A Conceptualist Argument for the Existence of God

Over two decades ago Thomas Morris and Christopher Menzel published their widely influential paper “Absolute Creation,” provoking a brainstorm of thought among philosophers on the relationship between God and abstract objects.1 Extending as far back as Plato (Sophist 246a-c), the topic overall is not by any means new. Nonetheless, fresh insights on the nature of abstracta have been breathed into the metaphysics of theism, providing new resources for the project of natural theology—in particular the so-called conceptualist argument for God’s existence. Alvin Plantinga foresees the argument’s barebones:

Suppose you find yourself convinced that (1) there are propositions, properties, and sets, (2) that the causal requirement is indeed true [i.e., a causal relationship between an external object of knowledge and the knower], and (3) that (due to excessive number or excessive complexity or excessive size) propositions, properties, and sets can’t be human thoughts, concepts, and collections. Then you have the materials for a theistic argument.2

Although Plantinga mentions this argument on a number of occasions, he unfortunately never elaborates much more than the above. I hope to sketch, in slightly more detail, a possible formulation of the conceptualist argument, or at least show in the endnotes that there are more than enough resources available in contemporary philosophy to furnish one. In so doing I also hope to show how the existence and nature of abstract objects bodes trouble for naturalism.

First and foremost, what are abstract objects and why bother arguing over their existence? Relevant literature on abstract objects can be divided into two categories: epistemological and metaphysical.3 By focusing only on a single object of thought and disregarding areas of peripheral awareness, the former category explores the notion of “abstract object” as merely an object of mental abstraction. The latter category, on the other hand, seeks to explore issues of ontological significance—do abstract objects exist? If so, what are they and what explains their existence? The two categories often overlap, but most of what follows will deal only with the metaphysical sense, focusing in particular on the ontological status of abstract objects.

Perhaps the best way to understand what an abstract object is is to provide examples rather than state necessary and sufficient conditions for their existence. Entities such as propositions, laws, relations, values, universals, logical and mathematical objects (numbers, sets, lines, shapes), fictional objects (characters, storylines, fantasy worlds), pieces of art, possible worlds, etc. are all commonly cited as abstracta. Abstract objects are typically contrasted with concrete objects, or substances, such as tables, trees, baseballs, and, if they exist, immaterial persons such as Angels and God. Gideon Rosen4 contrasts paradigmatic cases of each:

ABSTRACTA CONCRETA
   
Classes Stars
Propositions Protons
Concepts The electromagnetic field
The letter A Stanford University
Dante’s Inferno James Joyce’s Copy of Dante’s Inferno

Even though clear examples of abstracta are not hard to come by, a good analytic definition of ‘abstract object’ is. Several criteria have been offered, but three stand out. First, abstracta are causally impotent. The number two, for example, cannot cause any effects. They are essentially acausal. This first criterion entails the second: abstract objects are unextended and immaterial. Philosopher Corey Washington captures this nicely when he asks, “When’s the last time you bumped into the number one? When’s the last time you slipped on the concept of truth? Or saw a justice sitting by the side of the road?”5 It is worth mentioning that this criterion doesn’t necessarily mean abstracta are non-spatiotemporal. For example, if the truth-value of tensed propositions are relative to when they’re uttered, this implies propositions are in some sense temporal. Or again, it is hard to make sense of a property (e.g. a universal) being exemplified by two different particulars without using spatial referents. But this suggests that properties are, at least in some sense, spatial. To avoid the bottomless issues relevant to linguistics and universals, I prefer to identify abstract objects simply as immaterial and unextended rather than non-spatiotemporal.

Lastly, at least some abstract objects are necessarily existent. That is to say, some abstracta exist in all possible worlds. For example, propositions which are broadly logically necessary, such as “whatever has a shape has a size” seem to exist and be true in all possible worlds. Even propositions whose truth-values are subject to change seem necessary. For example, the proposition “there are human beings” could have been false, but it is hard to see how it could have been non-existent. Moreover, if propositions are necessary, then numbers can plausibly be taken as necessary also. Neil Tennant has persuasively argued that certain necessary propositions such as “There are n Fs” and “The number of Fs is n” incur ontological commitment to at least one number; namely, 0. If 0 exists, says Tennant, it must exist in all possible worlds.6 Why does Tennant think 0 must exist in all possible worlds? Briefly ponder the following proposition:

(P) There is no possible world such that there are no things that are not self-identical

It follows, according to Tennant, that (P) entails the existence of 0 as the number of things that are not self-identical. Indeed, the necessity of a whole host of abstract objects seems entailed by a similar concept; the impossibility of a null-world.7 To see why, Thomas Morris asks us to try to imagine a world in which nothing exists:

If there could be such a world, it would be a world, or state of affairs, in which the number of things that exist would be properly numbered by the number 0. The number 0 would be instantiated, or exemplified, precisely by the absence of anything else. But then it would have to exist to be exemplified, or to number the things that exist.8

Now if the number 0 exists in all possible worlds, then there is at least one number in all possible worlds, which implies the number 1 also exists in all possible worlds. In fact, that the number 1 exists necessarily seems to be true a priori of possible worlds, in that it will always designate the number corresponding to whichever world is possibly actual (i.e., the number 1 exists in the actual world because it refers to the number of worlds that are actual). But if that’s the case, there are at least two numbers in all possible worlds, 0 and 1, which implies there are three: 0, 1, and 2. And so on. So long as numbers necessarily exist, we can say properties and relations necessarily exist, too. For numbers have properties such as being even and stand in relations to other numbers, such as being more than. Morris summarizes:

The net result of such reasoning is that it is plausible to suppose that such abstract objects as numbers, properties, and propositions necessarily exist as a sort of formal framework of reality, providing necessary conditions for the possibility of any world.9

Why bother arguing over the existence of such entities? J. P. Moreland has pointed out that much more goes into the success of a worldview than mere logical consistency. He observes “it sometimes happens that some metaphysical commitment, though logically consistent in a strict sense with competing, broad world views is, nevertheless, more plausible and at home in one rival compared with the other.”10 Abstract objects, arguably, are a prime example of such a commitment. More to the point, abstract objects are more at home in a theistic weltanschauung than a naturalistic one.

But what sort of naturalism does abstracta supposedly rub against? There are of course varying degrees of naturalism, some strong and some modest. Stronger varieties, such as those committed to a hard-core materialism or only to objects existing within space and time are not just uncomfortable bedfellows of abstracta, they do not share the same bed at all. The vindication of one is the falsification of the other. Hence Howard Robinson’s remark that “materialist theories are incompatible with realist theories…the tie between nominalism and materialism is an ancient one.”11 But more sophisticated versions of naturalism try to make room for abstracta. Take Graham Oppy’s, for example:

(N) a. There are no entities which are causally related to things hereabouts but which are not spatially related to things hereabouts (hence: no souls, no spooks, no entelechies, no gods), and b. there is no sufficiently good reason for believing in the kinds of entities which are denied to exist in a.12

There is no prima facie inconsistency between (N) and the existence of abstract objects. However, there are good reasons for thinking there is at least tension between the two—a burden theism doesn’t bear. In other words, even if one could draft a realist ontology of abstracta compatible with naturalism, a better, more-at-home account can be offered by theism. So what accounts are there to consider and into which worldview does the preferred account most comfortably fit?

According to W. V. O. Quine there have traditionally been three main positions regarding the ontological status of abstract objects: nominalism, platonism, and conceptualism.13 Each of these three positions can take different forms, but the trichotomy is sufficiently exhaustive and so should be beyond dispute. Nominalism, as previously mentioned, denies the existence of abstract objects, or at least eschews ontological commitment to them. On the other hand, platonism affirms their existence, though as independently existing realities. Abstract objects exist inexplicably a se; they’re just sort of “out there” as part of the necessary furniture of the universe. Conceptualism can be thought of as a middle ground between nominalism and platonism. A conceptualist would say that abstract objects indeed exist but are better understood as grounded in the mind of an agent. We can represent each view respectively in the premise

(1) Abstract objects either a. do not exist, b. are independently existing realities, or c. exist as mental concepts

Given (1), we can begin to outline a conceptualist argument for theism with

(2) Abstract objects a. exist, and b. are not independently existing realities

Establishing (2) can be achieved by either offering positive arguments for it or by refuting its negations, nomonalism (1a) and platonism (1b). What follows is a brief sketch of some of the arguments and strategies one might using on behalf of (2). Taking their negations in turn, why think

(2a) Abstract objects exist

is true? The chief considerations here are indispensability arguments. Indispensability arguments attempt to show that abstract objects are indispensable to our experiential framework. Such arguments work as a sort of reductio strategy against nominalism, or (1a). Whether it be in mathematics, nomology, linguistics, or some other essential framework by which we experience and explain the world, the dispensing of abstract objects would have disastrous consequences.14 Consider propositions. According to truth-maker theory, truth obtains when some truth-bearer ‘captures’ or ‘corresponds to’ some state of affairs. For example, if Mary is watching television then what makes that true is some truth-maker (the state of affairs consisting in Mary watching television) to which some truth-bearer relates. Several candidates have been offered for appropriate truth-bearers, the most plausible being propositions (in this case the proposition “Mary is watching television”). The proposition“Mary is watching television” is true if and only if Mary is watching television. But if we dispense of propositions as real entities, as nominalism says we should, then we are left with nothing in which to ground truth. But this is absurd.14 It is up to the nominalist to find an escape route either by rejecting truth-maker theory or by offering an explanation of how to preserve truth despite this problem.

Numbers and other mathematical entities such as sets factor most heavily into indispensability arguments. It is on this point that Quine, who has been called “the leading advocate of a thoroughgoing form of naturalism,”16 abandoned his nominalist programme and embraced the existence of abstracta as indispensable to our best scientific theories. Quine, along with Hilary Putnam (who is himself no friend of realist commitments), later advanced what has become one of the most influential indispensability arguments.17 The idea of their argument is roughly the following:

(2a.1) We ought to have ontological commitment to entities that are indispensable to our experiential framework
(2a.2) Abstract entities are indispensable to our experiential framework
(2a.3) We ought to have ontological commitment to abstract entities

Though this argument has been influential, both of its premises have been called into question. Hartry Field, for example, argues that mathematical objects are ontologically dispensable, in a strict sense, but are nonetheless indispensable to our experiential framework as a consistent body of principles. Insistent on having it both ways, Field defends a version of nominalism called fictionalism, where abstract objects are more-or-less useful fictions.18 Propositions do not really have truth values, but are rather judged useful via consistency within a system or theory. If Field is right, then not only is (2a.1) false, but (2a) would be as well. Fictionalism may seem radical at first, but it should not be too quickly dismissed. For one thing, fictionalism arguably represents the most plausible form of nominalism to date. Moreover, fictionalism may actually prove to be an attractive option for theism in lieu of reconciling necessary abstracta with God’s aseity.19

These virtues considered, however, fictionalism is perhaps too radical. The most jarring problem is its renunciation of obviously true descriptive states of affairs the truth of which seem to be grounded in certain abstracta (“2 + 2 = 4” or “two entities x and y can have the same shape and size”). For according to fictionalism, such abstracta are merely useful fictions. Indeed, it will be hard to take seriously a view which maintains that propositions such as “4 is divisible by 2”, “Jones believes that [some proposition p]”, or “triangles have three sides” are not just false, but literally truth valueless! Consequently, the truth value of the proposition “something cannot both exist and not exist at the same time” is ontologically equivalent to the truth value of the proposition “2 + 2 = 5”. What are usually taken to be necessarily true propositions such as the former assume no privileged status over and above necessarily false ones such as the latter. So long as they operate consistently within their respective systems, even contradictory propositions could be said to accurately describe the way the world is.

