A book review of Peter Boghossian's Manual for Creating Atheists

Though it took me about a month longer than expected to get a chance to read philosopher Peter Boghossian's new best-selling book, A Manual for Creating Atheists, with foreword by noted skeptic Michael Shermer, I did get time this weekend to check it out. I think you should do so too, and I'm going to take a moment to tell you why.


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It's not a trivial problem, and I'm not by any means the first to notice, that the highly influential and important work done by the famous New Atheists, most notably Richard Dawkins, Sam Harris, Daniel Dennett, and the late Christopher Hitchens, and those less famous, has led to a situation in which huge numbers of people have left their faith traditions, embraced reason and informed skepticism, and became motivated to help others to do so as well--and yet one in which many of these people want for a clear way to do it. Peter Boghossian's book spells out one such method that promises to be very effective, likely to be more potent than legions of people following in the footsteps of the so-called New Atheists, myself among them. In that regard, reading Boghossian's Manual is like finding a critical, but lost, piece of a challenging jigsaw puzzle and placing it right where it has been needed.

Boghossian turns the focus of the conversation about religious belief directly at its support mechanism: faith. Indeed, the book would be more accurately, though more poorly, titled "A Manual for Creating People who Reject Faith Because They Have a Better Epistemology." That is to say that Boghossian is striking not at the religious believer's specific beliefs or the harms thereof, but instead he's trying to pave a road that leads them to see that they don't know those things that they only "pretend to know."

Turning the focus of the conversation in this way raises the bar and lifts discussions between believers and skeptics out of the ubiquitous quagmire that is squabbling about particular faith-based claims, the societal benefits versus harms of religions, debates about evidence and what constitutes it, and other red herrings. Discussions based upon Boghossian's methods, instead, are ideally open-ended dialogues in which seeds of reasonable doubt are sown in soil that is being made fertile, a state that Boghossian describes as one of "doxastic openness."

Contrary to what many may expect, the brilliance of Boghossian's approach is that it does not rely upon trying to change anyone's mind or even on getting them to think in this way or that. "It's not about what I think; it's about what you think," is a frequent refrain from Boghossian that has totally changed how I think about talking with people of faith. It's both humbling and exciting to find out that it's very likely that I've been doing much of it wrong.

Instead of arguing or trying to convince, Boghossian's method is Socratic, and he does a fantastic job presenting it, both in theory and in practice. A shining gem, in fact, of this book is the fairly extensive collection of actual dialogues, which Boghossian calls "interventions," that the author has had with people, exemplifying the very methods he's attempting to teach. Reading his presentation of the Socratic method "as a conversational intervention to liberate people of their faith" (p. 107) has already changed how I talk with people and has fundamentally reorganized how I want to proceed in future conversations on these sorts of topics.

Boghossian, however, doesn't shy away from controversy or the refreshing kind of firmness that the New Atheists popularized. In fact, the entire book carries the thesis that faith is a kind of cognitive virus, a sickness, that needs to be cured so that we can live in a healthier society. Much of his language and the structure of his method is borrowed from epidemiology and addiction counseling--sure to be insulting to those afflicted. He also reserves a chapter to thrust directly at far-left accommodationists, obscurantists, and relavitists that is not only sure to be unpopular in certain circles but also is sorely needed. He does these things expertly, exerting a clear confidence that lacks arrogance, backing his case substantively.

This book is an important one. Its lessons are excellent for changing the manner in which we engage in discourse over charged topics that rely upon faith, religious, political, or otherwise, and they're likely to be a key element in a new Enlightenment. It's been far too long that our discussions about faith, and really then about how we come to claim to know things, are messy arguments instead of productive dialogues, and it's far past the time when we should accept faith-based statements as a viable method for claiming knowledge about the world.

Boghossian is offering us powerful medicine, and we owe it to ourselves to take it. Reading this book is the first step.

Highly recommended without reservation.

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Get A Manual for Creating Atheists here, and start helping to change people's faulty reasoning methods to better ones!

Morality and the Christian God, A Sam Harris video mash-up

Since I've been poking around a little bit at teaching myself some basic video editing, I made my own video mash-up for Sam Harris's "Morality and the Christian God," per the invitation he published on his blog.

