As Salamu Al 'ala man itaba'a Al-Houda
Peace be Upon All those who seek guidance, Peace be Upon the Messenger of Allah his noble Household and Companions.
…
To
start off the Islamic argument for the Existence of God is one of the
better known and respected arguments in scientific & philosophical
circles.
Dr James Watson he American molecular biologist, best
known as one of the co-discoverers of the structure of DNA, wrote in an
article about Islamic theology the following:
The kalām
Cosmological Argument, is one of the better-respected arguments for the
existence of God. Because its validity is not controversial, because it
aligns with the most prominent scientific theories of the universe, and
because it agrees with general philosophical insight concerning
properties of infinities, it is one of the more interesting pieces of
religious philosophy. It can be stated as follows:
Quote:
(1) Whatever begins to exist has a cause of existence.
(2) The universe began to exist:
(2.1) Argument based on the impossibility of an actual infinite:
(2.11) An actual infinite cannot exist.
(2.12) An infinite temporal regress of events is an actual infinite.
(2.13) Therefore, an infinite temporal regress of events is an actual infinite.
(2.2) Argument based on the impossibility of the formation of an actual infinite by successive addition:
(2.21) A collection formed by successive addition cannot be actually infinite.
(2.22) The temporal series of past events is a collection formed by successive addition.
(2.23) Therefore, the temporal series of past events cannot be actually infinite.
(3) Therefore, the universe has a cause of its existence.
|
…
The Kalam Cosmological Argument and Infinite Regress
Inshallah
we will continue with the Islamic argument for the existence of God.
Unfortunately the first few posts will be complicated bringing many
advanced philosophical and scientific arguments. As the post advances
though we are hopeful that we can also provide easier proofs for the
layman to understand as well inshallah.
Some have objected to the
KCA (Kalam Cosmological Argument) such as J. L. Mackie and Graham Oppy
but we will inshallah examine their argument and respond by using the
research paper by Dr. James Watson.Objections to the Kalām Cosmological Argument
J.
L. Mackie criticizes the kalām argument (KCA from here on) in his
posthumous The Miracle of Theism[10]. Beginning with (2.2), Mackie says
KCA proponents show a prejudice against actual infinities. In the
medieval versions, this argument addressed the impossibility of
traversing an actual infinite. “Since an infinite distance cannot be
crossed, if the past were infinite, then today would never arrive. But
this is obviously absurd, since today has arrived.[11]”
Mackie
claims that the arguers make the unwarranted assumption that the
universe had an infinitely distant starting point and then claims that
it is impossible to traverse that infinite. But, in taking infinity
seriously, there would be no starting point at all – not even an
infinitely distant one. So any point in past time is only finitely far
from the present.
Addressing (2.1), Mackie says that transfinite
mathematics refute the premise that an actual infinity cannot exist.
Just because properties differ from infinite sets to finite ones does
not entail any contradiction or absurdity with regard to the
actualization of those infinities.
Quote:
…our
normal criteria for smaller than and equal to fail to be mutually
exclusive for infinite groups. For finite groups to be smaller than
means that the members of one group can be correlated one to one with a
proper part of another group; to be equal to means that the members of
the two groups can be exactly matched in a one to one correlation. These
two criteria are mutually exclusive for all finite groups, but not for
infinite groups. Once we understand this relation between the two
criteria, we see that there is no real contradiction.[12]
|
Mackie
further rejects on the grounds that there is no a priori reason to
accept it. Craig quotes: “…there is a priori no good reason why a sheer
origination of things, not determined by anything, should be
unacceptable, whereas the existence of a god [sic] with the power to
create something out of nothing is acceptable.[13]” Mackie claims that,
despite the fact that astronomy and physics support a finite past for
the universe, given Cantorian formulations with infinite numbers and
that there are, a priori, no good reasons to accept, the KCA is a flawed
argument.
Graham Oppy agrees with Mackie and claims that the KCA
is full of unwarranted assumptions[14]. Proponents of the KCA generally
accept the validity of transfinite mathematics, but they also ask
whether it is possible for these infinities to be instantiated in the
actual world. Oppy admits that he does not understand exactly what is meant by this question.
The logical possibility of infinities referred to in transfinite math
entails their possibility in some possible world. So what is the
question meant to address? Oppy offers three ways of interpreting the
question,
Quote:
One
suggestion is that the question is whether there are any infinities in
the actual world. Another suggestion is that the question is whether it
is possible for there to be any infinities in the actual world. And a
third suggestion is that the question is whether it is possible for
there to be any infinities in any world.[15]
|
Oppy
takes the third suggestion to be out of the question since proponents
of the KCA accept the logical consistency of transfinite infinities. The
KCA takes it for granted that the first suggestion is not true, stating
it without argument. Oppy concedes for the sake of argument, but points
out that Mackie does not.[16]
Therefore, Oppy says a stalemate
exists between those who accept the existence of actual infinities, like
Mackie, and those who do not.
