Ok, here’s episode 3 of the Mile-High Reviews, William Lane Craig’s presentation of the kalam argument. As an usual disclaimer, I’m neither a philosopher nor a theologian…nor a physicist, thinking about it.

The cosmological argument is a family of arguments that seek to demonstrate the existence of a Sufficient Reason or First Cause of the existence of the cosmos. The roll of the defenders of this argument reads like a Who’s Who of Western philosophy: Plato, Aristotle, ibn Sina, al-Ghazali, Maimonides, Anselm, Aquinas, Scotus, Descartes, Spinoza, Leibniz, and Locke, to name but some.

…Couldn’t you finish your introduction before tripping a fallacy?!? Because that appeal to authority is all good and well, but geocentrism can claim as lofty a list of defenders.

Then, like Mr. Pruss, Mr. Craig listed the various kinds of cosmological arguments and then went on to describe the kalam argument’s history. Following that historical digression, he started with the kalam argument’s outline, to wit:

10. Everything that begins to exist has a cause.
20. The universe began to exist.
30. Therefore, the universe has a cause.

Before treating each step separately starting, for the needs of his presentation — in his words, “since some attempts to subvert (10) are based upon cosmogonic theories – the discussion of which would be premature prior to their introduction in our treatment of (20)” — by the part about the second step.

So first is step two, which he begins by presenting a few traditional arguments against, beginning by the argument from the impossibility of an actual infinite, which he describes thus:

211. An actual infinite cannot exist.
212. An infinite temporal regress of events is an actual infinite.
213. Therefore, an infinite temporal regress of events cannot exist.

Ok, so he starts by some definitions: an actual infinite is the likes of ℵ₀ or ℵ₁. That is, a defined infinite — well, transfinite really — quantity in contrast to ∞ which is a potential infinite.

Unfortunately, Cantor’s notion of a set as any logical collection was soon found to spawn various contradictions or antinomies within the naive set theory that threatened to bring down the whole structure. As a result, most mathematicians have renounced a definition of the general concept of set and chosen instead an axiomatic approach to set theory, by means of which the system is erected upon several given, undefined concepts formulated into axioms.

At the end of the day, all mathematics deals in axiomatic systems about concepts though, so I’m not sure where that “renouncement” comes from: any concept in mathematics is defined relative to a given axiomatic system.

Next up is exist, which he defines as “instantiated outside a mind”, that is if things outside the mathematical theory defining the concept matches up with it. Fair enough, even if he gives the game away a bit that he uses this one because mathematical existence is perfectly happy with an actual infinite (in his words, «We thereby hope to differentiate the sense in which existence is denied to the actual infinite in (211) from what is often called “mathematical existence.”»), it’s still fair enough. The modality he goes for exist is metaphysical modality, which is broader than broad logical possibility (i.e. strict logical possibility augmented by the meaning of terms in the sentence within the scope of the modal operator) which, as he admits, cuts both ways: that makes an argument based on it terribly subjective compared to ones based on strict — or indeed broad — logical possibility but he needs the less narrow modality for his argument to take off so one does what one has to… The upside though is that it also makes the argument more resilient to punts such as the states of affairs his argument goes against not being proven logically inconsistent, there’s always that.

Then, defining event, which is defined as a change of non-zero duration. And, on his redefinition of the question in (212) (paraphrasing, is there a first event not preceded by a duration equal to it) — specifically the footnote “This criterion allows that there may be events of shorter duration prior to the first standard event. By stipulating as one’s standard event a shorter interval, these can be made arbitrarily brief.” — doesn’t he runs head along against quantum mechanics which state that no event as above defined will take less than a Planck interval? But that’s not overly important if the ability to make events previous to the first standard one as short as needed isn’t used though.

