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I am having trouble with the word precisely in this sentence and generally, probably, in any mathematical context.

A statement reads

The points within the parallelpiped determined by $\boldsymbol{a}$, $\boldsymbol{b}$, and $\boldsymbol{c}$ are precisely the endpoints of vectors of the form $\alpha \boldsymbol{a} + \beta \boldsymbol{b} + \gamma \boldsymbol{c}$ where $...\quad$ [emphasis mine]

Does the use of the word precisely have any logical or set-theoretic meaning: does the word imply a two-way set inclusion, i.e., that "points in the parallelpiped have the form $\alpha \boldsymbol{a} + \beta \boldsymbol{b} + \gamma \boldsymbol{c}$" and that "points of the form $\alpha \boldsymbol{a} + \beta \boldsymbol{b} + \gamma \boldsymbol{c}$ are contained in the parallelpiped"?

If so, how would I have known this? Is there an available list of such modifiers that are used in mathematical contexts?

ryang
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AndJM
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    In the example above, you can omit it without loss of... precision. – Mauro ALLEGRANZA Aug 07 '25 at 10:19
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    "Precisely" does not have a technical connotation. Here it's used for its meaning in English: specifically, as synonymous with "exactly". – Sassatelli Giulio Aug 07 '25 at 10:23
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    I would suggest here that it is being used to emphasise a one-to-one correspondence. I.e., not only are these points in the parallelepiped, but there are no other points - this is an exhaustive description of all points in the set. The meaning of the sentence doesn't change if the word is removed, but it's used for emphasis. – Dan Rust Aug 07 '25 at 10:38
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    The meaning of the sentence doesn't change if the word is removed, but it's used for emphasis. To add to what @Dan Rust said, and to show limitations of English (and presumably other languages), the meaning of the sentence DOES change if, in addition to "precisely" being omitted, "the endpoints" is changed to "endpoints" (i.e. omit "the"). For various reasons such as this, it's helpful in verbal explanations to err on the side of too much explicitness (or alternatively, provide an example that would serve to eliminate possible alternative meanings present in natural language). – Dave L. Renfro Aug 07 '25 at 10:49
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    @DaveL.Renfro I think you should post your comment as an answer. – Ethan Bolker Aug 07 '25 at 10:58
  • @Ethan Bolker: OK, my comment is now an answer. I might add to it later, but if I do, probably not for at least the next 6 to 8 hours, as I'm about to leave home now for 3 hours, back for at most an hour, then leave again for 2-3 more hours. – Dave L. Renfro Aug 07 '25 at 11:10

3 Answers3

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(slightly edited version of a former comment)

In a comment @Dan Rust said: The meaning of the sentence doesn't change if the word is removed, but it's used for emphasis.

To add to what Dan Rust said, and to show limitations of English (and presumably other languages), the meaning of the sentence DOES change if, in addition to "precisely" being omitted, "the endpoints" is changed to "endpoints" (i.e. omit "the"). For various reasons such as this, it's helpful in verbal explanations of precise mathematical things (and here I'm using "precise" in a different way, further showing how muddy things can be) to err on the side of too much explicitness (or alternatively, provide an example that would serve to eliminate possible alternative meanings present in natural language).

Dave L. Renfro
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I'm pretty sure the meaning here is that points in the parallelepiped are exactly the same as endpoints of vectors of the form...

While the word "precisely" can mean the opposite of "vaguely", there is this other common usage that is meant to signal an if-and-only-if statement, e.g., "$P$ is a point in the parallelpiped if and only if it is an endpoint of a vector of the form...". So "precisely" there could also be colloquially rendered as "no more and no less (than)".

user43208
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The points within the parallelpiped determined by $\boldsymbol{a}$, $\boldsymbol{b}$, and $\boldsymbol{c}$ are precisely the endpoints of vectors of the form $\alpha \boldsymbol{a} + \beta \boldsymbol{b} + \gamma \boldsymbol{c},$ where $...$

Does the use of the word precisely have any logical or set-theoretic meaning: does the word imply a two-way set inclusion, i.e., that "points in the parallelpiped have the form $\alpha \boldsymbol{a} + \beta \boldsymbol{b} + \gamma \boldsymbol{c}$" and that "points of the form $\alpha \boldsymbol{a} + \beta \boldsymbol{b} + \gamma \boldsymbol{c}$ are contained in the parallelpiped"?

Yes, precisely! The given sentence means

  • Points are within the parallelpiped precisely when they are of the form $\alpha \boldsymbol{a} + \beta \boldsymbol{b} + \gamma \boldsymbol{c}$
  • Points are within the parallelpiped if and only if they are of the form $\alpha \boldsymbol{a} + \beta \boldsymbol{b} + \gamma \boldsymbol{c}$
  • For every point $P,$ $P$ is within the parallelpiped if and only if there exist scalars $\alpha,\beta,\gamma$ in $[0,1]$ such that $\vec{OP}=\alpha \boldsymbol{a} + \beta \boldsymbol{b} + \gamma \boldsymbol{c}.$

"Precisely when" and "if and only if" are both Mathlish for "is equivalent to".

In set notation:

  • The set of points within the parallelepiped is $\;\{\alpha \boldsymbol{a} + \beta \boldsymbol{b} + \gamma \boldsymbol{c}\mid \alpha,\beta,\gamma\in [0,1]\}$
  • The parallelpiped contains precisely the points in $\;\{\alpha \boldsymbol{a} + \beta \boldsymbol{b} + \gamma \boldsymbol{c}\mid \alpha,\beta,\gamma\in [0,1]\}.$

Note: in the bullets above, omitting the word "precisely" is tantamount to omitting the words "and only if": they each mark the introduced condition/set as exhaustive, that is, as a complete characterisation of the points within the parallelpiped.
(In contrast—to reiterate the comments above—omitting only the word "precisely" from the given sentence doesn't alter its meaning, because its second instance of "the" is already signalling—as you've pointed out—that set of the points within the parallelpiped is an improper subset of the set of points of the form $\alpha \boldsymbol{a} + \beta \boldsymbol{b} + \gamma \boldsymbol{c}.)$

ryang
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