In considering the second
suggestion, Oppy says, “in order to distinguish this claim from the
third suggestion, it seems that we shall need to interpret it to be
asking whether the existence of infinities is compatible with the actual
laws of physics…”[17] But since this ceases to be an a priori question
and thus an a posteriori one, and since “we do not yet know what are the
laws of nature,”[18] we are in no position to judge the question.
Oppy
finds the arguments for (2.2) much more interesting. Oppy does not seem
to agree with Mackie’s claim that KCA proponents misunderstand
infinities by claiming an infinitely distant starting point. Proponents
have clarified that the KCA refers to sets of the type *ω, when
referring to the past regress of an infinite amount of time. William
Lane Craig explains,
Quote:
For
in this case the past would be like the second version of Zeno’s
Dichotomy paradox, in which Achilles to reach a certain point must have
traveled across an infinite series of intervals from the beginningless
and open end, with this exception: in the case of the past, unlike the
case of the stadium, the intervals are actual and equal. The fact that
there is no beginning at all, not even an infinitely distant one makes
the difficulty worse, not better… For the past to have been ‘traversed’,
would be equivalent to saying someone has just succeeded in enumerating
all the negative numbers ending at 0. But this seems to be
inconceivable; as G. J. Whitrow urges, a collection of order type *ω is
simply not constructible.[19]
|
But
Oppy says that this is merely one way to construe a beginningless set.
He gives a prima facie tip-o-the-hat to the idea that *ω sets cannot be
constructed. But goes on to say that there are other types of
beginningless sets that admit of some ability to be traversed. For
example, {1, 2, 3,…3, 2, 1}. Given other forms of consistent,
beginningless sets the second sub-argument is unsound.
However,
since this new set has a starting point we must examine the case
further. If we simply raise the objection based on this set, it would
only prove that, whether time was infinite or finite, it would have a
starting point; but this is what the KCA wants to prove. The KCA needs a
starting point because a beginningless set, one that has no first
member, cannot be traversed. Oppy claims this is tautologous. “But what
does this mean? Well, as far as I can see, it means that it is a
legitimate objection to infinities which have no first member that they
have no first member!”[20] He says that it is no argument against the
actualization of infinities that they have no first member, but merely
more prejudice against actual infinities. “Once we grant…that Cantorian
set theory reveals that worlds with actual infinities are logically
possible, there can be no good a priori argument against actual infinite
temporal sequences.”[21]
Support for the Kalām Argument
William
Lane Craig is one of the staunchest supporters of the KCA and rebuts
Oppy’s claims of prejudice and unwarranted assumptions in a series of
articles. Craig does not feel that Oppy has accomplished anything toward
supporting Mackie’s objections. That Cantorian set theory is logically
consistent does not imply that it is possible in some worlds. There is a
subtle distinction that Oppy fails to address:
Quote:
But
how does Cantorian set theory show that there are possible worlds in
which there are actual infinites? And even if there are, how does that
show that an actual infinite is ontologically possible? The issues
involved here are more subtle than Oppy seems to realize. He states,
"[Craig] concedes that infinite set theory is a logically consistent
system; consequently, it seems that he concedes that there are logically
possible worlds in which various 'infinites' obtain." But it is by no
means obvious that this second alleged concession follows from the
first. The validity of this inference depends on how broadly one
construes the logical modality involved in one's possible world
semantics.[22]
|
According
to Craig, Oppy fails to take into account the distinction between
‘strictly’ logical and ‘broadly’ logical. To say that a system’s logical
consistency in first-order logic indicates that it is true in some
possible world is to construe the system much more broadly than is
normally considered in possible world semantics. Alvin Plantinga, in The
Nature of Necessity, provides some background on modal construal.
Plantinga explains,
Quote:
…Kareem
Abdul-Jabbar’s being more than seven feet tall is a state of affairs,
as is Spiro Agnew’s being President of Yale University. Although each of
these is a state of affairs, the former but not the latter obtains, or
is actual. And although the latter is not actual, it is a possible state
of affairs; in this regard it differs from David’s having traveled
faster than the speed of light and Paul’s having squared the circle. The
former of these last two items is causally or naturally impossible; the
latter is impossible in that broadly logical sense.
