Wednesday, June 19, 2013

Two Inadequate Arguments for a Finite Past

In this post I will consider two arguments which have at times been brought up in connection with the Kalam Cosmological Argument (KCA), which I will call the "subtraction argument" and the "argument from traversing an infinite," the former of which I have heard from Dr. William Lane Craig. The KCA goes as follows:

1. Whatever begins to exist has a cause.
2. The universe began to exist.
3. So the universe has a cause.

The arguments in question are designed to defend the second premise, which is presumably implied by the past's being finite. I should note that I think the second premise is true and there is strong evidence in favor of its truth. Alexander Pruss has given an excellent argument here, to which I have heard no compelling reply. I also think there is very strong scientific indication of the premise's truth, which Craig has adequately demonstrated. I just don't think these two arguments demonstrate its truth.

The "argument from traversing an infinite" goes something like this:

1*. If the past were infinite, one would have to cross an infinite temporal distance to get to the present moment.
2*. If one had to cross an infinite temporal distance to get to the present moment, then one could not get to the present moment.
3*. So if the past were infinite, then one could not get to the present moment.
4*. But we are at the present moment.
5*. So the past is not infinite.

The argument requires some unpacking. First of all, to say the past is finite is to say there was a beginning of time, and to say the past is infinite is to say there was no beginning of time. Second, 'temporal distance' means the length of time between one moment and another. There is a perfectly good way to define finite temporal distance. If we take our measure of time as a second, we can assign the current time the number 0, the time one second ago -1, the time two seconds ago -2, and so forth. To find the temporal distance from one time t1 to another t2, we take the number assigned to t1 and the number assigned to t2, and take the absolute value of the difference between the two. For instance, take the time 1000 seconds ago. To find the temporal distance from that time to the present time you take the absolute value of -1000 minus 0, which is of course 1000 seconds. Pretty simple.

However, problems begin to arise when we start to talk about an "infinite temporal distance." This phrase is ambiguous, and depending on which interpretation of this phrase we take it will either cause problems for premise 1* or for premise 2*. First, the phrase could mean something analogous to the way finite temporal distance has been defined above. However, infinity is not a real number, so you simply cannot define an infinite temporal distance the same way as above. There is no number "-infinity" from which you can subtract, say, -5. So if this is what is meant, then premise 1* appears to be false, since no real sense can be given to an infinite distance in this way.

On the other hand, crossing an infinite temporal distance could just mean that the set of all the numbers assigned to the seconds is infinitely large. This makes perfectly good sense of the phrase, but then in that case it is not clear why premise 2* is true. As Thomas Aquinas points out, there being an "infinite temporal distance" in this sense is perfectly consistent with all the temporal distances from the past to the present being finite, where "temporal distance" is defined as it was earlier:

"Passage is always understood as being from term to term. Whatever bygone day we choose, from it to the present day there is a finite number of days which can be passed through. The objection, however, is founded on the idea that, given two extremes, there is an infinite number of mean terms." [ST Ia q.46 a.2]

So for instance, the distance from the present to one second ago is 1 second, the distance from the present to two seconds ago is 2 seconds, etc. and so on forever and ever back into time. Hence, no matter how far you go back in time, the distance in the way I've defined above from any given past moment to the present will be finite, and thus you will only have to cross a finite number of seconds to get to the present moment. But of course any finite number of seconds can at least in principle be crossed; hence, premise 2* is false.

So much for the "traversing an infinite" argument then. The "subtraction argument" goes something like this:

1'. If the past were infinite, then an actual infinity would be possible.
2'. If an actual infinity were possible, then one could perform subtraction on infinities.
3'. But if one can perform subtraction on infinities, then one will get contradictory results.
4'. So if the past were infinite, then one would get contradictory results.
5'. So the past is not infinite.

This seems to be one of the arguments William Lane Craig gave in his debate against Peter Millican. Let me first point out an ambiguity in the phrase "actual infinity," after which I'll assess the argument under each interpretation. Here are two possible meanings of the phrase "actual infinity":

(ACT1) An actual infinity exists just in case for some time, at that time there exist distinct concrete objects such that the size of the set containing all and only them is infinite.

(ACT2) An actual infinity exists just in case there is some set containing only distinct concrete objects whose size is infinite.

Some elaboration is in order. First off, both definitions presume when speaking of actual infinities that we are dealing with concrete objects. While my understanding is that Craig does not believe there are infinitely many numbers (he is a nominalist), presumably his argument doesn't presuppose this view; Craig only wants to rule out the possibility of infinitely many concrete objects. Now as for the definitions themselves, the difference between (ACT1) and (ACT2) is that in (ACT1) you only have an actual infinity when all the concrete objects exist at the same time. In (ACT2) you could have finitely many objects at t2, finitely many at t1, and so forth, yet if you take a set containing concrete objects from different times, and the times go back to infinity, you will still have an actual infinity. So both of these definitions make fine sense. However, the assessment of the argument will depend on which interpretation we take.

