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| aleph_0, smallest infinite cardinal (Photo credit: Wikipedia) |
Let's start by considering the set of all positive integers {1,2,3,4,...}. Notice that there are infinitely many numbers in this set, and yet the difference between any pair of numbers is always finite. Even though we have an infinite set, there is no member of that set that is infinitely big, there is no integer of size infinity. Of course if we include negative numbers (and zero) to get all integers {...,-3,-2,-1,0,1,2,3,...} the same holds true. We have a set which limits to negative infinity in one direction and positive infinity in another, and yet every number in the set is some finite distance away from zero. This type of thing is very common when we deal with infinities and it demonstrates that precision and care is very important. Your first instinct when considering this infinite set is that there must be some point in that set which is infinitely far away from zero. However, by looking at it from this perspective, I hope it is easy to see that it's not quite right. (Although what you can say is that for any number N, there is some integer which is more than distance N away from 0)
While arguing with apologists, I will commonly see it very casually mentioned that an infinite regress is impossible. Although the explanation is typically quite sparse, I've luckily had a few good conversations with apologists lately where they have tried to explain themselves more fully on this topic. The first such conversation took place on my blog in a comment from The Rational Zealot
Let’s say the past is represented by negative numbers, zero is the present, and positive numbers are the future. Let’s say you never start counting but have been counting from an infinite past. An infinite amount of time later, you are still counting negative numbers. An infinite amount of time after that, you are still counting negative numbers. An infinite amount of time after that? Still negative numbers. To say otherwise means you haven’t really been counting from negative infinity, but have changed infinity into a number.So the situation we find ourselves in is an infinite regress. Every moment has a moment before it. If we take the entire timeline at once we have an infinite number of moments. I would argue that there is nothing wrong with this, there is an infinite past, so what? Every moment has an infinite number of moments preceding it, and yet the distance between any two moments is finite. Strange for sure, but there is no contradiction here. Let's look at The Rational Zealot's argument one step at a time.
Let’s say you never start counting but have been counting from an infinite past.I'm with you so far, we have been counting forever into the past, there was no start.
An infinite amount of time later, you are still counting negative numbers.Here is where he loses me. An infinite amount of time later from when? It seems that we have assumed a starting point at negative infinity (we'll explore what this means in a moment). He continues
An infinite amount of time later, you are still counting negative numbers. An infinite amount of time after that, you are still counting negative numbers.I believe this is the key problem to this conversation every time I have it. Every point on the timeline is a finite distance from the current moment. There is no point on the timeline from when I can count for an infinite amount of time and still land on the timeline. Again, every pair of points on the timeline are a finite distance apart. So what does it even mean to count for an infinite amount of time?
Sometimes for convenience we will compactify the space and put a point at infinity. In the case of a line we will probably put a point at negative infinity and another point at positive infinity. These two points are not standard, they are instead a mathematical abstraction. They are special points very different from the mundane numbers on the rest of the timeline. One way to look at this is that starting at negative infinity in the compactified space is the same as saying that you have always been counting without a start in the non-compactified space. Similarly, saying you will end on positive infinity in the compactified space is the same as saying that you will never stop counting in the non-compactified space. It is sometimes nice to translate "I will count forever" into a more manageable form.
If we do allow these points at infinity and we allow ourselves the ability to actually count an infinite number of moments, there are three possibilities as far as I can tell
- You start on the timeline and after an infinite number of moments you are at positive infinity
- You start at negative infinity and after an infinite number of moments are at positive infinity
- You start at negative infinity and after an infinite number of moments are anywhere on the timeline
To say that we count for multiple infinities worth of counting and still are at negative numbers makes no sense.
To say otherwise means you haven’t really been counting from negative infinity, but have changed infinity into a number.Ultimately, I believe that the mistake that has been made is to assume that this special point at infinity exists, but then treat is as a standard point in some ways. The easiest way to solve this problem is to not allow this point at infinity. Only consider normal points, even though there are an infinite number of points, there is no first point and any pair of points is a finite distance from one another. The complaint evaporates because there is no start, there is no point from which we can count an infinite amount of time and still be in negative numbers.
