The Kalam Fallacy and the Initial Singularity
The religious apologist William Lane Craig has repeatedly asserted that Big Bang cosmology supports the thesis that the universe (viz., spacetime and matter) had an absolute beginning. To sustain this claim, Craig needs an outdated version of cosmology that predicts there was an initial singularity prior to the Big Bang. There are several reasons to reject this view (i.e., that of an initial singularity), but I'll accept it here for the sake of argument and assess whether it supports Craig's Kalam cosmological fallacy or not.
Timeless Singularity
One of the conclusions of Craig's Kalam fallacy is that the universe had a non-temporal cause (which is taken to be God). However, since this concept is hard to understand, Craig tries to elucidate on the idea by drawing an analogy from cosmology. He writes:
"The initial singularity is not considered to be part of physical time, but to constitute a boundary to time. Nevertheless, it is causally connected to the universe. In an analogous way, we could say that God's timeless eternity is, as it were, a boundary of time which is causally, but not temporally, prior to the origin of the universe." (Craig & Sinclair, "The Kalam Cosmological Argument," p.196)
Aron Lucas, rightly puzzled by Craig's assertion, wrote the following: "The idea seems to be that if a timeless singularity can be causally related to the universe, then so can a timeless God. But if Craig is willing to say that the Big Bang singularity is outside of space-time, and is therefore spaceless, timeless, changeless, and beginningless, then why doesn't he just say that the singularity, rather than God, is the cause of the universe? He cannot argue that the singularity is also in need of a cause, because according to his definition of "begins to exist," the singularity did not begin to exist. So if the singularity causes the rest of the universe, and it doesn't need a cause itself, what room is left for God?" (Review of The Blackwell Companion to Natural Theology)
Philosopher Erik Wielenberg mentioned this hypothesis as well: "Lots of stuff is on the table that doesn't involve God at all... Suppose that once upon a time there was this timeless, infinitely dense, dimensionless physical object which we can call 'singularity' and this thing exists timelessly. At a certain point – 13.8 billion years ago – an event occurs that causes the singularity to, in Craig's terminology, enter time. And what is this first event? It might be [that] this object begins to expand, so we could call this event 'first expansion'. ... So, the picture here is [that] instead of a god... it's the beginning of the expan[sion] of this... singularity. That, of course, is what the Big Bang is – this thing expanding. Space-time and all of its contents – the universe – emerge ultimately from this object singularity." (Kalamity at Creation? | Dr. Erik Wielenberg)
Philosopher Quentin Smith went one step further and defended this view: "I argue that the timeless [and] uncaused... being that causes spacetime’s beginning to exist is not God but a spatially zero-dimensional point. This point contingently has the property of being the big bang singularity from which our maximal spacetime (“the Friedmann universe”) exploded and began to expand." (Quentin Smith, Time Was Created by a Timeless Point, p.96)
Elsewhere, Quentin elaborated on this idea a bit: "The theory of the infinite singularity implies that the space of the singularity topologically transforms into the three dimensional space of the universe at the big bang explosion. It is a familiar notion in the mathematical discipline of topology that a space with a topology of a point can assume the topology of a finite 3D space. The topological transformation of the 0D space to the 3D space is precisely the big bang explosion." (Atheism, Theism and Big Bang Cosmology)
And, indeed, many cosmologists agree that time ends at singularities:
"A straightforward application of general relativity... shows that the enormous mass and energy crushed together at the black hole's center causes the fabric of spacetime to suffer a devastating rift, to be radically warped into a state of infinite curvature – to be punctured by a spacetime singularity. One conclusion that physicists drew from this is that since all of the matter that has crossed the event horizon is inexorably drawn to the center of the black hole, and since once there the matter has no future, time itself comes to an end at the heart of a black hole." (Brian Greene, The Elegant Universe, p.156)
"Then within a Planck time (10⁻⁴³ s) before that the models all predict a singularity, where time stops." (Lee Smolin, The Singular Universe, p.402)
"This breakdown of Einstein’s equations at the birth of the universe is known as the Hawking-Penrose singularity, after the two iconic physicists who generated the theorem. Time stops at this singularity—there is no “before”; clocks freeze." (Laura Mersini-Houghton, Before The Big Bang)
"According to the Penrose Singularity Theorem, which he proposed in 1965, a time-like singularity will occur within a black hole whenever matter reaches certain energy conditions. At this point, the curvature of space-time within the black hole becomes infinite, thus turning it into a trapped surface where time ceases to function." (Matt Williams, What Is A Singularity?)
