Kalam II: Philosophical arguments for the beginning of the universe

Now I’ll deal with Craig’s philosophical arguments for the claim the universe had a beginning. The first argument goes like this (Reasonable Faith pp. 116-120):

  1. An actually infinite number of things cannot exist.
  2. A beginningless series of events in time entails an actually infinite number of things.
  3. Therefore, a beginningless series of events in time cannot exist.

Craig’s argument for (2) is that if the universe never began to exist, the number of past events is infinite. This is problematic, especially given that Craig believes Christians will be rewarded with eternal bliss in heaven. Craig seems to need it to be true that past events are real in a sense which future events are not, a controversial philosophical point.

This is because if past events are not real in the relevant sense, Craig’s argument fails. However, if future events are real in the relevant sense, and Craig is also right about premise (1), he will have inadvertently proved the future must be finite, and therefore there will be no eternal reward or punishment in heaven. (One of philosopher Wes Morriston’s papers has more on this.)

But it’s pretty clear there are no good reasons to think Craig is right about (1). Craig tries to support this claim by citing advanced mathematics, and when I first encountered Kalam, I actually found that part of the argument convincing. Only later, when I took a graduate-level course in logic, which covered Cantorian set theory, did I realize what nonsense this is.

In his debates, Craig likes to say things like, “mathematicians recognize that the existence of an actually infinite number of things leads to self-contradictions.” I’ve never heard him explain which mathematicians these are. In fact, the mathematics of infinity is a well-developed and perfectly consistent branch of mathematics.

In particular, contrary to what Craig often claims, Cantorian set theory does not say that when you take infinity minus infinity,  you get contradictory results. Set theory does not say infinity is a single thing that can be subtracted from itself. Instead, set theory deals with a great variety of possible infinite sets. In set theory, you can take “A set minus B” where A and B are both infinite sets, but the result depends on which infinite sets A and B are.

That was very abstract. So here’s a thought-experiment often used to explain set theory: Hilbert’s Hotel. Imagine a hotel with an infinite number of rooms, all of which have guests in them. Here are two things that might happen (if you ignore practical problems, including the laws of physics):

  1. All the guests leave. An infinite number of guests have left, and no guests remain.
  2. All the guests in the even-numbered rooms leave. An infinite number of guests have left, and an infinite number of guests remain.

Strange, isn’t it? And Craig frequently calls Hilbert’s Hotel “absurd” and claims it is therefore impossible. But contra Craig, it involves no contradictions. In one situation, the guests do one thing, leading to one result. In the other, the guests do another thing, leading to another result. Hilbert’s Hotel is perfectly consistent, as is Cantorian set theory.

I’m not the first critic of Kalam to see no impossibility in Hilbert’s Hotel. Craig cites a number of philosophers who accept that Hilbert’s Hotel is possible. His response to them? Complain they haven’t given a reason to think the hotel is possible (p. 119).

This is just an example of Craig’s annoying tendency to make unsupported claims and then demand his critics disprove them, and it’s an absurd way to argue. If Craig is going that way, why not just announce God exist, demand atheists prove otherwise, and be done with it?

Craig’s other philosophical argument that the universe had a beginning goes like this (pp. 120-124):

  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.

The question here is what Craig means by this. If “a collection formed by adding one member after another” Craig means “a collection formed by starting with nothing and then adding one member after another,” then premise (1) begs the question, because if the past is infinite, it has always been infinite. It did not start from nothing.

On the other hand, if “a collection formed by adding one member after another” means “a collection that has been added to by adding one member after another,” then premise (2) is false, because if you add to an infinite collection by adding one member after another, it will still be infinite.

  • mnb0

    So a number is not a thing according to Craig. Then what is it?

    “In fact, the mathematics of infinity is a well-developed and perfectly consistent branch of mathematics.”
    Exactly. It can be found on Wikipedia.

    http://en.wikipedia.org/wiki/Infinity

    It’s not hard.

  • Dunc

    If Craig is going that way, why not just announce God exist, demand atheists prove otherwise, and be done with it?

    How many books can you get out of that?

  • sc_13130ced6394732ac8bdb91abb5485e0

    I don’t see relativity mentioned.

  • OverlappingMagisteria

    It seems to me that presenting Zeno’s “paradoxes” would disprove Craig’s premise that an infinite number of things is not possible.