Humblesmith provided a very similar complaint over on his blog where he discusses an example of an infinite string of dominoes that is falling over. At first I thought the addition of physical objects would make things more complicated (where did the dominoes come from, etc), but if we ignore those problems, having dominoes set up does wind up being instructive. He was talking about various problems with such a setup and we get to his third point
This is essentially the same objection, we have an infinite string of dominoes, which have been falling forever, and therefore the falling cannot get to the current position. But we have a similar problem, it seems to me that what is happening is they are trying to put a domino at that "point" at negative infinity, flick it, and say it can never hit any of the other dominoes. But there is no domino at negative infinity, the fact that we have an infinite number of dominoes just means that every domino has one before it. To say that this string has been falling forever simply means that every domino has been hit by the one before it. Let's again, highlight the real misunderstanding hereThird, as we observe this string of dominoes falling over, if it were infinite, we must ask ourselves “how did the falling get to me?” If the line of dominoes were infinitely long, it seems the falling would always be an infinite distance away from me. The atheist might reply, “Well, the ones currently falling over have to be somewhere. It just so happens that it is next to you.” But this misunderstands infinites. If the line were truly infinite, then the falling would always, at all instances, be an infinite distance away from any one point on the line. Pick any domino, and the falling would have been an infinite distance away. Since the falling is happening in sequence, it is impossible to select a domino where the falling is not an infinite distance away. The dominoes are always falling, but never arriving anywhere, which is an absurdity.
(Photo credit: Wikipedia)
If the line were truly infinite, then the falling would always, at all instances, be an infinite distance away from any one point on the line.No, this is incorrect. Every domino is on the line, so if any dominoes are falling then the falling is somewhere on that line. If the falling is an infinite distance away from the domino that I am standing next to, then which domino is falling? Remember, every domino is a finite distance from the one I'm standing next to, if falling is happening it has to be happening to some domino. Either the falling is some finite distance from my domino, or no dominoes are falling and it doesn't really make sense to say that falling is happening at all. Falling can't be happening an infinite distance away because there is no domino at distance infinity from mine.
In both cases here, the key is to ask what is meant when things are happening "at infinity". If we count for an infinite amount of time and are still stuck at negative numbers, then when did we start? If a domino is falling an infinite distance away, where is that domino? The problem here seems to stem from the same counter-intuitive notion that we can have an infinite number of points such that every pair of points is actually a finite distance from one another.
Ugh! Clearly I have failed to effectively communicate, but I do take some solace in that the topic is infinity (which is infinitely difficult). The overarching point is that infinity is not a number and cannot be treated as such. It’s a concept or idea or mathematical tool. I can see why you take exception to the phrase “from negative infinity” as it implies a from point…..there is no point to count from, which was kinda my point (although poorly communicated) since negative infinity has no limit on how negative negative infinity can be. We seem to agree on this, which leads me to my question: Can you show that an infinite regress is logically valid in reality (outside of a math tool)?
ReplyDelete“Every moment has an infinite number of moments preceding it, and yet the distance between any two moments is finite.”
This statement is not completely true. The distance between any two moments may be finite, but not always. If there are an infinite number of preceding moments (an actual infinity) then it is possible to have two points infinitely far apart; otherwise, there were not an infinite number of moments. I think you are mixing up a potential infinity with an actual infinity.
Happy Holidays!!
Rational Zealot, thanks for returning to continue this back and forth, I certainly am enjoying it :)
Delete"The distance between any two moments may be finite, but not always. If there are an infinite number of preceding moments (an actual infinity) then it is possible to have two points infinitely far apart"
I disagree, I think it is possible to have an infinite number of moments, and yet every pair of points is a finite distance from one another. I can see two potential spots that we might be having a problem. Let me know what you think of them.
1. The integers are infinite, and yet any pair of integers are a finite distance apart.
I justify this in my first paragraph. Would you agree that this is a property of the integers? If so, great. If not, where do you think I'm going wrong in my first paragraph?
2. An infinite regress maps perfectly onto the integers.
As far as I can see, in infinite regress means that every moment has a moment before it, just like every integer has an integer before it. It seems like a perfect one to one map to me.
It seems to me that if both of these facts are true, then an infinite regress would have the property that I'm talking about, we have an infinite number of moments and yet any pair is a finite distance away from one another.
Rational Zealot:
DeleteAnother thought, you said:
"If there are an infinite number of preceding moments (an actual infinity) then it is possible to have two points infinitely far apart; otherwise, there were not an infinite number of moments."
I'm not quite following here, I'm wondering if you can elaborate. Where would these 2 points be that they are an infinite distance apart? Or perhaps that is not the way you would like to frame things, perhaps I need to be looking at the situation from a different perspective. If this is the case, how can I look at things in order to see two points which are infinitely far apart from one another?
Thanks, and happy holidays
“thanks for returning to continue this back and forth, I certainly am enjoying it :)”
DeleteI enjoy reasonable discussions with reasonable people!
My original example of counting from negative infinity was an attempt to show what would happen if one tried to traverse negative infinity to the present. You rightly pointed out that when I used the phrase “from negative infinity” that there is no “from” or “point”. In showing how my example was false, you correctly demonstrated that one cannot count from negative infinity. It’s logically incoherent, but this is what you must do in order to have an infinite past.