I know that the composition style is similar to what a couple of other people did before me, but I feel that their videos are too long to possess the kind of punch needed to really get Harris's message to a wide audience. In fact, I found that I needed a lot of willpower to sit through the full length of those videos, which is not a good recipe for a viral video. Therefore, I made mine much shorter, just under two and a third minutes.

To accomplish that goal, I had to edit Harris's original audio file pretty heavily, cutting it and then reordering the remnants so that they would still present a cohesive narrative, hopefully carrying a lot of punch. To be fair to any critics, I agree--Harris's words as composed originally are very powerful and better--but I don't think they are amenable to a video of this composition style given that they run for nearly eight minutes. Perhaps the RSA Animate style Harris suggests on his blog would be more potent for the full audio track.

So, I hope you check it out and enjoy it, and if you do, please share it. To be clear, I don't stand to gain anything from it other than the benefits of living in a world where more people have heard and hopefully considered these thoughts of Sam Harris.

Questions about Infinity 1: Infinite Goodness, Etc.

I've put out an open call on Twitter for questions about infinity, especially as they relate to God, offering to blog answers to those questions.

I received the following question:

I frequently don't understand the meaning of "infinitely X". For example, is "infinitely good" even meaningful?

This question is a pretty fun one, and it is one that is fairly nuanced in its own way.

The simple answer

To be completely straight, the answer to this question, in the context meant, is "probably not." We should be cautious, though, with an answer like that because infinity can be understood to have a few different meanings, and in one of those meanings, "sort of" is nearer the mark for an honest reply. Particularly, I think we all have a general and qualitative sense of what is meant by "infinite goodness," and that is the state of being unsurpassable in its good quality. Metaphorically, then, it definitely has some meaning, but whether this phrase literally means anything or not is less clear.

If we understand infinity to be taken as an actual infinity, meaning an infinitely large quantity that is somehow manifest, I would suspect that the answer is "probably not." We don't know for sure if our universe is infinite in scope, thus admitting infinities in the physical, and all meanings of actual infinities would have to count abstractions if it isn't. Even so, X, say "goodness," could be an abstraction, but since something like goodness seems to be necessarily rooted in physical experiences of sentient entities, it's unlikely that something could be "infinitely good."

Infinite goodness

Before getting on to other possible meanings of infinity, though, the idea of infinite goodness needs some elaboration. Goodness, plainly, is a difficult thing to measure, and since "infinite goodness," as it immediately seems to us, implies a quantity of goodness, some kind of metric is needed to give meaning to this turn of phrase. Of course, much can be hidden in this complexity, as is often the case, and so it is easy to pass off phrases like "infinite goodness" without an easy rebuttal.

Even if we suppose that we can measure goodness in some salient way, though, what would be meant by (actual) infinite goodness. By definition, in this context, infinite means that if we were to take some part away from this collection (of goodness), it would not be diminished (in goodness). Inversely, if we were to add something to this measure of goodness, say another act that could be deemed good, we would not actually increase the total goodness present. This, I think, presents an odd state of affairs, and where theology comes into it, it leaves open a door to nonsense.

To wit, on the claim that God is infinitely good, if measured by that which he has come to make be good in this world, if some act of goodness were to be undone or left undone at God's responsibility, God's goodness would not be diminished. On the other hand, if God were to do more good than we currently see, it would not increase his goodness at all. This would seem to justify a great deal of utter moral atrocity while leaving God perfectly good. Of course, to caring, feeling beings like ourselves, it would also render God's goodness perfectly hollow and meaningless, divorced from anything we can identify with the word "good." This strikes me, then, as rendering the notion of the "infinite goodness of God" as an empty concept, mere words said to convey some non-literal meaning.

Getting more complex

"Infinitely" can mean more than the literally infinite idea captured in actual infinity, though. It could apply to potential infinities as well. A potential infinity is, in essence, the unlimited capacity for more. Like with numbers, no matter what number we're thinking about, we can think of larger numbers than that. In this case, "(potentially) infinitely good" would mean that however good we conceive of something like God being, he possesses the capacity to be more good--and not just in a way where we are approaching some maximum limit of goodness that cannot actually be reached (a supremum of goodness just outside the set of possible states of goodness). In this case, we should see as reasonable and possible as large an increase in goodness as anyone might imagine.

