XFiles: To infinity (and be wrong!)

(Book: On Guard, by William Lane Craig. Chapter 4: “Why did the universe begin?”)

We’re on a trip back to 12th-century Persia. Dr. Craig needs someone to rescue the Christian God from the perils of scientific advancement, and he thinks he has found a champion in a Muslim philosopher named Ghazali. (It’s amazing that Dr. Craig is able to successfully sell Muslim philosophy to conservative post-9/11 American Christians, don’t you think?) According to Ghazali, whatever begins to exist has a cause (premise 1), the universe began to exist (premise 2) and therefore the universe has a cause (conclusion).

Being a medieval Muslim philosopher, Ghazali did not have access to discoveries about particle physics and so on, so to prove premise number 2, he (and Dr. Craig) rely on philosophy. The argument is a bit flawed, however, in that it relies on a fundamental misunderstanding of what an infinity is. Curiously enough, it also contradicts the Christian doctrine of eternal life.

Ghazali argued that if the universe never began to exist, then there have been an infinite number of past events prior to today. But, he argued, an infinite number of things cannot exist… Ghazali recognized that a potentially infinite number of things could exist, but he denied that an actually infinite number of things could exist….When we say that something is potentially infinite, infinity serves merely as an ideal limit that is never reached. For example, you could divide distance in half, then into fourths, then into eights… The number of divisions is potentially infinite, in the sense that you could go on dividing endlessly. But you’d never arrive at an “infinitieth” division. You’d never have an actually infinite number of parts or divisions.

If there cannot be an actually infinite number of events in the past, then there also cannot be an infinite number of events in the future. Believers can get around this by saying that most of those events are only potential events because we haven’t experienced them yet. According to Christian cosmology, however, God exists outside of space and time, and is therefore not subject to the “haven’t experienced it yet.” All times are supposed to be immediately real to God, and thus if there cannot be an infinite number of them, then sooner or later there must come a day that will be the last day in the life of “immortal” believers and even God Himself. This means that, according to Dr. Craig’s arguments in Chapter 2, Christian life is devoid of all true meaning, value and purpose.

Sorry, Christians.

The fundamental mistake that Dr. Craig makes is a failure to really grasp a point that he alludes to above: there is no “infinitieth” division or anything else. Infinity means “not finite,” i.e. not constrained by limits, not a fixed quantity. There’s an old joke that says two plus two equals five, for sufficiently large values of two. (Sounds like something an engineer would say, eh?) The number two is a fixed quantity, a finite amount. It has limits. If it’s going to be two, then it cannot be more than two nor less than two. It’s a fixed quantity.

Infinity, by contrast, is not a fixed quantity, and that’s where Ghazali and Dr. Craig get themselves confused. I’m sure they know that, in dictionary terms, infinity is infinite, i.e. not finite, i.e. not a fixed quantity. Yet when it comes to practical applications, they expect it to have the same behaviors and characteristics of a finite quantity. They treat it like it was just some very large—but still fixed—number.

The way in which Ghazali brings out the real impossibility of an actually infinite number of things is by imagining what it would be like if such a collection could exist and then drawing out the absurd consequences.

Dr. Craig’s favorite example is the “Hilbert Hotel” imagined by mathematician David Hilbert. See if you can spot all the ways Dr. Craig assumes that infinity is a very large finite number.

[L]et’s imagine a hotel with an infinite number of rooms, and let’s suppose… that all of the rooms are full… There isn’t a single vacancy throughout the entire infinite hotel; every room already has somebody in it. Now suppose a new guest shows up at the front desk, asking for a room. “No problem,” says the manager. He moves the person who was staying in room #1 into room #2, the person who was staying in room # 2 into room #3…and so on to infinity. As a result of those room changes, room #1 now becomes vacant, and the new guest gratefully checks in, but before he arrived, all the rooms were already full!

Then an infinite number of guests show up, and the manager solves the problem by moving 1 to 2, 2 to 4, 3 to 6, 4 to 8, and so on, leaving all the odd-numbered rooms vacant. Since there are an infinite number of odd numbers, everybody gets a room even though the hotel was already full. Absurd, right?

One of the qualities of finite numbers is that you can compare two finite numbers to see if they are equal. You can say, for example, that the number of rooms in a finite hotel is equal to a finite number of guests (assuming one guest per room, for simplicity). Another quality of finite numbers is that you can subtract one from another. You can say, for example, that the number of guests is equal to the number of rooms, and therefore there are zero empty rooms left over.

