336
votes

Accepted

### 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 ...

Community wiki

84
votes

### 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 ...

Community wiki

50
votes

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

This might not quite count, but if you start with a principal $G$-bundle $f:P\rightarrow B$, there are two natural ways to put a $G\times G$ structure on the bundle $P\times G\rightarrow B$ given by $(...

Community wiki

43
votes

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

This will probably not be considered a serious mistake, but maybe it counts:
According to Dray, Manogue if you ask the following question to scientist:
Suppose the temperature on a rectangular ...

Community wiki

36
votes

### When was the "arrow notation" for functions first introduced?

Despite the claims reported from Wikipedia and the “Earliest Uses” site, this notation certainly started much before Hurewicz-Steenrod (1940; 1941) or Ore (1935, p. 416; 1936) for, respectively, ...

35
votes

### How does one write the "gothic" letters ($\mathfrak{g}$) in handwriting?

Andrew Stacey's homepage includes a wonderful guide on drawing fraktur here.
Here's a copy of the video illustrating it there:
P.S. Adding more features to the above can often be helpful in ...

34
votes

### If d/dx is an operator, on what does it operate?

Not sure why this question is back on the front page, but I just wanted to add that the situation seems to be clarified by temporarily generalising to higher dimensions and to curved spaces, i.e., by ...

34
votes

Accepted

### Was Jacobi the first to notice the ambiguity in the partial derivatives notation? And did anyone object to his fix?

An extensive review of the history is given by Florian Cajori, The History of Notations of the Calculus.
Q1: Yes, it does appear that Jacobi was the first to explicitly state this ambiguity.
Q2: The ...

32
votes

Accepted

### The letter $\wp$; Name & origin?

Apparently first introduced by Weierstrass in Winter 1862/63 lectures published by H. A. Schwarz (1881, 1885, 1892, 1893), §9:
Mit der Sigma-Function $\mathfrak Su$ ist die Pe-Function $\wp u=\wp(u\...

31
votes

### 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, ...

Community wiki

27
votes

### 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 ...

26
votes

Accepted

### Why are Thompson's groups called $F$, $T$ and $V$?

The "$F$" actually stands for "free homotopy idempotent", since $F$ is the universal group encoding a free homotopy idempotent (the endomorphism sending each standard generator $...

25
votes

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

One huge advantage, to my mind, of the bracket notation is that it admits 'blanks'. So one can specify the notation for an inner product as $\langle \ , \ \rangle$, and given $\langle \ , \rangle : V \...

Community wiki

24
votes

### 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 ...

Community wiki

21
votes

Accepted

### 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 ...

20
votes

Accepted

### 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 ...

20
votes

### 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 ...

Community wiki

18
votes

Accepted

### 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.

17
votes

### 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 ...

Community wiki

16
votes

Accepted

### 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 ...

16
votes

### 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 ...

16
votes

Accepted

### 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 ...

16
votes

Accepted

### How does one write the "gothic" letters ($\mathfrak{g}$) in handwriting?

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

Community wiki

16
votes

### 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,
...

Community wiki

15
votes

### 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 ...

15
votes

Accepted

### 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 ...

15
votes

### 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 $(x,y)$.
...

Community wiki

15
votes

### 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 ...

Community wiki

14
votes

Accepted

### 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 ...

14
votes

Accepted

### Who introduced the notation for $\beth$ numbers and when?

Charles Sanders Peirce is credited with the beth notation ℶ, first introduced in a December 1900 letter to Cantor. Apparently, this was then forgotten for half a century.
I reproduce the relevant ...

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