101–110 of 226 posts
So, here's to math keeping our imagination limber and extending our ideas of what's real.
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What specifically doesn't sound right?
The claim is that every bb(n) is computable but I don’t think you can compute bb(6) without knowing which machines won’t halt. That doesn’t seem like a finite calculation? But given the answer, I suppose you could write a program that just returns it. This seems to hinge on the definition of “computable.” It’s an integer, so that fits the definition of a computable number. My mistake.
Then HH the function itself is not computable, but the numbers 0 and 1, which are the only two outputs of HH are computable.
Integers themselves are always computable, even if they are the output of functions that are themselves uncomputable.
Just models, useful but flawed abstractions.
I'm an enthusiastic Cantor skeptic, I lean very heavily constructivist to the point of almost being a finitist, but nonetheless I think the thesis of this article is basically correct. Nature and the universe is all about continuous quantities; integral quantities and whole numbers represent an abstraction. At a micro level this is less true -- elementary particles specifically are a (mostly) discrete phenomenon, but…
You know it wouldn't be possible for us to tell the difference between a rational universe (one where all quantities are rational numbers) and a real universe (one where you can have irrational quantities). The standard construction for the real numbers is to start with the rationals and "fill in all the holes". So why even bother with filling in the holes and instead just declare God created the rationals?
I'm an enthusiastic Cantor skeptic, I lean very heavily constructivist to the point of almost being a finitist, but nonetheless I think the thesis of this article is basically correct. Nature and the universe is all about continuous quantities; integral quantities and whole numbers represent an abstraction. At a micro level this is less true -- elementary particles specifically are a (mostly) discrete phenomenon, but…
You know it wouldn't be possible for us to tell the difference between a rational universe (one where all quantities are rational numbers) and a real universe (one where you can have irrational quantities). The standard construction for the real numbers is to start with the rationals and "fill in all the holes". So why even bother with filling in the holes and instead just declare God created the rationals?
Earlier quoted context omitted.
The human mind can't work with a real number any more than it can infinity. We box them into concepts and then work with those. An actual raw real number is unfathomable.
I don’t know about you, I can work with it just fine. I know its properties. I can manipulate it. I can prove theorems about it. What more is there? In fact, if you are to argue that we cannot know a “raw” real number, I would point out that we can’t know a natural number either! Take 2: you can picture two apples, you can imagine second place, you can visualize its decimal representation in Arabic numerals, you can…
At least at this stage I think it relates to whether you believe "the universe"/reality is a sort of momentary collection of the currently-existing things. Vs seeing reality as the set of all things that might obstruct "me" or any entity from doing something.
To me, even if the wall is invisible it's still a wall
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Please say more, I don't see how you can be _skeptical_ of those ideas. Math is math, if you start with ZFC axioms you get uncountable infinites. Maybe you don't start with those axioms. But that has nothing to do with truth, it's just a different mathematical setting.
I loosely identify with the schools of intuitinalism/construtivism/finitism. Primary idea is that the Law of the Excluded Middle is not meaningful. So yes, generally not starting with ZFC. I can't speak to "truth" in that sense. The skepticism here is skepticism of the utility of the ideas stemming from Cantor's Paradise. It ends up in a very naval-gazing place where you prove obviously false things (like Banach-Tars…
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The quantity of matter and the quantity of electricity are discrete, but work, time and space are continuous, like also any quantities derived from them. There have been attempts to create discrete models of time and space, but nothing useful has resulted from those attempts. Most quantities encountered in nature include some dependency on work/energy, time or space, so nature deals mostly in continuous quantities, o…
> but work, time and space are continuous I'm under the impression that all our theories of time and space (and thus work) break down at the scale of 1 plank unit and smaller. Which isn't proof that they aren't continuous, but I don't see how you could assert that they are either.
It’s fairly easy to go from integers to many subsets of the reals (rationals are straightforward, constructible numbers not too hard, algebraic numbers more of a challenge), but the idea that the reals are, well real, depends on a continuity of spacetime that we can’t prove exists.