“...an initial “Big Bang” singularity [is] where time begins... [and] a final “Big Crunch” singularity [is] where time comes to an end. ... Thus to ask what happened before the initial singularity is the same as asking what happened before an eternal universe came into existence. A meaningless question, surely.” (Frank Tipler, The Physics of Immortality, pp. 101, 102)
“If a physical infinity appears in a theory like Einstein's, where the fabric of space and time is determined by the physical density of matter within it, then it requires that... time [is] destroyed at the places where infinite densities appear.” (John Barrow, The Infinite Book, p.103)
Elaboration
It is fair to say that Quentin Smith is the man who contributed most to this theory, so it is worth quoting his paper on this topic at length here. He explained and defended his proposal in the article Time Was Created by a Timeless Point, (pp. 114-117):
"The big bang singularity is temporally and spatially zero-dimensional. The metric tensor, which is defined on each point in spacetime, is not defined on the big bang singularity, which thereby is not a spacetime point. The metric describes the curvature of spacetime, and since the singularity is an isolated point, it has no curvature; that is to say, the notion of curvature is undefined on the singularity, which is what physicists mean when they say the point has “infinite” curvature. Methods of attaching the singularity to spacetime and defining the metric tensor on the singularity as the first instant of time, t = 0 (rather than as a metrically undefined timeless point), have run into the problem of being counterexampled. As I have indicated earlier, the two procedures for attaching the singularity as the first instant of time, the B-boundary procedure and the G-boundary procedure, have been shown to be unacceptable definitions of a singularity attached as a first instant of time.
But this failure still leaves one able to attach the singularity as an unmetricated spatial point that is topologically attached to certain types of spacetime, most notably, the type of spacetime that general relativists believe we occupy, a Robertson-Walker-Friedmann spacetime. This was first noticed by one of the general relativists who argued that the B-boundary and G-boundary methods fail, Robert Wald. Wald notes that adding an unmetricated singular point to an otherwise metrically well defined spacetime 'would allow one to talk in precise terms of a singularity as a place even though the metric is not defined there. However, while this could be done by hand in a few simple cases like the Robertson-Walker or Schwarzschild spacetimes, severe difficulties arise if one tries to give a meaningful general prescription for defining a singular boundary.'
John Earman’s recent argument that the big bang singularity in no sense “exists” is invalid. He believes that counterexamples to the definition of a general relativistic singularity, counterexamples that consist in possible spacetimes acquiring physically impossible features if an existent singularity is added to them, suffice to show that our universe does not have an existent singularity. However, Earman’s conclusion that singularities are not existents, since there is no adequate definition (either the B-boundary definition or the G-boundary definition) that applies to all general relativistic singularities, is a non sequitur. If there is no adequate definition of a game that applies to all games, it does not follow that there are no chess games.
There may be no defining essence that is common to all singularities, and “singularity” may not have a univocal meaning. In the case at hand, there can be a big bang singularity in a Robertson-Walker-Friedmann spacetime (which is the spacetime in which we live, according to general relativists) even if there is no adequate definition that applies to all singularities. As I mentioned above, Robert Wald, one of the physicists who showed there is no definition that applies to all singularities, argues that one can nonetheless have a singularity in some cases, one of the cases being our Robertson-Walker-Friedmann spacetime. Since the equations of big bang cosmology predict an initial singularity, the prediction of the singularity requires a realist interpretation if the equations are given a realist interpretation. Physicists correctly note that “the initial singularity is a consequence of the equations of general relativity.” A justification for deciding to interpret all of the consequences of the equations of general relativity realistically but this one consequence would be that this consequence has no coherent realist interpretation. But if the Friedmann equations, which general relativists believe describe our spacetime, have an incoherent consequence, how could they even be possibly true, let alone actually true (or highly confirmed)? The singularity has the spatial topology of a point (even though it has no spatial metric and does not exist in time), and it is topologically connected to the metrically well defined spacetime."
