Why is infinite regression held to be impossible?

[Qoeleth]

Thanks for the interesting responses. One which struck me was the difference between an infinite progression (e.g. starting at one and counting to infinity) and an infinite regression (starting at infinity and counting back to one- which is impossible).

Now, I suppose the universe must be spatially infinite- otherwise there would be a ‘wall’ at some point marking the end of the universe, which would seem foolish (what would be on the other side of the wall?) Now, if causes can function in a spatial as well as a temporal medium (i.e. a cause can be synchronic with its effect), the spatial infinitude of the universe (if this is conceded) implies the possibility of an infinitude of synchronic causes?

 

[Qoeleth]

Fair enough. I will provide such a proof, without Hilbert’s help.

There are 3 types of infinities:
(1) The potential infinities (the infinity used in mathematics)
(2) The actual infinity (a real, physical infinity)
(3) The descriptive infinity (describing an attribute as unlimited)

The OP is giving us examples of the first type of infinities, mathematical relations (pi & 1/3) that are examples of potential infinity. He is then asking us why these cannot be extrapolated to an actual infinity. To measure something, you have to contain it in some fashion: with mass, it needs to fit on the scale; with length, it needs to fit between the ends of the meter-stick; with luminosity, the light rays need to fit on the bolometer; and so on. In order to measure infinity, we would need something bigger than infinity to contain it. Something bigger than infinity is a contradiction of the definition of infinity.

I will try to give an example of a physical infinity- the size of the universe.

 

[TheTrueCentrist]

Fair enough. I will provide such a proof, without Hilbert’s help.

There are 3 types of infinities:
(1) The potential infinities (the infinity used in mathematics)
(2) The actual infinity (a real, physical infinity)
(3) The descriptive infinity (describing an attribute as unlimited)

The OP is giving us examples of the first type of infinities, mathematical relations (pi & 1/3) that are examples of potential infinity. He is then asking us why these cannot be extrapolated to an actual infinity. To measure something, you have to contain it in some fashion: with mass, it needs to fit on the scale; with length, it needs to fit between the ends of the meter-stick; with luminosity, the light rays need to fit on the bolometer; and so on. In order to measure infinity, we would need something bigger than infinity to contain it. Something bigger than infinity is a contradiction of the definition of infinity.

But wouldn’t that only imply that we couldn’t completely measure a physical infinity? If there were an infinite length, for example, we could measure parts of the length, but not the whole. It does not imply that such an infinity does not exist.

Moreover, consider this drawing by Escher. Inside the circle, the angels and demons grow smaller as they approach the edge of the circle, until they become infinitely small at the boundary. If an angel at the center flew towards the edge, it would become smaller and smaller, never reaching the edge of the circle. Indeed, a “line” drawn across the diameter of the circle would be infinitely long according to the angels and demons inside the circle. We could measure that line. Does that mean that the line is not infinite? Depends on whether or not you are constrained by the rules of that circle.

 

[MarcoPolo]

The infinite regression requires nothing

In this statement, “infinite regression” itself is your cause/god.

 

[SonCatcher]

The creationist arguments and its variants rely on supposing that an infinite regression is impossible. I wonder, why precisely is such an infinite series held to be impossible.

If one places two mirrors facing each other, there is one mirror the image of the other, which contains its own image, which contains the image of the other, which contains its own image, et cetera ad infinitum.

If one divides 10 by 3, the answer is 3.3333333333333, etc. ad infinitum.

The value of pi, is held to be 3.14159, etc., etc.

If become aware that I think, I think that I think, and I think that I think that I think, etc. ad infinitum.

Are each of these not infinite regressions?

Nope. They are numbers. One is rational, the other is not.

 

[WesleyF]

His objection to Aquinas is really stated in only one sentence: “it is also true that there yet remains an infinite amount of time prior to any given day in an infinite past that must have passed before that day.”

Notice the bolded word. As Aquinas states in the text the author is objecting to, “[p]assage is always understood as being from term to term.” So, in claiming that an infinite amount of time must have “passed” before any given day is reached, the author is committed to saying there is an extreme infinitely before any given day. That’s obviously false though, if time regresses infinitely, there is no first extreme.