    In order to walk from point A to B, I have to pass through an infinite number of points on the way. Passing through an infinite number of points would be an infinite number of events and, according to Craig, would be impossible. Therefore, it should be impossible to ever travel between two points.

    Obviously it is possible, so there is something wrong with Craig’s assumption

    • http://skepticsplay.blogspot.com miller

      AFAIK WLC believes Zeno’s paradox is a real paradox, and that points aren’t real. All on the basis that this is more intuitive!

  • schmeer

    I’m very underwhelmed by WLC. He does not seem to be capable of logical thought. What a huge disappointment. I’ve heard a lot about him, but hadn’t read any of his arguments. I have read theologians that, while unconvincing, were much better than this.

  • Andrew G.

    Craig seems to need it to be true that past events are real in a sense which future events are not, a controversial philosophical point.

    Philosophy of time is apparently Craig’s other main speciality; and there, he does indeed defend the A-theory position (in which past, present and future are qualitatively distinct). In the philpapers survey, A-theory and theism are somewhat correlated, which isn’t really a surprise. B-theory is more popular with philosophers of physical science.

  • sc_13130ced6394732ac8bdb91abb5485e0

    Relativity states time is not constant: Which means that time can vary between points of ~nil to high@ (X)E -2(G/Tc)

  • Aviatrix

    If by “things”, Craig means physical objects, his first assertion would be true within a finite space (leaving quantum weirdness ou of it). But unless I’m missing something, I don’t see any reason an infinite number of things could not exist within an infinite (or infinitely expanding) space – or infinite number of discrete spaces (cue the theoetical multiverse).

    • josh

      As with time, if space is infinitely divisible then you have an infinite number of points within a finite volume. I would love to see someone try to work out a theology of non-divisible spacetime but I think it’s significantly beyond WLC’s pay grade.

  • Brad

    Yeah, the whole “infinity isn’t real” thing seems more like a profound misunderstanding of mathematics than a profound argument for God.

    After all, if I slice it in just the right way, I could have an infinite amount of distinct “moments in time” between one second and the next! (1/2 + 1/4 + 1/8 + …) And yet, the next second inevitably gets here (because the series sums to 1). Sure, that is kind of hard to get your brain around, but that’s a problem with our brains, not a problem with the universe, nor a proof of God.

  • Kevin

    Of course, the Kalam was proposed in an attempt to prove the existence of Allah.

    One wonders why Craig isn’t a Muslim. It’s his favorite argument, after all.

  • L. Jones

    There are a lot of hidden assumptions (or maybe hidden syllogisms) and other defects in Craig’s argument, compounded by the fact that Craig doesn’t really make clear what he means by the words “finite” and “infinite.”

    The assertion that an infinite past has something to do with the Hilbert’s hotel scenario requires the inference that if the past is infinite, then there is a sequence {x_n} of past times, where for each natural number n, x_n is a time in the past. As far as I know, Craig never even acknowledges this. Making such an inference without mathematical justification commits the so-called fallacy of forming an actual infinite from a potential infinite, while appealing to mathematics requires a choice of set theoretic axioms, which Craig’s rejection of Platonism makes difficult, if not impossible.

    In the Blackwell Companion to Natural Theology article on the Kalam, Craigs discussion of finite vs. potentially infinite vs. actually infinite seems to imply that given a set S which is not finite in the sense that it is in bijection with an initial segment of the natural numbers has a subset in bijection with the natural numbers. Of course, this is false in some models of set theory, but Craig doesn’t even seem to be aware of this.

    While I agree with Hallq’s choice to use the Reasonable Faith source, it’s amusing that the longer, more “scholarly” article in the Blackwell Companion actually makes the argument look worse.

  • L. Jones

    One more thing: it’s not known whether the usual “mathematics of infinity” is consistent. For all we know, even the axioms of Peano Arithmetic could be inconsistent, contra Craig’s assertion at 1:17 in this video: http://www.youtube.com/watch?v=NOrlIOm6eGM

    Actually, the video raises some interesting questions:

    1. How does Craig know the axioms of Peano Arithmetic are true?

    2. What does Craig mean by “the axioms of Peano Arithmetic are true”?

    3. How can Craig make the assertion that the “the axioms of Peano Arithmetic are necessary truths” and claim to reject Platonism?

