Is infinity a necessary concept?

[FermisPairOfDucks]

TL;DR

Is infinity worth the trouble that it brings?

How fundamental is the concept of "infinity" to mathematics? Is it just a necessary idea to say that a limit has converged? I sometimes worry about the thought experiments that people come up with:
1) monkeys recreating Shakespeare
2) there must life out there because space is "infinite" so the Drake Equation doesn't apply
3) "infinite" series of positive numbers converging to -1/12

I realize that there are many useful ideas that come of out theories of infinite sets, but in the age where computers can really allow us to look at small numbers (but not necessarily "infinitesimals") in a way where we can say that they are "close enough" to get "accurate enough" answers, do we really still need this concept?

I've heard that there are some mathematicians out there who are trying to recreate important results without referring to infinity, but I've also heard that this is a very "fringe" pursuit.

Thanks to anyone who can help me clarify my thoughts on this.

 

[anuttarasammyak]

1 and 2 are not troubles for me. 3 is not correct saying.
$$1+2+3+4+… \neq \zeta (-1) =-1/12$$

 

[Matterwave]

I mean, the fact that the world of mathematicians generally work with infinities on a daily basis should tell you something about its utility.

You might be interested in Finitism: https://en.wikipedia.org/wiki/Finitism

 

[Nugatory]

TL;DR: Is infinity worth the trouble that it brings?

How fundamental is the concept of "infinity" to mathematics?

There isn’t any single “the concept” of infinity in math. There are multiple concepts, all of which have rigorous definitions that don’t create any logical difficulties. The problems that you’re seeing (infinite monkeys and Shakespeare, the zeta function and -1/12, infinite possibilities imply that anything is possible, …) come from not paying attention to these rigorous definitions. For example, consider the paradox of Hilbert’s Hotel which is full but always has room for another guest: it’s hard to explain coherently if we say that there are an infinite number of rooms, but if we were being precise we would instead say that there is a one-to-one mapping between the rooms and the natural numbers and the paradox is easily resolved.

One point that laypeople often don’t fully appreciate: the $\infty$ symbol does not represent a number, whereas arithmetic operations like addition and multiplication operate on numbers. Thus, we can’t multiply a number by infinity (as we do with infinite monkeys each having a finite probability of producing Shakespeare) without first defining what that operation means - and typically the answer is “not 100% but undefined “.

 

[Matterwave]

There are multiple concepts, all of which have rigorous definitions that don’t create any logical difficulties.

If by logical difficulties you mean logical contradictions then yes. We build our systems of mathematics on the principle that we can't have actual logical contradictions in them. If we found axioms which prove contradictions then we need to rework those axioms.

That being said, you certainly do sometimes get "difficulties" as I would read the word (not necessarily a contradiction but hard to reconcile with our general notions of things). For example the Banach Tarski paradox being a consequence of including the Axiom of Choice. Not a mathematical contradiction. But I would use the word "difficult" for it.

 

[Dale]

One point that laypeople often don’t fully appreciate: the ∞ symbol does not represent a number

This is often true even in number systems like the hyperreals that actually do have infinite numbers.

 

[Dale]

TL;DR: Is infinity worth the trouble that it brings?

How fundamental is the concept of "infinity" to mathematics?

I would say that the most fundamental concept in mathematics is the natural numbers. And since the set of all natural numbers is infinite, I would say that the concept is pretty fundamental.

If you tried to make the set of all natural numbers finite then even something as basic as addition of natural numbers fails. So, yes, it is worth the trouble.

 

[FactChecker]

TL;DR: Is infinity worth the trouble that it brings?

How fundamental is the concept of "infinity" to mathematics? Is it just a necessary idea to say that a limit has converged? I sometimes worry about the thought experiments that people come up with:
1) monkeys recreating Shakespeare

This doesn't really require infinity, just huge enough numbers to get it done.

2) there must life out there because space is "infinite" so the Drake Equation doesn't apply

Space is not infinite.

3) "infinite" series of positive numbers converging to -1/12

I assume you mean convergence to 1/12. A finite series of positive numbers never totals below 0.

I realize that there are many useful ideas that come of out theories of infinite sets, but in the age where computers can really allow us to look at small numbers (but not necessarily "infinitesimals") in a way where we can say that they are "close enough" to get "accurate enough" answers, do we really still need this concept?