A possible
world, then, is a possible state of affairs—one that is possible in the
broadly logical sense. But not every possible state of affairs is a
possible world.[23]
|
And, also important to the distinction,
Objects
or individuals exist in possible worlds, some like Socrates existing in
only some but not all possible worlds, and others, like the number
seven, existing in every world. To say that an object x exists in a
world W is to say that if W had been actual, x would have existed; more
exactly, x exists in W if it is impossible that W obtain and x fail to
exist.[24]
Notice that it is “objects or individuals”
that exist in possible worlds. Given these distinctions, actual
infinities and properties of transfinite math do not necessarily exist
in some possible world. Their independent coherence determines that.
‘Socrates’ and ‘Spiro Agnew’ are independently coherent objects
described in a situation as possible or impossible, but with a symbol or
a vague term it is unclear how to determine possibility without
reverting to mere definitions. To attribute properties to a symbol and
place it in a situation where contradictions between the properties and
the situation cannot be derived is to create the possibility of
instantiation. Despite the fact that infinities are capable of being
introduced in complex theories and attributed properties that limit
their functions (such as subtraction and division in cardinal
arithmetic), it is the individuals themselves, the infinities, that are
defined in such a way as to reject instantiation, not the consistent
mathematical systems. Transfinite arithmetic is not an ontological
argument for infinities. For Oppy to rely on transfinite arithmetic in
support of the possibility of an actual infinity is to miss the point of
what is required for modal instantiation.
Craig points out that
Plantinga has criticized Mackie on just this point. “…for the resources
of first order logic do not permit us to deduce a contradiction from
propositions like ‘2+1=7’ or ‘Some prime numbers weigh more than Jackie
Gleason,’ but we should not regard such propositions as therefore
possible.”[25]
‘Broadly logical modality’ typically indicates a
concept of possibility that is “narrower than that of strictly logical
possibility (which characterizes a proposition just in case it is not
the negation of a thesis of first-order logic, for example) but broader
than physical possibility (which characterizes a proposition just in
case it does not violate a law of nature)…”[26]
Therefore, in failing to note this distinction, Oppy has failed to
provide any evidence that the logical consistency of transfinite numbers
necessarily lends to their instantiation in some possible world. And
even if we define a “sphere of accessibility containing strictly logical
possible worlds,” as does Oppy, and concede the possibility of strictly
logically possible worlds, a logically consistent system such as
infinite set theory does not imply that an infinite is ontologically
possible.[27]
A Note On Instantiation
D. M. Armstrong sets out a criterion for instantiation:
Quote:
C1:
“The Possible is restricted by the actual in the sense that all
possibilities are composed from actually existing elements, but actually
existing combinations form a subset of possible combinations.”[28]
|
Armstrong
is defending his version of naturalism against primitive modality. He
is basically saying, “…the only universals which can exist are those
which are instantiated in the natural world…”.[29] If infinity is taken
to have the predicate un-completeable totality, we certainly have record
of objects that, given their form constitute ‘always incomplete,’ the
natural numbers being the most obvious. However, predicates like ‘the
cardinal number of N’ do not seem to exist by any empirical method.
Therefore if one is a naturalist, one can accept Armstrong’s thesis
concerning the possibility of instantiating a modal predicate for
infinity, and thus reject the possibility of an actually completed
infinity.
Craig does not get off so easily. He is an
anti-naturalist and holds a theory of universals much closer to the
classic version: namely, there are uninstantiated modal properties. This
means that the logical possibility of an infinity’s being complete,
would require the acceptance of the existence of an infinity in some
possible world. Craig wishes not to accept this and must argue as to why
it is unreasonable to do so. He gives us little or no reason to this
effect in the exchange with Oppy, though he relies on the reader’s prior
knowledge of an article by Plantinga concerning essential properties,
which we will examine later. For now, let us see why Craig might be
right.
Peter Suber provides some interesting insights into the
way infinite set theory is conceived among mathematicians. In, “A Crash
Course in Mathematics of Infinite Sets,” Suber highlights something to
which we cling intuitively, “Now you know how many natural number there
are: א0. But this is not profound. So far we’ve only invented a name (numeral) for the number of natural numbers.”[30]
…
The
“so far” seems an obvious indication that he will flesh out later why
we can do so and what this magnificent number could be. But he never
does this except in relation to larger infinities. The only
understanding he or anyone else has concerning a completed infinity is
the א0’s
relationship to exponentially higher infinities. The argument is that
there must be a higher infinity than cardinal because between any two
points there is necessarily a third. But, given that we know this, why
postulate the cardinal number in the first place. All the other numbers
existed all along, every point between 1 and 2 is an infinity equal to
the natural numbers. But the same goes for counting by tens (10, 20, 30,
40…), all these are equal to the infinite number of cardinals; just as
an infinite number of cardinal infinites is equal to the exponential
increase of one cardinal. But this is merely a mathematical way of
saying there are infinite numerals and the natural numbers are a smaller
subset, however, they are also infinite, but a smaller infinite. The
question remains how could there be a ‘smaller’ infinite if a basic
definition of infinite is that it is never complete?
