| > 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. |
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| ▲ | freetime2 3 hours ago | parent | next [-] | | 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. | | |
| ▲ | samplatt 3 hours ago | parent | next [-] | | And then the next kid says "infinity plus two", which is a perfectly acceptable progression, and the cycle starts again. | | |
| ▲ | rmunn 2 hours ago | parent [-] | | 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). | | |
| ▲ | ndriscoll 2 hours ago | parent [-] | | 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. |
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| ▲ | p1necone 3 hours ago | parent | prev [-] | | INFINITY PLUS 1 | | |
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| ▲ | jcgrillo 3 hours ago | parent | prev [-] | | > 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. |
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