An Amended Modal-Epistemic Argument for God’s Existence

Several years ago I was introduced to a clever and fascinating argument, developed by a philosopher named Emanuel Rutten, which attempts to demonstrate the existence of God from two key premises: (i) that anything which is possibly true is possibly known, and (ii) that it is not possible to know that God does not exist, from which it logically follows that (iii) God exists. The argument has some intuitive appeal to me, though I was initially skeptical about the second premise (skeptical, that is, that the atheist could be persuaded to accept the second premise). I had also heard certain criticisms of the argument which seemed to present nearly insuperable objections to it; although I started working on responses to those objections, I eventually moved on to other philosophical inquiries leaving this argument (and my many notes on it) to gather proverbial dust on my old hard drive. Recently, however, I decided to revisit the argument and use a variation on it in the context of a semi-formal online debate. I was shocked by my interlocutor’s reaction; although he had not been shy about sinking his teeth into every other argument I had presented for theism (from the cosmological argument from contingency, to the transcendental argument from the laws of logic, to a version of the moral argument, to the modal-ontological argument), I received radio-silence when presenting this argument. After several days of him reflecting upon the argument, he eventually rejoined by saying that he couldn’t think of a single criticism, but that he was convinced the argument was bad for some reason he was unable to articulate. This made me want to revisit the modal-epistemic argument for God’s existence and see if it couldn’t be salvaged in light of certain criticisms of which I am aware.

The basic intuition behind Rutten’s argument is that reality’s being intelligible is somehow connected to, and explained by, the existence of a God-like being. This same intuition seems to lurk behind Bernard Lonergan’s argument for God in the nineteenth chapter of his magnum opus, Insight, where he made the tantalizing claim (for which he argued at great length) that “if the real is completely intelligible, God exists. But the real is completely intelligible. Therefore, God exists.”1 There is also a subliminal connection here, I think, even to C.S. Lewis’ argument from reason. The same intuition is also bolstered, to some extent, by Fitch’s paradox, which is a logical proof developed by the philosopher and logician Frederic Fitch in 1963. Fitch was able to prove, using prima facie uncontroversial assumptions, that “necessarily, if all truths are knowable in principle then all truths are in fact known.”2 This philosophical finding was taken to be paradoxical by many, but it sits exceptionally well with the theist who affirms that omniscience is exemplified by God. What these observations show, I think, is that the intuition behind Rutten’s argument is widely shared (at least among theists) and may be well motivated.

The bare-boned sketch of Rutten’s argument can be outlined as follows:

  1. All possible truths are possibly known (i.e., if there are logically possible worlds in which P is true, then there will always be a subset of such worlds in which P is known).
  2. It is impossible to know that God does not exist.
  3. Therefore, God necessarily exists.

It has to be said straight-away that this is an over-simplified formulation of his argument; we will come, in due course, to his more measured articulation of the argument, but the rough sketch provided by this syllogism will help us lay the groundwork for the actual argument.

Rutten stipulates the following relatively modest definition of God, for the purposes of his argument; God is the personal first-cause of the world (where the world is the whole of contingent reality). Since that logically implies that God is incontingent, I will abbreviate this as ‘IPFC.’ He also specifies that, for the purposes of the argument, he means the following by knowledge: “A conscious being… knows that proposition p is true if and only if p is true and the being, given its cognitive situation, cannot psychologically but believe that p is true.”3 More precisely, for any P, if some conscious being B cannot psychologically help believing that P is true, then P satisfies at least one of the following four conditions for B: “(i) The proposition is logically proven; (ii) the proposition is obviously true, i.e. intuitively self-evident; (iii) the proposition is grounded in indisputable experience; or (iv) the proposition is based on indisputable testimony.”4 This makes it obvious that Rutten means that something is known if and only if (a) it is true, and (b) given some conscious being’s cognitive situation, that being, whose cognitive faculties aren’t malfunctioning, cannot psychologically help believing that it is true. In what follows I will refer to this peculiar kind of knowledge as knowledge*, instances of knowing satisfying these conditions as knowing*, et cetera.

The first premise seems to flow directly out of the perennial philosophical commitment to the world’s intelligibility. Arguably, to be intelligible the world has to be the kind of thing which is knowable* in principle (if not always to us, due to some limitations of our cognitive faculties, then at least to some logically possible intellects with different cognitive faculties). This philosophical presumption has, Rutten hastens to note, “led to extraordinary discoveries”5 in science. In fact, it seems to be a fundamental pillar of science itself, for science is predicated on the assumption of the world’s intelligibility. The second premise also seems prima facie plausible; it is, somewhat ironically, appealed to confidently by many agnostics and some atheists.

The argument is, in its rough form, susceptible to a myriad of informative objections. Consider, for instance, the possibly true proposition: “God understands my reasons for being an atheist.”6 The proposition, although plausibly possibly true, is not knowable – for knowledge requires belief, but no atheist can believe the proposition. Similarly the proposition “there are no conscious beings”7 may be possibly true but is also not rationally believable. To avoid these kinds of counter-examples Rutten stipulates that his first premise should only quantify over rationally believable propositions. He thinks it is reasonable to exclude rationally unbelievable propositions, and that this way of restricting his first premise is not ad hoc, for it seems intuitively plausible that all rationally believable possible truths are knowable. Requiring the propositions of the relevant sort to be both (possibly) true and rationally believable navigates the argument away from obvious counter-examples. There are other counter-examples, however, and Rutten discusses some. First, consider a proposition like “‘John left Amsterdam and nobody knows it.’”8 This seems possibly true and obviously unknowable, even though it could be argued to be rationally believable. To deal with objections like this Rutten introduces a distinction between first-order propositions and second-order propositions; first-order propositions, he says, are directly about the world, whereas second-order propositions are about people’s beliefs about the world. Rutten then decides to limit the first premise of his argument to truths expressed by first-order propositions. In this way he blocks cute objections from propositions like ‘there are no believed propositions.’

Then he states his argument9 more formally in the following way (I have changed the wording very little, and added nothing of consequence):

1. If a rationally believable first order proposition is possibly true, then it is knowable* (first premise),
2. The proposition ‘IPFC does not exist’ is unknowable* (second premise),
3. The proposition ‘IPFC does not exist’ is rationally believable (third premise) ,
4. The proposition ‘IPFC does not exist’ is first order (fourth premise),
5. The proposition ‘IPFC does not exist’ is not possibly true (from 1, 2, 3 and 4),
6. The proposition ‘IPFC does not exist’ is necessarily false (from 5),
7. The proposition ‘IPFC exists’ is necessarily true (conclusion, from 6).

The third premise is either true, or else atheism is irrational. The fourth premise is self-evidently true. The fifth premise follows logically from 1,2,3 and 4. Six follows logically from five. Seven follows logically from six. So the key premises are 1 and 2. The first premise is very plausible insofar as its negation would imply that reality is not intelligible, but to deny that reality is intelligible seems absurd. That reality is intelligible (if not to us then at least in principle) seems to be a fundamental commitment of epistemology. However, if reality is intelligible, then for any first-order rationally believable proposition P, if P is possible then P is possibly known*. Can we know this premise in the strong sense of knowledge used within the argument? Maybe (e.g., perhaps it is obviously true, i.e. intuitively self-evident), but that’s also irrelevant; all we need is to ‘know’ it in the more general sense (i.e., having a true justified belief – allowing for whatever epistemology you’d like to use in order to qualify ‘justified’) in order to know (as opposed to know*) that the conclusion is true. 

The second premise is plausible given that, for the purposes of the argument, ‘knowledge’ is defined as satisfied just in case at least one of the four stipulated conditions are satisfied. However, God’s non-existence cannot be logically proven (if it can, then obviously this and all other arguments for God’s existence are worthless). On atheism, the proposition that God does not exist is not self-evidently true. On atheism, the proposition ‘God does not exist’ cannot be grounded in indisputable experience. On atheism, the proposition ‘God does not exist’ cannot be believed on the basis of indisputable testimony. It follows that the second premise is true. So, the argument looks sound, at least at first blush.