The best way to support (211) is by way of thought experiments that illustrate the various absurdities that would result if an actual infinite were to be instantiated in the real world. Benardete, who is especially creative and effective at concocting such thought experiments, puts it well: “Viewed in abstracto, there is no logical contradiction involved in any of these enormities; but we have only to confront them in concreto for their outrageous absurdity to strike us full in the face”

That the consequences of a given concept go against our common sense is not exactly that good a tool in an universe where quantum mechanics is a good way to describe some phenomena though, just saying. It’s true that Hilbert’s Hotel seems absurd, but then, so does the double-slit experiment.

Partisans of the actual infinite might concede the absurdity of a Hilbert’s Hotel but maintain that this case is somehow peculiar and, therefore, its metaphysical impossibility warrants no inference that an actual infinite is metaphysically impossible. […] Thus, thought experiments of this sort show, in general, that it is impossible for an actually infinite number of things to exist in reality. At this point, the actual infinitist has little choice but, in Oppy’s words, simply to “embrace the conclusion of one’s opponent’s reductio ad absurdum argument” (Oppy 2006a, p. 48). Oppy explains, “these allegedly absurd situations are just what one ought to expect if there were … physical infinities” (Oppy 2006a, p. 48). […] Rather the question is whether these consequences really are absurd. Sobel similarly observes that such thought experiments bring into conflict two “seemingly innocuous” principles, namely,

(i) There are not more things in a multitude M than there are in a multitude M′ if there is a one-to-one correspondence of their members.

and

(ii) There are more things in M than there are in M′ if M′ is a proper submultitude of M.

Mind, real infinitists have another choice: admit the absurdity, like Mr. Oppy, but go for broke, embrace it to the fullest and go “so what?” in answer to that appeal to consequences. I mean, we are in a universe where, at latest physics, adding to a physical quantity doesn’t change it, time can act like space, and dead-alive cats are a sane thought experiment amongst other silliness. What’s one more absurdity?

The metaphysician wants to know why, in order to resolve the inconsistency among (i)–(iii), it is (ii) that should be jettisoned (or restricted). Why not instead reject or restrict to finite multiplicities (i), which is a mere set-theoretical convention? More to the point, why not reject (iii) instead of the apparently innocuous (i) or (ii)? It certainly lacks the innocuousness of those principles, and giving it up would enable us to affirm both (i) and (ii). Remember: we can “have” comparable infinite multiplicities in mathematics without admitting them into our ontology.

Pragmatism. Or, to be less cheeky, what use is it to throw out (iii) beyond allowing a metaphysical cosmological argument? Because, let’s be serious: for all its claims to go to what underlies everything metaphysics is a modelling venture, same as physics or mathematics. If a given set of principles (aka axioms) is useful, it’s getting used even if the constituent concepts don’t get instantiated in reality (cf. complex numbers). Besides, throwing out (iii) can be seen as exactly as arbitrary as restricting (ii): after all, (ii) is a mere rule from minds hopelessly mired in the finite, human-scale world and throwing out (iii) is only the expression of a metaphysician of limited imagination’s easily bruised sensibilities. (Yes, I’m being of bad faith. But eh, goose, gander and all that.)

In this case, one does wind up with logically impossible situations, such as subtracting identical quantities from identical quantities and finding nonidentical differences.12

---

12. It will not do, in order to avoid the contradiction, to assert that there is nothing in transfinite arithmetic that forbids using set difference to form sets. Indeed, the thought experiment assumes that we can do such a thing. Removing all the guests in the odd-numbered rooms always leaves an infinite number of guests remaining, and removing all the guests in rooms numbered greater than four always leaves three guests remaining. That does not change the fact that in such cases identical quantities minus identical quantities yields nonidentical quantities, a contradiction.

That’s assuming that properties holding for finite sets ought to hold for infinite ones (i.e. in Mr Craig’s example, for B ⊂ A, card(A \ B) = card(A) – card(B)) though, which is far from given. Besides, the passage from finite to infinite sets is far from the only place where that happens: square matrix’s multiplication isn’t commutative whereas real numbers multiplication is. Natural numbers have a lowest element whereas integral numbers don’t. Et cætera, et cætera. One thing though: I was under the impression that Mr. Craig was ignoring mathematical existence and going for a metaphysical one. What are we doing in that transfinite mathematics jaunt?

