Why start at x, y, z - User contributions [en-gb]

Commutative diagrams

2021-08-30T13:46:02Z

Benrbray: add section about parallel arrows in commutative diagrams

[[Category:Ambiguities]]

== Bound and Free Variables ==

In category theory, reading commutative diagrams requires an understanding of which variables are bound and which are free. Some authors choose to annotate their diagrams with universal quantifiers like $ \forall x,y,z $ to indicate the set of free variables, implying the rest are bound.

When such annotations are not included, the diagram alone is not precise enough to decode the intended logical statement. To read a commutative diagram, we must infer the bound and free variables from context.

== When Commutative Diagrams Don't Commute ==

When encountering commutative diagrams for the first time, many students are told that whenever there is more than one path between two objects in a commutative diagram, the corresponding morphisms must always commute. A notable exception occurs when there is a pair of parallel directed edges between two objects, as in the following [https://en.wikipedia.org/wiki/Equaliser_(mathematics) equalizer] diagram. Without additional context, parallel arrows in a commutative diagram are not necessarily the same, even if they may participate in other commutative paths.

[[File:Commutative-diagram equalizer.png|frame|center|This is a typical commutative diagram depicting the universal property of the [https://en.wikipedia.org/wiki/Equaliser_(mathematics) equalizer] of two morphisms $ f, g : x \rightarrow y $. Here, $f$ and $g$ are two different paths between the objects $x$ and $y$, but in fact $f$ and $g$ are assumed to be distinct morphisms.]]

For example, consider the diagram below. The statement "this diagram commutes" is equivalent to the statement "$ h \circ f = h \circ g = k \circ f = k \circ g $". It is not necessarily implied that $ f = g $ or $ h = k $.

[[File:Commutative-diagram two-parallel.png|frame|center|A commutative diagram with three objects (x, y, z). There are two parallel morphisms (f,g) from x to y, and two parallel morphisms (h,k) from y to z.]]

File:Commutative-diagram two-parallel.png

2021-08-30T13:40:51Z

Benrbray:

A commutative diagram with three objects (x, y, z). There are two parallel morphisms (f,g) from x to y, and two parallel morphisms (h,k) from y to z.

File:Commutative-diagram equalizer.png

2021-08-30T13:29:56Z

Benrbray:

A commutative diagram depicting the universal property of the equalizer $(e,\lambda_x)$ of two morphisms f,g in a category.

Use of different typefaces to convey meaning

2021-07-13T03:20:03Z

Benrbray:

[[Category:Unpleasantness]]
[[Category:Variable Names]]

Mathematicians love using a different font, or style, to get a semantically different symbol, instead of using diacritics or a different letter entirely.

It can be hard to differentiate instances of the same letter rendered in different styles, or to miss the difference altogether.

In computer representations of mathematics which only use plain text, this strategy doesn't work at all, unless you're able to type in Unicode characters.

When handwriting mathematics, many of these differences are hard or impossible to produce, so there are different conventions to produce the same meaning.<ref>[https://loopspace.mathforge.org/CountingOnMyFingers/Calligraphy/ Old Pappus' Book of Mathematical Calligraphy]</ref>

Sometimes, the alternate font looks very different to the standard one, and produces false friends (Fraktur).

==Examples==

A bold lower-case letter for a vector: $ \mathbf{a} $. In handwriting, a common convention is to underline the letter instead.

Upright instead of italic for matrices: $ \mathrm{A} $. Very hard to reliably produce in handwriting.

Script font for measure spaces $\mathscr{M}$ or categories $\mathscr{C}$, especially when the calligraphic font ($\mathcal{M}$ or $\mathcal{C}$) is also in use.

==References==
<references/>

Probability theory

2021-07-12T17:25:57Z

Benrbray: /* Probability Distributions */

[[Category:Unpleasantness]]
[[Category:Ambiguities]]

Probability theory makes a number of pragmatic abuses of notation. There are a lot of symbols, such as the magic function $ P $ (sometimes written $\mathbb{P}$ or $\mathrm{Pr}$), which appear to accept almost any expression as an argument!

Unlike other areas of mathematics, it is the names of variables that matter, rather than the order in which they appear.

==Random Variables==

Variable names matter!