One immediate reaction to this argument is to suggest that it can be parodied by a parallel argument for atheism by substituting the second premise for: 2.* The proposition “God exists” is unknowable*. However, this is naïve; in at least one possible world in which God exists, plausibly God knows* that the IPFC (i.e., himself) exists, but in no possible world where no IPFC exists can anyone know* that no IPFC exists. As Rutten explains:“on the specific notion of knowledge used for the argument… logical proof, intuition, experience and testimony exhaust the range of knowledge sources, and none of them suffices to know that God does not exist.”10

Years ago now I heard one very interesting objection which I will try to reproduce as fairly as my memory and skill will allow. The objection basically maintains that if God could know* that the IPFC (i.e., God) exists, then it is possible for at least one atheist in at least one logically possible world to know* that the IPFC does not exist. Rutten suggests, in the paper, that “God’s knowledge that he is God – if possible – is an instance of (iii) (or (ii)),”11 meaning that it is either “obviously true, i.e. intuitively self-evident; [or]… grounded in indisputable experience.”12 But what experience could possibly establish the indubitability of being the IPFC? For any experience you can imagine having (if you were God), it seems logically possible that it is the result of an even greater being who created you with the purpose of deceiving you into thinking that you are the IPFC. What about intuitive self-evidence? Well, if it is possible for God to simply look inward and, through introspection, discover his relations (for, to be the IPFC is to bear certain relational properties, such as that of being first-cause), then why can’t there be a logically possible world in which an atheist introspects and discovers that she lacks any relation to an IPFC? If it is logically possible for the IPFC to introspectively survey its own relational properties, then why can’t a logically possible atheist do the same?

I think the best answer to this is to note that it may be possible to introspectively discover at least some of one’s essential properties (as opposed to merely accidental properties). I can know, by rational reflection, that I exist (cogito ergo sum), that I am a thinking thing, that I am either contingent or not omniscient, et cetera. I can also deduce from what I discover as self-evident through introspection that other facts happen to be true, such as that there exists something rather than nothing. So, coming back to God, perhaps God can know by introspection that he is incontingent, personal, and has some uniqualizing properties13 (that is, properties which, if had at all, are had by no more than one thing) etc. – and perhaps that means that he can deduce that he is the only being which could be an IPFC in principle, and that he is an IPFC just in case a contingent world exists. But, he could plausibly know* from indisputable experience (of some sort) that a contingent world exists. Therefore, he could deduce and know* that he is the IPFC. If atheism were true, no being would have, as an essential property, a lack of any relation to an IPFC. Lacking a relation cannot be an essential property, so there’s no reason to think it could be introspectively discovered that one lacks a relational property to the IPFC. Moreover, unless the atheist can actually produce (perhaps with the aid of premises introspectively discovered as self-evident) a logical proof that the IPFC does not exist it seems they cannot know* that no IPFC exists. So while this objection is extremely interesting, I do think that it fails; it is reasonable to maintain that, possibly, God knows* that the IPFC exists, and it does not plausibly follow that an atheist possibly knows* that no IPFC exists.

Another objection might come from considering large facts. Take, for instance, what Pruss has called the Big Conjunctive Contingent Fact (BCCF),14 and let’s take the sub-set of that fact which includes only first-order, rationally affirmable facts (for simplicity, I will abbreviate this as the BCCF*). The BCCF* is plausibly comprised of infinitely many conjuncts, and at least is possibly comprised of infinitely many conjuncts. Is this possible truth, the BCCF*, possibly known? I think it is possible so long as there is possibly a being with an infinite capacity for knowledge (or else, perhaps, an actually infinite number of beings with some finite capacity for knowledge not all of which are such that a discrete set of first-order rationally affirmable truths would have been beyond its ken). But, assuming there cannot be an actually infinite number of beings, doesn’t that presuppose something like theism, by presupposing the possible exemplification of omniscience (here we assume that BCCF*⊃BCCF, and that any being which knows the BCCF* also knows all analytic truths)? After all, the Bekenstein bound15 is generally taken to imply “that a Turing Machine with finite physical dimensions and unbounded memory is not physically possible.”16 However, it seems senseless to suggest that there could be a physical object (like a brain, or some other kind of computer) which is actually infinitely large. Therefore, doesn’t the first premise presuppose something like theism insofar as it presupposes the exemplifiability of omniscience or at least an intellect with an actually infinite capacity for knowledge? That would make the argument ostensibly circular.

First, the IPFC needn’t be omniscient even if it knew the BCCF*. Second, and more importantly, the IPFC isn’t being presupposed to be omniscient, or even knowledgeable enough to know the BCCF*. Third, a being’s being omniscient is necessary but insufficient for the truth of theism. Fourth, I’m not sure whether it is senseless to talk about infinitely large physical objects, or (actually) infinitely many beings, but I am relatively sure that most atheists have a vested interest in allowing for those kinds of possibilities in order to avoid conceding important premises in some (Kalaam) cosmological arguments. So this attempted charge of subtle circularity seems wrong.

[I should grant this this last objection could be accused of being a straw man erected by none other than myself; to that I just briefly want to say that I had originally thought that there may be an objection here, but as I tried to write the objection down clearly it seemed to crumble in my hands. Having already written it out, and having found it interesting to reflect upon it whether or not it is a viable objection at all, I decided to keep it in this final draft.]

I’m sure there are other possible objections which I would have been better able to iterate or anticipate had I done so years ago when this argument, and some objections to it, were still fresh in my mind. However, my sense is that that will do for an introduction to the argument. What makes this argument really exciting, I think, is that it, as Rutten notes, “does not fall within one of the traditional categories of arguments for the existence of God. For it is not ontological, cosmological or teleological. And it is not phenomenological either, such as for example the aesthetic or moral argument[s] for God’s existence.”17 The argument, whether sound or unsound, is doing something genuinely novel, at least for the analytic tradition of the philosophy of religion.

Rutten ends his short paper on an optimistic note which may be appropriately appended here, and this is where I will end my short excursus:

As I mentioned in the introduction, I propose to refer to the argument as a modal-epistemic argument. Ways to further improve it may be found, just as has been done with arguments in the other categories. I believe that if this happens, the prospects for the argument are rather promising.”18

1 Bernard Lonergan, Insight: A Study of Human Understanding, Collected Works of Bernard Lonergan, vol. 3, ed. Frederick E. Crowe and Robert M. Doran (Toronto: Toronto University Press, 1992), 695.

2 Brogaard, Berit and Salerno, Joe, “Fitch’s Paradox of Knowability”, The Stanford Encyclopedia of Philosophy (Winter 2013 Edition), Edward N. Zalta (ed.), URL = <https://plato.stanford.edu/archives/win2013/entries/fitch-paradox/&gt;.

3 Emanuel Rutten, “A Modal-Epistemic Argument for the Existence of God.” Faith and Philosophy (2014), 3.

4 Emanuel Rutten, “A Modal-Epistemic Argument for the Existence of God.” Faith and Philosophy (2014), 4.

5 Emanuel Rutten, “A Modal-Epistemic Argument for the Existence of God.” Faith and Philosophy (2014), 14.

6 Emanuel Rutten, “A Modal-Epistemic Argument for the Existence of God.” Faith and Philosophy (2014), 7.

7 Emanuel Rutten, “A Modal-Epistemic Argument for the Existence of God.” Faith and Philosophy (2014), 8.

8 Emanuel Rutten, “A Modal-Epistemic Argument for the Existence of God.” Faith and Philosophy (2014), 9.

9 Emanuel Rutten, “A Modal-Epistemic Argument for the Existence of God.” Faith and Philosophy (2014), 10-11.

10 Emanuel Rutten, “A Modal-Epistemic Argument for the Existence of God.” Faith and Philosophy (2014), 2.

11 Emanuel Rutten, “A Modal-Epistemic Argument for the Existence of God.” Faith and Philosophy (2014), 5.

12 Emanuel Rutten, “A Modal-Epistemic Argument for the Existence of God.” Faith and Philosophy (2014), 4.

13 Alexander R. Pruss, “A Gödelian Ontological Argument Improved Even More.” Ontological Proofs Today 50 (2012): 204.

14 Alexander R. Pruss, “The Leibnizian cosmological argument.” The Blackwell Companion to Natural Theology, ed. W.L. Craig and J.P. Moreland (2009): 24-100.