Let's deal with the first definition, (ACT1). Given our definition of actual infinity in (ACT1), premise 1' does not appear to be true, or at least not obviously true. It is consistent with holding that the past is infinite that at each time there are only finitely many concrete objects. And if you believe only objects in the present moment exist, then 1' is definitely not true. The fact that there were objects at each time in the eternal past in no way implies an infinite collection of simultaneously-existing objects.

My main concern is with 2' and 3' though. Take 2', since it is also ambiguous to a certain extent. The problem is it is not immediately clear what is meant by "perform subtraction on infinities." Craig acknowledges that the ordinary operation of subtraction is not defined for "infinity". If he did mean this, i.e. the ordinary operation of subtraction, then clearly 3' would be true but 2' would have no support at all. So it cannot mean the ordinary operation of subtraction.

Now, Craig gives us an example to support 2' and 3', and this may help us understand what he means. Suppose we have an infinite number of coins. Then we can take away all the coins except three of them. And in this sense we can be said to perform subtraction on infinity, i.e. taking away some number of things from an infinitely large collection. This definition of performing subtraction on infinities makes sense.  But then why is 3' true? Craig says, considering our infinite number of coins, that you can take away infinitely many coins and be left with 3 coins, and thus infinity minus infinity = 3; but you can also take away infinitely many coins and be left with 2 coins, and thus infinity minus infinity = 2; hence, 2 = 3, which is our contradiction.

The problem with this argument is that it runs on an equivocation: We agreed that we are not using "performing subtraction" or "minus" in the normal sense of the arithmetical operation, since this just makes no sense. So "infinity minus infinity = 3" must simply be shorthand for saying "taking away infinitely many objects from an infinite collection leaves us with 3", and similarly with "infinity minus infinity = 2." But then if "2 = 3" means that 2 is identical to 3, then it certainly does not follow that 2 = 3; all that follows is that you can take away an infinite number of things and be left with 3, and also take away an infinite number of things and be left with 2, and this is certainly not a contradiction! It only looks like a contradiction when we are illicitly inferring "2 = 3", as if the phrase "infinity minus infinity = 2" were using "minus" and "=" in the same way as "5 minus 3 = 2." It would be like if I had infinitely many pennies and dimes, and I said, "infinity minus infinity = a penny, infinity minus infinity = a dime, so a penny = a dime." Clearly I am making an illicit inference here, and for the same reason Craig's argument makes an illicit inference as well.

Now, this whole time I have been working under the assumption that throughout the argument "actual infinity" is meant in the sense of (ACT1). But under interpretation (ACT2) the situation is even worse, since it is not clear 2' is true. It seems that in order to "subtract" infinitely many coins in the sense defined above, all of them must exist at the same time. But if "actual infinity" is taken in the sense of (ACT2), then it is not required that all of the infinite number of coins exist at the same time, and thus 2' has no support. And of course, with the exception of premise 1', all the same criticisms I have just given apply equally well under (ACT2). So, interpreted charitably, the argument seems to be a failure, with the primary problem being in premise 3'.

I should note one more time that, in spite of all my criticisms of these two arguments, I think there are good reasons for thinking the KCA is sound. I just don't think these are among them.

6 comments:

elliottroland said...

[I've had to break this comment into two, sorry.]
First off, cool blog, really enjoying reading it. I also really like Pruss' grim reaper argument (specifically because it's independent upon the theory of time we adhere to).

Now with regard to this post, I myself haven't read Craig's defenses of the argument from the impossibility of actual infinities, so I won't comment on that one. I do, however, find his argument from the impossibility of forming the infinite via successive addition to be quite compelling, so I thought I'd make a few comments about that.

First, as far as I'm aware, the formulation you gave isn't how Craig's formulating it these days. Perhaps it's how he formulated it a while back, so I'm not gna say you're wrong, but in the Blackwell Companion to Natural Theology paper (with Sinclair), Reasonable Faith (3rd ed.), and his and Moreland's Philosophical Foundations for a Christian Worldview, the argument is formulated as such:

1) The series of events in time is a collection formed by adding one member after another.
2) A collection formed by adding one member after another cannot be actually infinite.
3) Therefore, the series of events in time cannot be actually infinite.

He doesn't mention "temporal distance" at all. Not that your second interpretation of the term is wrong. He agrees that when speaking about an infinite past, we're not talking about an infinitely distant point from which we start counting. Rather, he understands infinite past merely as there being infinitely many days (or seconds, hours, minutes, whatever) prior to the present day.

