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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?

Harshit Rajput
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    It makes more sense to me than your suggested replacements do. It feels like proper English to me. – Sue VanHattum Sep 13 '23 at 20:26
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    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. – user22788 Sep 14 '23 at 00:55
  • @user22788 Did the needful. – Harshit Rajput Sep 14 '23 at 11:29
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    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? – Michał Miśkiewicz Sep 15 '23 at 11:12
  • Why do you aks this question? In English, it does not look like an abuse, but maybe in other languages, that might be the case. Do you have the situation that it sounds like an abuse in another language (like your mother tongue)? – Dominique Sep 22 '23 at 08:50
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    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"? – Stef Sep 27 '23 at 08:40

6 Answers6

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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.

Natalie Clarius
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  • 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. – Harshit Rajput Sep 13 '23 at 21:26
  • 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 $x$' – Harshit Rajput Sep 13 '23 at 21:41
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    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... :) – paul garrett Sep 13 '23 at 23:30
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    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 = ....". – Natalie Clarius Sep 14 '23 at 00:59
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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".

Michael Bächtold
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    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." – Justin Hancock Sep 15 '23 at 11:19
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    @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". – Michael Bächtold Sep 15 '23 at 12:34
  • @MichaelBächtold Re: the '2023' example: nice analogy. – ryang Sep 15 '23 at 13:56
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    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." – Justin Hancock Sep 15 '23 at 14:31
  • @JustinHancock I find your objection interesting but still need to reflect on it. Some preliminary thoughts/questions: I don't understand which additional information is conveyed by your Spider-Man sentence, that wouldn't be conveyed by the simpler sentences "Spider-Man has superpowers". And returning to math: I wouldn't know which additional information is conveyed by saying "for all values of $x$, $x^2\geq 0$" as opposed to just saying $x^2\geq 0$ (in the common context where $x$ is assumed real). They seem equivalent to me. – Michael Bächtold Sep 19 '23 at 13:05
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    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...) – Michael Bächtold Sep 19 '23 at 13:06
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    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. – Michael Bächtold Sep 19 '23 at 13:07
  • Here's my interpretation with "variable quantities." The statement "for all values of $x$, $x^2 \geq 0$" asserts that the squares of the values of the variable quantity $x$ are all non-negative. If we formalize variable quantities as functions, then we are saying "for each $t$ in the domain of $x$, $(x(t))^2 \geq 0$." – Justin Hancock Sep 20 '23 at 00:53
  • And it seems that, in some contexts, e.g. if $x$ is "arbitrary", we are to assume that the image of $x$ is "as large as possible." Thus "all real values of $x$" would be all real numbers. There's an interesting passage in this 1857 algebra textbook that says this: "in the identity we are dealing with we predicate of $x$ continuity, making it thereby a symbol of quantity in the most general and unrestricted sense it can possibly be conceived in." – Justin Hancock Sep 20 '23 at 01:02
  • However "Spider-Man" is not an arbitrary variable, and thus "all versions of Spider-Man" is something like the set {Peter Parker, Miles Morales, ...}, and this is clearly not the same as "all versions of Bart Simpson," which could just be the singleton {Bart Simpson}. – Justin Hancock Sep 20 '23 at 01:16
  • @JustinHancock concerning your interpretation of variable quantities: when you write "for each $t$ in the domain of $x$, (x(t))^2\geq 0" you are explicitly mentioning $t$ in the statement after the "for each..." (in "$(x(t))^2\geq 0$"), while in the original formulation no value was visible in $x^2\geq 0$. So although I have no problems understanding your re-interpretation, I don't find it settles the original question of what we are saying with "for all values of $x, x^2\geq 0$". Note also that $x$ is free in you re-interpretation and that it's equivalent to the assertion $x^2\geq 0$. – Michael Bächtold Sep 20 '23 at 05:22
  • That quote from the algebra texts sound interesting, but I cannot make much sense of it. Concerning the Bart Simpson counter argument: I suspect it's possible to find some fictitious character which sometimes has super powers and other times not. Maybe Hulk? – Michael Bächtold Sep 20 '23 at 05:23
  • Let us continue this discussion in chat. – Justin Hancock Sep 20 '23 at 13:16
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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.

