Talk:Real number/Archive 4

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Continuous quantity

It seems to me that "continuous quantity" is not a common phrase, at least not in the mathematics books I see that talk about real numbers. I was wondering:

1. Is there is any good reference in a contemporary book for the term "continuous quantity"? In particular, in a book that is not about history or statistics?
2. As we all know, there are some quantities in physics that are measured with complex numbers, rather than real numbers. Are these not "continuous quantities" as well? It seems to me that the identification of real numbers with continuous quantities only works when continuous quantities themselves are identified as signed magnitudes on the real line, at which point the first sentence here might as well cut out the middleman. But this takes us back to #1.

— Carl (CBM · talk) 22:28, 9 May 2018 (UTC)

• I still think that referring to the "real line", with or without link, in the first sentence, as kind of "introduction" for the notion of reals, is a logical flaw of circularity. I am convinced that there must be abundant citations about reals forming "the" or "a" continuum, a term I would prefer to talking about "continuous quantity". I am averse to using the term "measurement" in connection with reals, even when I disagree that physicist make measurements of "complex numbers". All these mathematical constructs are just models in the physicist's worlds.

Here is my effort to contribute to the discussion:

Real numbers are mathematical constructs, defined in a way that satisfy most needs in arbitrary continua. They may/can be envisioned as distances or points on a line.

Eleuther should have left sleeping dogs lie. Should Eleuther have let sleeping dogs lie? rephrased for below 08:20, 10 May 2018 (UTC) Purgy (talk) 06:43, 10 May 2018 (UTC)

No, I disagree that it is circular. The line is the underlying notion and is intuitively well-specified. The ancients understood what it was; they just didn't know how to work with it. You don't need the reals to specify what the line is. When you have the concept of the line, and elaborate it in the only possible correct way, you will necessarily recover the reals. --Trovatore (talk)

I am fond of the "line", especially of the blank-faced Euclidean version (curtly behind "continuum", but immensely before "continuous line/quantity"), possibly even when linked (ridiculously?) to "line (geometry)#In Euclidean geometry", but I ferociously oppose to "real line", linked or unlinked, the use of which is, I insist, circular. Purgy (talk) 10:36, 10 May 2018 (UTC)

Sleeping dogs are still dogs. Eleuther (talk) 07:01, 10 May 2018 (UTC)

And incidentally, the correct word here is "let," not "left." Eleuther (talk) 07:32, 10 May 2018 (UTC)

@Purgy: as Trovatore says, people measured distances along lines for a long time, so there is no circularity in saying that this is one thing that real numbers can measure. Your "very bold" edit included the phrase "quantity along a line" which has, as far as I can tell, no meaning at all. The thing that real numbers measure along a line is distance, not some unnameable "quantity along".

But my question still stands: is there any modern source that defines continuous quantity? If not, I think we should avoid it in the first sentence as an overly specialized term. — Carl (CBM · talk) 12:25, 10 May 2018 (UTC)

Related to being bold - bold edits are good when they work towards consensus. On this page (previous section), several people have indicated they are in favor of talking about the real line as a way of understanding the real numbers, so simply removing references to the line is not likely to work towards consensus. It's also worth remembering that the first sentence of the article is not constructing the real numbers -- it is not talking about Dedekind cuts or Cauchy sequences, etc. Instead it is trying to identify the real numbers, so the reader knows what we are talking about. That kind of identification may well have some circularity to it. — Carl (CBM · talk) 12:34, 10 May 2018 (UTC)

The mention of a line in the first sentence is a reference to a mental representation (intuitive representation). Even if this mental representation is commonly accepted, there is no reason to impose it to everybody in the first sentence of an article. On the other hand, it should be useful to say why the concept has been introduced, and why real numbers are not simply called numbers. Therefore I suggest for the first sentence:

Real numbers are the mathematical formalization of the non-integer numbers, which are of everyday use.

This proposal is a first draft, that may and should be improved. In particular, I do not like the use of "non-integer". D.Lazard (talk) 12:59, 10 May 2018 (UTC)

That's a good idea to think about. I would replace "integer" with "rational", since that is the bigger leap (cf. the history behind the irrationality of the square root of 2). The other thing is to get a phrasing that does include the integer and rational numbers among the real numbers. One challenge is finding a sentence that does not apply to the complex numbers just as well (e.g. measurable quantities, continuous quantities). How to do that without referring to a line is something to think about. — Carl (CBM · talk) 13:49, 10 May 2018 (UTC)

I think we should hold the line on "line". That's how the reals are typically introduced to children, and for once it's an excellent choice. The ancients weren't disturbed about the irrationality of √2 for algebraic reasons — if they had been, they would have just said, OK, 2 doesn't have a square root, maybe surprising, but that's how it is. But they could see that there had to be a square root of 2, because it was the length of the hypotenuse of an isosceles right triangle with unit side.

It would be a serious mistake not to mention the line in the first sentence. --Trovatore (talk) 17:24, 10 May 2018 (UTC)

I think this excludes complex numbers.

∗Real numbers are  a  the more correct? 07:10, 11 May 2018 (UTC) mathematical formalization, which that more correct? 07:10, 11 May 2018 (UTC) rounds off the numbers, which are of everyday use. They may/can be envisioned as distances or points on a line.

Please, just don't make it the "real line" again. Purgy (talk) 17:34, 10 May 2018 (UTC)

There's a linguistic or rhetorical infelicity in defining the reals in terms of the "real line", not really a logical circularity I think, but it definitely doesn't sound good. I hadn't realized that was the sticking point for you. I agree that "line" is better than "real line". We can point to the real line article somewhere later, if we want a link to that article (and if indeed that article shouldn't just be merged here anyway). --Trovatore (talk) 17:43, 10 May 2018 (UTC)

As for defining "real number" in terms of "number", no, I don't like that at all. "Number" is a very fuzzy word; means all sorts of things. Defining "real number" in terms of "number" is a much worse circularity than defining it in terms of "line".

"Line" also means all sorts of things, but I think most readers have come across the Euclidean concept — a line is infinitely thin (that is, zero-width), infinitely long in both directions, consists of points of zero size, and has no holes in it. That description is enough to make the reals well-specified, with a couple of quibbles. That's the concept we should be starting with. --Trovatore (talk) 17:50, 10 May 2018 (UTC)

IMO, "of everyday use" excludes the complexes for everybody. By the way, a reader who knows of the complexes knows enough mathematics for not needing any explanation of what are the reals. Thus we do not have to take care of the complexes in this sentence. Here, understandability by the layman is much more important than logical and mathematical correctness. As "fuzzy words" are not really a problem for the layman, if he think understand them, they are not a problem here. D.Lazard (talk) 18:02, 10 May 2018 (UTC)

Well, if "fuzzy words" are not a problem, then "line" shouldn't be.