“Infinity” is just a word and no one knows what it means. To say that there is an extreme infinitely before any given day is to say that there is no extreme. So yes, his is right that there is no first extreme. So it’s not at all odd that he admits it in the next sentences.

Oddly, he admits this only a few sentences later:

“the first extreme (the present day being the second extreme) is missing, so there can be no passage to the present in the first place.”

So, it seems he’s committed himself to a contradiction: in an infinite past, there is and there is not an extreme infinitely before any given day. The former claim is what his objection amounts to, and the latter is a statement of his I cited.

We know for a fact that days prior to this day have indeed passed. Yesterday passed, the day before yesterday passed, and so on…But, if passage is from term to term, then the other extreme must be achievable. Therefore, the universe must have a beginning. But if the universe has no beginning, then we are unable to find the first term and that leads to a contradiction.

Aquinas misunderstands the argument when he says that the premise is based on an infinite number of mean terms between two extremes. That’s not the argument. The argument is that since we know for a fact that days have passed, we therefore should be able to trace the first extreme. If it cannot be traced, then we have a contradiction.

So if the universe had no beginning, then from the current day, we can keep counting one day backwards, but we will never reach the other extreme even though we know that the days have passed. That’s the contradiction.

 

[Alterum]

Sure, I agree with that, why do you think I was misreading Aquinas? My remarks don’t seem contrary to any of that. With the BVG Theorem, it’s controversial on what theory of time ‘beginning’ should be understood. If we work with B-theory, it’s a beginning like the first line of a segment is a beginning: doesn’t require a cause.

Actually, it was the guy you quoted who was misreading Aquinas, and also the author he quoted. (Sorry about that.) I meant to direct this at him.

 

[Perplexity]

WesleyF:

Your argument begs the question. It assumes that time could only have made passage to the present if there was a first term from which time made passage to the present.

This is how you’re able to say that if time regressed infinitely, an infinite amount of time must have been traversed to the present: you’re assuming there must be a first term. Of course we’ll get an absurdity from an infinitely regressing past if we assume this. But, it’s precisely this assumption that you’re being called to demonstrate; because we couldn’t make that assumption if the past did regress infinitely.

 

[ivebeenshown]

WesleyF:

Your argument begs the question. It assumes that time could only have made passage to the present if there was a first term from which time made passage to the present.

This is how you’re able to say that if time regressed infinitely, an infinite amount of time must have been traversed to the present: you’re assuming there must be a first term. Of course we’ll get an absurdity from an infinitely regressing past if we assume this. But, it’s precisely this assumption that you’re being called to demonstrate; because we couldn’t make that assumption if the past did regress infinitely.

The ‘first term’ would be infinite and thus could never have passed. Of course there must be a first term, because we know that there is a past! The ‘first term’ is the past itself!

 

[Alberti_Devoveo]

I will try to give an example of a physical infinity- the size of the universe.

The size of the physical universe is actually quite finite. The size of the potential universe (i.e., what the universe is expanding into) is also very much unknown.

 

[TheTrueCentrist]

The ‘first term’ would be infinite and thus could never have passed. Of course there must be a first term, because we know that there is a past! The ‘first term’ is the past itself!

So then the ‘first term’ is infinitely long. Is that a problem?

 

[Alberti_Devoveo]

But wouldn’t that only imply that we couldn’t completely measure a physical infinity? If there were an infinite length, for example, we could measure parts of the length, but not the whole. It does not imply that such an infinity does not exist.

Supposing your objection is true, we would never know if it were infinite in length or not. We would need an infinite amount of time to measure the entire infinite line, segment by segment. Still the same problem occurs: you need to be bigger than infinity to measure it.
What I failed to do, due to my being in a hurry to catch the bus, was extend this to the universe. The observable universe is of known, finite size (on the order of 28 billion parsecs in diameter) with a finite number of atoms (on the order of 10[sup]80[/sup]). You cannot get infinity from a finite value. Thus, a physical infinity does not exist.

Moreover, consider this drawing by Escher.

This is already starting to go bad. Citing a drawing by a guy who is famous for making physically impossible three-dimensional scenarios by drawing it in two dimensions as an example for infinity? Hardly plausible. Escher was able to do one thing very well: make optical illusions.

