Friday, December 17, 2010

Omniscience Again cont.

This is the last of 17 posts, which are collectively Eternity, etc.
......There is something counter-intuitive about the suggestion of the previous post, of course (even on the modern view of arithmetic). If B is the biggest Beth that has been constructed, then my suggestion denies that 2-to-the-power-of-B exists, where 2-to-the-power-of-B is the cardinality of P(X) when X has cardinality B. Were there no such B, my suggestion would deny that the union of the existing sets has an actual cardinality, on the modern view of the natural numbers (on the older view, it would deny the existence of M + 1, where M is the biggest natural number divinely constructed). Either way, my suggestion is effectively that there are true statements that God did not know but which were bound to be true and which, if we could come to know them, we would most naturally say had always been true. Intuitively, that seems to fall short of divine omniscience.
......Nevertheless, we know from section III that what can seem, with hindsight, to have been timeless truths may not have been. And according to section VII it is logically, not just physiologically, impossible for anyone to say or know all such things. Such statements therefore belong to an indefinitely extensible totality. Would it therefore be more accurate to talk of possible statements here? Maybe not [i], but it’s certainly logically possible that our intuited shortfall is due to our being dependent creatures. For us, even physics is immutable, but God is certainly the ground of metaphysical possibility. And He may well be the ground of all meaning and value. So the counter-intuitiveness of divinely created mathematics may prove, upon reflection, to be no more conclusive than the counter-intuitiveness of Divine Command Metaethics [ii]. After all, my suggestion does not deny that, in the time we took to think of 2-to-the-power-of-B, God had already constructed it [iii].
......So, to recap, God’s omnipotence conflicts with His timelessness, according to section VII, unless we deny arithmetical Realism, or deviate further than Presentism does from standard logic. And under Presentism, even such an omnipotent God could be necessarily omniscient. So what follows from God being necessarily omniscient—and our freedom being libertarian—is primarily disjunctive. Either God has timeless knowledge of the future—if that does cohere with our freedom being libertarian—but Realist arithmetic is paradoxical, or Realist arithmetic is divinely constructed and time is Presentist, or some other option. So even if they regard God as necessarily omniscient, Perfect Being Theists who take a libertarian view of free will—and regard the future as (partially) real—should not reject Open Theism.
......[i] Statements are basically possible assertions (see note ii of Eternity), but a possible statement is not necessarily just a statement. Similarly, one might be unable to say something in French, and yet be able to learn (more) French, so that one would be able to be able to say it. It is of course hard to tell how apposite that analogy is, for the language (so to speak) of God’s thoughts.
......[ii] Lois Malcolm, “Divine Commands,” in Gilbert Meilaender & William Werpehowski (eds.), The Oxford Handbook of Theological Ethics (Oxford Univ. Press, 2005), pp. 112–29.
......[iii] Suppose (see note v of Possible Worlds) that a God who could change had made our 4-dimensional world in an instant. Then some biggest Beth, say B, would be known by Him at all (of our) times. But then we might use “2-to-the-power-of-B” as a definite description of a Beth that He does not know, at any (such) time, which hardly coheres with His being the greatest conceivable being. By contrast, a Presentist God would most plausibly be learning arithmetic too quickly for us to describe any such number.

