Physical implications of Kalām argument

[ddt]

That might be true, although neither current theory nor any experimental evidence support it. And even in a universe like that, one can still construct infinite sets: the set of all ordered sets of points, for example.

Now I'm beginning to wish I had studied set theory properly yet. Is there any short version of how that is done, or some link that explains it in more detail?

I studied set theory, and I'm puzzled too by sol's comment. We are arguing about a universe with a finite number of points (call it N), aren't we?

Then the number of possible orderings on all points is also finite. Let's see:

Take two points x and y. They can be either ordered as x < y, or x > y, or not - that's 3 possibilities. There are N * (N-1) / 2 possible unordered pairs of x and y, so the number of orderings is bounded by 3 ^ (N(N-1)/2). I say bounded, because not all combinations are valid; an ordering must adhere to the law of transitivity:

if x < y and y < z, then x < z

So there's a finite number of possible orderings of N points. If you include the orderings of subsets of those points, for each size n of such a subset, there's a finite number of ways to get a subset of size n, and per subset you get the same upper bound for the number of orderings as above (with n instead of N), so - all in all - you still get a finite number of orderings.

Did I miss something?

 

[stup_id]

That is not likely to be the number of partciles in the universe, just the observable universe?

Thanks, there's a huge difference in that, I'm not familiar however as how they've reached this number, I hope is not one of those "fun facts" such as "we only use 10% of our brain"

Prime numbers?

Yep yep, I know that, but that's why I made the dinstinction of "physical" things, you know... made of matter can there exist and infinite amount of things of such?

 

[sol invictus]

Now I'm beginning to wish I had studied set theory properly yet. Is there any short version of how that is done, or some link that explains it in more detail?

Nothing fancy - see below.

That's annoyingly inconvenient... Is there no way that singularities could—even just in principle—be used to test if space is quantized on any level?

It's very hard to make completely general statements. Certainly in any given theory in which space and time are discretized, there are certain potentially observable consequences. Usually those consequences get milder and milder as the discreteness scale goes to zero, but there are some surprising exceptions - black hole horizons (and singularities, if you could access them) are often among them.

I studied set theory, and I'm puzzled too by sol's comment. We are arguing about a universe with a finite number of points (call it N), aren't we?
<snip>

You're assuming that each point can only appear once. I probably should have said "list" or "sequence".

But anyway, it's a rather nonsensical discussion, because (as has been pointed out) it's very unclear what it means to place the mathematical restriction of finiteness on things the universe contains. Does the universe contain numbers? If so, does it contain infinite sequences of numbers? If not, why not? Does it contain spacetime points? What about sequences of spacetime points, as I had in mind?

 

[blutoski]

I may have stated #1 badly. Is it clearer if we restate it as "The universe cannot contain an infinite number of anything, whether elementary particles, points in spacetime or other objects", or does that make it too fuzzy?

OK, but now #1 appears to be three orthogonal premises glommed together (it is possible for some or all parts to be true or false independently):

• #1a: the universe does not contain an infinite amount of matter
• #1b: the universe does not contain an infinite number of coordinates (points in space)
• #1c: the universe does not contain an infinite number of moments (points in time)

premise #1c seems the most relevant to premise #2, and there are two orthogonal re-wordings that further clarify meaning:

• #1c/i: the universe does not contain a moment that is infinitely large or small
• #1c/ii: the universe consists of a limited number of moments

Now: premise #1c/i is the interpretation I was using when I said it was a duplicate of premise #2. I still think this interpretation is a duplicate.

However, if the actual meaning is clarified by premise #1c/ii, then its truth does not imply that the universe must have a beginning and end. Consider the scenario where the universe has a meagre three points (moments) in time: a point infinitely far in the past, a point right now, and a point infinitely far in the future. So, we have a finite number of points in time, but in a universe that nevertheless is infinite in duration.

 

[blutoski]

Just read the OP. Or, here, I'll cut and paste it for you:



Pretty straightforward, don't you think?

I think so...

... so why did you say ([in this post]) that we were evaluating the first premise when it's clear from the quote you just excerpted that Dorfl has asked us not to? (if we're assuming it is true, this means we don't have to evaluate it)

 

[Yoink]

I think so...