==Probability Distributions==

The magic function $ P $ (sometimes written $\mathbb{P}$ or $\mathrm{Pr}$) can accept almost any expression as an argument! Consider:

* When $X$ is a random variable, $ P(X) $ is its probability distribution.
* When $X$ is an event, $ P(X) $ is the probability of that event with respect to some implicit underlying distribution
* When $X$ and $Y$ are random variables, $ P(X|Y) $ is the conditional probability distribution of $X$ given $Y$, while ( P(Y|X) \) is the conditional probability distribution of $Y$ given $X$.
* $P(A,B,C|X,Y,Z)$ may or may not be equivalent to $P(B,A,C|Y,Z,X)$, depending on who you ask.
* Sometimes we use $ P(X_1, X_2, X_3) $ to mean the probability of observing a sequence events $X_1$, $X_2$, and $X_3$ in that particular order, in contrast to $ P(X_2, X_1, X_3) $.
* In Bayesian statistics, $ P(X|Y;\theta) $ or $ P_\theta(X|Y) $ means "the probability of X, given Y, parameterized by $\theta$".

==Expected Value==

See Carpenter 2020, [https://statmodeling.stat.columbia.edu/2020/02/05/abuse-of-expectation-notation/ "Abuse of Expectation Notation"]

Probability theory

2021-07-12T17:24:51Z

Benrbray: /* Probability Distributions */

[[Category:Unpleasantness]]
[[Category:Ambiguities]]

Probability theory makes a number of pragmatic abuses of notation. There are a lot of symbols, such as the magic function $ P $ (sometimes written $\mathbb{P}$ or $\mathrm{Pr}$), which appear to accept almost any expression as an argument!

Unlike other areas of mathematics, it is the names of variables that matter, rather than the order in which they appear.

==Random Variables==

Variable names matter!

==Probability Distributions==

The magic function $ P $ (sometimes written $\mathbb{P}$ or $\mathrm{Pr}$) can accept almost any expression as an argument! Consider:

* When $X$ is a random variable, $ P(X) $ is its probability distribution.
* When $X$ is an event, $ P(X) $ is the probability of that event with respect to some implicit underlying distribution
* When $X$ and $Y$ are random variables, $ P(X|Y) $ is the conditional probability distribution of $X$ given $Y$, while ( P(Y|X) \) is the conditional probability distribution of $Y$ given $X$.
* $P(A,B,C|X,Y,Z)$ may or may not be equivalent to $P(B,A,C|Y,Z,X)$, depending on who you ask.
* Sometimes we use $ P(X_1, X_2, X_3 $ to mean the probability of observing a sequence events $X_1$, $X_2$, $X_3$ in that particular order.
* In Bayesian statistics, $ P(X|Y;\theta) $ or $ P_\theta(X|Y) $ means "the probability of X, given Y, parameterized by $\theta$".

==Expected Value==

See Carpenter 2020, [https://statmodeling.stat.columbia.edu/2020/02/05/abuse-of-expectation-notation/ "Abuse of Expectation Notation"]

Commutative diagrams

2021-07-12T17:17:35Z

Benrbray: basic summary of commutative diagram ambiguities

[[Category:Ambiguities]]

In category theory, reading commutative diagrams requires an understanding of which variables are bound and which are free. Some authors choose to annotate their diagrams with universal quantifiers like $ \forall x,y,z $ to indicate the set of free variables, implying the rest are bound.

When such annotations are not included, the diagram alone is not precise enough to decode the intended logical statement. To read a commutative diagram, we must infer the bound and free variables from context.

Probability theory

2021-07-12T17:06:29Z

Benrbray: tag with unpleasantness and ambiguities categories

[[Category:Unpleasantness]]
[[Category:Ambiguities]]

Probability theory makes a number of pragmatic abuses of notation. There are a lot of symbols, such as the magic function $ P $ (sometimes written $\mathbb{P}$ or $\mathrm{Pr}$), which appear to accept almost any expression as an argument!

Unlike other areas of mathematics, it is the names of variables that matter, rather than the order in which they appear.

==Random Variables==

Variable names matter!