15 See: “Bekenstein Bound,” Wikipedia, accessed March 24,2017. https://en.wikipedia.org/wiki/Bekenstein_bound

16“Bekenstein Bound,” Wikipedia, accessed March 24,2017. https://en.wikipedia.org/wiki/Bekenstein_bound

17 Emanuel Rutten, “A Modal-Epistemic Argument for the Existence of God.” Faith and Philosophy (2014), 28.

18 Emanuel Rutten, “A Modal-Epistemic Argument for the Existence of God.” Faith and Philosophy (2014), 28.

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Notes on a Transcendental Argument from Logic

Nearly ever since I was first exposed to transcendental argumentation through listening to that famous debate between Greg Bahnsen and Gordon Stein,1 I have retained the intuition that there is an interesting potential argument from the fact that there are necessary propositions (necessary, that is, simpliciter) to the conclusion that there is a necessary mind. While the analysis of what it means to be a necessary mind will fall short of the God of perfect being theology or classical theism, it will still provide a being which so resembles God that it significantly undermines atheism. This being may not have all the superlative attributes, but it will be a metaphysically necessary immaterial spaceless timeless being with an intellect (and whatever that entails), et hoc omnes intelligunt Deum. However, to avoid the charge of using St. Thomas’ famous phrase in order to paper-over the chasm between my conclusion and full-blown theism, I will state the conclusion more modestly in terms of which the good old reverend Bayes would approve. Enjoy;

1) There are laws of logic.
2) Logical laws are identical to necessary propositions (exempli gratia [P v ~P])
3) Therefore, there are necessary propositions.
4) Propositions are not real entities which exist mind-independently, but are mind-dependent (i.e., there is no proposition for which there is not at least one subvenient mind).
5) A necessary truth is a truth which obtains in all logically possible worlds.
6) Necessary truths are either grounded in at least one contingent mind, or at least one incontingent mind.
7) There are logically possible worlds without any contingent minds.
8) Therefore, there must be at least one necessary mind.
9) If there is at least one necessary mind then it is a being with intellect (plausibly knowing all necessary truths), which is immaterial (spaceless, timeless) in nature.
10) The conditional probability of theism is, ceteris paribus, greater than the conditional probability of not-theism on the condition that there is at least one metaphysically necessary immaterial being with intellect. 
11) Therefore, theism is probably true, 
ceteris paribus.

There are plenty of points at which one could still object to this argument, but it seems to me that most objections are philosophically more costly than the conclusion. One might also just accept the conclusion but deny that, in fact, things really are equal (i.e., cetera non sunt pariba) in this case. For instance, the objector could insist that there are no propositions which are ‘necessary’ in the sense required here (that is, necessary simpliciter – not a merely model-dependent necessity). They might also insist, for some odd reason, that there are not possible worlds without contingent minds, or that those worlds are possible in a merely model-dependent way while other possible worlds are possible simpliciter. That would be pretty wild. Another might argue that the existence of a metaphysically necessary immaterial mind doesn’t raise the conditional probability of theism at all (maybe because the probability of theism is ‘0’ – or because it is ‘1’). Somebody could, of course, deny the major premise, that there are laws of logic. Somebody may also insist that laws of logic are not identical to the propositions which express them (though that seems to reify them so much as to put the objector, for other reasons, in the near occasion of belief in theism anyway). Alternatively one may think that each premise on its own seems more plausibly true than false, but that the collection of them together seems to have a upper-bounded probability of lower than or equal to 0.5, and that would be a principled way to object.

Edit*: it occurs to me that there’s no way of which I’m aware to really set an upper-bound on the probability of a conclusion. What the objector could say, then, is either that the conclusion just seems to be no more likely than 0.5 (notwithstanding the plausibility of the individual premises), or that the premises collectively set a lower-bounded probability on the conclusion of less than or equal to 0.5, in which case the argument fails to be compelling.

To be fair, this argument of mine very likely draws significantly from the influence of James N. Anderson and Greg Welty,2 whose argument seems, to me, much better than what often passes for responsible argument among presuppositionalists (among whom, I should take a moment to clarify, I adamantly do not count myself).

1 For those interested, you can find the audio of the debate, and the transcript (because the audio is really not great) at the following two links: https://youtu.be/ZLZdOGCE5KQ?t=34s and http://www.brianauten.com/Apologetics/apol_bahnsen_stein_debate_transcript.pdf

2 James N. Anderson and Greg Welty, “The Lord of Non-Contradiction: An Argument for God from Logic” Philosophia Christi 13:2 (2011). http://www.proginosko.com/docs/The_Lord_of_Non-Contradiction.pdf

A Semantic Problem with Platonism

Previously noted sympathies notwithstanding, I have grave and seemingly intractable problems with Platonism. Perhaps the most severe of these follows from Christian Theism, which suggests that there is one necessary being, God, without whom nothing which exists would exist (in the sense that all other things which exist are ontologically dependent upon God). This is the confession of the central creeds of the faith, starting with the Nicene-Constantinopolitan creed (325-381 A.D.), referred to affectionately by Catholics simply as the symbol of faith. There are, of course, (in my view, quisling) children of the Church who argue that the “all” in “all things visible and invisible” does not quantify over universals, but I think that interpretation exceptionally dubious. However, this is inside baseball at its worst, and bound to leave those uninterested in theological minutia bored or irritated, if not entirely lost.

There is, however, one problem I have with Platonism which is at once subtler, less indirect and more accessible than my principal objection. I have not yet developed this line of thought, and I am unacquainted with any literature which successfully fledges this out into a respectable argument (on that note, if anyone is aware of sources which further develop the thought I am about to present, I would welcome their reading recommendations), but I mean, here, merely to register a suspicion; to gesture, in a vague and lackadaisical way, in the general direction of a possibly indissoluble difficulty. As such, I abandon any pretense to having found a proof (in the form of a compelling falsifier) of anything and submit the comparably modest suggestion that I think I have found a problem. With that caveat, let me invite the reader into the weeds.

There is, I suspect, an under-appreciated difficulty with the Platonist’s claim that universals ‘exist.’ This, as I interpret it, is the central claim of Platonism; Platonism, if it signifies anything, signifies that for any x, if x is a universal then x exists. Symbolically:

(∀x)(Ux⊃Ex)

(Where Ux means “x is a universal” and Ex means “x exists.”) This helps to differentiate Platonism from other competing views, such as neo-Meinongianism.[1][2] The definition of full-blooded Platonism goes further than this, perhaps, but it certainly signifies no less than this.

Let us bracket, for the moment, concerns about using ‘exists’ as though it were a (first-order) predicate. I note in passing, however, that if one insists on existence being a second-order predicate indicating that the thing to which it applies has at least one first-order property, then platonic forms will have properties, and there an interesting puzzle arises, for all (first-order non-vacuous standalone) properties are universals, thus implying that universals may be properties of universals. Indeed, there may be cases where two (or more) universals are symmetrically related to each other as each other’s properties (each one being a property of the other(s)).[3] This is all both interesting and moot, for even if all properties are universals, not all universals are properties, and the argument is, as far as I can see, compatible with any (metaphysical or semantic) analysis of ‘existence.’

It should also be appreciated that some views on universals may carry the implication that existence is a first-order predicate after all. I am not an expert on neo-Meinongianism, but it seems, on its face, to entail that existence is a property (for it maintains that there are actual non-existent objects, as well as actual existent objects).[4] The Stanford Encyclopedia of Philosophy entry under Alexius Meinong does, however, note the following:

“Meinong’s distinction between judgments of so-being and judgments of being, combined with the indifference principle that being does not belong to the object’s nature (so-being), reminds one of Kant’s dictum that being is not a real predicate. Meinong did not accept the ontological argument either, and argued that “being existing” is a determination of so-being and can in a certain sense be properly accepted even of the object “existing golden mountain,” and, say, even of the object “existing round square,” whereas “existence”, which is a determination of being, will no more belong to the one than it does to the other (1907, §3; 1910, §20, 141 [105]).”[5]

So perhaps it is unclear whether Meinong’s view, properly interpreted, does imply that existence is a first-order predicate. In any case, it may have this implication, and that suffices for maintaining that, for all we now, Platonism may have this implication as well. For the purposes of this post, therefore, I ask that the reader give me some leeway in allowing me to speak as though existence is a property.

