Mawson’s
beliefs, Cantor’s paradox
Martin C. Cooke
April, 2009
Abstract
Mawson has argued that the
God of open theism would be liable to
make mistakes because, unlike an atemporal God, He would not infallibly know
everything about the future. God being maximally good and knowledgeable, Mawson
concludes that He transcends time as well as space. But such a lack of
infallible knowledge would be, if not simply false then metaphysically
insignificant under presentism (and presentism is not unlikely). Mawson’s analogical
argument for the consequent liability does not indicate any such significance (or
unlikelihood) because it contained independent lacunae. Furthermore, when
comparing different conceptions of God to see which is the greatest we should consider,
not just the actual future but all possible creations. And the logical evidence
alone then indicates that the most powerful God is eternally learning—and hence
able to change, whence scriptural evidence would more strongly indicate open
theism (and empirical evidence presentism)—because Cantor’s paradox indicates
that transfinite arithmetic is indefinitely
extensible. Such a God might also be classed as omniscient, although that
is little but semantics.
1. Necessary omnipotence
God would be ‘subject to
the vagaries of luck [...] for His goodness’ under open theism, according to Mawson (2008, 49), recently reiterating his
(2005, 35–48) well-known argument against open theism. That argument really is
worth repeating, because it fades into an argument for open theism (see section four) under the light of logic (see
section six).To begin with, open theists do, as Mawson (2008, 36) says, think
that God has kept His—or Her (or Their)—options open, to some extent. I shall,
for simplicity, be using ‘God’ to name the perfect person (or Trinity) who
created this universe ex nihilo. God
understands this universe completely, has absolute power over it, and is
necessarily good. Since God’s self-knowledge is presumably also perfect, many
would conclude that He is omniscient, omnipotent, and perfectly good. But
although Mawson (2008, 41) takes that conclusion to be our starting-point—and
Rhoda (2008, 226) would agree—there is an immense conceptual difference between
the world as this universe and the
world as all that exists apart from
God. And the question of God’s omniscience is complicated by tangential
semantic issues (see section seven).
Nonetheless, God is
evidently unimaginably powerful (if
he exists), and for various reasons open theists generally agree with Mawson
that God is (metaphysically) necessarily omnipotent.
Anselmian theists like us believe that God is the greatest possible being, although it is perhaps not perfectly clear what
that means, as Nagasawa (2008) recently noted. E.g. what if one God is more
omnipotent but less omniscient than another? My conclusion will be that
although Mawson’s God is completely
omniscient (timelessly), and mine is arguably incompletely omniscient (at all times), mine is more powerful (although both are omnipotent),
and hence greater. Incidentally, another reason why I accept that conclusion is
that open theists have access to better theodicies (whence our God has the
greatest conceivable goodness), but I shall not be defending that much more
complicated claim here. What is relatively clear—and hence my starting-point—is that the greatest possible being is
metaphysically necessary, with all other metaphysical possibilities finding
their ground in His omnipotence.
God is impeccable, for
example, and infallible and—in some
sense—eternal: Let us call Him ‘everlasting’ if, as open theists like myself believe,
He has the power to change, and ‘timeless’ if, as atemporalists like Mawson believe,
He is not liable to change. Note that both ‘everlasting’ and ‘timeless’ connote
an unfading immortality. Mawson prefers the terminology ‘temporal’ and
‘atemporal’, but the former connotes limitation, e.g. it can also mean secular
or civil, as opposed to sacred or spiritual; and it even sounds like
‘temporary’ and ‘corporal’. Connotations are not unimportant—suppose I used ‘vital’
and ‘frigid’ instead—and I would rather use ‘temporal’ and ‘everlasting’ to
distinguish between human and divine experiences of changeability (see section
two). But it is of course more important that, since it is conceivable that
very little is metaphysically possible, so we Anselmian theists ought perhaps
to think of God as having the greatest conceivable omnipotence (see section
four).
And note that open theists
also agree with Mawson (and some
other atemporalists) that our wills are free, in a libertarian sense: ‘On the
libertarian view of free will (as it is standardly construed), all that has to
be true for you to be free in the future in your choice to do X is that you
have the power at that time to do something other than X’, as Mawson (2008, 40)
put it. Now, van Inwagen (2002, 171–3) worries that the freedom of our wills is
to some extent a matter of luck. But his argument seems to be invalid under presentist open theism (see section
two); and since O’Connor (2007) recently answered Mele’s (2006) rather similar
worries, so I shall take such claims to be false in what follows, although they
would of course yield Mawson’s (2008, 49) aforementioned conclusion relatively
straightforwardly.
Where open theists most
significantly disagree—with each other (see Rhoda 2008) and hence, some of
them, with Mawson—is over what Mawson (2008, 37, 40) calls ‘the reality of the
future’, and in the next section I describe how the conceptual possibility of presentism undermines Mawson’s first argument.
Mawson argued that were open theism true, God would not infallibly know everything
about the future, whence He would be liable to make mistakes. In section three I
criticise Mawson’s analogical argument for the latter conclusion, which is false
(e.g. via God’s omnipotence), and then I refine his main Anselmian argument, by
considering all possible creations (section four), not just the actual future,
and by considering arithmetic (section five) in light of Cantor’s paradox
(section six).
I conclude that the most
powerful God is everlasting, although I end by returning to the relatively
tangential question of His omniscience (in section seven). According to Mawson
(2005, 35, my emphasis) ‘a being is omniscient
just if it is the case that for all
statements, if a statement is true, then that being knows that it is true’. But there are various ways in which that
‘knows’ might be analysed, and different ways may well suit different
conceptions of God. Swinburne (2001, 192–220) considers some apposite analyses;
and Prichard’s (2007) thoughts on the strength of the connection between a true
belief not being accidentally held and it counting as knowledge are especially
relevant. But we will only be comparing totalities of truths, to begin with, so
let’s say that to be omniscient is, for all statements, to believe a statement
(only) insofar as it is true.
2. Future contingents
Mawson’s (2008, 37) argument
that our quotidian beliefs imply the reality of the future was essentially as
follows. You will, when you finish reading this sentence, either continue with
the next one, or else you won’t, and whichever it is that you actually do, if
you happen to believe now that you’ll do it, then you have—it seems—a true belief
about the future. And you may now know that your belief was correct (or not). Had the future not been real, even then (as
the future), such a belief could hardly have been true. The truth of our beliefs
is, after all, their correspondence with reality.