    4. (This one refers to the whole video) Why would someone with a (supposedly) sophisticated understanding of mathematics and the philosophy of mathematics appeal to Peano Arithmetic to justify the assertion 2+2=4?

  • eric

    So, WLC seems to have moved on from ignoring modern physics (in part I) to ignoring modern physics AND mathematics (in part II). Can’t wait for part III.

    if future events are real in the relevant sense, and Craig is also right about premise (1), he will have inadvertently proved the future must be finite, and therefore there will be no eternal reward or punishment in heaven.

    This is also the very first response I thought of, and is quite simple: rejecting infinity means no eternal life for anything – God or human. Chris, has WLC responded to it? Either via you, Wes Morriston, or another critic? If so, what did he say?

  • Kevin

    You might want to give an example of easy ways to add and subtract infinite sets.

    For example, here are four finite sets A, B, C, and D:

    A = [1,2,3,4,...]
    B = [2,3,4,5,...]
    A-B = [1]

    Subtracting is simply taking out the values that they share from the first set.

    C = [1,3,5,7,...]
    D = [2,4,6,8,...]
    C + D = [1,2,3,4,...]

    Adding is simply combining the values of each set.

    And now we see that while A-B leads to a finite set, C-D would lead to an infinite set (They share nothing in common so C would remain the same). It isn’t that complicated.

    • ‘Tis Himself

      Infinities are counter-intuitive.

      There are an infinite number of points on a line. There are an infinite number of arcs which pass through each point. Some infinities are larger (more infinite) than others. See http://planetmath.org/AlephNumbers.html for a somewhat more rigorous discussion.

      • Kevin

        So, why are infinities counter-intuitive? I realize that I have a lot of schooling in math, but I wouldn’t expect infinities to startle anyone of mediocre intelligence. They probably won’t understand the syntax for aleph numbers, but I’d bet that they know that there are an infinite number of points on a line segment.

        • http://youcallthisculture.blogspot.com/ vinnyjh57

          If there is a problem with an infinite regress, can it really be solved by positing an infinite being?

          • Kevin

            Relevance?

            And to answer your question, no. You can’t end a never ending process within that scope*. Its a simple contradiction in terms. Also, ‘infinite being’ doesn’t mean anything to me; it sounds like something that Deepak Chopra would say.

            *You can bypass infinite processes by going outside of its scope. For example, you can end a program that is doing an infinite loop by restarting your computer (or IDE). However, by definition, you can’t make a program that has an infinite loop that doesn’t loop infinitely.

        • ‘Tis Himself

          The idea of an endless series isn’t counter-intuitive. What is counter-intuitive is the concept of cardinality of infinite sets. That aleph-0 sets can be treated as being smaller than aleph-1 sets (sorry, having trouble with the html), essentially that one infinite set is smaller than another infinite set, is counter-initiative.

  • http://skepticsplay.blogspot.com miller

    In my reading of William Lane Craig, he makes two distinctions: a) potential and actual infinities, and b) mathematical consistency, and actual possibility.

    WLC pretends that the first distinction is a mathematical distinction when it’s not. His definitions are extremely vague and obviously pre-Cantor. For my own amusement, I once tried to make his definitions rigorous, but the only way I can think to do it is using possible world semantics rather than pure math.

    WLC makes the second distinction in response to mathematicians who point out that infinities are (as far as we know) consistent. He thinks that even if it is consistent, it is still absurd and therefore can’t actually exist. But how can it be absurd in reality when it’s not absurd in mathematics? The mathematical consistency clearly indicates that it is not actually absurd, just counterintuitive.

    • eric

      He thinks that even if it is consistent, it is still absurd and therefore can’t actually exist.

      Ah, good old argument from incredulity, how I’ve missed you.

      That’s pretty much what a lot of these theological proofs come down to, isn’t it? “If my premises are wrong, the world would feel absurd to me. So they must not be wrong.”

      • Reginald Selkirk

        Yes, in a nutshell: Craig’s arguments on infinity are dressed up as reductio ad absurdum, but he never actually shows a contradiction; he instead subtitutes an argument from incredulity.

    • Rick Taylor

      There is a distinction between a possible and actual infinity (more accurately, viewing infinities as potentially infinite or actually infinite).