You are rightly pointing out flaws in my examples in regards to infinity, since the point of the examples is to show the flaws of infinity in reality, but by showing how my examples are not possible you are helping my case! What does it matter if you can measure two finite points on the infinite line? It doesn’t resolve the problem of the preceding infinity before any point on that line nor show how you can traverse infinity. Can you logically demonstrate that infinity can be traversed? If one cannot traverse infinity, how can there be an infinite past?
"you correctly demonstrated that one cannot count from negative infinity. It’s logically incoherent, but this is what you must do in order to have an infinite past."
DeleteFirst we must decide what we even mean when we say "count from negative infinity". If it means to count down with no beginning, then I think I have described how this can be done, every point has a point preceding it, no matter how far back you go, there is a point before that one, so we have been counting since before that time.
The language does get a bit difficult, but I think that is partly because "counting" implies a beginning. What does it mean to count without a starting point? I'm not sure, and hence the wording gets a bit confused. Nevertheless, I see no reason there can't be an infinite past, every moment has a moment preceding it.
"What does it matter if you can measure two finite points on the infinite line?"
Because your argument relied on two points on the line which are infinitely far apart. If no such pair of points exists then the argument doesn't work.
"It doesn’t resolve the problem of the preceding infinity before any point on that line nor show how you can traverse infinity."
What exactly do you mean by "traverse infinity"? To say you are traversing something implies a starting point and an ending point, but again, there is no starting point here. For me to demonstrate how to traverse infinity, I would have to have it explained exactly what meant here.
“Because your argument relied on two points on the line which are infinitely far apart. If no such pair of points exists then the argument doesn't work.”
DeleteYeah, that’s what I get for posting something after midnight on Christmas Eve. You were right; any two points on an infinite line would be a finite distance apart. I even talk about this on the infinity blog post I did earlier this month. I apologize for my mental lapse…not the first, nor the last. :0
Let’s see if a different direction can help explain my confusion. Start at t=0 and continue to add years into infinity. You never get to infinity because at any point in time, you have a finite distance back to zero. The infinity is potential, but never becomes a true infinity (or actual infinity). When you flip the chain of time on the zero axis and form an infinite regression, you have turned the potential infinity into an actual infinity. This is a different ball game. With an infinite past, you have to pass through (or traverse) an infinite number of years before arriving at the present, but you are saying you cannot come from negative infinity. This seems like a contradiction to me.
“every point has a point preceding it, no matter how far back you go, there is a point before that one, so we have been counting since before that time.”
How do you move forward if there is always one more moment before that time? That’s like running on a treadmill….every step taken is replaced by on more step. You said that every point in time in the infinite past has a finite distance to the present, which is true, but there are an infinite number of prior moments. Drawing an infinite number of finite lines does not help you pass through an infinite number of years. This is circular reasoning; infinity can exist if infinity can exist.
Infinity discussions tend to get circular after a while…maybe because a circle is a type of infinity? Interesting discussion!
"With an infinite past, you have to pass through (or traverse) an infinite number of years before arriving at the present"
DeleteI'm happy with this. My issue with "traverse infinity" is that it seems (at least to my brain) to suggest a starting and ending point, so it subtly suggests starting at negative infinity. However, to clarify it as having gone through an infinite number of points in the past sounds good to me.
So now, we seem to be on the same page, if there is an infinite past, then every point has an infinite number of points preceding it. I don't think this is a problem, every point in time has an infinite history, so what?
You seem to be trying to answer my "so what?" question with your treadmill analogy, but I'm not sure I really understand. What do you mean by "every step taken is replaced"? This seems to only make sense if we are trying to hunt backwards until we get to the start, but there is no start here, so we shouldn't expect to be able to get back to the beginning (because there is no beginning).
"This is circular reasoning; infinity can exist if infinity can exist."
It is true that I am assuming an infinite past and trying to show that things logically work out under that assumption. But that's fine, I'm not trying to assert that there IS and infinite past, merely that an infinite past is a valid possibility.
Your side, on the other hand, is trying to demonstrate that an infinite past is logically impossible. So your task is harder, you need to assume that there is an infinite past and demonstrate that a contradiction occurs. As far as I have seen, this has not been accomplished.
The treadmill example is to demonstrate the +1+1+1+1+1+1…..nature of infinity. There’s always one more step prior to anywhere.
DeleteIt’s always difficult to convince someone to change beliefs, which is true for everyone. It’s not difficult to demonstrate that infinity in reality leads to various logical absurdities. Believing that an actual infinity is possible in the real world carries baggage that must also be embraced.
I’ve enjoyed the discussion and reading your blog! I also finally finished Part 2 of my multi-verse series that discusses how the multi-verse is like Hilbert’s Hotel.