A Platonist, as here understood, is committed to the existence of universals, and universals are those things which can be said of many. Existence, however, can be said of many. Existence is, therefore, a universal, and the Platonist is committed to its existence. But now we draw nearer to the problem. How is it that one platonic form can be a constitutive property of itself? Can existence be a property of existence? If existence must be said to exist, either it will be said to exist in some non-univocal sense, or else the statement will become transparently bankrupt of propositional content. In the first case, something may be said to exist either equivocally or analogously (the only alternatives to univocity). If equivocally, I defy (with nearly hubristic confidence) anyone to make heads or tails of the statement. On the other hand, analogous predication, being already difficult to make good sense of, leaves me, here, feeling as nauseous as I imagine it must feel to be lost at sea. At least with Theism I can make some headway with this philosophically abstruse doctrine, since there is a paradigmatic exemplar to be intimated (along with some reasons for suspecting that the created order would intimate its creator, in much like the way structural realists in the philosophy of science believe scientific theories intimate reality). How, though, can we make sense of analogously predicating predicates of predicates, much less predicating predicates of themselves? How can first-order properties have first-order properties which, themselves, have their subjects as first-order properties? Analogy does nothing to lubricate the discussion at this point.

Am I too infected with Theism to see what sense this could make? Even if we turn to a close (and theistic) cousin of Platonism, namely ‘absolute creationism,’[6] (according to which platonic forms do exist, but (necessarily?!) proceed necessarily from God as creatures), we find nothing which alleviates the perplexity. In fact, it adds to the perplexity by introducing the so-called bootstrapping problem, for there are properties which, in order for God to create them, God would already have to possess (if existence is a property, then it serves as a fine example; another example is the property of powerfulness, which God would need in order to create the property of powerfulness).

So where does all this leave us? Here, I’m afraid, my thinking proceeds with less precision than I am comfortable with, and with embarrassing, though seemingly unavoidable, obviousness. This is precisely why I proceed with such caution, as though clumsily feeling my way through a thick fog. I avoid committing myself with any rigidity to this point. Nevertheless, if I am right then Platonism turns out to be highly sophisticated gobbledygook. At least this will be true of wholesale Platonism (as opposed to constrained or qualified forms of Platonism, such as those prefixed with terms like ‘mathematical,’ ‘prepositional,’ ‘evolutionary,’ et cetera).

Commentaria welcome.

[1] Maria Reicher, “Nonexistent Objects,” in The Stanford Encyclopedia of Philosophy, edited by Edward N. Zalta (2015), accessed November 26, 2016. http://plato.stanford.edu/archives/win2015/entries/nonexistent-objects/

[2] Johann Marek, “Alexius Meinong,” in The Stanford Encyclopedia of Philosophy, edited by Edward N. Zalta (2013): http://plato.stanford.edu/archives/fall2013/entries/meinong/  adds: “… in the appendix to his 1915 (p. 739–40) Meinong himself interprets such incomplete objects as platonic universals without being (see also 1978, 368), and he also states there: “what words mean [bedeuten] is the auxiliary object, and what they designate [nennen] is the target object” (1915, 741).”

[3] Existence is a property of Being, and Being is a property of Existence, no? This is unclear due to my total lack of clarification (through conceptual analysis) of these terms, but it seems intuitive enough for the moment. I cannot see why there couldn’t be some relatively clear-cut case of this pernicious symmetry.

[4] I believe Vallicella argues that it does somewhere in: William F. Vallicella, A Paradigm Theory of Existence: Onto-theology Vindicated. Vol. 89. Springer Science & Business Media, 2002.

[5] Johann Marek, “Alexius Meinong,” in The Stanford Encyclopedia of Philosophy, edited by Edward N. Zalta (2013): http://plato.stanford.edu/archives/fall2013/entries/meinong/

[6] Thomas V. Morris and Christopher Menzel. “Absolute creation.” In American Philosophical Quarterly 23, no. 4 (1986): 353-362.

Math, therefore God?

“Monsier! (a+bn)/n=x, donc Dieu existe; répondez!”[1]

Thus (allegedly) spoke the mathematician Leonard Euler when, at the invitation of Russian Empress Catherine the second, he confronted Denis Diderot in a (very short) debate on the existence of God. Diderot, who was not very good at math, was dumbstruck; he had absolutely no idea how to even begin responding to such an argument. In fact, he couldn’t even understand the argument, and Euler knew it! The court laughed him literally out of town (he promptly asked the Empress for leave to return to France). The formula, of course, is entirely meaningless, and may have been sleight of hand on Euler’s part (making his argument mathemagical rather than mathematical). Additionally, the anecdote has survived only in sparse notes (of dubious historical relevance) here and there with probably varying degrees of accuracy, so it is anyone’s guess what Euler actually meant. This amusing anecdote does, however, invite us to think about what arguments there could be, in principle, from mathematics for the existence of God.[2] Without offering much commentary on how promising these arguments are, I want to distinguish three viable (or, at least, viably viable) types of arguments which could be constructed.

The Argument from Mathematical Beauty

Although the formula Euler originally spouted off didn’t signify anything of mathematical (or philosophical) consequence, the beauty of Euler’s equation, eiπ + 1 = 0, gave rise to the apocryphal anecdote that Euler argued “eiπ + 1 = 0, therefore God exists.” There is (mathematicians tell us) a sublime mathematical beauty in this equation, and there is no obvious or intuitive reason why it is true. What is so special about this equation? One savvy commentator I ran across online put it so nicely I feel compelled to quote him:

“It’s a sort of unifying identity in mathematics, containing each of the fundamental operations (additive, multiplicative, exponential) and each of the fundamental constants (e, i, pi, 1, 0) combined in a theorem that united trigonometry, analysis, and algebra and geometry. It’s really an amazing identity, and the proofs for it are diverse and fascinating…”[3]

It has, thus, been called the origin of all mathematics. Keith Devlin is purported to have said:

“like a Shakespearean sonnet that captures the very essence of love, or a painting that brings out the beauty of the human form that is far more than just skin deep, Euler’s equation reaches down into the very depths of existence.”[4]

Its elegance cries out for an explanation, but that explanation has proved so elusive that a desperate appeal to God begins to look almost reasonable, even to (some) mathematicians.

What should we make of this sort of argument? It seems on its face to be about as prima facie (in)admissible as any other argument from beauty. However, this argument may have more to recommend it than meets the eye. In particular, mathematical beauty has an uncanny predictive ability, at least in the sense that the more beautiful the mathematical formula, the more likely it is to describe the fundamental structure of the real world. Robin Collins has noted, for instance, that:

“To say that the beauty of the mathematical structure of nature is merely subjective, however, completely fails to account for the amazing success of the criterion of beauty in producing predictively accurate theories, such as Einstein’s general theory of relativity.”[5]

John Polkinghorne, in a lecture I recently had the pleasure of listening to (via podcast), said something similar though with less economy of words:

“It isn’t just [to satisfy] an aesthetic indulgence that theoretical physicists look for beautiful equations; it is because we have found, time and again, that they are the ones which actually do describe… a true aspect of the physical world in which we live. I suppose the greatest physicist I’ve known personally was Paul Dirac, (who held Newton’s old chair… in Cambridge for more than 30 years, who was one of the founding figures of quantum theory, [and] unquestionably the greatest British theoretical physicist of the twentieth century) and he made his great discoveries by a relentless and highly successful lifelong quest for mathematical beauty. Dirac once said ‘it is more important to have beauty in your equations than to have them fit experiment.’ Now he didn’t mean by that that it didn’t matter at the end of the day whether your equations fit the experiments (I know no physicist could possibly mean that), but what he meant was this: ok, you’ve got your new theory, and you use the solution and you find it doesn’t seem to fit what the experimentalist is telling you – now there’s no doubt that’s a setback, but it’s not absolutely necessarily fatal. Almost certainly, you will have solved the equations in some sort of approximation, and maybe you’ve just made the wrong approximation, or maybe the experiments are wrong (we have [known that] to happen even more than once in the history of physics – even in my lifetime I can think of a couple examples of that), so at least there’s some sort of residual hope that with a bit more work and a bit more luck you might have hit the jackpot after all. But, if your equations are ugly, there’s no hope. The whole 300-year history of theoretical physics is against you. Only beautiful equations really describe the fundamental structure of the world. Now that’s a very strange fact about the world… What I am saying to you is that some of the most beautiful (mathematical) patterns that our pure mathematical friends can think up in their studies just thinking abstractly… are found actually to occur, to be instantiated, in the structure of the world around us.”[6]

So mathematical beauty satisfies the empirical desideratum of predictive power in the sense that the more beautiful the mathematical expression, the more likely it is to describe reality.