Also, the
use of infinity in Calculus is on par with that of Aristotle because,
with the infinitely small, things ‘tend toward’ zero, which means that
zero is the stopping point and any non-extended point on your trajectory
from a starting point to zero is perfectly willing to act as an
indication of the direction you are heading: toward zero. There is
nothing that calculates the cardinal number of points between your
starting point and zero; it is merely potential. The same goes for the
infinitely large. Numbers, propositions, possibilities tend toward
infinity, but are inexhaustible.
Therefore, if one is a
naturalist, if you accept Armstrong’s theory of universals you are
forced to accept the infinity statements of the KCA. On the
other hand, if you take Craig’s position, you need to find something
between cogent talk of infinities in mathematics and logical
possibility. From the classic point of view, after defining the terms
appropriately, it may be possible to speak of infinities, yet
contradictory to treat them as completed.
An Aside Concerning Modal Possibility
Craig
briefly describes properties generally attributed to ‘broadly logical
possibility.’ Continuing from the previous quote, he says:
Quote:
Actualists
like Plantinga and Stalnaker construe the possibility of the abstract
objects which are possible worlds to consist in their instantiability
and hold that the framework of possible worlds is grounded in these
abstract objects’ possessing the modal property of being possibly
instantiated. Broadly logical possibility/necessity is therefore
frequently identified with metaphysical possibility/necessity. A state
of affairs which is strictly logically possible may, in fact, be
metaphysically impossible, incapable of being instantiated. [31]
|
This
is the crux of Craig’s argument against the actualization of
infinities, but it involves some intricacy. Craig says that broadly
logical modality is typically left undefined, except as construed above,
and then examples are given to show their uniqueness. However, no
examples are given for the impossibility of ontological instantiability
of logically consistent states of affairs. This leads us to question our
typical notion of taking ‘logically consistent’ to mean ‘logically
possible.’
Craig seems to be relying on Plantinga’s argument for
the ability to identify modality de re via modality de dicto, though he
does not indicate this directly. Roughly the argument states that
existence need not be considered a property for determining essential
properties of an object, and that if it can be shown that “de dicto
modal properties determine whether x has P essentially.”[32] Given the
complexity of this argument, we may substitute a simpler argument along
the same lines.
We will take it that infinity (i) has the
property of (P) incompleteness (u). (This is incompleteness in the
quantitative sense, rather than the Anaxagorian ‘unboundedness,’ or
qualitatively incomplete.) If we do not take it this way, it would be
finite, the word ‘infinite’ would not exist, and discussion would be
over. Therefore we must investigate to what degree infinity has the
property of incompleteness, that is, we must determine the sense of our
predicate. Our range includes the traditional categories of necessity,
possibility, essentially, and contingently. Some, like Armstrong, say
that no object has a property necessarily but that all objects
necessarily have some property or another. This means that it could not
have the property essentially. Therefore, if it is the case that [Piu],
then it is contingently. This means that it could have been otherwise
and that in some possible world infinity does not have the property
incompleteness, and it may be the actual world. But if infinity does not
have the property of incompleteness in this world, we have merely
chosen the wrong word to define it. We are actually working with a
finite sum. We are merely noting that in some possible world there
exists a number of objects that has no coherent ending point, but we are
not saying that it is the case in this world. But this is senseless;
even the mathematicians who chose to use the word infinite claim that it
is not characterized by any known numeral.
If infinity possibly
has the property of incompleteness we are stuck with the same quandary.
If it is possible that [Piu] then it is not the case in every possible
world, but need only be so in one possible world. If it were the case
that [Piu] in every possible world, then it would be so necessarily and
therefore ‘i’ would have ‘u’ essentially. If then, [Piu] in this, the
actual world, then in all other possible worlds it may be the case that
[~Piu] and the word would then be meaningless in those possible worlds.
This is the most likely alternative to essentialism for infinity. But if
it is the case then in the actual world, [Piu] necessarily in this
world and any additional predicate that contradicts this predicate makes
the term incoherent.
Given that infinity is properly a
predicate of mathematical sets and we have dispensed with the use of
‘infinite’ as a qualitative whole, this argument involves the invocation
of second order properties. Despite the skepticism concerning their
efficacy, they serve to indicate the sense in which the property
‘infinite’ is used in various contexts. Here especially, if it is the
case that it has some property or other essentially, then a
contradictory attribution is unacceptable.
