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In my backstory, I was introduced to the geometric concept of infinity rather young, through reading about the inversive plane. In the course of learning calculus, I'm pretty sure I formed a concept of $\pm \infty$ lying at the endpoints of the real line (although they weren't real numbers themselves), and understood limits in terms of that.

By the time I was introduced to the extended real numbers, it was simply putting a name to the prior concept, and providing a framework to work with them in a clear and precise fashion. (and similarly for the projective real numbers)

Fast forward 20 years later, and through interactions with people here at MSE, I find there is a lot of antagonism towards the concept of the extended real numbers. I don't mean things like "it would be confusing to teach them in introductory calculus" -- I mean things like "the extended reals $\pm \infty$ are best thought of in terms of limits rather than as actual points" or even "thou shalt not develop a concept of $x$ approaching something as $x \to +\infty$" as well as some patently false claims (e.g. "$+\infty$ cannot be a mathematical object; it can merely be a a 'concept'").

I had previously brushed off those opinions, but they seem pervasive enough that I felt I should ask the titular question: is there any good reason for this antagonism? Or is there any good reason to avoid understanding calculus in terms of the extended reals (when they are suitable objects to do so)?

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The main (only?) reason against using the two-point compactification of $\mathbb R$ (you can also get along with a single $\infty$ that glues the ends together!) is that $\mathbb R\cup\{-\infty,\infty\}$ is not a field. Also, it's topology is not given by the standard metric (though the space is metrizable) –  Hagen von Eitzen Feb 23 at 15:19
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@user72694 I disagree. It's quite clear to me that the OP refers to the two-point compactification of $\mathbb{R}$ (and at one place the one-point compactification), the question is not at all about infinitesimals or surreal numbers or whatever non-standard analysis one prefers. –  Daniel Fischer Feb 23 at 15:46
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I think that the issue is that a lot of people feel that mathematics is composed of numbers, and that you can divide, subtract, multiply and add numbers. But the points at infinity do not obey the usual rules. And people, as people usually are, have an inherent dislike to things that contradict their internal values (that were usually put there by previous indoctrinations of their parents and teachers). So there is a lot of fuss about introducing formally and fully things which are not numbers into the real numbers. –  Asaf Karagila Feb 23 at 15:47
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I have indeed indeed seen this sort of antagonism towards such notions. We can see this in the terminology of ideal points in projective geometry.I also agree with Asaf that people seem to violently protest these things because we cannot perform the standard arithmetic operations with the extended reals. Indeed I have said to student "You may have been told that infinity is not a number. Whoever told you that is wrong. Indeed infinity is a number, it just is not a real number. Since it is not a real number, we cannot perform all of the arithmetic operations with it the same way we usually do". –  Baby Dragon Feb 23 at 16:04
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Seeing many different points of view is confusing to a beginning student. After 2-3 years studying math, then differing points of view (such as extended reals) can be introduced fairly easily. –  GEdgar Feb 24 at 0:58

6 Answers 6

up vote 22 down vote accepted

The extended reals are a way of thinking, but hardly the only way. It is normal to think of a line as having no endpoints, rather than as ends that meet.

The reason you see antagonism is that this site is frequented by learners of mathematics, that often have very muddled ideas about infinity. Introducing the extended reals and allowing $\infty$ to be a number would only add to their confusion. Saying that $+\infty$ is a concept and not a number is a simplification made for calculus students to help them understand the definition of $\lim_{x\to \infty}f(x)$. Mathematics educators regularly simplify things for beginners, and this is an excellent example.

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I wrote my comment above, and then I found out that your answer is very similar. I fully agree with the argument here! –  Asaf Karagila Feb 23 at 15:48
    
I think this is a good answer. I have elsewhere (twice) linked to my notes on honors calculus (which I construe as being a kind of elementary real analysis). I do not introduce the extended reals when teaching freshman calculus because it would add another conceptual layer to what is already the most confusing part of the course (limits). On the other hand I sometimes act as though I have introduced extended real numbers: e.g. when evaluating improper integrals I have no qualms about writing $e^{-\infty} = 0$ (although others reasonably might). –  Pete L. Clark Feb 23 at 22:33
    
Not much related, but isn't the infinity treated as a number in complex analysis? The Riemman sphere and all that. –  jinawee Feb 23 at 23:23
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@jinawee: yes, that is the one-point compactification of the complex plane, or in more geometric terms: the projective closure of the complex affine line. It think it is somewhat related to what we're talking about, yes. –  Pete L. Clark Feb 23 at 23:26

You have the special set $\mathbb{R}$. Addition and multiplication are defined and you have a total order on it.
So, for all $a, b \in \mathbb{R}\,$ $(a+b)$, $ab$ in $\mathbb{R}$ too, and it is true that $a\leq b$ or $b\leq a$ is true.