... so why did you say ([in this post]) that we were evaluating the first premise when it's clear from the quote you just excerpted that Dorfl has asked us not to? (if we're assuming it is true, this means we don't have to evaluate it)

I was responding to your question "Which one are we evaluating. There's four there." "Evaluating" is a pretty vague term. I assumed you meant it in the sense of "which one are we trying to understand the implications of" or "which one is under discussion here." If you meant it to mean "which one are we meant to testing to see if it is true" then I don't know why you asked the question at all: the answer would obviously be "none of them."

 

[blutoski]

I was responding to your question "Which one are we evaluating. There's four there." "Evaluating" is a pretty vague term. I assumed you meant it in the sense of "which one are we trying to understand the implications of" or "which one is under discussion here." If you meant it to mean "which one are we meant to testing to see if it is true" then I don't know why you asked the question at all: the answer would obviously be "none of them."

Maybe we'll need Dorfl to help us understand what, exactly, he's asking us to do. I'm pretty sure he's not asking us to discuss the implications of a demonstrated true existence of God, per premise #4. That's a big topic, and probably also not really appropriate for the 'science' area in the forum (that would be best suited in the 'religion' area).

I thought he was asking for us to review the merit and naturalistic implications of the argument, with the assumption that premise #1 was true.

 

[Yoink]

Maybe we'll need Dorfl to help us understand what, exactly, he's asking us to do. I'm pretty sure he's not asking us to discuss the implications of a demonstrated true existence of God, per premise #4. That's a big topic, and probably also not really appropriate for the 'science' area in the forum (that would be best suited in the 'religion' area).

I thought he was asking for us to review the merit and naturalistic implications of the argument, with the assumption that premise #1 was true.

I think what he's asking is "what would be the implications if the premise in #1 were true." I think he wants us to ignore the rest of the argument entirely.

 

[blutoski]

I thought he was asking for us to review the merit and naturalistic implications of the argument, with the assumption that premise #1 was true.

With that in mind, I still think the primary weakness of the version in the original post (or alternative rephrasings in later posts) is the non sequitur between math concepts that represent limited modelling abstractions from reality vs our understanding of reality itself.

I had a read through the Wikipedia summary of the original argument: [Kalam cosmological argument] and the objections do seem familiar to my thoughts on the subject.

In particular, that there is no compelling reason to accept the first premise as a mathematical truth, and even if it were true, there is no reason to believe that a mathematical truth projects into reality.

The other obvious objection that I overlooked relates to what Dorfl put in the position of a fourth premise (Kalam cosmological argument actually only has two premises) which ties the principle of a beginning of the universe to a creator. The weakness being that if we want to say that the creator did not have a creator, we must accept that premise 1 ("Everything that begins to exist has a cause.") is both true and false, which means the argument appears self-contradictory. This was Bertrand Russel's reason for rejecting it.

It appears to be a fancy version of "argument from contingency" with superfluous reference to 'sets'.

 

[blutoski]

I think what he's asking is "what would be the implications if the premise in #1 were true." I think he wants us to ignore the rest of the argument entirely.

Well, assuming that, and based on the clarifying re-wording provided that stripped away the debatable stuff about mathematical set theories and just focussed on the claim that: there is a limited amount of matter in the universe, that there is a limited amount of height, width, depth, and time...

I don't think this would have much of an implication for anything else, and for what it's worth, when stated this way, it's probably true. I'm pretty sure the amount of matter in the universe is finite. On the other hand, I'm pretty sure spatial distance is infinite in all directions, although we may find that the local spacetime properties have a perimiter. (Could this be defined as a boundary to our universe?)



But the re-wording Dorfl provided in a more recent post is not the Kalam cosmological argument. The Kalam cosmological argument does say very specifically that mathematical models do map or project to the real universe (eg: that infinite sets can't exist in mathematics, therefore, no concept in the universe can have infinite values). That's a very important supporting premise and distinguishes the Kalam Cosmological argument from other Cosmological arguments that just claim that the universe must be physically and chronolocically bounded a priori.

This really is philosophy.

 

[Dorfl]

Oops.