Wednesday, December 15, 2010

Omniscience Again

This is the sixteenth of 17 posts, which are collectively Eternity, etc.
......You may be wondering how, if the Open God is forever acquiring arithmetical knowledge (see previous post), He could ever be omniscient (or how His understanding of His options could ever be perfect). It would not even help us here to think of omniscience in Swinburne’s terms, because however much arithmetic God knows it is logically—indeed, metaphysically—possible for Him to know more (according to section VII); and nor could He know all the interesting Beths (that could ever exist) [i], because the smallest Beth that He did not know would be of some objective interest.
......Nevertheless, such Perfect Being Theists as Augustine and Duns Scotus took the Platonic Forms to be divinely created (in view of God’s omnipotence) [ii], and similarly, Open Theists might take arithmetic to be divinely constructed [iii]. Suppose that arithmetical statements are true or false only when divinely proved or refuted (respectively). That process could not always be instantaneous, according to section VII, and of course, not yet knowing something that is not yet true would not obstruct omniscience. And while most of us think of arithmetic as timeless, by “arithmetic” we ordinarily mean finite arithmetic, and on the modern view such a God could have always known all of that (instantaneously constructed in His primal state). Indeed, He could have always known all the Beths that are not, for us, unimaginably large.
......My suggestion is therefore that God, being epistemically omnipotent, constructs each and every modal consequence of the concept of a thing (which He understands perfectly), and in particular the cardinalities of possible creations (doing so endlessly because such is the nature of that concept). He knows all the Beths that exist. Before He constructs a Beth, it has only a potential existence. And eventually (and arbitrarily quickly) He knows any Beth that could ever exist. And His understanding of such possible Beths is perfect (cf. how we could understand the essence of an arbitrary natural number, even on the older view of arithmetic).
......[i] For a similar suggestion, see Menzel, “God and Mathematical Objects,” pp. 93–4 n. 42.
......[ii] For Augustine, see Sorabji, Time, Creation and the Continuum, p. 252. For Duns Scotus, see Gunton, The Triune Creator, pp. 118–9. For more details, see Copan & Craig, Creation out of Nothing, pp. 173–80.
......[iii] Menzel, “God and Mathematical Objects,” defends such a view, called “theistic constructivism” by Copan & Craig, Creation out of Nothing, p. 191.

Monday, December 13, 2010

Cantor’s Paradox again

This is the fifteenth of 17 posts, which are collectively Eternity, etc.
......Because of all those unions (see previous post), our collection is a nested hierarchy of sets, whose cardinalities the Beths are defined to be. And so because our collection is not quasi-spatial, nor are the Beths, collectively. So even on the modern view, cardinal arithmetic is indefinitely extensible [i]. And while that result is of a kind with Cantor’s Paradox [ii], it is the belief that cardinal arithmetic is timeless that makes it paradoxical. The atemporalist faces some tough choices, because the truths known by a timeless God would be collectively quasi-spatial, rather than variable.
......But an everlasting God could acquire arithmetical knowledge endlessly. And Presentist time is merely our natural reification of the possibility of change, not a real dimension. And under Presentist Open Theism, that possibility originates with the greatest conceivable being’s power to change. So such a God would have enough time to know each arithmetical truth, and to know it arbitrarily quickly. Time would then be indefinitely extensible, and absolutely continuous [iii], in the sense that for any duration, and for any Actual Infinite cardinal number (that could ever exist), that duration has more than that many instants (possible instantaneous changes).
......It seems, then, that only a God with the power to change is, for every Actual Infinite cardinal number, able to know all about a possible world of so many things, and hence able to create such a world perfectly freely (with a perfect understanding of His options). So the argument at the end of section VI becomes an argument that God is, since omnipotent, not timeless. And note that the informality of this rather mathematical section does not make that a weak argument. Formal proofs can only prove theorems within axiomatic systems, and since the justification of such axioms is necessarily informal, so informality also suits a more direct argument about metaphysically possible creations.
......[i] For more details, see W. D. Hart, “The Potential Infinite,” Proceedings of the Aristotelian Society 76 (1976): 247–64; Alvin Plantinga & Patrick Grim, “Truth, Omniscience, and Cantorian Arguments: An exchange,” Philosophical Studies 71 (1993): 267–306; Stewart Shapiro & Crispin Wright, “All Things Indefinitely Extensible,” in Agustin Rayo & Gabriel Uzquiano (eds.), Absolute Generality (Clarendon Press, 2006), pp. 255–304; Nicholas Rescher & Patrick Grim, “Plenum Theory,” Noûs 42 (2008): 422–39.
......[ii] For Georg Cantor, sets were consistent Actual Infinite collections. But he thought that all Potential Infinite collections presuppose Actual Infinite collections, much as mathematical variables range over fixed domains. So he thought of collections like that of all the sets (Cantor’s Paradox) or all the cardinal numbers as Actual Infinite but inconsistent. For more details, see Michael Hallett, Cantorian set theory and limitation of size (Clarendon Press, 1984), pp. 24–48. Of course, taking inconsistency on the chin like that is a high a price to pay for Realism (whence the foundation of mainstream mathematics is now an axiomatic set theory). But even if Potential Infinite collections do depend upon something being Actual Infinite, that might be a power (see note iv in Cantor's Paradox) or a length (see following note) rather than a collection.
......[iii] For such continua, see my “To Continue with Continuity,” Metaphysica 6 (2005): 91–109; Philip Ehrlich, “The Absolute Arithmetic Continuum and its Peircean Counterpart,” in Matthew Moore (ed.), New Essays on Peirce’s Mathematical Philosophy (Open Court, forthcoming).