Second, fictionalism may itself fall victim to indispensability arguments. Take Tennant’s argument for the necessary existence of numbers (which can also serve as another argument for (2a)). Even fictionalist accounts cannot avoid quantifying over entities in their reductive analyses, which Tennet says commits them to the existence of numbers. Tennant writes:

Once the language of arithmetic has been adopted…one incurs commitments. These are, first, to the number zero, as the number of non-self-identical things; and, thereafter, to each natural number n in turn, as the number of numbers preceding n. In any world in which one uses a rich enough first-order language—with the identity predicate, the existential quantifier, negation and the numerical term-forming operator #—one has (on reflection) to acknowledge the existence of zero.20

If Tennet’s argument passes, then fictionalism must be false. His argument can be outlined as follows:

(2a.4) There is no possible world such that there are no things that are not self-identical
(2a.5) 0 is the number of such things that are not self-identical
(2a.6) Therefore, 0 exists in all possible worlds

Does Tennet’s argument pass? (2a.4) is self-evidently true, so what support does Tennant give for (2a.5), the argument’s main premise? On the face of it, (2a.5) seems to beg the question against the nominalist by introducing the existence of a number into the premise. But what else could the number zero be replaced by? In Tennet’s words, “one cannot be thinking of ‘0’ as a term which might denote some person, or physical object.” He concludes

Why can one maintain [(2a.5)]? One can only consider the question whether 0 exists by framing the thought x(x = 0) in a language one of whose sentences can be thus regimented. Moreover, one has to be thinking of 0 as a number, that is, thinking of “0″ as a term which, if it denotes anything at all, denotes a number.21

The insight behind Tennant’s argument can be formalized into an analytic proof.22 Many philosophers have pointed out that, contra Kant, existential statements are not always synthetic. An analytic proof for the existence of 0 would run thusly:

(2a.7) a exists ≡ (∃y) a=y
(2a.8) (number x) F(x)=0 ≡df ¬(∃x)F(x)
(2a.9) F(x) is xx

Statements entailing the existence of analytic truths such as (2a.4) entail there is something ‘=’ to nothing. But obviously nothing does not exist. So what is there to be equal to? Well, the number 0 itself. Therefore,

(2a.10) (∃y) y=0

It is plausible to think, then, that the number 0 exists and is indispensable to our experiential framework. Even if partial fictionalism is granted (e.g., applicable only to mathematical entities), the fictionalist still bears the burden of extending his theory to incorporate other abstracta if realism is to be avoided. Indispensability arguments such as these prove to be a powerful force against nominalist theories.

What else can be said on behalf of (2a)? Perhaps a brief appeal to intuition favoring (2a) can be made. Upon considering the competing realist and nominalist theories, I get a feeling of unreality about the whole debate. The existence of abstracta is so obvious that realist theories often appear to be guilty of proving the obvious via the less obvious and the complex reductive analyses proffered by nominalist theories are so counterintuitive as to be rejected outright. I think it is the overwhelming consensus that abstract objects do exist, save only the few ever-present philosophers who voice their denials louder than their competitor’s arguments and the crowd’s intuitions. At any rate, (2a) appears to be on solid ground. But what about (2b)?

(2b) Abstract objects are not independently existing realities

A good argument for (2b) often overlooked is that it commits us to the existence of an actual infinite (0). If abstract objects are independently existing realities, then each is a definite and discrete entity. But there are an infinite number of abstract objects. Just think of natural number series alone. Throw in sets and the number of propositions expressing possible states of affairs and we’re compiling infinitude on top of infinitude. The problem is that an actual infinite number of things generates absurdities. To simply illustrate, imagine we have a library with an actual infinite number of books. Wesley Morriston concisely summarizes the problem:

Let m = the number of books in our infinite library, n = the number of odd-numbered books, and o the number of books numbered 4 or higher…
  (mn) = infinity, whereas (mo) = 4.
But,
  n = o (since both n and o are infinite)
It follows that we get inconsistent results subtracting the same number from m.23

The conclusion is that we obviously couldn’t perform such operations in the actual world, so an actual infinite cannot exist. Ergo, platonism, or (1b), must be false. The argument could be outlined:

(2b.7) An actual infinite cannot exist
(2b.8) If (1b) is true, then there is an actual infinite number of abstracta
(2b.9) Therefore, (1b) is false

The great German mathematician Abraham Robinson reasoned similarly, confessing “I cannot imagine that I shall ever return to the creed of the true Platonist, who sees the world of the actual infinite spread out before him and believes that he can comprehend the incomprehensible.”24

What else might be wrong with (1b)? No doubt the most frequent argument lodged against (1b) is known as the epistemological objection.25 If abstracta are independently existing realities as (1b) holds and are also causally effete (as no philosopher to my knowledge denies), then it would be impossible for us to have knowledge of them. To have knowledge of something independent of us entails a relation of sorts; one where information about some object of knowledge can pass from it to us or vice versa. But how can this be if abstracta exist a se and unextended? (1b) therefore renders abstracta epistemologically inaccessible. Where O is some object of knowledge (say, an abstract object), the argument can be summarized:

(2b.1) If O is external to S, S can have knowledge of O only if there is some causal relation R between S and O
(2b.2) O is such that it cannot enter R
(2b.3) If (1b) is true, then O is external to S
(2b.4) Therefore if (1b) is true, then S cannot have knowledge of O
(2b.5) But S has knowledge of O
(2b.6) Therefore, (1b) is false

The best escape route for the platonist would be to refute (2b.1) by offering an account of how we can have knowledge of O that does not involve R. “After all,” the platonist might insist, “the very origin of the term ‘abstract object’ comes from their being ‘abstracted’ from our concepts.” Two replies can be made. First, so long as these “abstractions” remain derivatives of our concepts and do not somehow become independent, free-floating entities, the conceptualist will welcome this view as being similar to his own. Second, so reducing abstract objects to epistemic affairs only is a common nominalist strategy entirely inconsistent with plationism. So what could the platonist suggest as a non-causal relation between us and abstracta? Some form of intuition will be the most likely candidate here, serving to grant us access to abstracta in much the same way perception does concreta. Maybe our minds are such that we simply grasp or intuit the existence of abstracta in the appropriate circumstances. But this route fares no better. The chief difficulty for this approach, according to Alvin Goldman and Joel Pust, is that intuition doesn’t seem capable of delivering the information needed, and so can’t be the source of our knowledge of abstracta. They ask

Is there any reason to suppose that intuitions could be reliable indicators of a universal’s positive and negative instances (even under favorable circumstances)? The problem is the apparent “distance” and “remoteness” between intuitions, which are dated mental states, and a nonphysical, extra-mental, extra-temporal entity. How could the former be reliable indicators of the properties of the latter?26

It will have to take more than intuition to cross the epistemological gap between us and abstracta. To make matters worse, no other candidates appear even remotely plausible. Thus Moreland’s comment that “[such] an attempt to account for knowledge of abstract objects will have to be given in terms of some mysterious, even mystical, aphysical ‘grasping connection,’” which is philosophically repugnant, especially for a contender for (N) (no souls, no spooks, no entelechies, no gods).27 To this effect others have mentioned a sort of “argument from queerness” against platonism. Platonic entities are so foreign to and different from everything else in a naturalist ontology that special pleading against non-reductive theories threatens. In an oft-quoted passage, the late atheist J. L. Mackie wrote:

If there were [abstract objects], then they would be entities or qualities or relations of a very strange sort, utterly different from anything else in the universe. Correspondingly, if we were aware of them, it would have to be by some special faculty of … perception or intuition, utterly different from our ordinary ways of knowing everything else.28

Platonist theories of free-floating immaterial objects knowable only via mystical experience prove adverse and excessive to (N).

Moreover, if there are sufficiently strong arguments in favor of conceptualism, they could count as evidence against both (1a) and (1b). Unfortunately most philosophers see no arguments here strong enough to sufficiently eliminate an alternative view such as platonism. However, this point is not entirely lost for at least two reasons. First, it can be argued that conceptualism has more scope than (1a) and (1b), as it nicely accounts for precisely the desiderata they are at odds with (more on this later). Second, it is the testimony of some philosophers that conceptualism has more intuitive support than its alternatives, or at least enough to make it rationally acceptable in the absence of compelling pros and cons.29 I’ve found this intuition shared by laypeople also. In my discussions about the ontological status of abstract objects with people wholly unfamiliar with philosophical parlance, I often receive the eager opinion that “they’re like, in your brain or mind” before I even mention conceptualism as an option. One instance was even with a Jr. High student! Plantinga gives a similar report: “It also seems plausible to think of numbers as dependent upon or even constituted by intellectual activity; indeed, students always seem to think of them as ‘ideas’ or ‘concepts’, as dependent, somehow, upon our intellectual activity.”30 Admittedly, this doesn’t prove that conceptualism is true, but I certainly think it is a consideration in its favor and hence out of favor with (1a) and (1b).

Furthermore, in a situation like this we might be able to justify our inference (to conceptualism) by appealing to the Principle of Credulity expounded by Richard Swinburne: certain beliefs with which agents find themselves are—in the absence of counter evidence—probably true; the mere fact that you have a belief is grounds for believing it.31 These points notwithstanding, there just might be a positive argument for conceptualism after all. Recent thought on the nature of intentionality may provide us one. Intentionality is the property of being of or about something (of-ness or about-ness). Many abstracta seem to be characterized by intentionality. For example, thoughts, as propositions, are of and about things. Possible worlds and (and perhaps states of affairs) consist of and are about descriptions of reality. The curious thing about intentionality is that it is characteristic exclusively of mental phenomena, so much so that it has been dubbed ‘the mark of the mental.’32 The implication is obvious: if intentionality is characteristic exclusively of mental phenomena and some abstracta have intentionality, then by modus ponens abstracta are characteristic of mental phenomena. This argument has the potential to be developed in much more detail (and, to some extent, already has33). What has been mentioned so far suffices to show how (2) could be met. From (1) and (2) it follows that

(3) Abstract objects exist and are mental concepts

So far any philosopher could accept our running argument regardless of whether God plays a role in their metaphysic. That is not to say, however, that (3) is metaphysically-neutral. Ardent naturalist David Armstrong, a main voice in the realism/anti-realism debate, is not dull to this point:

I suppose that if the principles involved [in analyzing and explaining reality] were completely different from the current principles of physics, in particular if they involved appeal to mental entities, such as purposes, we might then count the analysis as a falsification of Naturalism.34

Atheist philosopher Kai Neilson agrees:

Naturalism denies that there are any spiritual or supernatural realities. There are, that is, no purely mental substances and there are no supernatural realities transcendent to the world; or at least we have no sound grounds for believing that there are such realities for perhaps even for believing that there could be such realities. It is the view that anything that exists is ultimately made up of physical components. … There are no purely mental realities in a naturalistic account of the world.35

Armstrong’s and Neilson’s characterization of naturalism is clearly incompatible with (3). The defender of (N), on the other hand, can yet consistently maintain (3), though not without considerable tension (for reasons outlined above). But here the logical consistency between (N) and (3) ends by adding the explicitly theistic premise

(4) If abstract objects exist and are mental concepts, they must be concepts of a necessary, omniscient mind

What can we adduce for (4)? Why do we need a necessary, omniscient mind? The school of thought known for grounding abstracta in human minds is known as psychologism, traditionally associated with John Stuart Mill’s System of Logic. Psychologism, however, infamously stands as one of the most thoroughly refuted views in this field, beginning with the analyses of F. G. Frege and Edmund Husserl.36 Among phychologism’s insufferable difficulties is its inability to account for the necessity and sheer volume of abstratca. It would do precious little to explain the necessity of one entity in terms of another entity that is itself not necessary. Human minds are wholly inadequate for the task. Say some abstract object O is the concept of some human mind at time t1. Surely there must have been times titn< t1 such that there were no human minds that had O as a concept. It would be less congenial to say that at t1 O came into existence than to say O must have existed as a concept of a necessary mind during titn. This argument runs backwards as well: given their voluminous and complex nature, there must be abstract objects that have not yet nor will ever be concepts in human minds.