Bad Objections and Good Responses
Craig objects that the singularity cannot be the cause of the universe, since it does not have any real ontological status; it is a mere mathematical fiction that appears only in cosmological models. It is a "mere conceptualization ontologically equivalent to nothing." He thinks so because the singularity is hypothesized to be "infinitely dense." As Craig explained:
“This event that marked the beginning of the universe becomes all the more amazing when one reflects on the fact that a state of 'infinite density' is synonymous to 'nothing.' There can be no object that possesses infinite density, for if it had any size at all it could still be even more dense.” (William Lane Craig, The Existence of God and the Beginning of the Universe)
Claiming that this potential Infinity Paradox indicates there was nothing "prior" to the Big Bang is a huge unjustified leap (a non-sequitur) as was pointed out by philosopher Wes Morriston:
"This argument of Craig’s need not detain us for long. There are at least three quite obvious – and decisive – objections to it. (i) In the first place, “infinite density” is not synonymous with “nothing,” and the “initial singularity” that figures in Craig’s statement of the Big Bang theory is not simply nothing at all. A mere nothing could not begin expanding, as the infinitely dense “point universe” is supposed to have done. And even if it lacks spatial and temporal spread, the initial singularity would have other properties – for example, that of “being a point.” It would therefore be a quite remarkable something, and not a mere nothing. So, [this argument] is obviously false. (ii) ... No one would suppose that it follows from the fact that there can be no round squares, that “round square” is synonymous with “nothing.” But neither should anyone suppose it follows from the fact (assuming it is a fact) that there can be no infinitely dense objects, that “infinite density” is synonymous with “nothing.”" (Creation Ex Nihilo and the Big Bang, pp. 25-26)
Morriston, then, goes on to suggest that one could accept the existence of a singularity and yet reject the idea of infinite density: “Even those who believe there was an initial singularity do not hold that it possessed infinite density. They suppose instead that the singularity had no density, since it had zero volume. For a helpful explanation, see Milton K. Munitz, Cosmic Understanding, p.111.” (p.32)
I found a discussion of Munitz’s argument in a paper by Quentin Smith: “When it is said that the big bang singularity has an infinite density, infinite temperature, and infinite curvature, it is not being said that the singularity has parts or properties that map onto a set with an aleph-zero or aleph-one cardinality. ... Consider the phenomenon of density, which is the ratio of mass to unit volume (density=mass/volume). 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/0, 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. Although philosophers frequently misunderstand this use of the word ‘infinite’ by physicists, this usage has been clearly grasped by Milton Munitz in his recent discussion of the big bang theory. He notes that
‘the density of a homogeneous material is mass per unit volume-for example, grams per cubic centimetre. Given both a zero value and the conservation of the mass-energy of the universe [at the big bang singularity], no finite value can be given to the ratio of the latter to the former (it is forbidden to divide by zero). This is normally expressed by saying that the density becomes infinite. It would be more accurate to say the standard meaning of ‘density’ cannot be employed in this situation. The density cannot be assigned a finite measurable value, as is the case in all standard applications of the concept.’” (Atheism, Theism and Big Bang Cosmology)
Quentin made the same point in another paper: “The singular point has “infinite mass-energy density” in the sense that the measure of its density has zero for its denominator, not in the sense that it has aleph-zero density. Density is the ratio of mass-energy to unit volume, e.g., grams per cubic centimetre. Given the conservation of mass-energy at the singularity, there would be a large number of grams per zero cubic centimetres, since the singularity has zero volume. However, n/0 is a mathematically meaningless expression since division by zero is impermissible. The expression “the singularity is infinitely dense” means that the concept of density is inapplicable at the singularity. Likewise “the singularity has infinite curvature” means the concept of curvature is inapplicable, since a point cannot be curved.” (Time Was Created by a Timeless Point, p.116)
In the article A Discussion of the Kalam Argument, Greg Scorzo addressed other objections:
"Even if we accept the causal rule, the truthfulness of the Big Bang theory, the impossibility of an actual infinite, Craig's definition of causation, and his insistence that this sort of causation applies to the universe as a whole, we do not arrive at the conclusion that God is the cause of the universe. Rather, we arrive at the conclusion that the initial Big Bang singularity is the cause of the universe.
Craig anticipates this objection, responding to it by denying the reality of the singularity and attempting to deduce, on a priori grounds, the existence of a personal, supernatural creator [by u]tilizing the principle of determination... Craig objects to the reality of the singularity on ontological grounds. Craig believes that the initial cosmological singularity is not an actual existent, but merely a mathematical idealization whose ontological counterpart is nothing. As Craig writes,
'The break down of the laws of physics and the attendant unpredictability is perspicuous in light of the fact that nothingness possesses no physics... Simply put, an object that has no spatial dimensions and no temporal duration hardly seems to qualify as a physical object at all, but is rather a mathematical conceptualization.'
