It isn't actually declaring anything about the universe that isn't already known (merely that will will be referring to a concept by a specific term.)