Interestingly I think this kind of consideration can motivate a scientist (and perhaps even a die-hard empiricist, and/or a naturalist) to believe in the objectivity of aesthetic properties. In fact, unless they find a plausible evolutionary account for why our brains should be calibrated so as to recognize more beauty in the abstract mathematical equations which, it turns out, describe reality, than we find in other equations, there will be a residual mystery about the eerie coincidence of mathematical beauty and accurate mathematical descriptions of physics. An eerie coincidence the queerness of which can perhaps be mitigated by admitting the objectivity of aesthetic qualities.

However, the puzzling queerness of that eerie coincidence can only be (or can most plausibly be) ultimately alleviated if the universe is seen as the product of a (trans-)cosmic artist. If behind the fundamental structure of the universe there lies an intellect with aesthetic sensibilities (in some sense), then that would explain why the world showcases the mathematical-aesthetic qualities it does at the level of fundamental physics even when there is no (obvious?) reason why it should have. That, though, begins to look quite a lot like Theism.

The Argument from the Applicability of Mathematics

This segues into the next kind of argument from mathematics, which concerns the applicability of mathematics to accurate descriptions of the fundamental structure of the physical world. For the purposes of this argument beauty is entirely irrelevant. What is surprising, and in need of an explanation (according to this argument), is that the physical world would turn out to be describable in the language of mathematics (and here we are not simply referring to the basic truths of arithmetic, which are true across all logically possible worlds). William Lane Craig has become the most well-known proponent of this argument, and his articulation of it is relatively succinct.

“Philosophers and scientists have puzzled over what physicist Eugene Wigner called the uncanny effectiveness of mathematics. How is it that a mathematical theorist like Peter Higgs can sit down at his desk and by pouring over mathematical equations predict the existence of a fundamental particle which experimentalists thirty years later after investing millions of dollars and thousands of man-hours are finally able to detect? Mathematics is the language of nature. But, how is this to be explained? If mathematical objects are abstract entities causally isolated from the universe then the applicability of mathematics is, in the words of philosopher of mathematics Penelope Maddy, “a happy coincidence.” On the other hand, if mathematical objects are just useful fictions, how is it that nature is written in the language of these fictions? In his book, Dr. Rosenberg emphasizes that naturalism doesn’t tolerate cosmic coincidences. But the naturalist has no explanation of the uncanny applicability of mathematics to the physical world. By [contrast], the theist has a ready explanation. When God created the physical universe, he designed it on the mathematical structure he had in mind. We can summarize this argument as follows:

  1. If God did not exist, the applicability of mathematics would be a happy coincidence.
  2. The applicability of mathematics is not a happy coincidence.
  3. Therefore, God exists.”[7]

I am not sure of this argument’s philosophical quality, since it seems to me that it may be a metaphysically necessary truth that a logically possible world be amenable to mathematical description of some kind. For instance, it certainly seems true that whatever the geometry of space happens to be, there’s no necessary fact of the matter, but it also seems true that if the geometry of space isn’t Euclidean, it may be hyperbolic, or elliptic, (or maybe something else, je ne sais quoi) but it has got to be something, and what it happens to be may, therefore, not cry out for any more explanation than any other quaint contingent fact about the world.[8] However, maybe I’m mistaken about this; maybe the argument is, in fact, just as viable as other teleological or ‘fine-tuning’ arguments are.

Argument from Mathematical Truth

Finally, the third kind of argument I can think of would go something like this: mathematical truths, like all truths, have truth-makers. These truth-makers will have to be metaphysically necessary on pain of mathematical truths being contingent – but it seems obvious that mathematical truths are necessary truths, that they hold across all logically possible worlds. Now, Nominalism about mathematical objects is incompatible with the commitments we just outlined (unless one adopts Nominalism about modal properties as well), and so seems implausible (or, at least, less plausible than it otherwise would have been in virtue of this incompatibility). Platonism also, however, seems to be problematic. Between Platonism and Nominalism, there is a wide range of views including Divine Conceptualism (according to which mathematical objects exist as necessary thoughts in the necessary mind of God), Theistic Activism, Scholastic Realism[9] and many others besides. In fact, a variety (and perhaps a majority) of the accounts of abstract objects on offer today presuppose the existence of God in different ways.

This opens the way to at least two arguments we could construct for the existence of God. First, we could argue that one of these accounts in particular is most plausibly correct (such as Greg Welty’s Theistic Conceptual Realism),[10] and work our way up from there to the implication that God exists. Second, we could take the disjunction of all the accounts of abstract objects which require the existence of God and argue that (i) if any of them is correct then God exists, but (ii) it is more plausible than not that at least one of them is correct, from which it follows (iii) it is more plausible than not that God exists.

So, there we have it, three kinds of arguments from mathematics for the existence of God; a transcendental argument (from beauty), a teleological argument (from applicability), and an ontological argument (from necessity). Could there be others? Maybe, but I suspect that they will all end up falling into one or another (or maybe at least one) of the general categories I tried to outline here. I admit that I didn’t outline them as general categories very well, but that exercise will have to wait for another day when I have more time to blog to my heart’s content.

As a quick post scriptum; if Euler had any substantive argument in mind and wasn’t merely mocking Diderot for his lack of mathematical aptitude, which of these three kinds of arguments would he most likely have had in mind? It’s hard to say, of course, but my best guess is that if he had anything in mind at all, it would fall into the third category. He may have been thinking that the fact that mathematical and purely abstract (algebraic) ‘structural’ truths exist at all requires some explanation, and this explanation must be found in God. This is just a guess, and I make no apologies for it – I am happy to think that Euler was just teasing Diderot, but I am equally happy to entertain the thought that if Diderot had not immediately asked to leave (because of his embarrassment), Euler may have been able to elucidate his point.

[1] Gillings, Richard J. “The so-called Euler-Diderot incident.” The American Mathematical Monthly 61, no. 2 (1954): 77-80. http://www.fen.bilkent.edu.tr/~franz/M300/bell2.pdf

[2] Notice that these are not to be confused with mathematical arguments per se; they are merely arguments from mathematics, in the same way as you might have arguments from physics (the argument from cosmological fine-tuning, the Kalam, etc.) for the existence of God which are not intended to be scientific proofs of God’s existence, but scientifically informed philosophical proofs/arguments for God’s existence.

[3] Russel James, Why was Euler’s Identity Supposed to be a Proof for the Existence of God, https://www.quora.com/Why-was-Eulers-identity-supposed-to-be-a-mathematical-proof-for-the-existence-of-God; Note that he finishes the quoted paragraph with the words “but It has nothing to do with god whatsoever.” I have left this out not because I think he is wrong, or to misrepresent his position, but because it has nothing to do with the formula and everything to do with the propositional attitude he adopts with respect to the question of whether the formula is any kind of reason to think there is a being like God.

