Terribly, terribly irrelevant, but:
Er, I meant “earlier”, of course.
Hmmmmmm . . . which seems to imply that if you add 0+0+0+0+0… + 0+0+0+0+0… + 0+0+0+0+0…, it can add up to any number you want, even infinity itself. This is the hazard of casually kicking around concepts like “infinity.”
But the premise of the OP has been discussed before, and is fraught with gaps in logic. Starting out with the phrase “Given infinity.” Infinity is not at all given; there’s no reason to assume that we live in any kind of “infinite” universe. And even if we do, there’s no reason to assume that anything is necessarily inevitable. Is it inevitable that there exists a place where 18-wheelers grow on vines, fully manned, gassed up and stocked with cargo . . . just because it can be imagined? Just like “infinity,” “inevitability” is a term not to be tossed around lightly.
I just want to put forward the idea that the common assumption that “the universe is alone, unique, finite, and containing finite things” is not something to throw around lightly either. There is no reason to assume that there are NOT an infinite number of universes, that our universe is finite, or that when you look closely as a little patch of space-time or quantum field it isn’t a frothing fractal containing infinite universes within it. We simply do not know. But to me, philosophically, it feels somehow biased and parochial to assume that “finite” is somehow more natural than “infinite.” To me it seems like “finite” is the stranger, forced, unnatural idea. That’s just my 2 cents.
True. Scientific American, for instance, had an article on mainstream “constructive mathematics,” and it seemed to be a perfectly reasonable, although slightly alternative, approach.
My friend is an “extreme” constructionist, and is willing to declare, from a thundering crag of Parnassian omniscience, that 10^230 is “not a number” since it does not represent anything that could ever possibly be made, counted, built, shown, or otherwise constructed.
Specifically, he says that there is a number n, for which “n+1” is “not a number.” This destroys one of the most vital properties of numbers, and makes groups, fields, and algebra impossible.
But, then, everyone to his weird. I’m a member of the Flat Earth Research Society!
Again, that’s true of the sensible variety. It’s like the difference between the strong Gaia hypothesis and the weak Gaia hypothesis, or the strong and weak Anthropic principles. A moderate form of constructive mathematics is instructive. The strong form is merely a dogma.
I apologize for tarring everyone with my friend’s brush.
Obvious joke: “See? Construction has drawbacks as well as advantages!”
(I miss the old Alpha Beta supermarket chain!)
A) Your obvious joke pleases me.
B) For the sake of offering up labels, your friend sounds like an “ultrafinitist”. Which is incidentally a perspective I am also rather sympathetic to (though from Wittgensteinian, not physical, motivations; after all, it is in fact true that everything we do in mathematics, including our calculations with ostensible humongous objects, is actually done with only feasibly-sized utterances). From the sound of it, I personally wouldn’t be sold on your friend’s particular flavor of ultrafinitist philosophy, but I also wouldn’t necessarily call the mathematical/logical framework which arsises out of this insensible, just different from tradition, in ways which could be enlightening (again, as always, so long as an interest in ultrafinitism didn’t blind one to the possibilities in studying other math-games, including many of the traditional ones, as well). Ultrafinitism intrigues me greatly, particularly as it has typically been difficult to give it a satisfactory formal account. But, yes, it goes far beyond most “constructivists”.
C) Pedantically, I suspect your friend would not feel groups, fields, algebra, etc. were impossible. They’d happily accept the field of size 7 and the symmetric group on 4 and so on. But, yes, I agree that not allowing oneself to talk about the group of integers and addition, if that is indeed something your friend bars themself from doing in any sense, is rather stifling.
I seen a peanut stand /And heard a rubber band /I’ve seen a needle that winked its eye / But I been done seen about everything / When I see an elephant fly.
I never claimed the universe is finite or alone either; to the extent of our actual knowledge, either assumption is unwarranted. Like you said, we simply do not know.
We are getting into completely the wrong existential theory here, because presuming that all that is imaginable is inevitable would include the conceivable eventuality that existence would eventually produce or concoct a means by which existence itself, in all quantum permutations, could be eradicated. As our continuing existence proves that this cannot be the case, it precludes any possibility that the theory is valid.
shadyginzo: To take another shot at this kind of self-referentiality, there can’t be a “place” out there that has sent public ambassadors to all of the other places. We know this, because there isn’t a public ambassador from them here.
I don’t think it weakens the notion irreparably to allow the “places” out there to be causally isolated from each other.
The first two words of your post are, “Given infinity”.
Why do you think that infinity is a given?
I cannot see anything that would remotely be qualified as “infinite”, outside of mathematics, and even there, it is an imaginary construct.
OffByOne, if you’re responding to the OP, note that this thread was started around 3 years ago, by someone who hasn’t posted in about a year.
I think that given infinite amount of time and space in which for things to happen, the number of things that will indeed happen is infinite, but so is the number of things that will not happen.
The number of digits of pi is infinite and there are an infinite number of sequences of numbers of various finite lengths that can be found within it, but the number of sequences of digits that do not appear within it is also infinite.
AHunter3: has that been demonstrated mathematically? Are there actually known sequences of digits (in whatever base arithmetic you choose) that are proven not to appear anywhere in an expansion of the value of pi?
Certainly there are sequences that do not appear in any finite subset of the infinite expansion, but is it known that there are finite sequences of digits that never appear? I was very much under the impression that the opposite was the prevailing idea – although I do not know if it has ever been proven.
The digits of pi are so remarkably “normal” as to appear all but indistinguishable from random. The digit “5” appears almost exactly once in ever ten digits – just as it would in a random sequence. (And so on, for every test for distribution known.)
In contrast, the cosmos has “rich” zones and “poor” zones – galactic centers and intergalactic vacuum. This anisotropy could (in theory) be a reason why “everything” doesn’t happen. It’s the ratio of interesting to uninteresting zones that is the key, and nobody knows this.
Pi (I believe) is almost perfectly isotropic. (Or do I mean isotopic?)