Many human ideas, from the Christian hope of 'eternal life' to the mathematical 'set of all whole numbers', require the concept of infinity. Indeed, our current scientific model of the universe still includes infinity, in that space-time is said to locally resemble a real metric space (excluding singularities).
On the other hand, the 20th century saw the standard model get a good deal more restrictive, with a lot of infinites falling out: Einstein's cosmic speed limit, the Big Bang with its implication of a universe of finite diameter, and the quantisation of matter-energy, with an attempt to classify all matter into a finite number of elementary particles.
With this in mind, suppose that space-time in our universe can be divided into finitely many 'pixels' which can be split no further (different positions within the same pixel are always exactly the same as each other). Suppose further that, if the universe is not deterministic, the probability of a particular particle lying in a particular pixel can take one of a finite number of values (eg it must be a multiple of 1/5). Finally, suppose that if there are other universes out there, there are finitely many of them and each is as finite as our own.
The totality of all that is, was and ever could be is thus reduced to a finite set of possibilities, and infinity is merely an abstract concept people have come up with, like the complex numbers, with no direct connection to reality. Is this possible? How would we know? And what philosophical implications would it have?
Originally posted by AcolyteMany models that incorporate infinity lead to real paradoxes, i.e. logical contradictions, which is evidence that those infinities do not in fact precisely reflect reality.
The totality of all that is, was and ever could be is thus reduced to a finite set of possibilities, and infinity is merely an abstract concept people have come up with, like the complex numbers, with no direct connection to reality. Is this possible? How would we know? And what philosophical implications would it have?
So to answer your questions..
Is it possible? Yes, and for some cases it is even known.
How would we know? By the presence of infinity-related contradictions.
One philosphical implication is that one must strive to not mistake one's model of reality for reality itself.
Originally posted by AcolyteThe implication that jumps out at me is that if everything is finite, then theoretically there might have been more of anything. I don't know where that goes, but it's kind of interesting.
Many human ideas, from the Christian hope of 'eternal life' to the mathematical 'set of all whole numbers', require the concept of infinity. Indeed, our current scientific model of the universe still includes infinity, in that space-time is said to locally resemble a real metric space (excluding singularities).
On the other hand, the 20th century saw ...[text shortened]... lity. Is this possible? How would we know? And what philosophical implications would it have?
On the flip side of this, one of the biggest problems I have with a lot of conceptions of an eternal paradise is that it is eternal, because given an infinite amount of time, I could, following arbitrarily long procrasination, do anything logically possible with no constraints (for example, I'm not clever enough to play a game of perfect Go on an infinite board, but given infinite time, I could follow every possible game to its 'conclusion', while leaving time to meet everyone who will ever exist). In fact, other than about the order in which to do things, I would have no major decisions to make, a situation I'd find boring enough that I'd rate it short of paradise.
Originally posted by royalchickenWell this is assuming Heaven would just be like life on Earth. I always imagined Heaven, if it were to exist, would be eternally being in a state of total ecstacy, bliss, and happiness. It wouldn't matter what you were doing, because you'd enjoy it so tremendously all the time forever.
On the flip side of this, one of the biggest problems I have with a lot of conceptions of an eternal paradise is that it is eternal, because given an infinite amount of time, I could, following arbitrarily long procrasination, do anything logically possible with no constraints (for example, I'm not clever enough to play a game of perfect Go on an infin ...[text shortened]... decisions to make, a situation I'd find boring enough that I'd rate it short of paradise.
This would be how I'd imagine Heaven, if it existed. I don't think it does though.
Originally posted by royalchickenNot really - if there are infinitely many things to do, you need only do a subset of them which takes up whatever eternity you've been allotted, and in fact there may not even be time to do all of them. For example, if you resolve to spend one minute contemplating each real number individually, an ordinary sigma-finite eternity just wouldn't be enough. So you would have some prioritising to do.
On the flip side of this, one of the biggest problems I have with a lot of conceptions of an eternal paradise is that it is eternal, because given an infinite amount of time, I could, following arbitrarily long procrasination, do anything logically possible with no constraints (for example, I'm not clever enough to play a game of perfect Go on an infin ...[text shortened]... decisions to make, a situation I'd find boring enough that I'd rate it short of paradise.
However, it's possible that there are only finitely many interesting things open to human comprehension, and that all of these put together would only take finite time to fully understand. For example, it's possible that after a finite time you would understand Go completely, and so seeing any further Go games would be pointless.
Of course, some people (including many Christians) believe in a rather abstract afterlife in which the honoured dead bear little resemblance to their formerly living equivalents, having no memories of life and no particular desires. How this is different from destruction (which is the most popular alternative to belief in a concrete Hell) is not entirely clear to me.
Originally posted by AcolyteThere are only countably many things I can do though (Or are there? Consider moving my arm; that could be interpreted as doing uncountably many things.). For example, I don't think it's meaningful to talk about contemplating each real number, because any contemplation I can think of requires the number to be definable, in which case I've only got a countable subset of the reals to contemplate. Given this, the worst I'd be faced with is figuring out how to order the small actions that make up each of the large (possibly infinite-step) actions I undertake. (I'd have to consider some sequence of Go moves and then meet some group of people, then resume Go-playing.)
Not really - if there are infinitely many things to do, you need only do a subset of them which takes up whatever eternity you've been allotted, and in fact there may not even be time to do all of them. For example, if you resolve to spend one minute contemplating each real number individually, an ordinary sigma-finite eternity just wouldn't be enough. S ...[text shortened]... which is the most popular alternative to belief in a concrete Hell) is not entirely clear to me.
Originally posted by royalchickenExactly. The very paradox I had in mind in my first post was that which arises when choosing a real number at random from some real interval.
For example, I don't think it's meaningful to talk about contemplating each real number, because any contemplation I can think of requires the number to be definable, in which case I've only got a countable subset of the reals to contemplate.
My bad; I'm saying two different things. Each possible position in infinite Go is representable by an infinite sequence of 0s, 1s and 2s (a given point can be empty, have a black stone or a white one, and the set of points is countable). Thus the set of infinite Go positions corresponds to the real numbers, so considering infinite Go positions is equivalent to doing exactly what I said I can't do.
Originally posted by AThousandYoungOne of the many things I'm looking forward too is music.
Well this is assuming Heaven would just be like life on Earth. I always imagined Heaven, if it were to exist, would be eternally being in a state of total ecstacy, bliss, and happiness. It wouldn't matter what you were doing, because you'd enjoy it so tremendously all the time forever.
This would be how I'd imagine Heaven, if it existed. I don't think it does though.
Since it is a timeless place, music will be interesting.
Kelly