Sorry to carry over a question from the last thread on infinity, but I'll just copy paste it here since maybe you missed it:
Delete"Sorry to jump in here. Was trying to follow along and I am missing something in your reasoning. Maybe you can correct me where I am going wrong, but this is how I was following your argument: Everything needs a cause and can’t come from nothing. Then you say things with beginnings cannot have infinities. You then jump to say that God has no beginning and is an infinity, but don’t really justify why this is or even how this might be different than a universe without a God. I think I am misunderstanding your argument, because the way I read it, it sounds like special pleading. Thanks!"
@John - “God has no beginning and is an infinity”
DeleteThere are three types of infinity; potential, actual, and transcendental. I described potential and actual earlier. A transcendental infinity is not a quantitative infinity (1+1+1+…), but a qualitative infinity. It is beyond human comprehension, but not a series of 1+1+1+…. For example: omniscience; God knows everything that can be known. Now the follow-up question, which is what I think you are getting at, is how can God not have a beginning if an infinite regression is impossible? Time began at the Big Bang and is perceived by motion or change. Without motion/change, there is no time. Without time, there is no series of 1+1+1+… Without time, there is no infinite regression.
Another question.. Why does that "beginning" have to be God? Why can't it be something else? (Thanks for taking the time to answer btw) :)
Delete@John - Time had a beginning at the Big Bang. Time cannot cause itself to begin, so something outside of time is needed to start time. It’s either a material timeless cause or immaterial timeless cause. Remember, time is a result of motion/change so something that is material and timeless is also motionless and static. The material is frozen, so there’s nothing that can start the motion. Plus at the Big Bang matter/energy also began so this also points to an immaterial timeless cause…..a mind.
DeletePremise 1: “Time had a beginning at the big bang.”
DeleteOkay. I actually agree with this.
Premise 2: “Time itself can not cause itself to begin, so something outside of time is needed to start time.”
Why is this? We don't even know what caused the big bang. But alright, I'll bite. Like the last argument everything needs a cause, except God, right? (special pleading).
Premise 3: “It's either a material timeless cause or immaterial timeless cause”
Why does it have to be “immaterial”? And what is this “frozen material”? There's no real justification because we really don't know if there was a “cause”, but I'll allow it.
Premise 4: “There's nothing that can start the motion.”
But doesn't this contradict your premise premise 2?
Conclusion: “Matter/energy also began so this also points to an immaterial timeless cause.. a mind”
If we accept that it was something “outside of time” how does this jump to a “God” or even a mind? This conclusion does not follow the premises. The last part is even just a big jump with no real justification. I think this is begging the question.
Just my two cents.
@John – When I read your response, my initial reaction was “Did he read what I wrote?” Then I recognized the argumentative style….taking lines out of context and then straw-manning them. That makes for a rather dull discussion. Please see Mr. Hausdorff’s manner as a model for having a discussion with someone who holds different beliefs.
DeleteOkay.. Well, that's how arguments are made. You have premises and you draw conclusions from them. Your argument makes no logical sense. Sorry. They are just as you wrote them, word for word.
DeleteIf you feel this isn't the case, you might need to break down your premises and explain them a bit more because as you wrote it they do not in fact lead to your conclusion. Sorry. This isn't a "trick" because I did take your argument point by point without straw manning it.
Delete@John: You lost me a little bit with this:
Delete"Premise 3: “It's either a material timeless cause or immaterial timeless cause”
Why does it have to be “immaterial”? And what is this “frozen material”? There's no real justification because we really don't know if there was a “cause”, but I'll allow it."
I'm not sure if you are getting ahead of yourself here or what, but I definitely see what RZ means when he asks if you read what he wrote. immaterial vs. material seems like a reasonable dichotomy to me. and RZ tries to justify why he thinks it needs to be immaterial next.
@Rational Zealot: You've clearly crammed a big idea into a small comment, so I have some thoughts/questions:
1. couldn't there be a different time concept in the multiverse, or perhaps each universe within the multiverse has it's own isolated timeline?
2. I'm not sure I understand how immaterial plays a role? Why could an immaterial thing do stuff without time's arrow but a material thing couldn't? Or put another way, why couldn't a purely material universe/precursor universe have gotten things started for us instead?
3. "at the Big Bang matter/energy also began" Do we really know this? I was under the impression that we don't really know what happened before the planck time. Isn't it just that we know we can go really far back and all the matter in the universe was crammed into a very small space? I'm really stretching my physics knowledge here, but I was under the impression that this was far from settled.
I suppose so.. I guess we don't really know of anything that is "immaterial", but I suppose there's the stuff we know about "material" and there might be a possibility that there's something else, but we really don't know.. Then again, we don't really know much about the big bang except after the first .000....001 seconds (no idea how many zeros). The rest is just guessing.