Saturday, December 11, 2010

Cantor’s Paradox cont.

This is the fourteenth of 17 posts, which are collectively Eternity, etc.
......You may be familiar with N (see previous post) from school mathematics. Such informal sets are basically collections that are quasi-spatial, in the sense that their members coexist (insofar as they do exist) altogether. Given any spatial collection—e.g. some ordinary objects in a room—any sub-collection of them is clearly also spatial; and similarly, a definitive property of our informal sets is that every conceivable sub-collection of such a set is itself quasi-spatial, is a subset [i].
......Surprisingly, the modern view (of arithmetic as timeless) offers little support to atemporalism, as follows. To say that two collections have the same cardinality—the same cardinal number of members—is to say that the members of each collection could all be paired off, one-to-one, with those of the other [ii]. And for any set, S, if the collection of all its subsets is also quasi-spatial, then that collection—including (for simplicity) the so-called improper subsets, S and the empty set—is the powerset of S, say P(S). And according to Cantor’s Diagonal Argument [iii], P(S) always has a greater cardinality than S.
......In particular, the cardinality of P(N)—which Peirce called “Beth-1”—is greater than the cardinality of N, which is Beth-0. And the cardinality of P(P(N)) = P-squared(N) is Beth-2, which is greater than Beth-1. And so on; for each natural number n, P-to-the-nth-power(N) has Beth-n members. And the union of N and all those P-to-the-nth-power(N) is the collection of all their members. For each n it contains at least Beth-(n + 1) members. So its cardinality, say Beth-omega [iv], is greater than Beth-n for every n. And if that union is also a set, say U, then by Cantor’s Diagonal Argument, P(U) has an even greater cardinality, Beth-(omega + 1) [v],
......We might expect that union to be a set, because Beth-0 being an Actual Infinite number means that all those Beth-0 sets coexist quasi-spatially (like a row of houses, whose contents therefore coexist similarly). So the next union might be of U and all the P-to-the-nth-power(U). But by continuing in that way, taking powersets and unions as far as is logically possible [vi], we cannot end up with a set because from any set we could have continued further in that way. We have, then, a collection that is not quasi-spatial, being generated by a process that cannot be completed (as a matter of logical necessity). Continued...
......[i] By contrast, if the natural numbers are forever growing, according to the rule of add 1 repeatedly, then only those sub-collections that are similarly specified by a finite rule exist in the same kind of way.
......[ii] The natural numbers are finite cardinal numbers. And N has the same infinite cardinality as the subset of just the even numbers because n can be paired with 2n for all natural numbers n. There are, in an obvious sense, more natural numbers than even numbers, but cardinality is fundamental to our number concept; Shapiro, Thinking about Mathematics, pp. 133–8.
......[iii] If S and P(S) had the same cardinality, there would be one-to-one mappings from S onto all of P(S). Let M be one such mapping, and let a subset of S, say D, be specified as follows: For each member of S, if the subset that M maps it to contains it then D does not contain it, and otherwise D does. The problem is that since D differs from every subset that M maps the members of S to, D differs from every subset of S, whereas D is by definition a subset of S. That is, D is contradictory, and so there is no such M, which means that S and P(S) do not have the same cardinality. But for each member of S, say m, P(S) contains {m}, and so the cardinality of P(S) is greater than that of S.
......[iv] Omega is the first ordinal number after the natural numbers. Ordinal numbers generalize counting numbers as such beyond the natural numbers (whence their use indexing the Beths).
......[v] Such ordinal addition corresponds to a rearrangement of the natural numbers, e.g. from their natural order (to which omega corresponds) to 2, 3, ..., 1.
......[vi] We could also take unions of Beth-1 sets, since Beth-1 is Actual Infinite; and similarly, Beth-2 sets, etc.