On the surface, this seems pretty reasonable, if a little outlandish. I do not think it is, though, because it brings us back to the question of what defines "goodness." If defined in terms of the experiences of sentient beings, it strikes me as a definition that requires a universe able to spawn and hold an unending number of such beings. Maybe that's the case, but then tying that notion of goodness back to a single entity like God gets pretty strange, particularly since any positive amount of goodness (and thus far from maximum possible goodness for any) spread out over (potentially) infinitely many sentient beings would render an "infinitely good" God that still left a lot to be desired.

More importantly, then, it raises the question of what infinite goodness would mean for God. Would such an infinitely good God be beholden to create a maximal amount of good for each being, a large amount, a little, or goodness that just barely tips the balance toward net-good? I think most of us recoil at the notion that "infinitely good" can be used to describe a God that so orders the universe such that every sentient life suffers exactly enough to just barely break even on the good versus not-good spectrum of total life experience. Indeed, many of us reject the idea of a God that allows whatever suffering in real life to be made up for with infinite reward in some promised afterlife.

Getting even more complex

There's another possible meaning for this phrase, "infinite goodness." Take, for example, the notion that God and everything God allegedly has done maximizes goodness in the universe. Technically, this isn't "infinite" goodness necessarily, but the question was about whether or not this phrase has any meaning, this being one of them--proposed, for instance, by Kurt Gödel's ontological argument (see Dot, Dot, Dot) for the existence of God. Could "infinitely good" here actually mean "maximally good," with people using a faulty means of communicating this ideal? Evangelical Christian apologist William Lane Craig seems to think so.


Well, consider what it would mean for God if it were the case. Such a God would have to have so-ordered the universe so that a single additional or different act of any kind on his part would necessarily make it worse off. This commits us, then, to a remote God that is not in any way involved in our affairs unless via the particularly sick way in which he has ordered the world to where we must ask (or beg) for his intervention in order for him to give it to us but in which he has predetermined it to be so all along.

We could step back one step and go to the claim that God's "infinite goodness" really means a "potential maximum goodness," which hopefully starts feeling very gerrymandered in the usual theological way. By this I mean that God has ordered the universe so that if we all use our free wills (NB: It begs the question to assume we have these, but that's a different topic for a different discussion) the way he hopes we will, making no errors, then maximal goodness will be achieved. This may be theologically, and perhaps even philosophically, defensible, but what a hamper God is in, and what an opportunity pastors have to blame people for the troubles they and others face in life.

This, of course, is the kind of "infinitely good" God many people believe in: one that made things so that they would be maximally good if people would just stop screwing everything up. There's a reason those without belief reject this idea as repugnant: it is repugnant.

Of course, we could go on and on and on with excuses for why God's "infinitely good" nature doesn't mean what "infinitely good" means, and we could probably find scripture or the writings of early theologians to back up every word of it, but what a disingenuous waste of time that would be.

Other properties that aren't goodness

To be complete, we should wonder about other properties, like knowledge or power, often said of God to be infinite. By the given definition of the (actual) infinite, I think it divorces them from reality, God with them. This, of course, is a major theme within Dot, Dot, Dot.

Another point to make, though, is that even if God is potentially infinitely able or knowledgeable, this seems to commit God to having an actually infinite set of abilities or knowledge. I provide what I think is a very compelling argument for this claim near the end of Dot, Dot, Dot, and so interested readers can find it there. Since that actualizes infinity in God, we're back to the situation in the previous paragraph: God is divorced not from the universe but from reality itself. This isn't a cute theological hack either, I'm claiming with it that such a God is imaginary, existing only via the minds of people.

"As Christians, we worship an infinite God."

The key to an effective sermon may very well be to get right enough of the periphery to be able to slide in the nonsense with authority, or at least we might guess that to be the case judging by this sermon by Pastor John Van Sloten from New Hope Church Calgary on "God, Infinity and Mathematics" from their "God @School" sermon series. It's worth noting that this video was posted just this week, so it's reasonable to conclude that it was probably preached somewhat recently. Check out the video if you've got fifteen minutes or skip around a little bit to get a sense of what kinds of things they're saying. The sermon itself starts about a minute in.