Delete@Hausdorff
Delete1). Yes, there would be a different time concept in the multiverse. Relativity says that the perception of time is relative to the motion of the observer, so if the multi-verse exists, the time in our universe is perceived differently than time in other universes. If there is no motion, there is no time which is why I said that something that is both material and timeless would be motionless or frozen.
2). Since time began at the Big Bang, the cause must be timeless and changeless (change causes/requires time). An immaterial thing, such as a mind, is not physical so it is not frozen. An omnipotent mind that is not bound by a physical body can choose to create. Could a previous universe kick off our universe? This moves the questions back a level and starts the infinite regression of past universes/causes. There’s also the Second Law of Thermodynamics to consider which is also problematic for the universe to have an infinite past.
3). I believe most of the models, except for the oscillating model, have matter/energy in our universe beginning at the Big Bang. The popular models seem to be either string theory or the freak quantum fluctuation (aka from ‘nothing’).
"Yes, there would be a different time concept in the multiverse"
DeleteGreat! But then you said
"Since time began at the Big Bang"
I would say our timeline began at the big bang. You said yourself that there could be different time in the multiverse. So it wouldn't have to be motionless because it wouldn't have to be timeless.
“I would say our timeline began at the big bang. You said yourself that there could be different time in the multiverse.”
DeleteIf there is a multi-verse, both are correct….our timeline would have a beginning and there would be a different time in the multi-verse. If the multi-verse had a beginning, then you still have to deal with time having a beginning and a timeless cause (unless there is a multi-multi-verse). The alternative to a timeless cause is the infinite regression of past time or causes.
“So it wouldn't have to be motionless because it wouldn't have to be timeless.”
I was responding to the question as to whether or not the multi-verse can be a material timeless cause. Something that is both material and timeless is frozen since there can be no motion without time. If the multi-verse is the cause, it cannot be timeless.
"The alternative to a timeless cause is the infinite regression of past time or causes."
DeleteAgreed, which I still haven't seen a reason why an infinite regression is a problem.
"I was responding to the question as to whether or not the multi-verse can be a material timeless cause. Something that is both material and timeless is frozen since there can be no motion without time. If the multi-verse is the cause, it cannot be timeless."
I see, so this part of the argument essentially says "if timeless and material then frozen". In that case my objection doesn't apply because it is addressing a different case, where the multiverse isn't timeless.
Although now that I think about it, I have another thought, why would an immaterial being not be frozen in a timeless universe? Another way to ask the same question is: how does the material nature of a thing contribute to it being affected by time? Take for instance our souls (which I assume you believe we have), they are immaterial, correct? Yet aren't they subject to the passage of time?
(damn, I feel I'm taking an already long conversation off on a whole new tangent. Feel free to ignore this last musing of mine unless it really catches your interest)
“how does the material nature of a thing contribute to it being affected by time?”
DeleteThe material is made up of particles that are moving. Motion or change causes time. God is not made of moving particles.
“Take for instance our souls (which I assume you believe we have), they are immaterial, correct? Yet aren't they subject to the passage of time?”
Yes, the soul is immaterial and currently bound by a material body. Let’s say my body evaporates and all that is left is my mind. Is that still subject to time (which I believe is your question)? This isn’t something I’ve given a lot of thought to, but on the surface I’d have to say probably, our souls are still bound by time. A difference between our minds and God as a mind is that we need to think, which corresponds to a series of changes. Even though there would be no moving particles, a series of thoughts would cause some form of time (although I suspect it would be different than what we have now). God knows everything and does not need a series of thoughts, so he can be changeless and timeless.
"the soul is immaterial and currently bound by a material body"
DeleteOh sure, our souls would be linked with our bodies. Fair enough.
"Even though there would be no moving particles, a series of thoughts would cause some form of time (although I suspect it would be different than what we have now)"
I buy this as well, it would seem to also apply to various kinds of ghosts and spirits and lesser gods that we can read about in various pieces of fiction as well. I'm not entirely sure what you mean when you say it would be different, though.
"God knows everything and does not need a series of thoughts, so he can be changeless and timeless."
So does God not have thoughts? I'm not sure I really understand this. God doesn't think? You say that God is a mind, but what does it even mean to have a mind that doesn't have thoughts?
Also, how can he be changeless? He's omnipresent, right? If he creates the universe, doesn't he necessarily change himself as well? How can he be in a place that doesn't exist. If he is in a place, and the place changes doesn't he necessarily change as well?
“So does God not have thoughts? I'm not sure I really understand this. God doesn't think? You say that God is a mind, but what does it even mean to have a mind that doesn't have thoughts?”
