# Tag Info

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### What are the benefits of writing vector inner products as $\langle u, v\rangle$ as opposed to $u^T v$?

Mathematical notation in a given mathematical field $X$ is basically a correspondence $$\mathrm{Notation}: \{ \hbox{well-formed expressions}\} \to \{ \hbox{abstract objects in } X \}$$ between ...

### Has incorrect notation ever led to a mistaken proof?

Here is an example from set theory. Set theorists commonly study not only the theory $\newcommand\ZFC{\text{ZFC}}\ZFC$ and its models, but also various fragments of this theory, such as the theory ...

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### Has incorrect notation ever led to a mistaken proof?

Ramanujan's notebooks are an interesting case study. As discussed in detail in Chapter 24 ("Ramanujan's Theory of Prime Numbers") of Volume IV of Bruce Berndt's series Ramanujan's Notebooks, ...

### Examples of improved notation that impacted research?

There is a notation that had an immediate and profound impact on research in algebraic topology, later algebraic geometry, and was eventually adopted by all areas of mathematics: the introduction of ...
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### How does one write the "gothic" letters ($\mathfrak{g}$) in handwriting?

Technically, in mathematics we use "Fraktur" letters, not "Gothic"... These were introduced into mathematics 100 years ago by German mathematicians--in those days it was included ...
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### History of the pullback corner notation

In an email to me dated 17 February 1992, Peter Freyd said: I was using a different notation in 1974 in lectures at Montreal. A high school teacher named Butler suggested the right-angle. It was an ...
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### History of the notation for substitution

Some early examples of the form $[t/x]$ are due to Haskell Curry. See: Haskell Curry & Robert Feys & William Craig, Combinatory Logic. Volume I (1958), page 54: Let $a$ and $b$ be obs and ...
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### Mathematical objects whose name is a single letter

A lovely story of $\nabla$ can be found on Wikipedia Introduced by Hamilton in 1837, his initial notation was ◁. Quotes from Wikipedia: The name comes, by reason of the symbol's shape, from the ...
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### Notation for "the" left adjoint functor

In EGA 0.1.5.2-3 (from the 1971 Springer edition) the right adjoint and the left adjoint of a functor $F$ are denoted by $F^{\rm ad}$ and ${}^{\rm ad}\!F$, respectively.
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### Mathematical objects whose name is a single letter

Amir Alexander's 2014 book Infinitesimal says that John Wallis introduced the symbol $\infty$ for infinity (p.280). I asked on HSM, "Where did John Wallis get the idea for $\infty$?," and user ...
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### When did the abuse of notation $y=y(x)$ start?

Regarding the original question of who started literally writing $y=y(x)$ or something like it, which I understand Jacobi didn't do in the quoted 1840 paper: Cayley (1859, p. 3), where $\Omega$ is ...
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### Examples of improved notation that impacted research?

This might be too old to qualify but I have always felt that the decimal notation is a wonderful thing! From a modern perspective, when it is taught to everyone at a very young age, it might be hard ...
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### Why aren‘t op and co switched?

The problem is that for a long time there were only 1-categories and hence only one kind of duality, and sometimes people called it "op" and sometimes "co". Colimits and ...
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### How does one write the "gothic" letters ($\mathfrak{g}$) in handwriting?

In Lie theory you just add an underline below the letter.

### Examples of bad notation and its consequences

In my opinion, bad notations are often a confusing notations, having different possible meanings. For example, the use of parenthesis for too many things can be confusing: argument of functions, ...

### History of the pullback corner notation

If it is of any use, on page 251 of Taylor's "Practical Foundations of Mathematics", shortly after introducing pullbacks, he writes: Pullbacks are often indicated with the right angle symbol, which ...
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### Whence "Durchschnitt" and "Vereinigung"?

An extensive discussion of the origin of "Menge" is given in Earliest Known Uses of Some of the Words of Mathematics (scroll down to "Set and Set Theory"). Cantor's (1880) Über unendliche linear ...
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### What are the benefits of writing vector inner products as $\langle u, v\rangle$ as opposed to $u^T v$?

Inner product is defined axiomatically, as a function from $V\times V\to k$, where $k$ is a field and $V$ is a $k$-vector space, satisfying the three well-known axioms. The usual notation is $(x,y)$. ...

### Examples of bad notation and its consequences

Suppose that $A$ is an oracle; then it is standard to write $\mathsf{P}^A$ for the complexity class $\mathsf{P}$ relativized to $A$. As I have mentioned elsewhere on MO, this is incredibly confusing ...
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### Who invented diagrammatic algebra?

I believe diagrammatic algebra started with 19th century invariant theorists. See Section 2 of my JKTR article "On the volume conjecture for classical spin networks" as well as Section 8 of this ...