Moreover, it’s a bit too optimistic (even for the philosopher!) to think, as Abraham Robinson remarked, that we can “comprehend the incomprehensible,” such as the hierarchal order of sets or some immensely complex mathematical formula. The existence and truth of abstracta such as these is obviously independent of our cognizing them. Quentin Smith, for example, imagines an infinitely complex conjunct of all true propositions as itself a single proposition. Such a proposition, Smith argues, could only be the accusative of an omniscient mind. In addition to propositions, Plantinga suggests reasons for thinking the same is true of sets, numbers, and properties as well.37 Alternatively, It is not hard to see how the theist’s version of conceptualism would be immune to the standard objections to psychologism. It is also not hard to see how the theist’s version of conceptualism would be a hand-in-glove fit with precisely the desiderata these other theories of abstracta are at odds with: knowability despite causal impotence, unextendedness, and necessity. But if (3) is true and other theories like psychologism that try to “drag everything down to earth out of heaven and the unseen” cannot bear the argument’s weight, then plausibly

(5) Therefore, a necessary, omniscient mind exists

which turn entails ~(N). But the theist shouldn’t be too hasty. It remains to be shown that the conclusion is coherent. If demonstrated that (4), for example, is in some way incoherent, the conceptualist argument would be unsound (here one thinks of Morris and Menzel’s theistic activism). No doubt the theist will also have to spell out in much more detail the version of conceptualism needed. For example, even if the theist establishes conceptualism, it still needs to be said how abstract objects become concepts in human hinds. One can predict the Christian’s appeal to the imago Dei within us, the blueprint of which includes knowledge of abstracta. Moreover, the inherent fallibility and error in our best efforts at reasoning with abstracta would not be surprising given the cognitive consequences of sin. Robert Adams entertains these possibilities:

If God of his very nature knows the necessary truths, and if he has created us, he could have constructed us in such a way that we would at least commonly recognize necessary truths as necessary. In this way there would be a causal connection between what is necessarily true about real objects and our believing it to be necessarily true about them. It would not be an incredible accident or an inexplicable mystery that our beliefs agreed with the objects in this.38

Forthcoming efforts on these and similar issues are already underway.39 At any rate, the existence and nature of abstract objects might prove to be either at odds or flatly inconsistence with naturalism whereas no such difficulties loom for theism. Indeed, abstract objects may provide the tools for a promising theistic argument. I therefore agree with eminent atheist philosopher Quentin Smith that “the conceptualist argument … may be interpreted as contributing to establishing the rational acceptability of theism.”40


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  1. Thomas Morris and Christopher Menzel, “Absolute Creation” American Philosophical Quarterly 23 (1986): 353-362. Several other relevant pieces are Menzel’s “Theism, Platonism and the Metaphysics of Mathematics” Faith and Philosophy 4/4 (1987): 365-382; “God and Mathematical Objects” in eds. Russell W. Howell and W. James Bradely, Mathematics in a Postmodern Age (Eerdmans, 2001). Brian Leftow, “A Leibnizian Cosmological Argument” Philosophical Studies 57 (1989): 135-155; “Is God an Abstract Object?” Noûs 24 (1990): 591-598; “God and Abstract Entities” Faith and Philosophy 7/2 (1990): 193-217. Scott A. Davison, “Could Abstract Objects Depend Upon God?” Religious Studies 27 (1991): 485-497. Matthew Davidson, “A Demonstration Against Theistic ActivismReligious Studies 35 (1999): 277-290. Michael Bergmann and Jeffrey Brower, “A Theistic Argument Against Platonism (and in Support of Truthmakers and Divine Simplicity)” Oxford Studies in Metaphysics 2 (2006): 357-386. No doubt a primary influence behind all of this was Platninga’s 1980 Aquinas Lecture, “Does God Have A Nature?”.
  2. Alvin Plantinga, Warrant and Proper Function (Oxford, 1993), 121 fn. 25. See also his “How to be an Anti-RealistAPA Proceedings and Addresses (1982), 47-70, where a version of conceptualism is defended in some detail. Realism is often called platonism and anti-realism, nominalism. But some realist theories demur with platonism and some nominalist theories are not quite anti-realist. Moreover, I will use the terms “entity” and “object” interchangeably, though mindful of E. J. Lowe’s useful distinction in “The Metaphysics of Abstract Objects” Journal of Philosophy 92/10 (1995): 509-524. Furthermore, I leave the notion of ‘concept’ undefined, though aware of the desperate need of clarifying its relation to abstracta.
  3. See Ted Sider’s impressive though incomplete “Bibliography on Abstract Entities”.
  4. Gideon Rosen, “Abstract Objects” in Stanford Encyclopedia of Philosophy. See also the entry “God and Other Necessary Beings” by Matthew Davidson.
  5. Does God Exist?” a debate between William Lane Craig and Corey G. Washington on February 9, 1995 at the University of Washington.
  6. See Neil Tennant, “On the Necessary Existence of Numbers” Noûs (1997): 321. George Bealer, “UniversalsJournal of Philosophy 90/1 (1993): 5-32. Sets with contingent members seem to be an exception to the criterion of necessary existence.
  7. On the notion of a null-world and God’s relationshipto propositions, see the excellent exchange between Richard Davis and Brian Leftow in Religious Studies 42 (2006). See especially Davis’s article, “God and Counterpossibles, 371—391.
  8. Thomas V. Morris, Our Idea of God (InterVarsity, 1991), 109
  9. Ibid, 110.
  10. J. P. Moreland, “Naturalism and Libertarian Agency” Philosophy and Theology 10/2 (1997): 353.
  11. Howard Robinson, Matter and Sense (Cambridge, 1982), 50. On the incompatibility of realism and naturalism, see J. P. Moreland, “Naturalism and the Ontological Status of Properties”, in ed. William Lane Craig and J. P. Moreland, Naturalism: A Critical Analysis (Routledge, 2000), 67-109.
  12. See his Review of: Naturalism: A Critical Analysis. I specifically use Oppy’s definition to preempt the sort of hand waving he does in his review, dismissing the entire book on the basis that the contributors didn’t define naturalism to his liking!
  13. W. V. Quine, “On What There Is”, in ed. Stephen Hales, Metaphysics: Contemporary Readings (Wadsworth, 1999), 212. What Quine calls realism I’m calling platonism.
  14. See Torsten Wilholt, “Think about the Consequences! Nominalism and the Argument from the Philosophy of Logic” Dialectica 60/2 (2006): 115–133.
  15. Surprisingly Josh Parsons does not consider this in his “There is no ‘Truthmaker’ Argument against NominalismAustralasian Journal of Philosophy 77 (1999): 325-334.
  16. Alex Orenstein, W. V. Quine (Princeton, 2002), 1; 55-56. Contrast Quine’s earlier views expressed in his paper “Steps Toward a Constructive Nominalism”, coauthored with Nelson Goodman: “We do not believe in abstract entities. No one supposes that abstract entities—classes, relations, properties, etc.—exist in space-time; but we mean more than this. We renounce them altogether…Any system that countenances abstract entities we deem unsatisfactory as a final philosophy.” Idem., 55.
  17. On Quine’s (and Putnam’s) indispensability argument see Mark Colyvan, “In Defence of Indispensability” Philosophia Mathematica 6 (1998): 39-62. I reword their argument to broaden its scope. To defend the argument so-worded would call for precision on what exactly “experiential framework” means and why we must have ontological commitment to those entities indispensable to it. See also Mark Colyvan’s Stanford Encyclopedia of Philosophy’s entry, “Indispensability Arguments in the Philosophy of Mathematics”.
  18. On fictionalism see Hartry Field, Realism, Mathematics and Modality (Blackwell, 1989). Mark Balaguer, Platonism and Anti-Platonism in Mathematics (Oxford, 1998); “A Theory of Mathematical Correctness and Mathematical Truth” Pacific Philosophical Quarterly 82 (2001): 87-114. Jody Azzouni, Deflating Existential Consequence: A Case for Nominalism (Oxford, 2004).
  19. Richard Swinburne takes this route in The Christian God (Oxford, 1994), 96-116. On this topic in general see Paul Copan and William Lane Craig’s excellent chapter “Creatio ex Nihilo and Abstract Objects” in Creation out of Nothing (Baker, 2004), 167-195. See also Craig’s concise summary of the fictionalist position in his “Current Work on God and Abstract Objects”.
  20. Tennant, “On the Necessary Existence of Numbers”, idem. A standard fictionalist rendition would be(#) For any sort of entity F, There are n Fs iff according to the numbers fiction, The number of Fs = nBut this is precisely the sort of language that Tennet claims incurs ontological commitments. For more on this point against fictionalism, see Daniel Nolan and J. O’Leary-Hawthorne, “Reflexive Fictionalisms” Analysis 56 (1996): 26-32. See also Matti Eklund’s, “Fictionalism” in the Stanford Encyclopedia of Philosophy.
  21. Tennant, “On the Necessary Existence of Numbers”, 322.
  22. Argument adapted from John Baggaley, “The Ontological Argument for the Existence of God” 25
  23. Wes Morriston, “Craig on the Actual Infinite” Religious Studies 38 (2002): 151.
  24. Quoted in Reuben Hersh, What Is Mathematics, Really? (Oxford, 1999), 42. For more on this point against platonism see William Lane Craig, The Kalam Cosmological Argument (Wipf and Stock, 1979. ed. 2000), 88-94. However, this modus tollens is another’s modus ponens: some claim the existence of abstracta is proof positive of an actual infinite. On this see J. P. Moreland, “A Response to a Platonistic and Set-theoretic Objection to the Kalam Cosmological Argument” Religious Studies 39 (2004): 373-390.
  25. For powerful presentations of this argument, see Paul Benacerraf’s “What Numbers Could Not Be” Philosophical Review 74 (1965): 47-73; “Mathematical Truth” Journal of Philosophy 70 (1973): 661-79. More recent and exhaustive is Colin Cheyne’s Knowledge, Cause, and Abstract Objects: Causal Objections to Platonism (Springer, 2001). Benacerraf et al. have also advanced what is called the uniqueness objection against platonism. Frankly I am not able to make much sense out of that argument.
  26. Alvin Goldman and Joel Pust, “Philosophical Theory and Intuitional Evidence” in eds. Michael R. DePaul and William Ramsy, Rethinking Intuition: The Psychology of Intuition and Its Role in Philosophical Inquiry (Rowman & Littlefield, 1998), 185.
  27. J. P. Moreland, Universals (McGill-Queen’s, 2001. ed. 2005), 121. For more on the incompatibility of platonism and naturalism, see Jarrold J. Katz, Realistic Rationalism (MIT, 1998), 25-83. Laurence Bonjour, In Defense of Pure Reason (Cambridge, 1998). One important rejoinder to the epistemological argument I did not mention is Mark Balaguer’s. Balaguer argued that if every possible entity e in some category C existed, then we can have knowledge of extra-mental entities without a causal relation simply by imagining or “dreaming up” any e from C. Given that every e from C exists, we must be cognizing at least some actual entities of which C is composed. Now if we assume that every possible abstract object that could exist does exist, then it follows that the ones we believe exist do exist. See Mark Balaguer, Platonism and Anti-Platonism in Mathematics, 49. Three responses are in order. First, it is interesting to note Tennet’s argument for the primacy of zero, if successful, rebuts Balaguer’s argument because it demonstrates that it is false that every possible mathematical entity exists. Second, it is doubtful that the act of “dreaming up” or imagining some e, even if that e existed, would be sufficient for genuine knowledge of e to obtain. And third, even if we granted that every e that could exist does exist and that our act of imagining some e is sufficient for knowledge of e, we would still run into the problem of the actual infinite. For if every e that could exist does exist, there must be an actual infinite number of es. For a more detailed criticism of Balaguer’s argument, see Colin Cheyne, “Problems with Profligate Platonism” Philosophia Mathematica 7 (1999): 164-177.
  28. J. L. Mackie, Ethics: Inventing Right and Wrong (Penguin, 1977), 38. Mackie is speaking specifically of objective values here, but such would themselves be abstract objects.
  29. See Alvin Plantinga, Does God Have A Nature? (Marquette, 2003), 127-140; “How to be an Anti-Realist”, 67-68. Quentin Smith, “The Conceptualist Argument for God’s ExistenceFaith and Philosophy 11 (1984): 45-49.
  30. Alvin Plantinga, “Two Dozen (or so) Theistic Arguments,” as printed in ed. Deane-Peter Baker, Alvin Plantinga (Cambridge, 2007), 213.
  31. See his The Existence of God (Oxford, 2nd ed. 2000), 303-315. See also Alvin Goldman and Joel Pust, “Philosophical Theory and Intuitional Evidence”, 181.
  32. Tim Crane, “Intentionality as the Mark of the Mental” in ed. Anthony O’Hear, Current Issues in Philosophy of Mind (Cambridge, 1998), 229-252.
  33. Richard Davis is pioneering work in this area. See his recent “God and Modal ConcretismPhilosophia Christi 10/1 (2008): 57-74.
  34. David M. Armstrong, “Naturalism, Materialism, and First Philosophy” Philosophia 8 (1978): 262.
  35. Kai Nielson, “Naturalistic Explanations of Theistic Belief” in eds. Philip Quinn and Charles Taliaferro, A Companion to Philosophy of Religion (Blackwell, 1997), 402.
  36. G. Frege, The Foundations of Arithmetic, trans. J. L. Austin (Northwestern, 1980). Edmund Husserl, Logical Investigations (Routledge, 2001).
  37. Quentin Smith, “The Conceptualist Argument”, 40. Alvin Platninga, “Two Dozen (or so) Theistic Arguments”, section I arguments (a)-(c). Op cit., 210-213.
  38. Robert M. Adams, The Virtue of Faith (Oxford, 1987), 218.
  39. See especially Greg Welty’s, An Examination of Theistic Conceptual Realism as an Alternative to Theistic Activism (Oxford, 2000). John Byl, “Theism and Mathematical RealismProceedings of the Association of Christians in the Mathematical Sciences (2001): 33-48. Vern S. Poythress, “Reforming Ontology and Logic in the Light of the Trinity: An Application of Van Til’s Idea of Analogy” Westminster Theological Journal 57/1 (1995): 187-219; “Creation and Mathematics; Or What Does God Have To Do With Numbers?” The Journal of Christian Reconstruction 1/1 (1974): 128-140.
  40. Quentin Smith, “The Conceptualist Argument”, 48.