[4] Paul J. Nahin, Dr. Euler’s Fabulous Formula: Cures Many Mathematical Ills, (Princeton University Press, 2011), 1. https://books.google.co.uk/books?id=GvSg5HQ7WPcC&pg=PA1&redir_esc=y#v=onepage&q&f=false

[5] Robin Collins, The Case for Cosmic Design, (2008), http://infidels.org/library/modern/robin_collins/design.html

[6] John Polkinghorne, Science in the Public Sphere, http://www.veritas.org/science-public-sphere/

[7] William Lane Craig, Is Faith in God Reasonable? William Lane Craig vs. Dr. Rosenberg, http://www.reasonablefaith.org/debate-transcript-is-faith-in-god-reasonable

[8] I am really, honestly, no more sure of this counter-argument than I am of the argument. For those interested, please do check out the debate between Craig and Daniel Came on the Unbelievable? Podcast, which you can also find here: https://www.youtube.com/watch?v=nn4ocx316dk

[9] J.T. Bridges defends this view: https://www.youtube.com/watch?v=eFU1BKxJf1k

[10] See: Greg Welty, “Theistic Conceptual Realism,” in Beyond the Control of God: Six views on the Problem of God and Abstract Objects, ed. Paul Gould, (New York: Bloomsbury Academic, 2014), 81-96.

Naturalism and Supernaturalism

What, exactly, is Naturalism? The naïve definition would go: Naturalism is the belief that there are no supernatural entities. What, though, are supernatural entities? The go-to example would be God, but that’s an example rather than a definition. As far as definitions go, a typical place to start is to say that a supernatural entity is anything which is empirically undetectable, or not verifiable/falsifiable by the scientific method. However, plenty of unquestionably scientific beliefs are in things which are not strictly falsifiable (such as the existence of our universe), and a ‘scientific’ view of the world often involves commitment to beliefs which aren’t strictly verifiable (such as the legitimacy of inductive reasoning, or the reality of the past). Moreover, this definition entails that moral values, the laws of logic, the fundamental principles of arithmetic (and all mathematics), aesthetic qualities, facts themselves (as model-independent truth-makers), propositions (whether necessary, contingent, or necessarily false), the (noumenal) external world, and even purely mental phenomena (eg. qualia), will all be supernatural. Science itself, it turns out, is replete with presumptions of supernaturalism according to the stipulated definition.

Alvin Plantinga once defined Naturalism as the belief that there is no such being as God, nor anything like God. I used to think that this definition was serviceable, but I have come to see that it invites some of the most egregious difficulties of all. Buddhists and Mormons may qualify as Naturalists on this definition, and mathematical Platonists may not qualify as Naturalists! Surely that can’t be right. A definition of naturalism on which it turns out that Joseph Smith is a naturalist and Frege a supernaturalist cannot be right. The notorious difficulty of defining Naturalism should now be evident. What once looked like a trivially easy task now appears to be a herculean feat; how are we to draw the line between the natural and the supernatural? To echo (mutatis mutandis) a famous saying of St. Augustine: if nobody asks me what Naturalism is, I know, but if you ask me, I do not know.

One could always suggest that the term ‘Naturalism’ has no definition precisely because concepts have no definitions. Wittgenstein’s famous suggestion that concepts like ‘GAME’ have no definition,[1] and Quine’s famous skepticism about analyticity,[2] are just two of many factors which have contributed to the recent retreat from ‘definitions’ in the philosophy of concepts.[3] This trend has led to the wide embrace of prototype theory, theory-theory, and other alternatives to the classical theory of concepts. If we must give up on definitions, it seems to me that we must largely give up on the project of analytic philosophy, and that makes me considerably uneasy; but then, I’ve always been squeamish about anti-rationalist sentiments. It may turn out we can do no better than to say something like that Naturalists adopt belief systems related by a mere family resemblance, but which cannot be neatly subsumed under one definition. I, however, (stubborn rationalist that I am) will not give up on definitions without a fight.

On the other hand, if Naturalism cannot be defined then those of us who wish to remain analytic philosophers can just cut our losses and accuse self-identifying naturalists of having an unintelligible worldview; one the expression of which involves a fundamental theoretical term for which no clear definition can be given. In other words, when somebody claims that Naturalism is true we can simply retort: “I don’t know what that means, and neither do you.” What kind of rejoinder could they give? Either they will provide us with an acceptable definition (so that we’ll have finally teased it out), or they will have to reconsider the philosophical foundations of everything they believe they believe. Win-win by my count.

In the meantime, let’s try on some definitions for size. Here’s one:

P is a naturalist =def. P is an atheist who believes that all that exists is discoverable by the scientific method.

This definition is bad for several reasons. To begin with, it isn’t clear that a Naturalist need be an atheist; why couldn’t they be a verificationist,[4] or a Wittgensteinian? It seems, at first blush, sufficient that one not believe that “God exists” is a metaphysical truth, but then it also seems wrong to say that an agnostic can be a naturalist. An agnostic is agnostic with respect to supernatural entities, but a naturalist is not. So we’re left in a quandary with respect to the first half of our definition.

The second half doesn’t fair much better. Apart from the fact that scientists routinely commit themselves to the reality of entities which are beyond the scope of strictly empirical discoverability (such as the existence of alternative space-times in a multiverse), there is an puzzle involved in stating what, precisely, qualifies as scientifically discoverable. For instance, many of the fundamental entities in particle physics are not directly empirically observable (they are, in fact, often referred to as ‘unobservable entities’), but we have good reasons to think they exist based on the hypothetico-deductive method (i.e., we know what empirical effects they would have if they did exist, and we can verify those). However, that amounts to having good scientific and empirical motivation for believing in unobservable entities. Is it impossible to have good scientific and empirical motivation for believing in ghosts, or numbers, or God? W.V.O. Quine famously stated that if he saw any empirically justifiable motivation for belief in things like God, or the soul, he would happily accept them into his ontology. In fact, in a move motivated by his commitment to his Naturalized Epistemology,[5] Quine did eventually come to accept the existence of certain abstract objects (namely, sets). Quine leaves us with two choices: either we say that even Quine wasn’t really a (metaphysical) Naturalist in the end, or we find a way to allow Naturalists to believe in things like numbers, moral values, aesthetic facts, and other things which we don’t usually think of as ‘Natural’ entities. I suggest we make use of the notion of scientific/empirical motivation; in other words, we should make room for Naturalists to work out an ontology motivated by a scientific view of the world. The only danger I foresee in that move is that if even belief in abstract objects can be scientifically motivated, it seems as though belief in God, or anything, might turn out to be possibly scientifically motivated. Nevertheless, let us consider a second definition:

P is a naturalist =def. P believes that “God exists,” interpreted as a metaphysical statement, is untrue, and that the only entities which exist are the entities to which the acceptance of a literal interpretation of science commits us.

The first half of this definition seems fine to me, so that’s some progress. The second half is problematic because it implies that constructive empiricists, for instance, are not naturalists; the constructive empiricist agrees with the scientific realist that the statements of science should be literally construed/interpreted, but that when we accept a scientific theory we commit ourselves only to (i) the observable entities posited by the theory, and (ii) the empirical adequacy of the theory. Since the constructive empiricist adopts an agnostic attitude towards unobservable entities, none of them would qualify as naturalists on the above definition. In fact, anyone who adopts any version of scientific anti-realism (including the model-dependent realism of Stephen Hawking and Leonard Mlodinow, or even structural realism) will be disqualified from the running for candidate naturalists.

Let’s try a third:

P is a naturalist =def. P believes that “God exists,” interpreted as a metaphysical statement, is untrue, and P believes in some of, and only, the entities to which a literal interpretation of science commits us.

A possible problem with this definition might be that it threatens to include solipsists (though it isn’t clear what in science, interpreted literally, would commit anyone to the existence of persons). Perhaps we should replace “entities to which the acceptance of a literal interpretation of science commits us” with something like “entities to which our best understanding of science commits us.” That might be problematic since what the best understanding of science is seems up for debate. Perhaps it should be changed to: “entities to which a legitimate interpretation of science commits us.”

P is a naturalist =def. P believes that “God exists,” interpreted as a metaphysical statement, is untrue; P believes in some of the entities to which a legitimate understanding of science commits us; P does not believe in any entities belief in which cannot be motivated by a scientific view of the world (with the possible exception of God – caveat in casu necessitas).