Second, he defends the second premise there by noting that when you're successively adding one member after another, you always have a finite number. He notes that even if you turned this around (that is, count down from infinity instead up towards it), the problem wouldn't disappear, for if one can't count up to infinity, then surely one can't count down from it. We can see this, particularly, in the case where we're considering reaching a specific day in time. If the past is infinite, then before we can reach any specific day, there'll always a day prior to it that we need to reach first, so as a result we can never reach any day (in a finite past the sole exception to this is the first day). But of course this is absurd, since here we are! Here are his words from Reasonable Faith: "...before the present event could occur, the event immediately prior to it would have to occur; and before that event could occur, the event immediately prior to it would have to occur; and so on ad infinitum. So one gets driven back and back into the infinite past, making it impossible for any event to occur. Thus, if the series of past events were beginningless, the present event could not have occurred, which is absurd."

elliottroland said...

Third, unless Ive misread you he's addressed your main complaint a number of times. Consider his statement and response in On Guard: "Some critics have responded to this argument by pointing out that even in a beginningless past, every event in the past is only a finite distance from the present. Compare the series of negative numbers: ..., -3, -2, -1, 0. It's beginningless; nevertheless, any number you pick, say, -11 or -1,000,000 or whatever, is only finitely distant from zero. But the finite distance fro any past event to the present is easily crossed, just as you can count down to zero from any negative number you pick...

This objection commits a logical fallacy called the "fallacy of composition". This is the fallacy of confusing a property of a part with a property of the whole. For example, every part of an elephant may be light in weight, but that doesn't mean the whole elephant is light in weight!

In the case at hand, just because every finite part of a series can be crossed or counted doesn't mean the whole infinite series can be crossed or counted... The question is not how any finite part of the past can be formed by adding one event after another, but how the whole, beginningless past could be completed by adding one event after another."

I think the point can be illustrated again with the idea of successively counting up to infinity starting from zero. It is true that every number will eventually be counted (say you're counting a number per second, then after n seconds (for any positive integer n), the number n is counted), but it is not true that eventually all the numbers will be counted, since at any point there'll always be another number to count.

awatkins69 said...

Hey Elliott, thank you for reading and commenting! This was a bit of a longer post so I'm glad at least someone was able to make it through it. I apologize but I only have time for a couple quick remarks.

First, I actually didn't get the first argument from Craig and I wouldn't attribute it to him directly. I actually got it from Bonaventure and read it on a Scotist blog, here: http://lyfaber.blogspot.com/2007/12/bonaventure-against-eternity-of-world.html The point being it wasn't really directed at Craig in particular.

As far as Craig's actual argument which you've formulated, tentatively I am worried about the second premise. It seems to me that, in the defense of this premise, the word 'infinity' is being misused here in ways similar to the ones I criticized in the second half of my post. I think we'll need to get much more clear about phrases likes "actually infinite" and "adding by successive addition" and so forth; they may seem to have clear meanings on the face of it but as I hope this post has shown they actually are quite ambiguous! And I would suggest that when the meanings are clearly drawn out the premise will either not seem plausible or the conclusion will not follow.

As regards the charge of the fallacy of composition, I don't think I am committing it. I am arguing that in order to get to the present moment, all I need to be able to do is get from every day in the past to the present moment. If I can do that even given an infinite past then you can't object to an infinite past by saying we'd never get to the present moment. So, if this does not count as crossing an infinite series, then I have to first ask what it means to cross an infinite series, and second why it would be necessary to cross an infinite series in this sense to get to the present moment rather than just have to be able to get from every past moment to the present.

At the risk of sounding repetitive, like I've said, I do find the Kalam's premise to be adequately supported both by Pruss's Grim Reaper argument as well as the scientific evidence which Craig has brought forward.

elliottroland said...

I think I can help clarify the terms.

"Successive addition" means "the accrual of one new element at a (later) time" (Blackwell Companion to Natural Theology paper)

"Actual infinite collection" is meant in a sense closer to your ACT2. As we reach successive days we add them to the set.

So you can understand the action being performed in (2) as you starting with an empty (or singleton) set, and each day you add a number to the set that isn't in the set already. The second premise says that there will never be a day on which your set has infinite cardinality (actually infinite), even though the set's cardinality tends towards an infinite cardinality without reaching it (potentially infinite).

You could think of it as the sequence of natural numbers (n_i)_i where n_i = i. No element in this sequence is infinite (they're all natural numbers after all), but the sequence nonetheless tend diverges to positive infinity.

By "infinite past" I simply mean that the today was preceded by an infinite amount of days. No infinitely far starting point needed.

I fail to see how the statement "in order to get to the present moment, all I need to be able to do is get from every day in the past to the present moment" is any different from the statement "even in a beginningless past, every event in the past is only a finite distance from the present...But the finite distance from any past event to the present is easily crossed".