29 Responses to “A Conceptualist Argument for the Existence of God”

  1. Lifeguard Says:

    Whoa. Let me catch my breath. I’ve stopped by countless times since your last entry waiting specifically for this post. So let me begin by saying thanks and that I found it well worth the wait.

    Although I cannot imagine how you could have written it more clearly, I think amateurs like me may need a little time to gobble it all down and let it digest. That being said, I think I understand the general contours of the argument much better, and, after a few more readings, I’ll be sure to start peppering you with questions again!

    In the meantime, I just wanted to let you know I’ve read it through and to pay you a compliment on an especially well written post.

  2. mattghg Says:

    Awesome post, Chad. I’ve already linked to your earlier discussion of this argument before, and am chuffed that you’ve updated and expanded it.

  3. Emanuel Goldstein Says:

    Of course, someone could accept everything you have written and still go to hell.

    Like the swine at atheism sucks, you have distorted the gospel.

  4. Chad McIntosh Says:

    EG,

    If there’s any obvious distortion of the gospel here, it has come in the form of a visceral condemnation without apparent reason. Don’t forget that someone could accept everything you apparently do and still go to Hell…

    “But I say, if you are even angry with someone, you are subject to judgment! If you call someone an idiot, you are in danger of being brought before the court. And if you curse someone, you are in danger of the fires of hell.” (Matt 5:22)

  5. Mark Jiang Says:

    Emanuel,

    Do you mind explaining how Chad has distorted the gospel?

  6. mattghg Says:

    Emanuel,

    Huh?

    Chad,

    Have you seen this discussion over on DI2? It seems very relevant to this project.

  7. Heath Says:

    Don’t look now, Emmanuel, but your self-righteousness is showing…

    Faith is worthless apart from reason.

  8. Dmitry Chernikov Says:

    You write that abstracta are causally inefficacious and also that they must have the passive power to come to be known by minds. I think that’s an excellent refinement, because a thing with no power at all is nothing; it does not exist. But what causes abstract things to have this power? Do we have direct access to those entities via intuition? Are we reading the mind of God in Whom abstracta reside?

  9. MG Says:

    This was an excellent overview of the issues, more thorough than any I’ve read to date. Sorry for being so late to read it.

    Here’s a question: are there any abstract objects that have intentionality? Are any abstract objects “of” or “about” something else? If so, then could we argue directly that they are mental entities, given that the only entities we know of with intentionality are mental ones? This seems a promising route, but I wonder what examples there are.

  10. Lifeguard Says:

    I told you I’d come back to this post once I had a chance to fully digest it. While I think I have a handle of the argument, I have to admit it still exceeds my grasp of the issue as an amateur in philosophy. Still, a very interesting post, and I can see how your ideas lay a ground work for the “accept ability” of theism, though I won’t yet concede proof!

  11. Chad McIntosh Says:

    Well, the question then is, What would you take to be “proof” of something?

  12. Lifeguard Says:

    Interesting question. My thinking on the issue of “proof” has evolved a bit. Back when I believed, at rock bottom, I felt that when faced with the question of why we exist, you got a feelin in your bones– either that god exists or that god doesn’t exist and matter is all there really is. The question of arguments for and against, the proofs and what not, have more to do with articulating that intuition than anything else. I see value in that– I even see value in debates about god’s existence– but I find that these arguments are far more likely to degenerate into something pointless than end up being constructive.

    Regarding the conceptualist argument, again, I think I see the outline of the argument and basically understand it, but it doesn’t change my mind. It’s rational, but I don’t find my own worldview any less rational and it makes more sense to me. Maybe I’m simply too dense to understand that god exists or something, but, as far as I recall, God wants us to BELIEVE in him as opposed to be CONVINCED of his existence based on abstract arguments.

    Me? I don’t believe. I have yet to come across an argument or a worldview that makes more sense to me than an atheistic naturalism. It’s not that I don’t read or that I have just stopped trying to understand the issues. In the last six months I’ve read C.S. Lewis as much as I’ve read Daniel Dennett. Dennett just makes more sense to me. What I will say, is I’m committed to finding the right answers and being truthful to myself wherever I am and wherever I end up on this journey.

    If an all knowing, all loving god does exist, then I don’t think he would hold that against me.

  13. One Brow Says:

    These virtues considered, however, fictionalism is perhaps too radical. The most jarring problem is its renunciation of obviously true descriptive states of affairs the truth of which seem to be grounded in certain abstracta like “2 + 2 = 4” or “two entities x and y can have the same shape and size.” For according to fictionalism, such abstracta are merely useful fictions. Indeed, it will be hard to take seriously a view which maintains that propositions such as “4 is divisible by 2”, “Jones believes that [some proposition x]”, or “triangles have three sides” are not just false, but literally truth valueless.

    Well, that would depend upon the rules set up to establish the useful fiction, would it not? After all, isn’t “useful fiction” in some ways another term to describe a formal system? We would find a fiction where 4 is not divisible by 2 to be not very useful, but that does not impart some sort on inherent falsity, especially since the notions of “true” and “false” are in and of themselves merely useful fictions.

    If Tennet’s argument passes, then fictionalism must be false. His argument can be outlined as follows:

    (2a.4) There is no possible world such that there are no things that are not self-identical
    (2a.5) 0 is the number of such things that are not self-identical
    (2a.6) Therefore, 0 exists in all possible worlds

    Does Tennet’s argument pass? (2a.4) is self-evidently true, so what support does Tennant give for (2a.5), the argument’s main premise? On the face of it, (2a.5) seems to beg the question against the nominalist by introducing the existence of a number into the premise.

    On the contrary, 2a.5 is an instance of the use of a useful fiction to describe a situation, and is true under our usual understanding of 0. It is 2a.6 that introduces the concept of numbers being real, with no justification. 2a.6, without the extra assumption, would read: 0 is useful in every possible world.

    The grounds for 2a is completely circular, in that you need the assumption the number is real to derive it.

    Wesley Morriston concisely summarizes the problem:

    Let m = the number of books in our infinite library, n = the number of odd-numbered books, and o the number of books numbered 4 or higher…
    (m – n) = infinity, whereas (m – o) = 4.
    But,
    n = o (since both n and o are infinite)
    It follows that we get inconsistent results subtracting the same number from m.19

    The conclusion is that we obviously couldn’t perform such operations in the actual world, so an actual infinite cannot exist.

    This argument merely proves that subtraction is not well defined for infinite numbers. Since the definition of infinite is a set that is the same size as a proper subset, this is hardly surprising.

  14. Chad McIntosh Says:

    Well, that would depend upon the rules set up to establish the useful fiction, would it not? After all, isn’t “useful fiction” in some ways another term to describe a formal system? We would find a fiction where 4 is not divisible by 2 to be not very useful, but that does not impart some sort on inherent falsity, especially since the notions of “true” and “false” are in and of themselves merely useful fictions.

    I can’t help but ask the question, then, of whether you consider the statement “the notions of “true” and “false” are in and of themselves merely useful fictions” to be a useful fiction. The point is this: what possible rules to determine what are useful fictions could you establish that aren’t themselves dependent upon useful fictions?

    On the contrary, 2a.5 is an instance of the use of a useful fiction to describe a situation, and is true under our usual understanding of 0. It is 2a.6 that introduces the concept of numbers being real, with no justification. 2a.6, without the extra assumption, would read: 0 is useful in every possible world.

    The grounds for 2a is completely circular, in that you need the assumption the number is real to derive it.

    As I point out in n.17, even the fictionalist re-wording involves quantification. “But they’re not quantifying anything per se,” you might object. But this begs the question against (2a) because the very question with respect to (2a) is whether a fictionalist can escape ontological commitment over premise (2a.4). As I briefly summarized elsewhere, I don’t think it can be done. What revisionist expression do you offer?

