Noah's Ark

[luke warm]

Even in death, Hilbert is probably a smarter guy than I am so I won't be explaining why he is wrong. But if you have a context for the quote I could try thinking about it. Maybe something got lost in the translation from the German. I suppose there are only finitely many rocks on the face of the earth, and maybe only finitely many in the universe, but I doubt that Hilbert spent much time reflecting on such a matter. I don't know what he said about the actual infinite or the potential infinite. I am sure that I have never before heard either of these phrases.

 

The story goes that Hilbert was attending a mathematics lecture and the speaker began discussing Hilbert Space. Hilbert turned to the guy next to him and asked "What's a Hilbert Space?".   Words sometimes have a local meaning that outsiders have no idea of. "Hilbert Space" is now universal. "Potential infinity" is new to me.

apparently hilbert spent more time contemplating these things than we might think... the context can only be found, to my knowledge, in hilbert's "on the infinite" which appeared in "philosophy of mathmatics" in 1964... the quote in question read, "The infinite is nowhere to be found in reality. It neither exists in nature nor provides a legitimate basis for rational thought... The role that remains for the infinite to play is solely that of an idea."... hilbert in no way denied the importance of cantor's work, even saying (in the same place), "No one shall be able to drive us from the paradise that Cantor has created for us."

 

this, imo, shows the difference between an actual vs. a potential infinite - the actual infinite is only conceptual, it can only exist in the mind... if, in nature, an actually infinite number of things could exist, there would be all kinds of absurdities.. this is what hilbert's hotel was supposed to show... unlike hilbert's space (whatever that is), i doubt he'd have denied knowledge of his hotel...

 

it is much like (if i have understood your and helene's explanation) matching {a,b,c} with {1,2,3}, with the difference being we're matching guests to hotel rooms... if an infinite number of rooms were each occupied by one person and if another person showed up asking for a room, he could be accommodated by simply moving each guest from his already occupied room to the next... guest 1 to room b and so forth... subsequent addition of guests to the hotel can be done conceptually only... such a hotel is not possible in reality since, in transfinite arithmetic (correct me if i'm wrong), subtraction is not allowed because of inconsistencies encountered... if this hotel existed in reality, however, a person could conceivably check out if he chose (i.e. the total could be subtracted from)

 

that was, i'm sure, a poor effort to explain the philosophical difference between actual and potential infinites

[helene_t]

Time is not real. The infinite only exists as a contruct in mathematics.

Time not real? What's that supposed to mean? That timespace has four dimensions and we just by convention chose a particular coordinate system and call one of the coordinates "time"? Or something else?

[TimG]

The infinite only exists as a construct in mathematics.

I think this is what Jimmy means by "actual infinite", the mathematical construct.

 

Jimmy's "potential infinite" is the real world's "unimaginably large". If you keep adding one grain of sand to your collection, the collection will get very large, but can always be described by a finite number.

 

Jimmy's view of "always", "forever" and "eternal" are as real world concepts of time and thus are "potential infinite" -- that is they can be a very long time, but no matter how far backwards (or forward) you go, the amount of time will be measurably finite.

 

(I'm not offering an opinion regarding these concepts, just trying to simplify the definitions in order to make the discussion more understandable.)

 

How'd I do, Jimmy?

[luke warm]

The infinite only exists as a construct in mathematics.

I think this is what Jimmy means by "actual infinite", the mathematical construct.

 

Jimmy's "potential infinite" is the real world's "unimaginably large". If you keep adding one grain of sand to your collection, the collection will get very large, but can always be described by a finite number.

 

Jimmy's view of "always", "forever" and "eternal" are as real world concepts of time and thus are "potential infinite" -- that is they can be a very long time, but no matter how far backwards (or forward) you go, the amount of time will be measurably finite.

 

(I'm not offering an opinion regarding these concepts, just trying to simplify the definitions in order to make the discussion more understandable.)