I actually don't agree that readers who know about the complex numbers don't need any explanation about what the reals are. The reals are a subtle and non-obvious concept. Their most salient distinguishing feature is their topology, which is what we should be trying to get across (informally) right up front. Just because you're comfortable with −1 having a square root doesn't mean you've thought about the topology of the reals. --Trovatore (talk) 18:20, 10 May 2018 (UTC)

May I point to the effort of avoiding introducing "reals" as "numbers", but instead as a "formalization" of something in "everyday use", similar to the notion of "line" in this here context. I share the opinion of Trovatore that mentioning the line is helpful for most readers, and is no obtrusion, regardless if it helps against complex numbers or not. Purgy (talk) 18:07, 10 May 2018 (UTC)

Here are a few thoughts:

1. The real numbers are not a "formalization". There are formalizations of the real numbers but the real numbers themselves are not a formalization, unless every mathematical object is somehow a formalization (in which case there's no reason to mention it).

2. Saying that the real numbers "round off the numbers" does not make sense to me. "Round off" means to round a number to an approximation, i.e. we can round off π to 3.1.

3. The word "number" on its own does not have much meaning. It could mean an ordinal number, surreal number, natural number, or many other things.

— Carl (CBM · talk) 20:37, 10 May 2018 (UTC)

And here some more:

1. Reals ARE a formalization of an indeed very abstract idea, and the reason for mentioning this here is not that math is about formalization, but here it is exactly about "the formalization that ..." Roughly: I do not believe in real numbers in any philosophical reality, besides as reals-in-themselves, they are no measurements, neither real nor complex, ...
2. I humbly ask for linguistic help in finding the right wording for the meaning of "completion", which I had in mind, and wanted to circumscribe in a most accessible way by "round off".
3. Well, ... I think the phrase "everyday use" excludes some of the incriminated deviances, and integers are a nice object to continue a "rounding off"/completioning process, which might have started at the naturals.

Purgy (talk) 07:10, 11 May 2018 (UTC)

Purgy, for question 2, I think the idiomatic phrase you want is "rounding out," rather than "rounding off." Then there's no danger of some dumbhead thinking you're talking about removing precision. Or, of course, you can just cay "completion." Cheers, Eleuther (talk) 06:09, 12 May 2018 (UTC)

The reals are not a "formalization" of anything. They aren't formal in the first place. I'm not sure what you're trying to say, exactly, but "formalization" is definitely the wrong word.

I thought maybe you wanted to say that they're an "idealization" — that is, you start with a messy group of empirical observations, and you abstract out an underlying idea, one that perhaps none of the observations exactly match, but which simplifies discussing what they have in common.

But then I noticed you say they're a "formalization" of a "very abstract idea" (what idea?) so maybe you actually want the other direction from idealization — say, "realization" or "instantiation".

I'm not sure exactly what word you want because I haven't quite understood what you're getting at. But "formalization" is definitely not it. --Trovatore (talk) 06:40, 12 May 2018 (UTC)

Again I oppose: I hold dear that mathematicians, talking about reals, do talk about formal objects from the formalization of an idea, and I am convinced that a large majority of potential readers consider this very idea (take any construction of reals out there) a "highly abstract one". I do not want to enter the philosophical realm discussing whether numbers are a realization or an abstraction. I do admit that a certain axiomatic system describing/generating(?) the reals is one "realization" or "instantiation" of them, but I think discussing a meta-equivalence of axiomatic systems for the reals-in-themselves is way beyond the scope of this article. The category of "is an idealization" is similar to "is beautiful" and not in my focus here. Maybe I could get some consent against your categorical But "formalization" is definitely not it. from D.Lazard, it was not me, who introduced Real numbers are the mathematical formalization of the non-integer numbers, which are of everyday use. as a first draft. I just consider this a good approach, and tried to work upon:

Real numbers are the mathematical formalization, that rounds out the notion of numbers, which are in everyday use. They may/can be envisioned as distances or points on a line.

This is the current version of my suggestion, based on D.Lazard's draft and critique a perceived. Purgy (talk) 07:40, 12 May 2018 (UTC)

No, sorry, the reals are not a formalization of anything. There are formal theories that talk about the reals. But you must not confuse the formal theories with their objects of discourse.

Look, I'm not saying you can't be a formalist. If you're a formalist, then you may well take the position that the objects of discourse don't exist. But not existing is not the same thing as being a formalization.

The formalization talks about the reals, whether or not they exist. But the reals themselves are not a formalization. --Trovatore (talk) 07:54, 12 May 2018 (UTC)

You must not confuse "formal theories" with formal systems. "Formal" and "formalization" have been used in mathematics a long time before the introduction of the first formal systems. This being said, I agree that, formally, this is not the reals that are a formalization, this is the concept of reals, which was formalized by the introduction by Dedekind and others of a formal definition (many occurrences of formal, I assume them even I have no formal definition of them :-). Thus a formally correct first sentence would be "The concept of real number is a formalization ...". However, such a formulation would be a little pedantic for the first sentence of the lead, and "The real numbers are a formalization ..." seems acceptable, as, for users of mathematics, mathematics is essentially a formalization of the methods of modelization. —D.Lazard (talk • contribs) 09:08, 12 May 2018 (UTC)

No, the reals are not a formalization, period. The concept of the reals is not a formalization either. There is no reason at all to mention formalizations in the first sentence. --Trovatore (talk) 09:20, 12 May 2018 (UTC)

As I'm sure you know, D., and I'm less sure about Purgy but probably, it is standard in mathematics to talk like a Platonist whether you are one or not. This is the convention we should follow here. In the lead, or at least the early part of the lead, we should describe the reals uncritically as objects. What exactly that means is really a topic for a philosophy-of-math article more than for this one, though the reals are an interesting enough dividing line (there are people who are realist about the natural numbers but formalist about the reals, for example) that some discussion might not be out of place. I don't really think it belongs in the lead though. --Trovatore (talk) 09:36, 12 May 2018 (UTC)

"The reals are not a formalization, period." Which sources allow you to making such an authoritative assertion. Authoritative assertions are of no value here. You are confusing your opinion with truth. D.Lazard (talk) 09:59, 12 May 2018 (UTC)