    This argument merely proves that subtraction is not well defined for infinite numbers. Since the definition of infinite is a set that is the same size as a proper subset, this is hardly surprising.

    This was precisely what Morriston went on to argue, which misses the point entirely. For the point is that were the actual infinite to describe reality, then these absurdities would result. But because the actual infinite is so defined, it can’t possibly describe reality.

  15. One Brow Says:

    Sorry for the delay in response, but I wanted to read up more on fictionalism to make sure I did not misrepresent the argument. We amatuers need to be careful. :)

    I can’t help but ask the question, then, of whether you consider the statement “the notions of “true” and “false” are in and of themselves merely useful fictions” to be a useful fiction.

    It is a statement concerning abstract concepts, which are not real things. So I see no reason to think of that statement otherwise. Pretty much all of logic, philosophy, and mathematics strike me as such.

    The point is this: what possible rules to determine what are useful fictions could you establish that aren’t themselves dependent upon useful fictions?

    I can’t think of any. That’s basically asking to define abstract terms in non-abstract terms.

    As I point out in n.17, even the fictionalist re-wording involves quantification. “But they’re not quantifying anything per se,” you might object. But this begs the question against (2a) because the very question with respect to (2a) is whether a fictionalist can escape ontological commitment over premise (2a.4). As I briefly summarized elsewhere, I don’t think it can be done. What revisionist expression do you offer?

    Well, let me back up a bit. Going all the way back to 2a.1, I would first argue that 0 is dispensable. There is a difference between useful and essential, and man lived for a long time with numbers, yet without the number zero (and of course, even longer without numbers at all).

    Also, looking at 2a.4 a little more closely, do you mean “In every possible world, there are no things that are not self-identical”?

    With that out of the way, lets look a little deeper at what 0 is: the number of the empty set. Would you say that the empty set is a real object as well?

    Finally, getting back to Tennet’s argument directly, I suppose my disagreement is with the notion that, since we commit ourselves to certain abstract notions whenever we engage in abstract dialog, those abstact notions have an ontological reality. Egaging in abstraction has already removed us from necessity of existing.

    This was precisely what Morriston went on to argue, which misses the point entirely. For the point is that were the actual infinite to describe reality, then these absurdities would result.

    I guess I just fail to see the absurdity. It must be my mathematical training. There is nothing absurd about subtraction being ill-defined for infinity, just as there is nothing absurd about division by zero being undefined.

    But because the actual infinite is so defined, it can’t possibly describe reality.

    Oddly, it would seem to be a perfect description of some parts of reality to the orthodox Christian theist. For example, since God has an infinite amount of power available to use in Creation, whenever God performs any act, there is no reduction in God’s power. In fact, you could even say that angels are infinitely powerful, but that the power of God is a strictly larger size of infinity (there are more infinite numbers than there are finite numbers).

  16. Chad McIntosh Says:

    Thank you for your reply, One Brow.

    It is a statement concerning abstract concepts, which are not real things. So I see no reason to think of that statement otherwise. Pretty much all of logic, philosophy, and mathematics strike me as such. I can’t think of any. That’s basically asking to define abstract terms in non-abstract terms.

    If it is a statement not about real things, but merely useful fictions, then you admit your response to me is truth-valueless? To which I’m sure you’d reply, “well, strictly speaking, yours is, too.” Then on what grounds do we decide who’s right?

    Well, let me back up a bit. Going all the way back to 2a.1, I would first argue that 0 is dispensable. There is a difference between useful and essential, and man lived for a long time with numbers, yet without the number zero (and of course, even longer without numbers at all).

    This to me seems to beg the question. You can’t just assert that “and man lived for a long time with numbers, yet without the number zero (and of course, even longer without numbers at all)” without argument. I understand that you “would first argue that 0 is dispensable” (my italics), but you haven’t provided any argument for that conclusion. I provided two for the indispensability of 0. Note that even if you refute those, your position would still not win by default. You have to also offer a revisionist expression for (2a.4) that doesn’t commit you to 0.

    Also, looking at 2a.4 a little more closely, do you mean “In every possible world, there are no things that are not self-identical”? With that out of the way, lets look a little deeper at what 0 is: the number of the empty set. Would you say that the empty set is a real object as well?

    Yes to both.

    Finally, getting back to Tennet’s argument directly, I suppose my disagreement is with the notion that, since we commit ourselves to certain abstract notions whenever we engage in abstract dialog, those abstact notions have an ontological reality. Egaging in abstraction has already removed us from necessity of existing.

    This is Harty Field’s point stated nicely. But if we have independent reasons to suppose those do have ontological reality, such as the arguments for 0, then we have to deal with them first.

    I guess I just fail to see the absurdity. It must be my mathematical training. There is nothing absurd about subtraction being ill-defined for infinity, just as there is nothing absurd about division by zero being undefined.

    There is no absurdity with those things with respect to mathematics. The problem surfaces when we try to use such operations to describe reality. For example, no one would say imaginary numbers denote real, independently existing things despite their usefulness in mathematics. Or take simple subtraction, for example. Suppose you have three marbles. Even though it would make perfect sense mathematically to subtract a larger quantity from a smaller quantity, 3 – 5 = -2, such would not make any sense with respect to the marbles. The same is true of the concept of the actual infinite. Prominent mathematicians such as David Hilbert agree.

    But because the actual infinite is so defined, it can’t possibly describe reality.

    Oddly, it would seem to be a perfect description of some parts of reality to the orthodox Christian theist. For example, since God has an infinite amount of power available to use in Creation, whenever God performs any act, there is no reduction in God’s power. In fact, you could even say that angels are infinitely powerful, but that the power of God is a strictly larger size of infinity (there are more infinite numbers than there are finite numbers).

    Now we’d be equivocating the actual infinite with an infinite of another sort. No theist would maintain that the actual infinite applies to God, for God is not composed of an actually infinite number of members or parts. Rather, when theists speak of God’s infinity, especially regarding the divine attributes, they denote a certain quality of them, not quantity.

  17. One Brow Says:

    If it is a statement not about real things, but merely useful fictions, then you admit your response to me is truth-valueless?

    Do you mean “truth” as in:
    1) consistent within the fiction itself, according to the rules of the fiction, with no regard for its use exterior to the fiction,
    2) having some ontologically real thing called truth (which I am not even sure exists),
    3) while being a part of a fiction, it represents a statement that extremely effectively models reality, to a degree that the results of the model are highly reliable, or
    4) ???

    I would answer 1) yes, 2) no, 3) sometimes, and 4) we’ll see.

    To which I’m sure you’d reply, “well, strictly speaking, yours is, too.” Then on what grounds do we decide who’s right?

    Maybe it’s as interesting to have the discussion even if neither of us can prove ourself right or th other wrong. I have no believe that I will alter beliefs, but I do think we wil both be able to gain from the conversation.

    This to me sems to beg the question. You can’t just assert that “and man lived for a long time with numbers, yet without the number zero (and of course, even longer without numbers at all)” without argument.

    If you like, feel free to interpret that as “and man lived for a long time with knowledge of numbers, yet without knowledge of the number 0, (and of course, even longer without knowledge of numbers at all)”. Do you dispute that these are matters of historical/anthropological fact?

    I provided two for the indispensability of 0.

    I agree that 0 is indispensible for the use of philosophical argumentation at the level with which we are conversing.

    Yes to both.

    So, nothingness exists.

    There is no absurdity with those things with respect to mathematics. The problem surfaces when we try to use such operations to describe reality. For example, no one would say imaginary numbers denote real, independently existing things despite their usefulness in mathematics.

    Unless I grossly misunderstood them, electrical engineers would disagree. A two-dimensional fields effect is often best described using complex (aka imaginary) numbers.

    Even though it would make perfect sense mathematically to subtract a larger quantity from a smaller quantity, 3 – 5 = -2, such would not make any sense with respect to the marbles.

    That absurdity does not come from the properites of the model, but the application thereof. Your arguments against an actual infinity come from the nature of the model itself. It makes no sense mathematically to subtract infinity, so why should it make sense in the real world? If your argument breaks real infinities, it breaks the model, because it applies to both.

    Prominent mathematicians such as David Hilbert agree.

    No doubt you can find prominent mathematicians on both sides of this debate, as well as many others. An appeal to authority does not impress me. Now, if you can find an example of the notion of infinity that works in the model, but breaks in reality, that would be more convincing.

    Now we’d be equivocating the actual infinite with an infinite of another sort. No theist would maintain that the actual infinite applies to God, for God is not composed of an actually infinite number of members or parts.

    I don’t recall referring to God’s “parts” or counting any aspect of God. However, I’m wiling to listen: is God considered to have infinite power? What “quality”, as opposed to quantity, of power does this refer to?

  18. Chad McIntosh Says:

    Do you mean “truth” as in:
    1) consistent within the fiction itself, according to the rules of the fiction, with no regard for its use exterior to the fiction,
    2) having some ontologically real thing called truth (which I am not even sure exists),
    3) while being a part of a fiction, it represents a statement that extremely effectively models reality, to a degree that the results of the model are highly reliable, or
    4) ???

    I would answer 1) yes, 2) no, 3) sometimes, and 4) we’ll see.

    What I mean by “truth” is that which corresponds to reality; reality being that which objectively exists. Your view seems obviously inconsistent, for if you suspend judgment on or reject (2), you have no grounds for accepting (1) or anything else. This is what I was trying to point out. Perhaps I can better illustrate by jumping off your claim that

    (P) The notions of “true” and “false” are in and of themselves merely useful fictions

    which is just to say

    (P’) For any proposition p, p is such that it is truth-valueless

    which removes any grounds you might have for asserting (P) truthfully, for (P) itself would be among the truth-valueless ps in (P’). In other words, your view turns out to be self-refuting. Even if (P) weren’t self-referentially inconsistent (I just don’t see how that couldn’t be—indeed, (P), and especially (P’) seem necessarily false), then we still couldn’t proceed meaningfully. Anything you or I could say to one another would be vacuous (truth-valueless). The discussion would be pointless because literally “neither of us can prove the other wrong”. Ah! But it’s not pointless because “it’s interesting to have the discussion” and it is possible that “we will both be able to gain from the conversation”, you say. I appreciate your optimism, but I submit that if (P) is true, then I can find a whole host of more important things to be doing than reading and typing truth-valueless utterances.

    If you like, feel free to interpret that as “and man lived for a long time with knowledge of numbers, yet without knowledge of the number 0, (and of course, even longer without knowledge of numbers at all)”. Do you dispute that these are matters of historical/anthropological fact?

    Yes, I do. But note that it would be fruitless to dispute the ontological status of 0 historically or anthropologically, or even epistemologically (which is what I’m sure you’re getting at), because whether or not man has been aware of the existence of the number 0 in times past is simply irrelevant to 0’s existence. You’d be confusing epistemology with ontology. This can be seen by considering the fact that if at one point man was unaware of 0’s existence or even unaware of his cognitive reliance on 0, 0 itself would be no more dispensable to his thought or in reality.

    Unless I grossly misunderstood them, electrical engineers would disagree. A two-dimensional fields effect is often best described using complex (aka imaginary) numbers.