If this line of argument were successful, it would, indeed, be a tour de force since it would show mathematical thought from Aristotle to Gauss to be not merely mistaken or incomplete but incoherent in this respect. But the objection is not successful. For the claim that a physical distance is, say, potentially infinitely divisible does not entail that the distance is potentially divisible here and here and here and… Potential infinite divisibility (the property of being susceptible of division without end) does not entail actual infinite divisibility (the property of being composed of an infinite number of points where divisions can be made). The argument that it does is guilty of a modal operator shift, inferring from the true claim

(1) Possibly, there is some point at which x is divided

to the disputed claim

(2) There is some point at which x is possibly divided.

First, that wouldn’t be as much of a prowess as Craig’s implying in that cross between an argument from authority and one from antiquity: Gödel already proved that the bog-standard conceptualisation of arithmetic over natural numbers used by such luminaries — Peano’s — is incoherent and that doesn’t bother mathematicians much after all. Second, I don’t know why but I’m under the impression that Mr Craig added a modal operator here just to sink Rucker’s, Sorabji’s et al argument: given Craig’s description of a potential infinite re. divisibility (being susceptible of division without end), what is that “possibly” doing in (1)? And without that possibly in (1), what’s the problem for going from (1) to (2)? Hell, I’d argue (2) is a better statement of that property in the first place… Now, if Craig were a constructivist, then sure it is there in (1) because potential infinity as stated isn’t actually constructive: how do you get that infinite? But then I doubt Craig is one because a cosmological argument is hardly constructive either: how do you get to that god?

The question arises whether on the A-Theory the series of future events, if time will go on forever, is not also actually infinite.

I love how Mr Craig smuggle in his hypothesises: here, that A-theory of time (roughly, there’s a global clock) is the applicable one for he stays well away from B-theory of time and the series of future events. Because if B-theory (roughly, everything have their own clock) with its ontological parity between past, present and future is the applicable one, a potentially infinite time means an actually infinite series of future events.

Then the second argument for part 2 is an argument from the impossibility of the formation of an actual infinite by successive addition, where Mr. Craig funnily forgets to tell that his demonstration only holds in a A-theory of time. He gives away the plot in a footnote in answer to a counterargument of the “formation of an actual infinite” part†:

17. Richard Gale protests, “This argument depends on an anthropomorphic sense of ‘going through’ a set. The universe does not go through a set of events in the sense of planning which to go through first, in order to get through the second, and so on” (Gale 2007, pp. 92–3). Of course not; but on an A-Theory of time, the universe does endure through successive intervals of time. It arrives at its present event-state only by enduring through a series of prior event-states. Gale’s framing the argument in terms of a “set of events” is maladroit since we are not talking about a set but about a series of events which elapse one after another.

On B-theory though, per Craig’s own presentation of it, there’s no difference between a set and a series of events.

Moreover, Ghazali asks, will the number of completed orbits be even or odd? Either answer seems absurd. We might be tempted to deny that the number of completed orbits is either even or odd. But post-Cantorian transfinite arithmetic gives a quite different answer: the number of orbits completed is both even and odd! For a cardinal number n is even if there is a unique cardinal number m such that n = 2m, and n is odd if there is a unique cardinal number m such that n = 2m + 1. In the envisioned scenario, the number of completed orbits is (in both cases!) ℵ₀, and ℵ₀ = 2ℵ₀ = 2ℵ₀ + 1. So Jupiter and Saturn have each completed both an even and an odd number of orbits, and that number has remained equal and unchanged from all eternity, despite their ongoing revolutions and the growing disparity between them over any finite interval of time. This seems absurd.

Sigh… Mr. Craig had a nice rebuttal of the notion that his argument was similar to the long rebutted Zenonian paradox of the Stadium and here he had to say that tripe…parity is a property of integers. Transfinite numbers aren’t members of ℤ, so talking of the parity of ℵ₀ pretty much makes as much sense as talking of the parity of ⅓ or π*.