 

How'd I do, Jimmy?

pretty damn good... this should show why a successive addition of temporal events isn't possible if space/time were an actual infinite... if there were a starting point however (i.e. a beginning), we can so add

[jdonn]

All this talk about the Hilbert quote, I found this on the wikipedia article of Hilbert's hotel:

 

Because the Hilbert's paradox is so counterintuitive, it has often been used as an argument against the existence of an actual infinity, for instance an argument for the existence of God posed by the Christian philosopher William Lane Craig is roughly as follows;

 

Although there is nothing mathematically impossible about the existence of such a hotel (or any other infinite object), intuitively no such object could ever exist, and this intuition is a specific case of the broader intuition that no actual infinite could exist. Since a temporal sequence receding infinitely into the past would constitute such an actual infinite, time must have "started" at some point. Since "time" cannot be started by any temporal thing, and every action must have a cause, this cause must be God.

I guess my question is, how can someone honestly believe intuition proves anything? Or even moreso, how it can disprove anything that is not mathematically impossible?

[luke warm]

All this talk about the Hilbert quote, I found this on the wikipedia article of Hilbert's hotel:

 

Because the Hilbert's paradox is so counterintuitive, it has often been used as an argument against the existence of an actual infinity, for instance an argument for the existence of God posed by the Christian philosopher William Lane Craig is roughly as follows;

 

Although there is nothing mathematically impossible about the existence of such a hotel (or any other infinite object), intuitively no such object could ever exist, and this intuition is a specific case of the broader intuition that no actual infinite could exist. Since a temporal sequence receding infinitely into the past would constitute such an actual infinite, time must have "started" at some point. Since "time" cannot be started by any temporal thing, and every action must have a cause, this cause must be God.

I guess my question is, how can someone honestly believe intuition proves anything? Or even moreso, how it can disprove anything that is not mathematically impossible?

because hilbert's hotel is an example (if i understand helene and ken) of set theory where {1,2,3... } are guests and {a,b,c... } are rooms... if my understanding is correct, a guest in hilbert's hotel would be unable to check out, for example he could not leave to catch his flight to new orleans, because subtraction is not allowed and a guest who left would be subtraction... but in a real hotel, guests aren't usually held hostage... so intuitively (and, it seems, mathmatically), such a hotel is not possible in reality, iow transfinite arithmetic only applies conceptually (see the hilbert quote above)

[TimG]

The infinite only exists as a construct in mathematics.

I think this is what Jimmy means by "actual infinite", the mathematical construct.

 

Jimmy's "potential infinite" is the real world's "unimaginably large". If you keep adding one grain of sand to your collection, the collection will get very large, but can always be described by a finite number.

 

Jimmy's view of "always", "forever" and "eternal" are as real world concepts of time and thus are "potential infinite" -- that is they can be a very long time, but no matter how far backwards (or forward) you go, the amount of time will be measurably finite.

 

(I'm not offering an opinion regarding these concepts, just trying to simplify the definitions in order to make the discussion more understandable.)

 

How'd I do, Jimmy?

pretty damn good... this should show why a successive addition of temporal events isn't possible if space/time were an actual infinite... if there were a starting point however (i.e. a beginning), we can so add

What temporal events must be added?

[jdonn]

so intuitively (and, it seems, mathmatically), such a hotel is not possible in reality

LOL, uh, you do know that "it seems" is just another way to say "intuitively", right?

 

I mean, similar to as you often say, it's either mathematically impossible or it's not. If you can't prove that it is then how it seems proves nothing.

[helene_t]

All this talk about the Hilbert quote, I found this on the wikipedia article of Hilbert's hotel:

 

Because the Hilbert's paradox is so counterintuitive, it has often been used as an argument against the existence of an actual infinity, for instance an argument for the existence of God posed by the Christian philosopher William Lane Craig is roughly as follows;

 

Although there is nothing mathematically impossible about the existence of such a hotel (or any other infinite object), intuitively no such object could ever exist, and this intuition is a specific case of the broader intuition that no actual infinite could exist. Since a temporal sequence receding infinitely into the past would constitute such an actual infinite, time must have "started" at some point. Since "time" cannot be started by any temporal thing, and every action must have a cause, this cause must be God.

