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I chose to ask this question on MESE because I think it's not about mathematics per se but more about how it should be communicated.


Quantified statements in mathematics are often written for instance, as,

For all real numbers $x$ and $y$, $x+y=y+x$

where $x$ and $y$ are variables whose domains are the set of all real numbers.

Even though I understand here that the expression "For all real numbers $x$ and $y$..." means that all the elements in the domains of the variables $x$ and $y$ satisfy $x+y=y+x$, I feel the expression "For all real numbers $x$ and $y$" by and of itself doesn't convey that same meaning. Here is why I feel like that:

  • Case 1. If by writing $x$ in the expression "For all real numbers $x$" we're referring to the variable i.e. the symbol then the expression "For all real numbers $x$" is just like saying "For all real numbers (and then abruptly referring to) * symbol x * ". I mean the symbol "$x$" is not all real numbers. So what sense does it make to follow For all real numbers by "$x$"?
  • Case 2. If by writing $x$ in the expression "For all real numbers $x$" we're referring to an unspecified element in the domain of "$x$", then "$x$" only refers to a single element (just unspecified). So even following "For all real numbers" by $x$ won't make sense because $x$ is a single real number (unspecified); it is not all real numbers.

As an analogy to explain my point further, say we take a set $S$ whose members are all men in a particular region. Also, let's assume that all these men are married. If we consider the pronoun "he" as a variable, then writing something as "For all men $he$, $he$ is married" won't make sense right? We're trying to say that all men in $S$ satisfy the open sentence "He is married", so wouldn't something like "For all men in the domain of $he$" or "For all replacements of $he$" make more sense?


It seems to me like "For all real numbers $x$....." is a contraction for something fuller such as:

For all real numbers in the domain of "$x$" and "$y$", $x+y=y+x$

or

For all significant replacements of "$x$" and "$y$", $x+y=y+x$

So my question is, is writing only 'For all things $x$' instead of something like 'For all things in the domain of $x$' an abuse of language in mathematics?

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    $\begingroup$ It makes more sense to me than your suggested replacements do. It feels like proper English to me. $\endgroup$
    – Sue VanHattum
    Sep 13, 2023 at 20:26
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    $\begingroup$ Your question would be improved by explaining in more detail why you do think "For all x" is an abuse of language. In other words, your point is hard to grasp for non-experts of mathematical logic. $\endgroup$
    – user22788
    Sep 14, 2023 at 0:55
  • $\begingroup$ @user22788 Did the needful. $\endgroup$ Sep 14, 2023 at 11:29
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    $\begingroup$ I just want to mention that the phrase "for each real number $x$" (with each replacing all) is also used; it could be more to your taste, maybe? $\endgroup$ Sep 15, 2023 at 11:12
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    $\begingroup$ If you don't like "for all x" because you're uncomfortable with x "representing" more than one value, perhaps you'd prefer "for every x"? $\endgroup$
    – Stef
    Sep 27, 2023 at 8:40

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No, there is no abuse of language here.

$x$ and $y$ are placeholders that stand for individual numbers, and your second suggestion captures this: For each number we can insert in place of $x$ and $y$, the statement holds. The "For all real numbers $x$" part specifies where the instances come from. The values $x$ and $y$ take are in the domain of the real numbers; $x$ and $y$ themselves are not the domain, so your first suggestion actually has it slightly backwards.

The meaning of the variables is something like "Imagine an arbitrary number - let's call it $x$ ... - then we have: $x + ... = ...$ ". We use letters like $x$ and $y$ because giving these abstarct placeholders names just makes it easier than refer to than "it", especially as soon as there is more than one variable involved.

