I don't understand why this misconception happens:
"If a number is irrational then the decimal expansion goes on forever and never repeats.
"Therefore every finite sequence of numbers must appear somewhere."
This is clearly, demonstrably false, and yet some people cling to it, dismissing any attempt to explain why it's not true.
They're not stupid, so:
* Why do they believe it?
* Why do they reject the demonstration that it's false?
I don't understand ...
@ColinTheMathmo it does seem plausible though.
Confusing infinite sequence with equal distribution with irrational number where distribution is unknowable and may not contain certain digits at all. Type 1 thinking shortcut.
I think we’re in Monty Hall territory.
@ColinTheMathmo @francis I think the certainty must lie in their definition of infinity, and it's a definition that seems to me to be shared by a lot of professional physicists:
"infinity is the size of all possibilities". Are there alternate realities? An infinite number you say? Obviously they must contain every conceivable variation of me, and of Earth, etc.
That's not true, of course; every reality could be the same, for instance. "But then you don't need infinity, that's only one." There may be a kind of conceptual size optimization there that brains do automatically. Perhaps.
But physicists talking about infinite realities, and/or infinite spatial universes, seem to make this conceptual error quite often.
They may make caveats about the obvious exceptions, like the above one where all are identical, but the *real* "infinite universes" tends to be assumed to be "all possible universes".
P.S. I have tried to wrap my mind around what new possibilties are possible (or required) if the number of (ergodic) realities / universes is ℵ 1 rather than ℵ 0 . It can definitely be a confusing topic.
@dougmerritt
I think this primarily stems from the "many worlds" interpretation of quantum mechanics, in which wave function "collapse" spawns one new universe for every eigenvalue of the measured operator. So, in that specific and explicitly defined sense, the idea really is "every possibility".
But then people talk imprecisely and extend this idea in ways that aren't warranted.
@ColinTheMathmo @francis
@jswphysics @ColinTheMathmo @francis
Indeed, no question, but that goes to the heart of it: what is "every possibility"? This started with the question of numbers being normal and related qualities. What if, after the 10^(10^200)'th decimal point of the decimal representation of pi, there were no longer any 0 digits or 1 digits or something?
What if in all those "possible worlds", I (or you or Colin or someone) never occurs, we're only in this reality?
There is a strong tendency for people to assume, and not even notice they are assuming, that "every possibility" might exclude certain things, in the absence of proof otherwise. It's a slippery topic.
@dougmerritt
In "many worlds", it means precisely all of the eigenvalues of the measured operator. No ambiguity in that narrowly-defined context.
Any extension beyond that context has no guarantee of being well formed.
@ColinTheMathmo @francis
@jswphysics @ColinTheMathmo @francis
Quite, and I didn't mean to slight the profession of physicist earlier, experimental or theoretical: I didn't say so, but I meant "when they are explaining things to a mass audience", not when they are writing research papers and textbooks.
In books written for the popular audience, and in tv shows explaining things like many worlds, and so on, but especially about multiple universes, rarely will you see an explanation of "possibilities are limited to/defined by eigenvalues of measured operators, and here, let's spend an hour discussing measurement in modern physics".
The expert in those scenarios almost necessarily gives the wrong idea, and the lay person already thinks in terms of the wrong idea.
But also, all this changes when we move sharply away from measurement in our current reality. In this current discussion we're also talking about things like speculation that there are infinite universes "just like ours" but unreachable from ours because their dimensions are unrelated to our dimensions, but people (expert and otherwise) have speculated or even asserted that they exist nonetheless, and then we wonder, "what are they like"?
In *that* context we all have to be very careful. It's not simply mostly well-understood "many worlds" anymore. Those universes may or may not be ergodic within a speculative Hilbert space of "all possible qualities of universes" (and in what precise sense? which qualities?)
So I was claiming that *that* is where it is slippery to conceptualize, for physicists and non-physicists alike. It's easier to be rigorous about the digits of pi, but then one must be enough of a mathematician to avoid naive thinking about that.
@dougmerritt
Thankfully, as an experimentalist, I am permitted to sidestep most such questions 😂
@ColinTheMathmo @francis
@francis I think it's confusing vague concepts of infinite and randomness. There are true statements that are fairly close to these false ones, and the differences are easy to miss if you don't think carefully and precisely.
I'm just taken aback by how people can be so adamant that statement S is true in the face of clear demonstrations that it is not.
So I'm trying to find where their certainty comes from.