Inside the circle, the angels and demons grow smaller as they approach the edge of the circle, until they become infinitely small at the boundary. If an angel at the center flew towards the edge, it would become smaller and smaller, never reaching the edge of the circle. Indeed, a “line” drawn across the diameter of the circle would be infinitely long according to the angels and demons inside the circle. We could measure that line. Does that mean that the line is not infinite? Depends on whether or not you are constrained by the rules of that circle.

I am astounded that you are even considering this as a “truth.” The image is finite in size, and nothing tends to infinity here. The smaller & smaller angels & demons can only shrink down to the size of the tip of the pencil point, which is several hundred thousand atoms big (definitely not infinitely small). The line is most certainly not infinite because it can be contained in the circle, which is measurable because it is of finite size.

 

[TheTrueCentrist]

Supposing your objection is true, we would never know if it were infinite in length or not. We would need an infinite amount of time to measure the entire infinite line, segment by segment. Still the same problem occurs: you need to be bigger than infinity to measure it.

Indeed, we couldn’t be sure. In fact, natural limits like the speed of light would make its measurement completely impossible. However, that does not mean an infinite length is impossible, only that we couldn’t be certain it was actually infinite.

What I failed to do, due to my being in a hurry to catch the bus, was extend this to the universe. The observable universe is of known, finite size (on the order of 28 billion parsecs in diameter) with a finite number of atoms (on the order of 10[sup]80[/sup]). You cannot get infinity from a finite value. Thus, a physical infinity does not exist.

It is not impossible for things to exist beyond the observable universe. Note the name: “observable universe,” it implies it is the subset of the universe which is observable. There could be infinite dimensions just passing through our little bubble of observation.

I am astounded that you are even considering this as a “truth.” The image is finite in size, and nothing tends to infinity here. The smaller & smaller angels & demons can only shrink down to the size of the tip of the pencil point, which is several hundred thousand atoms big (definitely not infinitely small). The line is most certainly not infinite because it can be contained in the circle, which is measurable because it is of finite size.

Your refusal to consider the situation does not invalidate my point; Escher’s drawing is just an illustration. I am saying that our entire universe could be a sphere 1 meter in diameter with rules similar to those in Escher’s circle. However, since we are inside and subject to its rules, we would find it to be infinite. We could travel at light speed forever and never reach the edge.

 

[ivebeenshown]

So then the ‘first term’ is infinitely long. Is that a problem?

Yes, it is. An infinitely long past could never have passed because it was infinite in length.

 

[TheTrueCentrist]

Yes, it is. An infinitely long past could never have passed because it was infinite in length.

There are no two events that are separated by an infinite amount of time. Think of the timeline like the set of real numbers, ℝ. You can choose any two numbers in that set and there will *always *be a finite difference between them. Therefore, when you say “an infinite amount of time must have passed” I ask “since what?” and as soon as an answer is provided, the time that has passed is no longer infinite.

However, since there is no beginning to time (represented by ℝ,) if you chose any time, t, then the length of time less than t, (-∞, t) is always infinite regardless of what value of t you select (note that -∞∉ℝ) You say that this is impossible, but if we actually do have an infinite timeline, it is simply a natural property of that timeline.

In conclusion, there are no infinite differences in time, and there is always an infinite amount of time less than any particular time.

 

[Alberti_Devoveo]

Indeed, we couldn’t be sure. In fact, natural limits like the speed of light would make its measurement completely impossible. However, that does not mean an infinite length is impossible, only that we couldn’t be certain it was actually infinite.

It is not impossible for things to exist beyond the observable universe. Note the name: “observable universe,” it implies it is the subset of the universe which is observable. There could be infinite dimensions just passing through our little bubble of observation.

For sure, the number of atoms in the universe is finite at 10[sup]80[/sup]. You cannot get an infinite value of anything from a finite value. No object surely can be of infinite dimension (mass, length, etc).
As to the non-observable universe: that is only a potential infinity, and not an actual infinity. We have no idea as to the extent of the reaches of the entire universe, and likely we never will. Your statement of “there could be infinite dimensions…” is completely hypothetical and not an assertion that it is, physically, the case.