But you might know now
that your belief was correct without
it having initially corresponded with reality. In general, had you supposed
that S will P (e.g. that Sue will pray) and then said, about that supposition,
‘I was right’, you would normally be taken to have said only that S did P (even if you had already answered ‘Will S
P?’ as though you had been asked about S’s intentions and opportunities). And in
fact, it is precisely because truths correspond with reality that, insofar as
reality was changing (as it seems to), so the truth-value of your belief might
have changed, e.g. from indefinite to
true (or false), as you continued to read (or not). The possibility of calling
that initial truth-value ‘indefinite’ corresponds to the second of Rhoda’s (2008,
231) three varieties of open theism. With its three truth-values, this variety
deviates from bivalent logic. But whether or not logic itself demands such
deviations—e.g. via the Liar paradox, or Russell’s paradox—such deviations need not be illogical. For a brief
overview of such deviations see Priest (2001, 117–138), and for an up-to-date
introduction to this particular question see Brogaard’s (2008) reply to
MacFarlane (2003).
The first of Rhoda’s varieties
corresponds to calling the initial truth-value ‘true’ (or ‘false’), and the
third to calling it ‘false’. See Rhoda et
al (2006) regarding that third variety, and note that insofar as ‘S will P’
means that S will definitely P, it is indeed false if S might not P (whether or
not S does P). I consider the first variety below because there are, I think,
senses in which each of those assignments is correct. But the second variety
captures the most apposite sense under presentism. Presentism is essentially,
as Rhoda (2008, 234) put it, ‘the view that only what exists now has any reality’. Of course, time is
real in the sense that things do
change, and that it is not true that there is no tomorrow, etc. But there is also
a sense in which tomorrow does not
exist, not yet, and if presentism is true then that sense is important, is
metaphysically significant.
If presentism is true
then, before you continued reading, there was—or rather, there may have been (as follows)—nothing for
your belief to correspond to. Now, maybe
the content (not just the cause) of your belief included such things as, for
example, your tendency at that time to continue reading, or God’s intentions,
etc. But even so, insofar as your belief was not suppositional its truth could hardly have been able to indicate
the reality of the future, rather than that tendency’s strength, or His intentions,
etc. So, to undermine Mawson’s first argument I need only that presentism is
not obviously implausible under theism.
To begin with, we meet the
concept of time when we relate to the regular changes of other continuants, such as clocks
or the earth, moving according to the laws of physics. A continuant (e.g. a
tree, a chair, a dog) is an object that is completely
present whenever it exists (e.g. I may lift up the whole dog) and which
continues to be the same thing even
as its properties change (e.g. I may put the same dog down again). Most of our
quotidian beliefs concern continuants; and one can, I think, become directly
aware that one is a real continuant. We
communicants are the paradigm continuants; and it is because our ordinary logic
is naturally suited to such subjects as ordinary continuants—and to predicates
that are being effectively univocal
(whence our natural clarification procedures)—that there is the danger, as we
logically reflect upon time, that we inappropriately objectify the temporal dimension
and find ourselves philosophically confused by this thing called ‘time’. It is
partly because we so naturally objectify whatever we are thinking about—inevitably
as something like a continuant—when thinking
reflectively, that non-classical logics have been of such use to analytical
philosophers (e.g. see Priest 2001). And it is because we are continuants—because
our essential natures inevitably persist unchanged so long as we exist—that we
have an essential consistency. So
although Mawson (2008, 38) concluded that ‘given that the world’s having a
future at all is dependent on God’s freely choosing to sustain it from moment
to moment [...] then He does not have infallible knowledge of the future of the
world in any respect’, note that God has the power to do many things that He can know for sure that He would never choose to do (cf. Ockham’s
distinction between absolute and ordained powers). In other words, God has a
certain necessary consistency, which
means that His infallible knowledge of the future of the world need be limited
only by the freedoms He has granted to His creatures.
As we try to understand
change—to see what’s going on—we
naturally visualise it by analogy with the spatial dimensions that we more
obviously live within. The temporal dimension is a very natural, quasi-spatial
representation of change. As Stewart (2007, 226) says: ‘Time is a nonspatial
variable, so it provides a possible
fourth dimension, but the same goes for temperature, wind speed, or the
lifespan of termites in Tanzania.’ And it is precisely because such
representations are so natural that metaphysicians should take special care with the spatial connotations
of our talk about time. (And not just time; for a nice example of spatial
intuitions leading us astray in logical space, see section one of van Inwagen
2002.) The following, for example, is supposedly a dilemma for presentists: The
present is either a mere instant, or else it has some duration. But under
presentism, instants are merely parts of various pictures of how continuants
might change (e.g. in jumps); and under theism, if there was only the changing
present it would not be thin—as though the future rained down upon the surface
of the past, and only that surface was real—but rather rich, God Himself being
fully present there, or rather—since it is not so much that He is located only
there—changeability being ultimately an aspect of the divine essence (e.g. via
His omnipotence).
Mawson thinks that an
open-theistic God would be inside
time. But that would, under presentism, be like being inside a story about
oneself. The strength of your sense
of being inside time is arguably due to your being a dependent creature,
subject to natural laws and in particular facing physical death (in a world
that determines much of the sense of your words). Rather than thinking of a
presentist open-theistic God as inside (as opposed to outside) time we should,
since both kinds of God are spatiotemporally omnipresent (see below), more
accurately think of Him as having (as opposed to lacking) the ability to
change. Less analytically—more metaphorically—we could also think of Him as
being effectively atemporal up to his
first actual change (were there one). And we might even think of Him as presently
temporal, insofar as He is intimately
involved with us (to the point, perhaps, of kenotic crucifixion).