I guess my question is, how can someone honestly believe intuition proves anything? Or even moreso, how it can disprove anything that is not mathematically impossible?

Maybe Wikipedia is not Being fair to Craig, but as described in the Wiki article the argument is total rubbish.

 

First, who says the World isn't absurd?

 

Second, who cares if it would be absurd if someone got a room in a full hotel as long as it doesn't actually happen although it was possible? For all we know we live in one of an infinity of universes, so even if each was constrained not to contain hotels with an infinity of rooms, there would still be an infinity of hotels in the whole multiverse.

 

Third (and this is most crucial), how can one extrapolate from hotel rooms to time? Even if Hilbert's Hotel cannot exist (which I disagree with, but OK), why can't there be an infinity of time points?

 

Fourth, who says everything has a cause?

 

Fifth, even if there is only a finite number of time points, who says one had to be the first? Why do they have to be strictly ordered?

 

Sixth, so the argument goes that God existed before time existed? Now this seems totally absurd to me! But wth, maybe the world is absurd, what do I know, maybe the flying spaghetti monster and several even more absurd gods actually exist!

 

Seventh, even if we accept that something must be the cause of the beginning of time, I don't see how that something has to have any resemblance to any of the gods conceived of by religion. It could be some quantum fluke that by an improbable accident turned on time in the hitherto timeless proto-universe. Or some angry geek pressing ctrl-alt-delete on the computer that runs Fluffy's universe-simulation-software. Or .....

 

I personally don't see how the existence of something in nature that must necessarily be modeled by means of infinite sets could ever be confirmed. I find it more plausible that the non-existence of an infinite number of particles, or maybe even the non-existence of an infinity of points in spacetime, could be argued on the basis of experimental data.

 

However, such an argument would require insight in physics. Forming a firm opinion about such issues solely on the basis of an armchair argument is silly.

 

On the other hand, that spacetime is most mathematically conveniently modeled by means of infinite sets, is beyond dispute.

[PassedOut]

Before this thread, I was never able to make sense of most of the arguments people made for the existence of god. I've found this thread helpful, and am glad that Jimmy stuck with it until I understood. I now see that my strict interpretation of certain words like "infinite" and "logical" impeded my understanding. I didn't realize that others did not share the same understanding.

 

For example, I've never been able to comprehend the position that time proceeding backwards infinitely is logically impossible, for that certainly is not the case when one uses the term "logic" in the strict sense that I do. And because my intuition always told me that time never had a starting point, it was hard to grasp that anyone seriously considered it to be impossible.

 

Considering that in the history we know, for every moment in time there is always moment - 1, it always seemed simplest to continue inductively. In fact, I have a hard time getting my head around the notion that time might have started with the big bang, even though I know we have no information before that.

 

Maybe it does all come down intuition and whether or not one considers counting time and counting marbles as necessarily the same. Maybe if my intuition were different, I'd believe in a god too (shudder). Anyway, thanks.

[helene_t]

Considering that in the history we know, for every moment in time there is always moment - 1, it always seemed simplest to continue inductively. In fact, I have a hard time getting my head around the notion that time might have started with the big bang, even though I know we have no information before that.

 

Maybe it does all come down intuition and whether or not one considers counting time and counting marbles as necessarily the same. Maybe if my intuition were different, I'd believe in a god too (shudder).

Lol, who knows.

 

I share your view that infinite time (or alternatively: cyclic time) is easier to grasp. It is not something that really bothers me.