==Probability Distributions==

The magic function $ P $ (sometimes written $\mathbb{P}$ or $\mathrm{Pr}$) can accept almost any expression as an argument! Consider:

* When $X$ is a random variable, $ P(X) $ is its probability distribution.
* When $X$ is an event, $ P(X) $ is the probability of that event with respect to some implicit underlying distribution
* When $X$ and $Y$ are random variables, $ P(X|Y) $ is the conditional probability distribution of $X$ given $Y$, while ( P(Y|X) \) is the conditional probability distribution of $Y$ given $X$.
* $P(A,B,C|X,Y,Z)$ may or may not be equivalent to $P(B,A,C|Y,Z,X)$, depending on who you ask.
* In Bayesian statistics, $ P(X|Y;\theta) $ or $ P_\theta(X|Y) $ means "the probability of X, given Y, parameterized by $\theta$".

==Expected Value==

See Carpenter 2020, [https://statmodeling.stat.columbia.edu/2020/02/05/abuse-of-expectation-notation/ "Abuse of Expectation Notation"]

Probability theory

2021-07-12T17:05:35Z

Benrbray: Created page with "Probability theory makes a number of pragmatic abuses of notation. There are a lot of symbols, such as the magic function $ P $ (sometimes written $\mathbb{P}$ or \(\math..."

Probability theory makes a number of pragmatic abuses of notation. There are a lot of symbols, such as the magic function $ P $ (sometimes written $\mathbb{P}$ or $\mathrm{Pr}$), which appear to accept almost any expression as an argument!

Unlike other areas of mathematics, it is the names of variables that matter, rather than the order in which they appear.

==Random Variables==

Variable names matter!

==Probability Distributions==

The magic function $ P $ (sometimes written $\mathbb{P}$ or $\mathrm{Pr}$) can accept almost any expression as an argument! Consider:

* When $X$ is a random variable, $ P(X) $ is its probability distribution.
* When $X$ is an event, $ P(X) $ is the probability of that event with respect to some implicit underlying distribution
* When $X$ and $Y$ are random variables, $ P(X|Y) $ is the conditional probability distribution of $X$ given $Y$, while ( P(Y|X) \) is the conditional probability distribution of $Y$ given $X$.
* $P(A,B,C|X,Y,Z)$ may or may not be equivalent to $P(B,A,C|Y,Z,X)$, depending on who you ask.
* In Bayesian statistics, $ P(X|Y;\theta) $ or $ P_\theta(X|Y) $ means "the probability of X, given Y, parameterized by $\theta$".

==Expected Value==

See Carpenter 2020, [https://statmodeling.stat.columbia.edu/2020/02/05/abuse-of-expectation-notation/ "Abuse of Expectation Notation"]

Symbols for names

2021-07-12T16:33:01Z

Benrbray:

[[Category:Unpleasantness]]
[[Category:Variable Names]]

There are many conventions to do with naming things. This page might split into several when it gets big enough.

==Symbol exhaustion==

Mathematicians tend to use single Latin or Greek letters to name things, when they can. A common consequence of this is that the author runs out of symbols, so needs to look to other alphabets or [[use of different typefaces to convey meaning]].

Once you've exhausted all the alphabets you know, you should take a long hard look at yourself, but you might feel tempted to start using non-letter symbols.

This question on academia.stackexchange.com<ref>[https://academia.stackexchange.com/questions/170398/what-are-the-potential-pitfalls-of-using-non-greek-non-european-characters-as-sy "What are the potential pitfalls of using non-Greek/non-European characters as symbol in scientific writing?"]</ref> asks whether it's OK to use symbols from other character sets for names. The highest-scoring reply points out that using more symbols makes the material harder to read.

==Starting at $x$==

For reasons that aren't completely clear, $x$ is normally used for the first variable, then $y$ and $z$. After that, some go to $u$, $v$ and $w$.

Florian Cajori says that Descartes is responsible for starting at $x$<ref>[http://archive.org/details/historyofmathema031756mbp/page/n399/mode/2up?view=theater Section 340. in ''A History of Mathematical Notations''] by Florian Cajori</ref>. The start of the alphabet should be used for known quantities, and the end for unknowns. Cajori gives a few dubious explanations for why $x$ is more common than $z$.

An alternate tack is to use Greek letters $\alpha$, $\beta$, $\gamma$, $\delta$, and so on: the convention seems to be to use those in the standard order.

==References==
<references/>

Symbols for names

2021-07-12T16:32:36Z

Benrbray:

Prime symbol in variable names

2021-07-12T16:31:48Z

Benrbray: summarize ambiguities around using prime symbol in variable names

[[Category:Variable Names]]
[[Category:Ambiguities]]

The prime symbol ($\prime$) is often used to distinguish between variable names. For example, given a variable $ f $, some authors might use $ f' $ as a separate variable name, which may or may not be related to the original $ f $. If more than one variation is needed, it is not uncommon to see two or more tick marks, as in $ f, f', f'', f''', f'''', \dots $.

