## Tuesday, May 02, 2006

### On Traversing an Actually Infinite Past

In comments on an earlier thread, both Ocham and Brandon express perplexity over my defense of the impossibility of traversing an actually infinite past. Before responding, I must first elaborate a bit on the potential / actual infinite distinction. The distinction goes back to Aristotle and is pivotal in his response to Zeno's paradoxes. Basically, what it amounts to is this.

A potential infinite is a collection or quantity that can increase without bound but is always actually finite. For example, I can begin counting 1, 2, 3, ... and keep going, but I'll never reach an 'infinitieth' number. At every point I am at some finite number. Similarly, I can take a line segment and divide it, thus yielding two line segments. I can then divide it again (3), and again (4), and so forth. At every point what I have is an actually finite number of line segments. The notion of a potential infinite is the one involved in the mathematical notion of a limit. Thus, we say, for example, the limit of 1/x as x approaches infinity (i.e., increases without bound) is zero. We don't evaluate the limit by setting x equal to infinity - 1 over infinity is undefined - but by imagining the value of x getting larger and larger, indefinitely. In sum then a potentially infinite collection or quantity is indefinitely large, but always actually finite.

An actual infinite (of a given order of magnitude), by contrast, is a collection or quantity that has a proper subset of equal magnitude as the parent set. Since Cantor, the standard way to determine whether two (denumerable) sets are equal in magnitude is to place them in 1-1 correspondence with each other. Thus, the set of natural numbers {1, 2, 3, ...} is an actually infinite set because it has a proper subset (a set containing only, but not all, members of the parent set) that can be put into 1-1 correspondence with it. For example, the set of even numbers {2, 4, 6, ...} is a proper subset of the set of natural numbers, and both sets can be mapped on to each other without remainder. Thus, 1 maps onto 2, 2 onto 4, 3 onto 6, etc. The important point to note here is that an actually infinite collection is given all at once - it doesn't start out as a finite collection and then, by finite addition, become actually infinite. No, we posit actually infinite collections as infinite from the get-go. Thus, we refer to the set of natural numbers, or the set of prime numbers, and so forth, as completed totalities.

Now, one way of developing the kalam cosmological argument is as follows:
1. If there is no beginning to time (a first moment), then an actually infinite number of events has elapsed prior to now.
2. It is impossible for an actually infinite number of events to elapse.
3. Therefore, there is a beginning to time.
The rationale for the first premise is that no beginning to time means that before every state of world is another, different state. If only a finite number of events has elapsed, then there would be a state without a prior state. There would be a first state of the world, and a first event. By hypothesis, however, there is no first event; hence, the number of elapsed events must be infinite, not finite.

The rationale for the second premise is simply the idea that one can't turn a potentially infinite magnitude into an actually infinite magnitude by finite addition. This seems obvious: I can keep counting till the cows come home and beyond, count faster and faster and faster, but I'll never arrive at an infinitieth number. At every point in the process, the quantity is finite.

Now, Brandon objects:
If every traversal requires a beginning and an end, and an infinite past has no beginning, this is a problem only if we already assume that traversal of an infinite past would require traversal of infinite days. But on the infinite past view, every day in the past is finitely distant from the present; it's just that for every finitely distant day there's a day that is more distant. Thus this is true: For every day in the past, traversal of the days from that day to today is traversal of a finite number of days. The fact that there are infinite such days doesn't change this. This is true just as much as it is true that the fact that every integer is a finite distant from 1 is not affected by the fact that there are infinite integers.
The crucial assumption of this argument is this:

(A) Every day in an infinite past is finitely distant from the present.

The problem with this assumption is that it conflates the distinction between potential and actual infinites and thereby fails to take the idea of an actually infinite past seriously. (A) is clearly true when we're talking about a potential infinite. We start at the present and run through the time series in reverse, moving farther and farther into the past. Nevertheless, at any point we stop at, we're only a finite remove from the present. But if the distance from past event E to the present is actually finite, then we haven't yet captured the idea of an actually infinite past.

Similarly, the notion of ever larger integers being still a finite remove from 1 is that of a potential infinite, of a magnitude increasing without bound, not of an actual infinite.

The problem with an actually infinite past is that it requires us to posit the impossible, some event in the past that is at an actually infinite remove from the present. The proper analogy is not one of starting in the present and then receding ever further into the past, but of something like trying to count up from negative infinity to zero. It can't be done. And that's the point.

At 5/03/2006 1:12 AM, Ocham said...

I don't follow this at all. You say "The problem with an actually infinite past is that it requires us to posit the impossible, some event in the past that is at an actually infinite remove from the present." Why is this impossible? I posit

(*) There is a time t such that, for all n in the past, there is an m before n, but after t.

(Where the variables range over discrete time intervals such as days, years whatever). There is absolutely no contradiction in this, indeed, it is a standard set-theoretical formulation.

You follow this up with the argument "The proper analogy is not one of starting in the present and then receding ever further into the past, but of something like trying to count up from negative infinity to zero. It can't be done."

Of course it can't. That's follows from definition of t in (*). That's what it is saying. Where on earth is the contradiction or impossibility?

You say "There can't be a point in the past which is infinitely distant, because we could not get from that point to here". But the definition of "infinitely distant" is precisely of a point as defined above, which you can't get from by any number of steps, it being a kind of limit. That's how we define infinity.

Cantor says that the problem with all finitist proofs is that "from the outset they expect or even impose all the properties of finite numbers upon the numbers in question, while on the other hand the infinite numbers, if they are to be considered in any form at all, must (in their contrast to the finite numbers) constitute an entirely new kind of number, whose nature is entirely dependent upon the nature of things and is an object of research, but not of our arbitrariness or prejudices. "

And this is exactly what has happened here. You say there can't be a point that is infinitely distant, because you couldn't "get from" there to here. But you are imposing a property of finitude (of "getting from" there to here) on the very point in question.

It's as though you said there can't be an infinite number, because all numbers can be reached by counting. But an infinite number is precisely such as you can't reach by counting. "There are no infinite numbers, because all numbers are finite". Petitio.

At 5/03/2006 9:49 AM, Brandon said...