Such reasoning in support of the finitude of the past and the beginning of the universe is not mere armchair cosmology. P. C. W. Davies, for example, utilizes this reasoning in explaining two profound implications of the thermodynamic properties of the universe:

The first is that the universe will eventually die, wallowing, as it were, in its own entropy. This is known among physicists as the ‘heat death’ of the universe. The second is that the universe cannot have existed for ever, otherwise it would have reached its equilibrium end state an infinite time ago. Conclusion: the universe did not always exist. (Davies 1983, p. 11)

The second of these implications is a clear application of the reasoning that underlies the current paradox: even if the universe had infinite energy, it would in infinite time come to an equilibrium since at any point in the past infinite time has elapsed, a beginningless universe would have already reached an equilibrium, or as Davies puts it, it would have reached an equilibrium an infinite time ago. Therefore, the universe began to exist, quod erat demonstrandum.

That’s a fairly good argument, but it presupposes a non-cyclical universe though.

Space does not permit a review of the arguments for and against the A- and B-Theories of time respectively. But on the basis of a case such as is presented by Craig (2000a,b), we take ourselves to be justified in affirming the objective reality of temporal becoming and, hence, the formation of the series of temporal events by successive addition.

I wonder, is that standard operating procedure when pitching one’s own past articles?

Then we get a rather thorough presentation of various cosmological models, where I know enough to know that I’m out of my depth. But, if accurate, that would make Mr. Craig a better physicist than he is a philosopher…

The second, far shorter part, concerns the fact that the universe began to exist and contrarily to Mr. Pruss, Mr. Craig goes on to treat quantum mechanics straight away:

Sometimes it is said that quantum physics furnishes an exception to the claim that something cannot come into being uncaused out of nothing, since on the subatomic level, so-called “virtual particles” come into being from nothing. In the same way, certain cosmogonic theories are interpreted as showing that the universe could have sprung into being out of the quantum vacuum or even out of nothingness. Thus, the universe is said to be the proverbial “free lunch.” This objection, however, is based on misunderstanding. In the first place, wholly apart from the disputed question of whether virtual particles really exist at all, not all physicists agree that subatomic events are uncaused. A great many physicists today are quite dissatisfied with the traditional Copenhagen interpretation of quantum physics and are exploring deterministic theories like that of David Bohm. Indeed, most of the available interpretations of the mathematical formalism of QM are fully deterministic. Quantum cosmologists are especially averse to Copenhagen, since that interpretation in a cosmological context will require an ultramundane observer to collapse the wave function of the universe. Thus, quantum physics hardly furnishes a proven exception to (10). Second, even on the indeterministic interpretation, particles do not come into being out of nothing. They arise as spontaneous fluctuations of the energy contained in the subatomic vacuum, which constitutes an indeterministic cause of their origination. Third, the same point can be made about theories of the origin of the universe out of a primordial vacuum. Popularizers touting such theories as getting “something from nothing” apparently do not understand that the vacuum is not nothing but is a sea of fluctuating energy endowed with a rich structure and subject to physical laws. Such models do not, therefore, involve a true origination ex nihilo.

That would give a cause to virtual particles, but that would still open random happenstance as a valid cause for events but more on that later.

J. L. Mackie […] believes creatio ex nihilo raises problems: (i) If God began to exist at a point in time, then this is as great a puzzle as the beginning of the universe. (ii) Or if God existed for infinite time, then the same arguments would apply to his existence as would apply to the infinite duration of the universe. (iii) If it be said that God is timeless, then this, says Mackie, is a complete mystery.

