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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?

#11

Post by jcgrillo »

bmenrigh wrote: post #8

The article doesn’t really tell us what is gained by rejecting infinity. And in general, why not also reject zero, negative numbers, irrational numbers, complex numbers, uncomputable numbers, etc.? Seems like an article about quacks that can’t even agree on what the bounds and rules of their quackery are.

All indications seem to be that things are only lost, not gained. But that doesn't mean it doesn't hew closer to how things actually are. But if that's how reality actually is, then developing a rigorous understanding of it can only be a good thing, right?

Re: What can we gain by losing infinity?

#12

Post by freetime2 »

nkrisc wrote: post #9
freetime2 wrote: post #6

> 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. I'm hoping this is just bad writing from Quanta rather than something "ultrafinitists" truly believe. I really don't think it's that complicated. Even pre-s…

I’m pretty certain a finite number of pre-schoolers can only recite a finite number of numbers. Yes, they could on indefinitely, but will they ever?

They pretty quickly realize that there is no winning because you can always just say more numbers than the last kid - there is no biggest number. Usually something like "a hundred million million million million million and two", "a hundred million million million million million and three", etc.

And then someone, whose friend or older brother taught them the concept, blurts out "infinity". And after a quick explanation, everyone more or less gets it.

Re: What can we gain by losing infinity?

#13

Post by p1necone »

freetime2 wrote: post #12
nkrisc wrote: post #9

Earlier quoted context omitted.

I’m pretty certain a finite number of pre-schoolers can only recite a finite number of numbers. Yes, they could on indefinitely, but will they ever?

They pretty quickly realize that there is no winning because you can always just say more numbers than the last kid - there is no biggest number. Usually something like "a hundred million million million million million and two", "a hundred million million million million million and three", etc. And then someone, whose friend or older brother taught them the concept, blurts out "infinity". And after a quick explanat…

INFINITY PLUS 1

Re: What can we gain by losing infinity?

#14

Post by freetime2 »

p1necone wrote: post #13
freetime2 wrote: post #12

Earlier quoted context omitted.

They pretty quickly realize that there is no winning because you can always just say more numbers than the last kid - there is no biggest number. Usually something like "a hundred million million million million million and two", "a hundred million million million million million and three", etc. And then someone, whose friend or older brother taught them the concept, blurts out "infinity". And after a quick explanat…

INFINITY PLUS 1

Uncountable infinity

Re: What can we gain by losing infinity?

#15

Post by drpixie »

jcgrillo wrote: post #3

In school I developed a strong hunch that continuity and infinity are "convenient delusions" we have that allow us to process the otherwise horrific complexity of the world. Experiencing time, sound, or visual motion as continuous, rather than discrete signal inputs is so much simpler . Similarly, the mathematical tricks and shortcuts we can use on well behaved continuous functions are both "unreasonably effective" a…

I've always felt that to treat infinity as number is to commit a category error (aka type conflict), to confuse the process with the outcome of the process. Infinity has proven to be very useful, but usefulness doesn't make it always valid.

Re: What can we gain by losing infinity?

#16

Post by bmenrigh »

jcgrillo wrote: post #11
bmenrigh wrote: post #8

The article doesn’t really tell us what is gained by rejecting infinity. And in general, why not also reject zero, negative numbers, irrational numbers, complex numbers, uncomputable numbers, etc.? Seems like an article about quacks that can’t even agree on what the bounds and rules of their quackery are.

All indications seem to be that things are only lost, not gained. But that doesn't mean it doesn't hew closer to how things actually are. But if that's how reality actually is, then developing a rigorous understanding of it can only be a good thing, right?

Rejecting infinity is a purely philosophical stance that doesn’t teach us anything about reality.

There is a big difference between “infinity doesn’t exist” and “infinity doesn’t exist physically”.

I should also add that the resolution of zeno’s paradox in the form of calculus where and infinite set of steps can occur in a finite time (or infinite set of distance can span a finite total distance) is conceptually very simple and useful. Rejecting it as unphysical, or saying it must imply time or space come in discrete chunks, is not contributing to an understanding of reality unless the rejection also comes with a set of testable (in principle) predictions.

Re: What can we gain by losing infinity?

#17

Post by samplatt »

freetime2 wrote: post #12
nkrisc wrote: post #9

Earlier quoted context omitted.

I’m pretty certain a finite number of pre-schoolers can only recite a finite number of numbers. Yes, they could on indefinitely, but will they ever?

They pretty quickly realize that there is no winning because you can always just say more numbers than the last kid - there is no biggest number. Usually something like "a hundred million million million million million and two", "a hundred million million million million million and three", etc. And then someone, whose friend or older brother taught them the concept, blurts out "infinity". And after a quick explanat…

And then the next kid says "infinity plus two", which is a perfectly acceptable progression, and the cycle starts again.

Re: What can we gain by losing infinity?

#18

Post by Ifkaluva »

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, but that’s not a math question! There’s nothing logically inconsistent about infinite sets nor their axiomatizations.

Re: What can we gain by losing infinity?

#19

Post by jcgrillo »

bmenrigh wrote: post #16
jcgrillo wrote: post #11

Earlier quoted context omitted.

All indications seem to be that things are only lost, not gained. But that doesn't mean it doesn't hew closer to how things actually are. But if that's how reality actually is, then developing a rigorous understanding of it can only be a good thing, right?

Rejecting infinity is a purely philosophical stance that doesn’t teach us anything about reality. There is a big difference between “infinity doesn’t exist” and “infinity doesn’t exist physically”. I should also add that the resolution of zeno’s paradox in the form of calculus where and infinite set of steps can occur in a finite time (or infinite set of distance can span a finite total distance) is conceptually very…

> There is a big difference between “infinity doesn’t exist” and “infinity doesn’t exist physically”.

Is there? I think one could make a decent case for "nothing exists which doesn't exist physically[1]".

[1] https://plato.stanford.edu/entries/physicalism/

EDIT: you could even probably claim "nothing exists which isn't physically measureable" which may or may not be a stronger claim depending on your point of view.

EDIT AGAIN: rate limited by this dogshit website :D but I'll respond to this comment here:

> Which is exactly why I mentioned rejection of zero, negative numbers, etc. You can reject them, but doing so just throws away useful tools without gaining anything in return.

Yeah! I fully agree. I can see no obvious benefit to rejecting these powerful tools. However, important discoveries often happen in non-obvious directions, and exploring unexplored territory is generally worthwhile. So the fact that it doesn't seem immediately useful doesn't mean it's not worth trying!

Re: What can we gain by losing infinity?

#20

Post by gmuslera »

The first thing that came to mind reading the article is that you need only 60ish digits of pi to calculate the circumference of the universe with a resolution of a Planck length, or something like that. You can have all the digits you want, but at some point you are beyond what is possible in reality, and giving back wrong answers for what you are trying to achieve.
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