==Disadvantages==

In calculus, the prime symbol is usually used to indicate differentiation, making the use of the prime symbol ambiguous when naming variables.

In some fonts, the tick mark might be too small to read.

This naming convention is popular enough that programming languages like Haskell and Coq permit the single quote symbol <code>'</code> as a valid identifier. Code is normally displayed in a monospace font, where extra space before the single quote may make the tick mark appear visually separated from the identifier that it modifies.

In monospace fonts, whitespace around the single quote <code>'</code> often causes it to appear visually separated from the identifier that it modifies. As an example,

==Alternatives==

For Pairs:

* Primes: ($ f, f' $)
* Numbers: ($ f_0, f_1 $) or ($ f_1, f_2 $)
* Letter Pairs: ($ f, g $)

For Sequences:

* Primes: $ a, a', a'', a''', \dots $,
* Numbers: $ a_0, a_1, a_2, a_3 \dots $
* Numbers: $ a, b, c, d \dots $

Long variable names

2021-07-12T16:07:27Z

Benrbray:

[[Category:Ambiguities]]
[[Category:Unpleasantness]]
[[Category:Variable Names]]

Mathematicians normally stick to single-letter variable names. But when they don't, it can be hard to tell if there's a [[missing multiplication symbol]] or not.

Some people use an upright font for long variable names to distinguish them from single-letter variable names, drawn in italics (see [[Use of different typefaces to convey meaning]])

==Examples==

\[ speed = distance \div time \]

\[ \text{speed} = \text{distance} \div \text{time} \]

Is it OK to mix single-letter variable names and longer ones?

\[ x \text{speed} \]

==Why?==
<ref>[https://math.stackexchange.com/questions/24241/why-do-mathematicians-use-single-letter-variables Why do mathematicians use single-letter variables?] on math.stackexchange.com</ref>

Category:Variable Names

2021-07-12T16:06:47Z

Benrbray: Created page with "Conventions for assigning symbols or other identifiers to variables."

Conventions for assigning symbols or other identifiers to variables.

Main Page

2021-07-12T16:06:18Z

Benrbray: /* Categories */

[[Category:Admin]]

This is a collection of ambiguous, inconsistent, or just unpleasant conventions in mathematical notation, started by [[User:Christian Lawson-Perfect|Christian Lawson-Perfect]].

For each bit of notation, I want to collect examples, alternatives, and references to discussions about them.

The site's name is a reference to the question about why we start naming variables at 𝑥.

[[Special:AllPages|All pages]]

==Categories==

* [[:Category:Ambiguities|Ambiguities]] - notations which can be reasonably interpreted in more than one way
* [[:Category:Conflicting definitions|Conflicting definitions]] - definitions of terms or mathematical objects that don't have a single universally recognised meaning
* [[:Category:Handwriting|Handwriting]] - things that only or mainly come up in handwritten notation.
* [[:Category:Inconsistencies|Inconsistencies]] - conventions in notation that are inconsistent with each other, and conventions that could apply more generally, but for some reason don't.
* [[:Category:Language|Language]] - problems to do with mathematical language.
* [[:Category:Local variations|Local variations]] - conventions in notation that are particular to a place or group of people.
* [[:Category:Unpleasantness|Unpleasantness]] - conventions that just make you feel yuck.
* [[:Category:Variable Names|Variable Names]] - conventions for assigning symbols or other identifiers to variables.

* [[:Category:References|References]] - links to material that might help when filling in this site.
* [[:Category:Needs filling in|Needs filling in]] - pages that have maybe a link or two, but need filling in with text.

==How to contribute==

You can edit this site, once you've [[Special:CreateAccount|created an account]].

My aim is to describe conventions, without prescribing a correct notation.

While the tone of the site is informal, please avoid writing in a way that would make someone who follows a certain convention feel bad.

