Banach-Tarski and the Paradox of Infinite Cloning
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Banach-Tarski and the Paradox of Infinite Cloning
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Re: Banach-Tarski and the Paradox of Infinite Cloning
#2If I remember rightly there'a a Feynman anecdote where he points out that as the real universe is quantised this is a purely mathematical notion.
I used to riff with a friend that we were "the two members of the Banach-Tarski quartet." :)
Re: Banach-Tarski and the Paradox of Infinite Cloning
#3To me this is proof that infinity is something only present in our math and not in the universe.
Infinity is a nice approximation but it feels like wishful thinking that our universe or anything in it is infinite.
Happy to hear disagreements tho.
Re: Banach-Tarski and the Paradox of Infinite Cloning
#4To me this is proof that infinity is something only present in our math and not in the universe. Infinity is a nice approximation but it feels like wishful thinking that our universe or anything in it is infinite. Happy to hear disagreements tho.
What's your concept of time and its extension then?
Re: Banach-Tarski and the Paradox of Infinite Cloning
#5If I remember rightly there'a a Feynman anecdote where he points out that as the real universe is quantised this is a purely mathematical notion. I used to riff with a friend that we were "the two members of the Banach-Tarski quartet." :)
Current physical models don't have the universe itself (space-time) quantized - only matter is quantized. Even the planck time and planck length only represent minimal measurable distances/durations - the maths still assume that two things can be separated by fractional multiples of these.
That's not to say that physics requires infinities, but current models also don't disallow infinity.
Of course, actual infinity is outside the purview of science - there is no way to differentiate between infinity and something too big/small to measure, even in principle. Apparent paradoxes related to infinity, such as Banach-Tarski, don't change this, as they also require infinite precision to realize, making them impossible to test as well - even if a sphere is indeed made up of an infinity of space-time points, and even if we could manipulate those, we wouldn't be able, in finite time, to extract the necessary infinite subsets of points to create the two spheres from one.
Re: Banach-Tarski and the Paradox of Infinite Cloning
#6To me this is proof that infinity is something only present in our math and not in the universe. Infinity is a nice approximation but it feels like wishful thinking that our universe or anything in it is infinite. Happy to hear disagreements tho.
Well, infinity is inherently unscientific, as there is no way to scientifically differentiate between an infinite quantity and a really huge quantity (same for infinitesimals), in finite time.
What this means is that a universe that contains infinities is, even in theory, entirely indistinguishable (in finite time) from an universe that contains really large/small but finite quantities.
Re: Banach-Tarski and the Paradox of Infinite Cloning
#7To me this is proof that infinity is something only present in our math and not in the universe. Infinity is a nice approximation but it feels like wishful thinking that our universe or anything in it is infinite. Happy to hear disagreements tho.
Well, infinity is inherently unscientific, as there is no way to scientifically differentiate between an infinite quantity and a really huge quantity (same for infinitesimals), in finite time. What this means is that a universe that contains infinities is, even in theory, entirely indistinguishable (in finite time) from an universe that contains really large/small but finite quantities.
> there is no way to scientifically differentiate between an infinite quantity and a really huge quantity (same for infinitesimals), in finite time
That depends on the model of computation you pick, doesn't it?
Re: Banach-Tarski and the Paradox of Infinite Cloning
#8To me this is proof that infinity is something only present in our math and not in the universe. Infinity is a nice approximation but it feels like wishful thinking that our universe or anything in it is infinite. Happy to hear disagreements tho.
Mathematics doesn't really "exist" in the first place. It's more of a language that's rich enough and with enough logic to describe/approximate the laws of physics that actually do exist.
Re: Banach-Tarski and the Paradox of Infinite Cloning
#9I do enjoy articles like this. They are such good ways to make math compelling for laymen such as myself.
Re: Banach-Tarski and the Paradox of Infinite Cloning
#10If I remember rightly there'a a Feynman anecdote where he points out that as the real universe is quantised this is a purely mathematical notion. I used to riff with a friend that we were "the two members of the Banach-Tarski quartet." :)
Current physical models don't have the universe itself (space-time) quantized - only matter is quantized. Even the planck time and planck length only represent minimal measurable distances/durations - the maths still assume that two things can be separated by fractional multiples of these. That's not to say that physics requires infinities, but current models also don't disallow infinity. Of course, actual infinity i…
Physics doesn't require infinities, but it's scary how well QED/QFT approximates the g-factor (https://en.m.wikipedia.org/wiki/G-factor_(physics) ) for electrons given the amount of renormalization (cancelling of infinities) needed to estimate the true value in nature.