Thursday, December 09, 2010

Cantor's Paradox

This is the thirteenth of 17 posts, which are collectively Eternity, etc.
......This section is rather mathematical, but we can—indeed, should—begin with the simplest numbers, 1, 2, 3, etc. Mainstream mathematics has axiomatic set theory for a foundation, for such reasons as Cantor’s Paradox [i], and pure mathematicians are certainly free to explore any interesting formal possibilities. But we are primarily interested in possibilities insofar as they are (or might be) grounded in the God that is.
......The natural (or counting) numbers are the products of endlessly reiterating the addition of 1, starting with 1. They are clearly instantiated, because you and I are 2 people. Arithmetic is prima facie the science of such elementary metaphysical possibilities as the possibility of two individuals. Such arithmetical Realism may be difficult to justify atheistically [ii], but we may think of the natural numbers as existing amongst God’s thoughts, arising via His epistemic omnipotence from His perfect grasp of the concept of a thing, whence our informal 1, together with a concept associated with His omnipotence, such as possibility, whence informal addition and its endless reiteration. Note that Realist arithmetic can be discovered a priori under Theism because we instantiate the concept of a thing and were created in God’s image (and we might also be divinely inspired) [iii].
......The endless reiteration of the addition of 1 means that the natural numbers are (collectively) infinite. Many mathematicians have taken them to be Potential Infinite, as Aristotle put it [iv], or as J. S. Mill put it, indefinitely extensible. The addition of 1 is a definite process, but the natural numbers would have no fixed extension if the endless reiteration of the addition of 1 led to growth that could not even in principle be completed. But most of us think of arithmetic as timeless, and the modern view of the natural numbers is that they are (collectively) Actual Infinite, existing as an immutably complete collection, N = {1, 2, 3, …}. Continued...
......[i] Within an axiomatic set theory, Cantor’s Theorem says only that there is no such set of all such sets. For Cantor’s Paradox, see note iii of Divine Attributes.
......[ii] For some well-known problems, see Stewart Shapiro, Thinking about Mathematics: The philosophy of mathematics (Oxford Univ. Press, 2000), pp. 107–289; George Lakoff & Rafael E. Núñez, Where Mathematics Comes From: How the embodied mind brings mathematics into being (Basic Books, 2000), pp. 342–3.
......[iii] For more details, see Christopher Menzel, “God and Mathematical Objects,” in Russell W. Howell & W. James Bradley (eds.), Mathematics in a Postmodern Age: A Christian perspective (Eerdmans, 2001), pp. 65–97 (especially pp. 92–6).
......[iv] To see what Aristotle meant, consider an everlasting fruit-tree. The tree’s endless production of fruit is the ever-incomplete expression of its power to fruit. The total amount of fruit produced is always finite, but always increasing; it is unlimited—is Potential Infinite—because the tree’s power to fruit remains infinite. For more details, see Copan & Craig, Creation out of Nothing, pp. 200–10; Peter Fletcher, “Infinity,” in Dale Jacquette (ed.), Philosophy of Logic (Elsevier, 2007), pp. 523–85.