Actually, anyone who wants to say that the reals "are" a formalization is the one who should show sources. And even then it might be "undue weight". I think very few high-quality sources in English are going to describe the reals as a "formalization". --Trovatore (talk) 10:03, 12 May 2018 (UTC)

I don’t think mentioning the idea of “formalization” in the first sentence, is the right way to go here. For me, the most basic property of real numbers is that they measure distance along a line. This is born out, it seems to me, by the historical development of what a number is. Euclid, for example, identified numbers with line segments. So where we would say "x is a multiple of y", he would say "the line segment x is measured by the line segment y". It was the (shocking!) discovery that not every pair of line segments had a common measure (i.e. that their ratio was not always a rational number), that pushed the idea of number beyond rational numbers, in the first place. Paul August ☎ 10:08, 12 May 2018 (UTC)

In the Princeton Companion

I thought it might be good to look at a couple sources. In the Princeton Companion to Mathematics (edited by Timothy Gowers), they motivate the real numbers by examples such as the square root of two. Then they say (informally) "The real numbers can be thought of as the set of all numbers with a finite or infinite decimal expansion". Would that be a reasonable way for us to proceed here? I also found at least one calculus book that gives the same characterization. This seems like a different way to avoid any awkward "continuous quantity" phrasing, and it has the benefit of being sourced to a high quality publication. — Carl (CBM · talk) 13:58, 10 May 2018 (UTC)

I think that is not a good approach. It starts with the representation rather than with an intuitive description of what is being represented. And it does it specifically in base 10, whereas there is nothing special about base 10 when talking about the reals. --Trovatore (talk) 17:17, 10 May 2018 (UTC)

I agree with Trovatore. Decimal represtations of real numbers are important, but decimal numbers are a different topic from real numbers, and the first sentence of the article should not reinforce any ambiguity. Sławomir Biały (talk) 18:07, 10 May 2018 (UTC)

In addition I bother that decimal expansion taken as the fundamental idea could be interpreted by some readers as a green light to the negation of 0.999...=1. Boris Tsirelson (talk) 19:15, 10 May 2018 (UTC)

I think this is one of Gowers's hobby-horses, the fact that you can define the reals in terms of decimal representations, modding out by the appropriate equivalence relation. And it's true, you can. In that approach, you make 0.999... equal to 1 by definition. But it's not a very perspicuous approach, and I think not one of the more popular ones, for good reason. --Trovatore (talk) 19:19, 10 May 2018 (UTC)

To be fair, I believe they are also defined that way in many elementary algebra and calculus books. I would suspect it is probably one of the most popular approaches to defining the reals in settings below the level of real analysis, along with references to the real line. Dedekind cuts and Cauchy sequences are not going to appear in grade school texts. The point of the first sentence is to have something at that lower level. — Carl (CBM · talk) 20:39, 10 May 2018 (UTC)

Dedekind cuts and Cauchy sequences do not appear in grade-school texts, but lines absolutely do. --Trovatore (talk) 21:37, 10 May 2018 (UTC)

Maybe we should mention both approaches, with a note about their equivalence. Boris Tsirelson (talk) 21:33, 10 May 2018 (UTC)

Yes, but not in the first sentence. One sticking point for me is that it privileges base 10. We could make it not privilege base 10, but only at the expense of lots more verbiage that's not really going to fit there. --Trovatore (talk) 21:37, 10 May 2018 (UTC)

I am perfectly fine with referring to the real number line as the model for the real numbers; I was suggesting the decimal option as a possible alternative, but I don't really have a preference. I spent a few minutes looking at elementary algebra books, and they seem to either use the real line or decimals as their way to explain the set of real numbers. (As for privileging base 10 numbers, I think that may be a lost cause for a general audience article, unless there is still any society that uses non-base-10 in their everyday affairs.) I do think that an article like this should be particularly careful to be accessible. — Carl (CBM · talk) 21:41, 10 May 2018 (UTC)

Whether anyone uses non-base-10 is not really the point. The point is that base 10 is a representation, not the the thing being represented. The importance of other radices is not so much that anyone uses them, as that they show that a particular radix cannot be of the essence.

A fairly reliable rule of thumb for the distinction between recreational mathematics and "real" mathematics is, if it depends on the radix, it's probably recreational. --Trovatore (talk) 21:45, 10 May 2018 (UTC)

Another interesting discussion of the reals is to be found in the historical note to Bourbaki's General Topology, Volume 1, Section IV, which begins thus: "Every measurement of quantities implies a vague notion of real numbers." The line is scarcely mentioned in their historical discussion. (With some irony, I note that this was very closely connected to User:D.Lazard's point raised in the previous thread. Perhaps it's a French thing?) It's not clear that this gives us any good insights into how the first sentence of the article should be written, but I think the rest of the lead should focus at least somewhat on measurement and less on the real line per se. Sławomir Biały (talk) 10:10, 12 May 2018 (UTC)

I does not think that this difference of point of view is a question of geographic origin. It is rather a different between a pedagogic approach, which emphasizes on intuitive motivation for young people, and a working-mathematician approach, which emphasizes on applications and historical motivations (Dedekind and Cauchy were not concerned by geometry when they developed the modern concept of reals), D.Lazard (talk) 14:07, 12 May 2018 (UTC)

I was not serious in my speculation of the French connection. While I share Trovatore's view that the Princeton Companion is not ideal, and that the primary conceptual motivation for the reals should be the line, I think something about the practical importance for measurement, calculus, and the sciences should be drafted for the lead. Sławomir Biały (talk) 15:24, 12 May 2018 (UTC)

Real numbers and "the line"

I think that Trovatore is very wrong on this issue, which has occupied the last few talk sections, and in which he has been an active disputant. But he's not alone in this error -- he's just expressing an orthodoxy that has been drummed into several generations of high school and college students, at least in the US. I'm picking on him because he has defined his position so succinctly above, i.e., because he writes well. In particular, he has said above that

The line is the underlying notion and is intuitively well-specified. The ancients understood what it was; they just didn't know how to work with it. You don't need the reals to specify what the line is. When you have the concept of the line, and elaborate it in the only possible correct way, you will necessarily recover the reals.

This is gibberish. In the first place, what does he mean by "intuitively well-specified?" Whose intuition his he referring to here? His? Mine? Yours?