      It’s a philosophical distinction but it can have practical consequences. Intuitionists and constructivists deny that the concept of an actual infinity is meaningful, and it has radical consequences for how they think math should be done (for example, an intuitionist will deny we can assume an arbitrary real number is either positive, negative, or zero).

      But his is completely irrelevant to Craig’s argument. Even if we admit that the sequence of days in the past is merely potentially infinite and not actually infinite, it still won’t have a beginning. No matter how far back we go, we’ll still be able to imagine the day before that.

  • http://youcallthisculture.blogspot.com/ vinnyjh57

    Something has always bothered me about the first premise:

    Based on my knowledge and observation, I might justifiably say “All crows are black and have wings.” On the other hand, I wouldn’t be justified is saying “All crows that are black have wings” unless I had some knowledge of non-black wingless crows. The statement “All crows that are black have wings” implies a connection between blackness and wingedness that isn’t justified (at least in my experience).

    By the same token, I might say based on my knowledge and observation “Whatever exists has a beginning and a cause, but I don’t see how I can justify saying “Whatever begins to exist has a cause” unless I can claim to have some knowledge of things that have no beginnings and are uncaused.

    Does anyone with more philosophical training than I have see that as a problem for the first premise?

    • anteprepro

      [WARNING: I AM NOT A PHILOSOPHER NOR IS MY EXPERTISE PHILOSOPHY ADJACENT. TAKE WITH GRAIN OF SALT]

      As far as I can tell, it shouldn’t be (directly) problematic.

      If the premise “all crows are black and have wings” then it is necessarily true that “all black crows have wings” because all crows are black crows and since all crows have wings, all black crows should have wings as well. If the premise “whatever exists has a beginning and a cause” is true then “whatever begins existing has a cause” should similarly be true as well, since existing things have both a beginning and a cause, so all existing things with a beginning should have a cause. It should similarly be true that “all winged crows are black” and “whatever is caused has a beginning”.

      It is only as problematic as any other statement that relies on induction: If you are relying on inaccurate or incomplete observations, your premises are false. But the formation you are talking about is less susceptible to error. “All crows are black and have wings” is refuted by a wingless black crow, a winged white/colored crow, or a white/colored wingless crow. “All black crows have wings” can only be refuted by a black wingless crow. “Whatever exists has a beginning and a cause” is refuted by a caused beginningless thing, an uncaused thing with a beginning, or an uncaused beginningless thing. “Whatever began to exist has a cause” can only be refuted by an uncaused thing with a beginning.

      There really isn’t much wrong with the premise’s formulation, aside from it probably being a tactical decision to reduce the number of things that could possibly be presented as a counterargument. That’s hardly the most damning thing in the world, though.

      • http://youcallthisculture.blogspot.com/ vinnyjh57

        Of course the purpose of formulating the first premise as “Whatever begins to exist has a cause” is to avoid the problem of what caused God. However, if we have no knowledge of things that don’t begin or things that are uncaused, it seems to me “Whatever begins to exist has a cause” contains no more information than “Whatever exists has a cause.” The former reduces to the latter in the same way that “All crows that are black have wings” reduces to “All crows have wings” absent some justification for thinking that wingedness and blackness are connected. If I am correct, then the Kalam argument is no different than any other first cause argument.

        • Hairy Chris, blah blah blah etc

          He also plays somewhat fast & loose with what he means by existence… Does the universe exist in the same way that god supposedly does? He also seems to treat existence as an attribute which, even as a non-philosopher, seems to be completely arse-backwards as an entity needs to exist before you attribute any other qualities to it at all.

        • http://iacb.blogspot.com/ Iamcuriousblue

          Like everybody else here, I see the obvious problem with appeal to God as creator, as just taking the paradox of ultimate beginnings back another step, and bringing up the next question, “But where did God come from?” Bringing one back to the same question as you had with a naturalistic explanation, but with an unnecessary extra step.

          Naturally, a religious person who I was having this debate with on another forum, who was arguing the impossibility of an infinite chain of causality, had a very oddly reasoned response. It seems to his way of thinking that since there must be a beginning and a prime mover for that beginning, that there needed to be an “uncaused cause”, and that uncaused cause was God. Which seemed to me like a contradictory rationale built out of the need to justify the existence of God.