    I must not be explaining myself well. I am not disputing the fact that we can describe systems, especially theoretical ones such as dimensional fields, Euclidean and non-Euclidean geometries etc. with mathematics. I’m saying the mathematical terms themselves do not have an ontological counterpart in reality in and of themselves. This is admittedly how these fields, including set-theory, are understood by their practitioners, especially in mathematics. For example, Bernard Bolzano described his dealing with infinite sets as in the “realm of things which do not claim actuality, and do not even claim possibility” (Paradoxes of the Infinite [Routledge 1950], p.84). Cantor himself understood the actual infinite this way, for he made it clear that held the members of sets to be objects of thought or intuition. “Cantor’s infinities are abstract and divorced from the physical world,” concludes Abraham Robinson (“The Metaphysics of the Calculus” in The Philosophy of Mathematics [Oxford, 1969], p.163). This is echoed by Abraham Fraenkel, who maintained that among the various branches of mathematics, set theory “is the branch which least of all is connected with external experience and most genuinely originates from free intellectual creation” (Abstract Set Theory [North-Holland, 1961], p.240). It is not surprising then, as Rotman and Kneebone point out, that “the Zermelo-Fraenkel universe of sets exists only in a realm of abstract thought…the ‘universe’ of sets to which the…theory refers is in no way intended as an abstract model of an existing Universe, but serves merely as the postulated universe of discourse for a certain kind of abstract inquiry” (The Theory of Sets and Transfinite Numbers [Oldbourne, 1966], p.61). I hope my point is clear that I’m not denying the descriptive virtues of mathematics internal to theoretical models and systems. I’m saying such virtues are only internal to the systems themselves and vanish once we make external applications beacuse, as we’ve seen, they were neither intended nor are understood to be applicable to the external world. You even go on to admit this:

    That absurdity does not come from the properites of the model, but the application thereof.

    I don’t think I could have stated it better myself! So you admit, then, that we get absurd results when we apply the model to real objects. I think this is a sign of great progress!

    Your arguments against an actual infinity come from the nature of the model itself. It makes no sense mathematically to subtract infinity, so why should it make sense in the real world? If your argument breaks real infinities, it breaks the model, because it applies to both.

    The argument does not apply to both, One brow. You’re refusing to acknowledge the distinction between consistent abstract theory and reality itself. The model in and of itself is fine—insofar as transfinite numbers are used to help establish the conceptual legitimacy of the actual infinite—but the failure of transfinite mathematics to provide an acceptable model of inverse operations is not an internal problem, but an external one of ontological significance. Sure, the prohibition of inverse operations is mathematically justified when dealing with the actual infinite. But “the absurdities that emerge when we suppose the actual infinite is able to exist extramentally are due to an unresolvable conflict between the logical possibilities expressed in the relations of Cantorian mathematics and the factual possibilities that govern the range of possible real relations in the extramental world. Real things, subject to factual possibilities, cannot serve to instantiate an actual infinite because of conflicts that arise with respect to their real features.” (Mark Nowacki, The Kalam Cosmological Argument for God [Prometheus, 2007], p.130).

    Now, if you can find an example of the notion of infinity that works in the model, but breaks in reality, that would be more convincing.

    Frankly I am astonished by this remark, for my very point was that every model of transfinite arithmetic (Bolzano, Cantorian, Zermelo-Fraenkel neo-Cantorian, et. al.) is an example of the notion of infinity that works in the model, but breaks in reality! You essentially admitted this when you observed that “absurdity does not come from the properties of the model, but the application thereof”.

    I don’t recall referring to God’s “parts” or counting any aspect of God. However, I’m wiling to listen: is God considered to have infinite power? What “quality”, as opposed to quantity, of power does this refer to?

    I fail to see the point of this inquiry, for I already pointed out that the notion of the actual infinite, which the argument in my post is concerned with, is completely different from the notion of infinity concerning God’s attributes. When a theist says “God has infinite power,” he does not mean that “God’s power is a set such that a proper subset is equivalent to it”. Instead, he means something like “Wow, God is unimaginably powerful!” or “Wow, God has power to a very great degree!”, which shows that the notion of “infinite power” is just a way to express a divine quality of greatness.

    I hope to have shown how radical (P) is. In addition to that, though, I want to remind you that you’ve again chosen to remain silent on offering a revisionist expression for (2a.4) that doesn’t commit you to 0. Moreover, you ignored countering my two arguments for 0, but instead said of them “that 0 is indispensible for the use of philosophical argumentation at the level with which we are conversing,” which, ironically, is rather uninformative for the use of philosophical argumentation at the level with which we are conversing. Finally, you more or less conceded that it is not the model of the actual infinite that is the problem, but it’s transition into reality. I sincerely hope to have persuaded you on at least that last score. Thanks again for your thoughtful reply.

  19. One Brow Says:

    Do you mean “truth” as in:
    1) consistent within the fiction itself, according to the rules of the fiction, with no regard for its use exterior to the fiction,
    2) having some ontologically real thing called truth (which I am not even sure exists),
    3) while being a part of a fiction, it represents a statement that extremely effectively models reality, to a degree that the results of the model are highly reliable, or
    4) ???

    I would answer 1) yes, 2) no, 3) sometimes, and 4) we’ll see.

    What I mean by “truth” is that which corresponds to reality; reality being that which objectively exists. Your view seems obviously inconsistent, for if you suspend judgment on or reject (2), you have no grounds for accepting (1) or anything else.

    Well, it’s not always possible to determine the truth of any given statement within a model/fiction, but I can certainly determine for a variety of statements whether they are true within the rules of boundaries of the model by using the processes of the model.

    This is what I was trying to point out. Perhaps I can better illustrate by jumping off your claim that

    (P) The notions of “true” and “false” are in and of themselves merely useful fictions

    which is just to say

    (P’) For any proposition p, p is such that it is truth-valueless

    So, you’re argument is that when we create a model, we have no guarantee that it is an accurate depiction of reality? I concur. That’s where the “useful” part of “useful fiction” comes from. We don’t judge, or select, a model by a guarantee of correspondence to reality that can never be achieved. We judge it whether it is consistent within its own rules and useful in helping us deal with reality.

    I appreciate your optimism, but I submit that if (P) is true, then I can find a whole host of more important things to be doing than reading and typing truth-valueless utterances.

    We could compare our various models of reality, look at why we find them useful, and perhaps even incorporate small parts of each other’s model into one or more of our models. That could be useful.

    If you like, feel free to interpret that as “and man lived for a long time with knowledge of numbers, yet without knowledge of the number 0, (and of course, even longer without knowledge of numbers at all)”. Do you dispute that these are matters of historical/anthropological fact?

    Yes, I do.

    http://en.wikipedia.org/wiki/0_(number)

    One of the examples mentioned is 6th century India, where instead of a zero or the equivalent, they would leave a blank space in a positional number system.

    As for the former, three billion years ago our ancestors were bilateral “worms”. I’m pretty sure we did not use numbers then.

    But note that it would be fruitless to dispute the ontological status of 0 historically or anthropologically, or even epistemologically (which is what I’m sure you’re getting at), because whether or not man has been aware of the existence of the number 0 in times past is simply irrelevant to 0’s existence. You’d be confusing epistemology with ontology.

    Well, I am disputing whether 0 is dispensable. If you can do without an object epistemologically, that seems to qualify as dispensable.

    This can be seen by considering the fact that if at one point man was unaware of 0’s existence or even unaware of his cognitive reliance on 0, 0 itself would be no more dispensable to his thought or in reality.

    That would be a statement that would require proof.

    Unless I grossly misunderstood them, electrical engineers would disagree. A two-dimensional fields effect is often best described using complex (aka imaginary) numbers.

    I must not be explaining myself well. I am not disputing the fact that we can describe systems, especially theoretical ones such as dimensional fields, Euclidean and non-Euclidean geometries etc. with mathematics. I’m saying the mathematical terms themselves do not have an ontological counterpart in reality in and of themselves.

    Again, unless I misunderstood, we are talking about real, measurable, current effects that follow the rules of complex numbers.

    http://en.wikipedia.org/wiki/Complex_number

    This is admittedly how these fields, including set-theory, are understood by their practitioners, especially in mathematics. … I hope my point is clear that I’m not denying the descriptive virtues of mathematics internal to theoretical models and systems. I’m saying such virtues are only internal to the systems themselves and vanish once we make external applications beacuse, as we’ve seen, they were neither intended nor are understood to be applicable to the external world.

    Please don’t try to lecture me on how mathematicians understand their field. They are hardly of the unified position you are presenting. If I go out and find 2, or 5, or a dozen mathematicians who think Platonically and see no contradiction, would that end the appeal to authority line of argumentation?

    You even go on to admit this:

    That absurdity does not come from the properites of the model, but the application thereof.

    I don’t think I could have stated it better myself! So you admit, then, that we get absurd results when we apply the model to real objects. I think this is a sign of great progress!

    I was speaking of your example of subtracting 5 apples from 3. The fact that some aspects of a model are not useful means we suitably restrict the model when we are counting apples. The absurdity is the application, not the model.

    Your arguments against an actual infinity come from the nature of the model itself. It makes no sense mathematically to subtract infinity, so why should it make sense in the real world? If your argument breaks real infinities, it breaks the model, because it applies to both.

    The argument does not apply to both, One brow. You’re refusing to acknowledge the distinction between consistent abstract theory and reality itself.

    I prefer a capitalized “B”, if you don’t mind.

    Yes, you *can* take every argument you apply to how infinites can be meaningfully subtracted, and use that same argument within the mathematical model. If your arguments would make a real infinity absurd, they also make the model absurd.

    The model in and of itself is fine—insofar as transfinite numbers are used to help establish the conceptual legitimacy of the actual infinite—but the failure of transfinite mathematics to provide an acceptable model of inverse operations is not an internal problem, but an external one of ontological significance.

    Why are inverse operations essential to ontological significance (and presumably existence)? This is a real irony, given that you are arguing for the significance and existence of a number that fails to admit of an inverse operation. I offer a comparable conversation.

    Person A: You can’t divide five apples by zero, so zero can’t exist.
    Person B: That doesn’t make sense. Dividing by zero is not defined mathematically, so why would you expect it have a real application?
    Person A: But you can multiply by zero, and division is the inverse of multiplication. If you can perform an operation with a number, there is a commitment for the inverse operation on that number to be non-absurd, or the number can’t have ontological significance. I can’t divide five apples in zero ways, so zero can’t be real.

    I ask how your argument regarding the need for inverse operations when dealing with infinity differs from the argument of Person A.

    … (Mark Nowacki, The Kalam Cosmological Argument for God [Prometheus, 2007], p.130).

    I have yet to see a version of the Kalam argument that creates an absurdity exterior to the model that does not apply interior to the model.

    Now, if you can find an example of the notion of infinity that works in the model, but breaks in reality, that would be more convincing.

    Frankly I am astonished by this remark,

    I don’t mind. I know my point of view is very different from what most people are accustomed to, and it takes a while to understand it.

    I don’t recall referring to God’s “parts” or counting any aspect of God. However, I’m wiling to listen: is God considered to have infinite power? What “quality”, as opposed to quantity, of power does this refer to?

    I fail to see the point of this inquiry, for I already pointed out that the notion of the actual infinite, which the argument in my post is concerned with, is completely different from the notion of infinity concerning God’s attributes.

    The definition of infinite, at least mathematically, is “of the same size as a proper subset”. If God can use some of his power, and still have the same amount of power after the use, God’s power meets the definition of infinite.

    When a theist says “God has infinite power,” he does not mean that “God’s power is a set such that a proper subset is equivalent to it”. Instead, he means something like “Wow, God is unimaginably powerful!” or “Wow, God has power to a very great degree!”, which shows that the notion of “infinite power” is just a way to express a divine quality of greatness.