The real numbers are a number system that includes the integers and the rational numbers — Preceding unsigned comment added by Eleuther (talk • contribs) 09:35, 12 May 2018 (UTC)

This is really not the place to discuss it at length, but having been called out directly, I will give brief answers. I'm happy to discuss it further on my talk page, or yours, but any further discussion here ought to be justified by some direct connection to what should appear in the article.

"Intuitively well-specified" means that, if you've understood the intuition correctly, then your mental picture must correspond to the true Platonic reals. A line has zero width, infinite length in both directions, and can't be pulled apart into two pieces without breaking it. That's enough to specify the reals up to local topology, anyway. If you have understood the intuition, you can use it to distinguish between correct and incorrect formalizations.

"Whose intuition"? Correct intuition. There is a right and wrong about this, though it is not subject to proof in the ordinary mathematical sense. --Trovatore (talk) 09:56, 12 May 2018 (UTC)

"he's just expressing an orthodoxy" – wow... I'd say, yes, Wikipedia is generally expressing an orthodoxy in the first place.

"The real numbers are a number system that includes the integers and the rational numbers" — rather, one of such number systems; recall complex numbers. Boris Tsirelson (talk) 10:21, 12 May 2018 (UTC)

Shall we return to discussing the article please? Paul August ☎ 10:26, 12 May 2018 (UTC)

Hi, sorry, I seem to have saved this section inadvertently while it was still a very rough work in progress, and while it still didn't say any of the main things I wanted to say. Please ignore. Sorry again. Eleuther (talk) 22:25, 12 May 2018 (UTC)

I agree with keeping the geometric line in the lede description for "real number"---it has an intuitive feel for much of the reading lay public. But my guess is, the typical lay reader will not have any (good) intuitive feeling for the phrase "continuous quantity"; in fact, he/she will likely suffer a 'lost cause' or 'eyes-glaze-over' feeling upon encountering it. The linked (C-class) page 'Quantity' doesn't read in 'lay terms' until the last section---quantity#Further examples---for what it's worth; and there is no lay explanation in the article of any connection to real numbers.

Even I, long of a technical/engineering career, cannot relate to a singular "number" itself being described as a "continuous quantity"---algebraic variables excluded. That is, here the phrase seems to imply: the set of all real numbers, rather than a(one) real number. It's doubtful the phrase "continuous quantity" has any knowledge recognition among the lay public, and the linked page Quantity---which is chiefly concerned with describing & measuring variables of multitude or magnitude, continuum or discontinuity---does not speak to the phrase orto real numbers.

I offer the following for your consideration:

• (Current) In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.

• (alt-A) In mathematics, a real number is a value that can represent distance along a straight, continuous line, or geometric line.

• (alt-B) In mathematics, a real number is the value of a distance along a straight, continuous line, or geometric line, from the zero point to a given point of the line.

A follow-on descriptive sentence might read>

• Among its other properties, a real number takes either positive or negative values; it will be either a rational or irrational number, and not a complex number.

Both alternatives have the advantage that the reader can ignore linking to "geometric line" and still have an intuitive feel for: ".. a distance along a straight, continuous line". Thanks, Jbeans (talk) 03:07, 17 May 2018 (UTC)

A real number takes values? No, it is not a variable nor function, it does not take values; each real number is a value.

More important point: we understand why we emphasize distances along a line, but some readers do not (I guess). They are puzzled: just distances on a line? Not distances on the plane, or space? Not areas and volumes? Not angles? Not masses, voltages etc? I'd say, the first phrase cannot just say "what is a real number?" (this is desirable, of course; but maybe impossible); rather, it should introduce the reader into the distinction between "practical" idea of numbers (approximate, subtleties aside) and "theoretical" idea of numbers (continuum, irrationals etc). Boris Tsirelson (talk) 05:58, 17 May 2018 (UTC)

As said, I am a fan of using an informal notion of a line as guiding paradigm to the notion of the reals, but am rather skeptic about all the epitheta like continuous, straight, geometric, ... and especially "real". While I mostly agree to the above arguments, I find not much of an improving move in the suggestions.

Wholesale, I think that it is helpful to tell the readership that the real numbers are not very real (in the sense of 'physically manifest') in the real (in the sense of 'as encountered') world. Real numbers exist in the way they do exactly to satisfy (most) mathematicians' needs for rigor, so that they can be used without bothering about exceptions and quirks (irrational, constructible, algebraic, computable, transcendental, ???).

I.e., I like to start saying that reals can be (vaguely) found along a line or in other measurements, and that their overall usefulness abounds, in spite of the next, that their accessibility results from one of several "unwieldy realizations" of highly abstract and partly unintuitive ideas (the history, leading to the reals should be told). Purgy (talk) 09:14, 17 May 2018 (UTC)

True. Abstractions are a half of the truth. The second half, I'd say. And here is the first half of the truth: real numbers are all (integer or not) positive and negative numbers (and zero), widely used as possible results of measurements (geometric, physical, etc). For some readers, this is enough. Others turn to the second half. By the way: is the (rational!) number 1+10^-1000 a possible result of a physical measurement? This naive question can be a kind of a bridge from the first part of the truth to the second part. Possible values versus possible results of measurement (of, say, a distance, or a volume, or a mass); the same or not the same? Boris Tsirelson (talk) 10:32, 17 May 2018 (UTC)

Response to: Boris Tsirelson, Purgy, (rather than require chasing intercalations all over the place):

R1---Agree; the text of the follow-on sentence can be adjusted, such as:

• Among its other properties, a real number is either a positive or negative value; it can be either a rational or irrational number, but not a complex number.

R2---re improvements, there are two chiefly---proposed by both alternatives: {One} The link to Quantity (with its phrase "continuous quantity") should be dropped for cause that the discussion there does not describe any relationship between "real number" and"continuous quantity" (Pls, don't presume it's just fine---read for yourself). Neither is "real number" actually discussed in the linked article---it is mentioned only 3 times in 3 separate sections. (All this reading for no good explanation of the subject is simply harmful in terms of reaching (and informing) the lay reader.) ... {Two} New specific text is added, saying a real number can be visualized as: "distance on a straight, continuous line" (which is both a phrase and a geometric form that a maximal number of the lay public relates to, very worthwhile because---the need is: make the discussion accessible tothe lay reader). Citing this particular geometric form (a straight line) does not offend the value of other forms that contain real numbers---it merely takes best advantage of what's already out there and provides an efficient, intuitive connection to the lay public.