Sorry about disappearing with this discussion going on. What I'm asking for is specifically what physical, observable effects it would have on the universe if #1 were true. If #2 and #3 would somehow observably affect our universe (even in just a negative way, such as forcing it to not have an open geometry), then we can ask if they actually follow, assuming #1s truth—as well as what other things about the universe which also would have to be true.

It might have been better if I had left out steps #2 and onwards from the OP, and just asked what the effects of #1 would be, but I wanted to give some background for why I was asking the question anyway.

 

[Dorfl]

OK, but now #1 appears to be three orthogonal premises glommed together (it is possible for some or all parts to be true or false independently):

• #1a: the universe does not contain an infinite amount of matter
• #1b: the universe does not contain an infinite number of coordinates (points in space)
• #1c: the universe does not contain an infinite number of moments (points in time)

My attempt to restate the Kalām argument was pretty bad, I admit. I'll try to see if I can find any better way of putting it.

premise #1c seems the most relevant to premise #2, and there are two orthogonal re-wordings that further clarify meaning:

• #1c/i: the universe does not contain a moment that is infinitely large or small
• #1c/ii: the universe consists of a limited number of moments

Now: premise #1c/i is the interpretation I was using when I said it was a duplicate of premise #2. I still think this interpretation is a duplicate.

However, if the actual meaning is clarified by premise #1c/ii, then its truth does not imply that the universe must have a beginning and end. Consider the scenario where the universe has a meagre three points (moments) in time: a point infinitely far in the past, a point right now, and a point infinitely far in the future. So, we have a finite number of points in time, but in a universe that nevertheless is infinite in duration.

I'm not really sure what it means to talk about the "size" of a moment, or the distance between them. Doesn't that require there to be a second, non-discrete time for the discrete time to be measured in?

It seems to me that if a universe contained three moments in time, then that universe's length in time would be three moments, period. Talking about the length between them seems like discussing the "size" of each square on a tic-tac-toe grid.

 

[Dorfl]

It's very hard to make completely general statements. Certainly in any given theory in which space and time are discretized, there are certain potentially observable consequences. Usually those consequences get milder and milder as the discreteness scale goes to zero, but there are some surprising exceptions - black hole horizons (and singularities, if you could access them) are often among them.

That's cool. So now I just need to find a naked singularity...

You're assuming that each point can only appear once. I probably should have said "list" or "sequence".

But anyway, it's a rather nonsensical discussion, because (as has been pointed out) it's very unclear what it means to place the mathematical restriction of finiteness on things the universe contains. Does the universe contain numbers? If so, does it contain infinite sequences of numbers? If not, why not? Does it contain spacetime points? What about sequences of spacetime points, as I had in mind?

True. #1 turned out to be a lot more ambiguous than I thought at first glance. And I'd rather not derail this into a philosophical discussion of which things "exist" and which do not. There was a reason I posted this in Science and not Philosophy.

 

[Roboramma]

Ah. So just one premise? Which premise are we evaluating? There's four there.

From the OP:

1. No infinite set can exist in the physical world.

And:

Whether the argument is sound or not is already being discussed in the other thread. What I wonder about is the physical implications that the premise #1 would have, if it were true. For example, it seems to imply that the universe is finite in both space and time—requiring a big crunch—and that space and time are both quantized.

Are there any other implications that #1 would have, and are they correct, as far as we know?

Bolding is mine.

 

[Monketi Ghost]

*smiling*

Best, most informative section of the forums.

 

[blutoski]

From the OP:


And:



Bolding is mine.

I appreciate that, and that's how I based my assumption that I was to assume #1 was true.

But in a later post Yoink said that we were to evaluate #1 for truth, rather than assume it was true. I was trying to clarify with Dorfl before proceeding, as I didn't want to try to answer a question he wasn't asking in the first place. It appears that Yoink just misunderstood what I meant by 'evaluate'.

I was using the meaning that is typical in critically evaluating an argument, because the original post had structured the question in the form of an argument.

I think this confusion was resolved a few posts ago.

 

[blutoski]

My attempt to restate the Kalām argument was pretty bad, I admit. I'll try to see if I can find any better way of putting it.

I'm not sure that's necessary... based on your further clarifications, it looks like we're not dealing with the Kalam argument anyway. The Kalam argument is metaphysics/philosophy.