Wednesday, December 08, 2010

English Numbers

Hartley Slater in The Reasoner 4(12), 175–6, tried to show, from the fact that the number of elements in the empty set is zero, that zero is not, as a matter of English grammar, the empty set—and in general, that numbers are not sets—because we don’t say that the number of elements in the empty set is the empty set. But things aren’t quite that simple.
......To begin with, Slater’s example of zero—which is often defined to be the empty set in pure mathematics—was an unfortunate choice, because mathematicians introduced zero relatively recently. Consequently English remains rather ambivalent about its status. There being no elephants in this room, for example, it’s false that there are a number of elephants here. So from it being true that there are zero elephants here, surface grammar might seem to indicate that zero isn’t even a number (a cardinal number). But zero is of course a number (the number of elephants in this room, the number of elements in the empty set).
......For another example, the numbers one, two, three etc. correspond to the positions first, second, third and so forth. And since no sequence has an element before the first one—that’s what ‘first’ means—so, in that ordinary sense, there’s no zeroth element, and so again, surface grammar seems to indicate that zero isn’t a number (an ordinal number). Nonetheless, there’s a more mathematical sense in which whenever an element is indexed by 0, it’s a zeroth element.
......In many mathematics textbooks there’s a Chapter 0, for example, containing the set-theoretic basics. Of course, such chapters don’t amount to much evidence that mathematicians take numbers to be nothing more than sets. Mathematicians make the standard identification of numbers with sets in order to prove theorems from set-theoretic axioms. They are thereby following in the footsteps of those who did geometry by proving theorems from Euclid’s axioms. And surely few if any geometers thought that there was nothing more to space than Euclid’s axioms. Space was rather the obvious space around us, of which Euclid’s axioms were taken to be true (and obvious enough to be the premises of proofs).
......Now, even if the space that we see around us is Euclidean—having been constructed as such by our brains from our sensory input—it’s surely not unlikely that what Aristotle meant by ‘space’ is non-Euclidean. So, similarly, even if our concept of zero comes (for example) from reifying the definitive property of an absence, it doesn’t follow that it’s impossible that Euler’s ‘0’ referred to an empty set. Indeed, the standard empty set can be an urelement—can be anything that has no members (where membership is an axiomatic primitive)—because its job within standard set theory is simply to have no members, and so in that sense (at least) zero can be an empty set.
......But more to the point, Slater’s argument may beg the question. That’s because if ‘the empty set’ was a definite description of zero then we could say that the number of elements in the empty set is the empty set (for all that we wouldn’t usually). After all, we can say that the number of ones in zero is zero. In general, for natural numbers n, the number of ones in n is n. Perhaps it would be more natural for us to say that two twos are four (for example), and hence that the number of twos in four is two. But such equations all follow from the natural numbers—most obviously those greater than 1—being essentially sums of ones, which seem to be some sort of collection, perhaps not unlike sets of points in that, while their elements are in obvious ways identical, they are distinguished in ways that derive from their origins (as positions in space, in the case of points).
......Two twos are four because any two things plus another two things are four things. And in English, there being a number of things of some kind is just there being some things of that kind. So again, surface grammar indicates that numbers—most obviously those greater than 1—are some sort of collection. And we might expect mathematicians to be the experts on what exactly numbers are. So, since mathematicians prove theorems about numbers from set-theoretic axioms, we’ve some evidence that numbers are sets.
......Still, such evidence is compatible with numbers being axiomatic sets only in a rather abstract way (cf. how the integers with addition are an abelian group). Slater’s argument was based on surface grammar, so it was presumably that numbers are not sets in some more obvious sense. So note that collections in the usual, informal sense can be variable, like a stamp collection, or non-variable, like a chess set. A fundamental question in this area is therefore whether mathematicians have discovered that numbers behave like sets—at least to the extent that the natural numbers are, collectively, non-variable—or whether they’ve just tended to assume that (even though we can’t so easily assume that cardinal or ordinal numbers are non-variable, in light of the famous set-theoretic paradoxes).
......Mathematicians don’t prove the standard Axiom of Infinity—which asserts the existence of a set containing one element for each natural number (amongst other axioms giving such sets the properties one would expect of non-variable collections)—but rather prove theorems from that axiom (along with the others), or work from some other foundation. Philosophical arguments are therefore needed, to assess whether the standard axioms are giving us a scientifically adequate description of the natural numbers or not. But arguments based on surface grammar are unlikely to be of much help in this area. After all, they can’t even show zero to be a number. (For a more apposite sort of argument, see my 2003: ‘Infinite Sequences: Finitist Consequence,’ The British Journal for the Philosophy of Science 54, 591–9, and my 2010: ‘Two Envelopes, two paradoxes,’ The Reasoner 4(5), 74–5.)

