June 4, 2009

  • GabrielPeter & "Potential Infinity vs Actual Infinity"

    Intro:

    I recall way back when I was startled that Christian apologist William Lane Craig (WLC) advocated that an actual infinity was impossible even though he believed in a god with infinite attributes (link).  Little did I know, they make some weird distinction that I don't think holds up to scrutiny.   You can tell me though.  Gabe lays out WLC's basic argument in the first section below.

    Gabe wrote (link):

    Let's say that I had an infinite number of marbles and I wanted to give you an infinite number of marbles.  I could just go ahead and give you all my marbles.  Then I'd have zero marbles.  Another way I could do it is to give you all of my odd-numbered marbles.  Then I'd have all the evens, you'd have all the odds, and we'd both have an infinite number of marbles.  I still have the same number of marbles I had before I divided them up!  I could also give you all of the marbles numbered from four and higher.  Then you'd have an infinity number of marbles and I'd have three left.  So here are our equations...

    Infinity - infinity = zero
    Infinity - infinity = infinity
    Infinity - infinity = three

    We have subtracted an identical number from an identical number and ended up with three different results.  Doing math with infinity presents some contradictions.  That's because the idea of an actual infinity is only conceptual.  It only exists in our minds.  Mathematicians can deal with infinite quantities and infinite numbers in the conceptual realm, but it can't happen in the real world.  We do not have an infinite past in a physical universe.  It's impossible.

    So why is it okay to believe in an infinite God if according to logic infinity doesn't work?  Because this logic is only prescribed within the bounds of time and physics where our universe exists.  God does not exist in time.  God exists outside of the laws of time and matter.  There is universe that has yet to exist after this moment, but God exists beyond this moment.  He was, He is, and He is to come.  God is the nonphysical infinite entity who brought our physical universe into existence.  It is the only way to explain how we came to be.  The most logical conclusion -- between believing in an infinite universe or an infinite God -- is to believe in the infinite God.

    I responded:

    Um, seems to me you've just proven that god is only conceptual and exists in your mind.  Nonphysical entity seems to = not existent entity or an entity without any of the properties of existence, but for some bizarre reason we're still pretending like it means something to say it "exists" anyway.  It's like saying a literal singularity actually exists, when by definition we're not actually talking about anything.  It seems incoherent. 

    Ben


    Gabe responded:

    @WAR_ON_ERROR - As I'd said to a commenter in the previous post, these are not "proofs."  Nothing metaphysical can ever be proven.  Rather, they are arguments that show believing in God is perfectly logical.  In fact, I'd say it's more reasonable than believing, "We just exist."

    Couple things here.  First, I read you saying that because God doesn't have any physical properties, then He must not exist.  However, if God had physical properties, He'd violate that infinity in a physical universe isn't possible.  If I'm incorrect in breaking down your argument then do clarify.  I see a bit of Russell's version of Occam's Razor playing out here, but I don't want to jump on that if it's not what you're trying to say.

    Second, you're not considering the complete argument made in the previous post.  In order for there to be a something, there must be at least one state that is self-existent and does not derive its existence from something else.  It must be nonphysical -- it cannot have the properties of our physical existence.  Without it, we wouldn't exist.

    Since we are not a product of a physical infinite universe, we must be a product of a nonphysical infinite cause.  That's not conceptual.  It's following the logic to its conclusion.  Believing that the physical universe can be infinite is only conceptual.  Believing that God is eternal and caused it is quite logical.  I believe that God has revealed Himself to us, that He's more than just a concept, and He can be found if you want to find Him.

    (I added to this response from my previous.  I apologize if you've received it twice.)


    I responded:

    @gabrielpeter - Yeah, no problem.  I didn't happen to catch your last post on this topic though I did just go back and read it (not the comments though). 

    We can set aside the incoherency of nonphysical entities and focus on just your justification.  You said:

    "Since we are not a product of a physical infinite universe, we must be a product of a nonphysical infinite cause.  That's not conceptual.  It's following the logic to its conclusion."

    In your marble illustrations you claim you are subtracting the same number of marbles (and you are), but you aren't subtracting the same actual marbles in each situation.  If you gave me the evens then you gave me the evens.  You did not give me the odds.  For the contradiction to work, you'd have to have given me the evens and at the same time not given me the evens.  Clearly that's not the case. 