Infinity, however, is not a finite number. It does not obey the same set of rules as a finite number, because it is not a fixed quantity. Thus, you can have an infinite number of guests in a hotel with an infinite number of rooms, and still have an infinite number of empty rooms available, allowing the manager to do his “magic.” You might think that infinity minus infinity must be equal to zero, since they’re both the same quantity, but that’s a misperception—you’re thinking of both infinities as being fixed quantities that you can compare and subtract.

I skipped over a section where Craig tries to dispense with Cantor and modern mathematics in order to make his point, but I think now would be a good time to go back and look at that.

It’s very frequently alleged that this kind of argument has been invalidated by developments in modern mathematics…

These developments in modern mathematics merely show that if you adopt certain axioms and rules, then you can talk about actually infinite collections in a consistent way, without contradicting yourself. All this accomplishes is showing how to set up a certain universe of discourse for talking consistently about actual infinities. But it does absolutely nothing to show that such mathematical entities really exist or that an actually infinite number of things can really exist. If Ghazali is right, then this universe of discourse may be regarded as just a fictional realm…

This from a guy who disparages materialists for not believing that souls and spirits are real! Discoveries that you can establish by rigorous application of mathematics are just “fiction,” but angels and demons and entire realms suggested by superstition and wishful thinking—they’re all real. And if you don’t believe it, then you must be a materialist.

Ah well. One of many points that Dr. Craig is overlooking here is that there are two alternatives open to us. One, we can embrace a mathematical understanding that lets us discuss infinity in a meaningful and consistent way. Or two, we can embrace a mathematical understanding that FAILS to allow us to discuss infinity in a meaningful and consistent way. By rejecting modern mathematics (option 1) as “fiction,” Dr. Craig is explicitly selecting the second alternative, and the “absurdities” he finds are merely a reflection of his poor choice of mathematical frameworks.

Think about it. What Dr. Craig is saying is that there cannot be an actually infinite number of integers greater than zero. In the real world, you can’t actually have numbers progressing onwards towards infinity. Somewhere out there is some finite positive integer that is The Biggest Number (because if there weren’t, that would be an actual infinity). But can such a number really exist? If you could add one to it (as finite numbers allow you to do), then it wouldn’t be the biggest number. The plus-one version would be the biggest number. But if you add one to that number…?

This kind of difficulty is what is properly called absurd, because the contradiction arises even though you’re correctly treating finite numbers like finite numbers. Dr. Craig’s “absurdity,” by contrast, is merely a reflection of the fundamental error that arises when you try to treat infinity like a finite number. It’s not that there’s anything deficient about infinity, it’s just that Dr. Craig is deficient in his understanding and application of mathematical principles.

Not all of his arguments are deficient, however. His second argument is that you cannot have an infinite number of events in the past, because if you did, you could not arrive at “now,” since you have an infinite distance to traverse before you get here. That’s not actually a bad argument, though it’s moot, given the origin of time at the Big Bang. He does a pretty good job of defending this against the “every point in the past is only a finite distance from the present” objection (fallacy of composition), though he doesn’t seem to appreciate the fact that a finite past leaves God without any time in which He could create time itself.

He got the main point right, though. Chalk one up for Dr. Craig. His next example fails again, however: suppose you have two planets that have been orbiting the same star for all eternity, and one takes twice as long as the other to complete one orbit. If they’ve been orbiting forever, which one has completed more orbits?

The answer is that the number of their orbits is exactly the same: infinity! (Don’t let someone try to slip out of this argument by saying infinity is not a number. In modern mathematics it is a number, the number of elements in the set [0, 1, 2, 3,  ...].)

Comparing two infinities as though they were fixed quantities. You can see it in his sly aside: infinity is a number, and not just any number. It is a specific number, the number of elements in the set of all natural numbers. By rejecting modern mathematics as a “fiction” that doesn’t actually exist in the real world, we are left with the kinds of numbers that are fixed quantities that can be compared to each other, leading to the absurdity of saying “2x = x” for x > 0. Yet the absurdity arises, not from any problem with infinity, but from the error of dismissing modern mathematics as some kind of fiction. Infinity is fine, only Dr. Craig’s math is deficient.

His final philosophical argument is to suppose you’ve met a man who is just counting down from infinity to zero. 3, 2, 1, 0.