This definition isn’t obviously problematic. It looks to be about as good as I can do, off the top of my head. Note that if this definition is successful, then we have also found the definition of supernaturalism, since (obviously) the definition of naturalism and the definition of supernaturalism bear a symmetrical relation of dependence to one another. This still has some notable disadvantages, including that naturalists will not be able to justify believing in moral facts unless they can generate motivation for believing in them given the resources of a scientific worldview. However, those disadvantages may just come with the territory; they may be the disadvantages not of our definition, but of the philosophy of metaphysical naturalism.

One final note; the term ‘supernaturalism’ has a bit of a bad rep because it is popularly associated with things like ghosts, energies, auras, mind-reading, witchcraft and (for better or worse) a variety of religious beliefs. Because of this many philosophers have opted for using synonyms such as ‘ultra-mundane’ to refer to things like moral facts, possible worlds, necessary beings, et alia. I don’t much mind which term is used, but one advantage to retaining the use of the term ‘supernatural’ is that it helps focus our attempt to define ‘natural’ and its cognates. If we had to define the terms ‘natural’ and ‘ultra-mundane’ it might be less apparent that whatever qualifies as unnatural is going to qualify as ultra-mundane, and vice versa.

[1] Ludwig Wittgenstein, Philosophical Investigations second edition, transl. G.E.M. Anscombe (Blackwell Publishers, 1999). http://lab404.com/lang/wittgenstein.pdf

[2] W.V.O. Quine, “Two Dogmas of Empiricism,” in The Philosophical Review vol. 60, no.1 (1951): 20-43.

[3] For more see: Stephen Laurence and Eric Margolis, “Concepts and Cognitive Science,” in Concepts: Core Readings (1999): 3-81.

[4] A verificationist, I mean, ‘about’ Theism.

[5] http://iweb.langara.bc.ca/rjohns/files/2015/03/Quine_selection.pdf

Banach-Tarski paradox, א Infinities, Infinitesimals, and the A-theory

I will offer an analysis of what is going wrong with the Banach-Tarski paradox suggesting that points, construed as infinitesimal surface areas, are nothing more than mathematically useful fictions. I will suggest that infinitesimals raise the same kinds of modally-prohibitive paradoxes in metaphysics as positing actually infinite quantities does (and for the same or similar reasons), and then consider an argument against the A-theory (in most of its forms) which can be purchased from these insights. I will then scout out some philosophical avenues available to the A-theorist.

The Banach-Tarski paradox is a famous mathematical paradox according to which it can be proved that if you divide the surface area of a sphere into little bits, and simply rearrange the bits appropriately, you can reconstruct two spheres each with the same surface area as the original sphere. In layman’s terms, you can prove (something just a shocking as) that 1=2.[1] To explain how it works, it may be worth calling to mind the various paradoxes associated with actual infinities.

Consider what it would be like to count upwards from -7 to infinity and stop only once you’ve arrived. Even if given an infinite amount of time you would never arrive, because no finite additions can sum up to a transfinite quantity. Subtract infinity from infinity, and what do you have? You have zero, but you also have infinity, and you also have 18.9801 (and every other real number); all of these are not just legitimate answers, they are mathematically correct answers. However, clearly 18.9801 is not equal to either zero, infinity, or anything else! Have a (Hilbert) hotel with an infinite number of rooms, all of which are occupied, and you want to check in an infinite number of new guests? No problem, just move every person from the room they are in (n) to the room with a room number equivalent to two times the original room’s room number (2n). Done; you’ve managed to move people around in such a way as to create an infinite number of vacant rooms without asking anyone to leave. Most of us (who are interested in this sort of thing) know the myriad paradoxes which arise from postulating even the possibility of an actual infinity. It seems relatively philosophically secure that there cannot be an א number of things (where א represents the first transfinite number, not to be confused with ∞ which symbolizes infinity taken as a limit rather than a quantity). If there are philosophically sophisticated caveats then so be it, but the point will remain that there are plenty of examples of things for which having an א number of them is clearly (broadly logically) impossible.

Let’s return, for a moment, to Hilbert’s Hotel, because it’s a particularly useful illustration. Suppose that the guest in room 3 checks out, while all the (infinitely many) other rooms remain occupied. The desk clerk decides that they want every room occupied, so they ask each person in room n (where n>3) to move one room over; that is, from room n to room n-1. That will fill up room 3, but the process will also leave no room empty because there is no room number n for which there is not an occupied room n+1. This works equally well for two dimensional shapes, such as circles; remove one ‘point’ from the circumference of a circle and you may have an infinitesimal gap, but simply move every other point along the circumference over (uniformly) by an infinitesimal amount and, voila, the gap is plugged and there will be no new gap. The trick in the case of the Banach-Tarski paradox is to apply the same reasoning to three-dimensional objects. For the best explanation of this paradox I’ve ever seen, (especially for readers who aren’t familiar with it, please make your life better and) check out Vsauce.

Alexander Pruss has noted on his blog that this result “is taken by some to be an argument against the Axiom of Choice.”[2] However, he argues that you can get the same paradoxical result in similar cases (and even in the same case) without the axiom of choice, so that the axiom of choice should be cleared of all suspicions. I agree (though I’m certainly no expert). Richard Feynman is purported to have said, upon being shown the proof, that “it’s fine you can do it with ‘continuous spheres’, since there’s no such thing. The important thing is you can’t do it with oranges, because oranges are made of a finite number of indivisible parts.” I think he is wrong about oranges (being actually comprised of indivisible finite parts, at least if the ‘parts’ are extended in three spatial dimensions), but his sentiment is appreciably insightful nonetheless.

The problem with the paradox, in my submission, is that it divides the surface of the sphere up into points. However, points on a sphere, like points on a line segment, are infinitesimals. This is precisely why (Aristotelians) say that line segments are not composed of points the way walls are composed of bricks, but, instead, points act as the limits between which a line segment is continuously extended. An infinitesimal is a quantity which is infinitely small. It is non-zero, but it is also smaller than any finite quantity. Sure infinitesimals are useful for doing things like infinitesimal calculus, developed by one of my all time favorite philosophers Gottfried Wilhelm Leibniz, but they remain, I believe, nothing more than useful fictions. To borrow a phrase from W.L. Craig;

“They are akin to ideal gases, frictionless planes, points at infinity, and other useful fictions employed in scientific theories.”[3]

If we are to accept the possibility of infinitesimal quantities in reality, then we will quickly run into paradoxes like the Banach-Tarski paradox (which, quite apart from being obnoxious to the rational intellect, seems to violate the law of conservation of matter and energy). Positing infinitesimals is just as paradoxical as positing sets of actually infinitely many discrete things (where ‘things’ is an ontologically loaded term). I am suggesting that infinitesimals are just as paradoxical as actual infinities, and, at bottom, for the same reason(s). In fact, I have this intuition that every argument for thinking that there cannot be any actual infinities (as opposed to potential infinities, where ‘infinity’ merely acts as a limit), admits of a parody for an argument against the existence of infinitesimals. I’m not sure I can rigorously prove it, but I think it’s very plausible.

It seems to me that there’s something conceptually parasitic about infinitesimals relative to infinities. They each conceptually supervene on each other symmetrically. To visualize this symmetry, consider plotting the function ƒ(x)=  1/x which will look like this:

[http://mathworld.wolfram.com/images/eps-gif/AsymptotesOneOverX_1000.gif]

The distance between the curved line and the x-axis (i.e., y=0) as x approaches (positive or negative) infinity is shrinking (or, at least, its absolute value is shrinking), and approaching an infinitely small non-zero measure. When X is infinite, the absolute value of the y-axis coordinate of the curved line (i.e., the distance between the curved line and it’s asymptote, here being the x axis) is infinitesimally small. This example helps to illustrate the point that the concept of an infinitesimal is bound up with the concept of infinity, so that in the absence of one the other would be inconceivable. That at least motivates the suspicion that if one turns out to be metaphysically impossible, so will the other.