Either one of the proposed alternatives plus the 'follow-on' sentence would serve the lay reader much better than the current lede sentence. Regards,Jbeans (talk) 13:48, 19 May 2018 (UTC)

In conclusion, re "the line"

Fellow editors, please review my two recent posts---they are in the section immediately above---regarding my two recommendations that follow. Please note the current lede sentence:

• "In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line."

I offer my concluding arguments for editing this sentence---as the following:

Page-article "Quantity" and particularly its phrase "continuous quantity"---as (nominally) discussed in that page---is faulty and unhelpful: 1) in that the article fails to inform the reader re the actual subject, ie, a "real number" as a "continuous quantity"; 2) it fails to make any meaningful connection between the two concepts; 3) and thus it is unhelpful to the lay person reading Wp and attempting to learn the basic parameters of real numbers; 4) the lay public does not have an intuitive grasp of any mathematical concept of "continuous quantity"---nor is the concept explained very well in Wp. :::: IMO, the (proposed) phrase "distance along a straight, continuous line" provides the most intuitive feel (for grasping the concept of "real number") to the largest number of the lay public.

Therefore, for a better description of "real number", I move that the current lede sentence be replaced with new lede and second sentences, as follows:

• In mathematics, a real number is a value that can represent distance along a straight, continuous line, or geometric line. A real number is either a positive or negative value; it can be either a rational or irrational number, but not a complex number.

{ Here the current text continues; and note, the phrase "geometric line", immediately above, is presented in apposition to the phrase "a straight, continuous line", and they are not redundant. }

Editors, please reply to both points, pro or con. Particularly---if you have found in "Quantity/continuous quantity", some specific text that helps the lay reader to better understand real numbers---I will be most grateful if you would please identify the text you see. (Please do not offer the John Wallis blokequote re ratios of magnitudes as real numbers, as here. Thanks & grins). Regards, Jbeans (talk) 02:29, 23 May 2018 (UTC)

No, this change would make the opening sentence significantly worse. "straight, continuous line, or geometric line" doesn't really make much sense. How can a line be not straight? or not geometric? What's a continuous line? Trying to cram in stuff about positive and negative, rational or irrational, and complex numbers so quickly isn't a very good approach either. The current opening paragraph already eases into the rest more gently. –Deacon Vorbis (carbon • videos) 03:37, 23 May 2018 (UTC)

Add: on second thought, some sort of rewording to make it clear that negative numbers are included might be good (a distance is often taken to be positive only, as the current version states). But the whole rigamarole that's proposed doesn't directly address that anyway. –Deacon Vorbis (carbon • videos) 03:46, 23 May 2018 (UTC)

I'm just going to pick on one little point, without commenting yet on the rest of the proposal. That is that I see no particular reason to mention a "straight" line. A Euclidean line, of course, is straight, but that seems to have almost nothing to do with its connection to the reals; a curvy line would do just as well. So at the very least I would leave that word out. --Trovatore (talk) 04:30, 23 May 2018 (UTC)

It seems, no one found a good one-line "definition" for real numbers. I guess, this is not possible. Here are some quotes:

• "A rational number or the limit of a sequence of rational numbers, as opposed to a complex number." [1]
• "The short, simple answer used in calculus courses is that a real number is a point on the number line. That's not the whole truth, but it is adequate for the needs of freshman calculus." [2]
• "All numbers on the number line. This includes (but is not limited to) positives and negatives, integers and rational numbers, square roots, cube roots , π (pi), etc." [3]
• "Real numbers are, in fact, pretty much any number that you can think of. This can include whole numbers or integers, fractions, rational numbers and irrational numbers. Real numbers can be positive or negative, and include the number zero. They are called real numbers because they are not imaginary, which is a different system of numbers." [4]
  

Boris Tsirelson (talk) 04:48, 23 May 2018 (UTC)

• Oppose. My opinion that the proposals by Jbeans are to my measures no improvement to the status quo is not affected by the recent statements by Jbeans. To me, the list by Boris Tsirelson indicates that it is not easy to find something better, and my preference of defaming the reals as something abstractly engineered, at least a little bit, has not changed. Negatives yes, complexes no, straight, geometrical, continuous, "number lines", ... are all constructs way above the accessible math for the targeted readership, and partly of no obvious help at all. I still like something along Trovatore's line and D.Lazard's formalization. Purgy (talk) 09:41, 23 May 2018 (UTC)
• Oppose. The proposal moves further away from a satisfactory beginning. The first sentence emphasizes the straightness of the line or its other (unspecified) geometric properties, which are irrelevant. The second sentence is completely redundant with the rest of the first paragraph. Sławomir Biały (talk) 11:13, 23 May 2018 (UTC)

It seems that no one is in favor that "straight, continuous line" be put (directly) before the reader. Regardless, please remember, two points were proposed; the 2nd was:

that the link "Quantity/continuous quantity" be deleted unless,

• " Editors, please reply to both points, pro or con. Particularly---if you have found in "Quantity/continuous quantity", some specific text that helps the lay reader to better understand real numbers--- ... please identify the text you see." (See my previous post, closing paragraph).

To date, no comments, pro or con, have been posted re the 2nd point---which raises the question: Is there anyone who will identify the text where: the link "continuous quantity" clearly narrates (to the lay reader) a meaningful connection to "real number"? Regards, Jbeans (talk) 02:38, 5 June 2018 (UTC)

@Jbeans, I do not think that the above discussion was targeted to express favor for some detail or linking of the status quo, but only shows no favor for any of your two suggestions, compared to this status. To me it is obvious that for the time being there is no agreed upon improvement visible.

Re your final question, I tend to the opinion (please, refer also Boris Tsirelson's punctuation) that for the highly sophisticated real numbers there might be no "clearly narrating" text available at all, but just some motivation leading to various axiomatizations, probably not perceived as "clear" by an uninitiated audience. Purgy (talk) 08:30, 5 June 2018 (UTC)

If the second part of the question concerns a lack of clarity in the article quantity, it should be asked at Talk:Quantity. I see nothing wrong here with referring to a real number as a continuous quantity in the text, as this is clear enough in natural language without the need to supply precise technical details. It does not even require a link to quantity, if that's your main beef. Sławomir Biały (talk) 11:02, 5 June 2018 (UTC)

Link Continuous Quantity vs just Quantity

@Trovatore:@Deacon Vorbis:@Purgy Purgatorio:@37.71.17.36: It seems to me that Trovatore agrees with Deacon Vorbis and IP that the pipe should not exist, based on the comment for the edit, but undid the removal of the pipe. This confuses me greatly. Is there a clear consensus to linking just Quantity or is there something still to be discussed? GiovanniSidwell (talk) 15:22, 13 June 2018 (UTC)