Now you define $\overline{\mathbb{R}}=\mathbb{R}\cup\{-\infty\}\cup\{+\infty\}$ (I will distinguish $+\infty$ and $\infty$).
You have a total order here: if $a, b\in \mathbb{R}$ than the same is true: $a\leq b$ or $b\leq a$, and for every $a\in \overline{\mathbb{R}}$ it is true that $a\leq+\infty,\,-\infty\leq a$.
But you do not have addition and multiplication. $3+(+\infty)$ could possibly be $+\infty$, but what about $(-\infty)+(+\infty)$? And division?

You may go deeper.
Let $\overline{\mathbb{R}}^*=\overline{\mathbb{R}}\cup\{\infty\}$. Now we don't have the total order but we can write not

$\lim_{x\to+\infty}x^2=+\infty$ and $\lim_{x\to-\infty}x^2=+\infty$

but

$\lim_{x\to\infty}x^2=+\infty$

It is good (is it?), but without total order it isn't even true, that every non-empty set has an upper bound. It is ... nasty.

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The real problem is that we lose additive inverses: for all $a,b \in \mathbb R$, we have $(a + b) + (-b) = a$. This breaks down if we allow either $a$ or $b$ to be $\pm\infty$. –  Ilmari Karonen Feb 23 at 17:27
    
@IlmariKaronen: One also loses many other related aspects of equivalence (e.g. a+b=a+c iff b=c). There may be uses for symbolic infinities, infinitesimals, etc. but they don't follow the normal rules of real numbers. –  supercat Feb 24 at 1:02

One can pinpoint the reason for such an antagonism to the notion of an infinite number (as opposed to cardinality) rather precisely due to the able work of the historian Joseph Dauben. Dauben wrote as follows:

Cantor devoted some of his most vituperative correspondence, as well as a portion of the Beiträge, to attacking what he described at one point as the ‘infinitesimal Cholera bacillus of mathematics’, which had spread from Germany through the work of Thomae, du Bois Reymond and Stolz, to infect Italian mathematics ... Any acceptance of infinitesimals necessarily meant that his own theory of number was incomplete. Thus to accept the work of Thomae, du Bois-Reymond, Stolz and Veronese was to deny the perfection of Cantor's own creation. Understandably, Cantor launched a thorough campaign to discredit Veronese's work in every way possible.

See pp. 216-217 in Dauben, J., 1980. The development of Cantorian set theory. From the calculus to set theory, 1630-1910, 181-219, Princeton Paperbacks, Princeton University Press, Princeton, NJ, 2000. Originally published in 1980.

These remarks make it clear that one of the sources of the hostility toward the concept of an infinite number is the attitude of Georg Cantor which has had a pervasive influence on the attitudes of contemporary mathematicians.

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Right, that explains antagonism in the late 19th century. What about in the early 21st century? –  Pete L. Clark Feb 23 at 22:13
    
Infinitesimals are routinely used when in settings where they don't cause any trouble, like in algebraic geometry. The definition of Zariski tangent space is in terms of infinitesimals, for example. –  arsmath Feb 24 at 10:32
    
@Pete, since there seems to be a consensus that this question is not about elementary extensions of R, I hesitate to answer it here. Perhaps someone could ask a similar question about extensions which would provide a more appropriate venue for an answer. –  user72694 Feb 24 at 15:47
    
@arsmath, if you can be more specific about what kind of "trouble" infinitesimals seem to cause outside of algebraic geometry, I could try to comment; see also my comment above. –  user72694 Feb 24 at 15:49
    
I suppose I understood your answer to be documenting that up through the late 19th century there was the feeling that nothing infinite in mathematics is "real". I didn't read it closely enough to see that it almost entirely concerns infinitesimals. So actually it doesn't seem so relevant to the question at hand, except to broadly document that the concept of infinity (in any form) has a long history of making people nervous. –  Pete L. Clark Feb 24 at 19:42

I'm not sure if explaining why the extended reals are natural is an appropriate answer to this question, but just in case:

As Baby Dragon and Bill Dubuque have pointed out, various notions of "compactifications" obtained by adding "points at infinity" are truly ubiquitous in mathematics, especially in geometry. The entire notion of projective geometry turns on adding these kinds of points, and it does not seem like much of an exaggeration to claim that the difference between affine and projective geometry is kind of the key to the city of algebraic geometry, a leading branch of contemporary mathematics.

Projectivization though would lead to the one-point compactification of $\mathbb{R}$ and we are talking about a $2$-point compactification. That shows up naturally by considering the order properties of $\mathbb{R}$: namely $\mathbb{R}$ is a Dedekind complete linearly ordered set whose order completion is precisely the extended real numbers: see e.g. this note for a take on this. So the extended real numbers are, from the perspective of order theory, extremely natural. They are also useful in calculus and analysis...

