Common Objections to Kalam Part 7: Singularities and Infinities

here I am breaking the silence of months with yet another Common Objection to Kalam. I’ve heard this objection explicated most often by the youtube Atheist community (a very vibrant community, to be kind). In all of his written works on the Kalam Cosmological Argument, William Lane Craig defends the second premise (i.e. the universe began to exist) philosophically by arguing for the absurdity (read: impossibility) of the existence of an actual infinite. And following this he defends the second premise by way of empirical confirmation utilizing Big Bang cosmology and modern astrophysics. He argues, rather persuasively I think, that the universe began in an infinitely dense hot state wherein the universe was condensed to a single point – marking the edge and thus beginning of space and physical time (and given Ockham’s Razor, we might as well say metaphysical time too).[1]

The objector who finds this particular objection persuasive to Kalam argues that there is an alleged contradiction between claiming that an actual infinite cannot exist and claiming that universe began to exist in an infinitely dense singularity. For if the mutakallimun[2] (Kalamist) is arguing (1) that actual infinities cannot enjoy extra-mental existence and (2) that the singularity is actually infinite, then there is indeed a contradiction.

Now suppose for a moment that the singularity is infinite in a Cantorian sense, this would not mean that the mutakallimun must abandon Kalam. Not by any means. The mutakallimun might very well argue that the singularity does not enjoy ontological reality; that is to say, the initial cosmological singularity is not an existent, but is equivalent to nothing (and because of the trickiness of the word ‘nothing’ amongst physicists, I’ll specify what is meant by this sort – really the only sort – of ‘nothing’: non-being).[3][4] Otherwise, the mutakallimun might abandon empirical arguments from Big Bang cosmology and stick with the strictly philosophical ones. This last option is ill advised.

But as with most of the common objections to Kalam, it is based on a misunderstanding. A misunderstanding of the nature of singularities as possessing infinite curvature, density and temperature. This is not meant in a Cantorian sense. Writing in “Theism, Atheism, and Big Bang Cosmology,” Quentin Smith lists three clarifications (of which I will list only two)[5] that further elucidate the point that the initial cosmological singularity is not infinite in a Cantorian sense. (1) As we approach the singularity, the values become higher and higher, “such that for any arbitrarily high finite value there is an instant at which the density, temperature, and curvature of the possess that value.”[6] (2) When the singularity is reached the values become infinite, but this in no way suggests that the singularity involves values such as aleph-null. Quentin Smith notes, “If the universe is finite, and the big bang singularity a single point, then at the first instant the entire mass of the universe is compressed into a space with zero volume. The density of the point is n/o, where n is the extremely high but finite number of kilograms of mass in the universe. Since it is impermissible to divide by zero, the ratio of mass to unit volume has no meaningful and measurable value and in this sense is infinite (emphasis his).”[7][8]

While a conclusion is not needed, one will be given: this objection is founded upon misunderstanding. As it has gone before, common objections don’t stand much of a chance of toppling the tower that is the Kalam Cosmological Argument. This does not mean the argument succeeds in providing warrant for belief in God simpliciter.[7] The opponent of Kalam might yet have a catapult of objections to offer, but it would seem that these objections are not common and not simple. It takes more than just a nudge of the will to knock down the mutakallimun’s fortress.


1. See Craig, William Lane, and James D. Sinclair. “The Kalam Cosmological Argument.” The Blackwell Companion to Natural Theology. Chichester, U.K.: Wiley-Blackwell, 2009. Print.

2. Taken as a cue from W.L. Craig’s The Kalam Cosmological Argument; meaning a defender of the Kalam argument.

3. Craig, William Lane, and Quentin Smith. “Atheism, Theism, and Big Bang Cosmology.” Theism, Atheism, and Big Bang Cosmology. Oxford [England: Clarendon, 1993. 258-261. Print.

4. My next post will be on the ontological status of the singularity.

5. The other clarification would have been redundant for our purposes.

6. Craig, William Lane, and Quentin Smith. “Atheism, Theism, and Big Bang Cosmology.” Theism, Atheism, and Big Bang Cosmology. Oxford [England: Clarendon, 1993. 209-10. Print.

7. Ibid. pg 210.

8. God simpliciter would be an agent that created the entirety of the spatio-temporal world ex nihilo.



  • Let’s bring some mathematics into this.
    Formally, mathematicians have what’s called the “archimedean property”. If S is a set that is also an ordered group or field (you can define a partial order on it) then S has the archimedean property if and only if for every y and every x greater than zero, there exists a natural number n such that y is less than n times x.

    On a side note: You might now think “Ok, so if S does not have this property, so if this doesn’t hold for some y, then y must be infinite”. That’s not true in the multi-dimensional case.

    Now, the scenario we have here is the following:
    We have a function F that maps the pair (x,y) to x/y, where x is greater or equal than zero and y is stricly greater than zero. If x is “the mass of the universe” and y is “the volume of the universe”, this is the well defined “density of the universe”-function. This function is continuous in the sense that if a sequence of xs converges to X and a sequence of ys converges to y, then the sequence of F(x,y) will converge to X/Y – but only if all the xs stay greater or equal to zero and the ys stay strictly greater than zero AND if Y is non-zero.
    This means that if we aproach the early stages of the universe and look at moments in time each prior to the other, then the ys will converge to zero (we actually don’t know what happens with the xs). But then the density is not well defined. So you could introduce the new convention “if you have positive mass and zero volume, that’s now called ‘infinite’ density”. But as you pointed out, that is semantics.
    But notice that we are trying to smuggle in that X is positive, that is to say: the mass in the initial state was greater than zero. Now you might think “Well, there IS mass in the universe and it stays constant, right?” That’s not entirely true. If you accelerate an object, special relativity will tell you that its mass increases. Also, since mass is proportional to energy and we assume that the initial state had zero energy, then there was zero mass (there are reasons to think the amount of positive energy in the universe .is equal to the amount of negative energy in the universe). Now, if there is zero mass in the beginning we could actually calculate the limit of x over y using l’Hopital’s rule. Here’s an example, the numbers are completely arbitrary:
    Suppose that the mass of the universe is three times the age of the universe (times a constant that does nothing but fix the units) and the volume is twice the age of the universe. Then the density at age zero is not defined because we would still divide zero by zero. But it approaches 3/2. So it need not to be infinite at all.

  • July 17, 2011 at 6:29 pm // Reply

    Seems sound, just a few questions:

    (1) For your first response, how does equating the singularity to nothingness solve the problem of infinity? I’m afraid I don’t quite see it.

    (2) So, if I understand correctly, the term infinite applied to the singularity doesn’t literally mean infinite? It just means that the numerical values are too immeasurable to have any sense of finiteness attributed to them? Just want to make sure that’s clear for me, I’m not really strong in physics.

  • It is nice to see you posting again, Alexander!

    As to your questions,

    (1) Well if the singularity did not in fact exist or was equivalent with nothing then it would not reach a state of infinite density and curvature, for it would not exist as a thing to have such properties. This would solve the problem by removing the singularity altogether.

    (2) The term “infinite” in this case is not used in the sense of a cantorian actual infinite. Otherwise, I do not know enough of the physics myself to say entirely what this means.

  • July 18, 2011 at 8:45 pm // Reply

    (1) Ah! Of course, I see it now, that’s actually pretty logical and straightforward. Not sure how I looked over that. Thanks.

    And thanks, it’s nice to be commenting here, helps sharpen my knowledge in PoR. :)

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