Why did mathematician construct extended real number systems, $\mathbb R \cup\{+\infty,-\infty\}$?

I know some properties cannot be defined with the real number system such as supremum of an unbounded set. but I want to know the philosophy behind this construction (extended real number system ($\mathbb R \cup\{+\infty,-\infty\} $) and projectively extended real number system ($\mathbb R \cup\{\infty\}$)) and why did mathematician want to do so? what are the beautiful properties they achieved? I want an answer with a philosophical point of view.

P.S. is there any books or notes or something which I can refer?

Solution 1:

I think the most extreme philosophical reason might be that mathematics is invented by mathematicians who are curious and inventive and invent things they find beautiful. Or, thinking as a Platonist, all those structures are out there in some sense and mathematicians exploring that world stumble on these extended structures and like spending time thinking about them.

In a narrower sense, many of these extensions are a kind of "completion". You need the negative integers in order be able to subtract, so you extend the natural numbers. You need the rationals to divide. You need the reals to have a square root of $2$ (actually, you need only the algebraics for that). You need the complex numbers to have a square root of $-1$ - and then you get the fundamental theorem of algebra as a consequence. (And you can extend the real numbers to include infinitesimal numbers, then do calculus with those instead of the usual treatment with limits.) So the extensions are meant to solve problems.

If you're then just curious you can look for multiplicative structures in higher dimensional Euclidean spaces, prove there are none in dimension $3$, find the quaternions in dimension $4$, and prove there are no more unless you give up associativity. That's an interesting story.

You extend the plane to add points and a line at infinity so that the axioms become neater and more symmetrical: two points determine a line, two lines determine a point. Then you get some nice theorems, and, if you're a painter in the Renaissance, you codify perspective.

In reality (if you'll allow the word) most of the extensions are not just "adding elements to structures". They are abstractions. Groups capture the idea of symmetry. Calculus captures the idea of change. Geometry and topology capture the idea of shape.

Edit in response to comment.

For the Platonist there's no distinction between the real world and the abstract one. All those fancy mathematical notions are real, out there somewhere. Just as real as the interiors of stars are to an astrophysicist. We explore them to discover how they behave. In both physics and mathematics, the things we explore are further and further from the part of the real world we can touch and see, but no less real for that.

By the way, not all mathematicians are Platonists. There are good philosophical arguments that claim humans invent mathematics, not discover it But I think most working mathematicians, whether Platonist or not, believe in the reality of their subject matter. They only differ about whether it's invented or discovered. Only outsiders say "that's abstract, not real".

Solution 2:

I forgot this when writing my comments to the question, but one reason is compactness.

For example, using the extended reals, the Extreme Value Theorem, "A continuous function on a compact interval is bounded", has the following corollary: "A continuous function on $\Bbb R$ with $\lim_{x\to\infty}f(x)$ and $\lim_{x\to-\infty}f(x)$ defined is bounded". Without the extended reals, we'd have to prove it separately.