In category theory, one sometimes denotes unique
arrows by ! in a commutative diagram.
In general two objects in a category may
have any number of morphisms between them
(including none). In most cases a diagram only highlights
a relationship between some arrows in the category. We
don't care about the specifics.
However certain categorical constructions (eg,
products — essential for Cartesian Closed Categories) require a unique arrow as part of the
definition. To call attention to the uniqueness of
the arrow, it's denoted by an exclamation mark.
Similarly, when reasoning formally,
mathematicians will write ∃!x. P(x) as a
shorthand for "There exists a unique
x such that P is true of it".
This is quite a powerful statement: in your entire universe of logical discourse there
is exactly one thing which has the given property.
Of course often mathematicians only care about uniqueness upto isomorphism. That is,
there might be another thing y with the
property, but you have a canonical way of converting
x to y and y to x, such that the two objects are equivalent.
The ∃! notation is not primitive, in fact,
you can define it (in an equational first order logic) as a kind of abbreviation:
∃!x.P(x)
iff
∃x.[P(x) ∧
∀y.P(y) ⇒
x=y]
Actually, philosophers (at least ones who work
on the foundations of mathematics) worry quite a bit
about uniqueness, equality and what you take as
a primitive notion. For example,
one could define equality by
x=y iff for all properties P, P(x) ⇔ P(y)
This is a meta-logical statement: you're quantifying
over all propositions in your logic. Essentially you've
just said that two objects are equal if inside the logic
you can not distinguish them. Which means that for all
intents and purposes, the object is unique within the
logic.
Most mathematicians are not troubled by such things.