Comment by freetime2 | Hacker News Reader

Comment by freetime2 | Hacker News Reader

> 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-schoolers, competing to see who can say the highest number, quickly learn the concept of infinity. Or elementary school students trying to write 1/3 as a decimal.

Of course you need to be careful mapping infinity onto the physical world. But as a mathematical concept, there is absolutely nothing wrong with it.

> Mathematicians can construct a form of calculus without infinity, for instance, cutting infinitesimal limits out of the picture entirely.

This seems like a useful concept that also doesn't require denying the very obvious concept of infinity.

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 everything is in fact finite.

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

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.

INFINITY PLUS 1

Uncountable infinity

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 write 0.9999...5, which falls into the same category error as "infinity plus one / infinity plus two".

Now, decades later, I get it better. The number 0.99999... is 9/10 + 9/100 + 9/1000 + 9/10000 + ..., which approaches 1 asymptotically the same way that 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ... approaches 1. Under many circumstances, you can treat that number as if it was 1, which neatly answers Zeno's Paradox. (Though beware of the limitations of that analysis: 1/n approaches infinity as n approaches 0, but 1/0 is not equal to infinity. Because 1/n approaches infinity only as n approaches 0 from the positive direction. If you look at the sequence 1/-0.1, 1/-0.01, 1/-0.001, etc. where n approaches 0 from the negative direction, that approaches negative infinity. A function that has two different limits as you approach the same number from two different directions cannot have its limit substituted like that).

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.

> semantics we attach to "=" when one of the sides is an infinite series

I would say that the semantics are about what an infinite series itself is, not about the equal sign. Once we have the common analytic notion of convergence of an infinite series, then the equality makes sense. The issue is that an infinite series is not an actual sum, but, formally, it is a sequence (of the partial sums). As you say, we represent the limit of the sequence of the partial sums with the same notation and only in the case that we have absolute convergence, but that's basically because we use the same notation for two different things (the sequence of the partial sums, and the limit of that). If we know we refer to the limit, I don't think there is any semantic complication with the equal sign.

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".

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?

24 is the highest number. Where you gonna go from there?

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