The complement of the set of normal numbers has measure zero. A normal number is what the author of the paper is talking about when referring to random numbers. I think most mathematicians, if they had to bet, would bet that sqrt(2) is normal. It is not known though. If it is normal then it’s digits are random.
I haven’t read the paper but I think the author is arguing that most real numbers don’t make sense physically. If sqrt(2) were shown to be normal and since it’s the hypotenuse of a right triangle of legs with length 1, I wonder what the author's response to this would be. Perhaps he’d say that such a triangle doesn’t exist physically.
Perhaps he’d say that such a triangle doesn’t exist physically.
I am certain that this is the case given that he's referring to the maximum information density density of space as one of the reasons why exact numbers make no physical sense. See https://en.wikipedia.org/wiki/Bekenstein_bound if you don't know about that limit.
More directly, the inability to represent exact lengths falls out of the Heisenberg Uncertainty principle. As an abstract mathematical concept, numbers could be anything. As a physically relevant concept, there are limits to how much precision can matter.
What does it mean to produce a physically realized triangle with length exactly 1? Some specific number of atoms in a specific place? How does that square with the uncertainty principle?
I haven’t read the paper but I think the author is arguing that most real numbers don’t make sense physically. If sqrt(2) were shown to be normal and since it’s the hypotenuse of a right triangle of legs with length 1, I wonder what the author's response to this would be. Perhaps he’d say that such a triangle doesn’t exist physically.