2 deleted 8 characters in body

I think displayed

Displayed formulas can serve two roles in a math paper: as abbreviations for text that would otherwise be unreadable, and as figures (or illustrations) that are referred to by the text but are not part of it grammatically. My opinion is that in the former case they should be punctuated, but in the latter they should not.

Here are some examples.

1) If $x$ and $y$ are points on a circle of radius $r$ then

$x^2 + y^2 = r^2$.

2) The points of a circle of radius $r$ satisfy the following equation.

$x^2 + y^2 = r^2$

3) The following diagram commutes.

(diagram without any punctuation)

I think these examples demonstrate the necessity of distinguishing the two roles a displayed equation can play. As Simon already pointed out above, there is no reasonable place to put a punctuation mark in a commutative diagram, presumably because a commutative diagram can't be read aloud. On the other hand, it's difficult to view the sentence in the first example as complete without a period at the end of the equation.

I suggest the following rule of thumb: if the formula can be removed from the text without breaking the flow of a sentence, then it does not need to punctuated. Otherwise, it should be punctuated as it would be if the symbols were expanded into words.

Many authors use a colon where I used a period in the second example and follow the equation with a period.

2') The points of a circle of radius $r$ satisfy the following equation:

$x^2 + y^2 = r^2$.

I don't consider this incorrect, but I do consider it a completely different sentence from 2).

I think displayed formulas can serve two roles in a math paper: as abbreviations for text that would otherwise be unreadable, and as figures (or illustrations) that are referred to by the text but are not part of it grammatically. My opinion is that in the former case they should be punctuated, but in the latter they should not.

Here are some examples.

1) If $x$ and $y$ are points on a circle of radius $r$ then

$x^2 + y^2 = r^2$.

2) The points of a circle of radius $r$ satisfy the following equation.

$x^2 + y^2 = r^2$

3) The following diagram commutes.

(diagram without any punctuation)

I think these examples demonstrate the necessity of distinguishing the two roles a displayed equation can play. As Simon already pointed out above, there is no reasonable place to put a punctuation mark in a commutative diagram, presumably because a commutative diagram can't be read aloud. On the other hand, it's difficult to view the sentence in the first example as complete without a period at the end of the equation.

I suggest the following rule of thumb: if the formula can be removed from the text without breaking the flow of a sentence, then it does not need to punctuated. Otherwise, it should be punctuated as it would be if the symbols were expanded into words.

Many authors use a colon where I used a period in the second example and follow the equation with a period.

2') The points of a circle of radius $r$ satisfy the following equation:

$x^2 + y^2 = r^2$.

I don't consider this incorrect, but I do consider it a completely different sentence from 2).