A refutation of an infinite regress

[John_Martin]

I’m wondering what Chainbreaker would consider an “actual infinity” to be. Our current models in physics assume that motion is continuous. Consequently, when you move through space (suppose we are considering only one-dimension for simplicity), the Intermediate Value Theorem applies, which essentially says that you pass through each point in space between your starting point and your endpoint. You cannot “teleport”.

Since each dimension of space is modeled by the continuum of real numbers, and there are infinitely many such numbers, you must pass through infinitely many points to move at all. If passing through a point constitutes an event, we have infinitely many events taking place in a finite amount of time. In other words, we found an “actual infinity”. You cannot use calculus at all if such infinities aren’t invoked.

Are you sure you have infinite events within a contained section of time? While motion is, indeed, continuous, it is also linear (rather than punctuated, rather than “digital”). And with inertia, there is no change in motion without some other cause of change (an event). Because of this, there may be considered to be a finite set of events occurring to an object in motion denominated (quantity) by its change in inertia, positive or negative acceleration or directional change, etc. Something like the acceleration due to gravity might be contrary to what I just said, because that specific cause of acceleration “appears” to be linear also, and therefore a continuous set of events in a given period of time. Since gravity is not fully comprehended, though, it is open as to whether it is continual versus periodic in its effect, perhaps.

Anyway, I think ChainBreaker is looking at what is open, rather than contained, timeline. And while we can imagine both infinite past events or causes or numbers, finally it is wisdom that stops and says, it is not actual.

 

[Oreoracle]

Are you sure you have infinite events within a contained section of time? While motion is, indeed, continuous, it is also linear (rather than punctuated, rather than “digital”). And with inertia, there is no change in motion without some other cause of change (an event). Because of this, there may be considered to be a finite set of events occurring to an object in motion denominated (quantity) by its change in inertia, positive or negative acceleration or directional change, etc. Something like the acceleration due to gravity might be contrary to what I just said, because that specific cause of acceleration “appears” to be linear also, and therefore a continuous set of events in a given period of time.

I think the crux of the matter is what one takes to be an “event”. I’ve debated this very topic on these forums before, so I’m confident that “event” will be redefined for each counterexample to the OP’s claim until there are no counterexamples left.

Naturally, the OP becomes the winner at that point by the rules of the Internet.

I’ll even make it easy for the OP to do this by suggesting a new definition for “event”. My example assumes that events do not come with a temporal quota–they can occur over arbitrarily small amounts of time. So if events are defined to require a minimum amount of time to transpire, this would disqualify my counterexample.

Anyway, I think ChainBreaker is looking at what is open, rather than contained, timeline. And while we can imagine both infinite past events or causes or numbers, finally it is wisdom that stops and says, it is not actual.

On this matter I will only say that, as far as I know, there is nothing inherently contradictory about an eternal universe. Some cosmologists believe that there was a Big Crunch which obliterated the previous incarnation of the universe, leading to a Big Bang from which our universe emerged.

And even if the universe isn’t eternal, there is a difference between observing that nothing is eternal and claiming that it is logically impossible for something to be eternal. It is logically possible, for example, that the universe could have different laws of physics, even though it does not.

 

[Gorgias]

What you say is simply not true.
One way of looking at ∞ is to add ∞ to the real numbers and thereby obtain a number that exceeds any real quantity.

That doesn’t seem to make sense. I think what you’re saying here is that, for some n in R (the set of real numbers), there exists a number m such that (n+∞)=m and m is not in R.

That doesn’t make any sense to me…

 

[Gorgias]

It’s logically impossible to count to an infinite. Not because it would take you an infinite amount of time to count, but rather it is because no matter how much quantity you have the amount you are left with is always a finite number away from where you started counting.

OK, fair enough. But who ever said that an infinite regress is ‘countable’? Besides which, the case you’re making is exactly the kind of case that demonstrates why an infinite regress is bad (such that, if we can prove that a given assertion leads to an infinite regress, we have to abandon the assertion as untenable)…!

 

[Oreoracle]

That doesn’t seem to make sense. I think what you’re saying here is that, for some n in R (the set of real numbers), there exists a number m such that (n+∞)=m and m is not in R.