Regarding spatiotemporal
omnipresence, I presume that both kinds of God are clearly spatially omnipresent (or immanent); and although the only real time under presentism is the
present, an everlasting God was, is or will be at all such times, while Mawson (2005, 48–51) argues well for his
God’s spatiotemporal omnipresence. But he (2005, 51) also adds that ‘of course
scientists are happy to talk of space and time as merely two aspects of a unity,
space-time’, so note that theoretical physicists are as happy to interpret
quantum mechanics via ‘many worlds’ (see below), mathematicians to define the number one to be the ZFC set
{{}} (see section five), neuroscientists to talk of conscious computers, and so forth. Such views, if not technical
rather than metaphysical positions, are highly implausible under theism. Under
the ‘many worlds’ interpretation, for example, you would be about to do
everything that was physically possible, splitting into infinitely many selves
as you did so, and the metaphysical
point of the ‘many worlds’ interpretation is that there is nothing metaphysically significant about any particular self. So,
it may therefore be worth noting that the debate about the compatibility of
presentism with relativistic physics
is merely on-going, as Bourne (2007, 141–203) shows.
Still, whether or not
presentism is true there is also a sense
in which your earlier belief was true (or false): Your belief was of what you
would actually be doing (or not). But we are here considering whether such
quotidian beliefs indicate the reality
of the future (in Mawson’s sense); and if presentism is true then the past,
present and future facts comprise a relatively artificial totality (see below), for all that it is typically called
‘the actual world’ (of the possible worlds). Should God know such truths about
what will have been the future, whose truth-makers do not yet exist? That is,
are such truths true in any metaphysically significant
sense? Mawson may well be right to think that, whether or not presentism is
true, if God’s ignorance of such truths would make Him liable to make mistakes,
then we should regard the answer as ‘yes’ (whence theists might reject
presentism). I shall therefore consider that argument in the next section. Regarding
the artificiality of the ‘actual’ world, note that the traditional
interpretation of quantum mechanics postulates collapses of wave-functions
representing current physical tendencies towards the possible outcomes of those
collapses, and all those physical possibilities
currently interact as such, which indicates the current reality of the non-‘actual’ (in this technical sense)—or
the unreality of the future (in
Mawson’s sense)—because most of those possibilities won’t have been how things
were actually to be.
Unlike most physicists,
theists need not reject that traditional interpretation just because of the
problem of who could collapse the
universal wave-function. And despite what many non-physicists imagine, such
probabilistic entanglements are not restricted to the realm of the very small.
Recent empirical confirmations of the Bell inequality have seen them extending
over 50 km, for example. Furthermore, cosmological evidence for dark energy and
matter, and also the overall uniformity and rippling, amounts to evidence for antimatter standing waves throughout the
(finite) universe; cf. atomic electron shells around a positively charged
nucleus. Such waves would be gravitationally repulsed into collapsed (resonant)
wavelengths across the entire universe simultaneously. But therefore their
explanatory power (e.g. the symmetry of matter and antimatter) has already been
rejected by those (e.g. most physicists) who have already rejected presentism. Metaphysicians
should be wary of the hidden presumptions in scientific theorising about the
empirical data, which is actually far from indicating the falsity of presentism
(which is implied by most of our quotidian beliefs).
Furthermore—and again, whether
or not presentism is true—it is not clear that the first variety of
open-theistic God (who freely chose to limit His foreknowledge) would know less than a timeless God. He would, for
example, know what it was like to be so limited, as well as unlimited (prior to
His choice), and to so choose. Similarly perhaps
one could only know what ‘pain’ meant if one had experienced pain (since to
successfully imagine pain is surely to experience pain). But even if He did
know less He would have been able to
know all about our futures, as well as able
to know less. So if Mawson is wrong about the liability to make mistakes, then
such a God might be more powerful than a timeless God; and, as Nagasawa (2008,
590) observed, the God with less omniscience but more omnipotence is perhaps the greater God.
3. Making mistakes
Mawson’s (2008, 47) argument
that an open-theistic God would be liable to make mistakes was essentially the
following scenario: An open-theistic God answers the prayers of Mrs Hitler by
miraculously saving her unborn baby from a probable miscarriage, because He
wants to increase the aggregate happiness of the world. Being open-theistic, He
was not to know that Mrs Hitler’s baby, Adolf, would cause that aggregate to
decrease. But that’s what happened, and so God’s desire for an increase was not
satisfied. And even if greater happiness had ensued, He would just have got
lucky. Such a God might happen to be perfectly beneficent, but a timeless God,
bodging up in no possible world, is necessarily perfectly beneficent.
Since God is maximally good,
so He is timeless, according to Mawson. Of course if, were God timeless, Adolf
Hitler would not have been so evil, then history would be some evidence that God is everlasting. But Mawson’s point was presumably
that something like that scenario
must occur under open theism. So although twin studies (and modern history)
suggest that God would probably have been able to know a lot about how Adolf
was likely to turn out simply by observing his foetus (and the wider world),
Mawson’s point requires only some
degree of uncertainty about how Adolf would turn out, which open theists would
accept there was. Still, since God was intervening miraculously to make the
world a happier place, one might wonder why He did not also physically improve
Adolf’s foetus while he was at it, e.g. to make it more like Mrs. Hitler’s
conception of the baby that she wanted God to save.
But the main problem with
Mawson’s argument is that it is quite unclear how such divine interventions as
we might expect under open theism would have to resemble that scenario. Now,
that may just be me being dense, but consider Mawson’s presumption that the
world’s aggregate happiness decreased, for example. The immediate consequence
of Adolf’s birth was merely more joy in the Hitler household. And God would
surely have intended to intervene further as necessary to ensure that
aggregate’s continued increase, had that been His motivation. So I wonder why
that decrease would have happened, in that scenario. After all, if by ‘world’
Mawson meant the whole of creation (not just this universe)—as he (2005, 10,
70) presumably did—then under any species of open theism God could have ensured
that aggregate’s increase simply by adding to His heavenly hosts. Now, perhaps
that scenario implicitly indicated that this universe alone was meant, since
the dead are arguably happier if, before they died, worldly woes turned them
heavenward. But if so then the posited motivation is even less plausible.
In any case, a more realistic
open-theistic motivation would have been, not so much that aggregate’s
continual increase, as its eventual perfection, e.g. everyone (or perhaps
only those who made the right choices) ending up in heaven. If an open-theistic
God had instead saved Mrs Hitler’s baby because He wanted to respond to her
prayer in a most loving way, then the satisfaction of His immediate desire—to
help her to relate to Him more fully—would not have depended upon how her baby
turned out. Indeed, it would not have depended upon any contingency. And while the satisfaction of His medium-term
desires—e.g. that she will soon go to heaven, having helped others there—would
plausibly depend upon such contingencies as her free choices, and those of
others, that of His long-term desires—e.g. that everyone will eventually end up
in heaven, having deserved to be there—might also be inevitable. Even those
open theists who do not believe that everyone will inevitably end up in heaven
believe that those who do not will only have themselves to blame, through the
exercise of their own free wills.