DeleteGod doesn’t need to process information the way we do. We have to think our way through things because we have limited knowledge. God is all-knowing and does not need to do this. It’s not that he didn’t have thoughts ‘prior’ to time, but he already had knowledge of everything that would happen, so there was no series of thoughts. All necessary thoughts were already there.
“Also, how can he be changeless? He's omnipresent, right? If he creates the universe, doesn't he necessarily change himself as well?”
There are differing views on this. Some believe God remains changeless and timeless after creation; others say he was changeless/timeless ‘prior’ to the creation of time and say he ‘stepped’ into time at the moment of creation. This would also depend on how you view time; whether the past/present/future are objective realities or if the past/present/future are fixed and time is merely an illusion (commonly known as A or B-theory of time). Omnipresent does not necessarily mean he physically fills every point in space. He’s not lurking inside of the pizza I’m eating. It can also mean he is observing or aware of every point in space at every moment. When it comes to change, God would already have foreknowledge of any change, but when a change happens he would be aware that the change has occurred, so yes, this would be a series of changes. I am of the opinion that ‘before’ time and after time are different ball games. God was changeless/timeless ‘prior’ to the creation of time, but I’m not sure how one could argue that there were/are no changes after creation (unless you are an extreme B-theorist).
These types of questions are getting to be above my pay grade, which I’m an amateur, so that’s not saying much. I haven’t spent much time thinking about time. :0
"These types of questions are getting to be above my pay grade, which I’m an amateur, so that’s not saying much. I haven’t spent much time thinking about time."
Deletehehe, I'm starting to feel the same way. :) Great conversation here, I think I'm getting to the point where I'm ready to reflect on it, and probably post more later on.
Great points Hausdorff. Infinity is one of these problems that are not always easy to comprehend, but think you did a great job here.
ReplyDeleteThanks Christian :) I do love talking about infinities. It's confusing but not impenetrable. That's why I like going over them occasionally, each time you pick up some new subtlety that you missed before.
DeleteYou might want to check out this: http://www.skepticink.com/tippling/2013/12/11/answering-for-william-lane-craig-a-series-from-notes-about-infinity-iii/
ReplyDeleteThanks :) I'm definitely gonna have to go check that out
Delete"The treadmill example is to demonstrate the +1+1+1+1+1+1…..nature of infinity. There’s always one more step prior to anywhere."
ReplyDeleteSure, but I don't see why that's a problem.
"It’s always difficult to convince someone to change beliefs, which is true for everyone. "
Agreed, I love to be able to convince you that infinity is okay in this context, but it's hard work =P
"It’s not difficult to demonstrate that infinity in reality leads to various logical absurdities."
such as? I know you have mentioned hilberts hotel, but I don't think that applies here. It's about an infinite amount of physical stuff that exists right now (in the form of a hotel). An infinite regress is about an infinite amount of time in the past. These are fundamentally different things. Demonstrating problems with one doesn't necessarily carry over to the other.
"Believing that an actual infinity is possible in the real world carries baggage that must also be embraced."
Like what? Perhaps some things I would be willing to accept and other things I won't.
@Hausdorff - “It's about an infinite amount of physical stuff that exists right now (in the form of a hotel). An infinite regress is about an infinite amount of time in the past. These are fundamentally different things.”
DeleteWhy is time different? Are you saying an infinite number of physical things cannot exist or is it because of the set physical structure of a hotel?
“Like what?”
Let’s say you have an infinite number of white marbles and an infinite number of black marbles mixed together. These two infinities added together form a larger infinite set. No surprises here; there can be an infinity that is larger than the other in math. If you were to remove an infinite number of marbles, how many would be left? There are an infinite number of possibilities. If you were to remove all of the white marbles, there would still be an infinite number of black marbles. If you were to remove all of the marbles, there would be zero marbles left. If you were to remove all but 1 white marble, there would be 1 marble left (and this could be repeated for 2 or 3 or ….). In each of these cases, an infinite number of marbles were removed (sometimes more than one infinity) yet the results range from zero to infinity.
Regarding the hotel, I guess I was thinking of the physical structure, where would it be, if you tell everyone in room x to go to room 2x there will be people having to travel arbitrarily large distances, etc. But honestly, the more I thought about it, the more I disliked my previous comment about it, because if we let those parts of it go it works out just fine. Many things are counter-intuitive, but that doesn't mean that infinity is impossible or illogical.
DeleteIt actually makes me think about the purpose of the hilbert hotel in the first place. The purpose is not to show that infinities are impossible, but rather that extreme care is needed. That infinity is not a number, and can't be carelessly treated like one. We often use the same words to talk about infinities as we do numbers, but they mean slightly different things. For example, you said
"Let’s say you have an infinite number of white marbles and an infinite number of black marbles mixed together. These two infinities added together form a larger infinite set."