Several things can be said in response to Craig's objection. First, just because an object possesses no time and no space, it does not follow that it is merely a mathematical conceptualization. Most philosophers would argue that abstract objects such as numbers, sets, and propositions are actually existent, despite the fact that they have no spatial or temporal dimensions. Even more puzzling is the fact that Craig's own conception of God entails that God is a being of no dimensions or duration, but Craig never refers to his God as a conceptual formalism. Why is it possible for a being of no dimensions or duration to obtain ontological existence, yet an object possessing these properties is merely a mathematical idealization?
Secondly, within Big Bang Cosmology, the initial singularity is depicted as the ontological consequence of the thermodynamic expansion of the universe. If Craig upholds a realist interpretation of the dynamic properties of the universe, his retreat into a formalist understanding of the singularity at the moment of creation is a bit suspicious. If Craig wishes to deny the ontological existence of the singularity and still remain within relativity theory, he must also deny the thermodynamic contraction of the universe which leads to the singularity. Craig cannot rely on a realist understanding of Big Bang Cosmology in order to [support] the second premise of his argument if he switches over to a conceptualist understanding of relativity when referring to the initial singularity. He should accept the ontological consequences of the Big Bang cosmological theory if he finds it the most substantiated physical cosmology available.
Craig gives an additional a priori argument in order to avoid a real singularity which could potentially be the 'cause' of the universe. Craig believes that a necessary and sufficient set of mechanical conditions existing from eternity couldn't possibly be the cause of the universe, since if that were the case, the universe would have always existed. As Craig writes,
'In fact, I think it can be plausibly argued that the cause of the universe must be a personal creator. For how else could a temporal effect arise from an eternal cause? If the cause were simply a mechanically operating set of necessary and sufficient conditions existing from eternity, then why would not the effect also exist from eternity? For example, if the cause of water's being frozen is the temperature's being below zero degrees, then if the temperature were below zero degrees from eternity, then any water present would be frozen from eternity. The only way to have an eternal cause but a temporal effect would seem to be if the cause is a personal agent who freely chooses to create the effect in time.'
This argument, aside from attempting to eliminate the reality of the singularity, also tries to deduce the personal attribute of the first cause. However, if one looks closely, several problems emerge. First, Craig is right that a mechanically operating set of necessary and sufficient conditions could not produce a temporal effect from eternity. However, Craig fails to notice that the singularity is not a mechanically operating set of conditions, but rather, a lawless and indeterministic point which can potentially emit any configuration of particles at any time with equal likelihood."
Greg is correct that a singularity is "lawless and indeterministic", as many cosmologists pointed out:
"Classical [General Relativity] places no restrictions on what non-deterministic influences can emerge from the naked singularities." (Determinism: What We Have Learned and What We Still Don’t Know, p.14)
"Anything can happen at a naked singularity. Cause and effect lose any meaning." (Joseph Silk, The Infinite Cosmos, p.207)
"Anything can come out of a naked singularity – in the case of the big bang the universe came out." (Paul Davies, The Edge of Infinity, p.161)
"Because general relativity would break down at a singularity, anything could come out of the big bang." (Stephen Hawking and Roger Penrose, The Nature of Space and Time, p.89)
"Anything at all can emerge from a singularity. The laws of physics do not apply to the split second in which energy or mass emerges from a singularity, and in principle a singularity could emit broken washing-machines, human beings or stars." (John Gribbin, After Cobe and Before the Big Bang, p.124)
"A singularity can be naturally considered as a source of lawlessness, because the spacetime description breaks down “there”, and physical laws presuppose spacetime." (M. Novello and Perez Bergliaffa, Bouncing Cosmologies)
"It seems that Einstein always was of the opinion that singularities in classical field theory [i.e., physics] are intolerable... because a singular region represents a breakdown of the postulated laws of nature." (Peter Bergmann, Some Strangeness in the Proportion, p.186)
"A singularity is a place where the classical concepts of space and time break down as do all the known laws of physics because they are all formulated on a classical space-time background." (Stephen Hawking, Breakdown of Predictability in Gravitational Collapse)
“One can think of a singularity as a place where our present laws of physics break down." (Hawking and Ellis, p.3)
"A space-time singularity... is not subject to the laws of physics." (Frank Tipler, Personal Correspondence)
Therefore, unlike freezing temperature (the cause) and frozen water (the effect), the singularity doesn't have to produce its effect, that is, the singularity can be present and yet the temporal universe may not be; this is because the singularity is completely lawless and non-deterministic; it doesn't have to create a universe. As Quentin explained: "I also share the view of these theists that this metaphysically necessary being contingently causes spacetime to exist (such that this necessary being exists in some merely possible worlds where it does not cause spacetime)... In some possible worlds, the point c exists timelessly but does not cause spacetime to begin to exist. It is not governed by any law that implies it causes spacetime if it exists." (Time Was Created by a Timeless Point, pp. 96, 115) Elsewhere he wrote: "The fact that the singularity gives rise to a three-dimensional space with matter does not imply that it is a physical law that the singularity give rise to something. There is no physical law in classical big-bang cosmology of the form “If there is a singularity, it must give rise to something at a later time.” ... There is no dispositional property of the instantaneous spatial point that manifests a physical tendency to explode into something else; the singularity could just as well have been followed by nothingness..." (Can Everything Come to Be Without a Cause?)
