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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?

#41

Post by [deleted] »

rmunn wrote: post #23
samplatt wrote: post #17

Earlier quoted context omitted.

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

When I was about ten, a math teacher once asked me whether the number 0.9999... (infinitely repeating) was different than 1. I said, with my child's intuition, that of course it was. He then challenged me to write down a number that was in between them, because if they were not the same number then there would be many (in fact, infinitely many) numbers between them. I couldn't, of course: the best I could do was to w…

[deleted]

Re: What can we gain by losing infinity?

#42

Post by srean »

ndriscoll wrote: post #26
rmunn wrote: post #23

Earlier quoted context omitted.

When I was about ten, a math teacher once asked me whether the number 0.9999... (infinitely repeating) was different than 1. I said, with my child's intuition, that of course it was. He then challenged me to write down a number that was in between them, because if they were not the same number then there would be many (in fact, infinitely many) numbers between them. I couldn't, of course: the best I could do was to w…

This is one of my life goals is to prepare my kids to troll their math teachers with the dual numbers and the claim that .999... is obviously 1-ε. Goal is to convince the teacher .999...≠1. Bonus points if they instead convince the teacher to doubt that complex numbers exist.

That would be both fun and correct.

It really comes down to what semantics we attach to "=" when one of the sides is an infinite series. The "equals to" sign that we have used prior to that mental exercise was for finite terms only, we had not had to deal with infinitely many terms before that leap in thought. So now we have to extend the notion in a way that is backward compatible.

A convenient one is it is equal to its limit if it exists.

Re: What can we gain by losing infinity?

#43

Post by archargelod »

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…

The obvious way to win in this game, that probably many kids discover is to define your number as "whatever number the other kid says, plus 1".

Re: What can we gain by losing infinity?

#44

Post by micheles »

I have always maintained that real mathematics starts when you address the infinite. I don't see how you can get anything interesting (like analysis, differential geometry, topology) without the assumption that the infinite exists.

BTW, the article is really badly written.

Re: What can we gain by losing infinity?

#45

Post by krackers »

Sad that the article doesn't mention wildberger (coincidentally similar last name), an (in)famous math youtuber that's been mentioned on HN several times before. He has a "rational trigonometry series" an approachable way to see how math would work in an ultrafinite setting.

Re: What can we gain by losing infinity?

#46

Post by don-bright »

Surprised Wildberger’s youtube channel wasnt in here.

People ask whats the point? For me the study of the infinitesimal vs finite has really helped me better understand issues of precision and approximation in computers. I feel like I know exactly why 1/3 plus 1/5 is not exactly 8/15 in my Calculator app. Or why points in my 3d object face are not coplanar after rotation. Or why games have weird glitches when your character is too far from origin point. Or why a spreadsheet shows rounding issues

Re: What can we gain by losing infinity?

#47

Post by marcus_holmes »

alex_young wrote: post #38
marcus_holmes wrote: post #30

Earlier quoted context omitted.

I think you missed the point. So, firstly, you have split the particle 5 times. That's not infinite times. You can split it more, so that would be 6 times. And more. Even if you could split it 1000 times, that's not infinity. The standard argument for infinity is that "you can always add 1 to any number, so there must be an infinity of them", and the refutation is that no matter how many times you add 1 to a number,…

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.

Re: What can we gain by losing infinity?

#48

Post by nkrisc »

jcgrillo wrote: post #10
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?

> Yes, they could on indefinitely Only if they live forever, which they won't. They can only count so fast, and there are only so many of them. Even if every atom in the observable universe was counting at, idk, 1GHz, that's still a finite number. The universe is not (as far as we know for certain) infinitely old. Time may extend infinitely into the future, or it may not. We don't know. So far as we know for sure eve…

Correct. Will they? No, they won’t, because they will die some day.

Re: What can we gain by losing infinity?

#49

Post by nkrisc »

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…

The game is to name a larger number. Is infinity a number? Some would say, “no”.

Eventually, one of the kids will name a largest number, because no one else will name another and the game ends with a largest number.

It is possible that aliens exist, so is that proof aliens exist?

Is is possible to create ever larger numbers, but is that proof that infinity exists other than as a fanciful idea in our minds?

Re: What can we gain by losing infinity?

#50

Post by jcgrillo »

nkrisc wrote: post #48
jcgrillo wrote: post #10

Earlier quoted context omitted.

> Yes, they could on indefinitely Only if they live forever, which they won't. They can only count so fast, and there are only so many of them. Even if every atom in the observable universe was counting at, idk, 1GHz, that's still a finite number. The universe is not (as far as we know for certain) infinitely old. Time may extend infinitely into the future, or it may not. We don't know. So far as we know for sure eve…

Correct. Will they? No, they won’t, because they will die some day.

Maybe an exceptional one will realize the task's futility and invent infinity as a way to rationalize giving up
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