What relevance does this have for the philosophy of time? Well, consider that on the A-theory there is such a time as the present. How long does the present last? What, precisely, is its magnitude, its duration? Let’s consider the following argument:

  1. If the A-theory is true, then the present is either infinitesimal in duration, or it is finite in duration.
  2. The present cannot be infinitesimal in duration.
  3. The present cannot be finite in duration.
  4. Therefore, the A-theory is false

Premise 3 can be established with Leibniz’ argument against the (logical) possibility of a physically indivisible element, or ‘atom’ (in the etymologically literal sense). For anything extended in three-dimensional space, however small, it will always be logically possible for me to divine it in two, even if I am physically incapable of doing so (due to some constraint, such as not having the appropriate equipment for the job, or maybe not even being able to develop any tool which could do the job). Physical impossibilities are not (all) logical impossibilities, and logically there is no constraint on how many times I could divide an object extended in space. To say that there is an object extended in space which is not logically possibly divided up into smaller constituent pieces is, according to Leibniz, incoherent. The exact same argument, mutatis mutandis, works against there being chronons (i.e., atomic chunks of time).

The denial of premise 2 is absurd given our observations that positing infinitesimals leads to modally unconscionable paradoxes like Banach-Tarski.

Ways out: I see four ways, not all of them equally viable, for an A-theorist to escape the conclusion of this argument.

First, they could challenge premise 3 on the grounds that, if there are chronons, then by definition they are entities which cannot be physically divided. The suggestion would be that the prima facie absurdity of a Chronon de dicto doesn’t entail the impossibility of a chronon de re. This dangerously dislocates rational intuition from epistemic reliability, but I can imagine extreme empiricists embracing this response.

Second, they could challenge premise 2 by arguing that positing any more than one real infinitesimal of any kind might be problematic, but that there’s no way to derive similar paradoxes from positing a maximum of one infinitesimal. In other words, perhaps paradoxes involving infinitesimals only arise when there are n infinitesimals, where n ∈ ℕ, and n>1. Multiply an infinitesimal by any natural number, or even a transfinite number, and you will still get an infinitesimal result, so it seems harder to show that from one infinitesimal you could derive some kind of contradiction.

Quick thought: Perhaps if there are rules/axioms such as (i) no infinitesimal can be larger or smaller than any other infinitesimal, but (ii) anything (other than 1) to the power of itself is larger than itself, you could derive a contradiction by taking an infinitesimal X, running it through Xx=Y, and then asking whether Y is larger than X, or the same size (it appears to be both). However, I don’t have the kind of facility in mathematics to be able to produce a rigorous proof that even a single infinitesimal would lead to some kind of contradiction or unconscionable paradox. Moreover, it isn’t entirely clear to me what relevance that kind of mathematical paradox would have for the metaphysical consideration at hand. In any case, the second challenge to premise 2 cannot be lightly dismissed.

Third, one could adopt a really wild philosophy of time, such as the Apresentism I wrote about in the last post (thus denying the first premise).

Fourth, one could deny the first premise by adopting what has been called a non-metric view of the present. This is the view preferred by William Lane Craig.[4] I have more than expended my allotted time for blogging and casual writing today, so I will leave this post here for now. I may return to the idea of non-metric present in the (near) future in another post.

[Ha, I don’t presently have time to write more. Get it?]

[1] For fun, check out and try to find the mistake in the following mathematical proof that 1=2 here: https://www.math.toronto.edu/mathnet/falseProofs/first1eq2.html

[2] http://alexanderpruss.blogspot.ca/2013/06/the-banach-tarski-paradox-and-axiom-of.html

[3] William Lane Craig, “Response to Greg Welty,” in Beyond the Control of God: Six Views on the Problem of God and Abstract Objects, ed. Paul Gould (A&C Black, 2014), 102.

[4] See: Craig, William. “The extent of the present.” International Studies in the Philosophy of Science 14, no. 2 (2000): 165-185.

Two Too Simple Objections to Open Theism

First, let’s agree to reject dialetheic logics out of hand; it will be taken as a non-starter for me, and, I hope, for you, if any argument were to proceed on the assumption that a proposition can be both true and false at the same time and in the same sense. It may be useful, at times, to proceed as though this were the case (I’m not denying the usefulness of paraconsistent logics), but it certainly cannot be literally correct. Such logical systems do not (and, by implication granting S5, cannot) describe the extra-mental structure of modality.  

Can God know the future on open theism? It is typically assumed that open theism involves a commitment to Presentism about time (according to which future events are not real, and so propositions about the future are not literally true). I am not sure that this is correct, since they may, perhaps, accept the growing-block theory of time instead, but that will land them in precisely the same predicament as Presentism will as far as my following objections are concerned. In any case, the open theist must accept some version of the A-theory other than the moving-spotlight theory of time (or other more esoteric theories of time which will allow for the reality of future events or states of affairs). God, on the open theist view, shouldn’t be able to know the future because there is no future to know.

It seems undeniable that if “P” is true, and if “P⊃Q” is true, then “Q” is true; that’s just good old Modus Ponens. Now, let’s take P to represent the tripartite conjunction: “the state of affairs S1 in the world will entail the subsequent state of affairs S2 just in case God does not intervene in the world at some time between S1 and S2 (inclusive of S1, not inclusive of S2) and God will not intervene in the world at any time between S1 and S2, and S1 describes the current state of affairs.” Let Q represent the proposition “in the future, S2 will be the case.”

Let us say that God knows P, and God knows that P⊃Q. Does God know Q? If not, He has a deficient grasp of logic. If so, then He knows at least some fact(s) about the future.

  1. If open theism is true, God cannot know the future.
  2. Possibly, God can know propositions like “P” and “P⊃Q.”
  3. If God can know propositions like “P&(P⊃Q),” then God can know propositions like Q.
  4. If God can know propositions like Q, then God can know propositions about the future.
  5. If God can know propositions about the future, then God can know the future.
  6. Therefore, open theism is false.

What will the open theist say? The most plausible response open to them, I think, is to deny premise 5. Generally we think of propositions about the future as having truth-makers which are future states of affairs, but it is conceivable that there be true propositions about the future which have, as their truth-makers, nothing beyond present truth-makers. Perhaps P is presently true, while Modus Ponens and P⊃Q are true presently (they may be timeless truths, so we avoid saying that they are ‘presently’ true, even if they are true presently). That might be a sufficient response. A second response might go like this: premise 1 should be restated as 1*: “if open theism is true, God cannot know the whole future,” and premise 5 should be restated as 5*: “If God can know propositions about the future, then God can know at least some of the future.” Obviously 6 does not logically follow from 1*-5*.

Here’s a second argument:

  1. If a proposition is meaningful, then it cannot fail to be true or false (where the ‘or’ is exclusive).
  2. There are meaningful propositions about the future which are not entailed by any presently available truths.
  3. Therefore, there are true propositions about the future which are not entailed by any presently available truths (they cannot all be false, for if P is false, then “P is false” is true).
  4. God is omniscient.
  5. A being is not omniscient if there are truths (i.e., meaningful true propositions) it fails to know.
  6. If open theism is true, there are meaningful true propositions about the future which God fails to know.
  7. Therefore, open theism is false.

The best responses to this argument which I have heard include (i) denying premise 2 altogether, or (ii) denying premise 1. The denial of premise 1 (given our assumed rejection of dialetheic logics) amounts to a rejection of the law of excluded middle (LEM), and that, my friends, is as good as a reductio against open theism. Rather, it is a reductio of open theism! Alternatively, to deny premise 2 (by denying the meaningfulness of propositions about future states of affairs not entailed by presently available truths), seems implausible given the fact that we all apprehend the meaning of sentences like “tomorrow Julie will eat worms in the playground again.” So, we have at least one relatively good, though simple, argument against open theism.

… Maybe there’s time for a quick third: suppose that epistemic justification means something like ‘true justified belief’ (and let’s, for the moment, ignore Gettier cases, just for simplicity). Now it looks like I can know propositions like P:”tomorrow I will finally propose to her,” even though it looks like God cannot know P! That’s another reductio ad absurdam to add to our growing list of reasons to reject open theism.

My mistake; obviously this last argument presupposes the ‘truth’ of propositions like P, but that’s the very object of contention, so my argument runs in, as they say, a circle of embarrassingly short diameter.

As to whether either of the former arguments will work, it seems to me that if the open theism is too deeply entrenched then the open theist will simply bite the bullet and accept the consequences of my arguments while maintaining open theism. However, at least the arguments can act as a warning to others to avoid the philosophical pit that is open theism.