Of course, sermons like this are a big part of what made me feel a need to write a book like Dot, Dot, Dot: Infinity Plus God Equals Folly. Who needs a book about infinity and God? Anyone who wants to inoculate themselves from the kind of nonsense presented in sermons like in this video, which being based upon mathematics have a high likelihood of being able to blind someone with science bullshit.

I don't know this, but I suspect we're going to be hearing more mathematics from pastors in the coming years. Too many people with too many voices know too much science for them to keep railing on it. Indeed, it chases many young people away to go anti-science for Jesus. Mathematics, though, is a subject that most people are shockingly ignorant of, it's awe-inspiring in its own weird ways, and it possesses enough similarity to theology--in that it's abstract and axiomatic--to really be able to wow an audience. Since the majorities of most audiences are likely to be both ignorant and afraid of mathematics, there won't often be a lot of checking for nonsense, and when eyebrows do rise, it's likely that the pastor will be forgiven his lack of expertise in a difficult subject just for the attempt. Thus, pastors will just need to get the peripheries right and talk about big, interesting things that they can somehow tie to God, like infinity.

This pastor isn't an expert either

The first minute of the video seems to be borrowed from a BBC documentary about mathematics and makes a valid point: you can't really count to "big" numbers. No one can. So here's something valid on the periphery, since this isn't going to be the pastor's point at all. Instead, the soft-spoken Van Sloten repeats his chorus throughout the sermon: "As Christians, we worship an infinite God."

I do not intend to give a blow-by-blow analysis of this sermon as that would be both boring and tasteless. (The pastor is clearly not an expert, so it feels a little like getting in a boxing match with a kid.) Though perhaps a bit pedantic, I do want to bolster that statement by noting that within the first minute he speaks, though, Van Sloten reveals his lack of expertise by talking about what we might call "big" numbers: the number googol and then two others, googolplex and Graham's number. The revealed lack of experience occurs blatantly when he mentions googol and then calls googolplex "the next biggest number up" and then says of Graham's number that "of course, there's even a bigger number than a googolplex." Indeed, there are lots of numbers bigger than the googolplex, and also bigger than Graham's number; almost all numbers are bigger than both.

Being inexpert in mathematics, though, is hardly a cardinal sin. In fact, it's quite common. We might celebrate his attempt to stand on a stage and preach mathematics appreciation if it weren't for the fact that he's using it to preach for some seemingly imaginary entity he calls "God," mathematics appreciation being incidental to the entire affair. This shouldn't be celebrated and will here be impugned.

Getting the periphery right

Van Sloten's sermon includes many accurate statements about mathematics: the definition of googol, the definition and details about googolplex, and the details about Graham's number being notable examples just within the first couple of minutes that he talks, noting the above comment, of course. Immediately, though, he's interjecting the idea of mystery into these "colossal" numbers by calling Graham's number a "mystery."

Of course, mystery will be a theme throughout his sermon, for as John W. Loftus has noted, "Faith is a parasite on the mysterious" (Outsider Test for Faith, p. 219). To really highlight the mysteriousness of Graham's number, in fact, Van Sloten notes the "celestial music" played in the background on the BBC documentary while talking about Graham's number. Though it goes without saying, celestial music has nothing to do with Graham's number outside of a documentary production room, and celestial music has nothing to do with "God" except in the minds of those who believe in such notions.

At any rate, Van Sloten goes to make a big point early on and quotes Graham to do so. In the BCC documentary, Ronald Graham notes that "his" number is no closer to infinity than is the number one (I have a chapter about this fact in Dot, Dot, Dot called "All Numbers and All Infinities Are Very Small"). Then he says it for the first time: "As Christians, we worship an infinite God" (emphasis mine). He says this three or four times, in one fashion or another, throughout the video, but just after saying it for the second time, he notes, "mathematics...can teach us about who our infinite God is." I quite agree.

Who is an infinite God?

This question, of course, is nonsense but makes for a nice transition. The right question is "what does 'being infinite' tell us about the idea called 'God'?" This, of course, is a major theme of Dot, Dot, Dot. It tells us that God is abstract.