A more detailed reply to
Mawson’s scenario might have to be based upon a demonstrably sound theodicy;
but to undermine the analogical argument based around that scenario it is surely
enough (see below) that the best theodicies do not appear to be implausible
under open theism. Indeed, they appear to be more plausible, as Rhoda (2007, 305) observed. If so then the
evidential problem of evil is, incidentally, not so much an argument against a
perfectly good omnipotent being, as an argument that the Anselmian God is
everlasting. Free will theodicies are quite popular, for example, but Mawson’s
(2005, 198–217) faces the problem that a timeless God’s ability to be sure
about outcomes is not so much better than the ability to ensure them as less
realistic, because a world in which everyone always freely chose well is an
apparently possible but non-actual world.
A clue as to why Mawson
thought that his scenario was
sufficient might be the thought that preceded it. If so, then the failings of
that thought may show why a more detailed (and tangential) reply is unnecessary
here. Mawson (2009, 45) was criticising the idea that God has false beliefs, which
he attributed to Swinburne (2001, 35), with the thought that since it was
highly unlikely, early in 1936, that Time’s
Man of that Year would be widely regarded, ten years later, as the most evil
man that ever lived, so a temporal God would, in 1936, have thought that highly
unlikely, and hence—according to Swinburne (according to Mawson)—have thought
that it would not happen. But to begin with, was that really highly unlikely? We know a little about the roller-coaster fortunes
of other Great Men in history, so we can make a guess about what God—perhaps King Saul’s God—would really have
thought. So while it is understandable that the man in the street would have
found it highly unlikely, would an informed sociologist or historian; would
God?
I think there are clearly
grounds for doubt; and furthermore, Swinburne (2001, 34–38) had been
considering what we generally count as one of our beliefs, not what God would think of as His beliefs (which He
presumably holds infallibly). Our languages get much of their power from their
versatility, and much of that from a ubiquitous vagueness that is not
ordinarily noticeable, but which means that paradoxes are often resolvable
through the partial clarification of a term. E.g. the Lottery paradox: I
believe of each ticket that it won’t win
and, whereas one ticket is bound to win, the standard conjunction of all those
italicised propositions is that none will. The usual resolution is to describe
my belief that it probably won’t win at least that precisely, as Swinburne
(2001, 37) observed. Mawson’s (2008, 46) suggestion that open theists make just
such a distinction—between the highly unlikely and the impossible—was therefore
a bit inapposite, because clarifying terminology (for academic purposes) is what
Swinburne (2001, 34–38) was explicitly doing.
To recap, were the future
real (in Mawson’s sense) there would be some truths about the actual future
that an open-theistic God would not (infallibly) know; but although there is,
even under presentism, a sense in which there are such truths, that sense has
not been shown to be metaphysically significant. Furthermore, even if Mawson could find a more successful scenario
(which seems unlikely), there was a fundamental problem with his Anselmian
reasoning, as follows.
4.
Anselmian comparisons
Mawson was comparing the
knowledge (and hence the beneficence) of an open-theistic God, with that of a
timeless God, to see which was greatest. Now, Mawson (2008, 43–44) made various
observations on the relationship between knowledge and power, but for example,
timelessly possessing knowledge is not obviously on a par with always having
it, because the latter is more obviously potentially useful. So for simplicity,
both knowledge and power will be considered below, although I suspect that either
might be considered alone. E.g. any piece of knowledge might be associated with a logically prior power to have such knowledge, while
having the power to choose to do
something implies having the knowledge
that one could do it.
The fundamental problem is
this: When comparing the knowledge and/or the power of an open-theistic God
(e.g. under presentism) with that of a timeless God, to see which is greatest,
should one conclude that God is timeless just because only then could He be
completely knowledgeable and/or powerful in respect of this universe, even were
that the case? I think that the answer is clearly ‘no’; but if that is unclear,
consider an enormous computer that can create complex virtual beings, about
which it would have a complete database, and over which it would have complete
control. And suppose some philosopher believes her maker to be the greatest possible
being, and that there are two conceptual possibilities to choose between: (i)
an everlasting God with limited foreknowledge, and (ii) such a computer,
uncreated and relatively transcendent. Although only the latter could be completely knowledgeable about our
Anselmian philosopher’s universe, it would hardly be more knowledgeable
than—and would clearly be less powerful than—such possible people as could have
built such a computer, and whose own maker could have been the former God.
So I think that one should
rather count each God’s infallible knowledge of and absolute power over any of His
metaphysically possible creations.
For both kinds of God there are presumably lots of possible creations. I agree
with Mawson (2005, 71, 158) that creation is contingent. Still, I suppose that
one might argue that if God deliberately created contingent things, then there
were on the one hand various live possibilities, and a divine choice between
them, and on the other hand an actuality amidst lots of counterfactuals, and no
such choice. In other words, one might
argue that were God unable to change, creation would not be contingent. But
even so, there would still be lots of possible creations for an everlasting
God. And not creating some of them would not be a failure of omnipotence, as
being unable to create them would; and whether or not they are actually
created, God would know pretty much everything about them. I say ‘pretty much’
because limitations to His knowledge might
come from indefinite extensibility (see section six), or from our being created
so in our open-theistic creator’s image that we could also be a little
original, etc.
Anyway, both kinds of God
are to be regarded as conceptually
possible for the purposes of such a comparison. Of course, our conception of what is metaphysically
possible depends upon our conception of the metaphysically necessary substance
(e.g. via God’s omnipotence), but we may nonetheless, as Mawson argues (2008,
40–41, 46), attempt such a comparison. And if God could be timeless, if a
universe like this one (appears to be) could have been created by a God who
completely transcends its temporal dimension, then an everlasting God could
surely create just such a universe instantaneously, because He would just have
to create things whose ways of changing (so to speak) were utterly different to
His own. And after all, those ways of changing might be quite different even
under presentist open theism; e.g. a body, or even the whole of creation could
conceivably remain completely still while time passed as usual, but if God as
well as creation ceased to change at all, then the flow of time (so to speak)
would also cease.
