Is infinity a necessary concept?

[dendros]

Well, that is not my objection but of those who reject infinity in math.
They seem to conflate our limits with math's limits.

 

[Mark44]

But to negate the infinity of the set of naturals, for example, is like negating math since addition with 1 guarantees that a number can always be increased so there is not a last natural number.

Now, some might say that you cannot actually perform this addition an infinite number of times, which is true.
This does not negate the actual infinity of the set of naturals but rather expresses that we have finite resources.

Did you instead mean to say "dispute the infinity of ..." rather than "negate"?

 

[dendros]

Well, finitism by definition is the negation of infinity.

 

[Matterwave]

No. I think it is part of the "Big Bang" expansion theory. But I will not say more, because I do not know more. I'll leave it to others.

Somewhat off topic to the thread, but just to close the loop on this: in our cosmological models the *observable* universe is finite due to the finite speed of light and the age of the universe being finite (~13.8Gyr). "The universe" has unknown status. Einstein's eqns admit both spatially finite (closed is the better term, there's no edge but the universe loops back in on itself) and infinite solutions. Our most basic data points to a roughly flat FLRW metric which would correspond to an infinite universe.

Note that even if the universe is finite in size/closed, we still model it using a continuous mathematical object (manifolds) on which we can do calculus (take limits, differentiate, etc.). So infinity still pops up in the sense that the continuum between (0,1) is infinite.

Indeed, it seems to me like Finitists have a lot of theory to develop if they want their mathematics to reach anywhere near the utility of standard mathematics.

 

[jeffn1]

When applied to the physical universe, it seems to me the concept requires some sense of continuity of the "thing" considered infinite. For example up until the '60s, we thought our Milky Way Galaxy was "the universe". And people argued that it was "infinite". Then we realized the we inhabit a particular galaxy and there are billions of other galaxies. It is likely that things morph into different structures at the largest and smallest scales. It seems unlikely that that concept of infinite properly applies to the physical world. It seems to me to be a "placeholder" or conceptual/mathematical tool. (Sorry, a bit off topic)

 

[Dale]

it seems to me like Finitists have a lot of theory to develop if they want their mathematics to reach anywhere near the utility of standard mathematics

Yeah, I mean you cannot even use the typical axioms of addition on the natural numbers.

 

[Jaime Rudas]

For example up until the '60s, we thought our Milky Way Galaxy was "the universe". And people argued that it was "infinite".

Actually, no. As early as the mid-18th century, Immanuel Kant and Thomas Wright proposed the "island universes" hypothesis—the idea that nebulae were, in fact, collections of stars similar to the Milky Way. This idea captivated a significant number of scientists, as illustrated by the famous Great Debate between Shapley and Curtis, which culminated in 1924 when Edwin Hubble demonstrated that Andromeda was, in reality, another galaxy.

 

[WWGD]

-1/12 is the value of the analytic continuation of the Riemann Zeta ( $\Sigma_n=1 ^ {\infty) \frac {1}{n^s }$ , at s=-1. Have @mathwonk or @fresh_42 verify it. The continuation doesn't have anice closed form, else the Riemann Hypothesis could be easily answered.

 

[Matterwave]

-1/12 is the value of the analytic continuation of the Riemann Zeta ( $\Sigma_n=1 ^ {\infty) \frac {1}{n^s }$ , at s=-1. Have @mathwonk or @fresh_42 verify it. The continuation doesn't have anice closed form, else the Riemann Hypothesis could be easily answered.

I covered this in posts 13 and 18.

 

[WWGD]

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.)

I think no, if you mean a finite universe of elements. Not sure if you need finitely- many predicates too. Then just checking finitely many conditions is your finite decision process to determine the truth.

 

[zzephod]

In understanding the truth of things, is it important that we can not perform the addition an infinite number of times? What if God can do it?
(I hesitate to put it that way and possibly sidetrack the discussion, but I don't see how the physical limitations of humans should limit our conceptual understanding. If a human could perform each addition in half the time of the prior addition, he could complete it quickly.)