    At best all we can say is that counter-intuitive things happen when dealing with infinity which is obviously something we are not used to.       

    Ben 


    Gabe responded:

    @WAR_ON_ERROR - Always glad to have you contribute.  I get what you're saying about breaking up the evens and odds, but I still don't see how that negates the set of equations.

    The problem still remains that after breaking up the infinite number of marbles I have and gave them to you, I still have the exact same number I started with.  It's infinite.  You also have an infinite number of marbles that came from my infinite.  You've gained and I've lost nothing.  Furthermore, we now have a new set of odds and evens.  Every other number in my even set now becomes odd, and every other number in your odd set now becomes even.  We could do the same thing again -- dividing up the odds and evens -- but you still wouldn't have any more than infinity and I wouldn't have any less than infinity.


    I responded:

    @gabrielpeter - :D

    Well perhaps instead of delineating the marbles by odds and evens which is subjective, we could delineate them as red and blue marbles.  If there is an infinity of red and blue marbles, you could give me all the definitively red marbles and we'd both still have an infinite number of marbles, but I'd have all the red ones and you'd have all the blue ones.  You could not then give me another infinity of red marbles. 

    That's only a support for the greater conclusion that that's just how infinity works.  You can't disqualify the existence of an actual infinity by having the expectations based on the ways that finite number sets work.  It's apples and oranges.  As I showed, there is no actual contradiction at the individual marble level. 

    Even a supposed "potential infinity" is an actual infinity if time also keeps going from a God's eye view.  Time would look like a fully complete ray from God's perspective, right?  If someone started making marbles and kept doing so as long as time persisted from their perspective, it logically follows that God can see there are an infinity of marbles down that ray of time. 

    If you believe you will breath air in heaven and that you will be immortal, it logically follows, from God's atemporal perspective, the immortal you in total has breathed an infinity of breaths already.  It's an unavoidable conclusion. 

    Ben


    kamikazegymnast0 interjected:

    @WAR_ON_ERROR - If I may interject for a moment:

    You make a good point about the odds/evens vs red/blue. However the math of the equation still holds up. Although I would no longer have any red marbles to give you, I would have the same NUMBER of marbles as I began with. Therefore the equation (infinity-infinity=infinity) still holds!

    Also, really cool perspective on potential infinity vs actual infinity! Honestly though, I don't think it's a concept that we can fully grasp...because we simply aren't infinite! It creates the paradox "Can God make a rock so big he can't move it?"...God is able to see infinity, so doesn't that make it less than infinity? If he can see the end, then there should be one more marble at the end...and one more after that...etc. So really all we really have to go on IS the potential infinity! Otherwise, anything that we could visualize/imagine/etc. would be less than true infinity! Lol I am quite sure that I didn't answer anything in this section, but I don't think that anyone truly can! I think that's one of the most amazing things about God...he can accomplish things that our minds can't even grasp!

    I'd love to hear your feedback!

    Gabrielpeter: sorry if I jumped into your convo! I just liked what I read, so I thought I'd throw in my two cents!

    I responded:

    @kamikazegymnast0 - Not a problem. 

    "Therefore the equation (infinity-infinity=infinity) still holds!"

    Yes, but it's a different infinity as I explained, so the equation is misleading (too simplistic).  It's really "an infinity - a particular infinity may still = infinity."

    "It creates the paradox "Can God make a rock so big he can't move it?"...God is able to see infinity, so doesn't that make it less than infinity?"

    If God can see a bear, does that make it not a bear?  Why would God's view affect the reality of the situation?  The "paradox" is created by presuming an actual infinity cannot exist.  If you allow for one to exist, then there is no paradox.  If you believe in immortality and in God's ability to know everything, then you are stuck with an actual infinity.  There's no getting around that, as I said. 

    "If he can see the end, then there should be one more marble at the end...and one more after that...etc."

