Infinite series are a staple in math; some of them are fundamental to calculations of certain extremely important constants such as $\pi$, $e$, $\gamma$, etc. In that sense and to that degree, they seem vital to mathematics.

On the flip side, they violate our intuitions and generate paradoxes (re Zeno's paradoxs).

I guess we have to do what we must do: look the other way when $\infty$ up ends our world and act as if it never happen(s/ed).

When a recursive process gets you closer and closer to an answer the more you apply it, and this process goes on step by step with no ending stipulated, then this is all about "infinity" that is needed for most mathematics. $\infty$ is a convenient way of saying this.

Foundations and set theory, however, go a significant step beyond into transfinite pastures. Occasionally a result in other areas of math require a cow chip or two from these leas.

Space is infinite. There is no end to it. The universe expands in it. Accelerated even. Will it come to a sudden stop because space ends?

What is the history of Infinity? — TiredThinker

Not as dry (or long & thorough) as the SEP article but pretty good nonetheless:
https://www.polytechnique-insights.com/en/columns/space/how-to-understand-infinity-in-three-steps/

Infinity is the spatial edition of eternity. It has a long history. An eternal history.

What is the history of Infinity? — TiredThinker

In Western thought, try the pre-Socratics, e.g. Thales, Anaximander - matter arises from "the limitless" - and for infinitesimal Zeno as Agent S says. There's a book about it: https://www.waterstones.com/book/infinity-in-the-presocratics/joseph-owens/l-sweeney/9789024711703.

In Eastern thought, try a whole new way of looking at the world that I don't understand. But somebody will.

What is the history of Infinity? I know it exists at least for the sake of math, but has anything ever been to indicate that anything about it goes on forever? — TiredThinker

I believe "infinite" was established as a fundamental principle of measurement, which would allow that anything and everything could be measured. There cannot be more items than can be counted, because there is no limit to how high we can count. There cannot be something bigger or smaller than what can be measured, because we can always come up with a bigger or smaller measurement.

"Infinite" only refers to something which "goes on forever" when we're talking about temporal extension. And then, it could only be said to go on forever under the assumption that time has no end.

Thomas Aquinas thought the world could be eternal. However he argued there are no actual infinities. But what if there were always humans then there would be an infinity of souls. It's accidental that animals or vegetables always existed rather than human. If there always were human souls then they would be counting marks for an actual infinity

Wave on the unit interval compresses to a line infinitely long.

Great work! And you even get paid for it! You have involved with the Colatz conjecture at any time in your career? It's proven up to 2^70 numbers! But how to prove it for all numbers? 2^70 is peanuts in the face of infinity. Do we have to calculate all individual orbits?

And you even get paid for it! — EugeneW

Not since I retired 22 years ago. :cry:

What is the history of Infinity? I know it exists at least for the sake of math, but has anything ever been to indicate that anything about it goes on forever? — TiredThinker

Infinity was first posited by Anaximander. The apeiron as the first principle is boundless. The first principle meaning the "beginning of everything". So, beginning here doesn't mean a start (a bound), rather infinity is the beginning and we couldn't posit anything prior to infinity.

To answer your question, is infinity necessary? Infinity is what we could only point to whenever we talk about the beginning of everything.

So no object goes on forever? Infinity only refers to repeated processes that are intended to go on forever but of course cannot because we ourselves are limited by time?

What I believe is that measurement is fundamentally unlimited. This does not mean that any measuring process will go on forever, but it could in principle go on forever. This allows that we can measure anything and everything. The measurement of time for example would go on forever, if time went on forever. But things that we measure are all finite, all limited, and nothing goes on forever, so we end up measuring everything we try to measure, and the measuring process never needs to go on forever.

[H]as anything ever been to indicate that anything about it goes on forever? — TiredThinker

Horizons. The circumference of a circle / sphere / torus / Möbius loop ... (E.g. circumnavigating / orbiting the Earth.)

