131–140 of 148 posts
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I'd say if our universe makes such computations possible, then that would be very interesting.
Absolutely! Similarly, if our universe allowed instantaneous travel and free energy, that would also be very interesting. Not holding my breath for either.
To 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.
So, while I can't "point" to an infinite number of things like I can point to 9 things or 3.62 things, I still think it exists.
I'm not sure how well this generalizes to all infinite cardinals, ordinals, or to transfinite induction/construction. It is certainly strange that Cantor's theorem (the cardinality of a set is strictly smaller than that of its power set) implies there are different sizes of "all" implicit in my usage of the word.
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I'd argue you can't cram any electrons into any finite space without infinite energy. But we're not really talking quantum physics, here.
I'm not sure what you mean - any battery you have would seem to contradict you - would you elaborate?
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> The Banach-Tarski theorem is a consequence of things we want Is it? I think the parent comment is saying: “maybe we shouldn’t want things that result in Banach-Tarski” Maybe it’s a hint that the underlying axioms we’ve selected aren’t exactly what we want. You’re right that we can’t pick and choose the results of our axioms, but we do explicitly get to pick and choose the axioms we start with. If we choose bad axio…
> Is it? I think the parent comment is saying: “maybe we shouldn’t want things that result in Banach-Tarski” OK, which axiom do you want to replace?
Actually selecting and proposing an axiom set is way outside my knowledge-base. My limited understanding is that the Axiom of Choice, in ZFC leads to Banach-Tarski, and if it’s removed Tarski doesn’t hold, but I don’t have nearly enough information to say if that’s worth exploring removing it.
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Ultimately this crowd wants to change the practice of mathematics in the real world, so they are very accomiadating. See https://golem.ph.utexas.edu/category/2021/06/large_sets_1.ht... for tackling the "large cardinal pissing contest" that is much of modern set theory. Your very statement is a good retreat from platonism with blinders, acknowledging the inherit "moral relativism" that there are many possible foundati…
> Ultimately this crowd wants to change the practice of mathematics in the real world, so they are very accomiadating PhD mathematician in industry here. The way I see it, foundations is to the rest of mathematics the way music theory is to music: it needs to be a describer, not a prescriber. (If I were less charitable I'd have said "ornithology is to birds"). > the mainstream formalizations have clearly failed in th…
That sounds nice, but breaks down when one thinks harder. Music is a little bit physical phenomena, a little more biological phenomena, and even more cultural phenomina. That's many layers at once, and theory has to conform to the evidence.
Math, is not science. This is no evidence external to reasoning. Different foundations / formal systems conclude different things.
At best, we can look at what matches existing working mathematicians mental heuristics and.....that's not ZFC, which admits all sorts of crap because it is untyped.
> On the contrary, ZFC has been a tremendous success in that most mathematicians don't need to worry about it at all.
That is how most mathematicians see it, but us in the type theory crowd see that as bad goalposts necessitated by the fact that ZFC is so clunky to work with --- of course one wants to declare mission accomplished and move on to other things as quickly as possible with a foundation like that.
Check out https://xenaproject.wordpress.com/ for a less heterodox approach, that nevertheless does use a type theoretical "user interface" and "kernel" (trusted foundation) for purely practical reasons. Basically, the idea is making making formalized mathematics not a huge burden necessitates a more ergonomic system than was needed 80 years ago without computers.
Can someone correct me if im wrong? What i see here is a splitting of the set of points in the sphere? However the set of points in the sphere is not really the sphere. A point has no volume so no matter how many you add together you don't get something with a volume. This seems more akin to splitting the natural numbers into odd and even numbers which are all equally large. The language that i see in this article an…
I'll be using spatial dimensions as a conceptual framework to tackle this exact issue in future videos.
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Sorry, I'm not getting it. A large use case for complex numbers is describing things that rotate, literally or not, like oscillations, waves etc. Trigonometry lies deeply in that math and the irrational number pi pops out left and right. An approximation of pi wouldn't cut it, would it?
Actually, for any practical use case or possible observation, there is an approximation of Pi that is good enough. The ancient Egyptians apparently did quite well in their architecture approximating Pi as 22/7 (3.(1428571)). You only need the exact number Pi if you want to measure something like the ratio between the length of a perfect circle and its radius with infinite precision. But you can't be sure your measure…
You're talking about measuring something in the real world. Measuring is basically counting how many thing a given reference thing fits into the thing you're measuring. I have no problems with the assumption that for all intents and purposes we live in a finite (space and time) physical universe and there is a maximum precision that will ever be necessary.
What I am talking about is that in order for the math we use to describe that universe to work out we need irrational numbers; otherwise you couldn't be able to prove theorems and whatnot. I think this makes the irrational numbers (e and pi in particular) quite fundamental tools and I don't care if the real world doesn't allow objects (or positions) to be measured with irrational numbers.
> there is no (known?) way to compute the ratio between the length of an ellipse and the properties of its foci.
There is no closed-form expression for the circumference of an ellipse. There is an infinite series though. Same for a circle; there is no closed-form for computing pi either!
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> Is it? I think the parent comment is saying: “maybe we shouldn’t want things that result in Banach-Tarski” OK, which axiom do you want to replace?
Me? I have no idea. I was just trying to catch and diffuse what seemed like a miscommunication between two people. Actually selecting and proposing an axiom set is way outside my knowledge-base. My limited understanding is that the Axiom of Choice, in ZFC leads to Banach-Tarski, and if it’s removed Tarski doesn’t hold, but I don’t have nearly enough information to say if that’s worth exploring removing it.
The Cartesian product of non-empty sets is itself non-empty.
The Cartesian product of sets S_1, S_2, S_3, ... is of course the set of tuples (s_1, s_2, s_3, ...) such that s_1 ∈ S_1, s_2 ∈ S_2, s_3 ∈ S_3, ... . An element of the Cartesian product is a tuple with one element drawn from each of the sets being, um, Cartesianly multiplied.
Thus, the Cartesian product of the three sets {1, 4}, {a, b}, and {@, 2} is the set {(1,a,@), (1,a,2), (1,b,@), (1,b,2), (4,a,@), (4,a,2), (4,b,@), (4,b,2)}.
The Cartesian product of the three sets {1, 4}, {}, and {@, 2} is {}, the empty set, because no tuples exist such that the second element of the tuple belongs to the set {} (the second Cartesian factor).
So all the Axiom of Choice asserts is that, if all of the Cartesian factors are nonempty, then a tuple exists with one element drawn from each of the Cartesian factors. The only way for it to be impossible for such a tuple to exist is if one of the factors itself has no elements.
It's a theorem for finite Cartesian products, so all the dispute is over infinite products.
It's probably also worth mentioning that the C in ZFC stands for the Axiom of Choice, which is an indicator that people have explored not using it. ZFC without the Axiom of Choice is ZF.
That part is easy - for each point on a unitary sphere move it to a point at position 2x ( ie. to a corresponding location on the sphere of the 2 units radius) - you've just doubled the volume, i.e. you've just built a 2 units radius sphere out of the points belonging to 1 unit radius sphere. Banach-Tarski of course more fun and illustrates much more than just volume.