I've put up a response to this on my own weblog. In summary the point is that you are right that the integers constitute a potential infinite, but this does not mean that the integers are finite. Far from it; it means that the integers are all finitely distant from each other, but are infinite as a whole. The distinction between potential and actual infinite isn't between the merely indefinite finite and the actually infinite; it's between an actually infinite set of potential things and an actually infinite set of actual things.

And this is why one can't reject Aquinas's response to the traversal argument simply on the basis of the infinity involved, because his point is that on the infinite past view, the days are actually infinite but do not constitute an actual infinite. Or in other words: on such a view, the past is not a traversed infinite distance, but infinite traversals of finite distances. Any number of these finite distances can be added together, but although there are infinitely many of them, they will never add to an infinite distance, but only to ever larger finite distances. Therefore in an infinite past there is no infinite to be traversed -- unless we have some reason, beyond the mere fact of infinity, that shows that an actually infinite set of finite traversals somehow constitute the traversal of an actual infinite.

At 5/03/2006 12:51 PM, Don Jr. said...

I agree with Alan here. Personally, I have never seen the response given by (I guess) Aquinas as convincing at all, neither at a superficial level nor upon deeper analysis. At the superficial level it posits, as Brandon says, that "in an infinite past there is no infinite to be traversed" which, no offense to Brandon, seems to be obviously bogus, indeed, flat out contradictory.

Brandon, you say, "For every day in the past, traversal of the days from that day to today is traversal of a finite number of days." I agree. You add: "The fact that there are infinite such days doesn't change this." Again, I agree. But no one is arguing that. No one is saying that you can't traverse a finite segment of an infinite. The argument is that you can't traverse an infinite, not part of it. You seem to be saying, though, that because one can traverse a part, one can traverse the whole. But that is fallacious reasoning.

Take an infinitely long course, that is, an endless course. Sure, one can traverse from meter 1 to meter 2 (that is, traverse a part), but that doesn't mean one can ever reach the end (that is, traverse the whole). Indeed, the course is endless, so necessarily one cannot reach the end because no end exists for one to reach.

The reverse is true for an infinite past, or something extending infinitely backwards rather than forwards. In that case the course, while it may have an end, is beginningless (that is, lacks a beginning). While you may not see that as an issue, it is an issue, an important one at that. For one can't even begin to traverse a beginningless series; it's not at all possible. Where can one start? In the middle? Then that's not the beginning; and if one begins there—that is, in the middle (if a middle can exist)—then one has not traversed the entire (infinite) series. What about the beginning? But it's beginningless, so one can't possibly begin there since it doesn't exist. Moreover, to say one can begin to traverse a beginningless series is a contradiction in terms (like immaterial matter).

Back to the finite traversal argument. One might say that an infinite past is composed of finite parts. Okay. (It's also, paradoxically, composed of infinite parts, but never mind that.) Then one might argue that the finite parts can be traverse. Okay. Thus, one concludes, that there is no problem (since the parts can be traversed). But in an infinite past, or any infinite for that matter, there's an infinite number of finite parts. So the problem remains. The above conclusion is like saying because I can finish bushing my teeth once I can finish doing it an infinite number of times.

Lastly, the number of events traversed—and it should be obvious that in an infinite past there will be an infinite number of events to traverse—is not a function of the direction one takes in traversing them. So any argument (pro or con) involving an actual infinite past would also apply to an actual (not potential) infinite future. If one takes your (or Aquinas's) argument, Brandon, and applies it to the future, one will see that, as Alan explained in his entry, "it conflates the distinction between potential and actual infinites." The crux of your argument in reverse would be: "For every day in the future, traversal of the days from that day to today is traversal of a finite number of days." Thus, when applied to the future it becomes increasingly apparent that Alan's criticisms are correct.

At 5/03/2006 1:03 PM, Ocham said...

Agree with B. And as I've said, there is a further confusion between the existence of the limit ordinal that A. calls "minus infinity" - note the sudden introduction of this term into the argument without any introduction.

Note that set theorists avoid the terms 'potential infinity' and 'actual infinity' as metaphysically laden and vague. If they have any correlate in set theory, it is between ZF-inf (potential) and full ZF (actual).

At 5/03/2006 1:07 PM, Don Jr. said...

Brandon, in your last comment you say, "the [infinite] past is not a traversed infinite distance, but infinite traversals of finite distances." Can't you see that that's just a semantic shift of the problem and not a solution? You're not claiming that it's infinite traversals of the same finite distances are you (even though that wouldn't matter)? Add up an infinite about of (equal) finite distances and what do you get? (If you're going to argue that the distances not be equal and that they can then sum to a finite about, then the past would be finite in that case, so then you wouldn't even be dealing with the issue at hand, namely, an infinite past.) Nevertheless, backwards or forwards, you can't complete an infinite number of traversals. So the problem still remains.

You also say, "Any number of these finite distances can be added together, but although there are infinitely many of them, they will never add to an infinite distance, but only to ever larger finite distances." If you read it carefully this comment alone should indicate to you that you are conflating potential and actual infinites.

At 5/03/2006 5:06 PM, Brandon said...

This will have to be a quick comment; commenting quickly on the topic of infinity is a bit dangerous, but here goes.

Don, Far from being a mere semantic shift, it's a very serious difference, one that is the key to understanding infinity at all.

The integers are infinite. But every integer is a finite integer -- there is no such thing as an infinite integer. Thus, there is all the world between saying that the integers are infinite and saying that there is an infinite integer. So it is with distances between days. If there is an infinite past, there are infinitely many finite distances between days that would have to be traversed, because there are infinitely many days. But this does not entail that there is any infinite distance between days that would have to be traversed, because there need be no day at infinity from which we have to count down to the present day. Rather, every day is finitely distant from the present; so there is no traversal of an infinite.

You bring up the point of adding; but I think you jump to a conclusion that would need more defending than you give it. For when we add up successive finite distances between days (e.g., the distance between yesterday and last week, and the distance from the week before last to that day, etc.), we always only get finite numbers -- it's just that for any finite number we get there is more to add to it. And this is not surprising; for the claim that there is an infinite past amounts to the claim that past days are such that for any n you choose there are at least n+1 days. This is not a conflation of potential and actual infinites; it's a potential infinite, i.e., an additive infinity of potential operations. That doesn't change the fact that it is actually infinite (see my post, linked to above).

On the assumption that the infinite cannot be traversed, it only follows that infinite traversals of finite distances can't be traversed if there is a traversal of the infinite traversals. But, again, there is no reason to hold this, any more than to hold that there is a traversal of infinite days. For this is the mistake that keeps being made: The claim that the past is infinite does not mean it has a beginning point at infinity, and we have counted down from there to 0. On the contrary, all it has to mean is that there is no beginning point, end of story, and from this it follows that there is no actual infinite to be traversed, only actual finites; but those actual finites taken together have a certain property, namely, that for any n of them there is at least n+1. This is why the past is called infinite. Your criticisms in the first comment only work if the sum of the finite distances reach back to an infinite starting point -- or, to put it in other terms, if they add up to an infinite number from which we have to count down to get to the present. But just as saying the natural numbers are infinite does not imply that there is an infinite number in the series of natural numbers, so saying that the past is infinite does not imply that there is an infinite number in the series of days in the past. Infinity is not part of the series from which we are counting down at all. Because it is not part of the series it is never traversed; the only things that are actually there to be traversed are the arbitrarily large finite distances in time. There need be no traversal of an infinite in a infinite past.

At 5/04/2006 6:34 AM, Tom said...

Integers are "infinitely many." But any one integer is "finite in number." That integers are infinitely many doesn't make counting at all impossible. I CAN count (1, 2, 3, 4...there), so I can traverse some integers. What I can't do FINISH counting all the infinitely many integers. This impossibility is what's being argued.

Tom

At 5/04/2006 9:52 AM, Don Jr. said...

Brandon, there is such thing as an infinite integer (see here, here, and here).

Nevertheless, I'll accept your statement that "the [infinite] past is not a traversed infinite distance, but infinite traversals of finite distances." (Note, though, that you're saying there is an infinite amount of traversals while others are merely saying there is an infinite amount of distance. The difference is semantical; it amounts to the same thing. Whether one has to traverse an infinite amount of distance or complete an infinite amount of traversals, the problem remains; linguistic modifications don't resolve it.) So you maintain that it is possible to complete an infinite amount of tasks, namely, traversals? Very well. How long does it take to complete these traversals? (I would really like to know your response to this.)

Assuming that the past is infinite, it follows, using your preferred language, Brandon, that we've completed an infinite amount of traversals (of finite distances). Okay. If we've completed them, if we've traversed the beginningless past, exactly when or how did we begin? I anticipate that you would say "We never began," or something similar (but please correct if I'm wrong on that). Okay. So we never began completing the infinite amount of traversals, but we did complete them? Is this correct?

Do you realize (honestly, I would really like to know) that your argument, if successful, would have to work for the future as well? That would mean that it would be possible to complete an infinite amount of traversals in the future (in addition to saying we've already completed an infinite amount of traversals in the past).

At 5/04/2006 11:41 PM, Ocham said...

> So you maintain that it is possible to complete an infinite amount of tasks, namely, traversals? >

Brandon is not maintaining this. There can be infinitely many finite traversals, without there being a single infinite traversal. Just as, in set theory, there can be infinitely many finite sets without there being a single infinite set.

This is key.

At 5/05/2006 12:54 PM, Don Jr. said...

Ocham, you're allowing yourself to get bogged down in semantics. Nevertheless, I'll just reword my statement to suit your (semantic) preferences.

Reworded Version: So you maintain that it is possible to complete "infinitely many finite traversals"? Very Well. How long does it take to complete these "infinitely many" traversals?

(Ocham, if you wouldn't mind responding to my other two paragraphs in my last post, in addition to the first paragraph, I would greatly appreciate it.)

At 5/05/2006 1:19 PM, Kevin Winters said...

A friend of mind, who is in mathematics, once made me repeat with her the following mantra: "Infinity is a concept, not a number." Thus, to refer to a point in time that is 'infinitely' long ago is to commit a category mistake: conflating a concept with a number, or treating the concept like a number. As there is no 'infinitieth' point of time, it is incoherent to make claims about it in terms of "traversing."

This also raises another (related) issue: if there is no 'starting point,' given an infinite past, and there is no such thing as an 'infinitieth' point in the past, then all discussion of "traversing an acutally infinite past" is incoherent; you do not traverse the infinity because the infinity is not a number, a 'starting' point, or a 'span of time.'

At 5/05/2006 1:25 PM, Don Jr. said...

Ocham, your "infinitely many finite traversals," which you approved of, is identical to my "infinite amount of tasks," which you criticized. The "tasks" in my statement referred to the "traversals of finite distances" (as Brandon put it). You (and Brandon), as you say, maintain that there are "infinitely many" finite tasks. In the statement you criticized I said there was an "infinite amount of" finite tasks. You're carping because I used "infinite amount of" rather than "infinitely many" and acting as if that has some ultimately significant relevance. Not only does it not have ultimately significant relevance to the current discussion, it has no relevance whatsoever. (Note that in my arguments I'm not objecting to the finite tasks but to the greater task of completing the "infinitely many" finite tasks.)

The only reason I am bringing this up again—and hopefully for the last time—is because I absolutely detest wasting half of a discussion on something completely irrelevant (in every possible way) to the discussion. Brandon wants "infinite amount of traversals" rather than "infinite amount of distance"; you want "infinitely many" rather than "infinite amount of." No offense, but this is beginning to get ridiculous. If you have a semantic preference of phraseology here, please list what is acceptable and what is not up front. I will kindly agree to use your phraseology. Forgive my tone here, but I just don't want to debate the use of "infinitely many" versus "infinite amount of" (or anything similar to that) as if it's of galactic importance; it's a waste of time, for everybody.

At 5/06/2006 2:26 AM, Ocham said...

Kw said
>>>
if there is no 'starting point,' given an infinite past, and there is no such thing as an 'infinitieth' point in the past, then all discussion of "traversing an acutally infinite past" is incoherent; you do not traverse the infinity because the infinity is not a number, a 'starting' point, or a 'span of time.'
>>>

Correct.

Dj
>>
You're carping because I used "infinite amount of" rather than "infinitely many"

No, because you used the word 'complete'. On the assumption that every completion implies a beginning as well as an end, why should the infinity of past times imply a completion? If every past period in time is a finite distance from the present, and if for every period there is an earlier period, it follows (a) there are infinitely many periods (b) there is no beginning period, hence no completion of an infinite period.

Perhaps your objection is that any period of time must have a beginning, hence an infinite period must have a beginning. In that case, the world must be finitely old. But you need to make this assumption explicit. And there is an objection to this: suppose the world is finitely old. Then can't we suppose there was an earlier point, before it began? Even if there wasn't, there is no contradiction in supposing this. Ergo, no contradiction in an infinite past.

At 5/06/2006 1:20 PM, Don Jr. said...

Ocham, if you wouldn't mind responding to the other two paragraphs in my previous post I would appreciate it.

Ocham, an actual infinite is a completed whole. If you're maintaining—and it is apparent that you are—that an "infinite past" would not constitute or entail a completed whole then, as Alan has indicated, you're conflating the concept of potential infinite with actual infinite.

If time ended today, Ocham, would not the concept of an infinite past entail that there had been "infinitely many periods" in time, as you say, or "infinitely many finite traversals"? Would that not mean that "infinitely many periods" in time or "infinitely many finite traversals" had been completed? If not, then what would it entail?

Looking back on some of the comments, this is another reason why the concept of an infinite past, even on the surface, seems ridiculous to me: "[O]n the infinite past view, the days are actually infinite but do not constitute an actual infinite." I see that in defense of this statement Brandon says, on his website, "[F]or instance, the potential divisions of a line segment are actually infinite." It should be obvious that this is, again, merely a semantic shift. Does he mean that the divisions actually exist and thus constitute an actual infinite? No. He means that they (i.e., the divisions) potentially exist and constitute a potential infinite, but rather than say that (i.e., that the divisions are "potentially infinite") he says they are "potential divisions [which are] actually infinite" which really means they are "actually potentially infinite" which means they are "potentially infinite." (Does anyone else find all these semantic shifts tiresome?) He then goes on to wrongly conclude that his phrase "actually infinite" is some meaningful category which is separate from potential and actual infinites; he does this by ignoring that fact that he was saying potential divisions were actually infinite, which means the actual divisions are potentially infinite, not actually infinite. The two categories are mutually exclusive; adding the phrase "actually infinite" is of no use whatsoever and only complicates/digresses the discussion, as it clearly has.

It should be obvious, though, from the start that the label "actually infinite" is simply, in the vast majority of cases, another way of saying something is an "actual infinite" just like the label "potentially infinite" is another way of saying something is a "potential infinite." In rare, confused cases, one might be using "actually infinite" as Brandon used it to mean that something potential is "actually infinite," but then one ought to just drop the actual in that case, so as to avoid confusing the matter, or simply reword the sentence to use the term "potentially infinite" or "potential infinite."

At 5/06/2006 1:31 PM, Don Jr. said...

Ocham, are you meaning to maintain that infinite past time or simply time up till now, up to the present, never began and never completed?

At 5/06/2006 5:14 PM, Mark Butler said...

"The rationale for the second premise is simply the idea that one can't turn a potentially infinite magnitude into an actually infinite magnitude by finite addition."

Certainly one may not turn a finite interval into a infinite interval using by adding a finite number of finite intervals.

However, one may easily do it by adding either an infinite number of finite intervals or a finite number of infinite intervals.

Excluding the latter two possibilities requires excluding the concept of an actual infinity a priori, which begs the question.

At 5/07/2006 2:52 AM, Ocham said...

>>
Ocham, if you wouldn't mind responding to the other two paragraphs in my previous post I would appreciate it.

Sorry, I didn't respond because I didn't understand the paragraphs. You use the word 'completed' a lot, in a way I don't understand. As I use that word , I mean that a period of time is completed when there is a start and an end point. All the arguments in your paragraph, if read in this sense, are question-begging. I don't mean they actually are question-begging, I mean that, read in the sense I use the word 'completed', they are.

>>
Ocham, an actual infinite is a completed whole.

There you go again! What do you mean by 'completed whole'? If completion requires a start and end point, there cannot be a completed period, in that sense. If every period of time is preceded by an earlier one, there can be no completion *in the sense I understand completion*. If there is another sense of the word 'completion', you need to define it carefully.

>>
If you're maintaining—and it is apparent that you are—that an "infinite past" would not constitute or entail a completed whole then, as Alan has indicated, you're conflating the concept of potential infinite with actual infinite.

I don't know what Alan means by 'potential' infinite and 'actual' infinite, as he hasn't responded to any of my questions in this discussion. Remember that 'potential' and 'actual' are terms used by the old medieval logicians. They are not used in set theory, which is very precisely defined using mathematical logic. There was a key point in Alan's original argument where he introduced the term 'minus infinity', previously undefined and unintroduced. It seemed to me his whole argument hangs on the idea of 'minus infinity', but until he defines the term, we cannot even understand the argument.

I mean by a 'potential infinite', where time is concerned, that any period (day, year, whatever) is preceded by an earlier one, and that every period is a finite distance from the present. I mean by an 'actual infinite' that not every period is a finite distance from the present, such that at least one period is infinitely distant.

Neither a potential nor an actual, in this sense, can be traversed. A potential cannot, because nothing exists that is a starting point infinitely away (since, by definition, every point is finitely distant). An actual cannot, because of the 'limit' problem.

>>
Ocham, are you meaning to maintain that infinite past time or simply time up till now, up to the present, never began and never completed?

Not necessarily, I'm just saying there is no contradiction involved in the idea that time never began, as it were, and that every period is preceded by an earlier one, ad infinitum

At 5/07/2006 1:59 PM, Don Jr. said...

Correction: In my previous post where I said that actual infinite and potential infinite were mutually exclusive, I meant to say, or should have added, that they are mutually exclusive and collectively exhaustive (in regards to types of "infinites").

Brandon, in your post "Beginningless Pasts" you mention Quentin Smith's paper, "Infinity and the Past." I will briefly discuss an error I think Smith makes in Section I of that paper. There are three types of collections: finite, infinite, and indefinite. Infinite collections are actually infinite, not potentially infinite, in that they involve determined wholes. Indefinite collections are variable and so potentially infinite; they are, as their name indicates, not definite wholes. Finite collections are (obviously) definite wholes which are finite.

Set theory is, as one might guess, only concerned with sets, that is, collections having a definite number of members, and thus, by necessity, set theory can't incorporate the potentially infinite; sets and set theory only involve definite, or actual, infinities and finite collections. In fact, sets are collective wholes and thus by definition invariable (not variable) and definite (not indefinite).

Smith, in Section I of the aforementioned paper, applies set theory to the set of past events while attempting to maintain that the set of past events doesn't constitute an actual infinite. In doing so he contradicts himself. He cannot assign cardinality to the set of past events and then maintain that that set is not an actual infinite. The set is either finite or infinite. It is obviously not finite. If it is infinite it is either actually infinite or potentially infinite. (I am not clear on what class he is officially willing to put the set of past events though, since I couldn't find it specifically stated anywhere. He obviously denies that it is an actual infinite, and it would necessarily follow that it would have to be potentially infinite, but I don't find him explicitly making that claim anywhere. Though I could be wrong, he seems to think that there is some other class of infinite, when in fact there is not.) Nevertheless, by assigning cardinality to the set of past events he designates it, whether he says so or not, as being an actual infinite, for potential infinites, as said before, are indefinite in size and can't possibly be assigned any cardinality; moreover, set theory isn't at all applicable to potential infinites, since they're not sets.

As an aside, Smith dismisses Whitrow's argument on incorrect grounds. Take the following line of reasoning: If the (infinite) divisions within a line are potentially infinite it is because they don't actually, or really, exist. If there are actual, or really existent, (infinite) divisions within a line they then would constitute an actual infinite. The use(s) of actual here don't amount to any (fallacious) equivocation of the term, and the reasoning is commonly accepted as correct. This line of reasoning, though, portrays the same manner(s) in which Whitrow uses "actually" and "actual." (To avoid any mistaken analysis of Whitrow's argument, such as that Smith is guilty of, one can simply insert "really" or "really existent" in the places where Whitrow's usage of "actually" or "actual" is as such.)

To clarify what was said in the last paragraph: simply because one uses a term in two different manners within a single argument doesn't mean that one has equivocated the term. For example: If my new dog is like (similar to) my old dog then I will like (have an affinity towards) my new dog; my new dog is like my old dog; therefore, I will like my new dog. There is nothing unsound about that argument (my never owning a dog notwithstanding), and no equivocation has taken place.

At 5/07/2006 2:51 PM, Don Jr. said...

Mark: However, one may easily [create an actual infinite through succession] by adding either an infinite number of finite intervals or a finite number of infinite intervals.

Can one ever get the point where one can shout in approval, "Finally! I've created an actual infinite!" and then go home satisfied? Note, though, that the reason one can't form an actual infinite through addition is not due to lack of time but because the concept of an actual infinite precludes it (see below).

Mark, your "latter two possibilities" are excluded by the very concept of a potential infinite; it by nature cannot become an actual infinite. The two are conceptually distinct. What you're advocating would be like trying to turn the immaterial into the material; they're completely distinct concepts. At any given point a potential infinite will be an indefinite collection which continually increases. If one chooses to terminate the collection at any given point it will be finite, not infinite. You can't amend an actual infinite onto a potential infinite; that violates the rules of sets, since a potential infinite isn't a set. A man running represents a potential infinite since he can potentially run forever; his steps are an indefinite collection that increases continually. But he cannot possibly create, with his steps, an actual infinite. Aleph-zero (or "infinity" if you like), the number of steps the running man would have had to take if he were to complete an actual infinite through succession, has no immediate predecessor and so can't possibly be arrived at through addition. Put another way: the number of units in an infinite collection, or the cardinality of an actual infinite, minus 1 is what? Whatever it is it's definitely not a finite number. There is no way the man can go from "I have completed 5 steps" (a finite number) to "I have completed an infinite amount of steps."

At 5/07/2006 7:10 PM, Mark Butler said...

I believe the distinction between a potential infinite and an actual infinite to also be a device that prejudges the question.

On a time independent view of the question, potential infinites are actual infinites. Arguing that a potential infinite is not equivalent to an actual infinite is like arguing that the limit of n as n increases without bound is not infinite.

Stopping n at some particular value divests the question of all interest - the limit of n as n approaches some number k is k. However infinity is not a number, at best it is a class of numbers, so the substitution of some number (or time) k for infinity implicit in a Kalaam argument is illegitimate.

What is the basis for assuming that all metaphysically real intervals must be constructable from a finite number of finite intervals? Is there any foundation for that assertion other than some sort of metaphysical intuition?

At 5/07/2006 10:53 PM, Brandon said...

I've been out of town, so I'm catching up on some of the discussion; here are some comments, but if I've missed anything crucial in reviewing the above comments, I apologize in advance.

Tom said:
That integers are infinitely many doesn't make counting at all impossible. I CAN count (1, 2, 3, 4...there), so I can traverse some integers. What I can't do FINISH counting all the infinitely many integers. This impossibility is what's being argued.

The question, I suppose, is why anyone would have to 'finish' counting all the infinitely many past days any more than anyone would have to 'finish' counting all the integers. You can begin counting them: yesterday was 1, the day before was 2, and so forth. By definition, there is no finishing in that direction. Of course, going in the opposite direction (counting down to 0), one never has an absolute beginning, and that seems to be what's being picked as the problem in the case of an infinite past; but that's not a problem for the infinite past insofar as it is infinite -- the parallel with the integers proves that. This was (if I'm remembering correctly) Ocham's very good point, somewhere above, that the responses to the infinite past supposition are like saying there must be a highest positive integer or lowest negative integer because otherwise we can't count down to zero from it. Such a response would miss the whole point. So the question is this: Since the problem can't be a matter of infinity as such, what additional aspect of an infinity of past days is supposed to generate these alleged contradictions?

Don, Jr.:

(While Aleph-null is sometimes called the 'smallest infinite integer', it is my understanding that this is colloquial and not technical; aleph-null is not an integer but the cardinality of the set of integers. Thus, all integers are even or odd. Is aleph-null even or odd? It is neither, because it is not an integer. And so forth. But, as you say, it's not a major issue here.)

I wasn't maintaining it is possible to 'complete infinitely many finite traversals', since, like Ocham, I don't know what you mean by that. But you cannot argue, for reasons already noted, that the existence of infinitely many finite traversals implies the existence of an infinite traversal. Such an inference simply is simply not valid if the problem is supposed to be simply one of infinites; so in virtue of what would it follow?

You are also clearly confusing what is meant by 'potential infinite'. The potential divisions of a line segment are obviously potential by definition; there are also quite literally infinite potential divisions of a line segment. They are, by every standard account, a potential infinite. This does not make them really finite, even indefinitely so; nor does it mean that the line cannot be divided, since any potential can be actual. In the case of an infinite past, the potential infinite is simply the same infinity as that of the integers, because the supposition of the infinity of the past one implies that the enumeration of past days (or minutes, or whatever unit of time you choose) is such that the first unit of time into the past corresponds to 1, the second to 2, and so forth, with all the integers matched to days. But the integers are a paradigmatic case of potential infinity, namely, the additive: counting numbers constitute one of the two major forms of potential infinity (the other being by division). The integers, however, are infinite for precisely the reason attributed to days: for any part taken, we can take a part outside of what is taken, i.e., counting back into the past, for any number n we reach there will always be an n + 1. If you are going to use the term in any other way, what you need is not to complain about semantic shifts but to justify your deviant usage of the term for this topic. It is also, I should point out, blatantly sophistical to claim that "The potential divisions are actually infinite" implies "The actual divisions are potentially infinite"; for instance, on Aristotle's account, if I understand it correctly, the potential divisions are actually infinite; but the actual divisions are not potentially infinite, because they cannot be infinite at all. There is an equivocation here: the actual divisions are 'potentially infinite' only in the sense Alan used the term, i.e., they are really finite but the potential divisions are infinite. But the potential infinity of potential divisions, on the contrary, is a genuine infinity -- it is called potential not because the things are actually but indefinitely finite, but because it is an infinity of potential things. It is this equivocation that is the real semantic shift here, and it gives the arguments from the impossibility of an actual infinite an apparent plausibility that cannot bear closer examination.

It is easy to see the perniciousness of this and other equivocations elsewhere. The integers are a potential infinite in the standard sense; but you deny that there can be a set whose elements are a potential infinite. This would, if taken strictly, simply be a denial of the existence of sets like aleph-null. It's only because you are importing a foreign understanding of the potential infinite as the indefinite finite that any of these problems with 'potential' and 'actual' infinites arise. You are merely muddying the waters, and in a completely unnecessarily way.

There are, therefore, two questions that need to be asked:

1. Is it possible for the past to be genuinely infinite?
2. If the past is genuinely infinite, what sort of infinity would it be?

And the answers to these questions are "It seems so, since the standard objections fail" and "Potential, because it would be the same infinity as the infinity of the counting numbers", respectively, for the reasons already noted. Whether you use the term 'potential' in the same way, the infinity involved is the same as that of the counting numbers, because that's how the days of an infinite past would be marked (the first day in the past is day 1, the second day 2, ad infinitum). If you are really interested in avoiding semantic shifts, this is the sort of infinity you have to argue is not involved in an infinite past.

At 5/08/2006 3:08 AM, Ocham said...

Nothing to add to Brandon's elegantly and carefully and knowledgeably expressed points. Except that a deep puzzle remains.

The potentialist's view is that the quantifier expression 'any time after now' ranges only over points in the finitely distant future, and that it is in the nature of time that every point in time is followed by a later point.

But every future tense statement if true can be converted to a past tense statement that will be true. E.g. if on Saturday it is true to say 'it will rain tomorrow', 'it rained yesterday' is true on Monday. So we can convert 'any time after now will be such that …' into 'any time after t was such that …' will be true. But the latter statement implies something being the case at a certain point in time that cannot be any of the points in time included in the range of 'any time after now', if the potentialist view is correct, since that point in time must lie after all the points within the range of the quantifier, hence not within the range of the quantifier. But the quantifier ranges over all points in time, if the potentialist view is correct. Hence the potentialist view is not correct.

On the other hand, if the potentialist view is not correct, then we can suppose there is a point in time which will never 'happen' for any finite being condemned to live out one day after another. So in what sense could it 'happen' at all?

At 5/08/2006 3:22 PM, Don Jr. said...

Brandon, I'm honestly completely baffled as to what your position is. You don't maintain that we've completed "infinite many finite traversals"? If past time were infinite and time ended today you would not maintain that we had completed traversing "infinitely many finite traversals"? What would you say then? You would say that there were "infinitely many finite traversals" (given infinite past time) but that we had not completed them? Is this correct?

If, however, you do maintain that our arrival at the present indicates that we have completed traversing "infinitely many finite traversals" then how long did it take for us to complete those traversals? And (only if, again, you maintain that our arrival at the present indicates that we have completed traversing "infinitely many finite traversals") do you also maintain that we have never began (since it would be a beginningless series) completing the "infinitely many finite traversals" but that we had completed them? Also, I would still like to know if you realize that your argument, if successful, would have to work for the future as well? That would mean that it would be possible to complete an infinite amount of traversals in the future, in addition to saying we've already completed an infinite amount of traversals in the past (again, this would only apply given that you maintain the whole completed thing).

Brandon, you say, "But you [Don Jr.] cannot argue, for reasons already noted, that the existence of infinitely many finite traversals implies the existence of an infinite traversal." To satisfy your semantic preferences I quit speaking about an "infinite traversal," so I'm confused as to why you keep bringing it up. I've agreed to use your preferred lingo here (i.e., "infinite traversals" or "infinitely many finite traversals" rather than "infinite traversal"). What's the problem? (I must say, though, that I honestly am in complete awe that you actually think using the phrase "infinite traversals" rather than "infinite traversal" makes or breaks your argument or is of any relevance whatsoever.)

"The actual divisions are potentially infinite" means one can actually divide a segment again, and again, and again, ad infinitum. And that is exactly equivalent, not semantically, but applicably—that is, in meaning—to "The potential divisions are actually infinite."

Brandon, I have absolutely no idea what you're trying to say when you claim that aleph-null is "colloquial" and "not technical," since I've never heard aleph-null spoken in everyday language yet have come across it in many "technical" writings. If you're trying to say that it doesn't appear in reality then I agree since I don't think actual infinities appear in reality. So I'm just lost as to what you were trying to say in regards to that.

Aleph-null is the cardinality of what set? The set of all positive integers. The set of all positive integers (and the set of all numbers) is an actual infinite, since set theory only deals with actual infinites. It's not that I deny that a set can be a potential infinite; it's not possible. These are not my rules. I'm not making this up as I go. A potential infinite is a concept. It represents an indefinite collection or series. Sets are definite; as such they can't possibly be indefinite. Not my rules. Look it up. (You're confusing the concept of counting all the integers with the set of all integers.) You say something about an "indefinite finite" in your last post too. "Indefinite finite"? That doesn't even make any sense.

In all honesty, I don't know how much progress can be made in this discussion since I have no idea what you mean when you say (or how you can even make) statements like "[I]n an infinite past there is no infinite to be traversed" and "[O]n the infinite past view, the days are actually infinite but do not constitute an actual infinite." You've pretty much stolen my reductio arguments against your view by turning them (to my disbelief) into arguments for your view. My only defense is to shout, "Look at what you're saying!" I'm also tiring of all the semantic quibbles and I truly, even after 22 comments, don't have any clue as to what your real position is other than that you think an "infinite past" is possible; but I have no idea what you think that entails. Therefore, I think I'm probably going to have to withdraw from the discussion. I can't in all honesty say I enjoyed it because I think we spent most of the time bickering over semantic issues. Oh well. Maybe Alan or someone else can pick it up where it left off.

At 5/08/2006 4:54 PM, Brandon said...

Don:

We need to be a little more careful here than you are in your above comment: I don't hold that the past is infinite. My position is Thomistic: while I don't think the past is infinite, I deny, unlike you and Alan, the claim that infinite past involves a contradiction. Causally there is a cause in virtue of which an infinite past is possible: God Omnipotent. The only way in which divine omnipotence can't extend to an infinite past is if infinite past involves a contradiction. It does not do so in virtue of being infinite, because the claim that the past is infinite is just that there is a day in the past corresponding to every counting number. There is no problem or paradox involved in the infinity of the counting numbers; therefore the mere fact that the past would be infinite poses no problem for the claim that the past is infinite. Therefore if there is any contradiction it must be in the claim that the past is infinite -- i.e., it must be something about the past, not about infinity, that causes the problem.

Again, I don't know what you mean by 'completed'. The claim that the past is infinite is not the claim that 'we' or that any particular physical thing has gone through infinite days; so if, by 'completed', you mean 'something in particular has gone through all these days', this is not a claim that need be made by those who hold that the past is infinite. If, on the other hand, you simply mean that we can take all past days together as past days, there is no more a problem of there being a 'completed' set of infinite past days in this sense than there is a problem of there being a 'completed' set of infinitely many integers -- 'completion' in that sense merely means the elements are such that they can be taken as forming a set. And it is not surprising that infinite past days would be 'completed' in that sense, because the claim is simply that each past day corresponds to a counting number.

I'm puzzled that you still aren't taking the obvious step of making sure that your 'refutation' of the infinite past claim isn't also a 'refutation' of the infinity of integers. If you did so you would realize that the distinction between infinite traversals of finite distance and a traversal of infinite distance is a very important one in talking about mathematical infinities (not only are there the parallels with integers, there are the parallels with infinite sets mentioned by Ocham); it is you who needs to justify the denial of the distinction in this case. Likewise, your exasperation over the claim that there is no infinite to be traversed in the infinite past is simply misplaced; there is no infinite to be traversed in the infinite sequence of integers, either, but that doesn't prevent the collection of integers from being infinite. You are confusing the infinity of the collection with infinity of distance between members of a collection. The integers are infinite, but there is no case of the latter among integers (and if you insist on the loose use of the term 'integer' that is sometimes used when people say that aleph-null is the first 'infinite integer', then we can just as easily confine ourselves to finite integers, because the infinite past claim is that the days of the past correspond to finite integers, which are all a finite distance from each other and yet form an infinite set).

As I've said before, contrary to your claim, 'potential infinite' does not mean 'indefinite collection'; otherwise it wouldn't be an infinite at all. In the case of the potential infinite by division (e.g., in a line), a line is infinitely divisible; that's a potential infinite. It is not infinitely divided; that would be an actual infinite. Potential infinites are as definite as actual infinites -- it is as definite that the line is divisible here, there, etc., as it would be if the line were actually divided here, there, etc. What makes it a potential infinite is not that it is 'indefinite' but that it is an infinity of divisibles rather than an infinity of divisions. The set of all positive integers is not an actual set in the relevant sense, as I pointed out; as William Lane Craig has noted, for instance, that there is an actual mathematical infinite just means that it makes sense to talk about infinites in the context of mathematics. It should not be confused, as you are confusing it, with other uses of the phrase 'actual infinite'. In any case, it doesn't matter whether you call the infinite past or the set of finite integers an actual infinite or not; the salient feature in both cases is that the set is infinite but there is no infinite distance between any given elements.

Of course it makes sense to talk about an indefinite finite; all one has to hold is that there are finites that are not defined by a sharp limit. This is the sort of thing Alan had called 'potential infinite' -- i.e., things that keep going but are always finite in multitude and magnitude. Since, by definition, they are always finite in multitude and magnitude, they are not infinite; what else would they be, other than finite? They just aren't definite finites, because they increase, and so don't constitute a single definite set.

I would ordinarily be sympathetic to the claim that most of this argument has been semantic bickering; but while you've claimed that, it seems to me to be fairly clearly false -- you haven't done anything, for instance, to motivate the claim that the distinction between infinitely many finite distances and an infinite distance (which is, as I and Ocham have already noted more than once, a good distinction in mathematics), is a bad one when we are talking about the infinite past; and even when I've explicitly pointed out the salient features of the infinites involved (whatever we choose to call them) you've ignored my pointing these out (that the set is infinite without any members of the set being infinitely distant from each other) and have focused on phrases like 'potential infinite' and 'actual infinite'. You have given, as far as I can see, not one reason to think that we cannot say of infinite past days exactly what we say of infinite sets, like the set of integers, in mathematics, where we can obviously say that Nonetheless, I've enjoyed the argument, because it has made more clear to me the reasons why people make the claim that infinite past is self-contradictory. Such a claim seems to me more confused now than ever before, and I am more certain than ever that it takes some rather serious equivocations to get to that claim, but I do have a greater appreciation for the reasons why the claim might be made.

At 5/09/2006 1:56 AM, Ocham said...

>>>Don (to Brandon)
I honestly am in complete awe that you actually think using the phrase "infinite traversals" rather than "infinite traversal" makes or breaks your argument or is of any relevance whatsoever.)

It's key to the argument. If I say 'for any x, however long, there is a y that is longer than x', it immediately follows that there is no longest x. Suppose there were. Then, because x is 'any' x whatsoever, there would be a y that is longer, and so x cannot be the longest. So, on the principal that if an arbitrarily chosen object is not the longest, there is no longest, it follows there is no longest.

Why is that relevant? The move from "there exist infinitely many finite traversals" to "there exists an infinite traversal" is not a valid step. Any more than the move from 'for any x, however long, there is a y that is longer than x' to 'there exists a longest x' is valid. What's interesting, as Brandon has noted, is that whereas we can see the latter is fallacious, the former isn't so obviously fallacious. There's something about the nature of time. As you note, how did we 'get here' in the first place? If there are infinitely many days in the past, how did 'we' arrive at this particular day.

>>>
do you also maintain that we have never began (since it would be a beginningless series) completing the "infinitely many finite traversals" but that we had completed them?

When you say 'completed them', do you mean 'completed each of them' which is correct, since every traversal (on the potentialist view) is finite, or do you mean 'completed all of them', which is not correct on the potentialist view, since the potentialist holds that every traversal is finite, and 'all of them' appears to refer to a traversal which is not finite?

>>>
If, however, you do maintain that our arrival at the present indicates that we have completed traversing "infinitely many finite traversals" then how long did it take for us to complete those traversals?

You persist in the assumption that, on the potentialist view, there is something corresponding to 'those traversals', i.e. all of them. This is precisely what the potentialist denies, for the potentialist holds that every traversal is finite. If every traversal is finite, and to every traversal there corresponds a larger traversal, it follows logically that there is no largest traversal. Every traversal whatsoever is smaller than at least one othere traversal, and that other traversal is smaller than at least one further one, and that further one is smaller than &c.

>>>
That would mean that it would be possible to complete an infinite amount of traversals in the future, in addition to saying we've already completed an infinite amount of traversals in the past (again, this would only apply given that you maintain the whole completed thing).

Again, the illicit move from 'we have completed infinitely many traversals' to 'a single infinite traversal has been completed'. What is fascinating about this argument is how strongly were are tempted to make this move. (Don't think I'm not tempted myself).

At 5/09/2006 1:36 PM, Don Jr. said...

I'm done debating (for reasons previously mentioned), but just to clarify: I never said—nor do I hold—that infinite past time is self-contradictory. I'm not sure where you got that from. In fact, I don't even hold that it's logically impossible. I hold that it's metaphysically or actually impossible, like something's coming into existence uncaused.

At 5/09/2006 1:51 PM, Don Jr. said...

Another clarification: My argument wasn't against the concept of infinite past time or the concept of "the infinity of integers"; it was against the actual (meaning, real) existence of infinite past time and the actual (meaning, real) existence of an actual infinite.

At 5/12/2006 11:22 AM, Ocham said...

Lots more material in the Logic Museum (though the exhibits are still 'under construction' in that they require better translations and suitable introductions. You might be interested, Alan, in a beautifully concise exposition of presentism given by Augustine - references are there in the link.

http://uk.geocities.com/frege@btinternet.com/time/eternity.htm

At 5/12/2006 10:52 PM, Alan Rhoda said...

Thanks, Ocham. I'll take a look at it. I've been tied up most of this past week with giving final exams and grading papers. Almost done.

At 5/20/2006 3:09 PM, C Grace said...

Has anyone thought to connect Kevin's comment
"Infinity is a concept, not a number."

with the comment in Alan's last post that an omniciencent being's knowledge "cannot be abstract. Abstractions always leave something out."

"an actually infinite collection is given all at once...we refer to the set of natural numbers, or the set of prime numbers, and so forth, as completed totalities."

The above quote contains a contradiction.

Alan's third paragraph seems to imply that an actual infinite is one that is bounded. This is what completed means. This is exactly the point at which infinity becomes a concept rather than an actually existing 'state' in reality. ('state' is probably not right but I am not sure what to put)

Therefore this definition of an actually infinite collection contains a contradiction. I would say there is no such thing, only a potentially infinite collection or an actual infinity.

At 5/20/2006 3:17 PM, C Grace said...

Sorry,
I should have said on my last post in the last sentence that any collection has to be a potential infinite. An actual infinite can never be a collection or an object, I think it could be only viewed as an act.

At 5/21/2006 3:26 AM, Ocham said...

I forget it's the exam time of the year.

Anyway, I've cleared up a lot of the Latin, and have made a crack at the English of Bonaventura's argument here

http://uk.geocities.com/frege@btinternet.com/latin/bonaveternitate.htm

There appears to be a whacking great quantifier shift fallacy lurking in there (before every time there is a time, thus before all times there was a time). But it depends whether I've translated right.

There are some interesting bits and pieces connected with this argument. There is another reply by Aquinas in Summa Contra Gentiles which I am currently translating. There is the famous condemnation of 1277 by the archbishop of Paris, where both the position that the world is eternal, and Aquinas argument that the finitude of the world is a matter of faith, not reason, were condemned and their authors threatened with excommunication.

(The condemnation of Aquinas' position was eventually revoked in view of the fact he was a saint).

These pages are all collected together in the Museum, though mostly under construction. Commenst welcomed.

Ockham

At 5/21/2006 2:36 PM, Don Jr. said...

If Kevin means that an infinite amount of something (as opposed to a finite amount of something) can't exist in reality, then I agree. If he's speaking theoretically then I disagree. We can talk about there being an infinite amount of cookies (without being nonsensical) just as much as we can talk about there being 5 cookies (cf. the first paragraph of this comment and the fifth paragraph of this comment).

By completed I think Alan means that actual infinites can't be created through succession; they would need to be determined wholes. That is to say, you can't turn a finite set into an actual set by adding one, and one more, and one more, and so on, until you create an actual infinite. Also, I think Alan would agree that actual infinites don't appear in reality, so in that sense it is merely a concept.