Learn English – Putting “foot in mouth” or “foot in it”

"Classical Mechanics is a well known field of Mathematics." Do you even listen to yourself...? – Najib Idrissi 19 hours ago

The user "Najib Idrissi" asks the author of the sentence Classical Mechanics is a well known field of Mathematics. to read that statement for themselves. Najib makes sure that the author will realize the scientific mistake in that sentence and, in short, "will come to their senses."

It's like your friend has told you something very oddly wrong, and you didn't expect him to say so. You tell him/her to listen to himself/herself, so that s/he will realize the mistake they've made. In this case, it seems very rudimentary to the mathematicians that "classical mechanics" are a physics' subject, and hardly related to math. So, "well known", as it means that "many are familiar with it being a part of mathematics" seems a very idiotic expression to them, as if the author didn't know what they were typing, or simply, talking about.

If you ask me, there is a bit of negative connotation in this sentence, but it wouldn't be considered as offensive. (As "offensive" is too strong for it)

In fact, the expression "listen to yourself" is commonly used in the area of philosophy when there's a speak of art of communicating between humans.

philosophy reference 1

philosophy reference 2

The third

And this is a nice example of a similar usage. (Though it contains the expression and not the exact question)

in Google books

Short answer: The cuts are visible marks to what he or she is feeling inside.

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The above usage seems more literal than some others.

My best guess is that to put a punctuation mark on something means

to bring to (visible) completion

or

to (visibly) complete

A sentence is not complete until a punctuation mark is "put on it".

A couple other uses (from Vocabulary dot com):

Evan Berry put a punctuation mark on the blowout by returning one of Tennessee’s four interceptions 100 yards for a TD in the closing seconds.

Notice this is talking about an exceptional running play that produced more points in the blowout game, "in the closing seconds".

The use you found regarding the basketball team is similar, since it was the last home game of their season.

It used to be the family dinner was a given; the punctuation mark at the end of each day, absolute and non-negotiable.

Notice at the end of each day. So we're in the realm of something that completes something.

Wagner's final shot of the match, a grunting effortful bouncer that clipped Ian Bell, was a worthy punctuation mark to a grand effort.

NB his final shot

The riot was symptomatic of the area’s sense of isolation and served as a punctuation mark on 10 years of restiveness in Miami.

All my examples come from the same vocabulary dot com link; I had to scroll through the examples. Some refer to 'literal' punctuation marks.

As far as the example with cutting,

It expresses emotional pain or feelings that I’m unable to put into words. It puts a punctuation mark on what I’m feeling on the inside!

Could it (cutting oneself) be thought of as somehow completing what the person is "feeling on the inside"?That's not too satisfactory. However, if you consider punctuation mark in the more literal sense, as visible marks, then this is perhaps what this usage is doing. The cuts themselves are visible marks on what he or she is feeling inside.

So, I'll just add this, to put a punctuation mark on my answer:

It’s now official what many people had predicted: former Alaska Governor Sarah Palin has ended her political tease — which helped her promote books and raise money for her political groups — and has formally announced that she is not going to run for the 2012 Republican nomination: ... The decision from Ms. Palin on Wednesday placed a punctuation mark on the Republican presidential field. Her intentions were the biggest remaining question in the race. [Emphases mine]

(The Moderate Voice, quoting The New York Times).