Remind me... which axiom set includes this "God" concept.

What you want is an appropriate theory/model of hyper-computation, which allows the completion of a countable infinity of operations in finite time... good-luck building a processor that realises such a model in this universe.

 

[FactChecker]

Remind me... which axiom set includes this "God" concept.

What you want is an appropriate theory/model of hyper-computation, which allows the completion of a countable infinity of operations in finite time... good-luck building a processor that realises such a model in this universe.

If your theory of reality is limited by computer processor speed, you need to think hard about what is real. If that is true, how was anything real before computers and humans existed?

 

[PeroK]

It's a significant step away from standard mathematics and logic to assume that the physical nature of the universe has anything to do with it. Or, the limitations of your computer!

Do we dismiss Euclidean geometry because the universe has 4D non-Euclidean spacetime?

This all seems like a massive step backwards. Mathematics is conceived beyond physical limitations. It is ultimately abstract, not physical.

 

[Dale]

What you want is an appropriate theory/model of hyper-computation, which allows the completion of a countable infinity of operations in finite time

Mathematica does such computations quite easily.

 

[Roberto Pavani]

Apologies in advance to the mathematicians here for my inevitable imprecision. I am not trying to give a formal definition of infinity. There are entire shelves of mathematics books devoted to the subject. I'm only trying to make an intuitive point.

One reason infinity seems fundamental to me is precisely that it is not an ordinary number.

For example, both $x$ and $\log x$ tend to infinity as $x \to \infty$, yet $\frac{\log x}{x} \to 0$.

So "going to infinity" does not identify a unique quantity. In fact, $x$ grows so much faster than $\log x$ that their ratio vanishes in the limit.

Mathematics has therefore developed entire theories, including limits, asymptotic analysis, cardinals, ordinals, and hyperreals, to distinguish different notions and scales of infinity.

To me, this is evidence that infinity is not merely a useful tool. It is a fundamental concept.

 

[A.T.]

In understanding the truth of things, is it important that we can not perform the addition an infinite number of times? What if God can do it?

Chuck Norris did it twice. He failed at his third attempt earlier this year. R.I.P.

 

[Roberto Pavani]

Another reason I find infinity fundamental is that, if we want to work with Hilbert spaces of functions, infinity seems unavoidable.

A finite-dimensional vector space admits a finite basis. By contrast, many function spaces used throughout analysis and physics are infinite-dimensional. They admit many different complete bases (Fourier bases, wavelet bases, eigenfunction bases, and so on), but every such basis is necessarily infinite.

In that sense, infinity is not merely a feature of a particular calculation or a limiting procedure. It is built into the structure of the space itself.

If we want to keep Hilbert spaces, we seem to have to keep infinity as well.

 

[DaveC426913]

Latecomer here.

I got the impression that the OP was more wondering if - rather than the very concrete entity of infinity, the concept of without bound was sufficient.

Eg.: Hilbert's Hotel can be handled using without bound - no need to invoke infinity. Because you never really get there.

It's all about the trip, not the destination. If your journey never ends, you don't really need to define anything about the destination.

Or is that a distinction without a difference?

 

[Roberto Pavani]

Perhaps I am missing something, but I am not sure that every occurrence of infinity can be replaced by "without bound".

For example,

$\int_0^1 x^2\,dx = \lim_{N\to\infty} \frac{1}{N}\sum_{k=1}^{N}\left(\frac{k}{N}\right)^2 = \frac{1}{3}.$

The final result is perfectly finite; nothing becomes arbitrarily large except the refinement parameter $N$ itself.

So I am not sure how naturally this can be expressed using only the notion of "without bound". To me, "without bound" seems to describe a process, whereas infinity in this context appears as part of the limiting construction that defines the object itself.

But perhaps I am overlooking something.

 

[DaveC426913]

To me, "without bound" seems to describe a process, whereas infinity in this context appears as part of the limiting construction that defines the object itself.

That concisely captures my argument, yes.