Whew! Why, we may ask, is he just finishing his countdown today? Why didn’t he finish yesterday, or the day before? After all, by then an infinite amount of time had already elapsed. So if the man were counting at the rate of one number per second, he’s already had an infinite number of seconds to finish his countdown. He should already be done! In fact, at any point in the past, he’s already had infinite time and so should already have finished. But then at no point in the past can we find the man finishing his countdown, which contradicts the hypothesis that he has been counting from eternity.

Comparing two infinities again, as though they were finite quantities. Well, at least he’s consistent. This line of reasoning does seem to have some merit in the sense of not being able to reach the present from an infinite past. Dr. Craig is just doing the math wrong.

The bottom line is that there are good reasons to suppose that time does not extend infinitely far back into the past. Unfortunately Dr. Craig’s arguments, and specifically his idea that there cannot be any real infinity, are not among them. Nor does Dr. Craig seem to appreciate the implications of having a finite past, because if time itself is finite, that makes it less likely the universe will require a supernatural creator. There’s less territory that a natural explanation needs to cover, and less reason to suppose that there was ever a time when the universe did not exist in some form.

Ghazali’s arguments try to show that the universe has a beginning, but by showing that the past is finite, all he achieves is to demonstrate that the natural explanation does not need to explain an infinite number of past events. There may be an origin (i.e. a minimum value for time), but there is no beginning, i.e. no transition from a point in time where the universe did not exist to a point in time where it did. For that, you need a time when the universe did not already exist. Even with medieval Muslim help, Dr. Craig is no closer to providing that evidence than he was when he started.

29 Responses to “XFiles: To infinity (and be wrong!)”

  1. Gord Says:

    Must confess, did not read the whole post. Just wanted to add a short note: al Ghazali was actually a terrifically interesting writer. Centuries before Descartes, he described the problem of doubt, in terms as clear as Descartes, or clearer (and yes, Descartes had read al Ghazali’s work). Interestingly, his solution was to pray for release from doubt… not the solution Descartes found, which is why we find Descartes revelatory, and not al Ghazali. But still… pretty damned interesting for an 11-century Islamic philosopher.

  2. 'Tis Himself Says:

    Infinity can be even more counter-intuitive. In set theory the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets. The name comes from the symbol used to denote them, the Hebrew letter aleph (א). The cardinality of the natural numbers is aleph-0, the next larger cardinality is aleph-1, etc.

    A line has an infinite number of points on it. Let’s designate this as aleph-0. An infinite number of arcs can be drawn through each point. Therefore the total number of arcs is at least aleph-1 (there could be a cardinality which is between the number of points and the number of arcs, so the number of arcs could be aleph-2 or higher). Some infinities are more infinite than other infinities.

  3. Andrew G. Says:

    The number of points on a line is already greater than aleph-0 (which is the smallest transfinite cardinal and the only countable one).

    aleph-1 is the cardinal number of the set of all countable ordinals; equivalently, it’s the cardinality of the first uncountable ordinal (usually designated omega-1 or big-omega). This is provably the smallest cardinal greater than aleph-0; any smaller cardinal is countable.

    The number of points on a line is 2^(aleph-0), usually designated C or beth-1 (where beth-0 = aleph-0, and beth-(N+1) = 2^(beth-N)). This number is not countable, and therefore is at least as large as aleph-1. (The operation 2^X for some cardinal X corresponds to the cardinality of the power set (the set of all subsets) of the set whose cardinality is X. 2^X is strictly greater than X for all cardinals, so there is no largest cardinal and the cardinals themselves do not form a set.)

    Whether or not C is equal to aleph-1 is not merely unknown, but known to be independent of the existing axioms of set theory; that is to say, if you introduce into set theory an additional axiom C = aleph-1, then it remains consistent (if it was already), likewise if you instead introduce the axiom C > aleph-1, then it is also still consistent.

    The argument that there is an infinite number of arcs through a given point therefore there are more arcs than points is false. In fact, the number of continuous functions from reals to reals (and therefore the number of continuous curves) is the same as the number of reals. Also, the cardinality of the set of all sequences of reals is still no greater. However there are larger sets: the number of functions (including discontinuous functions) from reals to reals is 2^C (therefore > C), as is the number of subsets of the reals (2^C by definition).

