Determined vs Uniquely Determined [duplicate]

Question

What is the difference between using the term ‘determined’ vs ‘uniquely determined’? The definition supplied by Google is

Mathematics: Specify the value, position, or form of (a mathematical or geometric object) uniquely.

It seems as though the uniqueness is built into the definition of the word determined, however I only ever see the phrase 'uniquely determined'. For example, should the following statement be interpreted the same, regardless of if we include 'uniquely' or not?

The value of a multilinear function at some point in the domain is (uniquely) determined by the values of the function at all combinations of basis vectors.

If the two statements are equivalent, is there a reason adding 'uniquely' is seemingly more common?

Answer 1

This terminology just means there is a choice of possible solutions.

For example, the angle $\theta$ is determined by $\sin\theta = 0$ to be $n\pi$, where $n$ is an arbitrary constant, but the angle $\theta$ is uniquely determined by $\sin\theta=0$ and $-\pi<\theta<\pi$ to be $0$. The integral $\int x^2=\frac13 x^3+C$ is determined up to a constant, but the integral $\int_0^1x^2$ is uniquely determined to be $\frac13$.

Answer 2

The number $i$ is determined by the equation $i^2 = -1$, but it is not uniquely determined. Indeed, $-i$ is another root.

The importance of the word "uniquely" is that it guarantees we can define a function associating to each input the unique object determined by some relation. In the reals for instance, for every positive $x$, there is a unique positive $y$ so that $y^2 = x$. This means we can define a function $\sqrt{x}$ sending each $x$ to the unique such $y$.

Whenever a relation does not uniquely determine something, we have to make a choice between the various options. For instance, a choice between $\pm i$ as the value of $\sqrt{-1}$. Experience shows that it's important to keep note of these choices, since they can often lead to more complicated behavior. Even in this toy example, there is no way to build a continuous $\sqrt{z}$ function on the whole of $\mathbb{C}$ -- if we try to make consistent choices in a way that guarantees $\sqrt{z}$ is continuous, then after walking around the origin once we'll see that we're forced to have made the other choice. See here for more. Already it's helpful to have the (high powered!) language of sheaf cohomology to study this example, and others like it.

In fact, objects which are "determined but not uniquely determined" show up quite often, and are usually the source of some difficulties. The algebraic closure of a field is determined, but not uniquely, and the various isomorphisms between the nonunique closures give rise to the (absolute) galois group of the underlying field! This is an object of intense study in number theory. Stacks, $\infty$-categories, and other high power tools exist in order to study these kinds of objects.

When we write "uniquely" in parentheses, then, we're saying that it's probably obvious to someone familiar with the material that an object is uniquely determined, and thus we don't need to worry about these subtleties. We can get a well defined construction, and call it a day. But even though it's probably obvious, it's still worth noting the uniqueness!

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I hope this helps ^_^