So where does this leave us with Craig’s argument? He says:
Quote:
Oppy,
like Mackie, seems to take a proposition’s freedom from inconsistency
in first-order logic to be indicative of that proposition’s being true
in some possible world. But this involves a notion of possibility which
is much broader than that normally countenanced in possible world
semantics.[33]
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The
propositions of infinite set theory that claim equal properties for
infinities of different sizes serve as an example here. One infinite can
be larger, even substantially larger than another infinite, yet they
are considered to be of equal value. For example, in Hilbert’s Hotel
with its infinite number of rooms, if there were no vacancies but an
infinite number of guests arrived needing rooms, the accommodation would
be effortless. Though this poses no problems in infinite set theory,
given that the sets to which it adds possess amorphous maximums, but it
creates some headaches for mathematics of distinct objects. Terms apply
in different ways in different types of mathematics. “This is because
our normal criteria for smaller than and equal to fail to be mutually
exclusive for infinite groups.”[34] However, in our world they are
mutually exclusive, even contradictory. Does this transfer to broadly
logical standards? In an interesting sense, smaller than = equal to on
Cantor’s scheme. What is needed is to show that there are necessarily no
possible worlds in which smaller than and equal to are not mutually
exclusive.
At minimum, it seems obvious that a world where
smaller than and equal to are not mutually exclusive could not be a
world that contained even one physical object. Smaller than and equal to
are conventions of language that describe physical things in addition
to conceptual ones, so they share a similarity with married and
bachelor. To say that one physical object is smaller than and equal to
another physical object simultaneously, and in the same sense without
qualification, is a misuse of language and a contradiction.
It
might be objected that we merely do not know of any objects on which
this contradiction would not rest. But is it because we do not know of
any ‘married bachelors’ that we call it a contradiction? If we devised a
theory in which a creature was so extravagantly virile or his marital
status was so surreally conceived as to consider him a ‘married
bachelor,’ would that lend any weight to its instantiability in a
possible world? Perhaps, though it does not seem possible with other
contradictions like round-squares. In the case of the bachelor, we would
have then altered the definitions such that they reject exclusivity.
But that seems to be what is done with infinite set theory. The infinite
is discussed in finite terms: an unlimited set, א0 is now limited to a symbolized finite sum smaller than a larger sum, א1.
But in doing so, the necessarily inexhaustible nature of infinity has
led to theories of talking about it that have nothing to do with
infinity. We have already seen that to add a predicate that contradicts
‘incompleteness’ makes the term incoherent. We must remove the predicate
altogether to add another, but then we are left with a finite sum,
rather than an infinite one.
To flesh out the coherence of the
objects of infinite set theory we have to reduce them to definitional
situations and determine their consistency. The natural numbers are
considered to be infinite because for any finite natural number, another
follows it. The process itself is infinite. The natural numbers are, by
definition, incomplete on any rendering. They cannot be rendered in
their entirety, regardless of human capacity. We could then define an
infinity as follows: a set of objects, conceptual or physical, the
number of which cannot be conceived (1) by continuous process of
accumulation, even granting the possibility of super-tasks and
indefinite time, or (2) as complete in its entirety as defined by a
single cardinality, belonging to that set, yet succeeding all other
members to the point that none follow it. Neither א
nor ω indicate a completed set, for doing so violates the very
definition of ‘infinite.’ Therefore, “An actual infinite [in the sense
of complete whole] cannot exist.” Therefore the infinities in set theory
are merely symbols representing logical potentialities (in Aristotelian
terms), not logical completions.
…
Further,
Craig finds the paradoxes of the infinite themselves to be a decisive
reminder that infinities cannot be instantiated. Any infinity, if
instantiated, would create a conundrum of contradictory situations. The
opponent of the KCA must show there to be a reason or possible situation
in which these problems would not ensue. Craig responds to Mackie,
“Rather than alleviating the difficulties entailed therein, Mackie has
merely specified an aspect of that system which supplies the conditions
which, if instantiated in the real world, would spawn the absurdities
like Hilbert’s Hotel or Russell’s Tristram Shandy paradox.”[35]
Except
for some quibbling, Craig’s attack on Oppy’s view of successive
addition (2.2) mainly concerns Oppy’s introduction of different types of
infinite sets, especially {1, 2, 3,…3, 2, 1}. Craig calls this
“bizarre.” Is this really a completable set? “If I started counting now,
when would I arrive at that second 3? Let us have no fictional
suggestions about counting progressively faster so that the infinite
super-task is completed in a finite time, for such scenarios are wholly
unrealistic.”[36] Such a set is necessarily incompletable in the case of
adding members. But Mackie does not dispute that a series type *ω is
unformable. He
argues that the KCA proponents mistakenly assume that the present
exists as a point succeeding the completion of a ω-type series, that is,
an infinite series that had a definite beginning point, though
infinitely far from the present. But Craig denies that the KCA makes any
such assumption and argues that no ω-type series can be formed by
successive addition.