    Is it possible for God to use so much power that He runs out? Runs down? Is there a limit? Most Christians I know would say there is no limit.

    A googleplex is unimaginably huge, but still finite. If you take one element from a set the size of a googleplex, you no longer have a set the size of a googleplex. Is that true of God, in your belief? If you honestly have no belief on that topic, I’ll accept that answer, but it does not match what I typically hear from Christians.

    I hope to have shown how radical (P) is. In addition to that, though, I want to remind you that you’ve again chosen to remain silent on offering a revisionist expression for (2a.4) that doesn’t commit you to 0.

    I believe I have already acknowledged that 0 is always useful.

    Moreover, you ignored countering my two arguments for 0,

    You mean 2a.5 (which I acknowledged that I accepted, within the frame that it meant 0 was useful in all worlds) and 2a.6 (which I said did not follow from 2a.4 and 2a.5 without an additional statement that requires the useful to be real)?

    but instead said of them “that 0 is indispensible for the use of philosophical argumentation at the level with which we are conversing,” which, ironically, is rather uninformative for the use of philosophical argumentation at the level with which we are conversing.

    Well, I just don’t see how “universally useful” implies “universally real”. After all, “conditionally useful” does not imply “conditionally real”. Further, since many beings at various levels get by just fine without the abstract concept of 0, it is not indispensable.

    Finally, you more or less conceded that it is not the model of the actual infinite that is the problem, but it’s transition into reality.

    No, apparently I was not sufficiently clear in my statement. The problem was in the translation of the model of subtraction to finite numbers to real apples that you presented. I see no problems in the translation of infinity to reality, period. The problems you espouse are also present in the model.

  20. Franky McGee Says:

    One Brow, when are you going to stop repeating yourself and deal with the real meaty issues?

  21. One Brow Says:

    One Brow, when are you going to stop repeating yourself

    When my objections are actually addressed?

    and deal with the real meaty issues?

    I don’t mean to overlook anything. Which meaty issue would you like for me to sink my teeth into?

  22. Chad McIntosh Says:

    G. K. Chesterton has observed how difficult it is at times to argue with someone whose mind isn’t slowed down by things that go with good sense… ;)

    It’s easy to lose sight of the real issues at stake in discussions like these (as opposed to, say, what you think of the kalam cosmological argument or the attribute of omnipotence), so let me summarize what I gather to be your case against the conceptualist argument as I have presented it, and then I’ll respond.

    I. Truth
    You’re first main contention concerns the nature of truth as correspondence. I didn’t anticipate this being much of an issue, but I can see now that the history of philosophy up to the 19th century isn’t appreciated by everyone…At any rate, your view of truth has severe problems, similar to those which plague the coherence theory. For example, in your last reply, you say

    We don’t judge, or select, a model by a guarantee of correspondence to reality that can never be achieved. We judge it whether it is consistent within its own rules and useful in helping us deal with reality…We could compare our various models of reality, look at why we find them useful, and perhaps even incorporate small parts of each other’s model into one or more of our models. That could be useful.

    But, as I pointed out, this approach obviously doesn’t hold water. For one, it implies that since it is possible that p coheres with my model of reality and ~p coheres with your model of reality, both p and ~p could, in some absurd and contradictory way, both be true. The only way out of this would be to bite the bullet and say that

    (P’’) p is true for S and ~p is true for S*

    but either (P’’) corresponds to how things are or it doesn’t. Which is it? I illustrated this above by noting the problems facing your (P) and (P’), which you didn’t respond to. Either way, your view is bankrupt. So far you have refused to even give an argument for why you think “the notions of ‘true’ and ‘false’ are in and of themselves merely useful fictions”. Perhaps you can privilege me with such.

    II. Indispensability of 0
    I submit that you’re yet to offer either a counterargument to the two indispensability arguments I mentioned or offer a revisionist expression of (2a.4) that doesn’t commit you to 0. You’ve mentioned reasons for thinking 0 is epistemologically dispensable, but as a debate about ontology and not epistemology, such are irrelevant to the discussion.

    III. The Actual Infinite
    The inconsistency of your comments on the actual infinite are painfully obvious (so much for coherence!). On the one hand you agree with me that that when it comes to mathematics, “The absurdity is the application, not the model.” On the other hand you “see no problems in the translation [application] of infinity to reality period. The problems you espouse are also present in the model.” This is textbook special pleading. “But the former comment was about your marbles illustration, and the second about the actual infinite”, you’ll say. But this misses the point because I’m arguing that both are instances where we have internal consistency at odds with external application. Why think the two are different?

    Thanks again for your comments.

  23. One Brow Says:

    G. K. Chesterton has observed how difficult it is at times to argue with someone whose mind isn’t slowed down by things that go with good sense…

    For better or worse, many people do indeed think that I lack good sense. I’ll try not to let it be too much of an issue. : )

    I. Truth
    You’re first main contention concerns the nature of truth as correspondence. I didn’t anticipate this being much of an issue, but I can see now that the history of philosophy up to the 19th century isn’t appreciated by everyone…

    Well, I only took a couple of introductory philosophy courses, and that was 25 years ago, so I am strictly an amateur.

    At any rate, your view of truth has severe problems, similar to those which plague the coherence theory. For example, in your last reply, you say

    We don’t judge, or select, a model by a guarantee of correspondence to reality that can never be achieved. We judge it whether it is consistent within its own rules and useful in helping us deal with reality…We could compare our various models of reality, look at why we find them useful, and perhaps even incorporate small parts of each other’s model into one or more of our models. That could be useful.

    But, as I pointed out, this approach obviously doesn’t hold water. For one, it implies that since it is possible that p coheres with my model of reality and ~p coheres with your model of reality, both p and ~p could, in some absurd and contradictory way, both be true.

    Not true in the sense of reflective of reality, just true in the sense of a workable proposition within that model of reality.

    The only way out of this would be to bite the bullet and say that

    (P’’) p is true for S and ~p is true for S*

    but either (P’’) corresponds to how things are or it doesn’t. Which is it?

    If you are interpreting S and S* to mean the models we have constructed to examine reality, obviously P’’ is correct in assessing the truth of the proposition within each model.

    I illustrated this above by noting the problems facing your (P) and (P’), which you didn’t respond to. Either way, your view is bankrupt. So far you have refused to even give an argument for why you think “the notions of ‘true’ and ‘false’ are in and of themselves merely useful fictions”. Perhaps you can privilege me with such.

    The notions of true and false are part of the model that we construct to examine reality. You can not point to a true, any more than you can point to a 2. You can classify a statement as true. From the vantage point of internal consistency, you can always create a model where a given proposition is true and a different model where that proposition is false. However, that can disregard the purpose in constructing the model, which is examining reality. For example, you can create a model where my fingers are touching a keyboard, or where they are touching an apple. Only the first is useful, because I don’t actually want to eat that thing. So, I assign the first statement to be true, and the second statement to be false.

    II. Indispensability of 0
    I submit that you’re yet to offer either a counterargument to the two indispensability arguments I mentioned or offer a revisionist expression of (2a.4) that doesn’t commit you to 0. You’ve mentioned reasons for thinking 0 is epistemologically dispensable, but as a debate about ontology and not epistemology, such are irrelevant to the discussion.

    I could have sworn I said that I agreed 0 is useful in all possible worlds. Under that understanding, I accepted 2a.4 and 2a.5.

    III. The Actual Infinite
    The inconsistency of your comments on the actual infinite are painfully obvious (so much for coherence!). On the one hand you agree with me that that when it comes to mathematics, “The absurdity is the application, not the model.”

    That was not all of mathematics, generally, but the specific case of negative numbers and counting apples. Negative numbers work in the model, but are meaningless in the application, so it is a bad application in that instance. There are many other applications were negative numbers work just fine, such as vectors.

    On the other hand you “see no problems in the translation [application] of infinity to reality period. The problems you espouse are also present in the model.” This is textbook special pleading. “But the former comment was about your marbles illustration, and the second about the actual infinite”, you’ll say. But this misses the point because I’m arguing that both are instances where we have internal consistency at odds with external application. Why think the two are different?

    Your absurdity complaints seem to be based that there is no well-defined notion of subtraction with infinite numbers. This is true in the model of numbers, so why should it be different in any theoretical infinite? When the prediction of the model carries over to any reality, this is not automatically an absurdity of the reality. Your claims strikes me no differently than a claim 0 can’t be real, because you can’t divide by 0.

    One more point: you mention being at odds with external application. What external application did you have in mind? Outside of some attribute of a putative God, such as power, I can’t think of any application you would consider real.

  24. Sonny Shi Says:

    great job Chad I like the class
    sooo…
    the infinite number of abstract objects can not be true because of there not being able to express an infinite number
    (like the hotel example)
    and because of that, Platonism is wrong.
    and that human life is not relevant to the number of abstracta there are in the universe or here on earth
    also that thinking is not an abstract object
    although propositions are
    so…
    if propositions are abstract, then saying them is not?
    or is saying them concrete because of the actual physical motion of talking
    or is the meaning abstract while the action of proposing an idea is not?

  25. Keith Brian Johnson Says:

    Regrettably, I only have time for a brief post, and I haven’t yet read the full post or all of its comments, but let me address the argument for the indispensability of zero and the argument against actual infinitudes.

    “(2a.4) There is no possible world such that there are no things that are not self-identical
    (2a.5) 0 is the number of such things that are not self-identical
    (2a.6) Therefore, 0 exists in all possible worlds”

    You (Chad) acknowledge that this seems to beg the question against the nominalist, sneaking in the existence of numbers, and indeed it does. (2a.5) is like saying, “Red is the color of cardinals,” and deducing therefrom that the color red exists. But red doesn’t exist; “red” is adjectival, not nominal–it’s descriptive. Yes, we can use “red” as a noun, but that just shows how we’ve agreed to use words; it doesn’t suddenly convert the color red into a “thing.”

    Similarly, the argument

    (2a.7) a exists ≡ (∃y) a=y
    (2a.8) (number x) F(x)=0 ≡df ¬(∃x)F(x)
    (2a.9) F(x) is x≠x
    (2a.10) (∃y) y=0

    doesn’t establish the existence of the number zero. As in the red cardinal example, “zero”–or any count of “how many”–functions as adjectival, not nominal. But even were it nominal, having a label–having a name and specifying what object or sort of object it would be applicable to, were that object or sort of object to exist–doesn’t bring an object into existence.

    (2a.8) establishes how we agree to use the symbol “0″–how we agree to use the word “zero”–so that when we find lacking any instances of something, we say there are zero of them–we say that the number of them is zero. Numbers are indeed part of our universe of discourse–”number” and “three” are nouns, all right–but that doesn’t make them part of the metaphysical universe. When we find that there are no objects unidentical to themselves, we agree to express this by saying that “the number of such objects is zero.” This is a linguistic convention; our word use does not bring the number zero–or any other number–into existence.

    (D) “There does not exist an x such that Fx” is not the same as (E) “There exists an x such that x is the number of x such that Fx, and x=0.” (D) says nothing about anything’s existence; (E) says a number exists. It’s convenient linguistically to have numerical terms that are nouns, so that we can equate the two sentences; but the kind of existence in (E) is not (or need not be) metaphysical existence.

    As to the argument against actual infinitudes…. I’m not at all a fan of reifying abstracta; I don’t think of abstracta as “things.” But I can’t see this argument from the impossibility of actual infinitudes (well, as actual as a Platonist might say abstracta were, anyway) as well-founded.

