# Confusion about articles before unique objects in math

I am a graduate math student, English is my second language but we have some courses taught in English.

I often see sentences like `There exists *a* unique map/morphism f such that...` (for example, when we define some objects by their universal property)

I am quite puzzled by this. We are talking about something unique, there is one and only one map that has desired properties. So based on what I was taught about articles, I would say that the should be used. However, it seems customary to write a and I must admit that when reading aloud, saying it with the sounds funny.

So is this a just a common mistake? And if it is correct, what is the justification for it?

• In general, you don't use "the" until you have identified the thing and have it at least conceptually within your grasp. "I know there's a pair of pliers somewhere in this box. ... Ah! Here's the pair of pliers!" Commented Apr 13, 2021 at 12:28
• Not a mistake. There are many morphisms such that blah. But there also is a unique morphism such that blah. Since the textbook is describing a (not-yet fully described) morphism, it uses the indefinite article, a. Commented Apr 13, 2021 at 12:44
• The same thing is done for mundane purposes as well. For instance recipe might say "break a medium egg into a bowl and beat it well". At this point any medium egg will do so the indefinite article is used. The recipe might then say "Add the egg to the flour and mix it in". At this point only the egg which had previously been selected and beaten will do and all the other eggs in the box, shop, farm or world are ignored so we use the definite article 'the'. Commented Apr 13, 2021 at 15:05

Using a is not a mistake but the beginning of the full description.

It is common to speak of the object with a and then go into the description which identifies it as unique. Then it is described as the or this mapping to include the details of the description.

If you were to use the to describe it from very the beginning it would be assumed that the description as unique has been delayed or made elsewhere.

• Yep, this is just standard English usage. "On top of the hill there is a castle. In the castle is a keep" etc. Commented Apr 14, 2021 at 9:40

When a mathematician says that There exists a `B` they know nothing else about `B` except that `B` must exist (in the appropriate mathematical sense of exist, which I won't go into here). Existence and non-existence can be proved without definiteness.

The appropriate article for something that has just entered the discourse without any information except its existence is the indefinite article. It's used for initial introductions.

• A man dressed in black entered the room.
• I saw a strange-looking dog this morning.
• Every field must have an additive identity.

Once an entity has been introduced to the discourse, it can be referenced with the definite article.

• The man in black looked around and left.
• I'd never seen the dog before.
• The additive identity must have certain properties.

Logically, the existential quantifier (, pronounced there exists or for some) is indefinite, while the universal quantifier (, pronounced for each, for every, or for all) is definite, though not necessarily presupposed to exist.

• Mostly agreed, but I don’t follow your claim that “the universal quantifier […] is definite”. In mathematical usage, at least to my ear, objects bound by universal quantifiers are only referred to by “the” if they’re definite for some other reason — e.g. they’re unique-if-they-exist, or a specific one has been previously introduced. On the other hand, universal quantification can be implicitly introduced by an indefinite article: “A complex polynomial can be factored…”, although it’s more common (and usually recommended) to be more explicit, eg. “Any complex polynomial…”
– PLL
Commented Apr 13, 2021 at 22:53
• Additional point: Before switching from "a" to "the", especially in mathematical examples such as the third one above, one should verify that the item referenced is unique. Commented Apr 14, 2021 at 12:36
• @AriBrodsky: I would not go quite that far. Grammatically, it's perfectly fine to refer to a previously mentioned thing using "the" even if the thing isn't unique, and mathematically this is also fine as long as the statements we make about "the" non-unique thing hold regardless of which one of possibly many alternatives is chosen. (In particular, an introduction to fields in abstract algebra may well refer to "the" additive identity of a field with a definite article even before proving that it must be unique, i.e. that no other element of the field can satisfy the same requirements.) Commented Apr 14, 2021 at 18:40