 

I can imagine myself turning religious because of personal spiritual experience or because of a likeness for some religious community. Factoring belief about scientific issues into it is something I find very weird. I mean, I really don't care how the universe came into existence, and I certainly wouldn't want to be a member of a religious society which required a particular belief of its members. I can much better understand how likeness for a particular kind of music, or moral issues, can define one's religious affiliation.

[luke warm]

What temporal events must be added?

temporal in the sense of historical happenings

Maybe Wikipedia is not Being fair to Craig, but as described in the Wiki article the argument is total rubbish.

well that settles it, then... by simply saying so you've accomplished, assuming the accuracy of the wiki article, in one sentence what others, some even doctors of philosophy, have attempted by writing volumes

Third (and this is most crucial), how can one extrapolate from hotel rooms to time? Even if Hilbert's Hotel cannot exist (which I disagree with, but OK), why can't there be an infinity of time points?

and i suppose you've done the same re: hilbert... perhaps he was wrong and you are right... since you think his hotel can exist, could someone check out of it?

Before this thread, I was never able to make sense of most of the arguments people made for the existence of god. I've found this thread helpful, and am glad that Jimmy stuck with it until I understood. I now see that my strict interpretation of certain words like "infinte" and "logical" impeded my understanding. I didn't realize that others did not share the same understanding.

i think you'll agree that hilbert was a mathematician of not inconsiderable talent... was his understanding of an actual infinite different from yours, and was he wrong?

[jdonn]

Maybe Wikipedia is not Being fair to Craig, but as described in the Wiki article the argument is total rubbish.

well that settles it, then... by simply saying so you've accomplished, assuming the accuracy of the wiki article, in one sentence what others, some even doctors of philosophy, have attempted by writing volumes

Helene even carefully noted that the article very well may be unfairly representing his view. But in any case, rather than resorting to sarcasm you might have considered responding to her particular points.

 

Why not just try to tackle this one?

I personally don't see how the existence of something in nature that must necessarily be modeled by means of infinite sets could ever be confirmed. I find it more plausible that the non-existence of an infinite number of particles, or maybe even the non-existence of an infinity of points in spacetime, could be argued on the basis of experimental data.

 

However, such an argument would require insight in physics. Forming a firm opinion about such issues solely on the basis of an armchair argument is silly.

 

On the other hand, that spacetime is most mathematically conveniently modeled by means of infinite sets, is beyond dispute.

[PassedOut]

i think you'll agree that hilbert was a mathematician of not inconsiderable talent... was his understanding of an actual infinite different from yours, and was he wrong?

Talented people go off on tangents now and then. Even Einstein made mistakes. Maybe even Bertrand Russell.

 

To be honest, I don't really understand "actual infinite" and don't see any reason to spend time working on it, given all the worthwhile things I've left to learn (and given the relative shortness of the portion of my life that remains).

 

However, I'll gladly read the Cliff Notes version you post here on that (and anything else you post). As I've said, I do appreciate the efforts you've made.

[TimG]

What temporal events must be added?

temporal in the sense of historical happenings

I still don't get it. Are trying to add the civil war into the time line?

[helene_t]

i think you'll agree that hilbert was a mathematician of not inconsiderable talent... was his understanding of an actual infinite different from yours, and was he wrong?

Talented people go off on tangents now and then.

Yes, but maybe Hilbert didn't.

 

I don't know if he said anything about the physical existence of infinite sets (Google only gives me some links to basic explanation of the Grand Hotel, and some religious propaganda).

 

But if Hilbert had an opinion about the issue, it might be a well-informed opinion. After all, he knew a lot about physics. Unlike people like Craig who suffer from the illusion that they can answer questions about the real world by pure armchair argument.

[blackshoe]

All this talk about the Hilbert quote, I found this on the wikipedia article of Hilbert's hotel:

 

Because the Hilbert's paradox is so counterintuitive, it has often been used as an argument against the existence of an actual infinity, for instance an argument for the existence of God posed by the Christian philosopher William Lane Craig is roughly as follows;

 

Although there is nothing mathematically impossible about the existence of such a hotel (or any other infinite object), intuitively no such object could ever exist, and this intuition is a specific case of the broader intuition that no actual infinite could exist. Since a temporal sequence receding infinitely into the past would constitute such an actual infinite, time must have "started" at some point. Since "time" cannot be started by any temporal thing, and every action must have a cause, this cause must be God.