  4. david Says:

    Gamow spelled out all the seeming absurdities of infinity nearly 50 years ago in the classic “One Two Three Infinity,” including, IIRC, the hotel paradox. But what’s the relevance to the khkhkhalam argument? Who benefits from a claim of infinite time? We know the time since the Big Bang is finite, indeed we know its duration with great precision. All this seems to be a great roiling of the sediment by Craig to defer approaching the fundamental weakness of the argument: that no conceivable physical evidence of a hypothetical cause could have survived through the infinite (!) temperature of the event itself. Hence, let there be a cause; its nature cannot be shown, and so cannot lend support to any possibly account of a creator. Craig apparently accepts modern cosmology, but at some point he has to attempt to derive the christian god from it, and at that point there will be blatant unsupported hand-waving.

  5. Tige Gibson Says:

    First of all we know that pi has an infinite number of significant figures. There is no last significant figure of pi. We can’t know what it is precisely because it does not exist, thus it is exactly like God.

    Second, the tangent of 90 degrees is the perfect example of what infinite means. You see, as you approach 90 degrees the result of tangent function projects exponentially larger, but if you perform the reverse, arctangent, what you are trying to do is figure out the length of the hypotenuse of the triangle using the angle. The hypotenuse is generalized as 1, but could be any length. In other words, of you are rotating a line fixed length around in a circle, at no point does the line explode to “infinity”. The word infinity is a placeholder for a value that we can’t know, another way of saying indeterminate. But we may have other ways of figuring out what the length of the line is. Again, this is like God: not as big as we thought, and by looking outside the Bible (the Christian arctangent function) we can determine that God is nonsense.

  6. eNeMeE Says:

    Even outside of set theory there are two infinities – countably and uncountably infinite (possible to map to the integers, not possible (such as the reals)).

  7. Len Says:

    Typo:

    This kind of difficulty is what is properly called absurd, because the contradiction arises from treating finite numbers like finite numbers.

    I think the first “finite” should be “infinite”.

    The answer is that the number of their orbits is exactly the same: infinity! (Don’t let someone try to slip out of this argument by saying infinity is not a number. In modern mathematics it is a number, the number of elements in the set [0, 1, 2, 3, ...].)

    Interestingly, when Dr Craig treats infinity as a number, it means that it is the infinitieth+1 number in the set (because 0 is also in the set) – ie, one more than infinity.

    • Deacon Duncan Says:

      @Len

      I think the first “finite” should be “infinite”.

      No, that’s correct as is. Treating finite numbers like finite numbers is correct technique, so when you apply this technique to a certain assumption (such as the assumption that there are no actual infinities) and you arrive at a contradictory conclusion, you know that your original assumption is invalid and that the apparent absurdity is not merely an artifact of using the wrong technique.

      • Deacon Duncan Says:

        Actually, on second thought I think my original phrasing is confusing, so I’ve rephrased it as follows:

        This kind of difficulty is what is properly called absurd, because the contradiction arises even though you’re correctly treating finite numbers like finite numbers. Dr. Craig’s “absurdity,” by contrast, is merely a reflection of the fundamental error that arises when you try to treat infinity like a finite number.

        I think that’s a bit clearer, isn’t it?

  8. Andrew G. Says:

    There’s a whole bunch of things wrong with Craig’s handling of infinities and similar arguments.

    For example, the fact of a bounded past does not imply that time has a “first moment”. One is a statement about the metric we use to measure time, the other is a statement about the topology of the early universe. For example the open interval (0,1) is bounded but has neither a smallest nor a largest point; it has the same topology as the unbounded real line.

    We do not know the topology of the early universe and we likely won’t know until we have a working theory of quantum gravity, and possibly not even then. Some cosmologists these days define the “big bang” as being the reheating at the end of the inflationary period; this reflects the fact that we have observational evidence for that, and theoretical justification for the existence of an inflationary period, but no good evidence or theoretical justification for anything older than that. Note also that current theories do not feature any past spacelike singularity. So the existence of an unbounded infinite past is once again an open question in spite of the Big Bang.

    The “counting down from infinity” argument fails in several ways. If you try and count down from aleph-0 by subtracting elements from a countably infinite set, you get nowhere because aleph-0 minus 1 is aleph-0. (And if you try subtracting successively larger natural numbers, then on the omega’th step you reach an undefined result, with all the preceding steps having the value aleph-0.)