    Again, this is the same error already addressed of applying the rules of finite numbers to an infinite set.  Obviously if we are dealing with immortality, there is by definition no "end."  Do you expect eternal life to stop?  Makes no sense.  If you think it continues and if you think God knows all that will be on it, it is an actual infinity.  I don't see how you can mystery your way out of that.  Good luck though, I guess.  But I get to claim anything you don't understand about atheism is just as mysterious. hehe  ;)

    Ben


    Gabe responded:

    @WAR_ON_ERROR - I don't believe I will "breathe air" in heaven because we honestly don't know what heaven will be like.  One thing is for certain -- it isn't a physical infinity.  It's something completely different.  We've been given word-pictures in the Bible, but since the finite human mind cannot comprehend infinity, it's not even a glimpse of what eternity holds.

    Nonetheless, we aren't truly "infinite" as it would be in spiritual terms because we had a beginning.  We will live forever if we are in Christ, but that doesn't make us infinite.  That just makes life un-ending.  The only infinite is God.

    In your example using red and blue marbles, the two infinities existed before they were divided.  There was an infinite number of red and an infinite number of blue, and after they were divided, there was still an infinite number of red and an infinite number of blue.  That delineation was already there.  All that changed was who possessed the different infinities.  However, in Craig's example, a separate infinity didn't exist before it was subtracted from the first.


    My closing:

    @gabrielpeter - The air breathing is a technicality.  Merely an example.  You will think thoughts in heaven right?  Surely, surely, surely there is at least one definitive action you can reason you will do over and over again for eternity even if you cannot be absolutely sure you know which it is. 

    And if God doesn't know what will happen, then he's not quite God is he?  Why couldn't he know?  Won't it happen?  Won't it ALL happen?  Which point in time of your immortality would he be unaware of?  Every single point in eternal time is known already by God or he doesn't know everything and that's not your definition of God. 

    "One thing is for certain -- it isn't a physical infinity."

    I'm sure you believe in a physical resurrection of Jesus, right?  Even though he had a suped up body, he could still be definitively touched, right?  Perhaps the story of doubting Thomas is a myth?  If not, that's not spiritual enough to wiggle out of the conclusive demonstration that from your own perspective you are logically forced to accept an actual infinity, in my opinion.

    "Nonetheless, we aren't truly "infinite" as it would be in spiritual terms because we had a beginning.  We will live forever if we are in Christ, but that doesn't make us infinite.  That just makes life un-ending."

    Rays are just as infinite as lines in geometry.  A yellow brick road that starts and goes off in one direction forever is infinite in that direction. 

    "In your example using red and blue marbles, the two infinities existed before they were divided.  There was an infinite number of red and an infinite number of blue, and after they were divided, there was still an infinite number of red and an infinite number of blue.  That delineation was already there.  All that changed was who possessed the different infinities.  However, in Craig's example, a separate infinity didn't exist before it was subtracted from the first."

    The odd marbles in the bunch of odd and even marbles could have been painted red by an infinite fleet of gnomes with paint brushes and the even ones could have been painted blue.  So that odd infinity definitely already existed in the example either way.  Does painting them change their number?  It doesn't matter whether we tagged things or not.  The mental assignment of even and odd was sufficient and the coloring was just my device to keep your mind on target and point out there's no contradiction. 

    It was not categorically different since in the original WLC conception, we could still say that we could not give someone else the exact same odd marbles that were previously given out since they are no longer in our possession.  We gave them away!  We have a whole new fleet of odd marbles in our odd and even bunch, but those were all previously the actual even marbles. 

    We could in fact do the same move with the color coated marbles by merely having our gnomes paint every other one of our new batch of only red marbles blue.  But they wouldn't be the same blue ones we gave out previously.  Hence there is no contradiction at the marble level as I've been saying and there's no way to change that.

    I think I've sufficiently made my case from my perspective on the main issue and the conversation requires going in several different directions on contentious issues.  I'm going to let this be my closing statement and we can move on to talk about other things some other time.  I'll be sure and post any follow up of yours on my repost of these comments (link) and leave it at that.  Fair deal?

    Ben


    Gabe's closing:

    @WAR_ON_ERROR - You state, "We have a whole new fleet of odd marbles in our odd and even bunch, but those were all previously the actual even marbles."  Yes, I'd said that, too.  I guess your example really isn't different than Craig's then.  Painting the marbles different colors or numbering odds and evens (or even adding gnomes), we still get the same result. 