Comments
It
is obvious that Oppy’s central task is to undermine the KCA’s strict
finitist metaphysics. He claims that, if we get rid of this unwarranted
notion, then the KCA falls apart. It is imperative that we examine such
claims, because if they undermine the KCA, they also undermine several
hundred years of physics and metaphysics.
First in this final
section, I will address Oppy’s objection to the claims concerning actual
infinities and then close with comments concerning the relevance of
these claims to the KCA. Oppy focuses his sights on Craig’s response to
transfinite math and its broadly logical possibility of instantiation of
infinities. Oppy merely states that he is not willing to concede the
point. His claims that the arguments are question-begging seem to hit
all around the point. He wants to say that the finitist metaphysic is a
presupposition, grounded only in prejudice. With regard to inverse
operations with transfinite cardinals, Oppy says, “so why should one who
thinks that Cantorian infinities might be physically instantiated lose
any sleep over these operations?”[37] The most obvious reasons are the
fact that the Cantorian system works on a different conceptual plane of
operation, that is, one in which subtraction and division “cannot be
defined,” and smaller than and equal to mean the same thing, and that
incompleteness seems an essential property of infinity at least in this
world.
Oppy’s defense of the ω+ω* set is baffling. The only
answer he can offer conjures the notion of topological space and then
invokes “supertasks”[38] to complete the traversal of an infinite in a
finite amount of time. Oppy rejected this series the first time
because it had a beginning, and that was what the KCA was trying to
prove. How does re-invoking it help his case? Where are the
‘other’ infinite types that do not fit the description of an
untraversable infinite, that he said existed in the first argument?
Again, Craig is charged with question-begging. “…[I]t seems to me that
all we have is the expression of a question-begging intuition.
Certainly, I have not been able to find anything in Craig’s writings
which give one who believes that an ω*-series can be formed by
successive addition a reason to change her mind.”[39] But really, what
more does he want? Infinities are, by definition, untraversable. Both
Aristotle and Cantor admit as much.
Oppy continually criticizes
the KCA proponents for holding a ‘strict finitist metaphysic.’ He
assumes that there is some reason to doubt that the universe, time,
sets, events,… whatever, can exist in complete, though infinite
packages. But is this the case? Certainly the advancements in
transfinite mathematics make room for these questions and demand a
certain amount of precision in our answers to them. To begin, we have
seen that a consistent system can be said to possess ontological import
in some possible world as long as its individual components possess real
possibility (though ‘real possibility’ tends to remain a fuzzy
concept). The individuals in our case are sets that contain an infinite
number of members. Though we have never discovered an actual infinite,
Oppy would say this is merely a posteriori and does not address the
modal possibility of these individuals. Granted. Also, to say that we do
not have a conception of what an infinite number of things is like is
merely untrue as far as Cantorian formulas are concerned; there are a
number of positive properties of infinities that abide by certain fixed
laws of transfinite math.
So what we need is an understanding of
infinity that resolves the problems of both temporal and ontological
construction implicitly in its definition. Whether such beasts can exist
at all is our question, but, as A. W. Moore explains, intuitionists
take it to be merely a matter of chronographic ineptitude that we cannot
construct an infinity. They invoke ‘super-tasks’ (sound familiar?) to
explain away the impossibility.
Quote:
Let
us call any story in which infinitely many tasks are performed in a
finite time a ‘super-task story’. Then one thing is surely beyond
dispute: that logically consistent super-task stories are there for the
telling. In one such story someone constructs all the natural numbers in
a minute. They spend half a minute constructing 0, a quarter of a
minute constructing 1, and so on ad infinitum. [40]
|
So,
the question is, does this make any sense? Moore says no. “It literally
makes no sense to describe anything as infinite in this or that
respect. We can only use ‘infinity’ to describe the endlessly nested
possibilities that (finite) things afford.”[41] This still sounds like
prejudice. However, Moore does not think that much else is available to
refute the intuitionist.
Quote:
But
the possibility that super-task stories are not coherent will
not—yet—help the intuitionist. For there can be no non-question-begging
way of explaining the incoherence if all we have to appeal to is pure
temporal structure and the fact that we are immersed in time. We need
some independent leverage.[42]
|
Moore
suggests that we conceive of infinites as Wittgenstein: “as abuses of
grammar, misappropriations of the language.”[43] Now we are back to
needing an understanding that resolves the issue of time.