    If you try and count down from omega, you fail at the first step because there is no “predecessor” operation for limit ordinals, only for successor ordinals.

    If you try and count down from omega represented as a surreal number, then you end up with a descending sequence of infinite surreals (not ordinals), until on the omega’th step you end up at 0, without ever having passed through any other finite number. (This is related to the fact that omega has no predecessor.)

    So whichever way, you never end up going … 3, 2, 1, 0. And also it makes no sense to ask “why finish now and not on the preceding step” because there is no preceding step; omega has no predecessor.

    Whereas if you consider the passing of time as being the equivalent of motion along the real line, then it is clear that an infinite past is no obstacle at all, since going from the point 1 to 2, say, works exactly the same whether you’re on the positive half-line or the full line. Furthermore, the number of points to the left of your current position doesn’t change, despite the fact that you traverse as many points in going from 1 to 2 as there are in the entire line.

    (The “potential” vs. “actual” infinities doesn’t fly either. The usual argument allows for limits of sequences to exist as “potential” infinities, but if you have limits of rational sequences, then you have the real numbers, and the cardinality of the reals is greater than the rejected “actual” infinity of the natural numbers.)

  9. Iain Walker Says:

    His second argument is that you cannot have an infinite number of events in the past, because if you did, you could not arrive at “now,” since you have an infinite distance to traverse before you get here. That’s not actually a bad argument

    What? Yes it is, because it makes the assumption that when we talk of an infinite number of past events, we’re talking about a series that starts in the past and progresses towards now. But that’s completely back to front – to speak of an infinite number of past events is to say that the series stretches backwards, as measured from now, and that there is no starting point. It’s to say that for every past time tn there is an earlier time tn-1.

    Craig is assuming that there is an earliest time in the sequence in order to argue against the claim that there is no earliest time. That’s some serious question-begging.

  10. Corkscrew Says:

    Andrew G: Beat me to it.

    Seriously, is there an “Infinity for Philosophers” textbook anywhere? Because I’ve seen a very large (albeit not infinite) case of the Just Not Getting It. This includes a relative who got a First in philosophy for a dissertation that, to a maths geek like myself, appeared to have a sodding great big hole in the logic.

  11. Corkscrew Says:

    “large … number of cases of Just Not Getting It”.

    Note to self: use Preview.

  12. MTiffany Says:

    His second argument is that you cannot have an infinite number of events in the past, because if you did, you could not arrive at “now,” since you have an infinite distance to traverse before you get here. That’s not actually a bad argument

    Actually it’s a transparently false argument. Change “infinite number of events in the past” to “infinite number of integers less than zero,” and “now” to “zero.”

    • Deacon Duncan Says:

      Hmm, an interesting rebuttal. What would you say if I point out that time has a direction, and integers do not? Suppose the only way you could arrive at zero would be by starting from -1, and the only way you could arrive at -1 is to get there from -2 and so on? Could you ever start counting? Or to put it in slightly different terms, suppose that from -100 you could only go to -99, i.e. there were no way to reach any number less than the one you started from. Can there be an infinite number of negative integers under those terms?

      • Andrew G. Says:

        Mathematically, that idea is nonsense – the existence of a “successor” relation implies the existence of a “predecessor” relation too, though it explicitly does not guarantee that predecessors exist for all entities or that they are unique.

        In the case of the ordinals, for example, every successor ordinal has a unique predecessor, while limit ordinals and zero have no predecessors.

        But here’s the thing: the moment you assert that the relation between instants of time is like the ordinal successor relation (as opposed to real or integer arithmetic), you have begged the question of the existence of a first moment in time, since the concept of “first”, i.e. zero, is fundamental to the ordinals.

        So it’s not legitimate to take some sort of ordinal successor relation between time instants as a premise and then argue for a first moment in time as a consequence.

        What’s more, even the concept of “instants” with its implication that things can be organized into discrete countable events is seriously problematic. For all the issues with uncertainty imposing a limit on how precisely something can be located in time, physicists still treat time (and other physical quantities) as continuous; thus implying an uncountable infinity of instants between any arbitrary times, and completely removing in the process any possible objection to the idea of an infinite past.

      • MTiffany Says:

        “What would you say if I point out that time has a direction, and integers do not?”

        I would say “that’s patent nonsense.” Inasmuch as time has ‘directions’ of past and future, integers (and numbers in general) have ‘directions’ also: positive and negative.

