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I’ve recently noticed that use of angle brackets for writing tuples, e.g. $\langle x, y \rangle$ instead of the usual round brackets in a few books I’ve been reading — Lawvere’s *Sets for Mathematics*, Mac Lane’s *Categories for the Working Mathematician*, Forster’s *Logic, induction and sets*, for example. I’ve also seen occasional use of it in Hartshorne’s *Algebraic Geometry*, but there round brackets seem predominant. Is there some subtle distinction between the two notations I’ve missed, and what might the reasons for *not* using round brackets be? Is this practice peculiar to a particular tradition in mathematics (say, foundations)?

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Some analysts (in a wide sense) write $\langle x,y\rangle$ (angle brackets), for $x$ and $y$ elements of a same set $X$, to denote the ordered pair element of $X\times X$. A more classical (to me) notation is $(x,y)$ (parenthesis), but, if for example $X=\mathbb R$ and $x\leqslant y$, the notation $(x,y)$ may refer to the open interval $\{z\in\mathbb R\mid x<z<y\}$. Hence the bracket notation might have been designed as a way to avoid the confusion. Bourbaki use the notation $]x,y[$ for open intervals and $[x,y]$ for segments. This notation, of frequent use in the mathematical literature written in French (and in others), removes the risk of confusion mentioned above.

I do not know how useful brackets are for objects like $\langle\mathbb Z,+\rangle$, since the objects inside the brackets are of a different nature.

I’m not an expert, but I’m quite sure the difference between angle and round brackets depends on the individual notation conventions of each author/paper/work.

Sometimes I see angle brackets instead of round ones without any particular meaning, while (in example) in the computability and complexity class I attended only round brackets were used for classic tuples: the angle brackets were used as a shortcut to mean the encoding for a turing machine of that tuple.

Unfortunately I have no clue about the origins of the angle brackets, it would be interesting.

Herbert Enderton –and other logicians– use angle brackets to indicate a structure, that is, a set with underlying functions, relations and constants. For example, we denote the usual natural numbers with addition, multiplication, an identity 0 for addition and an identity 1 for multiplication (in that order) as $\left\langle\mathbb{N},+^{\mathbb{N}},\cdot^{\mathbb{N}},0^{\mathbb{N}},1^{\mathbb{N}}\right\rangle$.

In general, for a structure $\mathfrak{A}$ with relations $r_1,\dots,r_i$ (each of a given arity), functions $f_1,\dots,f_j$ (each of a given arity) and constants $c_1,\dots,c_k$, an interpretation of this structure with domain $A$ is denoted $\left\langle A,r_1^{\mathfrak{A}},\dots,r_i^{\mathfrak{A}},f_1^{\mathfrak{A}},\dots,f_j^{\mathfrak{A}},c_1^{\mathfrak{A}},\dots,c_k^{\mathfrak{A}}\right\rangle$.

I don’t know wheter this notation came from algebra first, but, as commented above, it appears that some authors use angle brackes for algebraic structures in general (e.g. $\langle\mathbb{Z},+\rangle$, $\langle\mathbb{R},+,\cdot\rangle$).

Another use of angle brackets is to indicate an inner product in a vector space, to distinguish the inner product with other well-known operations between the elements of the space, for example, composition of automorphisms of a given set and multiplication of numbers in a ring, which are usually denoted with common syntactic concatenation ($fg$ for the composition of $f$ and $g$, $xy$ for the (ring-)product of $x$ and $y$).

I don’t think this is quite what you are asking about, but at the level of ordered tuples of elements, I would use parentheses to emphasize that I only care about the tuple as a point object, and angles to emphasize that I care about the tuple as a direction, or vector.

For example, I might have a line parametrized by $t$ with equation$$(x,y,z)=(x_0,y_0,z_0)+t\langle a,b,c\rangle$$

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