Were we within such a
universe (made instantaneously by an everlasting God), the transcendental—or as
we would see it, metaphorical—present at which our relatively atemporal God was
fully present would also include our past and future. So both presentism and
divine openness would be false, within such a universe. Incidentally, our God
would have no foreknowledge of our
future actions, so we might be free
agents within such a universe, much as Mawson (2008, 38) thinks we are. But it
is not my suggestion that our universe could really be like that. My point is
rather that, whether God is timeless or everlasting, He could create, and so is
completely knowledgeable about, such a universe.
And it should be clear
that the same could be said of any possible creation of a timeless God. There
might, conceivably, be things that God would only know all about if He actually
created them, in view of the aforementioned limitations, e.g. the possibility
of our being original, to some extent. But those possibilities are associated
with open theism. Still, we are presuming that (presentist) open theism is
conceivable, that a universe containing people like us could have been created
by God to have a future that is relatively open, in some metaphysically
significant sense. And a timeless God could not create such a universe, because
His knowledge of the future of any of His creations could not possibly
increase. Note that He could not know all about such a universe without being
able to create it, because He is omnipotent. That is, since God can create any
metaphysically possible universe, so He can only know all about the universes
that He can create. That is because a metaphysically impossible object does not
exist, so it cannot be known directly, and its properties include
contradictions that can be no more known than 2 + 2 = 5.
So it seems that an
open-theistic God is, at least under presentism, able to create (and hence
knows all about) all the possible universes of a timeless God, and more besides.
And while that most directly argues for the Anselmian God being everlasting, note
that if that was thought to be the case then we should have fewer problems with
such scriptural evidence for open theism as Mawson (2008, 36) began with, and
which Pinnock (2001) considers. One might object that a timeless God, being
able to know all about any temporal creation, would know all about whatever a
temporal God would be able to do. But an everlasting God is not so much inside
time as able to change. The non-instantaneous
creations of an everlasting God might not be like any of the temporal creations
that a timeless God could know all about. And arguably the simplest difference
arises with arithmetic, with the intrinsic properties of the whole numbers.
5. Whole numbers
The products of endlessly
reiterating the addition of 1, starting with 1, are the natural numbers.
Nowadays the natural numbers tend to include 0 too, but the older definition is
more suited to theism. Like most theists, I am a realist—as opposed to a
formalist—about arithmetic. Whereas formalist arithmetic is essentially a
formal-linguistic expression of the conceptual
structural possibilities associated with the human imagination (the standard
language being ZFC set theory), we might think of arithmetic as orientated
towards the metaphysical structural
possibilities that are ultimately grounded in the divine essence (e.g. via
God’s omnipotence). But whatever numbers are in general (e.g. objectifications
of structural possibilities), the natural numbers are specified by that endless
reiteration. So although we naturally intuit that each natural number exists
atemporally (insofar as it exists), there is also a sense in which they are,
collectively, rather open-ended, in view of the endless reiteration that
specifies them.
To get a feel for the
natural numbers as a whole, consider the rapidly increasing function that takes
each natural number, n, to nn. Let’s call that function
‘nin’, e.g. nin(2) = 22 = 4. For any function g, repeated applications of a function are denoted by gm + 1(n) = g(gm(n)), e.g. nin2(2) = nin(4) = 256. Let bob(n) = ninn(n), e.g. bob2(2)
= bob(256) = nin255(256256). So, with a few simple
definitions, we can get to the rather large bob101(2) very quickly. So just imagine the size
of the numbers that a whole page, or a whole book of such definitions would be
able to specify. Imagine a shelf of
such books, bob101(2) miles long (ignoring the physical
impossibility of that), defining some huge
natural number, say h, compared to which bob101(2) is miniscule.
Indeed, imagine a shelf of
length h, defining another number, j, and a shelf of length j, and so on,
because after bob101(2) repetitions of such shelf extensions, the
resulting numbers would clearly have got unimaginably
huge. There is not really any such thing as merely
finite. And even such huge numbers are miniscule compared to the bulk of the subsequent numbers (if
‘bulk’ is the right word). We might therefore find ourselves, with many
philosophers, taking seriously the admittedly unworldly thought that the
natural numbers are, collectively, not so much immutably complete as endlessly
growing. It was traditionally thought that their collection was indefinitely extensible (as Mill put
it), or ‘potential’ infinite, as Aristotle put it (see below). And we might ask
ourselves if they go all the way up to infinity; and if so, why none of them
are anything like infinitely big, and if not then how we could have them all. Note
that even if their collection is (at any time) incomplete, as a matter of
(metaphysical) necessity, we would still be able to refer to all of them, e.g. to say that all natural numbers are (or will be) sums of units; although in
view of the connotations of ‘all’, many would prefer to use ‘any’ there.
Aristotle’s analogy was
with a block of stone from which a statue will be carved, and which contains it
potentially. Unfortunately that makes it seem as if the ‘potential’ infinite
might actually be infinite one day, which is not intended (whence the
scare-quotes). When thinking of the ‘potential’ infinite it might be better to
think, not of the emerging statue but of the stone chips being endlessly
chipped away, from some big enough (or regenerating) block. The chips are to
such a block as the whole numbers are to God’s fundamental concept of a thing, or whichever concept yields the
whole numbers as structural possibilities, and which is implicit in His
creation of such definite individuals as souls that can be judged, each only for whatever each did, presumably. However, since it is tempting to think of the
original block as composed of at least all those chips stuck together—which
would make them ‘actual’ infinite—so an even better analogy might be an
everlasting fruit-tree and its fruit. Think of the endless production of fruit
as the expression of a logically prior power. What we begin with is not so much
a quasi-spatial block, as a unit—e.g. from the concept of a thing (or individual)—and addition—e.g. from the concept of the possible (or division)—and the power
to reiterate the addition of a unit. And if the total number of metaphorical chips
or fruit produced was ‘potential’ infinite, then the magnitude of that power (which must be ‘actual’ infinite) would be
incommensurate with that total.