That doesn’t make any sense to me…

You can add positive and negative infinities to the real numbers to get the extended real number system. This has limited applications, though, since you lose many of the properties of the real numbers by doing so. For example, the extended reals do not form a field since they lack a cancellation property, e.g., 1 + infinity = 0 + infinity doesn’t imply that 1 = 0. Several expressions also become “indeterminate forms”, such as differences and quotients of two infinities. So the extended reals aren’t closed under subtraction nor division.

A much better alternative is the hyperreal number system, which is a perfectly nice field, except for the fact that you lose the least upper-bound property. But I will defer you to Google on the subject of the hyperreals.

 

[Tomdstone]

That doesn’t seem to make sense. I think what you’re saying here is that, for some n in R (the set of real numbers), there exists a number m such that (n+∞)=m and m is not in R.

That doesn’t make any sense to me…

There is the set of real numbers, and there is the set of the extended real numbers. The set of extended real numbers is larger than the set of real numbers because it contains two additional elements, ∞ and -∞. The arithmetic for the set of real numbers is the same as before when you have two real numbers. However, the arithmetic is different when dealing with ∞.
See:en.wikipedia.org/wiki/Extended_real_number_line

 

[ChainBreaker]

I’m wondering what Chainbreaker would consider an “actual infinity” to be.

An actual infinity means that every event in an infinite regress has been realized. The number of changes that have happened are not potentially real and that in itself necessitates that an infinite number of events have been actualized. This is contradictory because that would mean that all finite quantities have been transgressed which is impossible since there is no objective quantity that can define that transgression.

 

[Gorgias]

An actual infinity means that every event in an infinite regress has been realized. The number of changes that have happened are not potentially real and that in itself necessitates that an infinite number of events have been actualized. This is contradictory because that would mean that all finite quantities have been transgressed which is impossible since there is no objective quantity that can define that transgression.

Still trying to understand your point, though…

Are you saying that claims to an infinite regress are illogical?

 

[ChainBreaker]

Still trying to understand your point, though…

Are you saying that claims to an infinite regress are illogical?

Well, i am not arguing that an infinite regress is logical.

 

[Oreoracle]

An actual infinity means that every event in an infinite regress has been realized.

Easy enough. Consider the usual setup for Zeno’s Paradox: A runner is stationed one mile from the finish line, and the race begins. Before he can reach the finish line, he must first reach the halfway point. But before he reaches the halfway point, he must get halfway to the halfway point, and so forth.

Zeno argued that this led to an infinite regress and, since he rejected infinite regressions, he concluded that motion is illusory. Assuming that you agree that motion is real and not illusory, it would seem that you are forced to accept the existence of infinite regressions.

Again, I suspect the real issue here is the question of what counts as an “event”. In this case, we have an infinite sequence of events, and each event is defined as getting halfway through completing the previous event. There is no “earliest event”, hence the regression. Can events be defined recursively in this way? Can they occur over arbitrarily small time intervals?

 

[Bahman]

Easy enough. Consider the usual setup for Zeno’s Paradox: A runner is stationed one mile from the finish line, and the race begins. Before he can reach the finish line, he must first reach the halfway point. But before he reaches the halfway point, he must get halfway to the halfway point, and so forth.

Zeno argued that this led to an infinite regress and, since he rejected infinite regressions, he concluded that motion is illusory. Assuming that you agree that motion is real and not illusory, it would seem that you are forced to accept the existence of infinite regressions.

Again, I suspect the real issue here is the question of what counts as an “event”. In this case, we have an infinite sequence of events, and each event is defined as getting halfway through completing the previous event. There is no “earliest event”, hence the regression. Can events be defined recursively in this way? Can they occur over arbitrarily small time intervals?

That is in fact a good example I was looking for. Infinite regress is possible if beginning (creation case) or end (Zeno case) is singular, meaning that the number of event grows in singular manner after and before these points respectively. Please read this.

 

[ChainBreaker]

Easy enough. Consider the usual setup for Zeno’s Paradox: A runner is stationed one mile from the finish line, and the race begins. Before he can reach the finish line, he must first reach the halfway point. But before he reaches the halfway point, he must get halfway to the halfway point, and so forth.