Without
answering again the argument from transfinite math, the best suggestion
that I can offer is that, implicit in the definition of ‘the infinite’,
is the notion of inexhaustibility. If an actual infinity exists,
instantiated in some objects or other, the whole of them would remain
untraverseable. As in one of Moore’s arguments, inexhaustibility is, “a
matter of there coming a point—some point or other—beyond which we
cannot exert ourselves…[and that would] remain beyond us however far we
extended our powers and abilities,”[44] including ‘super-task’ speed. If
this understanding is correct, then the possibility of an actual
infinite being formed by successive addition (or subtraction), whether
in ω or ω* or ω+ω* set-types, or even existing at all as completed or
whole, is incoherent. And this leaves untouched transfinite math’s claim
to consistency, as Moore points out, “…[L]ogical consistency does not
guarantee coherence—a point that intuitionists are especially keen to
emphasize.”[45] Yet at the same time it rejects ontological
instantiability in the sense of completing an infinite set, by
construction or numeration. Closing
There
are many objections to the KCA,[46] some concern the coherence of
cosmological singularity,[47] some concern the warrant to believe the
valid arguments of the KCA,[48] some concern the coherence of God’s
existence sans the big bang, that is without temporal succession of,
say, intrinsic divine thoughts leading to the act of creation,[49] and
some which primarily question the intuitive nature of the concept ‘ex
nihilo, nihil fit.’[50] But whatever the outcomes of those debates,
though I think they are very positive for the KCA, the issues of
infinity involved in the argument seem to present no problems for the
mechanics of cosmology. No one is having problems making their theories
work because an actual infinity must be constructed, and we are just
waiting on the tools to do so. In
fact, many physicists, mathematicians, and philosophers agree to the
claims made by the KCA, given their continuous attempts to debunk their
consequences. It is their application that makes these theoreticians
uneasy. Given their validity, they must be addressed in one form or
another, say, in the construction of a consistent theory of a
beginningless universe. But no one is running around trying to find
things that come into being out of nothing (except perhaps Quentin
Smith), and the evidence continually points to an initial cosmological
singularity.
__________________________________________________ ____
References:
[1]
A. W. Moore, The Infinite (London: Routledge/Taylor and Francis, 2001),
“[Cantor] was adament throughout his life that the whole idea of an
infinitesimal was demonstrably inconsistent,” p. 117; concerning the
paradoxes of the infinitely big, Cantor claimed some totalities “…were
too big to be regarded as genuine sets at all… There was no such set as
Ω,” p. 127; “There could not be any sets that were genuinely infinite,
p. 128, italics his; “…the truly infinite is that which resists
mathematical investigation,” p. 198.
[2] Ibid., pp. 36-44.
[3]
Its logic seems the least problematic of the theistic proofs. The
ontological argument suffers the pain of explaining “existence” as a
coherent predicate and the teleological argument has trouble avoiding
the naturalistic fallacy.
[4] In that it is a logical argument supporting the intuition that an infinity cannot be instantiated.
[5]
W. L. Craig schematizes the argument this way in “Professor Mackie and
the Kalām Cosmological Argument,” Religious Studies, No. 20 (1985), p.
367.
[6] As Mackie and Oppy will argue here.
[7] Quentin
Smith has argued this in “The Uncaused Beginning of the Universe,”
Philosophy of Science, Vol. 55, No. 1, (1988), pp. 39-57.
[8]
Paul Davies, God And The New Physics (New York: Simon And Schuster,
1983); Adolf Grünbaum, "The Pseudo-Problem Of Creation In Physical
Cosmology" in Leslie, J. (ed.) Physical Cosmology And Philosophy (New
York: MacMillan, 1990), pp.92-112.Grünbaum, "Creation As A
Pseudo-Explanation In Current Physical Cosmology" Erkenntnis 35, (1991),
pp.233-254. Steven Hawking, A Brief History Of Time (New York: Bantam
Books,1988).
[9] By ‘physical evidence’ I am referring to those
several independent confirmations of an inflationary universe, including
red shift and background microwave radiation. See Alan H. Guth, The
Inflationary Universe (Reading, MA, Perseus Books, 1997).
[10] J. L. Mackie, The Miracle of Theism: Arguments For and Against the Existence of God (Oxford: Clarendon Press, 1982).
[11] Craig, “Professor Mackie and the KCA,” p. 368.
[12] Ibid, italics his.
[13] Ibid, pp. 368-69, quoting Mackie, The Miracle of Theism, p. 94. “[sic]” belongs to Craig.