It's not clear to me there should be a link there at all, but what is clear to me is that piped links of the form [[bandersnatch|frumious bandersnatch]] are almost never a good idea. The link just points to bandersnatch, so the "frumious" has nothing to do with the link and should not appear in blue. The user who clicks on the link has no warning that the article is merely bandersnatch, and this is a violation of the least surprise principle. --Trovatore (talk) 17:31, 13 June 2018 (UTC)

Oh, now I see that I actually did the opposite of what I thought. Mea culpa. I'll fix it. --Trovatore (talk) 17:33, 13 June 2018 (UTC)

@GiovanniSidwell, already my revert was an oops. Mislead by the IP's edit summary I wrongly assumed, after only sloppily looking, that the word "continous" had been removed completely (cf. my summary). If it had not been for the possibility of misinterpretion as sarcasm, I would have even thanked Deacon Vorbis for his revert. I think that a hint to an intuitive notion of a continuum is essential for reals, and I am along the lines of Trovatore in not being sure, if at this point a link is appropriate at all. A "broader founding" of vagueness for "quantity" might not be really optimal. I apologize for having caused these troubles by my inadvertency. Purgy (talk) 07:03, 14 June 2018 (UTC)

Etymology in the second sentence

Re this edit: I agree with Deacon Vorbis that it's a point worth mentioning. But in the second sentence? I don't think it's that important. It strikes me that it distracts from the flow of thought, and should be deferred to a "History" or "Etymology" section.

(Speaking of history, this article is still missing any mention of Zeno, which as I have said before is a hard-to-understand omission. I should probably quit whinging about it and just try to do something about it.) --Trovatore (talk) 22:59, 18 January 2019 (UTC)

It certainly doesn't absolutely have to be in the second sentence, but I still think it should be in the lead. It's kind of an important note about the naming, that other numbers aren't any more or less real, but that it's just the name that got attached and stuck. (The article at Imaginary number does something similar, and I think that's good, for the same reason). –Deacon Vorbis (carbon • videos) 23:04, 18 January 2019 (UTC)

This sentence could easily placed after the first paragraph, either as a paragraph for itself or as a first sentence of the second paragraph. In the latter case, the transition with the second sentence could be assured by adding at the beginning of the second sentence something like "Following his idea of coordinates, ...". After all, the concept of number line consists of associating each point of a line with its Cartesian coordinate. D.Lazard (talk) 09:25, 19 January 2019 (UTC)

Construction of Real Numbers

Changes to this article are proposed, given new results provided in . The proposal is to include a construction of real numbers as an extension of natural numbers in the universe of sets.JPMural (talk) 20:24, 3 April 2019 (UTC)

The construction given in the paper may very well work. Its claim to canonicity seems much more in need of support, and not really subject to "proof" in a single paper. It's something that the mathematical community would need to come to a consensus about, and that would take years at a minimum, decades more likely. To uncritically cite the claims of a paper published this January for extraordinary assertions about the importance of results in the same paper — well, no, I don't think we're going to do that. --Trovatore (talk) 21:06, 3 April 2019 (UTC)

Agree. Paul August ☎ 21:08, 3 April 2019 (UTC)

(edit conflict) If nothing else, the statement you added that "Some modern constructions propose real numbers built directly from the structure of integers. However, it has been recently proven that real numbers are subsets of N." doesn't seem to be supported by the reference you give. There are many, equivalent constructions of the real numbers, and this paper just seems to give yet another, not make any sort of grand claim about the true nature of what the real numbers Really Are. –Deacon Vorbis (carbon • videos) 21:14, 3 April 2019 (UTC)

On second glance, maybe the author is actually making these sorts of claims, but to echo the points made above, this is more philosophical and probably not shared by the wider community. –Deacon Vorbis (carbon • videos) 21:19, 3 April 2019 (UTC)

The extraordinary claims made are fully supported in the paper. Maybe a few full review of the paper is in order.JPMural (talk) 09:35, 4 April 2019 (UTC)

Wikipedia policy WP:OR asserts that one can include in Wikipedia only results that are mentioned in secondary sources (this is necessary, but not sufficient, as notability is also required). So a full review is not a work for Wikipedia, and, for considering this result, one must wait that some reliable sources cite it.

Nevertheless, I have had a quick look on this article, and it seems not to be extraordinary at all: It is well known that the binary expansion of a real number may be identified to a subset of the integers (the set of the places where there is a one). So, the real numbers may be identified with a subset of $2^{\mathbb {Z} }.$ The main article claim is that the basic operations on reals can be realized by set-theoretic operations on subsets. This is not a surprise, and has essentially been used by computer scientists for hardware implementation of arithmetic. As I have read only the beginning of the article, I can have missed something, but, again, if there is more in the paper, we must wait a secondary source that explains the true new result, if any. D.Lazard (talk) 10:45, 4 April 2019 (UTC)

Yes. It is good to wait for a second source.JPMural (talk) 11:12, 4 April 2019 (UTC)

The paper makes reference to Paul Benacerraf's "What Numbers Cannot Be" as validation we have new results.JPMural (talk) 11:16, 4 April 2019 (UTC)

As D.Lazard explained above, that the results are "new" is not a sufficient reason for inclusion here. Paul August ☎ 13:25, 4 April 2019 (UTC)

"that the results are "new" is not a sufficient reason for inclusion here."

Thats clear, but youll have to see the paper to see what kind of "new results" we have. In my opinion they are well worth including, but its not just my opinion that counts so ill wait.187.189.208.183 (talk) 23:07, 7 April 2019 (UTC)

Well, it's not really our opinion that counts, either, at least not in theory. Naturally people's individual views are likely to color how they see the literature, and I don't think that can really be avoided.

That said, though it's off-topic, I'll indulge myself to say that I took a quick scan through the paper, and I don't see any real argument for canonicity. It kind of looks like it might go something like, this construction has this and this and the other nice properties, therefore it's the canonical one. I haven't checked the nice properties, but even assuming that part of the argument is correct, I'm just not really seeing the "therefore".