Zeno argued that this led to an infinite regress and, since he rejected infinite regressions, he concluded that motion is illusory. Assuming that you agree that motion is real and not illusory, it would seem that you are forced to accept the existence of infinite regressions.

Again, I suspect the real issue here is the question of what counts as an “event”. In this case, we have an infinite sequence of events, and each event is defined as getting halfway through completing the previous event. There is no “earliest event”, hence the regression. Can events be defined recursively in this way? Can they occur over arbitrarily small time intervals?

How does zeno’ parodox apply ontologically in terms of what i am arguing. How does zeno parodox show that an “actually infinite number” of events can exist in the past? Because what you are saying is that a runner is only half way a potentially infinite number of times.

Zeno’s parodox appears to be a mathematical problem as applied to events. I am dealing with ontological numbers, that is, i accept that there is an irreducible point at each index of chance. In zeno’s parodox there is no irreducible point and as such he divides a potentially infinite amount of times. Zeno parodox is like saying that the diameter of a circle never gets smaller no matter how small a circle gets but that doesn’t change the fact that there is no such thing as an infinitely small circle; it is only ever potentially that small.

Insofar as an actually infinite past is concerned Zeno’s paradox does not apply to my argument.

But if zeno’s parodox applied ontologically, we would never have reached the state or event that we are experiencing now.

 

[Oreoracle]

How does zeno’ parodox apply ontologically in terms of what i am arguing. How does zeno parodox show that an “actually infinite number” of events can exist in the past? Because what you are saying is that a runner is only half way a potentially infinite number of times.

I’m assuming that we are using “potential” and “actual” in the Aristotelian sense. An “actual infinity” would be an infinite number of events occurring, and a “potential infinity” is something that can grow without bound (but is at all times finite).

The sequence of events I described is an example of an actual infinity unless you argue as Zeno did that motion is impossible. This is an infinite regression because, even though infinitely many events occurred, no event was completed earliest.

Zeno’s parodox appears to be a mathematical problem as applied to events. I am dealing with ontological numbers, that is, i accept that there is an irreducible point at each index of chance.

Ontological numbers? Index of chance? I don’t have the faintest clue what you’re talking about, and neither does Google.

 

[ChainBreaker]

I’m assuming that we are using “potential” and “actual” in the Aristotelian sense. An “actual infinity” would be an infinite number of events occurring, and a “potential infinity” is something that can grow without bound (but is at all times finite).

An actual infinite has already occurred. It is complete, it has already been actualised. It is not occurring.

The sequence of events I described is an example of an actual infinity.

No it is not. Zeno is showing a person running half way a potentially infinite amount of times before reaching his goal and of course never does because he cannot transgress an infinite number. That’s a potential infinite.

 

[Oreoracle]

An actual infinite has already occurred. It is complete, it has already been actualised. It is not occurring.

It seems strange to me that you would concede that an actual infinity can happen but take issue with one “happening”. The idea that they can happen at all, complete or not, is usually the controversial part.

Okay, simply extend the runner analogy a little bit. Partition the track as before, but now extend it so that it is twice as long. Tell the runner to run through the first half of the track as usual. Once he reaches the halfway point, have him run half the remaining distance after standing still for a minute, then running half the remainder after the next minute, and so forth.

Since you admitted that the previous feat of running half the track is an infinity, surely this is also an infinity, since it includes more events. It is an infinite regression since there was no earliest event. And, as required, it is “occurring”, since the runner never reaches the finish line. It is analogous to an eternal universe: extending backwards forever with no earliest point, yet marching forward in time.

 

[Tomdstone]

It seems strange to me that you would concede that an actual infinity can happen but take issue with one “happening”. The idea that they can happen at all, complete or not, is usually the controversial part.

Okay, simply extend the runner analogy a little bit. Partition the track as before, but now extend it so that it is twice as long. Tell the runner to run through the first half of the track as usual. Once he reaches the halfway point, have him run half the remaining distance after standing still for a minute, then running half the remainder after the next minute, and so forth.