ryang
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  • '...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. – Harshit Rajput Sep 14 '23 at 12:02
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    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. – Harshit Rajput Sep 14 '23 at 12:06
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    [disappearing] @HarshitRajput I revised the answer. – ryang Sep 15 '23 at 09:54
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    @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). – ryang Sep 15 '23 at 10:58
  • '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. – Harshit Rajput Nov 04 '23 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.

user52817
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    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. – Justin Hancock Sep 15 '23 at 15:10
  • @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. – user52817 Sep 15 '23 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'"
Justin Skycak
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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.

Flater
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    Could you give full sentences from everyday english that are examples of what you wrote: "All [ is a real number] instances" and "[plural of real number] ? – Michael Bächtold Sep 26 '23 at 06:52
  • @MichaelBächtold "All people in cars" can be used both when there are multiple people per car or when there's specifically one person per car. The pluralization of "people" does not describe specifically one scenario and not the other. – Flater Sep 26 '23 at 07:24
  • So in this case the x would come after people? Like "All people, Bob, in cars"? Sorry for sounding silly, I still don't understand your answer. – Michael Bächtold Sep 26 '23 at 08:34
  • I also don't understand the difference between "plural of [real number ]" and "[plural of real number] " you are pointing out. Could you also make everyday sentence to illustrate that. – Michael Bächtold Sep 26 '23 at 11:15
  • @MichaelBächtold: You're unintentionally proving my point that what OP is focusing on is not a distinction that's really made in English. OP is interpreting "for all real numbers $x$" as "where $x$ is the set of all real numbers", and specifically rejecting the interpretation "where $x$ is any real number" (i.e. $x$ represents a single real number, but it could be any real number). As can be seen in the question when OP says: "I mean the symbol "x" is not all real numbers. So what sense does it make to follow For all real numbers by "x"?". – Flater Sep 26 '23 at 12:24
  • If it helps, I could've rephrased the first one as "where $x$ is the entire set of real numbers". – Flater Sep 26 '23 at 12:31
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    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. – Michael Bächtold Sep 26 '23 at 13:08
  • @MichaelBächtold: I'm saying that the order of operations is not as strict in everyday English (w.r.t. precisely what it is that you're pluralizing) compared to mathematics (where respecting the order of operations is highly important). You're not seeing the wood for the trees here. The specific phrase OP is using is only one example of a much larger phenomenon in spoken language: accounting for and working with technical inaccuracies. OP is trying to apply a stricter reading of the English language than a reasonable speaker of the language would apply, which is a fruitless endeavor. – Flater Sep 27 '23 at 23:45
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    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? – Michael Bächtold Sep 28 '23 at 05:50
  • @MichaelBächtold: My very first comment gave you a plain English example. Since then, I've also come up with better ways to rephrase OP's two interpretations (i.e. the one they accept and the one they don't), both in comment response to you and added to the answer itself to help improve the explanation. That last comment of yours contained nothing but a clever inversion of my words and a repetition of an already answered question, so I'm not sure what else you're looking for. – Flater Sep 28 '23 at 05:59
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    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. – Michael Bächtold Sep 28 '23 at 06:03
  • @MichaelBächtold For the purpose of the posted question and my answer to it "[all real numbers] [$x$]" is equivalent to turns of phrase such as "[all people] [who watch Macbeth]" in that there is no set expectation that we're scoping it to a single performance or multiple ones. All of this has been answered before. – Flater Sep 28 '23 at 06:16
  • @MichaelBächtold: "is not an example since there is no name being given to a single entity." Who said anything about this being related to the naming of entities? It seems to me that you're either inventing strawmen just to attack them or moving the goalposts to not have to acknowledge that your question has already been answered. I can no longer believe that you are asking questions in earnest. – Flater Sep 28 '23 at 06:17
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    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. – Michael Bächtold Sep 28 '23 at 07:31