Van Sloten states that "math is the perfect tool and language with which to engage the logical, reasoned, mathematical mind of God." I'm not at all sure what he means by this, or how he knows that God's mind is so ordered, but he comments that mathematics is "limitless," pointing out further that "while physicists can only go so far, mathematicians can imagine even beyond that with their math" (emphasis mine), meaning that mathematicians are not "fixed or somehow limited by material reality."

Okay, wow. Thanks, Pastor Van Sloten, for making my point while thinking of yours. Insofar as he is right, mathematicians aren't bound by material reality because we work with abstractions, and can imagine even beyond it. The key notions here, particularly when we get to the "mind of God" part, are the imagining and the working with abstractions.

Probably for this reason, along with his lack of mathematical expertise, much of what Van Sloten seems able to conclude about his infinite God is fluff. At one point, he states that what we have is an "infinite number of infinities all pointing to an infinite God," and shortly after he concludes that "we're made to engage an infinite mystery and tremble before it." This is all very poetic, but I don't think it's much more than theological twaddle--nowhere in the infinite number of infinities do I see a single reason to suggest that they point to any God.

Of course, much could be said about humans being "made to engage an infinite mystery," a dubious and probably patently false claim, and much more could be said about being made to "tremble before it." That commentary, though, would be less about mathematics than about the psychology of believers, I think. In that vein, I should note that he included a lot of "fear of God" talk just before the trembling bit, which raises the question of what kind of "New Hope" they peddle up there in Calgary.

It seems to me that Van Sloten is unable to conclude anything of any value from his investigation of God via the mathematical infinite, although he uses quite a lot of language sure to be charged with his audience, ripe for hoodwinking. The term "mystery" comes up several times, as noted, including once as a "mystery that draws us" and another time as we just said, an "infinite mystery" that we're "made to engage." He stresses words like "limitless" in a way that speaks to a non-technical double entendre, and even works in terms like "deliver," perhaps involving "celestial music" instead of banjos but with much the same connotation. Likewise, he suggestively emphasizes mathematically accurate phrases like "greater than what our universe can hold" (so far as we know and possibly probably), "unbounded by physical reality," "awe-inspiring," and "out of this world," but then near the end starts sliding this less toward mathematical accuracy and more toward his agenda with "holy and reverent." Infinity is holy? Really? (Indeed, he suggests that doing math, at least if done well, is holy. Well, I'll be damned.)

Given all that, I'm going to conclude that Van Sloten's analysis upon "As Christians, we believe in an infinite God" says exactly what I think it means: they worship an abstraction, and their use of the word "infinite" is symbolic and poorly understood. If it weren't so troublesome to the rest of us in so many serious ways, I'd say, "Good for them...," and leave it at that.

Get the periphery right and then lean upon authority.

If we're going to concoct a powerful sermon, though, it isn't enough to get the peripheral details right and then stuff in only fluff. It's critical to lean upon authorities.

Van Sloten quotes Galileo at length concerning his seventeenth century thoughts about God and mathematics. Similarly, Johannes Kepler's thoughts on God and mathematics are pulled in, as are Georg Cantor's, which will get some special attention from me later. He also quotes noted mathematicians Keith Devlin and Ian Stewart, which seems a bit peculiar given some of their other writing.

Specifically, he attributes to Devlin a quote that "mathematics makes the invisible visible," although this is actually the subtitle of a book by Devlin titled The Language of Mathematics. For my part, I don't think Devlin is talking about the same thing as Van Sloten with the word "invisible." Van Sloten quotes Ian Stewart as saying, "We encounter mathematics everywhere, every day, but we hardly even know it," and goes on to suggest that this is rather like how God is everywhere and communicates with us every day, though we hardly know it. I strongly suspect this isn't what Stewart was getting at.

It's worth noting that instead of quoting their own works directly, Van Sloten quotes Devlin and Stewart via a 2011 Christian apologetic work called Mathematics Through the Eyes of Faith by James Bradley and Russell Howell. As described on Amazon: "Ina [sic] new addition to the groundbreaking 'Through the Eyes of Faith' series, thenation’s [sic] top Christian professors approach mathematics from a Christianperspective [sic].," and the work is admittedly co-sponsored by the Council for Christian Colleges and Universities.