    When you first redefined the equation, you said, "infinity - a particular infinity may = infinity."  The slight-of-hand here is that we're not subtracting just any random infinity.  We are subtracting infinity from itself.  In this particular case, infinity - infinity = infinity.  But this is all still, what?  Conceptual.

    You mentioned that because I said infinity is conceptual, I've therefore proven that an infinite God is only a concept -- which was your first argument when I look back on it.  In hindsight, if I'd said, "You're applying the physical to the nonphysical," which is really all I needed to say, would we be here now?  ;)   God created time and space.  He transcends them both.  Once He's created the universe, He can enter time, which you touched on in part of your argument, but that's a different topic altogether.

    In summary, everything I need to say is contained within both of these posts, here and here.  I started typing something else and realized I'd just be rehashing the same thing.  I encourage you to read them again.  Don't stop at some point you disagree with something and formulate a new argument for yourself.  It's been thoroughly explained -- I promise.  I appreciate the discussion.  You never cease to challenge.  Always a pleasure.


    Outro:

    I'm glad I got around to addressing this issue since it is an important distinction that modern theists try to make.  It clearly doesn't work without seriously  backfiring unless you bury it in implausible ad hoc mysteries.  If you do that, to be fair, it seems atheism doesn't really need to address the theistic complaints about apparent philosophical contradictions after all.  If we can all agree that we don't really know how metaphysics works even when it really, really, really seems like there's a definitive contradiction, then why are we making arguments from that non-knowledge against anything?  "Atheism doesn't make any sense because...[insert supposed philosophical problem here]" is no longer valid coming from theistic philosophers.   If nothing else, I wish that's the lesson Christian thinkers would take away with them even if they don't get any of my points on understanding the basic nature of infinity. 

    I'm an agnostic humanist first and an atheist philosopher second, so having a worldview without a greater metaphysical conception (if I turn out to be wrong about something well outside the realm of human expertise) works perfectly well for me and my core beliefs about what is important in life.  I'm not so sure the Christian can say the same.  Feel free to disagree though! 

    Ben

Comments (23)

  • Do you think that if some type of physical matter and time have always existed, that we could arrive at the present?

  • Now I have a headache. "I keep trying to get out, but they keep pulling me back in". LOL

    Actual infinity +1 always makes me doubt that it is possible, but that is math. Infinity can best be described as 0, 0+0=0, 0-0=0.

    For us to talk about a being capable of observing infinity, we would need a context that we cannot get to, or it = 0. I remember one of intellectual spirits post a while back dealing with God Prime by Philosphical Model where he basically reasoned to not being able to say one coherant thing about the subject, and I tend to agree with him.

  • @musterion99 - Actually that is one thing I've heard in disproof of infinity, if we indeed have an infinite past, we could not arrive at any point in time, I believe Zeno had bunches to say about that. If you want a really big headache, read some of his stuff.

  • @musterion99 - It also takes an infinite series of half-way points to reach any object (we must first cross half the ground to the object, then half of the remaining ground, then half of that bit, etc.), and yet we always reach the object.  Infinity is not such a huge obstacle to overcome, it seems.

  • @Andrea_TheNerd - I don't believe that answers the question. I still don't see how if the past is infinite, that we could ever arrive at the present.

  • @musterion99 - Yes indeed.  The reason is that (as I've said twice before now in the post), is applying the rules of finite number sets to infinite numbers.  Your question presupposes that we began our journey and obviously that doesn't apply. It just means we've always been on it.  Where's the contradiction?

    Ben

  • @WAR_ON_ERROR - Aren't you presupposing that we didn't begin our journey?

  • @musterion99 - Nope.  I don't see a problem with having begun or not begun hypothetically.  Line segments make just as much sense as lines and rays in geometry without logical contradiction.  The whole context here is that the theist is making a claim that the line and the ray for some reason can't translate into physicalness and I'm rebutting their arguments against it.  That does not require that I have to know necessarily (or presuppose as you say) that our universe is actually infinite (or that it is really part of a greater infinite multiverse system).  It just means theists don't know it isn't that way.  I do not argue here for a positive position or claim that theism is necessarily false either. 