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  • $\begingroup$ The $x$ that just follows the expression 'For all real numbers', refers to the variable right i.e. the symbol '$x$'? (So that most accurately it should be written as '$x$'). If that is the case then doesn't 'For all real numbers '$x$' ...' sound like 'For all real numbers * the symbol x * ..'. I mean to say '$x$' is not forming a full sentence. It appears to me that something is missing like 'For all real numbers that can substitute '$x$'..' would be complete. $\endgroup$ Sep 13, 2023 at 21:26
  • $\begingroup$ Additionally, if we're making the name-object distinction then it's the names of real numbers that we're substituting for '$x$', not the real numbers themselves, so even the expression 'For all real numbers that can substitute $x$..' would be slightly incorrect, that's why I was inclined to saying something like 'For all significant substitutions of $x$' $\endgroup$ Sep 13, 2023 at 21:41
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    $\begingroup$ The typical failure to make name-object distinctions in math does cause some troubles occasionally, but systematically emphasizing that distinction is detrimental to doing math, I think. I do recognize the point! In logic and model theory this certainly becomes a non-trivial (basic?) issue. Anyway, "for all real x... blah" is what I would write, without attempting to explain the epistemological or ontological or nominalist or... issues. Yes, the notion of "variable" is a bit dicey... :) $\endgroup$ Sep 13, 2023 at 23:30
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    $\begingroup$ I think I can see what you mean now about it sounding weird, I had that feeling too initially. My answer to that is what I wrote in the last paragraph: Think of it as an abbreviation for "For any real number - let's call such a number "x" from now on - ...". But then we're not talking about it as just a symbol and hence don't need to put quotes around it, but we use it as something directly meaning that number, just like "it". It's really the same as "For any two real numbers - let's call them Peter and Karen - it will be the case that Peter + Karen = ....". $\endgroup$ Sep 14, 2023 at 0:59
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I'm not a native english speaker, but I agree that "for all numbers $x$ and $y$" sounds strange. Because in everyday language we don't use names after a "for all". For instance, we wouldn't say: "for all persons, Alice and Bob...". Nor can I think of a variation that makes this work: "for all two persons, call them Alice and Bob..."?

But I believe we can say: "for any pair of persons, call this pair Alice and Bob for the sake of argument..." or "for any two persons, call them Alice and Bob..." or maybe "for every two persons, say Alice and Bob..."

So maybe it sounds more natural if you say "for any/each/every" instead of "for all".

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    $\begingroup$ My conjecture is that this confusing language is a byproduct of the delegitimization of "variable quantities" that happened in the mid-20th century. This Google Ngram suggests that the sensible phrase "for all real values of" was replaced by "for all real numbers." $\endgroup$ Sep 15, 2023 at 11:19
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    $\begingroup$ @JustinHancock That's an interesting observation, thanks! I had originally written something about "for all real values of" in my answer, but then I realized that that wouldn't make sense either linguistically, since if I say for instance "for all real values of $x$, $x^2\geq 0$, I would still be referring to the variable $x$ in $x^2\geq 0$ and not to its values. A probably silly analogy might be: "for all days of the year 2023, 2023 was on average a hot year". $\endgroup$ Sep 15, 2023 at 12:34
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    $\begingroup$ I don't think "for all days of the year 2023" works as an analogy, because "2023" isn't the name of a variable that takes on different values. A multiverse-themed analogy to "for all values of $x$, $P(x)$" would be "for all versions of Spider-Man, Spider-Man has superpowers." $\endgroup$ Sep 15, 2023 at 14:31
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    $\begingroup$ It seem as though the $x$ in the "for all values of $x$, $x^2\geq 0$" examples is free (not bound), as is the case in the statement $x^2\geq0$. By "free" is just mean: we cannot arbitrarily rename x to something else and expect to get an equivalent statement. The same is true in your Spider-Man example: "For all versions of Bart Simpson, Bart Simpson has superpowers" is not obviously the same as your sentence, while in modern logic we are allowed to rename bound variables (avoiding capture...) $\endgroup$ Sep 19, 2023 at 13:06
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    $\begingroup$ So it seems that saying "for all values of x" is not the same as saying "for all x"... If I have time I'd like to look into the history of all this, since as you pint out, there was a clear preference for the former in the past. $\endgroup$ Sep 19, 2023 at 13:07
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The $x$ refers to the variable, i.e., the symbol '$x$', right? So, isn't

For all real numbers '$x$',...

a more accurate phrasing? If so, this sounds like

For all real numbers the symbol $x,$....