Other than by attempting to liken the "quote" from Devlin to a Bible verse (Romans 1:20), making suggestive inferences about Stewart's quotes, and salting his sermon with other verses, the only other authorities Van Sloten calls upon are two "math enthusiasts" from his own congregation, an engineer and a high school mathematics teacher. It is via their commentary and authority, and his own thoughts, that Van Sloten bridges the gap from "awe-inspiring" and "mystery" to "holy and reverent." Indeed, he usurps all of the mathematicians in the BBC documentary at the beginning of the clip and perhaps most, if not all others, by suggesting that "mathematicians all know they were on the edge of something holy" when talking about infinity. Quaerendo invenietis, as always, I suppose.

When in doubt, make it sound official.

Fluff is the stuff of sermons, though, and as a former boss used to tell me about how to bullshit through a tight spot: "When in doubt, just look official, and people won't question you." Van Sloten does this spectacularly at the point where he changes his tempo from misleading math appreciation to outright theological nonsense. I'll quote him.

Infinitely small numbers, going on forever. Infinitely large numbers, far greater than what our universe can hold. Infinity times two, infinite dimensions within which infinity can play out, infinity times infinity, infinity to the power of infinity! It's like there's no end to the study of infinity. An infinite number of infinities all pointing to an infinite God. Something about how wide and long and deep and high that makes up the nature of infinity that is awe-inspiring, not just to the mathematicians--out of this world, a mystery that draws us.

I don't even know what to say about this except to call it grade-A sermon fluff, and indeed it makes his direct segue to the theology, "being on the edge of something holy and reverent" when thinking about infinity, which he then connects via "reverent" to the "fear of the Lord." At any rate, it sounds official, but I'm calling it directly into question.

A couple of odds and ends

I'd like to wrap this up here, but I do want to mention Georg Cantor, one of the mathematicians Van Sloten leans upon in his sermon. Cantor gets special attention because Cantor is the mathematician who changed everything about how we study mathematics and especially infinity. He was also very devout, and this caused him some problems.

Cantor believed in God and, following dogma and popular belief, identified God with the infinite. Cantor's discoveries about infinity were fairly upsetting, though, to this dogma and belief, so much so in fact that Van Sloten (quoting Daniel Tammet's book Thinking in Numbers) notes that Cantor had to defend himself and his ideas from charges of blasphemy from the Vatican. Cantor tried to argue that his discoveries were indicative of God revealing more of his "infinite nature" and sincerely believed it, even if it did cause him to experience moments of questioning doubt. Doubts aside, Cantor believed that his work on transfinite mathematics was directly communicated to him by God. This attitude is a bit odd given that one of Cantor's chief discoveries is that given a set of any infinite size, one can use it to create a set with a size described by a larger infinity, making God as the ultimate infinity a difficult position to defend.

Cantor struggled immensely with all of this, defending both his work (upon which he felt his public reputation rested) and his commitment to God, and ultimately remained shaken for the remainder of his troubled life (much of it in an asylum) over ever having questioned God via his work, even momentarily. His story is quite a tragedy of faith.

Why this is important

This isn't the last we'll hear of "God and infinity" talk coming from Christians, and that's a huge part of why I bothered to write Dot, Dot, Dot at all. One needn't be intimidated by the mathematics in order to get familiar enough with the concepts at play to clearly understand why Van Sloten's sermon is hollow and misleading, and there may very well be a growing need to be able to help our faith-laden friends see it for themselves as well.

If Pastor Van Sloten or any of his congregation happen to notice this little blog post, I do hope they at least check out my short video including a brief excerpt of text from Dot, Dot, Dot, which I'll embed here in case that happy chance arises. I'd encourage them also to pick up a copy of the book for study in their God @School series.

Infinitely many possibilities? Pushing epistemology to the edges

Christian apologists bring up interesting questions from time to time, even if they do so for what are likely to be disingenuous reasons, like trying to cut the legs out from under science. I recently was asked the following question by apologist and philosopher Vincent Torley:

As I pointed out, "Our observations provide support for the hypothesis that the sun always rises at the same time every day – but they’re equally consistent with the hypothesis that the sun rises at the same time every day until the year 2050, after which it sails off into space, or the hypothesis that it rises at the same time until 1 January 2437, after which it turns into a green dragon. In short, there are infinitely many alternative hypotheses about the future path of the sum [sic] which are also fully consistent with the observations we’ve made to date. The question we need to ask ourselves is: why is it rational for us to single out just one hypothesis – the hypothesis that the sun always rises at the same time every day and always will – and ignore all the other hypotheses about the future course of the sun which are fully consistent with the evidence?"