This is probably incorrect, but it actually might be okay, it depends on what you mean by larger. Let's say A is the set of all of the black and white marbles, and B is just the set of all the white marbles. A is larger than B usually means that the number of elements in A is bigger than the number of elements in B. It's easy to see that adding a finite number of things to an infinite collection doesn't make the infinity bigger, but we can also show that combining two infinities doesn't make it bigger. (Hilbert's hotel is a great way to demonstrate this fact)
(Note: I'm assuming all infinities that we are talking about are countable infinity)
On the other hand, if you mean subset, then it is true that B is a subset of A, there are elements of A that are not in B, but every element of B is in A. But usually this isn't want we mean when we say that one set is larger than the other.
"there can be an infinity that is larger than the other in math"
That's true, but doesn't seem to be what we are talking about here. If we "add" two infinities together that are the same size we don't get a bigger infinity, we need different operations for that.
"If you were to remove an infinite number of marbles, how many would be left? There are an infinite number of possibilities."
Agreed, but why is this a problem? It shows that infinity is not a number, and that subtraction is not a valid operation on infinity. Doing "infinity minus infinity" or "removing infinity from infinity" is not enough information. You have to be more specific, and depending on those specifics you can get different answers. Take our example of an infinite regress. Suppose I assign to every second a number going backward from this moment. I can remove an infinite number of seconds in two ways
1. Remove every even second. So every odd second still remains and I still have an infinite regress
2. Remove every second that is more than 10 seconds into the past. So all that remains is 10 seconds.
This doesn't prove that an infinite regress is illogical, it shows that "removing an infinite number of seconds" is not a well defined operation.
To try an analogy in the finite world, suppose I said I have a set with 100 items and I am going to take some of it away. The operation I'm describing isn't clear, and therefore the result is anywhere from 0 to 99. This doesn't imply that having 100 objects is impossible, just that my subtraction operation is poorly defined. Once more information is given it all works out with no issues.
gah! Baby waking up. I will respond to the other comments as soon as I get the chance.
In relation to the three types of infinities, I must admit, this classification is something that I've never heard of before as a mathematician. Perhaps it is common in apologetics circles, but it's pretty new to me. After our discussion, I understand actual vs. potential. I would say "actual infinity" is just infinity. "potential infinity" is maybe finite but unbounded. Essentially, we are looking at a limit, actual infinite is if you let the limit finish, potential infinite is if you don't. Seems like a pretty reasonable thing to do as a limit is not going to be able to finish in reality (at least not the types of limits we are talking about here). On the other hand, it's a matter of perspective. Is there an infinite amount of time in the future of our universe? If we look at all of time together at once, then it is really infinity. If we ask if we can ever "get to" infinity, then no, we can get to an arbitrarily large finite number, but we can't get to infinity. We can never complete that limit. But I still think it makes sense to say there is actually an infinite future ahead in this universe.
ReplyDeleteAs to the transcendental infinity, I have no idea how to make sense of that, and apparently neither do you
"A transcendental infinity is not a quantitative infinity (1+1+1+…), but a qualitative infinity. It is beyond human comprehension"
Then why are we talking about it? How can you claim that God is this type of infinity when you admit that you don't even understand what this kind of infinity is? How is this even an answer?
To me this answer is equivalent to "God is magic, just don't worry about it". Totally unacceptable.
@Hausdorff – I found the term “transcendental infinity” on the Internet Encyclopedia of Philosophy. It’s not a term that I’ve seen elsewhere, but I think it’s useful in separating the word infinity when describing God vs 1+1+1+1+…..they are not the same. Here is the website.
Deletehttp://www.iep.utm.edu/infinite/#SH1a
“A transcendental infinity transcends human limits and detailed knowledge and might be incapable of being described by a precise theory. It might also be a cluster of concepts rather than a single one.”
“Could something like a multiverse be this kind of infinite?”
I don’t know…I haven’t really thought about it. Maybe? The term itself may be applicable, like if the multi-verse is incalculably large it may as well be infinite, even though it may not be an actual infinite. I’m not really sure, but I don’t want to confuse you by lumping God and the multi-verse together by using the same term. When describing God as infinite, I think of the omni’s; omniscient, omnipotent, omnipresent, etc. These are maximum qualitative statements.
“Do you know a place where I could read more about this kind of infinity?”
The Bible! :0 Sorry….had too.
Thanks for the reference. Have booked marked it for further reading when I have the time, but the following stood out to me
Delete"One difficulty here, though, is that it is unclear whether metaphysicians have discovered that God is identical with the transcendentally infinite or whether they are simply defining “God” to be that way. A more severe criticism is that perhaps they are just defining “infinite” (in the transcendental sense) as whatever God is."
