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) iffx.[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.

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