Circumflex to Distinguish Variable Names

2021-07-12T16:03:23Z

Benrbray: introduce variable name category

[[Category:Unpleasantness]]
[[Category:Variable Names]]

Some authors create new variable names by adorning well-known symbols (like $ A $) with the upright ($ \hat{A} $) and inverted ($ \check{A} $) circumflex. Depending on the font, these tiny diacritical marks can be hard to discern at a glance. A particularly egregious example is found in Hinze 2012, [http://www.cs.ox.ac.uk/ralf.hinze/LN.pdf "Generic Programming with Adjunctions"] (p. 5).

[[File:Hinze2012 adjunctions circumflex.png|frame|center|Hinze 2012, [http://www.cs.ox.ac.uk/ralf.hinze/LN.pdf "Generic Programming with Adjunctions"] (p. 5) creates two new variable names by adorning the letter A with upright and inverted circumflex symbols.]]

Circumflex to Distinguish Variable Names

2021-07-12T16:01:13Z

Benrbray: Introduce the upright and inverted circumflex convention, with an example.

[[Category:Unpleasantness]]

Some authors create new variable names by adorning well-known symbols (like $ A $) with the upright ($ \hat{A} $) and inverted ($ \check{A} $) circumflex. Depending on the font, these tiny diacritical marks can be hard to discern at a glance. A particularly egregious example is found in Hinze 2012, [http://www.cs.ox.ac.uk/ralf.hinze/LN.pdf "Generic Programming with Adjunctions"] (p. 5).

[[File:Hinze2012 adjunctions circumflex.png|frame|center|Hinze 2012, [http://www.cs.ox.ac.uk/ralf.hinze/LN.pdf "Generic Programming with Adjunctions"] (p. 5) creates two new variable names by adorning the letter A with upright and inverted circumflex symbols.]]

File:Hinze2012 adjunctions circumflex.png

2021-07-12T15:54:16Z

Benrbray:

Hinze 2012, "Generic Programming with Adjunctions" creates two new variable names by adorning the letter A with the upright and inverted circumflex symbols.

Use of different typefaces to convey meaning

2021-07-12T15:17:49Z

Benrbray: /* Examples */

[[Category:Unpleasantness]]

Mathematicians love using a different font, or style, to get a semantically different symbol, instead of using diacritics or a different letter entirely.

It can be hard to differentiate instances of the same letter rendered in different styles, or to miss the difference altogether.

In computer representations of mathematics which only use plain text, this strategy doesn't work at all, unless you're able to type in Unicode characters.

When handwriting mathematics, many of these differences are hard or impossible to produce, so there are different conventions to produce the same meaning.<ref>[https://loopspace.mathforge.org/CountingOnMyFingers/Calligraphy/ Old Pappus' Book of Mathematical Calligraphy]</ref>

Sometimes, the alternate font looks very different to the standard one, and produces false friends (Fraktur).

==Examples==

A bold lower-case letter for a vector: $ \mathbf{a} $. In handwriting, a common convention is to underline the letter instead.

Upright instead of italic for matrices: $ \mathrm{A} $. Very hard to reliably produce in handwriting.

Script font for measure spaces $\mathscr{M}$ or categories $\mathscr{C}$, especially when the calligraphic font ($\mathcal{M}$ or $\mathcal{C}$) is also in use.

==References==
<references/>

Use of different typefaces to convey meaning

2021-07-12T15:17:13Z

Benrbray: /* Examples */

[[Category:Unpleasantness]]

Mathematicians love using a different font, or style, to get a semantically different symbol, instead of using diacritics or a different letter entirely.

It can be hard to differentiate instances of the same letter rendered in different styles, or to miss the difference altogether.

In computer representations of mathematics which only use plain text, this strategy doesn't work at all, unless you're able to type in Unicode characters.

When handwriting mathematics, many of these differences are hard or impossible to produce, so there are different conventions to produce the same meaning.<ref>[https://loopspace.mathforge.org/CountingOnMyFingers/Calligraphy/ Old Pappus' Book of Mathematical Calligraphy]</ref>

Sometimes, the alternate font looks very different to the standard one, and produces false friends (Fraktur).

==Examples==

A bold lower-case letter for a vector: $ \mathbf{a} $. In handwriting, a common convention is to underline the letter instead.

Upright instead of italic for matrices: $ \mathrm{A} $. Very hard to reliably produce in handwriting.

Script font for measure spaces $\mathscr{M}$ or categories $\mathscr{C}$, especially when the calligraphic font ($\mathcal{M}$ or $\mathcal{C}$) is also in use.

==References==
<references/>