In the real world, things that were once considered so impossibly small that no computer would be built for it, eventually become possible. The sensitivity of the LIGO gravity wave detector comes to mind.

I've heard that there are some mathematicians out there who are trying to recreate important results without referring to infinity, but I've also heard that this is a very "fringe" pursuit.

There are a lot of things in pure mathematics that you might want to ignore and leave to others. This may be one of those. IMO, pure mathematics can get pedantic.

Thanks to anyone who can help me clarify my thoughts on this.

It is still important to know when a series will converge, versus those that will never converge. "Never" brings in "infinity".

 

[javisot]

do we really still need this concept?

We have very powerful mathematics thanks to dealing with the finite and the infinite; it would be less powerful if we only treated the finite as a mathematical object. Fewer tools, less depth, less rigor, etc.


(I have a question: in a context of "only finite mathematics" are incompleteness theorems still relevant? I would say no, but I prefer to ask.)

 

[dendros]

Considering that the set of integers (which are the simplest math entities) is infinite, I would say that yes, infinity is necessary in math.
But I'm not a mathematician.

 

[WWGD]

Op may want to consider the idea of compactification, where access to $\infty$ unboundedness is restricted or collapsed. In maybe the simplest example, the Real number line is turned into a circle, by collapsing the two infinities ( $\pm \infty$) into a single point. Notice you can similarly collapse through Algebraic quotients through rings like $\mathbb Z_p$, where you operate on infinite classes.

 

[WWGD]

So maybe Compact is an intermediate term, condition between finiteness and infinity. I can see a setup for a new cologne: " Between finite and infinite lies Compactness. By Channel ".

 

[Matterwave]

I assume you mean convergence to 1/12. A finite series of positive numbers never totals below 0.

I think the OP is probably referring to several pop-sci content (YouTube videos from Numberphile comes to mind) where it is "shown" that the sum of the natural numbers is $-1/12$.

One way (there are several, and different videos will use different methods) where this is "shown" is to take the infinite series:

$$\zeta(s)=\sum_{n=1}^\infty \frac{1}{n^s}$$

And then plugging in $s=-1$ so the infinite series reads like $1+2+3+...$
Of course, the infinite series is divergent there, but the Riemann Zeta function, being the analytic continuation of that infinite series has a definite value of $\zeta(s=-1)=-1/12$. The pop sci content often just then use this statement to say "the sum of the natural numbers is -1/12"...

There's a lot of technical issues that the OP is missing with such an example. The thing that makes this result weird is regularization (or tbh... sometimes just invalid math), not the infinite nature of the natural numbers.

 

[WWGD]

It's an issue of Analytic Continuation of a series, where analyticity is by default defined within open balls, though no valid reason that it holds only within a fixed radius. So through that technique, we expand the region where the Taylor series converges.

 

[Matterwave]

It's an issue of Analytic Continuation of a series, where analyticity is by default defined within open balls, though no valid reason that it holds only within a fixed radius. So through that technique, we expand the region where the Taylor series converges.

Do you mind rephrasing? I'm not sure I understand.

The series in my post is defined for an open region $\text{Re}(s)>1$

 

[FactChecker]

It's an issue of Analytic Continuation of a series, where analyticity is by default defined within open balls, though no valid reason that it holds only within a fixed radius. So through that technique, we expand the region where the Taylor series converges.

Even with analytic continuation, I doubt that there is any legitimate expansion at a point that allows that conclusion.

 

[Dale]

Even with analytic continuation, I doubt that there is any legitimate expansion at a point that allows that conclusion.

I agree. The analytic continuation of a series is only equal to the series in the region where the series converges.

This is not a problem with infinity. It is a problem with pop-sci entertainers

 

[Matterwave]

Wikipedia article for the $1+2+3+4+...$ that I think the OP was talking about: https://en.wikipedia.org/wiki/1_+_2_+_3_+_4_+_⋯

Note that I don't think anybody (even most pop-sci content creators) would say the series converges to $\frac{-1}{12}$. It is certainly divergent. They often just say it equals $\frac{-1}{12}$ (which might not be any better, but still).

 

[pinball1970]

TL;DR: Is infinity worth the trouble that it brings?

Thanks to anyone who can help me clarify my thoughts on this

It is worth checking Cantor out.

 

[Sagittarius A-Star]

Infinity is for example necessary to write transcendental functions via their Taylor series (if it exists).
$e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots$