On the Possibility of a Beginningless Past: A Reply to William Lane Craig

William Lane Craig has argued vigorously that, cosmological discoveries aside, it’s reasonable to believe on purely a priori grounds that the set of past events is finite in number.[1] He offers two main types of a priori arguments for this claim: (i) that it’s metaphysically impossible for an actually infinite set of concrete things to exist, in which case the set of past events can’t be actually infinite, and (ii) that even if such a set could exist, it’s impossible to traverse it even in principle. Craig doesn’t pursue this claim for it’s own sake, however. Rather, he does so as a means to demonstrating that a theistic god exists. He reasons that if the set of past events is finite, then the universe as a whole had an absolute beginning with the first moment of time[2]. But since nothing can come into existence without a cause, the universe as a whole has a cause. From here, he goes on to argue that such a cause must be timeless (at least sans creation), immaterial, immensely powerful, and a person of some sort.

I intend to show that one of Craig’s most popular versions of (ii) is unsound. In this essay, I’ll state this argument, prefacing it with an explanation of the concepts crucial to understanding it. Then, I’ll examine a common objection to his argument, along with Craig’s response to it, in order to shed light on an unstated assumption of the argument. Finally, I’ll show that the unstated assumption is false, and how this is fatal to his argument.


I
As I mentioned above, several concepts that are crucial for understanding the argument need clarification.[3] First of all, one needs a fairly perspicuous idea of a set and of a proper subset. A set is a collection of entities, called members of the set. The precise number of members contained in a set is its cardinal number. A proper subset is a part of another set, the former lacking at least one member which the latter contains, and which contains no other members (e.g., from a totally distinct set). More formally, a set A is a proper subset of a set B if and only if every member of A is a member of B, and some member of B isn’t a member of A. To illustrate: Suppose you have ten bottlecaps, five of which are from Pepsi bottles and five of which are from Coke bottles. Then we can call this the set of bottlecaps, the cardinal number of which is 10. Let’s call this set, A. Furthermore, the set of Pepsi caps (call it B) is a proper subset of A, since B consists in a collection of members that belong to A, and A has members that B does not (i.e., the Coke caps).

Another concept that plays an important role in the argument is that of a one-to-one-correspondence. This is a concept used to determine whether two sets have the same number of members (or, the same cardinal number). So there is a one-to-one correspondence between two sets, A and B, if and only if each member of A can be paired up with exactly one member of B, and each member of B can be paired up with exactly one member of A. To illustrate this concept, consider our set of bottlecaps. Now suppose that you didn’t know how to count, but you wanted to know if your had just as many Coke caps as you had of Pepsi caps. You could accomplish this task by pairing each Coke cap with each Pepsi cap, and each Pepsi cap with each Coke cap. If this can be accomplished with no remaining bottlecaps, then there is a one-to-one correspondence between the set of Pepsi caps and the set of Coke caps. If follows that the respective sets of bottlecaps have the same cardinal number.

The concept most important for our purposes is that of actual infinity. To obtain a grasp of this concept, consider the set of all the natural numbers (i.e., {1, 2, 3, …}). This set, as well as any set that can be put into a one-to-one correspondence with it, is an actually infinite set (It’s actually the “smallest” of the infinite sets, but we won’t be concerned with “larger” infinites here). An actual infinite has several interesting features. First of all, it is complete, in the sense that it has an infinite number of members; it is not merely increasing in number without limit. Second, any actually infinite set (of the “size” we’re here considering) can be put into a one-to-one correspondence with one of its proper subsets. This can be demonstrated by putting the set of natural numbers in a one-to-one correspondence with its proper subset of even numbers:

1 2 3 4…
2 4 6 8…

This example shows that a part of an actually infinite set can have as many members as the whole set! The cardinal number of an actually infinite set that can be put into a one-to-one correspondence with the natural numbers is called “aleph null” (let’s use ‘A0’ for brevity).

The final concept relevant to our discussion is order-type. I won’t talk at length about this concept here. Rather, I’ll barely do more than mention the order-types of certain sets containing A0 members. Four our purposes, it will suffice to know that sets can be sequentially ordered according to certain patterns or types. The order-type given to the set of natural numbers so ordered that, beginning with 1, each natural number is succeeded by the next largest natural number – i.e., {1, 2, 3, …} – is ‘omega’, or 'w’, and the set of negative integers so ordered that they are sequentially the opposite of w is w* (i.e., {…-3, -2, -1}). Sets with A0 members can have other order-types, however. For example, an A0 set can have the order type w+1 (i.e., {1, 2, 3, ..., 1}), or the order-type w+2 (i.e., {1, 2, 3, …, 1, 2}), etc. In fact, a set with A0 members can have the order-type w+w (i.e., {1, 2, 3, …, 1, 2, 3, …}), or the order-type w+w+w (i.e., {1, 2, 3, …., 1, 2, 3, …, 1, 2, 3, …}), etc.! To see this, recall that any set that can be put into a one-to-one correspondence with the natural numbers has a cardinal number of A0. But sets with the order-types mentioned above can be put into such a correspondence. So, for example, a set with the order-type w+1 can be put into a one-to-one correspondence with the natural numbers as follows:


1 1 2 3…
1 2 3 4…

Similarly, a set with the order-type w+w+w can be put into a one-to-one correspondence with the natural numbers as follows:

1 2 3…1 2 3…1 2 3…
1 4 7…2 5 8…3 6 9…

Therefore, since sets with such order-types can be put into a one-to-one correspondence with the natural numbers, it follows that their cardinal number is A0.

At this point, an interesting feature of certain sets with A0 members emerges. For consider any A0 set with an order-type other than w. For example, consider a set of A0 offramps on an infinitely long freeway, such that a distance of one mile separates each offramp from its predecessor and successor (except, of course, the first offramp, since it has no predecessor). Suppose further that the order-type of the offramps is w+1 ({1, 2, 3, …, 1}). The offramp assigned the first 1 would seem to be infinitely distant from the offramp assigned the second 1. Such a set has the interesting feature of being non-traversable in principle – it cannot, even in principle, be exhaustively counted through one offramp at a time. This is because it is logically impossible to count to a number that has no immediate predecessor. But the offramp assigned the second 1 has no immediate predecessor. Therefore, a driver on such a freeway could never reach the offramp assigned the second 1. Call this particular logical ban on traversing sets with w+1 or “higher” order-types ‘LB’.

Now it may be tempting to think that this consideration is decisive for the view that the past must be finite, since any set with A0 members can be ordered according to the order-type w+1. In this way, one might think, LB infects all actually infinite sets, and thus no set with A0 members is traversable. This reasoning can be expressed as follows:

1. A set is LB non-traversable if and only if it contains at least one member A0 distant from at least one of the other of its members.
2. Any set with the order-type w+1 is such that it contains a member A0 distant from at least one of the other of its members.
3. Therefore, any set with the order-type w+1 is LB non-traversable.
4. Any set with A0 members can be assigned the order-type w+1.
5. Therefore, any set with A0 members is LB non-traversable.

But this would be rash. For the inference from (3) and (4) to (5) is a non sequitur. For consider a set with A0 members that is assigned the order-type w. A set with this order-type is such that (i) no member is infinitely distant from any other member, and (ii) each member does have an immediate predecessor, as so is immune to LB. But any set with A0 members can be assigned the order-type w. So if the inference from (3) and (4) to (5) were valid, then by similar reasoning the following inference should go through as well:

3’. Any set with the order-type w is not LB non-traversable.
4’. Any set with A0 members can be assigned the order-type w.
5’. Therefore, any set with A0 members is not LB non-traversable.

But (5’) contradicts (5). Therefore, since the same pattern of reasoning yields contradictory results, it’s faulty. So, just because an A0 set can be assigned a non-traversable order-type, it doesn’t follow that such a set is non-traversable. What really follow from (3) and (4) is rather

5’’. Any set with A0 members can be assigned an LB non-traversable order-type.

Which, needless to say, doesn’t help to establish the finitude of the set of past events.

The arguments above suffer from another problem as well. For (2) is clearly false. To see this, consider again our infinitely long freeway. No suppose that it only has one lane, and that it has an infinitely long traffic jam. Finally, suppose that each car in the jam is assigned a number from the order-type w+1. Does it follow that there is a car A0 distant from any other car? No. For the car assigned the second 1 could be immediately in front of the first car. This illustration shows that the order-type assigned to a set of objects doesn’t necessarily affect the distances between its members. To drive this implication home, suppose that the cars of the traffic jam were assigned the order-type w. Would it follow that no car is A0 distant from any other car? Not in the least. For the first and second cars may be infinitely distant from one another. One might reply that we could just stipulate that the distances between the cars and the order-type assigned to the cars correspond. In such a case, each car would only be finitely distant from every other car when assigned the order-type w (e.g., the second car is 2 meters from the end of the traffic jam, the third car is 3 meters from the end of the jam, etc. [these are small cars!]). One could then reassign the cars with the order-type w+1, but then the correspondence between the order-type and the distances of the cars would break down. This is because no car assigned a number from the w order-type in our scenario is infinitely distant from any other car. Therefore, the second 1 of the newly assigned w+1 reordering would be assigned to a car that is only finitely distant from any other car. These illustrations show that (i) some sets assigned the order-type w+1 are such that no member is infinitely distant from any other, and (ii) we can know a priori than an A0 set of concrete objects cannot be reordered from w to w+1 in such a way that the distances between the members of such a set correspond to their order-type. Therefore, if a set of objects has A0 members (arranged linearly), it does not follow from this that it has members infinitely distant (whether in time or in space) from other members (and is therefore LB non-traversable).[4] Thus, to show that an A0 past is non-traversable, Craig must show that no A0 set with either the order-type w or w* (and is such that no member is infinitely distant from any other) is traversable. Let’s consider one of Craig’s main attempts to do this.
II
Craig advances an argument for the proposition that one cannot traverse a beginningless past and end at the present moment.[5] To do this, Craig assumes, for the sake of argument, that there could be a beginningless past, conceived as a set of events with the cardinal number A0 and the order-type w* (i.e., {…,. -3, -2, -1}), where each negative integer represents an event of the past. He then argues,

“…suppose we meet a man who claims to have been counting down from eternity and who is now finishing:…,-3, -2, -1, 0. We could ask, why didn’t he finish counting yesterday or the day before or the year before? By then an infinite amount of time had already elapsed, so that he should already have finished. Thus, at no point in the infinite past could we ever find the man finishing his countdown, for by that point he should already be done! In fact, no matter how far back into the past we go, we can never find the man counting at all, for at any point we reach he will already be finished. But if at no point in the past do we find him counting, this contradicts the hypothesis that he has been counting down from eternity.”[6]

Craig’s argument is a reductio ad absurdum, where we suppose that a beginningless past is possible in order to show that it entails a contradiction. The argument can be expressed as follows, with (1) as the premise set of for reduction:

1. The past is beginningless (conceived as a set of events with the cardinality A0, and the order-type w*).
2. If the past is beginningless, then there could have been an immortal counter who counts down from such a past at the rate of one negative integer per day.
3. The immortal counter will finish counting if and only if he has an infinite number of days in which to count them.
4. If the past is beginningless, then there are an infinite number of days before every day.
5. Therefore, the immortal counter will have finished counting before every day.
6. If the immortal counter will have finished counting before every day, then he has never counted.
7. Therefore, the immortal counter has both never counted and has been counting down from a beginningless past (contradiction)
8. Therefore, the past is not beginningless (from 1-7, reductio)


In short, Craig argues that the past must -- logically must -- have a beginning. For the very notion of traversing a beginningless past entails a contradiction. Craig’s underlying intuition here is that if the past is beginningless, then it must contain an actually infinite proper subset of events that was not formed by successive addition, and that this is absurd.

Critics typically attack (3), arguing that Craig mistakenly assumes that to count an infinite number of negative integers is to count all of them. However, critics of Craig’s argument point out that one can count an infinite set of numbers without counting them all.[7] For example, suppose our eternal counter just finished counting all the negative integers down to -3. Then it would be true that he has counted an infinite number of integers, and yet he has not counted all the integers. This can be demonstrated by the following one-to-one correspondence:

Days counted: -3 -4 -5…
Nat. numbers: 1 2 3…

In this case, the set of days counted has the cardinal number A0, since its members can be put into a one-to-one correspondence with the natural numbers. Yet he clearly hasn’t counted all the negative integers, since he has failed to count -2 and -1. Therefore, since counting an infinite number of things is not synonymous with counting them all, Craig’s (3) is based on an equivocation.

Craig has denied that he is guilty of this charge[8]:

“I do not think the argument makes this alleged equivocation, and this can be made clear by examining the reason why our eternal counter is supposedly able to complete a count of the negative numbers, ending at zero. In order to justify this intuitively impossible feat, the argument’s opponent appeals to the so-called Principle of Correspondence…On the basis of the principle the objector argues that since the set of past years can be put into a one-to-one correspondence with the set of negative numbers, it follows that by counting one number a year an eternal counter could complete a countdown of the negative numbers by the present year. If we were to ask why the counter would not finish next year or in a hundred years, the objector would respond that prior to the present year an infinite number of years will have elapsed, so that by the Principle of Correspondence, all the numbers should have been counted by now.

But this reasoning backfires on the objector: for on this account the counter should at any point in the past have already finished counting all the numbers, since a one-to-one correspondence exists between the years of the past and the negative numbers.”[9]

From this passage, we see Craig’s rationale for (3):

(R) The counter will have finished counting all of the negative integers if and only if the years of the past can be put into a one-to-one correspondence with them.

Furthermore, from the passage cited, we see that Craig thinks that the defender of an A0 past agrees with (R). But since the type of correspondence depicted in (R) can be accomplished at the present moment, it follows that the counter should be finished by now. Therefore, Craig’s opponent is committed to a view that entails the absurdity surfaced by the above reductio.
III
It isn’t clear that Craig hasn’t made his case, however. For consider the following scenario. Suppose God timelessly numbers the years to come about in a beginningless universe. Suppose further that He assigns the negative integers to the set of events prior to the birth of Christ, and then the positive integers begin at this point. Then the timeline, with its corresponding integer assignment, can be illustrated as follows:

…-3 -2 -1 Birth of Christ 1 2 3…

Suppose yet further that God assigned Ralph, an immortal creature, the task of counting down the negative integers assigned to the years BCE, and stopping at the birth of Christ. Call this task ‘(T)’. With this in mind, suppose now that Ralph has been counting down from eternity past and is now counting the day assigned (by God) the integer -3. In such a case, Ralph has counted a set of years that could be put into a one-to-one correspondence with the set of negative integers, yet he has not finished all the negative integers. This case shows that, while it is a necessary condition for counting all of the events that one is able to put them into a one-to-one correspondence with the natural numbers, this is not sufficient. For if the events that are to be counted have independently “fixed”, or, “designated” integer assignments set out for one to traverse, one must count through these such that, for each event, the number one is counting is the same as the one independently assigned to the event. In the scenario mentioned above, God assigned an integer to each year that will come to pass. In such a case, Ralph must satisfy at least two conditions if he is to accomplish (T): (i) count a set of years that can be put into a one-to-one correspondence with the natural numbers, and (ii) for each year that elapses, count the particular negative integer that God has independently assigned to it. According to Craig’s assumption (R), however, Ralph is supposed to be able to accomplish (T) by satisfying (i) alone. But we have just seen that he must accomplish (ii) as well. Therefore, being able to place the events of the past into a one-to-one correspondence with the natural numbers does not guarantee that the counter has finished the task of counting all the negative integers. In other words, (R) is false. But recall that (R) is Craig’s rationale for (3). Thus, (3) lacks positive support. But more importantly, (3) is false. This is because the scenario above is a counterexample to both (R) and (3). For (3) asserts that it is sufficient for counting down all the negative integers that one has an infinite amount of time in which to count them. But our scenario showed that one could have an infinite amount of time to count, and yet not finish counting all of the negative integers (e.g., one can count down to -3 in an infinite amount of time, and yet have more integers to count).

To sum up. We’ve looked at an argument that Craig repeatedly gives for the impossibility of a beginningless past. We then saw that one of its premises is false, in which case it is unsound. Thus, this argument, at least, cannot be used to offer a priori support for the key premise in his Kalam argument.


[1] See, for example, his The Kalam Cosmological Argument (London: Macmillan, 1979); Craig and Quentin Smith, Theism, Atheism, and Big Bang Cosmology (Oxford: Clarendon Press, 1995). See also Craig’s popular-level book, Reasonable Faith (Wheaton: Crossway Books, 1994).
[2] I should mention a wrinkle here: the possibility that the universe did not begin to exist with the first event of time, but rather existed eternally in a quiescent, eventless mode of existence “prior” to the first event. Craig addresses this worry in “The Kalam Cosmological Argument and the Hypothesis of a Quiescent Universe”, Faith and Philosophy 8 (1991), pp. 104-8.
[3] My discussion of the following set-theoretic concepts is indebted to J.P. Moreland’s Scaling the Secular City (Grand Rapids: Baker, 1987)
[4] The points and illustrations are similar to those made and conceded by Craig in “Reply to Smith: On the Finitude of the Past”, International Philosophical Quarterly 33 (1993), pp. 228-9.
[5] Actually, he advances two arguments for this proposition. One is a variation on the famous Tristam Shandy Paradox. In Craig’s construal of it, Shandy writes his autobiography from the beginningless past at the rate of one year of writing per day of autobiography. It seems that Shandy would never finish his autobiography, getting farther behind with each passing day. But since one can put the days of his life into a one-to-one correspondence with the set of past years, it (paradoxically) seems that he should have finished his autobiography by now. The other version is virtually the same as the one I consider here. It asserts that Shandy should be finished by now, irrespective of the rate at which he is writing. However, I won’t consider the former version here. See Craig and Smith’s Theism, Atheism, and Big Bang Cosmology, pp. 99-100, and Craig’s “Feature Review of Time, Creation, and the Continuum”, International Philosophical Quarterly 25 (1985), pp. 319-26. For a briefer exposition, see Craig, Reasonable Faith, pp. 98-9.
[6] Craig, Reasonable Faith, p. 99.
[7] This objection can be found in David A. Conway. “’It Would Have Happened Already’: On One Argument for a First Cause”, Analysis 44 (1984), pp. 159-66; Richard Sorabji. Time, Creation, and the Continuum (Ithaca: Cornell University Press, 1983), pp. 219-24.
[8] See, for example, Craig and Smith, Theism, Atheism, and Big Bang Cosmology, pp. 105-6; Craig, “Review of Time, Creation, and the Continuum”, p. 323.
[9] Craig, “Review of Time, Creation, and the Continuum”, p. 323.

27 comments:

Aaron Kinney said...

Holy cow exapologist, what a great post!!! You even got a great set of footnotes and references. Nice.

Excellent work. However, William Lane Craig makes some fatal flaws in his premises that may have been overlooked.

Allow me to quote:

"He reasons that if the set of past events is finite, then the universe as a whole had an absolute beginning with the first moment of time[2]. But since nothing can come into existence without a cause, the universe as a whole has a cause."

But the universe never "came" into existence. It itself is eternal. Time is temproral; it is a property of the universe. But the universe is not temporal, nor is it dependent upon time. And looking at Hawking's work on the Wave Function of the Universe theory, there is a kind of perpendicular time that he proposed that could have triggered the initial start of time as we know it. In short, there are naturalistic ways to account for the beginning of time as we know it without invoking a conscious agent who operates outside of time.

And finally, Craig's invoking of God to solve the "whatever began to exist has a cause" is merely special pleading. He refuses to assign the property of "eternal" to the universe (for unjustified reasons) yet then insists on assigning the property of "eternal" to an entity that he invokes (again, for unjustified reasons). Special pleading all around.

And since you were talking so much about sets and order, just for fun, and slightly of topic, you might want to read the material at http://www.everythingforever.com. Basically, it shows that "order" is actually found in two forms: sequential (1,2,3, 1,2,3,) and grouping (1,1,2,2,3,3), and DISorder is just a figment of our imagination. In other words, disorder doesnt exist and what we think is disorder is merely a mixture of sequential and grouping order. Its a fascinating website and makes one look at entropy in a whole new way. In addition, the site talks about infinity and infinite sets while still having boundaries. Since you did a lot of analysis of numbered sets, you might enjoy that site.

Anonymous said...

This excellently stated argument of yours intrigues me.

I believe your argument can be reduced to this: since counting an infinite number of things is not synonymous with counting them all, Craig’s premise that "the immortal counter will finish counting down from eternity if and only if he has an infinite number of days in which to count them" is based on an equivocation, for he could count an infinite number of days and still not count all of the days, right?

Your post shows you know a lot more about this subject than I do, but isn't Craig's whole question whether or not it's possible to count to infinity (backwards or forwards) begining at any integer, no matter how it's labelled (-1 or 1)? And when you arbitrarily stop the counting at the birth of Christ and then declare that the counter doesn't need to get to that day to get to infinity, isn't that Craig's whole point, that no matter how far back in time we go an immortal counter will already have finished counting to infinity?

Have I missed something here? Or, are you merely dealing with one particular formation of his argument?

NonProphet said...

Hi

There is a book you may be interested in by J. Richard Gott (Prof. Astrophysical Sciences at Princeton) called Time Travel in Einstein's Universe. It examines a number of different possibilities regarding time travel consistent with Einsteinian physics. Around a third of the book is dedicated to examining how the universe could created itself, through the mechanism of a 'closed timelike curve' (see here for a review which discusses this). I cannot claim to have a great grasp on the underlying physics (string theory etc.), but Gott's hypothesis would appear to be a novel approach to the 'origins' question which extends beyond the big bang without resorting to a supernatural explanation.

Here is the Amazon page.

exapologist said...

Thanks for the kind words, Aaron and John!

Aaron, I agree with you on just about every point.

John,
You're right that I'm just dealing with his "immortal counter" reductio here -- although I think that all versions offered by both Craig and Moreland have problems. Craig wants to show that the very notion of a beginningless infinite traversal entails a contradiction, viz., that such a traversal -- at any point -- would both be and not be done. My point is that the first conjunct of the contradiction -- that it would be done -- doesn't follow, and thus the very notion of a beginningless infinite traversal is *not* self-contradictory.

The picture that Moreland and Craig present about traversing actual infinites can be summed up in the following gisty way: Counting from 0 to "positive infinity" can *never* be completed; counting from "negative infinity" to 0 must *always* be completed.

The argument I discuss is of the "counting from "negative infinity to 0" type. Craig is saying, in effect, that if the past had no beginning *at all*, then the present moment should *always* have been behind us. My reply can then be seen as saying, "no it shouldn't".

P.S., I know that Moreland says the opposite -- that if the past is beginningless, then the present moment would *never* have arisen -- but that's because he's muddled on this point (or to be charitable, he was just speaking sloppily at the time).

exapologist said...

Hi Nonprophet,

Thanks for the info and the link! I'll be sure to check it out. I should say that I agree with both you and Aaron that there doesn't seem to be a good reason to think there's an absolute beginning to the series of events -- no need to posit a Big "Beginner".

exapologist said...

Hi John,

Having a moment to look back at your comment on the kalam post, I'm not sure if I construed your point correctly. Your main point was that a beginningless past is problematic merely from the fact that if such a past exists, then an infinite set of events exists prior to every event, right? If so, then I think it's a good idea for me to go into this point in some detail.

Although I agree that this is weird, this *just is* the view that the past is beginningless. The question is, then, what is it, exactly, that makes this problematic?

Well, an initial objection might be that (to quote Moreland quoting Craig) "you could never get a foothold in the series to even make progress" toward a count to the present moment. But notice that to say this just begs the question against the proponent of a beginningless past. It assumes without argument the following principle:

(*) In order to traverse or count through something, you must start at some point.

Well, if Moreland and Craig think that (*) is self-evidently true, then there's no need for them to go to the trouble of laying out the kalam argument, is there? All they have to do is trot out (*), and then all disputes about a beginningless past would be settled in one stroke!

In other words, if those on both sides of the debate accepted (*) from the get-go, then Craig and Moreland could bypass discussing the kalam argument and just say, "Well, if the past were beginningless, then the present was reached without a first event or starting point. But obviously, nothing can be reached without starting at some point; so you can see that the past had a beginning -- there was a first event."

But this *just is* the point under dispute; the proponent of a beginningless past is saying that you can traverse an actual infinite by never beginning and then ending at any arbitrary point. So to just *assert* principle (*) without arguing for it is to beg the question against the proponent of a beginningless past.

Sometimes I think that Moreland and Craig get confused about this by unconsciously slipping from the *appropriate* contrual of the past events as having an w* order-type (i.e., {..., -3, -2, -1}) -- in which there is clearly no first event but a last event (the present moment) -- to an *inappropriate* construal of past events: i.e., as one having either an w order-type ({1, 2, 3, ...}) or a 1+w* order-type (1..., -3, -2, -1}), where there must be a first event from which one can then "make progress" toward the present moment. But again, this is just to *beg the question* against beginningless infinite traversals that never begin -- i.e., the immortal counter has *always* been counting (i.e., he *never* started.

Now there may be interesting arguments that don't beg the question in this way, such as the "immortal counter" reductio I discuss in my post. It's just that, as I hope I've shown, at least one such argument doesn't succeed.

All the best,

exapologist

O'Brien said...

Coming from a mathematics background, I easily followed the discussion about countably/denumerably infinite and finite sets (which have Lebesgue measure 0), but I found the analogies and philosophizing dreadfully boring. You have piqued my interest in Craig's argument, though.

Wrightsaid said...

Exapologist, please, forgive my ignorance of some of the background for all of this, but I'm not sure I understand what you mean and I would very much like to understand. Let me point to the specific spot that confuses me.

In an infinite succession of moments (assuming that the universe never began to exist and was always "in time", progressing in moments), there is a kind of motion in a certain direction. Picturing this as a number line, the direction would be "forward" along the number line toward a predetermined end, let's say to zero.

If that is so, then the momentum of the successive moments, the direction toward zero, would seem to need to have some point of beginning, else it could never have got going, so to speak. In other words, it seems sensible to ask at what point time began to progress along our number line.

If the answer is, "There is no point at which time began progressing along the number line," then isn't that the same as saying that it never began progressing along the number line? And if it never began progressing toward zero, how can it ever reach zero?

Again, please forgive the sloppiness of my own thinking. If you could, I would like very much to understand another perspective. Thanks for your time.

exapologist said...

Hi Wrightsaid,

Thanks for your (very well-put) question:

To say that there was no *beginning* to the events is not to say that there was no *occuring* of events.

To come at it from another direction: one clear way to traverse something is as follows:

(i) Going from a state in which some sequence of events is *not* initiated to a state in which it *is* initiated, and then progressing on until the sequence is complete.

But, absent some hitherto uncovered incoherence in the idea, another logically possible way to traverse something is as follows:

(ii) *Always* being in a state in which some sequence of events is in progress, until the sequence is complete.

But if (ii) is logically possible, then saying that “there is no point at which time began progressing along the number line” is not the same thing as saying “time has never progressed along the number line”.

All the best,

exapologist

Wrightsaid said...

Thank you very much for your kind regard and willingness to answer my questions. I do not wish to seem slow in catching on but perhaps I am, though I am honestly trying. The explanations you gave are likely solid in some sense but I still have trouble. Allow me to explain...

Going from a state in which some sequence of events is *not* initiated to a state in which it *is* initiated, and then progressing on until the sequence is complete.

This sounds strangely like a beginning to me, which probably shows that I have not grasped it. From a state of uninitiation to initiation... even if this is an "initiation" of a state which has no beginning, the beginningless sequence of events itself would certainly have begun to exist if there were ever a time in which it was not initiated. Surely I am missing something. Can you tell me where?

Secondly, you said:

*Always* being in a state in which some sequence of events is in progress, until the sequence is complete.

I am likely hearing this wrong in my mind as well. Let us assume, as I think I must to understand, that there is a sequence of moments that has been progressing forward "always" in the sense that there never was a time when this progression began. If I think back to a spot on that line, let us say the number 10 followed by ten million zeroes, I will find that, still, we can say that the sequential progression has "always" been moving forward here, too. If I think back to 10 followed by 100 million zeroes, it is still the same. The sequence of moments stretches back as far as ever from any perspective in the past. The further back into the past I venture, the conclusion is still the same. No matter how far back I go, I am not any closer to where this progression of moments has been.

Now, thinking forward, it would also seem to follow that, no matter how long this sequence of moments has travelled, there is still an infinite distance on the number line it would need to go to reach zero, if we can truly say that it is covering the distance from infinity to the present.

Again, I have probably just misunderstood and if you are still interested, I would be extremely appreciative if you could explain this to me. Thank you for all your efforts.

exapologist said...

Hi Wrightsaid,

Sorry my post wasn’t clear enough. I’m battling against conflicting aims – clarity and thoroughness of discussion on the one hand, and daily tasks on the other. Unfortunately, priorities demand that I often have to sacrifice the former in order to accomplish the latter.

In any case, on to the discussion. First, you asked me to clarify my account of one way in which a traversal can be accomplished, which was:

(i) Going from a state in which some sequence of events is *not* initiated to a state in which it *is* initiated, and then progressing on until the sequence is complete.

About (i), you said:
“This sounds strangely like a beginning to me, which probably shows that I have not grasped it. From a state of uninitiation to initiation... even if this is an "initiation" of a state which has no beginning, the beginningless sequence of events itself would certainly have begun to exist if there were ever a time in which it was not initiated. Surely I am missing something. Can you tell me where?”

As far as I can tell, you’re not missing anything here. Such a manner of completing a traversal necessarily involves a starting point – a beginning. This is at least one logically possible way to traverse something. So far, we are in agreement.


I believe your second substantive point was in regards to my account of a second logically possible way for a sequence to be traversed, which was


(ii) Always being in a state in which some sequence of events is in progress, until the sequence is complete.

Regarding (ii), you said:

“Let us assume, as I think I must to understand, that there is a sequence of moments that has been progressing forward "always" in the sense that there never was a time when this progression began. If I think back to a spot on that line, let us say the number 10 followed by ten million zeroes, I will find that, still, we can say that the sequential progression has "always" been moving forward here, too. If I think back to 10 followed by 100 million zeroes, it is still the same. The sequence of moments stretches back as far as ever from any perspective in the past. The further back into the past I venture, the conclusion is still the same. No matter how far back I go, I am not any closer to where this progression of moments has been.



Now, thinking forward, it would also seem to follow that, no matter how long this sequence of moments has travelled, there is still an infinite distance on the number line it would need to go to reach zero, if we can truly say that it is covering the distance from infinity to the present.”

I like this argument very much – it’s very clever. J.P. Moreland gives a version of it in his book, Scaling the Secular City (among other places). Here’s one way to express it:

1. If the past were beginningless, then it’s impossible in principle to traverse the present all the way through the past.
2. If it’s impossible in principle to traverse something in one direction, then it’s impossible in principle to traverse it in the other direction.
3. Therefore, if the past is beginningless, then it’s impossible in principle to traverse the past all the way to the present.
4. But it’s not impossible in principle to traverse the past all the way to the present (after all, here we are!).
5. Therefore, that past is not beginningless.

My worry with Moreland’s argument is that (2) begs the question against one who is a proponent of (or even agnostic about) the possibility of a beginningless past. I grant that it’s impossible in principle to traverse an actual infinite by starting at some point – i.e., by traversal of type (i) above. But I have no good reason to think that it’s impossible in principle to traverse an actual infinite via a beginningless traversal – i.e., by traversal of type (ii) above. After all, such a type isn't handicapped with having to start at some point.

It’s of course granted that we may disagree about this, but if the Kalam argument isn’t to be question begging – i.e., if it’s rational to accept the premises without already accepting the conclusion – then Craig (and Moreland) need to furnish us with premises that would move someone from a state of not believing to believing that a beginningless past is impossible.

Regards,

exapologist

Anonymous said...

exapologist:
Your main point was that a beginningless past is problematic merely from the fact that if such a past exists, then an infinite set of events exists prior to every event, right?

John:
Yes it was.

exapologist:
Although I agree that this is weird, this *just is* the view that the past is beginningless. The question is, then, what is it, exactly, that makes this problematic?

Well, an initial objection might be that (to quote Moreland quoting Craig) "you could never get a foothold in the series to even make progress" toward a count to the present moment. But notice that to say this just begs the question against the proponent of a beginningless past.


Excellent! Excellent!

Wrightsaid said...

Thank you so much for being so quick to aid my thinking. I am not very familiar with Moreland's work so much of my part of this very helpful dialogue is going to come out sounding like the product of nontechnical, personal meditation on the subject. Thank you for your patience.

Let me gather my thoughts... I'll quote you first:

My worry with Moreland’s argument is that (2) begs the question against one who is a proponent of (or even agnostic about) the possibility of a beginningless past.

Perhaps a lack of much formal training in logic is hurting me and I use terms in ways that are too personal. I am not quite sure how Moreland is begging the question. As far as I can see, Moreland's idea, as you've relayed it, seems to be that an infinite series of numbers cannot, by nature, be traversed by means of simple addition or subtraction in either direction. Considering, in other words, that an infinite series of numbers is literally "endless," this implies that an end cannot be discovered, even by endlessly examining one breadcrumb at a time along a trail of breadcrumbs that make up an endless path... unless "endlessness" is a negotiable part of a definition of an infinite series.

Given this, what you seem to be objecting to is that Moreland is not even granting the "possibility" of traversing the endless series but that sounds to my admittedly untrained ears a bit like objecting that a person is not granting me the benefit of the doubt when I claim that my dog has lately been chewing on a one-ended stick. Should we assume, simply because some would like to suggest that this is at all possible, that they are championing anything coherent?

When you respond by saying that one way of traversing an infinite series can be described as, "*Always* being in a state in which some sequence of events is in progress, until the sequence is complete" sounds as though you were merely disagreeing, without explanation. As long as the progression has been from "always," an infinite series can be traversed. But that not only begs the question, as far as I can see, it also seems (at least to me) to be positing a logical absurdity.

Now, there is certainly the possibility that I've missed something (which is why I've asked to speak with you). What must be demonstrated, in other words, is where precisely the notion that this is all incoherent gibberish has really missed some vital aspect of the essence of a progression along an infinite series. I do not yet discern this explanation in anything you've written. What you have said is:

. . .I have no good reason to think that it’s impossible in principle to traverse an actual infinite via a beginningless traversal – i.e., by traversal of type (ii) above. After all, such a type isn't handicapped with having to start at some point.

Of course, an opinion, no matter how much conviction lies behind it, is not a demonstration. I can certainly appreciate that the idea of a beginningless progression, if coherent, would not be handicapped with a beginning. That is a tautology. But how can a beginningless progression reach some infinite point in the future when no such point exists? An infinite amount of time cannot exhaust an infinite distance of travel precisely because the inexhaustible cannot be exhausted by definition. We cannot reach the end of an endless series and it can hardly be begging the question simply to state a definition.

From what I understand of all of this (which perhaps isn't much), I think it would be best to disagree with William Lane Craig. The infinite series should not in the least have been exstinguished long ago. Rather, let's assume I were to take a road trip back in time endlessly into the past while another person were on a similar road trip moving into the future at a steady pace toward me from no point or place in the past (on a beginnless journey, if that makes sense). By the nature of an infinite, the distance between us should always remain the same: endless. No matter how far back I travelled, the two of us would never meet and we would be always the same distance apart, endlessly apart.

If we can never meet given our mutual efforts, how then can one catch up alone? If we imagine this as a number line, the same is true precisely because we would not be beginning from different sides of a number line (with a finite distance separating us). I would be beginning from one side (at zero) and my partner would be beginning from nowhere, at no point along the number line. Wouldn't he always be infinitely beyond me exactly because, in a sense, he is not really anywhere ahead of me to be reached at all?

In that case, and I certainly may be wrong, all it seems you have done is required of Moreland and others that they prove to you that it really is impossible to find the end of an endless series IF in fact one has "always" sought to do so. But should we assume that what is logically contradictory can in fact be accomplished so long as we're given enough time?

Again, thank you for all your efforts. I do hope this is not proving tiresome to you. I don't expect you to push your schedule aside for me, so please take your time.

christian lies said...

I visited a Free Church of Scotland service today.

My G/F who was brought up in the Wee Frees had horrified me with her stories of "Christianity" as taught by the FCS.

I was fairly mild - hardly any mention of God.

HYMS WERE SUNG SITTING DOWN ????????

They go on and on talking abour jesus the people must be bored outta their skulls.

Wrightsaid said...

Exapologist, I was thinking about this interesting problem and I believe I have worked out a good way of relaying the difficulties in a more specific way...

Let's take the illustration I used earlier of a person travelling "always" forward from the past toward the present. In the present, let's also imagine that we decided to take a trip back in time continually travelling so that we might reach the other time traveller in the past.

With the two of us progressing toward each other at the same exact rate of speed, if the infinite distance can be traversed as you say, then there should be a definite mathematical midpoint where we would eventually meet.

The time in human history where this experiment is undertaken will not change the reality that a midpoint should exist...

If someone is moving toward me from the infinite past and I am moving toward them from the present, can you tell me where we would meet? Where is the midpoint?

A possible objection might be that you are not sufficiently knowledgeable in mathematics to undertake the resolution of this difficult problem. But we must ask ourselves whether this problem is even resolvable by anyone.

Matthematically, can there be a midpoint between us and endlessness? Since endlessness is not a point or a number along our line, how can a precise mean be calculated?

exapologist said...
This comment has been removed by a blog administrator.
exapologist said...

Hi Wrightsaid,
I just noticed that you posted again as I was about to "paste" this post.

As to your newest post:
What you say is interesting, although it seems that nothing that lacks an endpoint could have a midpoint. It seems that it's a necessary truth that something has a midpoint just in case it has two endpoints.

As to your previous post:

If this discussion continues, can I ask that we limit it to one point per post? If so, you’d really be helping me out. Thanks a bunch.

Re: Premise (2) of Moreland’s version of the argument:

Me: My worry with Moreland’s argument is that (2) begs the question against one who is a proponent of (or even agnostic about) the possibility of a beginningless past.



Wrightsaid: I am not quite sure how Moreland is begging the question. As far as I can see, Moreland's idea, as you've relayed it, seems to be that an infinite series of numbers cannot, by nature, be traversed by means of simple addition or subtraction in either direction. Considering, in other words, that an infinite series of numbers is literally "endless," this implies that an end cannot be discovered, even by endlessly examining one breadcrumb at a time along a trail of breadcrumbs that make up an endless path... unless "endlessness" is a negotiable part of a definition of an infinite series.



Me: In premise (1), he’s asserting something that most everyone can agree on, viz., that “you can’t get there from here” with respect to a beginningless past. As you rightly point out, you can’t get to the end of something that has no end, by the very nature of the case. Of course, I’m not objecting to that.


In premise (2), he’s asserting that traversing in one direction is just as easy (or hard) as traversing in the other. One is possible just in case the other is possible. But you see, I don’t know if this is true. It’s certainly not self-evident, like premise (1). After all, there’s an asymmetry with respect to direction here – there’s an end in one direction; not so in the other direction. The worry, then, is that maybe this difference makes a difference with respect to how hard or easy it is to traverse an infinite. In other words, even if it’s impossible to traverse such an infinite by beginning at some point and then ending (after all, there is no end! This is the point you’ve got your finger on), perhaps it’s possible to traverse it by never beginning and then ending. After all, the latter has an end to reach, unlike the former.

So you see, the epistemic status of (2) is quite different from (1). The former is self-evident; the latter is not. In fact, I don’t see any reason to think it’s slightly more reasonable than not. In all honesty then, if this argument is aimed at me, then I am still left waiting for a reason to accept the conclusion.

Re: burden-of-proof issues:
I agree that I haven’t given you a positive reason to think that the past is beginningless. But I’m not trying to do that; nor do I have any intention of doing so.


Wrightsaid: Should we assume, simply because some would like to suggest that this is at all possible, that they are championing anything coherent?

When you respond by saying that one way of traversing an infinite series can be described as, "*Always* being in a state in which some sequence of events is in progress, until the sequence is complete" sounds as though you were merely disagreeing, without explanation. As long as the progression has been from "always," an infinite series can be traversed. But that not only begs the question, as far as I can see, it also seems (at least to me) to be positing a logical absurdity.


Me: I think that perhaps you’ve misconstrued what I’m trying to do when I talk about this other way to traverse an infinite. I’m not putting it out for you to accept, as though I were arguing for it. Rather, I’m doing something much weaker: I’m saying that here is something that for all I can tell is possible, and that no one has shown that there is something wrong with it. Philosophers thus distinguish between metaphysical possibility – i.e., a state of affairs that really could obtain – and epistemic possibility – i.e., a state of affairs that we can’t as of yet rule out as metaphysically impossible. I know that this sounds nitpicky, but it really isn’t. It’s distinction that makes a material difference.

In fact, many have argued that traditional theism is dead in the water without this distinction. For example, there’s near unanimity among philosophers who study such things that Alvin Plantinga’s version of the “freewill defense” against the logical problem of evil is the best there is. His defense hinges on the following claim:

Every free creature that God could have created is such that it goes wrong with respect to at least one moral action.

If this is metaphysically possible, then even given God’s maximal greatness, he literally cannot make a world with free creatures that never do evil. For there may well be possible worlds in which free creatures never do evil, but God can’t actualize them (because it’s partly up to the creatures as to what they freely do). But if so, then a key premise in the logical problem of evil is false, rendering it unsound.

Now, many philosophers have argued powerfully that such a claim is implausible. But if Plantinga’s goal isn’t to convert the unconverted on this, but rather to defend himself, then he doesn’t need to argue that it’s metaphysically possible; he only needs to trot it out, and say that he has no reason to rule it out – i.e., that it’s epistemically possible, and that until someone shows that it’s metaphysically impossible, he’s in his epistemic rights to reject the logical problem of evil.

Now I don’t think that a beginningless infinite traversal is as implausible as Plantinga’s claim, but that’s not the point. The point is that both claims are asserted as epistemic possibilities – states of affairs that we are as of yet unable to rule out as metaphysically impossible – and that as such, we’re not trying to convince the unconvinced that they’re metaphysically possible. They’re defensive claims, not an offensive ones.

Re: your argument against beginningless traversals:
Wrightsaid: From what I understand of all of this (which perhaps isn't much), I think it would be best to disagree with William Lane Craig. The infinite series should not in the least have been exstinguished long ago. Rather, let's assume I were to take a road trip back in time endlessly into the past while another person were on a similar road trip moving into the future at a steady pace toward me from no point or place in the past (on a beginnless journey, if that makes sense). By the nature of an infinite, the distance between us should always remain the same: endless. No matter how far back I travelled, the two of us would never meet and we would be always the same distance apart, endlessly apart.

If we can never meet given our mutual efforts, how then can one catch up alone? If we imagine this as a number line, the same is true precisely because we would not be beginning from different sides of a number line (with a finite distance separating us). I would be beginning from one side (at zero) and my partner would be beginning from nowhere, at no point along the number line. Wouldn't he always be infinitely beyond me exactly because, in a sense, he is not really anywhere ahead of me to be reached at all?



Me: I like your vivid illustration of Moreland’s argument. But to repeat some points above: (i) Grant it as agreed all around that you, the person going toward their friend from a starting point, can never reach them. (ii) But why think that your friend is “nowhere”? You seem to be assuming that to progress, you have to start at some point. But again, why, exactly, are we supposed to think this?

Also, it looks as though you’re misconstruing the nature of the relevant sort of infinite. To say that, e.g., a line is infinitely long, it doesn’t follow that any points on the line are infinitely distant from any other. For example, if the points are numbered and given an w order-type, with the second point one foot from the first, the third point three feet from the first, etc., no point is infinitely distant from any other.

To look at it from another angle: consider the set of natural numbers with order-type w: {1, 2, 3, …}. The set itself has infinitely members. But no number in the set is “infinitely distant” from any other. Rather, each number in the infinite set is only “finitely distant”. Sorry for the metaphorical language re: the numbers having distances, but I think the point comes across nonetheless. These sorts of points are addressed in my posted essay.

Wrightsaid: But should we assume that what is logically contradictory can in fact be accomplished so long as we're given enough time?



Me: I’m still waiting for the reason why it’s logically contradictory. Again, it had better not be because “we can’t get to the end of an endless line”. Because that’s granted, but of no help.

Regards,

exapologist

exapologist said...

Whoops! Looking over my post, I noticed some bloopers. I thought I got them all when I deleted a version of my last post, but one got away, viz., this one near the bottom:

(i) Grant it as agreed all around that you, the person going toward their friend from a starting point, can never reach them.

I should have said instead:

(i' )grant it as agreed all aroung that you, the person going backward in time, can never reach a beginning if it lacks one.

I actually think that you *would* meet at some point, due mainly to your friend's beginningless traversal -- at least this is epistemically possible.



Best,

exapologist

exapologist said...

At this point, it’s a good idea to summarize the main topics discussed within this thread of comments, and the main points discussed regarding these topics.

I’m agnostic about whether or not the past is beginningless. This is because I lack persuasive evidence for believing one over the other, and therefore suspend judgment on this matter. It just seems that the arguments I’ve read on this matter – including Moreland’s and Craig’s – aren’t persuasive. Thus, from where I stand, both a past with a beginning and a beginningless past are epistemically possible.

With regard to the latter part of the last claim, this means that I think that it hasn’t been shown that there is an explicit or implicit contradiction in the idea that before every event, there is another one. I conceive of such a past as one where the set of past, say, years is isomorphic with an A0 set of negative integers with the order-type w* -- i.e., {…, -3, -2, -1}, where ‘-1’ denotes the present moment. According to this conception, then, although the series of events as a whole is actually infinite, no event within the series is infinitely distant from any other (as opposed to an infinite past with the set of years isomorphic with, say, an A0 set of negative integers with order-type 1+w* -- i.e., {1…, -3, -2, -1}. On this conception of an infinite past, there is both a first event – the one assigned the 1 – and the first event is infinitely distant from all the other events. Such a past is of course non-traversable, since there is no immediate successor to the event assigned the 1, and no series is traversable if it contains an event without an immediate successor. But since the w*-way I’m conceiving of a beginningless past, no such problematic event exists within it, and thus doesn’t have this particular problem regarding traversing it.

Closely related to this claim is this one, which I also accept: it’s epistemically possible to traverse a beginningless past – for a beginningless series of events to elapse one after the other. Now some people who don’t think that such traversals are possible get hung up on at least one of the following two thoughts: (i) that such a traversal could never get going, since there’s no starting point within the series from which to even initiate such a traversal; (ii) that such a traversal could never be completed, because it would involve going from a finite set to an infinite set by successive addition. But this is impossible because this is equivalent to saying that one can get to the end of something that has no end, which is a contradiction. But both (i) and (ii) don’t seem persuasive.

With regard to (i): of course the traversal could never get going, but no one is claiming that it could. Rather, by the very nature of the case, a beginningless series has no beginning point from which it “got going”. It’s always been going, in the sense that prior to every event, there is another event the caused it. There is no transition from an eventless state to the first event, on such a conception of the past. Furthermore, while it’s true that in traversing such a series you never get to a point in which an infinite is traversed, this is only because an infinite series is already crossed at every point in a beginningless past – albeit a different and new one with each event.

With regard to (ii): of course you can’t get to the end of something that has no end. But this isn’t what’s involved in traversing a beginningless past. For traversing an infinite past does involve an end point – it just lacks a beginning point. So, for example, the set w* {…,-3, -2, -1} has the endpoint of -1. Thus, to say that traversing a beginningless past involves getting to the end of something that has no end is to conflate traversing an w* series {…, -3, -2, -1} with traversing an w series {1, 2, 3, …}.

Now I’ve considered three different arguments against the possibility of a beginningless past: one from Craig, one from Moreland, and one from Wrightsaid.

Craig’s argument is that a beginningless past entails that every event should already be behind us. But this argument falsely assumes that to count down an infinite number of negative integers entails counting all of them. But of course you can count down an infinite number of integers, and still have more to count. So, for example, an immortal counter could count the infinite set of negative integers {…, -5, -4, -3}, and yet still have more to count, viz., -2 and -1.

Moreland’s argument is that traversing an infinite of order-type w* is just as easy or hard to accomplish, irrespective of the direction you traverse it. Thus, since you can’t traverse it from the present all the way through the past, you can’t traverse it from the past to the present. But since the latter has been accomplished, it follows that the past had a beginning. But why think that it’s just as easy or hard to traverse it, irrespective of direction? If we were talking about the infinite series w*+w – i.e., {…, -3, -2, -1, 1, 2, 3, …}, then this makes sense. For then there’s a clear symmetry with respect to direction of traversal. However, this isn’t the case with an w* traversal – {…, -3, -2, -1}. In this case, there are several asymmetries that seem relevant to difficulty or ease of traversal, depending on direction. Thus: (i) Going forward, you don’t have to begin at some point; not so going forward.
(ii) Going forward, there is an endpoint to reach; not so going backward. And, most saliently, (iii) Going forward, an infinite traversal is already completed at every point; not so going backward. Prima facie, then, there seems to be good reason that Moreland is wrong about this: direction of traversal does seem to make a difference with respect to difficulty of traversal. Or, more weakly: Given these asymmetries, at the very least we must say that Moreland owes us an explanation as to why they have no bearing on ease or difficulty of traversing a beginningless past.

Wrightsaid has two versions of his argument. One is that if the past were beginningless, then it entails that if two people were traversing it from opposite “ends” of the timeline, then they could never meet. For (a) the friend going back in time must reach the “edge” of time, and there is no such “edge” by definition. And (b) the friend coming from the beginningless past should be nowhere to be found, as she never began traversing. But if they can’t reach each other even when people are traversing in both directions, how much less possible is it that one can do it in one direction, viz., from the past to the present?


The worry is that while it seems true that the friend starting at the present and traversing backwards will never reach an end – after all, there isn’t one! – the person traversing forward has an end in sight (and “edge”), viz., the present moment, and thus lacks the logical problem of traversing that their friend lacks. Also, this friend has already traversed an infinite temporal distant at every point in the past, and thus doesn’t suffer the difficulty of going from a finite distance to an infinite distance, as their friend does traversing from the present to the past. Thus, why think that the friend coming from the begininningless past won’t reach their friend from the present? The only reason I can think of is that one is assuming that the only way to traverse something is to begin at some starting point. But why think that? If this were self-evident, then there would be no need to bring up these arguments. We could just say that since an infinite past has no starting point, the past must be finite. Unfortunately, however, the claim that all traversals require a starting point lacks the self-evident nature of the claim that you can’t traverse to the end of something that has no end.

Wrightsaid’s second versionof the argument is that if the past were beginningless, then it entails that if two people were traversing a beginningless past from opposite “ends” of the timeline, then there both must be and couldn’t be a mid-point at which they met, which is a contradiction. So, by reductio, a beginningless past is impossible.

But the problem with this argument is that a beginningless past lacks an endpoint in one direction, and nothing that lacks two endpoints could have a midpoint.

A recurring claim is that the only way to traverse something is by starting at some point. But I’ve pointed out that there seems no good reason to think so. For while that kind of traversal is certainly possible (and actual!), a second way seems epistemically possible. Thus, the two epistemically possible options are:

(i) Going from a state in which some sequence of events is not initiated to a state in which it is initiated, and then progressing until the sequence is complete.

and

(ii) Always being in a state in which some sequence of events is in progress, until the sequence is complete.


In closing, I should point out that there is a good amount of recent literature that takes Craig to task on the Kalam argument in general and a beginningless past on the other. The most forceful critic is Wes Morriston, a philosopher at the University of Colorado, Boulder. You can Google him and go to his website, and download his papers on these points.


I’ve certainly learned a lot from this discussion, but I’m afraid that the school year is about to start, and I have to work on my dissertation and prepare for the courses I’ll be teaching. I hope to check in from time to time. Until then, I’ll let Wrightsaid have the parting shot, if he (right?) so desires.

Regards,

exapologist

Wrightsaid said...

This is an excellent conversation and I've gained a great deal from it. I am grateful...

I apologize for throwing out so much at once. I was collecting my thoughts on this subject and perhaps rambled a bit as I tried to find the best way of expressing the problem I was having. I have gathered them sufficiently and I believe I may understand the problem. If I can ask for your patience a little longer, I want to respect and honor your request to discuss this a little at at a time. However, I have to make an important comparison to make apparent where I think the crucial confusion lies.

You agreed that, "it seems that nothing that lacks an endpoint could have a midpoint. It seems that it's a necessary truth that something has a midpoint just in case it has two endpoints".

You also related, quite correctly, that no two points on the numberline are at an infinite distance from each other. True, very true. But I encourage you to consider afresh that this fact has no bearing on our problem. I would even say that the way you've attempted to use this mathematical relation seems guilty of the fallacy of composition in assuming that what is true of the parts of an infinite numberline is true of the whole. No two points are infinitely distant to be sure, but there are an infinite number of points.

In other words, what I see you doing is grasping perfectly the infinite nature of the problem when thinking in terms of travelling the distance from future to past. But when you consider someone travelling from past to future, you equivocate (if that is the right term) and conceive of the infinite numberline as being made up of a number of finite distances, concentrating specifically upon the distance between two points that aren't relevant to the whole. But I believe it is correct to say that an infinite line isn't simply made up of additions of finite distances. If it isn't, then your statement about finite distances between specific points doesn't resolve anything.

When you rightly consider the nature of the whole distance, you realize that no midpoint anywhere exists. But even if we think in terms of future to past, if your point that any two numbers, no matter how far apart on the number line, are only finitely separated is at all poignant to this discussion, then a midpoint between the present and an infinite past should exist. It doesn't precisely because an infinite is not merely a conglomeration of numbers only finitely distant from each other and I think you see this from one way but not another.

That's why you can say that a person "always" travelling from the past can reach the present but cannot, upon reaching the present, turn right around and travel back the same distance "always" into the past. The time traveller can't cover the road that he just covered precisely because, in one direction, you perceive the nature of an infinite and in the other, you distort it slightly and it causes you to arrive at contradictions.

For instance, if no midpoint exists between the infinite past and the present, then a midpoint obviously cannot be reached from either direction. That's true by definition. If it doesn't exist, it cannot be reached. If no midpoint can be reached, then the whole cannot be traversed, no matter how long someone tries. That also is true by definition. If one cannot reach midway, then one cannot reach beyond midway.

Yet, you've tried to have it both ways, explaining that from one direction, a midpoint exists and from another it doesn't, as though curiously midpoints mathematically pop into existence depending upon one's perspective or direction or time spent travelling. You say that, from one perspective (past-to-present), an infinite can be traversed because "no two points on an infinite line are at an infinite distance." You then seem to suggest that from the other perspective (present-to-past) the fact that no two points anywhere along the exact same line are at an infinite distance from each other no longer helps us. The whole cannot be traversed.

This reasoning should work the other way round because it is mathematically true regardless of perspective or direction or time spent travelling, but your change of conclusion depending upon these factors would seem to alter the mathematical relations of parts to whole.

Ultimately, it doesn't matter at all that no two points on the line are at a finite distance precisely because no two points on the line can ever represent the magnitude of the distance we're dealing with. And if no two points can deal with the magnitude, then the distance between any two points has nothing to do with the problem. That is why no midpoint can be calculated...

...and that is also why I talk about our past time traveller as existing nowhere (at no point) on our number line.

Wrightsaid said...

Exapologist, it looks like you're going to be busy and so I wanted to thank you for so patiently exploring this with me and helping me to clarify many things regarding the way both sides might look at this issue. You were very kind to set aside so much of your time and put so much effort into this.

I was in the middle of writing my post when you posted so I apologize as I was not able to take into consideration what you had just written regarding my "midpoint" argument.

It would probably not have changed my response in any significant way because, I must admit, I'm not sure I understood your rebuttal to it. I'll chalk that up to the lateness of the hour and my lack of sleep.

Again, thank you (and I'm a he).

Prup (aka Jim Benton) said...

I will not pretend to be able to follow this entire argument, but let me throw out some passing thoughts.

Much of it seems to me to be yet another variation of Xeno's Paradox which proves motion is impossible. (btw, doesn't the 'timeline' of necessity have to be 'C' rather than 'Aleph-null' and does this have any relevance to the argument?)

Can we argue that the obvious possibilities are exhaustive? Certainly I can conceive of others that seem to be logically undisproveable. (Not that I would accept them.) Thus the Universe could have begun to exist at any point of time from five billion years ago to five seconds ago complete with evidence of a past existence. The Universe could exist in an unreachable macro-Universe (I describe this as the 'high school experiment thesis,' that is that our Universe is the result of an experiment in Universe Creation in this macr Universe, one of millions done by the equivalent of High School students in a lab course in Universe Creation. Then there is the possibility -- if science rules out the 'pulsating Universe' that implies that the Universe will 'reverse itself' and come crashing in on itself to form a new 'primoridial atom' -- that once all the atoms of this Universe are dispersed, that each of them will have a sufficiently complex structure to serve as the new primoridial atom for the creation of a new Universe.

(Isaac Asimov had yet another, more reasonable, speculation involving the four-dimensional structure of space-time, but it was too clever to attempt to quote from memory and I no longer have the article in which he suggested it. If anyone remembers it, hopefully he can quote it correctly.)

The main problem, however, is that, even granting the existing of a creator, for the sake of argument, even granting the premise of Intelligent Design, this merely brings you to a Deistical God. Without showing that there has been specific interaction between this creator and his creation, you can not bridge the gap between the idea of a Deistical God and a Theistical God.

(In fact, and I hope to be expanding on this for a CotG post, there is no operational way of distinguishing between the following four possibilities:
a: a self-existent Universe
b: a Deistical God
c: a Theistical God who will, at some point interact with 'humanity' but hasn't yet
d: a Theistical God who has created the Universe for the benefit of bengs existing on another planet of another star, with us merely being an unnoticed by-product of this Creation.

Thus, the 'argument from first causes' even if acceptable -- needless to say I doubt this -- in fact proves nothing of what Theists attempt to use it for.

Wrightsaid said...

Hello, Mr. Benton... I don't know whether you were writing any of this to me, so I'll leave much of what you suggested for the author of the original article to respond to if he wishes.

Concerning the extent of the usefulness of such proofs, I agree with you that they lead only to what is often referred to as the ambiguous, "God of the philosophers." Nevertheless, that's not much of a criticism when the argument was never intended to lead one, in a single blow, to organized religion.

Usually, arguments of this kind are intended to form the basis of a cummulative case. Other lines of inquiry would be pursued after demonstrating that this vague theism is true in order to determine the more precise nature of the "God" that exists. In that case, whether arguments like Craig's prove what they set out to prove, no one should expect them to do more than they were intended for.

Prup (aka Jim Benton) said...

Wrightsaid:
You agree with me on something which, not only did I not say, but which was precisely the opposite to what I did say.

No, these arguments do not lead to an "ambiguous (theistic) 'God of the Philosophers'" or to a "vague theism." My entire point was that proof of the existence of a Creator -- presuming such proof is possible, which I deny -- only leads to proof of a Deistic God.

To get from a Deistic to a Theistic God requires evidence, not reasoning. You have to demonstrate an interaction between this god and humanity (and demonstrate that it is this god, or a messenger from him that is interacting and not an impostor). And you can't do this through reasoning.

(I will have more to say on this over the next few days, but once I finally got started on this, my refrigerator died, and that took precedence. Sorry, but i wasn't going to wait another day to put the basics down.)

exapologist said...

Hello Wrightsaid,

How are you doing? Thanks for your kind remarks. I’m enjoying our discussion as well.

I noticed that you posted a new set of remarks, and since I had a moment, I thought I’d reply.

Re: your last main post: It seems to me that there are at least three confusions here:

I.
In your “time-traveling twins” argument, it seemed to me that you were construing the relevant sort of infinite past as one having a set of events isomorphic with the order-type 1+w*, or at least some order-type other than w or w*. I thought this because you said that your friend traveling from a beginningless past should be “nowhere” – i.e., nowhere within the w* series. It was for this reason that I raised the point about the relevant sort of infinite having only finitely distant points. If you recall my posted essay, I went on at length on this point in section I, as it’s a common mistake in discussions of the Kalam argument (at one time, William Lane Craig thought he had an argument related to the Tristram Shandy Paradox that showed that a beginningless past entails that it has events infinitely far in the past. However, as I mention in footnote (4) of my posted essay, he has since abandoned that argument). As in the essay, my reason for bringing up this point in a previous post was to indicate that traversing a beginningless past isn’t ruled out a priori due to a logical ban on traversing points that have either no immediate predecessor, or no immediate successor (I label this logical ban on traversing non-w-ordered distances ‘LB’ in my essay).

However, in your most recent post, you make it sound as though I’m using the point as a premise in an argument for the possibility of traversing a beginningless past. That is, as though I were arguing:

1. Every proper subset of points on a beginningless line is finite and traversable.
2. Therefore, the whole set of points on a beginningless line is finite and traversable.

Obviously, to infer (2) from (1) is to commit the fallacy of composition. But equally obviously, I’ve made no such argument. My point in the earlier post (and in my posted essay) was the simple negative one that a beginningless past of the sort we’re considering isn’t ruled out by having events in the series of events that lack immediate predecessors or successors; it wasn’t the positive one exhibited in the inference from (1) to (2) above.

Interestingly, William Lane Craig makes this same mistaken attribution to the late J.L. Mackie. It’s instructive to point out where Craig goes wrong in this criticism of Mackie, as I think it’ll reveal an illicit assumption in our own discussion. So let me discuss the relevant part of the dialectic between Mackie and Craig.

Mackie replied to Craig’s kalam argument in his book, The Miracle of Theism. He took Craig to be arguing as follows:

1. To traverse a beginningless past, you must go from a state of not having traversed an infinite set of events to having fully traversed one.
2. It’s impossible in principle to go from a state of not having traversed an infinite set of events to having fully traversed one.
3. Therefore, it’s impossible in principle to traverse a beginningless past.

Mackie then replies that the argument wrongly construes a beginningless past as a past with a beginning infinitely distant from the present. But to take a beginningless past seriously, you must construe it as having no beginning at all – not even one infinitely distant in the past. But if the past is beginningless, then only a finite set of events needs to be traversed from any point in the past to reach the present.

Craig comes back and says that it’s Mackie, not Craig, who fails to take a beginningless past seriously. For Craig construes such a past as having no beginning at all – not even one infinitely distant from the present. But if so, then this makes the problem worse, not better. For then one couldn’t even get going to make progress to reach the present moment. Furthermore, Mackie’s point that each point in a beginningless past is only finitely distant from the present is irrelevant. For the issue isn’t whether any finite segment of a beginningless past can be traversed to reach the present, but rather whether the whole infinite past can be so traversed. To think that a whole infinite set can be traversed because each finite segment can be traversed is to commit the fallacy of composition.


What to make of this exchange? Mackie is correct, and Craig has misunderstood him. Let me explain point by point:

First Mackie is correct to say that Craig has misconstrued a beginningless traversal. For to say that the past is beginningless is to say that some infinite set of events or other has been traversed before every point in the past. But if so, then there can be no going from a state of not traversing an infinite to one of traversing an infinite in a beginningless past. The only way in which one could go from a finite to an infinite traversal is if you began at some starting point. And this is why Mackie says that Craig conflates a beginningless past (i.e., {…, -3, -2, -1}) with a past that had a beginning an infinite amount of time ago (i.e., {1, 2, 3, …} or {1, …-3, -2, -1}).

Second, in light of the previous point, we see why Craig is mistaken to say that Mackie has committed the fallacy of composition. For Mackie is not arguing that because every finite segment of a beginningless past is traversable, the whole infinite past traversable. Rather, he’s saying that since an infinite set of events has been traversed before every point of a beginningless past, there is only a finite set of subsequent events between that point and the present.

Now at this point, it’s tempting to think that if this is the way to interpret Mackie’s reply, then Mackie’s point is even less plausible than Craig makes it out to be. For if an infinite set is traversed before every event of an infinite past – in which case you never have to go from a finite to an infinite traversal at any point – then doesn’t this mean that the past has infinite proper subsets that weren’t formed by successive addition?

No, it doesn’t. For before each event of such a past, a different infinite set exists – each proper subset has been formed by successive addition. To think otherwise is to commit an illicit quantifier shift fallacy, from

1) Every point in a beginningless past is such that there exists an actual infinite set of events that existed prior to it.

to

2) There is an actual infinite set of events, such that it exists prior to every point in a beginningless past.


This is the same fallacy committed in the following piece of reasoning:

1) Every person is such that there exists a set of parents that directly begat them.

Therefore,

2) There exists a set of parents, such they directly begat every person.


In short, a beginningless past entails that an infinite set of past events exists and has been traversed prior to every event. But it's a new set every time – one that contains the prior set of events, plus one more. Thus, a beginningless past contains no "core" infinite proper subset that was not formed by successive addition, in which case it's not guilty of the inconsistency of both containing and not containing an untraversed proper subset of events.


II.
I don’t follow you when you say that an infinite line is made up of finite distances. This is exactly what it’s made up of – an A0 set of finite distances. That is, a set of finite distances that can be put into one-to-one correspondence with the natural numbers.

III.
Your “time-traveling twins” argument still has the problem with a lack of two endpoints. It’s hard to know where to begin here. First, you seem to agree with this, but then you seem to think this somehow makes things worse for traversing a beginningless past. If I understand you correctly, you’re arguing:

1. If a traversal is possible, then it has a midpoint.
2. A beginningless past has no midpoint.
3. Therefore, a beginningless traversal is impossible.

But I’m not seeing why a traversal must have a midpoint. It’s certainly not self-evident or analytic. However, the following is both self-evident and analytic:

1’. If a traversal from one endpoint to another is possible, then it has a midpoint.

However, since a beginningless past lacks one endpoint, (1’) irrelevant for the purposes of establishing a finite past. But this was to be expected all along. After all, geometers discuss lines with only one endpoint, and they don’t see themselves as talking gibberish. If your argument worked, then, it would be a tour de force, since it would overturn lots of established geometry!


Second, the correct conclusion of the “time-traveling twins” argument is that given that there’s no midpoint, that your friend is traversing from a beginningless past, and you’re traversing from a starting point, you will meet at some point, such that your friend has an infinite set of events behind them, and you’ll have a finite set behind you. There’s neither a contradiction here, nor any problem whatsoever here, as far as I’m able to tell.


Third, at one point, you say that I’m inconsistent by wanting to have it both ways – saying that a midpoint exists going one direction, but not going in the other direction. But nowhere do I say that a midpoint exists in either direction. So I don’t know where you’re getting that.

Fourth, I think you’re still hung up on thinking that in order to traverse something, you have to begin at some point. I think that this is why you say that your friend traversing a beginningless past is nowhere. After all, if they have to begin, then since there’s no beginning to a beginningless past by definition, your friend could never have made progress, in which case they should be nowhere on the number line. But as I’ve said repeatedly, it’s neither self-evident nor analytic that all traversals must have a beginning point – do you have an argument for this assumption? If the past is beginningless, then some infinite or other is already traversed before every moment. I’ve explained above why it’s a mistake to think there’s any incoherence in this idea.

Finally, you also seem to still think that direction of traversal has no effect on difficulty of traversal. It’s of course granted that this is true of finite sets, but infinite sets are a different kettle of fish. In my previous post, I gave three reasons to push the burden of proof onto the person who claims this: (i) from past to present, there’s an endpoint to reach; not so from present to past; (ii) from past to present, you don’t have to begin the process of traversing; not so from present to past; (iii) from past to present, you’ve already traversed an infinite before every point (in which case you don’t ever have to go from a finite to an infinite traversal of events!); not so from present to past. Now granted, these three things are weird, but being weird is not the same as being self-contradictory.

In closing, please allow me to make three final comments:

(i) Before discussing the argument further, can I recommend that you first read the relevant literature on the topic? I don’t mean this as an affront – I very much enjoy our conversations, and you seem to be a fine fellow. It’s just that I think our discussions would be more fruitful if we had the key literature before us as a sort of context-setting framework. I can send you a bibliography if you like, but here’s a short list of some of the most important authors:

Pro:
-William Lane Craig
-J.P. Moreland
-David S. Oderberg

Con:
-J.L. Mackie
-Graham Oppy
-Quentin Smith
-Nicholas Everitt
-Wes Morriston
-Paul Draper

Craig is the most forceful defender of the Kalam argument. Smith used to be the best critic of the argument, but Morriston has taken up that title since the late 90s. His papers on the Kalam argument are now contained in philosophy of religion anthologies side-by-side with Craig’s exposition of the argument, e.g., Raymond Martin and Christopher Bernard’s anthology, God Matters.

(ii) Given that the issues here are deep and subtle, perhaps it’s a bit optimistic to think that one is going to come up with a new, sound version of the Kalam argument where everyone else has failed – all with no graduate training in philosophy, and on couple of week’s worth of reflection on the matter? William Lane Craig wrote a book scrutinizing the cosmological argument in all of it’s important forms from Plato to Leibniz, and then wrote a book devoted to the Arabic, Kalam version of it. Perhaps, then, it might take a bit more work to get one’s bearing on the argument?

(iii) I deeply sympathize with you. I used to accept and defend the Kalam argument for many years. I used to make the same sorts of arguments and points that you’ve made in our discussion. It’s just that the more philosophical training I received, and the more I read and re-read the relevant literature, I gradually realized that all my assumptions that led me to accept the argument turned out to be unwarranted or mistaken upon closer scrutiny. I’ve debated and discussed the argument with other philosophers for years, and so I can assure you that I understand where you’re coming from.

Regards,

exapologist

Wrightsaid said...

Thank you for a very helpful response, exapologist. That does clarify quite a bit that I don't think I had perceived in your previous comments. I apologize that I haven't commented sooner but I didn't expect you were returning... I disocvered your remarks serendipitously while looking to discover whether anyone else might have remarked on it.

Let me say that I think you have misunderstood my intentions. I have no desire to defend the Kalam Cosmological Argument, an argument I reject for other reasons than we have taken up here. I have plenty of disagreements with William Lane Craig, so the idea that he is erring or missing the point would not cause me any trouble...

I never desired to defend the argument and I don't even have that much interest in it. Rather, I wanted to explore a single philosophical question with you that is to me separate from the Kalam itself which had captured my interest years ago. It would not be fair to say that I have only given this subject a "couple of week’s worth of reflection." I have read some of the relevent literature you cited and, of all those he's debated, am most familiar with Craig's exchanges with Quentin Smith, though I have certainly not covered the extent of all this as you seem to have.

Please, don't misunderstand. I don't think anything I've said here is a new idea in itself. I am not so arrogant as to presume that someone, somewhere (probably rather recently), has not already worked through all this.

It merely inspires in me the feeling of creative effort because I worked through something in a way I've not seen before. All the better, indeed, if someone of a genuine reputation did come up with the same argument. That would please me even more precisely because the examination can go further (and has) through the efforts of better minds than my own.

This may all be old hat to you and that is why I am grateful for your time. I simply know (and I think you would agree) that it is always a good idea to try to speak with and learn from those who have gone further down a certain path than you. We must take these opportunities when they arise, especially as our books can make very inadequate conversational partners.

There is more to say but, as I respect your request that I look further into the literature and, as this discussion has skirted off the edge of the DC blog and probably isn't attracting much attention, I suppose you should have the last word. Thank you once more for your efforts and your patience.

Wrightsaid said...

I don't want to be rude and ignore Mr. Benton. I apologize also to you, sir, as I should have returned sooner to check on any possible replies.

No, these arguments do not lead to an "ambiguous (theistic) 'God of the Philosophers'" or to a "vague theism." My entire point was that proof of the existence of a Creator -- presuming such proof is possible, which I deny -- only leads to proof of a Deistic God. To get from a Deistic to a Theistic God requires evidence, not reasoning.

I think we are talking past each other here. Although you seem to argue from the premise that "Theism" and "Deism" are somewhat opposing categories, Mr. Loftus does not seem to agree with this in his book. I have read other atheists who include Deism in the overall category of theistic philosophies. One would have to define "Theism" very precisely to exclude Deism, especially if Mr. Loftus is correct that one "can be a deist that allows for miracles" (Why I Rejected Christianity, 39). In that sense, then, the Kalam Cosmological Argument does lead to an "ambiguous" theism, especially as there is more than one version of Deism possible.

And I confess I don't know what you mean by "requires evidence, not reasoning." Surely even the interpretation of evidence requires reasoning. If you meant that the move from Deism to Theism is inductive, not deductive, how does that lead to the conclusion that "the 'argument from first causes' even if acceptable. . . proves nothing of what Theists attempt to use it for"?

But all "Theists attempt to use" the Kalam argument for is to establish precisely what the argument establishes (if true). It is other arguments that are relied upon to further a Christian apologetic, whether inductive or deductive. Is it your suggestion that inductive argumentation is useless or without merit?

If you mean to imply instead that Christianity simply cannot (logically) be verified inductively, that would seem to be true only IF GOD is a logically necessary falsehood, internally contradictory. I don't think that is your position as you granted the possibility that the Kalam could be true.

If you are merely saying that the evidence does not lead to GOD, that may be your opinion but it hardly reflects poorly upon the Kalam argument itself or those who use it simply because they do not agree with you.

In either case, I do not see the rationale behind your comment. Perhaps I have missed what you were demonstrating and, in that case, I would be happy to be corrected.