Quentin Smith also commented on Craig's arguments: "It might be objected that... the singularity is not a real physical state but a mathematical fiction... My response to this objection is that it is based on a misinterpretation of big bang cosmology, for this cosmology represents the singularity as real. For example, Penrose writes that "we think of the initial singularity as a single point [which] gives rise to an infinity of causally disconnected regions at the next instant," which entails that the point is earlier than the explosion and therefore real. But this response may miss the point of the objection, which is not that big bang cosmologists represent the singularity as unreal, but that the singularity is unreal, given reasonable principles for interpreting scientific theories... Craig notes that the big bang singularity is represented as having zero volume and zero duration and that this is sufficient reason to regard it as unreal. He asserts that 'a physical state in which all spatial and temporal dimensions are zero is a mathematical idealization whose ontological counterpart is nothing.' But Craig offers no justification for this assertion. Cosmologists find no difficulty in the concept of a space that has zero dimensions (a spatial point) and that exists for an instant and a simple assertion that a 0D space cannot instantaneously exist seems to be an expression of an unwarranted skepticism." (A Big Bang Cosmological Argument For God's Nonexistence)
Quentin repeated the same conclusion in another article: "Big bang cosmology represents the singularity as a unique sort of reality, a physical singularity, but it is represented as real nonetheless. This is evinced by the fact that past-directed spacetime paths in the early universe are not modeled on half-open intervals that approach arbitrarily close to but never reach the ideal limit, but on closed intervals one of the endpoints of which is the singularity. In the words of Penrose, ‘the essential feature of a past spacelike singularity [the big bang singularity] is that it supplies a past singular end-point to the otherwise past-endless timelike curve’. (A timelike curve is a spacetime path of a particle.) In the words of Geroch and Horowitz, converging past-directed spacetime paths are not commonly thought to merely approach with arbitrary closeness the same singular point but are thought to actually ‘reach the same singular point’, which requires the actual physical existence of the singular point. ... If Craig is to justify his claim that the assumption that [the singularity] is real is an illegitimate 'ontologising' of a mathematical construct, he must provide some reason to support this claim.. His recent and related claim that 'a physical state in which all spatial and temporal dimensions are zero is a mathematical idealisation whose ontological counterpart is nothing' is made with no effort to support it and should be rejected as an unjustified skepticism about a widely held scientific thesis." (Atheism, Theism and Big Bang Cosmology)
Craig is not the only apologist who tried to refute the idea that the singularity is (or was) a real object. Richard Swinburne stated something similar, as Quentin Smith explained in the article A Big Bang Cosmological Argument For God's Nonexistence:
"Richard Swinburne also believes that the singular point is a mathematical idealization. He provides an argument for this, namely, that it is logically necessary that space be 3D. Swinburne presents an argument against the logical possibility of 2D objects and suggests that analogous arguments can be constructed against 1D and 0D objects. He asks us to consider a two dimensional surface that contains two dimensional objects:
'…it is clearly logically possible that the two-dimensional "material objects" should be elevated above the surface or depressed below it...the logical possibility exists even if the physical possibility does not. Since it is logically possible that the "material objects" be moved out of the surface, there must be places, and so points, outside the surface, since a place is wherever, it is logically possible, a material object could be.'
Therefore, Swinburne concludes, if there exists a two dimensional object or surface there must also exist a third spatial dimension. Swinburne's argument instantiates the following invalid argument-form:
(1) Fx is logically possible (i.e. it is logically possible for x to possess the property F).
(2) C is a necessary condition of Fx.
(3) x exists.
(4) Therefore, C exists.