This is not the place to discuss it, but if you want to leave a note on my talk page clarifying that key step, feel free. --Trovatore (talk) 00:28, 8 April 2019 (UTC)

Sections 6 and 7 justify this. Reading those, and the following essay on Benacerraf's paper will help seeing this:JPMural (talk) 19:51, 8 April 2019 (UTC)

I don't think reaching the bar for some kind of representation of content requires all that widespread a conversation in the literature, but that proposed material is very new actively counts against inclusion in Wikipedia, which strongly favours material supported by secondary and tertiary sources. I recommend drafting the material you want included in your userspace and seek feedback on it, here and elsewhere, which might lead to discovery of related previous work. — Charles Stewart (talk) 07:02, 9 April 2019 (UTC)

I can see it is very wise to wait for second opinions in most everything. In mathematics, there may be an exception to this because if you prove something, its proven. All you have to do is read the proof.JPMural (talk) 17:50, 9 April 2019 (UTC)

Yes; but "canonicity" cannot be proven, just because it is not defined. Or did I miss its (mathematical!) definition somewhere? Boris Tsirelson (talk) 09:33, 1 December 2019 (UTC)

References

ℊ

In Unicode, the character ℊ is labelled ‘real number symbol’. As far as I can tell it was present, along with this label, in the very first version of Unicode. Does anyone know what is meant by this? Is there an area in mathematics that uses lower case g as a symbol for (a/the) real number(s)? — Preceding unsigned comment added by 2A02:A457:9497:1:F938:499E:52BC:CB4C (talk) 10:22, 25 April 2020 (UTC)

I remembre that in his article Mathematical typography Donald Knuth mentioned that, at some time, in the publications of the AMS, g did not have the same shape in superscript and on the normal line. One of the shapes was ℊ. The explanation, if any, could probably be found there. D.Lazard (talk) 10:43, 25 April 2020 (UTC)

I've consulted it and your memory is a blend of two independent observations of Knuth.

1. For years AMS used a slightly different font in subscripts and superscripts than on the baseline; he wasn't happy about this.

2. AMS went through several font transitions. At one point, they switched to Times Roman, but at first they opted to use a single-storey g in italic text like their previous font did, rather than the double-storey g of italic Times Roman. Neither looked script-like though.

In Latex using the default font, regular text gets a two-storey g and italic text gets a single-storey g, regardless of whether it appears in body text, a formula, subscript or superscript.

I think that makes this explanation unlikely, for now. — Preceding unsigned comment added by 2A02:A457:9497:1:F938:499E:52BC:CB4C (talk) 19:15, 28 April 2020 (UTC)

Blackboard vs. regular bold

@Trovatore: Greetings! Regarding this revert...MOS:BBB says "the LaTeX rendering (for example $\mathbb {Z}$) or the standard boldface must be used. As with all such choices, each article should be consistent with itself". Given that this article has an image that uses blackboard bold (File:Number-systems.svg), today I was making the rest of the article consistently use LaTeX instead of regular bold. Except, obviously, where the article is talking about the difference between the two. After this revert, it looks to me like this "Vocabulary and notation" section uses R in places where it could be using LaTeX for consistency with the image, sometimes does in fact use LaTeX where either style would be OK, and is also using the Unicode character ℝ, which is not allowed at all (except when discussing the character itself, which it looks like this passage isn't). Given that, does my edit make sense or am I missing something? Thanks! -- Beland (talk) 03:26, 11 January 2021 (UTC)

My concern is that we not give the impression that BBB R is "the" symbol for the reals. At least in origin, the symbol for the reals is R, and it's rendered on a blackboard using blackboard bold.

Now, it is true that BBB R often does appear in other places than blackboards, and it often does mean the reals. I could live with using it that way in the article, as long as we explain at first reference that it can also be R. --Trovatore (talk) 03:31, 11 January 2021 (UTC)

In the version with my changes, the intro does say at the end of the first paragraph: "The set of real numbers is denoted using the symbol R or $\mathbb {R}$." -- Beland (talk) 01:29, 17 January 2021 (UTC) (@Trovatore: -- Beland (talk) 19:45, 17 January 2021 (UTC))

The reals

I wanted to record here that "the reals" is not proper English, because "real" is not a noun. Also, there is a difference between "the real numbers" and "the set of real numbers"; the latter (or "the space of real numbers" or "the field of real numbers", as appropriate to context) should be preferred when referring to $\mathbb {R}$. Ebony Jackson (talk) 00:31, 8 February 2021 (UTC)

https://en.wiktionary.org/wiki/real#Noun --Modocc (talk) 00:54, 8 February 2021 (UTC)

I grant that it is sometimes used colloquially, and yes, now that you mention it, I can even find it in published dictionaries! Still, it is the kind of sloppy verbiage that I would not expect to find in a respectable encyclopedia (though someone will surely prove me wrong...) Ebony Jackson (talk) 04:23, 8 February 2021 (UTC)

I agree, and I think "reals" should be changed to "real numbers" throughout this article. This article also links to Tarski's axiomatization of the reals, which I think should be renamed.—Anita5192 (talk) 04:29, 8 February 2021 (UTC)

It is not "colloquial". It is perfectly standard usage in research mathematics, including in journal papers. Admittedly encyclopedic writing is a tad more formal even than research publications, but still I see no need to change it. --Trovatore (talk) 06:36, 8 February 2021 (UTC)

I agree, "proper English" is (in the appropriate context) whatever is commonly used. Paul August ☎ 14:10, 8 February 2021 (UTC)

Not only is it standard today, it has a long history with respectable authors. For example, Hamilton uses it in his Elements of Quaternions (1866):

Such scalars are, therefore, simply the Reals (or real quantities) of Algebra...p. 10

The Oxford English Dictionary has multiple quotes from math journals, math books, and popular books.s.v. 'real', n.2, B.4 --Macrakis (talk) 16:16, 8 February 2021 (UTC)

I'm happy with the "reals" in the plural (since I'm familiar with that usage), but I'm not familiar (so not happy) with it's use in the singular e.g. "x is a real". I don't see any such usage in my copy of the OED (1971), and I don't have a subscription to the online version linked to above. Can anyone point to such usage? Paul August ☎ 19:07, 8 February 2021 (UTC)

Yes, the OED agrees that it is "usually" used in the plural. The only case in the current version of the article that uses it as a singular noun is in the lead sentence. I am not sure how to rewrite that to both introduce the use of "real" as a noun, and not to add too much detail too early. I suppose we could remove "or real" and add a second sentence saying "The set of real numbers is also called the reals", but that seems a bit clumsy. What do you propose? --Macrakis (talk) 19:29, 8 February 2021 (UTC)

I'm not sure there's a graceful way to avoid singular "real" in the context of forcing (or why you would want to avoid it). Add a Cohen real; add a random real. It would sound really odd to say "add a Cohen real number". I suppose that's because you really aren't interested in it as a "number"; you're interested in the information it encodes. --Trovatore (talk) 19:36, 8 February 2021 (UTC)