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The difference between types of geometries is the key to the city of algebraic geometry? Now I finally know why I've been locked outside! :-) –  Asaf Karagila Feb 23 at 23:26
    
Asaf: There may in fact be several keys to the city, but yes: affine varieties are algebraically easy to study since they come down to ideals in polynomial rings. However they are (over $\mathbb{C}$) not compact, which is a real downer from the geometric/topological perspective. Projectivization comes to the rescue: the algebra of projective varieties (gradings, homogeneous ideals and such) is a bit more complicated than affine algebra, but the corresponding geometry more than makes up for it. –  Pete L. Clark Feb 23 at 23:30
    
@Pete Though I did not explicitly mention the 2-point compactification, it is in fact discussed in the Points at Infinity exposition that I linked to. This is a nice elementary exposition on this circle of ideas. –  Bill Dubuque Feb 23 at 23:34
    
Pete, thanks for the comment. I was making a very tired joke, though. :-) –  Asaf Karagila Feb 23 at 23:40
    
There is more than one way to extend real numbers, and none of them are "natural". One will have to chose what he doesn't mind to lose: infinite points destroy the field structure; complex numbers destroy the order. Also, the non-extended real numbers themselves are rather interesting to study, even if it sometimes sounds like "sure, a balloon is pretty dandy, but it behaves much more interesting when some of its points are removed". –  Joker_vD Feb 24 at 13:55

One negative point is that the use of $\pm\infty$ with $+$ and $\cdot$ is awkward to define. In textbooks, a list of relations like $\infty+\infty=\infty$ or $1/\infty=0$ is given, but usually this list is not exhaustive and it's up to the reader to define the rest himself. When definitions are left to the reader, it always leaves a nagging uneasiness (at least for me, and I guess also for many first year students who are newly introduced to the concept of rigor).

Also, to be consistent, theorems like $(x_n\rightarrow x\text{ and }y_n\rightarrow y)\Rightarrow x_ny_n\rightarrow xy$ would have to be proven for the extended real numbers, which would entail a few cases to distinguish.

So my antagonism (if you can call it that) is not against the extended real number line, but against lecturers and textbooks using all kinds of theorems about it without proving them or making it clear to the reader that there is something to proven. These concerns may seem trivial to more advanced mathematicians, but at the time they lead to me thinking that $\pm\infty$ were somewhat fishy.

In my Analysis course, $\rightarrow\infty$ was defined before $\infty$.

The definition of $x_n\rightarrow x$ is $\forall\epsilon>0:\exists n_0>0:|x_n-x|<\epsilon\text{ for }n\geq n_0$.

The definition of $x_n\rightarrow\infty$ is $\forall M>0:\exists n_0>0:x_n>M\text{ for }n\geq n_0$.

When I first encountered these definitions, they seemed two entirely different things to me, and, since I didn't know what to do with $\infty$ anyway, I refused to learn the second one.

It was only when I learned in General Topology that $\overline{\mathbb{R}}$ can be turned into a metric space such that $x_n\rightarrow x$ and $x_n\rightarrow\infty$ are just examples of the same metric space convergence, that I really felt secure about using $\pm\infty$.

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The extended reals are not a particularly natural way to think about infinity in the context of calculus. Historically, calculus was developed as the study of infinitesimals, and in some approaches (such as non-standard analysis, NSA), these infinitesimals are invertible, and their inverses are infinite. Other reasons to dislike the extended reals are that (1) they don't have properties that are closely analogous to those of the reals, and (2) introducing them is a lot of work for very little additional utility.

Historically, there was a lot of confusion and uncertainty about whether infinitesimals were even logically consistent. This was resolved by NSA ca. 1960, but education is conservative, and many people's training has been influenced by the feeling that was prevalent ca. 1880-1960 that limits were the only rigorous way to develop the calculus.

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I find "transfinite" to be a much better term for multiplicative inverses of infinitesimals. –  Asaf Karagila Feb 23 at 21:17
    
I'm not sure NSA is actually relevant: it just lets you restate the antagonism to "thou shalt not develop a concept of a standard point that the positive transfinites are near". –  Hurkyl Feb 23 at 22:09
    
I gave this a -1, though I wonder if I have just misunderstood your answer. But: yes, the extended real numbers are particularly natural when doing calculus: see e.g. Section VI.1.3 of math.uga.edu/~pete/2400full.pdf, and note that the material in this short section is used in the rest of the book. –  Pete L. Clark Feb 23 at 22:17
    
More specifically, to (1): yes, there are some things that you can't do in the extended real numbers; most of the section I linked to elaborates on that fact. (2): no, this was one of the easier concepts in the course, and it pays significant dividends: all of a sudden infima and suprema of subsets of $\mathbb{R}$ exist always, and this is extremely useful (for bookkeeping purposes, but bookkeeping can be important). –  Pete L. Clark Feb 23 at 22:17
    
I also agree that the remarks about infiniteimals and NSA seem not relevant here, unless the intention is to point out that people who object to "extended real numbers" are confused by the terminology and think they are objecting to infinitesimals/NSA. –  Pete L. Clark Feb 23 at 22:19

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