The fact that Swinburne's argument has this form becomes clear if we state his argument as follows:
(1A) It is logically possible for any object on a two dimensional surface to possess the property of moving above or below the surface.
(2A) A third spatial dimension is a necessary condition of any object on a two dimensional surface moving above or below the surface.
(3A) There exists an object on a two dimensional surface.
(4A) Therefore, there exists a third spatial dimension.
If (1A)-(4A) proves that objects on two dimensional surfaces require a third spatial dimension, then the following argument proves that there is a heaven:
(1B) It is logically possible for any human body to be resurrected after death and occupy a heavenly space.
(2B) Heaven is a necessary condition of any human body being resurrected.
(3B) There are human bodies.
(4B) Therefore, there is a heaven.
The fallacy, if the reader has not already grasped it, is the assumption that a necessary condition of an object possessing a certain property must be actual if the object is actual. This of course is not so; the necessary condition need be actual only if the object's possession of the property is actual. I conclude that Swinburne has given us no reason to believe that it is impossible for there to be a big bang singularity that occupies less than three spatial dimensions. Given that Swinburne's argument fails, and that no other arguments against the coherency of the big bang singularity have been presented (at least of which I am aware), the above considerations warrant the conclusion that there is no reason to deny reality to the big bang singularity."
Quentin, then, goes on to discuss a very curious and interesting question: "How can the entire mass of a finite universe be compressed into a point? The mass is 3D and the point is 0D, which involves a contradiction. But this is a misunderstanding. The mass as compressed into the point is not ordinary mass, 3D mass, but infinitely compressed mass, which means that it has lost its three dimensionality and assumed the dimensionality of the point it occupies. The assertion that at the instant of the singularity, n kilograms of mass is infinitely compressed in a zero volume, implies in part that (i) at this instant there exists no 3D mass, (ii) at this instant there exists only one 0D point, [and] that (iii) this point subsequently assumes the typology of a 3D space..." (Atheism, Theism and Big Bang Cosmology)
(Note: Craig objected to a physical substance being the timeless cause of our universe on the grounds that physical stuff is never changeless; quantum particles are always vibrating/fluctuating in space and fields. (Craig, p.107) Quentin replied that these fluctuations "cannot obtain on the quantum level in the singularity, since there is no quantum level in the singularity; the space-time manifold that quantum processes presuppose has broken down. The singularity is a violent, terrifying caldron of lawlessness." A Big Bang Cosmological Argument For God's Nonexistence)
A Real Problem for this Theory
While all of these apologetic arguments against the existence of gravitational singularities have fatal flaws, it seems that physicists do not interpret singularities as existent points or objects, as cosmologist Lee Smolin pointed out:
"It is important to dispel some false impressions about the cosmological singularity theorems which are widely spread due to misleading accounts in some popularizations. ... The singularity is not a moment of frozen time. The singular set is not in fact part of the spacetime geometry modeled by the metric. The singular set is a boundary which is a set of limit points of the spacetime geometry. There is no set in the spacetime geometry where time is not flowing. ... There is no event, force or influence which starts the universe evolving. The cosmological singularit[y] [is] simply [a] boundar[y] to the extension of a spacetime history to the past. There is nothing there, before the singularity, which starts the universe going." (The Singular Universe, p.404)
Prominent physicist Thomas Hertog provided a very clear and simple explanation: “The spacetime singularity of infinite curvature that awaits in the center [of a black hole] isn’t really a place in space but more like a moment in time—the last one. ... Penrose had shown that, according to relativity theory, time must end inside black holes. Stephen’s time-reversed argument demonstrated that in an expanding universe time must have a beginning. It is not that the big bang singularity sat there, like a cosmic egg, waiting to hatch a universe. The singularity rather signals the birth of [space and] time itself. ... Singularities are edges to spacetime...” (On the Origin of Time, Ch. 2 & 3)
