Why is there antagonism towards extended real numbers?

Question

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)?

Answer 1

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.

Answer 2

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.

Answer 3

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.

Answer 4

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...

Answer 5

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 $\pm\infty$.