Additionally, if we're making the name-object distinction, then it's the names of real numbers that we're substituting for '$x$', not the real numbers themselves,

In the sentence “For all real numbers $x,$...”, the variable/symbol $x$ (its entity, so to speak) is a placeholder for any real number, like $\sqrt2.$ Outside of an ontology discussion, I struggle to find the value in distinguishing between the names $x$ & $\sqrt2$ and the entities $x$ & $\sqrt2;$ enclosing them within quotations marks is also pointlessly meta. To wit: Harshit is both a referent and the referent's name, and in the following sentence, would you use quotation marks around the word and call attention to it?

  • For the candidate ‘Harshit’, an interview is scheduled.

Case 2. "For all real numbers $x$" doesn't make sense because $x$ is a single real number (unspecified); it is not all real numbers.

Thanks for pointing this out—this is the most interesting part of your Question! In a previous post, I expressed dissatisfation with the phrasing “for all $x$”, and hinted that I prefer to say “for each/every $x$”:

  • For all $x$ in $F,\, P(x)$ holds” sometimes sounds like the property $P$ might belong to $F$ as a whole rather than to its individual members: “for all members of the family, they have a house” (1 house in total? or 5?). Contrast with “for each/every member of the family, they have a house” (definitely 5 houses in total).

After rereading your Question, Michael's answer, and further rumination, I think it's finally clear that the root issue is as you are highlighting: in the phrasing “for all elements $x,\,P(x)$ holds”, every occurrence of $x$ is singular, so what's with the plural form “for all”? We could contrive that this phrasing is short for “for all elements, each of which we denote by $x,\,P(x)$ holds” but of course then we might as well just say “for each element $x,\,P(x)$ holds”.

Let's just acknowledge that the “for all $\boldsymbol x$” phrasing is Mathlish.

I am asking this question on MESE because I think it's not about mathematics per se but more about how it should be communicated.

Sorry for being so contrary, but this question is not about mathematics communication or teaching, but rather about mathematical writing, that is, communication in mathematics.

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  • $\begingroup$ '...not to say that it is a placeholder for a real number's name'. Isn't a variable (the symbol) a placeholder for the names of things? For example, we have the expressions "2+0=2", "3+0=3", "4+0=4",... so we use "x" to form a generalized expression such as "x+0=x". Here "x" is holding a place for the names of numbers, not the numbers themselves. By the way, I just made some edits to explain my point further, please consider reading them once. $\endgroup$ Sep 14, 2023 at 12:02
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    $\begingroup$ Also, I feel the use of proper language, both while teaching and writing mathematics, is a part of math education. Students won't be comfortable if teachers explain a concept well, but then write something that doesn't convey that same idea. This is what I felt with this question, so I considered asking it here. $\endgroup$ Sep 14, 2023 at 12:06
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    $\begingroup$ [disappearing] @HarshitRajput I revised the answer. $\endgroup$
    – ryang
    Sep 15, 2023 at 9:54
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    $\begingroup$ @HarshitRajput My point is that (precise) communication in mathematics is the practice of mathematics, so your Question IS about mathematics, rather than about mathematics/science communication (to the wider community). $\endgroup$
    – ryang
    Sep 15, 2023 at 10:58
  • $\begingroup$ 'In the sentence “For all real numbers x,...”, the variable/symbol x (its entity, so to speak) is a placeholder for any real number'. It just struck me that if the symbol x is a placeholder here, i.e. it holds a place for the names of numbers, then it should make sense to replace "x" in here by the name of a real number. However, the variable here is said to be bound, and bound variables can't be replaced by the names of things. $\endgroup$ Nov 4, 2023 at 14:27
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Do you have any qualms about "good old fashioned" first-order logic notation? In this language we would write something like $$ x,y\in \mathbb{R}\rightarrow P(x,y)$$ where, for example, $P(x,y)$ is $x+y=y+x$.