        I would also say that anyone who thinks that one can use ‘infinity’ as a starting point doesn’t understand what infinity is. It is not a number. It is a concept related to numbers, but it is not itself a number. There are an infinite number of even numbers, there is an infinite number of odd numbers, numbers evenly divisible 3, even an infinite number of numbers between 0 and 1, but infinity is not any one of those numbers.

    • josh Says:

      It’s a bad argument because it is a restatement of Zeno’s paradox. An infinite number of events does not imply an infinite measure of time or distance. You can divide a finite past into an infinite number of moments. Moreover, an infinite span of past time does not mean ‘you’ don’t get here. You are here. In the past, something else is there (possibly closely related to the current you).

  13. Corvus illustris Says:

    Sigh. When I first read this post–and the comments–I was reminded of Plato’s requirement that no one ignorant of geometry (or in this case, elementary cardinal arithmetic) enter his academy. Perhaps, I thought, what these clergypeople need is a mathematical tutor with unlimited amounts of patience who will lead them through some of the hard thinking about “infinity” that has gone on in the past 150 years or so; then they won’t have to sound so much like blithering idiots in public. But this is illusory. They don’t care about what might be correct, or what’s definitely wrong; they want some woo to peddle to their flocks of sheeple. So al-Ghazali (or any number of Greeks before him, or Aquinas after him) is fine–why not run Zeno’s paradox in both directions–just as prime matter/substantial form is fine for the RCs because it offers a conceptual framework for a physically meaningless “transubstantiation” in which they want to believe anyway.

    It would be good for Tige Gibson to talk to that hypothetical tutor, though; his confusion is remarkable, but not beyond redemption.

  14. Lord Griggs Says:

    As Kyle Williams notes, it’s one day after another forever.Potential time is the actual time. Successive addition means infinitity at work so that Craig contradicts himself.

  15. Martin Says:

    “His second argument is that you cannot have an infinite number of events in the past, because if you did, you could not arrive at “now,” since you have an infinite distance to traverse before you get here. That’s not actually a bad argument, though it’s moot, given the origin of time at the Big Bang.”

    I’d argue the exact opposite. There are cosmological theories where the Big Bang was not the origin of time, so the possibility of an infinite past should not be discounted.

    And the infinite traversal argument is bad, as shown here:

    http://www.philosophyetc.net/2006/04/unchanging-time-and-infinite-past.html

    I quote:
    We may posit an infinite space without supposing anything to cross it. Similarly, we may posit an infinite temporal dimension without supposing anything (the “moving ‘now’”?) to have traversed that.

    The problem lies in conceiving of “the passage of time” as being a kind of movement. We imagine the present-marker “starting” at the beginning of time, and moving forward into the future. But this picture belies a deep incoherence. It takes a second dimension – time – to move along some dimension. (Think of a graph plotting the change in y-axial distance against the x-axis of time.) But what is the present-marker moving through, as we track its changing temporal location? It can’t be moving through the first-order timeline, since that is rather what it is moving along. We need to posit another temporal dimension, a ‘meta-time’, in which it can traverse first-order ‘time’. This leads to infinite regress, and an absurd commitment to infinitely many temporal dimensions.

    We must conclude that there is no present-marker, or “moving ‘now'”. All times are on an equal ontological footing, the same way that all distances are.

  16. irritable Says:

    Part of Craig’s argument is that if there are an infinite number of “events” (I assume he means “instants of time”) before the present, then the present could not have been reached. We agree we have arrived at the present. Therefore, the past is not infinite.

    This argument seems illogical (at least, from the point of view of a layperson).

    If time is a dimension (like length), as Einstein and his successors appear to have demonstrated, then despite the fact that it is different from physical dimensions (so that we cannot re-experience the past at all and can only experience the future at the same speed as local clocks), it seems to follow that it should be treated like a physical dimension when dealing with “location” arguments.

    Therefore:

    1. If I stand at location x,y the X axis (arbitrarily, east/west) is unbounded (i.e. infinite) in either direction. Similarly the Y axis (arbitrarily, north/south) because each continuum has a direction but no bound.

    2. Applying Craig’s argument about the temporal dimension to physical dimensions, I could never have reached position x,y because I would need to have traversed an infinite number of locations to arrive there.

    3. However, if I step 1 metre to the left, I have stepped over an infinity of locations. I can keep dividing the distance I have stepped over by 2. That process can continue indefinitely: the sub-divisions of distance will approach, but never reach zero.