I'm not really sure what it means to talk about the "size" of a moment, or the distance between them. Doesn't that require there to be a second, non-discrete time for the discrete time to be measured in?

I'm not sure if this has meaning either. But you're asking us to assume it's true, right? So, tell us what to believe so we can answer your question.

It seems to me that if a universe contained three moments in time, then that universe's length in time would be three moments, period. Talking about the length between them seems like discussing the "size" of each square on a tic-tac-toe grid.

Why do you not think this is a philosophical discussion?

 

[Dorfl]

I'm not sure that's necessary... based on your further clarifications, it looks like we're not dealing with the Kalam argument anyway. The Kalam argument is metaphysics/philosophy.

Ok.

I'm not sure if this has meaning either. But you're asking us to assume it's true, right? So, tell us what to believe so we can answer your question.

I'm asking you to assume that time is quantized—if you agree that necessarily follows from there being a finite number of points in spacetime. I have no idea if that makes it meaningful to claim that a moment has a certain "size" though, beyond "one moment". If you think that it does, I would be interested in hearing why.

Why do you not think this is a philosophical discussion?

Well, I'll agree that the thread is sort of flipping back and forth between science, maths and philosophy, but I'm trying to avoid the latter as much as possible. My argument there was partly that a person inside a universe with quantified time would need to access some external non-discrete time to measure the "size" of a moment. So I didn't see how a statement about the size of a moment could be meaningful.

 

[blutoski]

I'm asking you to assume that time is quantized—if you agree that necessarily follows from there being a finite number of points in spacetime.

No, that doesn't follow.

I have no idea if that makes it meaningful to claim that a moment has a certain "size" though, beyond "one moment". If you think that it does, I would be interested in hearing why.

I don't think it makes sense to say that a moment has size. It's just a reference.

Well, I'll agree that the thread is sort of flipping back and forth between science, maths and philosophy, but I'm trying to avoid the latter as much as possible. My argument there was partly that a person inside a universe with quantified time would need to access some external non-discrete time to measure the "size" of a moment. So I didn't see how a statement about the size of a moment could be meaningful.

I was treating a moment as the temporal equivalent of a more familiar spacial coordinate: it has no size. It is a reference point along a line.

My point is that whether the moments are themselves finite, the line upon which they lay could be infinitely long. Two of the moments could be plus and minus infinitely far away from this moment now.




I'll move away from the moments thing, because my actual point is being obscured.

Instead, consider a case where there are four (a finite number of) atoms in the universe. The atoms are located infinitely far away from each other. The universe can still be infinitely large while containing finite number of real objects. What I'm saying is that a finite number of objects does not imply a bounded universe.

In the time dimension, the equivalent of matter is 'events.' So consider the case where there are three events in the universe: one infinitely far in the past, one infinitely far into the future, and the current event right now. Finite events, but infinite time.

 

[Dorfl]

No, that doesn't follow.

I don't think it makes sense to say that a moment has size. It's just a reference.

I was treating a moment as the temporal equivalent of a more familiar spacial coordinate: it has no size. It is a reference point along a line.

My point is that whether the moments are themselves finite, the line upon which they lay could be infinitely long. Two of the moments could be plus and minus infinitely far away from this moment now.

I'll move away from the moments thing, because my actual point is being obscured.

Then you can ignore this bit, if you feel it is going off-topic. But my point is that moments do not lie on a timeline. The timeline is made of moments. So for it to be infinitely long, there would have to be an infinite number of moments.

Instead, consider a case where there are four (a finite number of) atoms in the universe. The atoms are located infinitely far away from each other. The universe can still be infinitely large while containing finite number of real objects. What I'm saying is that a finite number of objects does not imply a bounded universe.

In the time dimension, the equivalent of matter is 'events.' So consider the case where there are three events in the universe: one infinitely far in the past, one infinitely far into the future, and the current event right now. Finite events, but infinite time.

Isn't "event" just the term for a point in spacetime? According to wiki:

In spacetime, a coordinate grid that spans the 3+1 dimensions locates "events" (rather than just points in space), so time is added as another dimension to the grid, and another axis.

That's why I claim that a universe which is not finite in both time and space will contain an infinite number of events.