I guess my question is, how can someone honestly believe intuition proves anything? Or even moreso, how it can disprove anything that is not mathematically impossible?

because hilbert's hotel is an example (if i understand helene and ken) of set theory where {1,2,3... } are guests and {a,b,c... } are rooms... if my understanding is correct, a guest in hilbert's hotel would be unable to check out, for example he could not leave to catch his flight to new orleans, because subtraction is not allowed and a guest who left would be subtraction... but in a real hotel, guests aren't usually held hostage... so intuitively (and, it seems, mathmatically), such a hotel is not possible in reality, iow transfinite arithmetic only applies conceptually (see the hilbert quote above)

The fact that discussion of Hilbert's Hotel do not address the question of guests checking out does not mean they can't - it just means that their checking out is not relevant to the question at hand, to wit, if a new guest arrives, and nobody has checked out, where do we put him?

[luke warm]

What temporal events must be added?

temporal in the sense of historical happenings

I still don't get it. Are trying to add the civil war into the time line?

yes

i think you'll agree that hilbert was a mathematician of not inconsiderable talent... was his understanding of an actual infinite different from yours, and was he wrong?

Talented people go off on tangents now and then. Even Einstein made mistakes. Maybe even Bertrand Russell.

 

To be honest, I don't really understand "actual infinite" and don't see any reason to spend time working on it, given all the worthwhile things I've left to learn (and given the relative shortness of the portion of my life that remains).

 

However, I'll gladly read the Cliff Notes version you post here on that (and anything else you post). As I've said, I do appreciate the efforts you've made.

fair enough... even though some mathmaticians have spent time on it, i understand it not being interesting to others

The fact that discussion of Hilbert's Hotel do not address the question of guests checking out does not mean they can't - it just means that their checking out is not relevant to the question at hand, to wit, if a new guest arrives, and nobody has checked out, where do we put him?

yes it does... i answered this above (and asked helene, ken, passedout, and other mathmaticians to correct my view if wrong)... checking in is no problem, we simply move each guest to another room... checking out is transfinite subtraction and, i think, not allowed

Maybe Wikipedia is not Being fair to Craig, but as described in the Wiki article the argument is total rubbish.

well that settles it, then... by simply saying so you've accomplished, assuming the accuracy of the wiki article, in one sentence what others, some even doctors of philosophy, have attempted by writing volumes

Helene even carefully noted that the article very well may be unfairly representing his view.

so did i... see the part starting with "... assuming the ..."?

Why not just try to tackle this one?

I personally don't see how the existence of something in nature that must necessarily be modeled by means of infinite sets could ever be confirmed. I find it more plausible that the non-existence of an infinite number of particles, or maybe even the non-existence of an infinity of points in spacetime, could be argued on the basis of experimental data.

 

However, such an argument would require insight in physics. Forming a firm opinion about such issues solely on the basis of an armchair argument is silly.

 

On the other hand, that spacetime is most mathematically conveniently modeled by means of infinite sets, is beyond dispute.

if i understood it i might... for example, the next to last sentence - i don't know that's what hilbert did, but i think he had at least a fair of insight into physics... i do think pointing to craig as an example of "armchair argument" is slightly preposterous (unless she meant me, in which case i agree with her)... as for the last sentence, how are we to understand the meaning behind "conveniently"?

[helene_t]

checking in is no problem, we simply move each guest to another room... checking out is transfinite subtraction and, i think, not allowed

I don't see why it shouldn't be allowed. Blackshoe is right - checking out is not the issue, the paradox is about checking in.