Tuesday, December 07, 2010

Possible Worlds

This is the twelfth of 17 posts, which are collectively Eternity, etc.
......As well as all those (correct) statements (see previous post), there are also a lot of truths about other metaphysically possible worlds [i]. Statements of the former kind naturally seem more important than truths about merely possible worlds; and I have not shown that not knowing the former would not make God liable to make mistakes. But balancing the possibility that they do is the possibility that such ignorance is required for our genuine freedom [ii]. And in any case, would it follow from God being maximally knowledgeable—as well as maximally powerful—that He is timeless even if He could only be completely knowledgeable about the future if He was timeless?
......Since the answer is no [iii], let us consider all metaphysically possible worlds, not just this one, under each candidate conception of God. And since God’s omniscience may be less certain than His omnipotence (see section II), let us consider His power as well as His knowledge. And in view of section IV, let us compare libertarian atemporalism with Presentist Open Theism, taking both those conceptions to be prima facie logical possibilities [iv].
......God being possibly timeless means that a world like ours could conceivably be the 4-dimensional creation of a God who transcends its temporal dimension. But it is therefore conceivable that a Presentist God could instantaneously make a 4-dimensional world that is similarly like ours—as it has been so far, and happening also to be that way in the future—but with a fourth dimension substantially unlike Presentist time [v]. So a world as ours would be were God timeless could conceivably be made by Him whether He is timeless or not. And He would be completely knowledgeable about it whether He made it or not.
......Similarly, for any possible spatiotemporal world that a timeless God could make, a Presentist God could conceivably make—and so would know all about—an isomorphic world. But Presentist Open Theism being possible means that this world might have a future that is open, in the sense that there are statements that are neither true nor false but which will be either true or false. And a timeless God could hardly make such a world, because for Him the future has to be completely real.
......Now, atemporalists may not regard that inability as detracting from His omnipotence, whether or not the creator of such a world would be liable to bodge things up in it. But there is also an argument that there are many more metaphysically possible creations if God is able to change (see section VIII), which uses what is shown in the next three posts, that there is no immutably complete totality of all the metaphysically possible whole numbers.
......[i] That is clearly so under Open Theism; and although the deliberate creation of something contingent seems to require several real possibilities to choose between, as well as a single actuality amidst counterfactuals, and hence some sort of change, creation is also taken to be contingent by Mawson, Belief in God, p. 71.
......[ii] Boethius famously argued that our freedom would not be limited by God’s knowledge of what we will be doing were that, not so much foreknowledge, as timeless knowledge (the analogy was with someone knowing what we do, not before we do it, but as we do it). (E.g. see Mawson, “Divine eternity,” pp. 38–40; Sorabji, Time, Creation and the Continuum, pp. 254–6.) Nevertheless, it would still have been true in the past that God knows (timelessly) all about the future. And to see why that might be a problem for libertarian atemporalism, consider a timeless God revealing truths about our future free actions to some of His saints in the remote past (and perhaps even on a distant planet). Is it obvious that our now being responsible for our actions could depend upon Him having done no such thing? (For more details, see Helm, Eternal God, p. 101 ff.)
......[iii] To see why not, consider someone choosing between Open Theism and the hypothesis that she was made by a transcendent computer, which has a complete database on (and complete control over) its creatures’ relatively virtual lives. Only the computer could be completely and infallibly knowledgeable about her whole life. But it would clearly be less knowledgeable (and powerful) than those who might have built an isomorphic computer, and who might have been created by an even more knowledgeable (and powerful) Open God.
......[iv] Cf. Mawson, “Divine eternity,” p. 46 n. 10.
......[v] If this world had been made like that, we should think of Presentism as false, because the analogical—as we should then see it—instant at which God was fully present would then include the past and future. Such a God would be neither everlasting nor timeless (see note vii of Divine Attributes cont.), but is also relatively implausible (see note iii of Omniscience Again cont.).

Sunday, December 05, 2010

Bodging Up cont.