    “Let m = the number of books in our infinite library, n = the number of odd-numbered books, and o the number of books numbered 4 or higher…
    (m – n) = infinity, whereas (m – o) = 4.
    But,
    n = o (since both n and o are infinite)
    It follows that we get inconsistent results subtracting the same number from m.19

    “The conclusion is that we obviously couldn’t perform such operations in the actual world, so an actual infinite cannot exist.”

    I don’t see that this conclusion follows. The idea that n=o is just wrong. The cardinality of n = the cardinality of o, true; but the operation of set subtraction is not the same as the ordinary operation of subtraction of numbers. The way the example would work in the real world, were there a real infinite library, is that removing just the even-numbered ones would leave the odd-numbered ones–an infinite “number” of them. Removing all of them after the first four would leave just the first four–a finite number of them. Neither of those is absurd. One cannot then say that n=o and therefore the same “number” of books should be left over; it simply does not follow from transfinite arithmetic.

    The real issue here, stripping away the window dressing, is that a set A may have the same cardinality as set B even though the set A is a proper subset of the set B. This is only true for infinite sets; so, of course, our ordinary intuitions about finite sets won’t work with infinite sets.

    If one wanted to apply the argument given, he would have to first come up with a more discriminating way of deciding a set’s “size” than its cardinality, so that, for example, the set of all even integers would have (intuitively) half the “size” of the set of all integers. We don’t have such a discriminating way of deciding set’s “sizes” now, for infinite sets, that I know of.

    (I note that William Lane Craig finds it simply absurd that Hilbert’s Hotel could be full but still accommodate infinitely more customers without doubling up in rooms. His sense of absurdity stems simply from his being unused to infinite hotels; all we have are finite hotels, so he’s used to how they work. If we lived in a world with infinite hotels, he’d be used to how *they* work, too, and probably wouldn’t find their workings absurd.)

  26. Keith Brian Johnson Says:

    I’m sorry–I don’t understand why “Mary is watching television” has to be a *thing* in order for truth to be grounded. Yes, “Mary is watching television” is true if and only if Mary is watching television; but that doesn’t mean that the state of affairs of Mary’s watching television *exists* as a *thing*; the noun phrase “state of affairs” is introduced simply as a way of talking about Mary’s watching television; it is only a linguistic device. The proposition “Mary is watching television” is a description of empirical reality; it is our way of talking about empirical reality–of talking about Mary’s watching of television. Our speech about empirical reality is true if and only if it accurately describes that reality; that doesn’t mean there is a thing called a “proposition” that stands in a relation to another thing called “empirical reality.” I’m not seeing why a correspondence theory of truth (with respect to empirical reality) would entail the *existence* of propositions as *things*.

    Somewhat similarly, I don’t see why “2+2=4″ could not be viewed as ontologically a “useful fiction” without entailing that it has no truth value. One might say that it is in fact grounded in empirical reality–that when we say that two plus two equals four, what we really mean is that anytime we have two discrete objects and two other discrete objects and put them together discretely (i.e., without merging, the way drops of water merge), we wind up with four discrete objects. Or one might say that it is a truth by fiat: One might say that we make certain arithmetic statements axioms, giving them truth by fiat, and then all arithmetic theorems have that same truth by fiat. Or one might say that the axioms are chosen so as to mirror empirical reality, so that they have the same sort of truth as “Mary is watching television,” and then describe theorems’ truth as deriving from that of the axioms. At any rate, I don’t see why “2+2=4″ has to be supposed to be an existing thing in order to have a truth-value.

    I’m not a Platonist (I don’t know whether I’m a nominalist or a conceptualist; I don’t think, though, that I have thoughts–i.e., I think but do not have things called “thoughts” residing in my head, in much the way in which my leg hurts [i.e., I hurt in-my-leg-ishly] but I do not have a thing called a “pain” residing in my leg), but I’m not sure that the Platonist can’t answer (2b1) pretty much as you suggest. Physicists didn’t have to have direct evidence of quarks in order to infer their existence. Similarly, the Platonist might say that he infers the existence of abstract objects from facts about the world. The inference is merely epistemological, but it is an inference *to* existence.

    Against psychologism, you write, “Say some abstract object O is the concept of some human mind at time t1. Surely there must have been times ti–tj prior to t1 such that there were no human mind that had O as a concept. It would be less congenial to say that at t1 O came into existence than to say O must have existed as a concept of a necessary mind during ti–tj. This argument runs backwards as well: given their voluminous and complex nature, there must be abstract objects that have not yet nor will ever be concepts in human minds.” But why would it be “less congenial” to suppose O’s coming into existence at t1 than to say that it already existed as the concept of a necessary mind? This is where I think that reifying mental phenomena is a problem: Instead of thinking of human beings as thinking and as hurting, the realist conceptualist with respect to abstracta thinks of human beings as having thoughts and as having pains (well, OK, pains aren’t abstract objects, but perhaps they’re illustrative), and then, thinking of them as *things*, has to wonder *how they came into existence*–and then concludes that human thinking wouldn’t suffice for the task of their creation (although why it strikes one as more plausible that an unevidenced necessary, omniscient mind would do the job better, I’m not sure). Similarly, thinking of as-yet-unthought-of abstracta as *things* that are not yet “contained in” any human mind invites the question of where they are (although they could simply be thought of as nonexistent until contained in some human mind). Yet, if abstracta are not thought of as *things*–if human beings abstract but do not think of things called “abstractions,” just as we do not have pains residing in our legs (where were those pains before we hurt?)–then the problem of how they came into existence vanishes. (Of course, the big problem I, and everyone else, I think, am left with, is how it is that material bodies are able to be aware at all–David Chalmers’s “hard problem.”) Similar considerations would apply to Quentin Smith’s infinitely complex conjunction of all true propositions (if such a thing did not turn out to be bedeviled by Russell’s Paradox-type problems), of course.

    The conclusion that there is a necessary, omniscient mind seems to be one that relies on thinking of abstracta (and of propositions) as *things*, and not only that, but as things whose existence cannot simply be accepted but must be accounted for in terms of an existing cause for being. It is a cosmological argument applied to abstracta (which, frankly, seems like a weaker argument than a cosmological argument applied to concreta, since it’s harder to question the existence as things of concreta than of abstracta). Yes, some propositions are eternally true, but that simply means that they will be true no matter who speaks them (or thinks them) or when or where they are spoken (or thought). Yes, some propositions are analytically true, but that just means that they will be true by virtue of their meanings no matter who speaks them (or thinks them), and so on.

    On to the comments.

  27. Keith Brian Johnson Says:

    Chad, I assume that One Brow could say that our statements about abstract objects were useful fictions without saying that that statement itself was only a useful fiction, in the following way: We agree to use words in certain ways, and those agreements are quite real, in that we really do make those agreements; and the statement that our statements about abstract objects are useful fictions is a statement about how we agree to use words. We agree to use certain nouns as though they were really naming actual objects, even though they aren’t, because we find it useful to talk that way. That’s a statement about how we agree to use language, not a statement about abstract objects. (If it also includes the statement that there are no abstract objects, then that, too, is not a statement about abstract objects, from his point of view, since he is asserting that there are no abstract objects at all; it is a statement about what there isn’t, not about what there is. And clearly, what there isn’t isn’t an abstract object: “Nothing” is a noun, but it does not name anything.)

    As to the argument using

    “(P) The notions of “true” and “false” are in and of themselves merely useful fictions

    which is just to say

    (P’) For any proposition p, p is such that it is truth-valueless,”

    it seems clear that in order to represent a useful fictionalist’s position, (P) would have to read more like, “The notions of ‘true’ and ‘false’ are in and of themselves merely useful fictions when applied to statements about abstract objects,” and (P’) would have to read more like, “For any proposition p about abstract objects, p is such that it is truth-valueless.” Then the useful fictionalist would still be OK, since (P) isn’t a statement about abstract objects but is instead a statement about how we will apply the notions of truth and falsity to statements about abstract objects.

    ———————-

    It’s missing the point to think that if there were actual infinitudes, the operation of subtraction could be performed upon them just as they can be upon actual finitudes. One could remove infinitely many books from an infinite library, choosing which infinite collection of books to remove; and which was removed would affect whether infinitely many books or only finitely many (and if finitely many, then how many) remained; but one wouldn’t apply the operation of subtraction to infinitudes. (You use the example 3-5=-2. But your own example, if it shows anything, shows that you can’t subtract five marbles from three marbles; it doesn’t show that you can’t have three marbles or five marbles. Why, then, should a problem with “subtraction of infinitudes” mean that you can’t have infinitely many of something?) And as for the failure to define an inverse operation–well, I’m not quite sure what addition would be for infinite sets, but if there were actual infinitudes, one could presumably take the infinitely many items in group A and put them in the same collection as the infinitely many items in group B, and then remove exactly those same items that started out in group A–the inverse operation of putting them into the same collection!–and be left with exactly those items in group B.

    ——————–

    (P’’) p is true for S and ~p is true for S*

    Naturally, one may reply that P” is true for S or S* if he is a relativist, but false for S or S* if he is not. A truth-relativist cannot assert the absolute truth of any of his claims, but can only assert their truth for him–including the correctness of truth-relativism. But I don’t think he’s inconsistent in so doing.

    The real problem is with why one should accept that truth is always truth for S, rather than correspondence with reality (or some such).
    —————-
    One Brow: I’m not sure you understand what is meant by “indispensability,” in what I take to be Chad’s usage. 0’s indispensability doesn’t mean that you can’t live without it; it doesn’t mean that you can’t have concepts without it; it doesn’t mean that you can’t do mathematics without it (although your Indian example, in which a blank space was left, showed that Indians did have the concept); what it means is that you cannot completely conceptually analyze reality without it. It means that it is indispensable to a proper understanding of the world. Specifically, it’s indispensable to a complete ontology that accurately captures all that exists.

  28. Timothy Underwood Says:

    There are a lot of problems I feel like exist here, but the biggest point at the moment, is that even given you show (3)

    I am not at all convinced that,

    “(3) Abstract objects exist and are mental concepts”

    and

    “It is the view that anything that exists is ultimately made up of physical components. … There are no purely mental realities in a naturalistic account of the world.”

    actually are clearly incompatible. It seems to me very, very possible we are using different meanings of the phrase “mental reality” and “exist”. To show a contradiction it is necessary to show that the terms are actually the same, or at least the same in a relevant manner.

  29. Chad McIntosh Says:

    Timothy,

    Thank you for the comment.

    I suspect you must have only skimmed the argument. Read the very next couple of sentences after the quote to which you allude:

    Armstrong’s and Neilson’s characterization of naturalism is clearly incompatible with (3). The defender of (N), on the other hand, can yet consistently maintain (3), though not without considerable tension (for reasons outlined above).

    I agree with you, and that’s why I use Oppy’s characterization of naturalism (which is (N) in the paper), not Neilson’s. My quoting Robinson, Mackie, Armstrong, and Neilson was not to argue for strict inconsistency between (3) and (N), but to highlight tension between the two. I make this quite clear all throughout the paper, such as when I say

    There is no prima facie inconsistency between (N) and the existence of abstract objects. However, there are good reasons for thinking there is at least tension between the two—a burden theism doesn’t bear. In other words, even if one could draft a realist ontology of abstracta compatible with naturalism, a better, more-at-home account can be offered by theism.

    It is only after (4) is introduced does strict inconsistency result.

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