I don't want to get into a long-winded discussion of something so common sense as why we wouldn't seriously bet anything we didn't wish to lose on a hypothesis like the sun turning into a green dragon on 1 January 2437, but I want to comment on why I've worded it this way--about betting. Hopefully in the process, I'll address the question in brief and move on to something more interesting to talk about that I'm not sure Torley realizes.

Betting

Why would I call this a bet? Because that's how we should be thinking about things. The certainty that philosophers classically chased with regard to epistemology--how we know things--is dead. Only when the topics are abstract, as in mathematics, philosophy, and theology, might it make any sense to talk about "certain" knowledge, and in those cases, certainty comes with a caveat: it's held as certainty given the axioms underlying the abstract framework in which the question makes sense. Humorously, theology is the only of these three fields in which even given certain abstractions, it seems unlikely that they can make statements of certainty--how can anyone be certain of anything to do with God?

In the real world, things are a little more difficult--or perhaps they're actually not, but we can't know that. Knowledge, it seems, is pretty clearly mental stuff, perhaps emergent phenomena upon the activity of neurons, etc., or something similar. We don't know reality, then, we know mental models of reality that are hopefully pretty good. The gap between the reality and the mental model that maps it out for us is what can be called an "epistemic gap," and the width of that epistemic gap tells us something about how good our knowledge is.

The best tool I'm aware of for measuring the width of an epistemic gap in many, many cases is a subjective probability assessment, which we can identify with a degree of confidence in the model we're using. That probability assessment can be read as a measurement how likely it is that reality is giving us something that seems to be explained by our model, and since it's a probability assessment, we could use it to make bets.

So let's talk about the green dragon and the sun for a minute this way, and hopefully readers can lead themselves to understand the matter for themselves. Would anyone put a serious bet upon the possibility that on 1 January 2437 the sun will turn into a green dragon? Would anyone not put a serious bet upon the sun "rising" (that is, the earth rotating so that the reader's locality turns to provide an unobstructed straight-line path to the sun)? What kind of odds would a bookie put against the sun failing to "rise" tomorrow, or any other day in the future?

If we ponder those questions seriously even for a few minutes, we get an immediate sense of why the green dragon thing is beneath consideration. Perhaps we could assess the probability that such a thing would happen and then create odds, but without even trying I'm quite confident in saying that such a probability assessment would be very, very small resulting in very, very long odds, made all the longer by the very specific date given. I'm also quite confident in saying that you are as confident as I am.

Our experiences, but more importantly a clear understanding of the physical processes involved in the rotation of the earth and the shining of the sun, among a few other things (like no good reason to believe in dragons, green or otherwise), provide us with a very, very narrow epistemic gap when dismissing the green dragon example as nonsense. We don't need to be certain, which would indicate having no gap. The gap that's there is so narrow that we don't even have to blink to jump over it and get on with our lives. It's fundamentally impractical to give it any of our concern, not least because literally nothing could give us certainty anyway.

Finally, the interesting part!

Having put aside Torley's nonsense and hopefully stoked our intuitions for how we can claim to know things, he does bring up one important and interesting point:

In short, there are infinitely many alternative hypotheses about the future path of the sum [sic] which are also fully consistent with the observations we’ve made to date. (emphasis mine)

Indeed, there are, especially if we just keep changing the date (and time, to arbitrary specification) for the sun to change into a green dragon--maybe with one toe on each foot, or two toes, or three, or four, or..., and in the context of Torley's example, we're not even limited to dragons and could make up anything. Infinitely many possibilities.