But it is of course
simpler to presume commensuration,
and the alternative to that traditional view of the natural numbers is that they
comprise a transfinite set, as Cantor
put it, which I shall denote by ‘N’. On that now standard view, the natural numbers are a bit like so many stars
shining altogether in some infinite space. Indeed, the conceptual possibility of
such a space was for Benardete (1964, 31) the major reason to think of the size
of N as a proper number. And intuitively, if N is indeed quasi-spatial, like
that, then each sub-collection, of just some natural numbers, could be expected
to be similarly quasi-spatial, to be a subset
of N. To see why, think of some ordinary objects in a room, forming a spatial
collection by virtue of occupying the same space. Any spatial part of that
collection—any sub-collection—will clearly be a collection of things coexisting
in the same spatial way. To call some infinite collection ‘quasi-spatial’ is
therefore to say that all conceivable
sub-collections of it are collections of the same kind. By contrast, if some infinite collection is thought of
as forever growing, according to some definite rule, then only those
sub-collections that are similarly specifiable by some definite rule would exist
in the same kind of way. Incidentally, in common with many writers Lavine
(1994, 77) means by ‘combinatorial’ pretty much what I mean by ‘quasi-spatial’
here; and in other words, the standard real numbers are isomorphic to the full second-order power-set (see section
six) of the natural numbers. By contrast, the traditional view is that most of
the standard real numbers do not exist as individual numbers (e.g. see Ormell
2006; and for more background Fletcher 2006). Note that ‘combinatorial’ and
‘full second-order’ presuppose something more than those words’ literal
meanings, to get their intended meanings there; something like a spatial
quality, I suggest.
The standard view gives us
a relatively simple picture; and perhaps there is an Anselmian argument that it
is accurate, since the standard view allows God more things, e.g. more real numbers, and hence more objects of
different sizes. An obvious problem with that thought is that perhaps one might
similarly argue that God can make
round squares, which would put Him well beyond ordinary logic, not just—as He
surely is—beyond our ordinary categories. Conceptually possible theories of
time, logic and arithmetic seem to push us up against the limits of Anselmian
comparisons; but let us, with Mawson, push on. The traditional view gives us
the indefinite extensibility of the whole numbers; what of the standard view? In
the following section I argue we get it on both views. The central argument is
based upon Grim’s (2000, 147–152) argument against an omniscient being, which
is more properly (I argue) an argument that the Anselmian God, whether
omniscient or not (see section seven), is everlasting.
6. Cantor’s paradox
For any quasi-spatial
collection, any set, say S, the
collection of all its subsets
(including S itself and the empty set) is called, if it is similarly
quasi-spatial, its power-set, say
P(S). Cantor’s diagonal argument shows that P(S) has a greater cardinality than S, where two collections have the same cardinality—the same cardinal number of elements—if all the elements
of one collection can be paired off one-to-one with all those of the other. That
argument is essentially a short reductio:
Suppose that S and P(S) have the same
cardinality. Then there is a one-to-one mapping, say M, from each element of S
to one of the subsets of S, which maps S onto P(S). Cantor’s diagonal procedure
specifies a subset of S, say D, according to the following rule: For each
element of S, if the subset it maps to under M contains it, then D does not
contain it, and otherwise it does, and D contains no other individuals. So D
differs from each subset that an element of S maps to under M. That is, D
differs from each subset of S. But D is by definition a subset of S (if possibly
an improper one). That contradiction means that there is no such M.
Consequently P(S) has a different
cardinality to S; and since to any individual, say E, in S there corresponds an
individual, {E}, in P(S), hence P(S) has a greater cardinality than S.
In particular, the
cardinality of P(N), or beth-1 (as Peirce called it), is greater than the
cardinality of N, beth-0, and less than beth-2, the cardinality of P(P(N)) = P2(N),
etc. Consider all the Pn(N)—each
with beth-n elements—for all natural
numbers n. Their union, the collection of all their elements, has a greater
cardinality than any of them, since for every natural number n it contains beth-(n + 1) elements. If it is also a quasi-spatial collection, we can
again consider its power-set; and similarly for any endless sequence of
power-sets. And by continuing in that way, as far as it is logically possible
to go, we must find ourselves with a collection—of all the beths—that is not
quasi-spatial, since otherwise we could continue further.
The beths comprise, as do
the more famous alephs (and the more academic ordinal numbers), not a set but a
so-called proper class. That result is widely regarded as a paradox because realist intuitions inclining us to the standard
view of the natural numbers seem also to incline us to think of all the
cardinal numbers as a quasi-spatial collection. For the idea that such proper
classes as the beths (or the alephs) could provide our analysis of the
‘potential’ infinite—whence the traditional view of the latter might help us to
understand the former—see Hart (1975–6), and more recently Rescher and Grim
(2008). Note that although my argument was informal, formal proofs of such
results can only really prove theorems within particular axiomatic systems, whence informality suits metaphysics (the source
of our axioms). Incidentally, cardinal numbers (e.g. the alephs) are defined to
be limit ordinal numbers in the standard systems, and when showing that the
ordinal numbers form a proper class the Burali-Forti paradox is easier to use than
Cantor’s paradox (e.g. see Drake and Singh 1996, 95, 113).
Even if the natural
numbers are not, the whole numbers are, in general, indefinitely extensible. Of
course, we do naturally think of whole numbers as atemporal, and indeed, of God
as atemporal. But if God was
everlasting then he would have had access
to an indefinitely extensible concept of a
thing (an individual, a subject, an object), and that does seems to be the concept that we instantiate. In our context of
Anselmian arguments, it is notable that indefinite extensibility arises with
something as elementary as transfinite (if not finite) arithmetic. But more
generally (and simply), were there a set of all the other sets there would be
(via the diagonal argument) more subsets than there are sets, whereas a subset
is a set, whence the sets form a proper class. And arithmetic being related to
logic, Cantor’s paradox is related to Russell’s paradox and the Liar paradox.
Standard mathematicians and logicians are strongly adverse to the ‘potential’
infinite, so it is significant that during the entirety of the twentieth
century they failed to find a more convenient resolution of Cantor’s paradox
(e.g. see Shapiro and Wright 2006).