This does seem to commit the fallacy of composition, for "to get to the present" if time is infinite, is to traverse an infinite amount of days. So what you're saying, if I understand, is "to traverse an infinite number of days, all I need to be able to do is traverse any finite number of days". But the infinity of the collection of days is the very thing that makes it disanalogous to the finite cases.

awatkins69 said...

Thanks for the clarifications, they are helpful. I found your example of the sequence of natural numbers to be a good illustration in particular.

On these interpretations, I'm actually curious about premise (1) then of his original argument you've presented. Take the phrase "a collection formed by adding one member after another." This could just mean a collection where some of the members are added after the others; on this interpretation I do not see why 2 is plausible or justified, since it's only plausible if we assume we start with a finite set of days, which on the infinite past theory we do not (on an infinite past there is and always has been an infinite number of past days). If on the other hand the phrase "a collection formed by adding one member after another" means we start from a finite set and add on more days by successive addition, premise 2 is eminently plausible but then premise 1 becomes question-begging.

Now, you said, "'to get to the present' if time is infinite, is to traverse an infinite amount of days"

I don't think that is true though. *Traversing* is a relation between an object and some temporal interval. For instance, "I [the object] traversed [the relation] the time between 2010 and 2012 [the temporal interval]." You can define traversing for finite temporal intervals (any finite amount of days), but it is not clear what exactly it means to say I traverse an "infinite amount of days." And my point is I do not even see why the believer in the infinite past is committed to our having to traverse an infinite amount of days. So, no, I am not saying "To traverse an infinite number of days, all I need to be able to do is traverse any finite number of days," since I believe it is unclear what it means to traverse an infinite number of days.

Maybe you can solve this problem of meaning and say by "traverse an infinite number of days" is meant "traverse the interval from every past day to the present, and the set of all past days is infinitely large." But then clearly, by this definition, you can traverse an infinite number of days by just being able to traverse all the distances from every past day to the present. For I am saying just this: For all times t, to get to the present moment from t, all I need to do is be able to cross a finite number of days. This is my premise. And since any finite number of days can be crossed, and the number of days is infinite (in the well-defined sense above), it actually does follow, with no fallacy of composition being committed.

I think this is in essence what I want to say and any further comment on my part would just be an elaboration on this. So I will let you have the last word on this one! Again, sincere thanks for commenting and for the helpful remarks.

elliottroland said...

I'm not sure I understand where the confusion lies, to be honest. On presentism (which is the theory of time this argument is predicated on), the universe moves through days one at a time. It doesn't skip days or traverse chunks at a time. What I mean by "traverse days" or "move through days" is that each day is reached after the previous one: it's day t, then day t+1, then day t+2, ... It doesn't require two terminal points since we're not talking about traversing a range, but individual elements. So perhaps my use of the word "traverse" is slightly unhelpful for you, sorry.

You say, "on this interpretation I do not see why 2 is plausible or justified, since it's only plausible if we assume we start with a finite set of days, which on the infinite past theory we do not (on an infinite past there is and always has been an infinite number of past days)."

The problem is, on presentism, the traversal of days is successive, in the manner I spoke about two paragraphs ago. When we say we've traversed an infinite number of past days, it means that we've generated an infinite number via successive addition, which is what (2) denies the possibility of. It's like counting all the negative numbers in order, ending at 0. It just isn't possible. It's not as if you start with an infinite set and then only add finitely many elements after that, since that is equivalent to starting a day finitely far in the past, which we've seen is irrelevant. There is no starting point (that is, the events are beginningless), but the fact remains that in order to accrue an infintie collection of days, you've somehow managed to form an infinite through successive addition. You might say that this is very strange, haha, all the worse for an infinite past then :P

By "traverse and infinite number of days" I mean "successively reach each day, in order, of an infinite collection". (we "reach" a day when that day is present). In the case at hand (that is, infinite past events) it is isomorphic to saying "successivley count all negative numbers in order".

I agree that if I meant "traverse the interval from every past day to the present, and the set of all past days is infinitely large" then it would be possible, since each of these is just a finite interval, and I have no problem with traversing finite intervals. But this would just be the same misinterpretation of the problem that came up in the fallacy of composition issue earlier.

Again, I urge you to think of the analogous problem of couting all the positive integers starting from 0. To be able to count an infinite number of positive integers is not that same as being able to count any given finite subinterval [0,n]. Or with negative integers, to be able to count all the negative integers in order is not the same as being able to count any finite interval [-n,0]. Likewise, to be able to pass through an infinite number of past days is not the same as being able to traverse any given finite past interval. This, I think, is what Craig means by composition.

Thank you for the discussion. I value your priority of coming to a proper understanding of the terms being used. I look forward to future posts :)