Thus, you are right back where I started from: an actual infinity is impossible because our universe is of finite dimension; a hypothetical infinity is permissible.

Your refusal to consider the situation does not invalidate my point; Escher’s drawing is just an illustration. I am saying that our entire universe could be a sphere 1 meter in diameter with rules similar to those in Escher’s circle. However, since we are inside and subject to its rules, we would find it to be infinite. We could travel at light speed forever and never reach the edge.

Supposedly, one way to think of our universe is to consider us, as three-dimensional beings, existing on the surface of a balloon (despite the surface being two-dimensional). There is no boundary, but it is of finite size. Who is to say that, according to Escher’s drawing, that if an angel were to travel from the center towards 30[sup]o[/sup] above the right side of the horizontal, it would not come out at 30[sup]o[/sup] below the left side horizontal? In that case, it is more like the finite-size but boundless balloon I mentioned previously. And, in that case, there still is no infinity.

 

[Cristiano]

For sure, the number of atoms in the universe is finite at 10[sup]80[/sup]. You cannot get an infinite value of anything from a finite value. No object surely can be of infinite dimension (mass, length, etc).
As to the non-observable universe: that is only a potential infinity, and not an actual infinity. We have no idea as to the extent of the reaches of the entire universe, and likely we never will. Your statement of “there could be infinite dimensions…” is completely hypothetical and not an assertion that it is, physically, the case.

Thus, you are right back where I started from: an actual infinity is impossible because our universe is of finite dimension; a hypothetical infinity is permissible.

Supposedly, one way to think of our universe is to consider us, as three-dimensional beings, existing on the surface of a balloon (despite the surface being two-dimensional). There is no boundary, but it is of finite size. Who is to say that, according to Escher’s drawing, that if an angel were to travel from the center towards 30[sup]o[/sup] above the right side of the horizontal, it would not come out at 30[sup]o[/sup] below the left side horizontal? In that case, it is more like the finite-size but boundless balloon I mentioned previously. And, in that case, there still is no infinity.

I am sorry but as I understand it this argument has some gaps. Your point about the limited mass in the observable universe and your conclusions about the lack of actual infinity in the observable universe make sense to me.
When you talk about the non-observable universe you talk about a potential infinity and that it denies an actual infinity. I think that your logic misses a step here. I think that the potential is not just in the infinity but also in our knowledge of such infinity. A thing can be potential and not actual, or we can either have or not have the potential to know something that is actual. If we have the potential to know that something is actual that does not imply that our knowledge will be actual.

Finally we might still have the impression that the non-observable universe is infinite while in reality it is finite.

 

[JDaniel]

The creationist arguments and its variants rely on supposing that an infinite regression is impossible. I wonder, why precisely is such an infinite series held to be impossible.

If one places two mirrors facing each other, there is one mirror the image of the other, which contains its own image, which contains the image of the other, which contains its own image, et cetera ad infinitum.

If one divides 10 by 3, the answer is 3.3333333333333, etc. ad infinitum.

The value of pi, is held to be 3.14159, etc., etc.

If become aware that I think, I think that I think, and I think that I think that I think, etc. ad infinitum.

Are each of these not infinite regressions?

If the universe itself contains these infinite regressions, why should there not be an infinite regression of causes?

Qoeleth:

It is important to understand the difference between actuality, which is, Reality, and that which is a reification, that is, an abstraction. Mathematics is a second form of abstraction, called, understandably, “mathematical abstraction” What is an abstraction? If you think it through, you discover that it is a hypothetical, it is imaginary. The hypothetical or imaginary is not - I repeat: NOT - actual, not reality.

Secondly, it is important to understand that an actual infinity is NOT a “set.” The mathematician can hypothesize it to be a set, but that should quickly dispel any rumor that it is actual.

Thirdly, infinity is un-actualizable. Think throughly about this word very carefully. Humans cannot positively define ‘infinity.’ All we can do is approach a vague idea of it from considering what it is not. For example, it is not an actualizable magnitude, or sequence of finite things or finite events. To think that it is (or, even could be) is to seriously misunderstand what ‘infinity’ conveys. It conveys the "unactualizable."