    4. Craig’s argument from implausibility (“Hilbert’s Hotel”) exploits a paradox of infinite sets – the “pairing” issue. The “pairing” paradox is advanced as support for the proposition that the time continuum should be treated, using “common sense”, as a set of individual instants which must have a beginning. (Much as Craig relies on “common sense” about causation and thereby elides his proposition that Reality is a “thing”, like its contents, so existence itself must have had a beginning and a cause). Infinities are paradoxical and counterintuitive but that’s because of our psychological tendency to treat infinities as special “numbers” rather than as descriptions of a process.

    5. As others have pointed out, you can’t deal with infinities or continua in this way. An “Infinity” involves a process of iteration – that is counting, dividing, or some other operation. Infinite iteration is perfectly comprehensible within the real world, as step 3 indicates. Craig’s argument employs Zeno’s paradox (as has been pointed out above). Achilles never catches the tortoise because, hidden in the premises of the argument, he is slowing down exponentially. Taken to its logical absurdity, in Craig’s argument, we never get from breakfast to lunch because there is an infinity of instants of time between those time locations.

    It appears that current cosmology is uninformative about T=0, the instant before the commencement of the Big Bang. Time = Zero seems to mean what it implies. There was no dimension of time, and there were no spatial dimensions. There is no “before” in that region, counterintuitive as that may seem in our normal environment. Hawking and Hartle suggest that, counting backwards, you approach but never reach T=0.

    Space time expanded after T=0, creating this universe. Whether T=0 was a “choke point” between two phases of our universe, or was the condition of a region no larger than the Planck volume in a larger environment (a Multiverse) or was part of some other state of affairs may be ultimately unknowable, either to mathematicians or astronomers.

    None of this assists Craig’s Kalam argument, which is based on his belief-based axioms:
    (1) reality had a beginning, and
    (2) despite (1), “prior to” that beginning it was “necessary” that a disembodied supermind existed.

  17. Skeptic Griggsy Says:

    Before Craig, Samuel M. Thompson^ taunted us naturalists with this: “But this desperate attempt to save naturalism and avoid theism fails because there can be no infinite series of actually existing things or events.Any sum of actual infinites,wheterh past or future existents,must be a finite number. To designate a sum or series as infinite is to deny that it is a series or sum of real existents.

    That we can construct an infinite series in arithmetic has no bearing on the question ofwhether there could have existed an infinite series of real causes.

    For rhe limits which differentiate a part from the rest of the continuum to which it belongs are conceptually determined; they ate not real distinctions which exist apart from each other in the thing itself.” He then states that this problem goes back to the time of Zeno.
    I thinks that that reveals the answer! And please, answer him!
    He finds that the composition fallacy does not apply as do others.

    Thompson ” A modern Philosophy of Religion,” p.318-321. Regnery

  18. Skeptic Griggsy Says:

    Craig, of course, is making the Henry Drummond the God of the scientific gaps; Lamberth’s the God of the explanatory gasp is that Aquinas’ primary cause and Leibniz’s sufficient reason both just amount to God did it- God wills what He wills- just a worthless tautology.
    Aquinas’ own superfluity argument boomerangs on his five ways and other ways! Percy Bysshe Shelley implicitly uses it thus: ” To suppose that some existence beyond, or above them [ the descriptions- laws- of Nature,S.G.] is to invent a second and superfluous hypothesis to account for what already is accounted for.” Theists would beg the question were they then to maintain that that is a category mistake.
    The Flew-Lamberth the presumption of naturalism also denies the need for that superfluity in that natural causes and explanations themselves are efficient, necessary,primary and sufficient. That neither sandbags nor begs the question, for it demands that theists overcome the presumption with evidence, not misinterpretation of evidence, just as Einsein overcame Newton.

  19. Skeptic Griggsy Says:

    I propose that instead of the Big Bang, we say the Big Transformation.
    Hoyle grieved for nothing as an atheist that his eternal creationism was unjustified as this transformation means the eternal Multiverse!

    Lord Griggs I have two WordPress accounts by mistake.

  20. Skeptic Griggsy Says:

    Deacon and others, what is your take on Thompson’s argument?
    What is your on mine?

  21. Skeptic Griggsy Says:

    Reblogged this on Hume Lives and commented:
    WLC


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