[14]
Graham Oppy, “Craig, Mackie, and the Kalām Cosmological Argument,”
Religious Studies, Vol. 27, No. 2 (June 1991) pp. 189-197.
[15] Ibid., p. 193, italics mine.
[16]
“[Craig] tells us that the proponent of the kalam argument is committed
to the claim that there are no infinities in the actual world;
however—at this point in his paper—he provides no further evidence for
the truth of the claim that there are no infinities in the actual world
beyond the thought that it would be absurd to suppose otherwise. Since
Mackie does not share this intuition…; at best we have a stalemate,”
Ibid., pp. 193-94.
[17] Ibid., p. 194.
[18] “Perhaps, on
the basis of our current knowledge of the actual laws of nature, we can
judge that it is fairly likely that there are no actual infinities;
however, it is hard to see that we have much reason to be very confident
about this,” Ibid.
[19] Craig, “Prof. Mackie and the KCA”, pp.
369-70. “Construction” here is taken in the normal sense, affirming that
the concept of infinity implies the inability to achieve it by finite
process.
[20] Ibid.
[21] Ibid., p. 195.
[22] Craig, “Graham Oppy on the Kalām Cosmological Argument,” Sophia 32 (1993), p. 2, citations omitted.
[23] Alvin Plantinga, The Nature of Necessity (Oxford: Clarendon Press, 1974), p. 44, italics his.
[24] Ibid., p. 46, italics his.
[25]
Craig, “Graham Oppy on the KCA,” p. 2. Craig directs readers to
Plantinga’s “Is Theism Really a Miracle?” Faith and Philosophy 3 (1986):
117.
[26] Ibid.
[27] Ibid.
[28] Steven K. McLeod, Modality and Anti-Metaphysics (Burlington, VT: Ashgate, 2001), p. 79.
[29] Ibid, p. 84.
[30] Peter Suber, “A Crash Course in Mathematics of Infinite Sets,” St. John’s Review, XLIV, 2 (1998), pp. 35-59.
[31] Ibid., italics mine.
[32] Alvin Plantinga, “De Re et De Dicto,” Nous, S. 69, V. 3, pp. 235-258).
[33] Craig, “Graham Oppy on the KCA,” p. 2.
[34] Craig, “Prof Mackie and the KCA,” p. 368, italics his.
[35] Craig, “Prof Mackie and the KCA,” p. 371.
[36] Craig, “Graham Oppy on the KCA,” pp. 3.
[37] Ibid., p. 17.
[38]
‘Super tasks’ are problematic in themselves, in that they presuppose
performing an infinite amount of tasks in a finite amount of time. But
given that the possibility of an actual infinity is at stake it seems to
beg the question to introduce them.
[39] Oppy, “Reply to Prof. Craig,” p. 18.
[40] Moore, The Infinite, pp. 213-14.
[41] Ibid. p. 215.
[42] Ibid,. p. 214.
[43] Ibid.
[44] Ibid., p. 213.
[45] Ibid., p. 214.
[46]
Three prominent objections are brought out in Graham Oppy’s article,
“Professor William Craig’s Criticisms of Kalam Cosmological Arguments by
Paul Davies, Steven Hawking, and Adolf Gruenbaum,” Faith and
Philosophy, Vol. 12 (1995), pp. 237-50.
[47] Quentin Smith
questions the nature of the initial cosmological singularity
traditionally postulated by physics and claims that, given the
unanswered problems, big bang cosmology actually contradicts theism,
“Atheism, Theism, and Big Bang Cosmology,” Australasian Journal of
Philosophy, Vol. 69, No. 1 (March 1991) pp. 48-66.
[48] As we
have seen Oppy takes a shot at this. John Taylor does also in “Kalam: A
Swift Argument from Origins to a First Cause?” Religious Studies 33
(1997) pp. 167-179.
[49] see Adolf Grünbaum, "The Pseudo-Problem
Of Creation In Physical Cosmology" in Leslie, J. (ed.) Physical
Cosmology And Philosophy (New York: MacMillan, 1990) pp.92-112, and
"Creation As A Pseudo-Explanation In Current Physical Cosmology"
Erkenntnis 35,(1991) pp.233-254.
[50] In addition to Graham Oppy,
Quentin Smith has criticized the rejection of something coming into
existence out of nothing. See "The Uncaused Beginning of the Universe,"
in Theism, Atheism, and Big Bang Cosmology, by William Lane Craig and
Quentin Smith (Oxford: Clarendon Press, 1993). Though in 1999 he changed
his contention from the universe coming into existence out of nothing
to the universe coming into existence as its own self-causal agent, “The
Reason the Universe Exists is that it Caused Itself to Exist,”
Philosophy 74 (1999), pp. 579-86.