Is your counter example not a process? As N increases without bound, the function approaches 1/3.

 

[Roberto Pavani]

Consider $\sin(1/x)$ on $(0,1]$.

For any $N \in \mathbb{N}$, let $\varepsilon=1/N$.

Then the interval $(0,\varepsilon)$ contains more than $N$ zeros of the function. In fact, it contains infinitely many zeros.

So the infinite aspect here is not associated with something extending arbitrarily far away. Instead, infinitely much structure is concentrated within an arbitrarily small interval.

This makes me wonder whether "without bound" fully captures what mathematicians mean by infinity.

 

[WWGD]

Similary, in compact spaces, infinite unions of sets in covers can be replaced by finite subcovers. Alternatively, an infinity of atoms is required for Banach Tarski, to turn a ball into into another of twice its Edit : volume.

 

[FactChecker]

There are many uses of "unbounded" N, where f(N)=x will be exact at some unknown large N. That is very different from a convergence that never exactly gives the right answer in a finite number of steps, but does in the infinite limit.
I have no objection to either situation. The objections seem pedantic to me.

 

[mathwonk]

This discussion seems to continue (here and elsewhere) without limit. However, at least in my lifetime, I suspect it will never be infinite.

 

[mathwonk]

......Or to put it another way, after all these years, Achilles has apparently still not quite caught up to the tortoise.

 

[PeroK]

Latecomer here.

I got the impression that the OP was more wondering if - rather than the very concrete entity of infinity, the concept of without bound was sufficient.

Eg.: Hilbert's Hotel can be handled using without bound - no need to invoke infinity. Because you never really get there.

It's all about the trip, not the destination. If your journey never ends, you don't really need to define anything about the destination.

Or is that a distinction without a difference?

This is wrong. If Hilbert's hotel is finite and it's full, then it's full. If it's not finite, then it's infinite (not finite). It can't be something in between. A single hotel can't be "arbitrarily large".

 

[PeroK]

Consider $\sin(1/x)$ on $(0,1]$.

For any $N \in \mathbb{N}$, let $\varepsilon=1/N$.

Then the interval $(0,\varepsilon)$ contains more than $N$ zeros of the function. In fact, it contains infinitely many zeros.

So the infinite aspect here is not associated with something extending arbitrarily far away. Instead, infinitely much structure is concentrated within an arbitrarily small interval.

This makes me wonder whether "without bound" fully captures what mathematicians mean by infinity.

You've given several good examples of why finite mathematics loses a lot of what we know as conventional mathematics. It's not just about the natural numbers. It's everything, except finite groups, geometries and combinatorics etc. Real-valued functions of a real variable involve uncountable infinity, which is completely outside the scope of finite mathematics.

 

[Matterwave]

It would be interesting if a mathematical Finitist could come and explain their position in a clear and coherent way.

I suspect most (all?) folks here (including myself) are not finitists and we're just beating a dead horse at the moment.

But of course if we take the view that mathematics is abstract and need not correspond to the world, then any set of axioms, as long as they do not form self contradictions, are valid fields of study and the differentiator can only be utility.

I wonder, for example, whether Finitists generally ascribe to the abstractness of mathematics.

 

[Dale]

This makes me wonder whether "without bound" fully captures what mathematicians mean by infinity.

I think the “without bound” works for the epsilon delta approach to calculus. And it probably works for infinite sequences and series. But I don’t think that it works for cardinality.

 

[WWGD]

It would be interesting if a mathematical Finitist could come and explain their position in a clear and coherent way.

I suspect most (all?) folks here (including myself) are not finitists and we're just beating a dead horse at the moment.

But of course if we take the view that mathematics is abstract and need not correspond to the world, then any set of axioms, as long as they do not form self contradictions, are valid fields of study and the differentiator can only be utility.

I wonder, for example, whether Finitists generally ascribe to the abstractness of mathematics.

Look up NJ Wildberger, he's got a YouTube channel.https://youtube.com/shorts/8uGQyxt_Pso?si=_dT3apQAHr12U4Pb