Cantor himself thought
that all ‘potential’ infinite collections presuppose ‘actual’ (or as I would
say, quasi-spatial) infinite collections, much as variables range over fixed
domains, and that the inconsistency implicit in that was an aspect of God’s ineffability
(e.g. see Hallett 1984, 25, 40–48). Of course, even the making of this universe
ex nihilo is hard to imagine, let
alone understand. But we are here entertaining the ineffability, not so much of
God, as of arithmetic. And even when faced with such contradictions as God does
and does not exist, one would ordinarily look for an equivocation such as God
not existing physically (when not incarnating) but rather transcendentally. I
think it is simply false that God
does not exist.
There is another
tradition, which Priest (2002) considers, but if one sides with such
paraconsistent logicians (or Cantor) when it comes to arithmetic, then
presumably one should also reason in a paraconsistent way (if at all) about
God’s maximal greatness. The problem with that, as far as Mawson’s arguments
are concerned, is that if one has a paraconsistent argument that God is
X—whence He may also not be X—then prima facie, one hardly has an argument that God really is X. (Note that
even if you only thought of paraconsistent logic as conceivably logical, you might have to accept its actual validity
via an Anselmian argument, if paraconsistent logic supports such arguments—and
if not then the aforementioned problem is so much worse—because paraconsistent
logic allows more things to be
possible.)
Anyway, most academic mathematicians
have rather aimed to retain consistency by treating pure maths as a language
game—not just formal but formalistic—and saying as little as possible about
proper classes. E.g. within some axiomatic set theory, Cantor’s theorem just says that there is no such
set of all such sets. But of course, that just passes the metaphysical buck to
those who apply such maths, for whom Cantor’s paradox still arises as such. In
particular, the problem for Mawson is that, since a timeless God would clearly
know all the individual truths known by Him altogether, so they would comprise some sort of ‘actual’ infinite
collection. And if we retain consistency, they would comprise an informal set.
But for each set there is the truth of what its elements are. There is also the
applied arithmetical truth of how many elements it has. And since there are too
many sets for there to be a set of all the other sets, so there are too many
truths—and in particular, too much transfinite (if not finite) arithmetic—for
there to be a set of all truths. That is, a timeless
God cannot be completely omniscient.
Indeed, He cannot even know all the truths that could each be known by Him,
since He could in theory know any set
of truths.
An everlasting God, on the
other hand, could know each of those truths eventually, because the openness of
his future allows His knowledge to be indefinitely extensible. He would have
time enough because, if time is an objectification of changeability, which is
an aspect of his omnipotence, which is maximally great, then His time (if not
our temporal dimension) could be expected to be absolutely continuous, in the
sense that for any duration, and every ‘actual’ cardinal number, that duration includes
that many instants. On the traditional view time could be ‘potential’ infinite into
the future, with events n years from
now for every natural number n, and
there would have been some first temporal event (or change), perhaps some unimaginably huge number of years ago.
But on both the traditional and the standard view, durations might have # = 1/0
instants. And arguably, # is the biggest conceivable ‘actual’ cardinal number
(contra the impression of Cooke, ibid, 103), because such a continuum would
have a ‘potential’ infinity of sub-collections of points. And God would be able to know each of those truths
because He is omnipotent, and because He would always have understood
completely the primitive arithmetical concepts (the unit and addition).
The entirety of transfinite arithmetic would be forever beyond His (or
anyone’s) full comprehension, as a matter of logical necessity, so calling Him
‘omniscient’ may seem wrong (although see the following section), but the main
thing here is that each of those
highly abstract truths would eventually—and arbitrarily quickly—be known by Him,
whereas if He was timeless then almost all arithmetic would be eternally
unknown. Consequently an everlasting God’s ability to learn is hardly compensated for by a timeless God’s
ability to already know, contra Mawson (2008, 44). Indeed, although we might be
better off if we knew more maths now, an everlasting God without our need for
improved technology, but with eternity to fill, might rather solve such puzzles
one by one forever, in which case His ability to do so would be for Him a power
(would be good). And arguably most significantly, an everlasting God would, for
any transfinite cardinal, be able to make a universe containing that many things.
So if God is maximally powerful
(and/or knowledgeable) then He is everlasting.
7. Potential omniscience
One might object that I
have only shown that under the conceptual possibility that God is timeless,
arithmetic would not be indefinitely extensible. But even were that the case,
modern maths would still amount to a lot of evidence that arithmetic is
indefinitely extensible, and hence that God is not timeless. And even if
arithmetic was not indefinitely extensible, and could all be known by a timeless God, an everlasting God would then
also be able to know it all. So even
that remote possibility is of little help to Mawson’s argument; and indeed, there
might still be other indefinitely extensible truths (e.g. we might be
sufficiently original), whence there might still be an Anselmian argument that
God is everlasting.
Or one might similarly object
that God cannot be eternally learning because He is omniscient, whence I must
be wrong about arithmetic, in ways not worth looking into. But that objection
presumes, not only that the Anselmian God is omniscient, which is less obvious
than one might imagine (see below), but also that an eternally learning God
could not properly be regarded as omniscient. And in order to justify the
latter presumption one would, after all, have to look into the metaphysical
foundations of arithmetic, if only to undermine the following reason why that
presumption is false.
An eternally learning God
could, I suggest, deserve to be called ‘omniscient’ if it was always be the
case that, for all the statements that (then) exist, He (then) believes each
statement only insofar as it is (then) true. My suggestion is basically that we
might take a statement to exist when
an expression is made possible by
some language existing. I am thinking of actual
expressions of propositions as what we actually make with such grammatical
arrangements of our current vocabulary as this. Perhaps, in theory, one could
translate everything expressible into a single super-language, in which case an
eternally learning God would never be omniscient. But it is certainly the case
that natural languages evolve; and what we express with them is, in all its
natural fuzziness, at least in part determined by the expressions that, while
being currently possible, we have chosen not to use. That indicates that any
such super-language could be naturally linguistically stratified, and that were
there no end to linguistic evolution, there would never be such a
super-language.