Fourthly, the only possible Infinity is that which transcends the limits of our understanding and finite structures. In other words, the only possible Infinity is God.

Fifthly, even a consideration of reductio ad infinitum is a hypothetical reduction. It can only exist as a vagary in our minds. There is a limit to the smallest size of a thing, or particle of physical matter, below which whatever might be said to be there would have zero dimensionality. That point below which no further dimensionality exists is Finitude, not Infinitude. And, in the other direction, consider the largest number you can conceive and one (unit), or more, can always be added to it, or subtracted from it. ALWAYS. There can be no FINAL number above which nothing more can be added, and from which nothing can be subtracted.

This leaves us with at least two distinctions between things we like to refer to as infinities.

(a) actual infinity, and
(b) potential infinity.

The first infinity cannot exist in physicality. Think this through: if it could, it would occupy absolutely every-thing, every-where, and every-when. If that were the case, there would be no time, as time requires motion/change and space, i.e., place within space. Infinity is boundless and boundary-less. The physical has boundaries. Matter has boundaries. Precisely because of such boundaries, motion/change can occur. Within such boundaries composites can occur - which may be self-moving, or mobile.

Even in your depiction of two mirrors, there comes a point at which the boundaries become irretrievably blurred. Just before that perfect blurring, at the outermost bondary of that perfect blurring, whatever is thought to be seen or measured, is Finite.

In mathematics, when certain equations are said to drop off to infinity, we are talking about the purely hypothetical, we are speaking of another sort of blurring. It cannot be seen or measured. For want of a better description, we say that at some point the calculation “dropped off to infinity.”

The second form of infinity can exist. It is that which is, or rather, could possibly/maybe/perhaps become infinite, hypothetically, abstractly. We’ll never see it, but, hypothetically, perhaps it might - if left to grow (by addition to it) long enough - become at some hitherto unactualized time in the future, infinite. As I said, not only will we never see it, neither will any of our ancestors see it. It will always remain only potentially infinite so that at any stopping point, it is merely finite - albeit a rather large finite structure.

Think it through a bit more.

God bless,
jd

 

[Alberti_Devoveo]

I am sorry but as I understand it this argument has some gaps. Your point about the limited mass in the observable universe and your conclusions about the lack of actual infinity in the observable universe make sense to me.

I am glad they make sense.

When you talk about the non-observable universe you talk about a potential infinity and that it denies an actual infinity. I think that your logic misses a step here. I think that the potential is not just in the infinity but also in our knowledge of such infinity. A thing can be potential and not actual, or we can either have or not have the potential to know something that is actual. If we have the potential to know that something is actual that does not imply that our knowledge will be actual.

There is no missed step here. While a potential infinity does not deny an actual infinity, we cannot conclude that the non-observable universe is actually infinite for one very critical reason: we cannot actually observe the non-observable universe (hence its name)! Thus, the non-observable universe is only a potential infinity, and not an actual infinity.

Finally we might still have the impression that the non-observable universe is infinite while in reality it is finite.

Of course! This is the reason why the non-observable universe is only a potential infinity, and not an actual infinity.

 

[ivebeenshown]

There are no two events that are separated by an infinite amount of time. Think of the timeline like the set of real numbers, ℝ. You can choose any two numbers in that set and there will *always *be a finite difference between them.

I understand and accept this premise.

Therefore, when you say “an infinite amount of time must have passed” I ask “since what?” and as soon as an answer is provided, the time that has passed is no longer infinite.

In this instance, there is no need to ask ‘since what’?

However, since there is no beginning to time (represented by ℝ,) if you chose any time, t, then the length of time less than t, (-∞, t) is always infinite regardless of what value of t you select (note that -∞∉ℝ) You say that this is impossible, but if we actually do have an infinite timeline, it is simply a natural property of that timeline.

In conclusion, there are no infinite differences in time, and there is always an infinite amount of time less than any particular time.

One natural property of a timeline is that each moment occurs after the other. One moment cannot pass until its preceding moments have passed. If an infinite amount of moments preceded a particular moment, the particular moment could never occur!