 

If the hotel had a policy of never leaving a room idle, the paradox could be rephrased: The guest in room 1 checks out. To keep all rooms occupied, they move the guest in room 2 to room 1, the guest in room 3 to room 2, etc.

[luke warm]

checking in is no problem, we simply move each guest to another room... checking out is transfinite subtraction and, i think, not allowed

I don't see why it shouldn't be allowed. Blackshoe is right - checking out is not the issue, the paradox is about checking in.

 

If the hotel had a policy of never leaving a room idle, the paradox could be rephrased: The guest in room 1 checks out. To keep all rooms occupied, they move the guest in room 2 to room 1, the guest in room 3 to room 2, etc.

well i sure don't know, helene... you'd know better than i, but is subtraction (checking out, in this case) allowed in transfinite arithmetic?

[helene_t]

aleph0 - 1 = aleph0

 

See? It works!

 

Seriously, transfinite arithmetics is about operations like 2^aleph0 > aleph0. Adding (or subtracting) ordinary integers from/to transifinites is not so interesting.

 

But surely you can remove a single element from an infinite set. The set of all integers which are not 7 may be constructed by removing 7 from the set of all integers. Nothing wrong with that.

[jdonn]

Of course you can remove an element from an infinite set. If you can add one to it, you can simply put it back exactly how it was before. That is removing an element, subtracting, whatever you want to call it.

[TimG]

What temporal events must be added?

temporal in the sense of historical happenings

I still don't get it. Are trying to add the civil war into the time line?

yes

Why isn't it already there?

[hrothgar]

i think you'll agree that hilbert was a mathematician of not inconsiderable talent... was his understanding of an actual infinite different from yours, and was he wrong?

Talented people go off on tangents now and then.

Yes, but maybe Hilbert didn't.

 

I don't know if he said anything about the physical existence of infinite sets (Google only gives me some links to basic explanation of the Grand Hotel, and some religious propaganda).

 

But if Hilbert had an opinion about the issue, it might be a well-informed opinion. After all, he knew a lot about physics. Unlike people like Craig who suffer from the illusion that they can answer questions about the real world by pure armchair argument.

Anyone else reminded of a classic scene from Annie Hall?

 

WOODY ALLEN: Tell him.

 

MARSHALL McLUHAN: -- I heard, I heard what you were saying. You, you know nothing of my work. How you ever got to teach a course in anything is totally amazing.

 

WOODY ALLEN: Boy, if life were only like this.

[Winstonm]

I have real trouble with the concept of infinite sets, and I believe the problem is due to word limitations rather than intellectual imagination. I also have a real problem with the construction of Hilbert's Hotel - if the hotel had infinite rooms it could never be full (unless the concept is just a mind exercise of imposssibilities but then it would be pointless, would it not?) - the very word full indicates a boundary and the word infinite defines a bound-less-ness.

 

My construct would be like this: two fruit trees each hold an infinite amount with each tree being only one type of fruit - one an apple tree and the other an orange tree - every time someone reaches his hand into either tree he is rewarded with either an apple or an orange, depending on which tree he choses.

 

But let's now add a variable - say that any person who choses oranges must pick them at a pace that is twice the speed of the apple picker - does that mean there must be a greater number of oranges?

 

The answer is no - neither tree nor any picker will ever run out of fruit so there will never be the boundary necessary for a finite comparison of total fruit yield of each tree.

 

The yield is relative.

 

All we can say is that at a certain point in time the orange tree has produced more fruit thus far than has the apple tree - but the amount of oranges being picked is relative to the speed of the apples being picked. If the apple picker started picking 3 apples at a time suddenly the orange productions would appear to have slowed. But no matter how much fruit of either type is being picked there is a boundless expanse of fruit left and it can never run out - so there is always the same amout of fuit left to be picked.

 

Thus, when considering infinity and time, a difference between infinite sets would be an impossibility if real infinity is the basis for the comparison - everything that precedes infinity is a relative measurement of what has occured or is occuring.