"The Bible"
haha :)
@Hausdorff - “If we ask if we can ever "get to" infinity, then no, we can get to an arbitrarily large finite number, but we can't get to infinity. We can never complete that limit.”
ReplyDeleteYet this is required for an infinite past. That is the difference between and infinite future and infinite past.
“To me this answer is equivalent to "God is magic, just don't worry about it". Totally unacceptable.”
We don’t understand everything about gravity and may never understand everything about gravity. This does not mean that gravity is magic or that we cannot learn a great deal about gravity. Saying that God is a transcendental infinity is a qualitative and relative statement. There’s a lot I don’t understand about God, but this doesn’t mean I am unable to learn anything about God.
"@Hausdorff - “If we ask if we can ever "get to" infinity, then no, we can get to an arbitrarily large finite number, but we can't get to infinity. We can never complete that limit.”
DeleteYet this is required for an infinite past. That is the difference between and infinite future and infinite past. "
I was going to respond to this last night, but I wanted to let it percolate on my mind a little bit longer. This is very interesting, it highlights the difference between an infinite past and an infinite future. An item (say a particle or whatever) has an infinite past and therefore has already experienced an actual infinity, whereas it has an infinite future which is just a potential infinity. So that is the difference (perhaps the only difference) between an infinite past an infinite future.
But it seems to me that a potential versus actual infinite future is really just a matter of perspective. What you quoted of me there, was from the perspective of an object in our universe with an infinite future. What did I mean by "get to"? I meant that there is no point in time that it can look around realize it has experienced an infinity. But of course not, from this perspective we are always cutting off the end of the timeline and forcing things to be finite. From it's perspective, it will always have only experienced a finite number of things, but it will always have an infinite future ahead of it. If we take an outsiders view, looking at the timeline as a whole, it will experience an infinite number of moments. From that view we can say that it will experience an actual infinity.
I would normally divide things into infinite and finite, everything has to be one or the other. There is either a finite future or an infinite future, has to be one or the other. If it's finite there needs to be an end point, call it x. If no such x exists, then there is infinite time. Potential infinity seems to be trying to shimmy in between these two possibilities, it's not finite, so no such x exists, but it's not really infinite either. The whole idea seems like it's trying to sweep something under the rug to me.
From the perspective of something on a timeline which is infinite in both directions, it could say that it has already experienced an infinite past, and it will experience an infinite future. It seems to me that any attempt to disprove an infinite past could be completely turned around to disprove an infinite future. The only way around this is the assertion that potential infinities are possible while actual infinities are not. I still don't see a justification for that.
@Hausdorff – “From it's perspective, it will always have only experienced a finite number of things, but it will always have an infinite future ahead of it. If we take an outsiders view, looking at the timeline as a whole, it will experience an infinite number of moments. From that view we can say that it will experience an actual infinity.”
DeleteYou lost me here. It has an infinite past but has only experienced a finite number of things? With no beginning, there is an infinite number of past finite moments, so how is the experience finite? I think I’ve missed something.
“There is either a finite future or an infinite future, has to be one or the other. If it's finite there needs to be an end point, call it x. If no such x exists, then there is infinite time.”
Conceptually, the future has the potential to be infinite. In reality, the series of changes leading to the perception of time is finite….the future infinity never becomes actual in reality and remains a potential in theory. You can never ‘get to’ infinity just like you cannot ‘come from’ infinity.
"You lost me here. It has an infinite past but has only experienced a finite number of things? With no beginning, there is an infinite number of past finite moments, so how is the experience finite? I think I’ve missed something."
DeleteHmm, what I wrote isn't very clear. Looks like some bad editing on my part. In the first part I was talking about something with infinite past. In the second, I switched to thinking of something with an infinite future instead but didn't make it very clear that I had done that.
"Conceptually, the future has the potential to be infinite. In reality, the series of changes leading to the perception of time is finite….the future infinity never becomes actual in reality and remains a potential in theory."
That's the idea that I was getting at. From our perspective this is true, there is never a time when something in the universe can say "I have experienced an infinite amount of time". But from the perspective of someone outside of time, say God's perspective, can see the whole timeline all at once and it is an actual infinity. Actual versus potential seems to me to be merely a matter of perspective.
@Rational Zealot: One last comment here (hope I'm not overwhelming your inbox too much). You said that transcendental infinity is beyond human comprehension. Did you mean we don't understand it, or we can't understand it? I took it as the latter, but your comparison to gravity perhaps suggests the former. Fair enough.
ReplyDeleteI guess my question would then by what do we know about transcendental infinities? We must know something about it to be able to assign the property to things (God in this case). Could something like a multiverse be this kind of infinite? If not, why not? Do you know a place where I could read more about this kind of infinity? transcendental numbers mean something in mathematics, which is getting in the way of my googling efforts.