Mr. Craig did a fair job batting away the first three objections — helped by Mackie missing the opportunity to ask whyever can’t the universe at large be timeless, with time an emergent property of its structure I feel — but the fourth one, I think, is a fail: “there is also an alternative that Mackie failed to consider, namely, (iv) prior to creation God existed in an undifferentiated time in which hours, seconds, days, and so forth simply do not exist”. If that’s a valid dodge for God, why in the nine hells wouldn’t that be one for the universe at large? That would solve a few problems re. entropy for cyclic universes…⁂

And the last one is regarding the Gott and Li hypothesis that the universe created itself, by dint of being stuck in a CTC (Craig’s presentation), which is dismissed by Craig thus:

Thus, the Gott–Li hypothesis presupposes the B-Theory of time. But if one presupposes such a view of time, then Gott and Li’s hypothesis becomes superfluous.

And then by asserting the truth of the A-theory of time. That’s, I feel, a misunderstanding of what question that hypothesis seeks to answer: as Craig himself admits, even B-theory of time has a question to ask re. the beginning of the universe. To wit, “Why is there (tenselessly) something rather than nothing?” and “Because the universe got itself stuck in a time loop” is an answer as any other.

Last part treats of the properties of that First Cause…which, mind, per his own presentation of some beginningless hypothesises might just be an energy source but let’s that pass. Given that just an energy source plugged into the universe and pumping energy in goes against what he wants to prove, he goes straight for personal, powerful, having agency, yadda, yadda, finishing his presentation with Aquinas’s quip that this is what everyone calls God. To which I snidely reply that Azathoth fulfils that description but they’re certainly not what Mr. Craig means by God. Or, less snidely, Brahma. I have one issue though with his presentation, specifically on why should it be personal:

Finally, and most remarkably, such a transcendent cause is plausibly taken to be personal. Three reasons can be given for this conclusion. First, as Richard Swinburne (1991, pp. 32–48) points out, there are two types of causal explanation: scientific explanations in terms of laws and initial conditions and personal explanations in terms of agents and their volitions. For example, in answer to the question, “Why is the kettle boiling?” we might be told, “The heat of the flame is being conducted via the copper bottom of the kettle to the water, increasing the kinetic energy of the water molecules, such that they vibrate so violently that they break the surface tension of the water and are thrown off in the form of steam.” Or alternatively, we might be told, “I put it on to make a cup of tea. Would you like some?” The first provides a scientific explanation, the second a personal explanation. Each is a perfectly legitimate form of explanation; indeed, in certain contexts it would be wholly inappropriate to give one rather than the other. Now a first state of the universe cannot have a scientific explanation, since there is nothing before it, and therefore, it cannot be accounted for in terms of laws operating on initial conditions. It can only be accounted for in terms of an agent and his volitions, a personal explanation.

Very well, but even admitting the contention that the first cause can’t be described by laws of the scientific kind, it doesn’t follow that it has to be personal: Mr. Craig opened random happenstance as a valid kind of cause in his rebuttal that virtual particles and the like are truly causeless. That would fit the bill too‡.

In the end, I find interesting that Craig’s dedicated only a short chapter to some roughly outlined objections by Grünbaum: he put in more space for physics than for objections. Hell, he put in more space for physics than for external supports!

Next up will be Collins for a teleological argument…late May?

*: Let’s be more accurate. It does make sense to speak of the parity of ℵ₀, but the definition of parity in the transfinite numbers — or, for that matter, of multiplication and addition — is different from the familiar one from ℤ so Craig’s sketched proof still doesn’t make much sense.
†: To be fair to Mr. Craig, he does admit a few pages later that his kalam argument relies heavily on the A-theory of time being the correct one.
‡: Also, that argument of Mr. Craig to eliminate scientific-style explanation from reckoning — which doesn’t eliminate just there being a limitless source of energy pumping in energy in a brane system or the like mind: no rules, no pesky conservation of energy — requires a “rules acting on conditions” conceptualisation of physical laws. On a “pattern descriptions” one like Carroll’s, that argument is on far shakier grounds.
⁂: More exactly, that solves any entropy- or actual infinite-related issues: no time, no problem with the Second Law of Thermodynamics. Also, if the universe at large is timeless — or in a time conception which doesn’t act like conventional time, which is the same here —, there’s no problems with it having been past-eternal: that doesn’t make sense in such a setting.