          BTW, the person had an equally odd argument for why there could be no such thing as a universe that’s eternally old. Because apparently if a chain of causality goes infinitely far back, it could never possibly reach the present and cause our current existence – there has to be a beginning point to reach the present. Which, again, strikes me as a very artificial and semantic argument. In any event, a moot point, since empirical evidence and deduction doesn’t support a steady state universe either. But I certainly wouldn’t dismiss the idea of a steady-state universe on something as silly as the “infinity paradox” argument.

          • Iain Walker

            BTW, the person had an equally odd argument for why there could be no such thing as a universe that’s eternally old. Because apparently if a chain of causality goes infinitely far back, it could never possibly reach the present and cause our current existence – there has to be a beginning point to reach the present.

            This is a misunderstanding of what is meant by an infinite past, which seems to be taken to mean that there is a starting point to the universe an infinitely long time ago. But to say that the universe is infinitely old is to say that there is no starting point – rather, it means that for each past time t(n) there is always an earlier time t(n+1). An infinite past is one that is measured backwards from the present, not one that is measured forwards from a starting point infinitely long ago. It’s a common mistake of treating “infinity” as if it were a normal number, leading to the assumption that “infinitely long ago” is a discrete point in the past from which the passage of time can then be measured, when it is no such thing.

          • Reginald Selkirk

            But to say that the universe is infinitely old is to say that there is no starting point

            Yes. Which means Craig’s argument on this point is analogous to: “Imagine yourself on a planet with no red tower. Go to the red tower.” Zeno would have run circles around Craig.

          • Reginald Selkirk

            Craig’s argument rephrased: If the past were infinite in time, we couldn’t have reached the present moment, because that would take, like, forever!

  • David Evans

    There is (at least) a countable infinity of true propositions about the natural numbers. For instance, the series:
    “2 is prime”
    “3 is prime”
    “5 is prime”
    “7 is prime”
    .
    .
    .
    is infinite and non-trivial – you can’t just derive each from the one before it.

    If God is omniscient, he must know all true propositions. Therefore his knowledge must include an actual infinity of propositions.

    Which is impossible, according to Craig.

    • http://twitter.com/blamer @blamer

      For now CH’s blog is just dismantling Craig’s arguments surrounding Genesis 1-3. We haven’t heard him argue how large minds can be and the maximum they can know… yet.

  • David Evans

    And another thing.

    If God is omnipotent, he can create an infinite number of things, or a universe with an infinite past. Is Craig denying God’s omnipotence?

    • http://twitter.com/blamer @blamer

      Again, we aren’t yet hearing about any limits to divine intervention. Apparently theologians commonly dodge criticisms of infinite omnipotence by conceding that the “omni” they have in mind is more precisely “as omni as logic permits”.

  • Rick Taylor

    As a mathematician, my first exposure to Craig reading the beginning of his debate with Armstrong where he brings up the Cantor’s hotel paradox. I found it both astonishingly unconvincing and his claims about what mathematicians believe appallingly dishonest. You do an excellent job in this post of explaining this at length.

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  • Greg G.

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

    Oh, yeah? What was God’s first thought?

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  • GGDFan777

    I’d maintain that an actual infinity can not exist and that therefore time also can not be infinite in the past. See this articles by philosopher Casper Storm Hansen of the university of Aberdeen titled “New Zeno and Actual Infinity”:

    http://core.kmi.open.ac.uk/download/pdf/5849860

    Abstract:

    In 1964 José Benardete invented the “New Zeno Paradox” about an infinity of gods trying to prevent a traveler from reaching his destination. In this paper it is argued,contra Priest and Yablo, that the paradox must be re-solved by rejecting the possibility of actual infinity. Further, it is shown that this paradox has the same logical form as Yablo’s Paradox. It is suggested that constructivism can serve as the basis of a common solution to New Zeno and the paradoxes of truth, and a constructivist interpretation of Kripke’s theory of truth is given.”

    And also this article by philosopher Laureano Luna titled “Ungrounded causal chains and beginningless time”:

    http://www.logika.umk.pl/llp/1834/5-1834zw.pdf

    Abstract:
    We use two logical resources, namely, the notion of recursively defined function and the Benardete-Yablo paradox, together with some inherent features of causality and time, as usually conceived, to derive two results: that no ungrounded causal chain exists and that time has a beginning.

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