The formalism of first order logic typically comes equipped with an infinite set of "variables" like $x_0,x_1,\ldots$ See nLab

Colloquially we would write "for all real numbers $x,y$ we have $x+y=y+x$". Or often people like $\forall x\in\mathbb{R}, \forall y\in\mathbb{R},\ x+y=y+x$.

You seem to be going down the rabbit hole of "types", and uneasy with issues related to symbols and variables. If so, then I don't think your question is pertinent to mathematics education.

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    $\begingroup$ I find it interesting to consider whether this was a case of new notation causing a shift in language. The symbol "$\forall$" was first used in 1935, whereas the phrase "for all real numbers" did not become widespread until later. $\endgroup$ Sep 15, 2023 at 15:10
  • $\begingroup$ @JustinHancock Very interesting! And thanks for the n-gram viewer. I do see some usage of the German "für alle reellen Zahlen" in the 1920s. which takes off again after 1935. $\endgroup$
    – user52817
    Sep 15, 2023 at 21:12
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"For all 'things' $x$" is just the shortest way to say any of the following, which are equivalent but progressively wordy and cumbersome:

  • "for all 'things' $x$"
  • "if you replace $x$ with any 'thing'"
  • "for every element $x$ that is a 'thing'"
  • "for any 'thing' that you replace $x$ with"
  • "if you let the domain of $x$ consist of all 'things'"
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edit - Stumbled on a succinct way to put it when writing the comments below:

You're trying to apply a stricter reading of the English language than a reasonable speaker of the language would apply. You're not wrong that your interpretation is a correct reading of that phrasing, but you are wrong when you claim that it is the only possible correct reading of that phrasing.

It is perfectly reasonable for "for all real numbers $x$" to be understood to mean "with $x$ being any possible real number value", as opposed to only being understood to mean "with $x$ being the complete set of all real numbers".


It would appear that the core issue you perceive here is that

all real numbers $x$

can only be interpreted as

$x$ is [all real numbers]

and not as:

All [$x$ is a real number] instances

This is an overly strict reading of the phrasing, and one that does not mesh well with how English is commonly interpreted. In informally spoken English, it is perfectly reasonable for

plural of [real number $x$]

to be considered synonymous with

[plural of real number] $x$

Is that formally correct? No, but English is not defined as rigorously as mathematics, and the informal understanding is that these two definitions are sufficiently equivalent to be understood the same way.

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    $\begingroup$ Could you give full sentences from everyday english that are examples of what you wrote: "All [𝑥 is a real number] instances" and "[plural of real number] 𝑥? $\endgroup$ Sep 26, 2023 at 6:52
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    $\begingroup$ I think I understand the OP similar to you, but I don't understand your answer. You seem to be saying that it is common in everyday english to say "for all" and then use the name of a single (but arbitrary) object. But I have never seen an example of such a sentence in everyday language, while you seem to be suggesting that it is common. $\endgroup$ Sep 26, 2023 at 13:08
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    $\begingroup$ I might not be able to see the wood, but you are unable to give a specific example of the general thing you are trying to explain. Are there any trees in your wood? $\endgroup$ Sep 28, 2023 at 5:50
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    $\begingroup$ The example with "people in cars" is not an example since there is no name being given to a single entity. It's like saying: "for all numbers greater than zero there is a multiplicative inverse." There is no x or any other name denoting a single number in such a sentence. The edit to your answer does also not help, since you are basically saying what others have said: in math it's common to understand "for all x" as "for any possible x". That ist true, but your original answer was claiming that such a reading is common in everyday english and you have not provided an example of that. $\endgroup$ Sep 28, 2023 at 6:03
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    $\begingroup$ When someone says "For all real numbers x: x^2>0" the x denotes a number, it's the name for a number, not the name for the set of all real numbers. I think we agree on that. When you say "all people in cars" the word people denotes a set of persons, not one persons. But I also give up this conversation, we are not getting anywhere. $\endgroup$ Sep 28, 2023 at 7:31

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