Well even if the singular usage occurs, it is apparently not common, (I don't think "Cohen real" necessarily counts as a use, since one could think of "Cohen real" as the noun), so I don't think it needs mentioning in the lead (frankly it sounds very odd to me there.) If it must stay, then explanatory details could (and should) be provided in a note, which should also include a source. I'd also be okay, instead, with including somewhere "The set of real numbers is also called the reals".Paul August ☎ 20:09, 8 February 2021 (UTC)

A comment: "the reals" and "the real numbers" are not exactly synonymous. The latter refers implicitly to the set, while the former refers to the whole structure. So "the multiplicative group of the reals" sounds better than "the multiplicative group of the real numbers". Moreover the latter is incorrect, as it should be written "the multiplicative group of the nonzero real numbers". So using "the reals" allows often a simpler wording, and not using it may lead to some pedantry if one want to be mathematically correct. D.Lazard (talk) 20:53, 8 February 2021 (UTC)

@D.Lazard: Yes, those are good points, though I would also say that it is quite easy to avoid the pedantry and to save even more space: "the multiplicative group of $\mathbb {R}$". Ebony Jackson (talk) 01:10, 9 February 2021 (UTC)

Generally bad to use symbols in place of prose, though, provided the prose is easily understood. Particularly given that not everyone notates the reals as $\mathbb {R}$; that symbol is in origin just a way of drawing an R on a blackboard, and is not necessarily distinguished from R, which can mean other things than the reals.

Long story short, we should keep "the reals". --Trovatore (talk) 01:16, 9 February 2021 (UTC)

I too prefer R; I think it looks better than $\mathbb {R}$. But the trend seems to be favoring $\mathbb {R}$. Ebony Jackson (talk) 01:33, 9 February 2021 (UTC)

My main point, though, was that we should spell out "the reals" in running text, rather than resort to a symbol. --Trovatore (talk) 01:36, 9 February 2021 (UTC)

By the way, what other things are commonly denoted R? Ebony Jackson (talk) 02:29, 9 February 2021 (UTC)

In any case, I am convinced by the OED references you all gave (going back to Hamilton!) that it is not wrong to write "the reals" for the set of real numbers. I can't say that I prefer that terminology, but this is a matter of taste. Thank you all, Ebony Jackson (talk) 03:01, 9 February 2021 (UTC)

We should also cite the OED here for this usage. If someone would provide a copy of the OED entry for this usage, I could do the cite. I'm interested to know what exactly the OED says here. Paul August ☎ 13:17, 9 February 2021 (UTC)

The cite is included in my comment above. The text of the entry is quite short. I've added a ref to the article, with the full text of the entry, which should be short enough to avoid copyvio. --Macrakis (talk) 20:42, 10 February 2021 (UTC)

Hmm, now that my attention as been drawn to it specifically, I am not convinced we need to say "or real" in the first sentence, particularly in boldface. I'm against removing the usage where it comes up naturally, but I don't really see a need for it there, in spite of the link from the disambig page. --Trovatore (talk) 23:23, 10 February 2021 (UTC)

Yes. Paul August ☎ 00:05, 11 February 2021 (UTC)

Agreed that real doesn't need to be in the first sentence. I just put it there because it seemed like the smallest reasonable incremental change. I do think it needs to be mentioned somewhere in the lead paragraph, though, and not just "the reals" as a synonym for "the set of real numbers". --Macrakis (talk) 18:22, 11 February 2021 (UTC)

I don't see why. It is a rare usage at best. If it needs mentioning at all it can be mentioned in a footnote. Paul August ☎ 21:00, 11 February 2021 (UTC)

It's not a "rare" usage in the area of academic math I was in, but I don't see any need to call it out specially. When it comes up, it's pretty obvious what it means. --Trovatore (talk) 21:42, 11 February 2021 (UTC)

The reason given for including "real" as singular noun in the first sentence was that the Real (disambiguation) page points to this page. But it is pointing to "Real numbers", and it also links to dozens of other phrases that include the word real, such as Real Colorado Cougars. Also, although the OED reference supports the use of "the reals", it indicates that the singular form is not typical.

So let's remove "real" from the first sentence, and move the OED reference to "the reals" at the end of the first paragraph. (P.S. to Trovatore: Out of curiosity, are you referring to descriptive set theory, or something like that?) Ebony Jackson (talk) 21:52, 12 February 2021 (UTC)

Agree, and yes, my research was largely in the area of Borel equivalence relations. --Trovatore (talk) 19:43, 13 February 2021 (UTC)

0

Is 0 a rational number? As per my classification every rational number has got some value .if I talk about (0/1) it doesn't have any value.

Eg.1/1=1 , 5/3=1.6...etc. But, 0/1= doesn't have any value. 14.192.30.162 (talk) 19:32, 17 April 2022 (UTC)

Zero is a rational number. Zero is the ratio of zero to one; that is, 0/1. Zero does have value. The value of zero is zero.—Anita5192 (talk) 19:50, 17 April 2022 (UTC)

Are real numbers real?

Should it be addressed that real numbers are not actually real. One can zoom Mandelbrot's set as far as one can and find the self similiarity but if you do that for eaxmaple to englisg coastline you come up with atomic level where the self similiarity ends. In fact if one uses planck lenght as the mneasurement every real lenght can be given by an integer. a square which side is planck length ha diaginal that has to be either plasnch length or twice that but there is no squareroot of 2 in reality as it is irrational and thus it's actual experssion would use all the energy in the universe and you still haven't got it.Linkato1 (talk) 20:27, 24 July 2022 (UTC)

This gets at very fundamental disputes in the philosophy of mathematics, on which we have no business taking a position in Wikipedia's voice. Mathematical realists consider real numbers to be real, albeit, as you observe, almost definitely non-physical. Mathematical formalists, roughly speaking, consider statements about them to be shorthand for claims about what can be proved. Intuitonists and constructivists have many varied approaches that we need not belabor here, even if I had the expertise to do so.

It would be possible to bring in some of this discussion, attributing various views to various schools, but the question is, why this article specifically? The same questions arise for lots of sorts of mathematical objects.

What might be worth discussing, if there are good sources, is that the word "real" does not mean that the reals are more real than, say, imaginary numbers (though it was likely thought at some point that it did mean that). --Trovatore (talk) 21:33, 24 July 2022 (UTC)