    All I'm showing here is that if a mainstream Christian theist denies the possibility of an actual physical infinity (and they do so to exclude the possibility of nontheism), then it logically backfires in their own belief system in the ways I've shown in the post. 

    Make sense?

    Ben

  • @WAR_ON_ERROR - Make sense?

    I'm not sure.  It seems like you're saying that even if materialism and time are infinite, then we could still arrive at the present. But isn't your conclusion based upon your assumption of time and it's relation to line segments?

  • @musterion99 - In response to your first statement, you are correct in your assessment of what I'm saying, but still apparently presuming that we began the journey (to maintain some kind of contradiction).  That's still applying the rules of finite numbers to infinite numbers.  It doesn't make sense to treat a hypothetical actual infinity as though it would be finite anymore than it does to treat a finite set like it is infinite.  Using one way to disprove the reality of one group of numbered things or the other makes no sense. 

    "But isn't your conclusion based upon your assumption of time and it's relation to line segments?"  I'm not sure what you are referring to. 

    Ben

  • I'm not an expert on this so it's hard for me to argue about the rules of finite and infinite numbers. It just seems to me that in order for us to be able to arrive at the present time, there must be a beginning. If there's no beginning and the past is infinite, I don't see how it's logically possible for us to arrive at the present. Of course as you know for theists, this is evidence that God is outside of time and created the beginning of time and the universe. Can that be proven? No, but it makes more sense to me than the alternatives.

  • @musterion99 - I get this in my inbox because it's on my xanga, so there's no real reason to @ me unless lots of people are talking.  Hope you don't mind me deleting your repeat comment.

    I'm not an expert either, but when dealing with an actual infinite past, to "arrive" here doesn't mean we had to begin.  We were always "becoming" at every previous point in time.  That's what infinity would mean if that were the case, so why would we suppose otherwise when dealing with the idea?  I can appreciate that infinity can be awkward to think about, but I see no contradiction here.  We'll just have to disagree.   

    Thanks for your comments.

    Ben

  • @WAR_ON_ERROR - I'm not an expert either, but when dealing with an actual infinite past, to "arrive" here doesn't mean we had to begin.

    That could be true but it makes no logical sense to me. I don't see how it could happen, unless time doesn't really exist. Even then, it's hard to comprehend that being the case. Stephen Hawking said - "Almost everyone now believes that the universe, and time itself, had a beginning at the big bang."
    The Kalam Cosmological argument  states:
    1 - An infinite number of days has no end.
    2 - But today is the end day of history(history being a collection of all days).
    3 - Therefore, there were not an infinite number of days before today(i.e., time had a beginning).

    Norm Geisler explains - "You can't add anything to something that is infinite, but tomorrow we will add another day to our timeline. So our timeline is undeniably finite. We know it had to begin at some point because only a finite amount of time could be passed for today to arrive. You can't transverse an infinite number of days. Thus time must have a beginning."

    He then goes on to say - "Some may say that infinite numbers can exist, so why can't infinite days? Because there's a difference between an abstract infinite series and a concrete one. The one is purely theoretical, the other is actual. Mathematically, we can conceive of an infinite number of days, but actually we could never count or live an infinite number of days. You can conceive of an infinite number of mathematical points between two bookends on a shelf, but you could not fit an infinite number of books between them. That's the difference between an abstract and a concrete. Numbers are abstract. Days are concrete."

    We were always "becoming" at every previous point in time.

    How do you know this and how does that solve the problem mentioned? If we were always becoming from an infinite past, I still don't see how we arrive at the present, of course unless you claim that time doesn't exist, which is nonsense in my opinion.

  • @musterion99 - "That could be true but it makes no logical sense to me."

    That's all I was arguing for that it could be true and that's why "becoming" solves the problem mentioned.  That term in application is consistent with dealing with the proposed infinite series whereas a beginning by definition is not.  Squares fit in square holes and circles fit in circular holes and "becoming" fits with an infinite past whereas beginning fits with a finite past.  These are just descriptions of what would be the case in the event they are the case.  Nothing discussed proves an infinite series can't be the case, as you apparently conceded.