Of course, arithmetic is
exceptionally pure and simple. But Cantor’s paradox does indicate that realist
arithmetic is indefinitely extensible, and hence that the notion of every
arithmetical truth existing already is an unrealistic extrapolation from our
quotidian beliefs about ordinary arithmetic. And note that our quotidian
beliefs can mislead us when it comes
to metaphysics. Although the sun will, I suppose, rise as usual tomorrow, for
example, such truths never falsified the Copernican
revolution. And although most of us think of arithmetic as timelessly true (in
all possible worlds), by ‘arithmetic’ we ordinarily mean finite arithmetic. And on the standard view of that an eternally
learning God could always have known all of it, and presumably all of the maths
that we could ever discover as well. Indeed, God might only learn arithmetic
when He does not have a mundane creation to deal with, in which case even transfinite arithmetic would be
relatively atemporal. And if the traditional view of the natural numbers is
correct, we need only replace such thoughts about very large cardinal numbers with thoughts about unimaginably huge
natural numbers.
Realist arithmetic may be
thought of as the result of God eternally learning absolutely all of the
structural consequences of the completely definite concept of an individual
thing. The necessity of knowing those consequences over time follows from their
indefinite extensibility, which follows from the intrinsic nature of the
primitive concepts. And while the idea of an arithmetical language being
indefinitely extensible might suit the traditional view of the natural numbers,
on the standard view it is at least arguably
more natural to consider a proper class of arithmetical languages, each of
which could be completely understood in an instant by an omnipotent God. We do
after all intuit quite strongly that arithmetic—of at least the ordinary kind—is
timelessly true. So the question of whether or not an eternally learning God is
omniscient seems to reduce to the question of how we conceive of propositions (cf. the question of
whether or not a God who cannot make a round square is omnipotent or not, which
reduces to our conception of logic).
But arguably our
conception of divine omniscience
ought to transcend that of propositional truth, in view of how one knows
oneself, one’s children, one’s compositions and so forth. Furthermore, even if
we did think of God’s arithmetical language as indefinitely extensible, we
should only have to choose between re-defining ‘omniscience’ or accepting that
an omniscient God was less powerful, less great
than an eternally learning God. And as aforementioned, the Anselmian God might
be, not so much timeless and omniscient, as everlasting and omnipotent. Open
theists generally think that God’s knowledge grows, as there becomes more for Him
to know. And there are other reasons why they might think of maximal knowledge (now)
as less great-making than infallibility (always), as follows.
God has authoritative
knowledge of creation, and perfect self-knowledge, of course. But knowing everything now is not obviously better, in
general, for an everlasting person, than being able to know anything she wants to know as quickly as she likes, just
so long as she knows enough to be completely free in respect of those wants and
likes, and if her choices were always good ones. Consider someone who knows
what phone numbers are, and where to look for them, and is always able to do so
as quickly as she likes. How could it be better to have memorised all the phone
books? And similarly, to be able to immediately initiate something that would
take some arbitrary time to complete is not obviously better than being able to
acquire that ability arbitrarily quickly. Indeed, whereas to be omnipotent is
(roughly) to be able to do anything of which it is not the case that it should
not be done, one who was necessarily unchanging could seem to be incapable of
doing anything at all.
But God is certainly omnipotent
under open theism, and so He could have ensured that only certain predetermined
events happened. He could even have ensured our free reactions to them, e.g. by
gracefully giving us only beatific visions. But He did not do that, and so He
has in that sense deliberately
limited the extent of His foreknowledge under all varieties of open theism. Furthermore, if God could only know
what we would choose, rather than
what we might choose, in all possible scenarios by actually putting us into
them and finding out, then since He could have so tested us in every
conceivable way (see below), so we do not think that His knowledge should be as great as it could have been.
On the standard view of such things, if God existed n years ago for each natural number n then He could have so arranged creation that, for any test
expressed by a possible sentence of modern English, it would always have been
the case that each of us would already have been so tested (infinitely many
times). But of course, God is too wise and good to have so chosen. Note that on
the standard view, God would have made no
decision without the wisdom that comes from having already made infinitely many similarly wise
decisions.
Rather than worrying about
defining ‘omniscience’ so that such limitations would not prevent God from
being omniscient, open theists might prefer to prioritise God’s power to understand
perfectly well whatever He wants to
understand, and the wisdom and freedom of His will, and only then consider the extent of His
knowledge. That extent would presumably be some
perfect quantity, if not a maximal one; but it is more important, if God’s
knowledge is forever changing in unpredictable ways, that His infallibility
arises from the perfect quality of his
understanding, from each of His actual beliefs being, of metaphysical necessity,
completely justified (e.g. via His omnipotence).
Incidentally, if none of
God’s beliefs are ever at all unjustified then an interesting example of a
family of theodicies that are completely unavailable to atemporalists (see
section three) follows from the possibility of some negative existential propositions being ‘known’ by Him no better
than we know that 44 = 256, despite such conceptual possibilities
as Cartesian demons who fool us about arithmetic, although much better than we
‘know’ that a flipped coin won’t land on its edge, to use Mawson’s (2008, 45) example.
God could nonetheless be omniscient, e.g. according to Mawson’s (2005, 35)
definition—although interestingly, not according to my own (via ‘insofar’)—but
some of that knowledge (as we might naturally think of it) would be
incompletely justified as a matter of logical necessity. And only if God could change,
and hence learn, could He make such true but partial beliefs more justified,
and hence fuller. The theodical import of that is that such learning might—although
the literature has as yet said little about this family of theodicies—be done
best by temporarily incarnating souls precisely this far from heaven. Perhaps, in view of the existing literature, I
should add that such a God would hardly be like Victor Frankenstein (or indeed,
Wayne Szalinski), since the explanatory power of such a theodicy comes from the
conceptual limitations imposed by His perfect wisdom and goodness.
But there are, after all,
lots of conceptual possibilities. Indeed, maybe God is timeless; and maybe we
can, even while not understanding arithmetic, understand that He is timeless,
e.g. by the way of negation, although that borders upon the paraconsistent realm
of Priest (2002, 22–23). But note that even the more definite thought of Davies
(1993, 141–167)—that change implies loss, and hence being limited, whereas God
is unlimited—has been answered implicitly by the above. Had God decided not to
create much that was contingent, perhaps to change only by eternally learning
mathematics, then none of His changes
would have been losses. He would only have gained ever more knowledge. And even
as it is, His ‘actual’ infinite power to know can only find its fullest expression by His knowledge
increasing in such an absolutely unlimited way. So, such arguments as Davies’
and Mawson’s fail to give us any positive
reason for believing that God is timeless. Mawson’s (2008, 48) picture of my
God as a character that Rick Moranis might play is witty, but why should my God
not be more like Jesus?
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