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What can we gain by losing infinity?

quantamagazine.org

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Re: What can we gain by losing infinity?

#51

Post by jcgrillo »

vlovich123 wrote: post #36
novia wrote: post #35

Earlier quoted context omitted.

yeah that seems fine. there's like no good reason to do that. are you trying to simulate reality or something? but my point still stands, choose whichever calculation you think is important to be able to do with Ω, defined as f(Ω), square it for good measure, and set that as the max, the min, and the number of numbers in between each integer. The total number of possible numbers will be ~2*f(Ω)⁴ which should be more…

AES256 already has more possible keys than exist atoms in the visible universe and that’s a pretty mundane thing. If you wanted to store all those keys, that’s even large. # of atoms in the universe turns into a very small very quickly when talking about permutations and permutations come up all the time (mathematical simulations, probability computations, etc). I really don’t understand what point you’re trying to m…

> It very well could be an infinite number of atoms and then what?

Where I get stuck with this is how might we measure that? Continuous measurements and infinite measurements are not something we can make. We fit continuous theories to discrete measurements--and the good ones fit really well!--but until we can measure it how can we actually know? I concluded we just can't, and we have to be OK with that.

Re: What can we gain by losing infinity?

#52

Post by alex_young »

marcus_holmes wrote: post #47
alex_young wrote: post #38

Earlier quoted context omitted.

Time has nothing to do with it. There are an infinite number of ways to divide anything. You don’t need time to prove that. Whatever number you think of you can divide by a larger number.

Yes, and that gets you to another number. Not infinity. You need an infinity of operations to create an infinity.

Create an infinity? What does that mean? Why would you need to do that?

Is there a limit to how many times something can be logically divided? If not, then there’s your infinity. It doesn’t require you to continue brute forcing it, just reason about it.

Re: What can we gain by losing infinity?

#53

Post by lostmsu »

alex_young wrote: post #52
marcus_holmes wrote: post #47

Earlier quoted context omitted.

Yes, and that gets you to another number. Not infinity. You need an infinity of operations to create an infinity.

Create an infinity? What does that mean? Why would you need to do that? Is there a limit to how many times something can be logically divided? If not, then there’s your infinity. It doesn’t require you to continue brute forcing it, just reason about it.

Maybe? Can you prove there's no limit? The default proof by induction requires postulate of infinity. (this statement is potentially incorrect, but takes across the point)

Re: What can we gain by losing infinity?

#54

Post by euroderf »

novia wrote: post #24

Take the approximate number of subatomic particles in the universe, call it Ω. Define the largest number as Ω² and the smallest number as -Ω², and define the number of decimal numbers between each integer number as Ω², evenly spaced. That should be more than enough numbers. Redefine Ω with each new discovery in physics. If this seems too conservative to you, like if for some reason you want to talk about the volume o…

At the bottom end we have the Planck length. How many cubic Planck lengths in the visible universe ? Anyone ? To paraphrase Bill Gates (allegedly), "(PlanckLengths/widthOfUniverse)*3 ought to be enough for anybody."

Re: What can we gain by losing infinity?

#55

Post by amelius »

Ifkaluva wrote: post #18

I don’t understand, and I hope it’s just bad writing. Certainly you can build a branch of mathematics without an axiom of infinity, and that’s fine, it’s math over finite sets. However, an axiom of infinity is independent, it doesn’t contradict anything in standard formalizations, and so it doesn’t make sense to say “infinity is wrong”. He may think the axiom of infinity isn’t satisfied by our real physical world, bu…

The problem with infinity is that it's a hack. It is basically the NULL pointer of mathematicians. An instance of a number that has a special meaning that breaks the abstraction of numbers.

If you want to do things with infinity, fine, but then do it properly and write things like lim x->inf (your expression with x here)

Re: What can we gain by losing infinity?

#56

Post by getnormality »

> To Zeilberger, believing in infinity is like believing in God. It’s an alluring idea that flatters our intuitions and helps us make sense of all sorts of phenomena. But the problem is that we cannot truly observe infinity, and so we cannot truly say what it is.

When the author says we cannot truly observe infinity, what does that mean? Infinity is a mathematical symbol we can observe. We can't observe infinitely many objects, but even if we could, it wouldn't be the same as observing infinity. You can't observe the number one by observing one stone.

I think there is some confusion in this article between symbols and what they can stand for, and I can't help but wonder if that same confusion is at the root of ideas like ultrafinitism.

Re: What can we gain by losing infinity?

#57

Post by gstrike »

novia wrote: post #35
vlovich123 wrote: post #33

Earlier quoted context omitted.

I want to count the number of possible permutations of the particles. We’ve now got a “larger” number than Ω will ever be able to represent by definition (even Ω² is minuscule by comparison).

yeah that seems fine. there's like no good reason to do that. are you trying to simulate reality or something? but my point still stands, choose whichever calculation you think is important to be able to do with Ω, defined as f(Ω), square it for good measure, and set that as the max, the min, and the number of numbers in between each integer. The total number of possible numbers will be ~2*f(Ω)⁴ which should be more…

[dead]

Re: What can we gain by losing infinity?

#58

Post by mcontrac »

Ifkaluva wrote: post #18

I don’t understand, and I hope it’s just bad writing. Certainly you can build a branch of mathematics without an axiom of infinity, and that’s fine, it’s math over finite sets. However, an axiom of infinity is independent, it doesn’t contradict anything in standard formalizations, and so it doesn’t make sense to say “infinity is wrong”. He may think the axiom of infinity isn’t satisfied by our real physical world, bu…

What people might not be understanding is that mathematics is inherently built... ZFC was pored over for years and eventually the community concluded it was a good system to (a) preserve most, if not all, of the mathematics that had already been done and (b) build more mathematics.

You can have gripes over whether or not pure math is compatible with the physical world but we're not exactly close to solving that problem... if we were, then physicists would have a much easier time lol

Re: What can we gain by losing infinity?

#59

Post by computably »

amelius wrote: post #55
Ifkaluva wrote: post #18

I don’t understand, and I hope it’s just bad writing. Certainly you can build a branch of mathematics without an axiom of infinity, and that’s fine, it’s math over finite sets. However, an axiom of infinity is independent, it doesn’t contradict anything in standard formalizations, and so it doesn’t make sense to say “infinity is wrong”. He may think the axiom of infinity isn’t satisfied by our real physical world, bu…

The problem with infinity is that it's a hack. It is basically the NULL pointer of mathematicians. An instance of a number that has a special meaning that breaks the abstraction of numbers. If you want to do things with infinity, fine, but then do it properly and write things like lim x->inf (your expression with x here)

> An instance of a number that has a special meaning.

Not really. There are infinitely many infinities. Infinite numbers are not particularly more special than real numbers, complex numbers, matrices, functions/operators, etc.

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