    I'm aware of the Kalam argument and was never saying the Big Bang was not the apparent beginning of our universe.  The Kalam argument just doesn't prove there's not more time and space on the "other side" of the Big Bang.  For instance, one hypothesis is that at the center of each black hole a new universe is born in another big bang.  For that universe, the initial "implosion" would be their apparent beginning, but wouldn't really be the "true" beginning if there even is one.  It's just ignorance all the way around until someone can prove something.  From my perspective it's just an unanswered question and not a refutation of theism or a proof of atheism.

    I believe time exists as a static dimension in addition to the other three dimensions of height, width, and depth.  Even if I were a theist, I would have to reach this conclusion since God is able to see the entire static 4D object from beginning to never ending.  The passage of time is just our relative perspective.  That doesn't make it any less real.  It's just a shift in perspective. 

    Ben

  • @WAR_ON_ERROR - For instance, one hypothesis is that at the center of each black hole a new universe is born in another big bang.

    If the gravity is so strong that nothing can escape, how could a new universe be born?

    For that universe, the initial "implosion"
    would be their apparent beginning, but wouldn't really be the "true"
    beginning if there even is one.

    Then you're just back to the problem of infinite regress.

    It's just ignorance all the way around until someone can prove something.

    Yes, I agree.

  • @musterion99 - If I'm not mistaken the strength of the gravity that makes it a black hole from our perspective is exactly the thing that empowers it in its new realm of expansion. 

    The problem of infinite regress is only a problem if you consider it a problem.  If the history of time/space is in fact infinite, then it is really just the fact of infinite regress.  Then you have to ask the same "how could we have gotten here" question which as I've been showing presupposes a situation (a beginning) that by definition wouldn't be the case.  It seems we are trapped in a potentially infinite loop on this point.  ;)

    Theists point to the Big Bang to show that even if there could be an infinite past, that in fact there wasn't.  That's why it is pointed out to them that they have to assume the Big Bang isn't just an apparent beginning for time and space.  Since you agree we are just ignorant as to the actual state of affairs, it makes no sense to argue from that ignorance as Kalam proponents attempt to do.

    Ben

  • @WAR_ON_ERROR -  If I'm not mistaken the strength of the
    gravity that makes it a black hole from our perspective is exactly the
    thing that empowers it in its new realm of expansion.

    I don't see why it would expand into a new dimension(is that even possible?) instead of just staying in the same one. But as we agree on, we don't know what happened.

  • @musterion99 - I dunno.  It sounds kinda like what happens when you start squeezing the end of one of those long balloons and if you aren't careful, you create a new pocket of air as it slips through your fingers.  Although in universe terms, it's more like the new dimension is inside out in relation to our space.  It's not like it shoots off the side like with a 3D balloon.  That's the best analogy I can think of right now, since space/time is something in and of itself. 

    Regardless, that is one of the options theorists like Lee Smolin are currently looking into (link) whether either of us understands it or not.

    Ben

  • @WAR_ON_ERROR - I dunno.  It sounds kinda like what happens
    when you start squeezing the end of one of those long balloons and if
    you aren't careful, you create a new pocket of air as it slips through
    your fingers.

    Well, the pocket of air is still in this dimension. It's all speculation. Thanks for the link.

  • @musterion99 - It was just an analogy.  :p

  • @Andrea_TheNerd - Actually, that we do in fact traverse an "infinite number of half-way points" in any movement provides good reason to believe, not that we actually are traversing an infinite distance, but that the notion of infinity is purely conceptual, existing only in our minds. In Xeno's paradox, there exists an infinite number of half-way points, precisely because to divide some number by two is, by definition, to always leave some mathematical remainder behind. thus, repeat this process as much as you like, there will always be some mathematical remainder. The question is, what reason can be given for thinking that this same rule can be applied to physical reality? 

  • @Ampersand - Holy ancient threads, Batman!  I'm drunk right now, and about 5 minutes away from pouncing on War_On_Error's lickably-delicious body, so I'll answer you later, if ever.

  • @Ampersand - Correct me if I'm wrong, but isn't xeno's paradox being applied to physical reality?  You don't seem to be negating what you think you are negating.  

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