Theist philosopher Jacobus Erasmus made a similar point: “A singularity is formally defined in terms of geodesic (or path) incompleteness and not as an object that exists apart from space-time (Wald 1984, p.215; Ellis et al. 2012, p.145). Accordingly, when a physicist states that “a singularity exists”, he/she means that “space-time has the property of being geodesically incomplete”. A geodesic in this sense refers to a space-time line or path along which a freely falling particle moves. If a geodesic has a finite, affine length and is not endless in either direction, it is incomplete. Thus, space-time is singular if it is time-like or null geodesically incomplete. ... [If] the universe has an initial singularity, [that would] mean that the universe has an edge or boundary. In this regard John D. Barrow (2007, p.39) declares that “[the Big Bang singularity] is the boundary of the Universe”. In this context, the phrase “boundary” does not refer to some existing abstract or concrete object, such as a wall or barrier with which an observer may collide with but, instead, it signifies that space-time is inextendible in at least one direction. As an analogy, one could think of the edge or boundary of a walking stick. A walking stick has a boundary or beginning point if it has a finite length… Thus, although some physicists speak loosely of a singularity as a “point” or “location”, this should not be taken at face value. As Tim Maudlin notes, depicting a singularity as a line or point may mislead “the incautious observer” because he/she might assume that “the singularity were some sort of thing”. However, “the singularity is an edge of space-time itself, where time-like curves simply cannot be continued” (Maudlin 2012, p.144).” (Is the Big Bang the Sole Cause of the Universe, pp. 5-6)
Philosopher of physics Daniel Linford explained it further: "I’ve compared singular space-times to the function f(x) = 1/x. The function f(x) has no value for x=0 because dividing by zero is forbidden. We can think of x=0 as a kind of barrier between the positive and negative values of x, where the barrier is missing its boundary. The boundary is missing because, again, there is no value of f(x) for x=0. Likewise, differential manifolds are not well-defined when the curvature becomes infinite. (Technically, there are many different ways of describing space-time curvature; in this context, I mean the Ricci scalar becomes infinite.) This is why Craig says that the singularities are “nothing”: they are not part of the space-time manifold and are, consequently, not part of reality at all, any more so than the point x=0 was part of the function f(x)=1/x. (At least this is the conclusion that one reaches if one takes General Relativity literally.)
We need to be careful. There are two different ways in which we can talk about singularities being “real”. If I say that Craig thinks singularities are real, but most physicists do not, I mean that Craig thinks there are points that are really missing from the space-time manifold and around which the curvature increases without bound. On the other hand, Quentin Smith has argued that singularities are real in the sense that, on his view, the space-time manifold does include the singular points.
....
I would say that Smith takes a view that most physicists (and philosophers of physics) would think to be nonsensical.
Smith claims that the metric becomes undefined for the singularity. In some sense, the metric can be naively computed for the singularity. For a simple example, let’s consider a flat FLRW universe that contains nothing but radiation. For flat FLRW universes, the metric has the form ds^2 = -dt^2 + a(t)^2 (dx^2 + dy^2 + dz^2). The function a(t) – what’s called the “scale factor” – roughly provides us with a length scale for measuring the universe’s expansion. In a radiation dominated universe, a(t) = t^(1/2). Now, t^(1/2) is a monotonically increasing function that passes through a=0 when t=0. That the scale factor becomes zero is what folks mean when they say that, according to the hot big bang model, the universe shrinks to zero size at some point in the past. And notice that as the scale factor becomes zero, ds^2 never becomes undefined. To see why the space-time manifold becomes undefined when the scale factor becomes zero, we need to look elsewhere.
(By the way, ds^2 is not the metric tensor, but the coefficient on the terms in ds^2 are the components of the metric tensor, so they encode the same information. And since ds^2 did not become undefined when t=0, ds^2 does not become undefined either. Moreover, as we will see, since the space-time manifold becomes undefined for t=0, there is a sense in which the metric can’t be defined there either – after all, a function cannot be defined for a non-existent point!)
There are a number of other mathematical objects that are computed from the metric tensor and so, in the case of FLRW space-times, from the scale factor. The Christoffel symbols (what’s sometimes called the metric connection) is computed in terms of partial derivatives of the components of the metric tensor and the various curvatures (e.g., the Riemann tensor, the Ricci curvature tensor, and the Ricci scalar curvature) are computed in terms of partial derivatives of the Christoffel symbols. In turn, the Einstein tensor is computed from the Ricci tensor and the Ricci scalar, and Einstein’s Field Equations involve the Einstein tensor.
In case that blizzard of mathematics blew right past you, here’s the important upshot. Various mathematical objects that express the curvature of the space-time manifold are calculated in terms of the metric tensor. And while the metric tensor may not be undefined for the singularity, these other objects do become undefined. Why do these other objects become undefined? In the case of, e.g., a flat, radiation dominated FLRW universe, the Ricci scalar curvature (for example) is proportional to 1/a(t)^2. So, when a(t) becomes zero, the Ricci scalar curvature becomes undefined because division by zero is undefined.