Since you admitted that the previous feat of running half the track is an infinity, surely this is also an infinity, since it includes more events. It is an infinite regression since there was no earliest event. And, as required, it is “occurring”, since the runner never reaches the finish line. It is analogous to an eternal universe: extending backwards forever with no earliest point, yet marching forward in time.

Set your time period. Whether it be one minute, five minutes, 30 minutes or whatever. This time period will be fixed throughout the experiment. The runner can easily reach the finish line in one time period, while covering an infinite number of consecutively smaller distance intervals and proceed beyond the finish line after one time period as follows:
The runner runs through the first half of the track in one half time period.
He then runs half the remaining distance in 1/4 the time period.
He then runs half the remaining distance in 1/8 the time period.
He then runs half the remaining distance in 1/16 the time period.
He then runs half the remaining distance in 1/32 the time period.
He then runs half the remaining distance in 1/64 the time period.
He then runs half the remaining distance in 1/128 the time period.
He then runs half the remaining distance in 1/256 the time period.
and continues thusly.
He then has run through an infinite number of consecutively smaller distance intervals in one time period but at the same time reaches the finish line in one time period and can then proceed to run further.

 

[Oreoracle]

Sure Tomdstone, but Chainbreaker wants an example where the infinity isn’t completed. So it was essential in my example that the runner never complete his overall task, yet still complete infinitely many events along the way without completing any “first” event.

 

[TarkanAttila]

Easy enough. Consider the usual setup for Zeno’s Paradox: A runner is stationed one mile from the finish line, and the race begins. Before he can reach the finish line, he must first reach the halfway point. But before he reaches the halfway point, he must get halfway to the halfway point, and so forth.

Zeno argued that this led to an infinite regress and, since he rejected infinite regressions, he concluded that motion is illusory.

To which The Philosopher (Aristotle) replied that to each distance there is a corresponding time. The shorter the distance between where the runner is and his goal, the shorter the time it will take to cover it. (See his answer to Achilles and the Tortoise.) Aristotle also distinguished between dividing something up infinitesimally, and a thing that is actually infinite in its quantity.

Also, Archimedes has actually shown that even an apparently infinitesimal set of changes can have a finite value, as he demonstrated with a parabola.

So the apparent infinite is not, of necessity, really and truly infinite. It only looks so from one angle - apparently, the angle of measurement, rather than the angle of existence. Therefore, time must be equal to 1, not to infinity, even if we seem to be able to chop it into 1/∞.

Can they occur over arbitrarily small time intervals?

Apparently, they must.

Sorites’s paradox demonstrates the difficulty of giving a value to space. (I figure it could just as easily be applied to time; perhaps it is a cousin to Zeno’s paradox.) As with Achilles and his tortoise, it seems the applications of, for example, fuzzy logic could apply. There are definite points where Achilles has not and has passed the tortoise. Then there are points where Achilles could pass the tortoise (say, any length of space shorter than Achilles’s foot). (Of course, at what point Achilles actually did pass the tortoise is a matter of fact, whereas Achilles and his tortoise are merely hypothetical. So this is all conjecture, anyway.)

Take it all together, time and space, and it’s not difficult to believe infinitesimally small divisions of time may have finite, definite properties.

 

[Tomdstone]

Sure Tomdstone, but Chainbreaker wants an example where the infinity isn’t completed. So it was essential in my example that the runner never complete his overall task, yet still complete infinitely many events along the way without completing any “first” event.

Chainbreaker also claimed that the runner never reaches his goal.

Zeno is showing a person running half way a potentially infinite amount of times before reaching his goal and of course never does because he cannot transgress an infinite number.

Obviously, this is not true and the runner can reach his goal and beyond, even though he does transgress an infinite number of consecutively smaller intervals.

 

[John_Martin]

In a few days it will be 2015. We have just lived through 2014.
How many years, then has creation existed (per our perspective) in this year? 1 year.

If creation happened an infinite number of years (our measure of time, which should not matter, since any perspective has its metrics of a specific measure of time), then how many years has it been?

The past has happened. It is actual, an actual quantity of years - so how many?
2013 - 2 years
2012 - 3 years
2011 - 4 years
etc.

This is not a matter of infinitely diminishing distances between points, but of years, fixed distances, that have been traversed, by Zeno and both Achilles and the tortoise.