I don't think the logic of the term infinite holds much value. That which is infinite cannot terminate (end), therefore it has to be eternal but then how did it become infinite? It seems to me, logically impossible to become infinite so infinite must have always existed if you throw the concept of linear time at it. Therefore that which is infinite cannot logically or physically exist within linear time, it is a metalogical concept. To me, it is just a convenient label for 'what we just don't understand yet.'
Picking a number between 1 and 2 seems to have an 'infinite' choice but within linear time, this is meaningless as you can choose a number between 1 and 2 and therefore satisfy the request to pick one.
I give the term the same cognisance as that of the god fable. 'Has its uses, but I see little value in it.'
In Science, the appearance of an infinity, normally indicates a flaw in the logic applied.

The circumference of a circle / sphere / torus / Möbius loop ... (E.g. circumnavigating / orbiting the Earth.) — 180 Proof

Aristotle dispelled this idea a long time ago. When you traverse a circle, you arrive back at the same point where you started. The circle does not go on forever. And, we cannot assume that there is no starting point because motion requires a cause.

motion requires a cause. — Metaphysician Undercover

I don't think so. Aristotle didn't know about inertia.

:up:

Space is infinite. There is no end to it. The universe expands in it. Accelerated even. Will it come to a sudden stop because space ends? — EugeneW

Einstein defined the universe as "finite, but unbounded". Ironically, "unbounded" is one definition of "infinite". Yet, Einstein's mathematical universe is depicted as a sphere, which would actually be finite in space, except that "space" is inside the sphere. Now wrap your mind around the paradox of unbounded space trapped inside a finite sphere. :joke:

Einstein postulated that the universe is finite in time (bounded at the big bang singularity), and unbounded in space -- i.e., if one could travel the four dimensional universe in a geodesic one would not find a boundary, and would end up at the starting point, just like the path of a geodesic on the 3-dimension Earth.
https://www.researchgate.net/post/Can-the-Universe-be-finite-and-unbounded-and-still-be-Euclidean



What Lies Beyond the Edge of the Observable Universe? :
So, in some ways, infinity makes sense. But “infinity” means that, beyond the observable universe, you won't just find more planets and stars and other forms of ...
https://futurism.com/what-lies-beyond-the-edge-of-the-observable-universe

But “infinity” means that, beyond the observable universe, you won't just find more planets and stars and other forms of ... — Gnomon

I think beyond the observable universe there are lots of planets. The universe extends beyond the horizon as Earth extends beyond the horizon. The universe seems flat but it is a closed 3d sphere. You will come back where you started (if you got the energy). Maybe the (4d) space in which it expands is infinite... Will we ever know? We can speculate! And that's the fun.

Is infinity necessary?

Hell yes for mathematics!!!

Just think about. The Natural numbers go from 1,2,3,... to number TT (Tired Thinker), when they end. And no bigger natural numbers exist than TT. Just think about that would do to the logic of mathematics.

Hence yes.

We do need infinity.

At least as an axiom, since we yet haven't understood the mathematical logic behind infinity.

Is infinity necessary? Hell yes for mathematics!!! — ssu

You bet. :cool: As an abbreviation for certain statements. For example $\sum\limits_{k=1}^{\infty }{{{s}_{k}}}\lt\infty$

There are even kinds of infinity. Not every infinity is the same.

There are even kinds of infinity. Not every infinity is the same. — EugeneW

Oh don't let me get started.

And there's a lot that we don't still know about infinity. If we knew everything, there wouldn't be things like the Continuum hypothesis yet still unanswered (or even the question so problematic for our logic to handle). It is totally possible that sometime in the future we will know more and the schoolbooks will teach about infinity in a totally new way.

Finding new things in math hasn't stopped yet!

The continuum hypothesis baffles me. How can a line have the same number of points as a plane? Or a plane the same number of points as a volume? Aleph one, that is. Has a line the same number of points as a point? Is a point the same as a line, as a plane, as a volume? They are continuous structures. But not the same. Still they have aleph one in common.

Finding new things in math hasn't stopped yet!
26m — ssu

Maybe it just began. It needs new approaches to prove (or disprove) the Collatz conjecture. You cannot simply apply the recipe (3n+1) to all n and see if it holds for all.