Now, there are a number of different things that one could mean if one said that singularities are real. One could mean:
1. The mathematics I described above provides us with a more or less complete theory about the early universe, so that there really is an open boundary to the universe. In this case, the view is not that the point at t=0 exists as a physical object of some sort but rather that the singularity in General Relativity’s mathematics tells us about some genuine feature of reality, i.e., the open boundary. This seems to be Craig’s position and is a position Smith disagree[d] with.
2. The Big Bang singularity is a real object of some kind. This object could not be a spatio-temporal object because the object has no spatio-temporal description. Smith defended this position in various places and Craig has denied this position.
The majority of physicists think both of these positions are deeply implausible. Physicists deny (1) because singularities are features in a classical theory that occur at precisely the place where we have independent reason to think a quantum mechanical successor theory is needed. Physicists deny (2) because (2) doesn’t appear to make any sense. Singularities are not genuine objects in any physical theory that I am aware of and, frankly, I don’t know what it would mean to say that a singularity was a genuine object. If one modified General Relativity by replacing singularities with objects of some kind, I think physicists would say that you’ve removed the singularity and not that you’ve identified a singularity as a specific object. This is what happens for coordinate singularities... In any case, General Relativity doesn’t provide a description of curvature singularities as objects within the theory and so Smith’s speculation would require some new theory that Smith has not provided." (Personal Correspondence)
In 2003, Quentin changed his mind and stated that the singularity does not actually exist: "No physicist holds that the Big Bang singularity actually exists. ... And why doesn't it exist? Well, it can't exist, because the definition of a singularity is of a self-contradictory entity. The singularity is supposed to be a zero-dimensional point. It's a spatial point, it doesn't have height, it doesn't have width, it doesn't have depth. It doesn't have 3 dimensions, 2 dimensions or 1 dimensions. It has zero dimensions! And this zero-dimensional point is supposed to be infinitely curved. Well, how could something that has no radius or size at all be curved whatsoever? It would be meaningless. You have to have sides to be curved in some way. But a zero-dimensional point has no sides. ... [Also], it would have infinite temperature. It would be infinitely hot. But temperature is the motion of molecules, or particles against each other. But the Big Bang singularity is a single zero-dimensional point. Nothing is moving. So it can't have infinite temperature. Temperature doesn't apply to it at all. And this zero-dimensional point is supposed to be infinitely dense. Well, it can't be infinitely dense, because it's got no matter in it. It's just a point. It's really nothing. ... And further, the contradiction is even worse what we know with this. That the matter is 3 dimensions of space. Height, width and depth. Well, this has zero dimensions! Zero "d"! So how could something with 3 dimensions fit inside something with zero dimensions? Well, it can't, it's a contradiction. So this singularity... doesn't exist! It's a part of the mathematical equations of Big Bang cosmology that physicists interpret as not corresponding to anything real."
Additionally, in the paper Collins on Cannons and Cosmology, Quentin and Paul Draper wrote: "The universe state that would exist at t0 if there were a first moment of time would be a zero-dimensional point in which there is packed all the three-dimensional matter of the universe; this is what physicists mean when they call that point "infinitely dense." But the existence of such a point is both a mathematical and a logical impossibility. Therefore, the singular state that supposedly would exist at t0 not only is not real but cannot be real. ... Despite [Quentin Smith's previous] arguments that the singularity at the physical time t0 is real, the fact remains that the mainstream belief of cosmologists is that... Friedman's equation is inconsistent with a realist interpretation of the big bang singularity. It is also worth noting that, while Smith's present view and those [of the past] are incompatible, it would be a mistake to think that Smith has changed his mind about the same set of ideas and arguments. Rather, further study of physical cosmology has led him to a more mathematically detailed and ontologically accurate understanding of the Friedman... cosmologies. (This sort of evolution of one's views is much more common in physics than in a priori philosophical inquiry.)"
Conclusion
So, to wrap it up, if the singularity is a timeless, non-deterministic and lawless zero-dimensional spatial point, then the Kalam certainly fails. But if the singularity is not existent as Smolin, Linford and, ultimately, Quentin argued (and if we concede the first and second fallacious premises as well as Craig's fallacious conceptual analysis), then the Kalam has a chance of succeeding.
Craig, Swinburne, Earman and other apologists presented no good reasons for thinking the initial singularity is not ontologically real and could not be the non-personal cause of space-time. Strong defenses were presented by Quentin and Scorzo. On the other hand, Smolin, Linford and other experts stated very clearly that a singularity is merely a "boundary" or "missing point" of space-time; not a real object.