This is the eleventh of 17 posts, which are collectively Eternity, etc.
......How closely would such interventions as we might expect under Open Theism have to resemble Mawson’s scenario (see previous post)? Why, to begin with, should the world’s aggregate happiness have decreased? The immediate consequence of Adolf’s birth was a little more joy in the Hitler household. And surely God would have intended to intervene further, as necessary to ensure that aggregate’s continued increase, if that had been His motivation. (Making such interventions would hardly cause further problems, as simply making evermore planets or heavens of inherently happy animals or angels might suffice.)
......But in any case, a more plausible motivation for an Open divine intervention would be that aggregate’s eventual perfection [i], e.g. by our becoming a communion of saints. Rather than the Open God intervening to ensure a baby’s birth [ii], He would more plausibly have answered Mrs. Hitler’s prayers in order to help her to relate to Him more fully [iii]. And had He done that, then the satisfaction of His desire would hardly have depended upon how her child grew up. It would have depended upon her free choices—to some extent (He would definitely have so helped her)—but that amounting to luck is generally rejected by libertarians [iv]. Furthermore, even if Mrs. Hitler did not respond by becoming a saint, the Open God could surely try again, and again. And His attempts might become irresistible, as Mrs. Hitler wised up, or perhaps she might become very undeserving.
......So in short, Mawson’s scenario did not show that the satisfaction of the Open God’s most beneficial desires—perhaps that everyone (who is not too undeserving) should end up somewhere heavenly forever—could not be inevitable. So we are left with no reason why we should think of the Open God (of any variety) as a “well-intentioned buffoon[v], rather than as Jesus [vi], and hence no reason why God should know all about the future (cf. end of section IV). So although there are statements about the future that would not be known by the Presentist Open God despite them being in a sense correct, that sense has not been shown to be significant enough to obstruct divine omniscience.
......[i] Keith Ward, Divine Action (Flame, 1990), pp. 134–9.
......[ii] Swinburne, Is There a God? pp. 114–5.
......[iii] Robert M. Adams, “Theodicy and Divine Intervention,” in Thomas F. Tracy (ed.), The God Who Acts: Philosophical and theological explorations (Penn. State Univ. Press, 1994), pp. 31–40.
......[iv] Timothy O’Connor, “Is It All Just a Matter of Luck?” Philosophical Explorations 10 (2007): 157–61.
......[v] Mawson, “Divine eternity,” p. 48.
......[vi] Richard Swinburne, Was Jesus God? (Oxford Univ. Press, 2008). If Jesus is, like us, a continuant, then it’s hard to see how he could be, not just the signature, but the identity of a timeless God. But clearly an everlasting God could incarnate as fully human, e.g. if we are essentially spirits that produce human minds because we animate human brains.

Friday, December 03, 2010

Bodging Up

This is the tenth of 17 posts, which are collectively Eternity, etc.
......Mawson’s main argument had the following preamble [i]. Early in 1936 it was, according to Mawson, vastly improbable that that year’s Man of the Year in Time magazine would be widely regarded ten years later as the most evil man ever. Mawson’s point seemed to be that if a temporal God would have known, early in 1936, that such a change was very unlikely, and if to find something so unlikely is to believe it false, which follows from Swinburne’s definition of “belief” according to Mawson [ii], then such a God would have believed that such would not happen, incorrectly (that man being Adolf Hitler).
......However, while the man in the street of early 1936 may well have found such a change unthinkable, surely the perfectly aware sustainer of the whole world might have known better. And in any case, while we do—and often should—believe things that, upon reflection, we would only regard as highly probable, the beliefs held by the Open God are presumably held infallibly. That latter observation was acknowledged by Mawson [iii], so let us now look at his main argument, which revolved around the following scenario [iv].
......The Open God answers the prayers of Mrs. Hitler by saving her unborn baby from a likely miscarriage, doing so because He wants to increase the aggregate happiness of the world. He does not know that her baby, Adolf, will eventually cause that aggregate to decrease. On the contrary, that Adolf Hitler will eventually produce terrible harm is known by Him to be fantastically unlikely. Still, terrible harm is what happened, and so His intervention did not result in greater happiness. And even if it had, He would just have got lucky.
......Of course if, were God timeless, so evil a man would never have been born, then history would be some evidence that God is not timeless [v]. But the idea behind Mawson’s scenario was that something like it must be possible under Open Theism, because the Open “God cannot will Himself to do anything under a description the truth of which depends on future free actions[vi]. He can in the case of His own actions, however, via His constancy (see section IV), and in my next post I wonder why He would need to in our case.
......[i] Mawson, “Divine eternity,” p. 45.
......[ii] One’s belief that p amounts to a belief that p is more likely than q for some q, according to Richard Swinburne, Epistemic Justification (Clarendon Press, 2001), pp. 34–6. Swinburne was examining the justification of human beliefs (of that form) however, not describing God’s beliefs (as such). Mawson observed that we tend to believe that a tossed coin won’t land on its edge, even though it might. But the Lottery Paradox—I believe, of each ticket, that it won’t win, so (logically) I should believe that none will, whereas I know that one will—was addressed by Swinburne; ibid, pp. 37–8. And a natural resolution is to describe my belief that it probably won’t win at least that precisely. And in general, logical reasoning seems to involve natural language clarification procedures whose aim is an adequate bivalence. E.g. the ambiguity mentioned in note 31 might be clarified thus: Either something will definitely happen, or else it is, if not impossible, merely possible.
......[iii] Mawson, “Divine eternity,” p. 45.
......[iv] Ibid, p. 47.
......[v] Indeed, a world in which everyone always made good choices is a prima facie logical possibility, so one might wonder why a timeless God would not have created such a world, were He perfectly beneficent. For more on the atemporalist free will defense, see Mawson, Belief in God, pp. 198–271. For more discussion of theodicies, see Omniscience and the Odyssey Theodicy.
......[vi] Mawson, “Divine eternity,” p. 47.

Wednesday, December 01, 2010

Theistic Presentism again

This is the ninth of 17 posts, which are collectively Eternity, etc.
......The collapse view of quantum mechanics (see previous post, also Modern Physical Probability) can help us to understand human agency [i]. But if wave-functions do not collapse then you would now be about to do everything that it was physically possible for you to do [ii], by splitting into lots of you (whereas you are a responsible agent). Furthermore, Perfect Being Theists—unlike most physicists—would not regard the question of what (or who) could collapse the universal wave-function as problematic.
......And our third reason—that there would be nothing to make all truths about the past true under Presentism—is similarly trivial under Perfect Being Theism, because God’s memory might serve as such a truth-maker [iii]. Conversely, Presentism can support Perfect Being Theism. E.g. it allows such Theists to “accept neither the timelessness nor the temporality of the being of God[iv], insofar as “temporality” means being inside the temporal dimension or being dependent upon temporal constraints [v]. Furthermore, Presentism allows perfect goodness to become omnipresent [vi].
......Presentism is not, then, obviously implausible under Perfect Being Theism. So although there is a sense in which your earlier belief about reading on (or not) may have been correct—perhaps it seemed so with hindsight—it does not follow that an omniscient God would have known what you were going to do. Under Presentism, such a belief would not have described what was originally real—the two possibilities—accurately enough for it to have been part of divine omniscience.
......Nevertheless, it seems to me that if a God who does not know all about how things will be would be liable to bodge things up, as Mawson’s main argument claims [vii], then an omnipotent being would know them.
......[i] Henry P. Stapp, “Quantum Interactive Dualism: An alternative to materialism,” Journal of Consciousness Studies 12 (2005): 43–58.
......[ii] Penrose, The Road to Reality, pp. 783–4, pp. 806–9.
......[iii] Alan R. Rhoda, “Presentism, Truthmakers, and God,” Pacific Philosophical Quarterly 90 (2009): 41–62. Incidentally, even the Presentist past is completely real in the sense that all statements about it have definite truth-values. Indeed, it is at least partially real in the more ontological sense that some continuants that did exist still do.
......[iv] Colin E. Gunton, The Triune Creator: A historical and systematic study (Eerdmans, 1998), p. 92.
......[v] With different senses—e.g. those of Padgett, God, Eternity and the Nature of Time—Presentism would allow us to accept both terms consistently.
......[vi] Paul Copan & William Lane Craig, Creation out of Nothing: A biblical, philosophical, and scientific exploration (Apollos, 2004), p. 162 n. 29.
......[vii] Mawson, “Divine eternity,” pp. 45–9.