This presents a problem with assigning a very low probability to each of them, as I discuss in Chapters 12 and 13 of Dot, Dot, Dot: Infinity Plus God Equals Folly. One of three situations must present itself since the total probability of all possibilities must converge to the value one and infinitely many "very low probabilities" can only add up to one under very specific circumstances. Either (1) Torley is wrong (me too) about there being a potentially infinite number of possible hypotheses, (2) most of the potential hypotheses get probability zero, almost surely, which is normally a taboo in Bayesian reasoning but can be handled using calculus, so far as I can tell, or (3) the probabilities must converge anyway, and so the probabilities within the space have to diminish (rapidly) according to some kind of schema.

This problem is actually interesting, whereas the question of how we get on living our lives calling narrow epistemic gaps "knowledge" is by comparison embarrassingly dull. My guess on this matter, for what it's worth, is option 2, but I don't actually know the answer to this question. Option 1 might be right too, or option 3, but I don't know how to begin to justify that one.

Even more interesting!

Back to Torley, though, to motivate a bit more, though we'll need to slog through some dull, obvious stuff again, if only to have addressed it plainly.

The question we need to ask ourselves is: why is it rational for us to single out just one hypothesis...and ignore all the other hypotheses...which are fully consistent with the evidence?

A point (that I'd classify as dull, but important) is that we're not doing that. We're not singling out a hypothesis and treating it differently, and we're not ignoring all of the others. What we're doing is using some method--Bayesian reasoning--to assign plausibilities to every possible hypothesis and letting the chips fall where they may. In this case, the possible hypothesis that we're right about the earth spinning so that we get a sunrise tomorrow--over and over again--gets a very high probability. Bullshit like green dragons gets a very low probability.

He might argue that we can't assess all of the other possibilities, but we can do so sufficiently without much work once we get one decent hypothesis. The total probability must be one, so if we have one possibility that we can assess as 90% likely, all of the other hypotheses combined have a total plausibility of only 10%. With a hypothesis like the sunrise being normal, the plausibility is so high for that hypothesis that only a tiny, tiny fraction of a percent is left for all of the other potential hypotheses combined, those having to share that tiny, tiny fraction in some way. In other words, we can immediately conclude that the vast majority of the rival hypotheses must have a negligibly low plausibility and thus do not have to examine them directly to know they can be dismissed.

Of course, we "ignore" possible hypotheses in proportion to how unlikely they are assessed. If we have a hypothesis with a very high probability assigned to it for good reasons, then we often call that hypothesis "the explanation" or "knowledge" or "what will happen," etc., and skip the song-and-dance about there being other very unlikely possibilities. Life's too short for all that.

We can get back to something interesting now that we see Torley's deep question isn't even hard--the answer is actually obvious for anyone that understands Bayesian reasoning and most people with common sense, even if they can't phrase it. The interesting bit relates to options 2 and 3 above. Maybe we do decide to ignore almost all (not all) other hypotheses, i.e. we give such low plausibilities to them, including zero, almost surely, that we effectively do ignore them.

It's worth noting that this isn't so insidious. We could note the above discussion about the total plausibility, but even without it, we never even think of almost all potential hypotheses. We'll never even invent names, nor could we, for all of the potential imaginary creatures that the sun could turn into on 1 January 2437. What plausibility do we give those hypotheses? Why do we ignore them?

In the second and third possibilities noted above, particularly in the second, we have to have some method by which we decide which hypotheses are admissible, i.e. given a non-negligible or non-zero (almost surely) plausibility, and which aren't. Finding a philosophical justification for such an epistemic paradigm is an actually interesting question, to be contrasted with Torley's abysmally boring one waiting outside these fabulous gates and looking in the wrong direction.

I can't answer that question, but I'm thrilled to be putting it out there and even to be exploring it. As I note in Dot, Dot, Dot, I don't think that these epistemic paradigms are even static entities. Indeed, I'd argue that some of the God hypotheses were defensibly within epistemic paradigms before a few hundred years ago and that they are not now.

Infinity and God

"Infinity and God," my first humble attempt at making a video so that my words might reach and help more people. Check it out, and if you like it, please share it!

A transcript of the text, which comes from Dot, Dot, Dot: Infinity Plus God Equals Folly:

For some centuries now, theologians have implied that infinity is intrinsic to God, and we should wonder why. The answer isn't so hard: because it has to be. When “"God"” was conceived, at least in the Abrahamic traditions, he lived on a mountaintop. When people climbed the mountain, God wasn't there to be met, and when they descended the other side to meet new people, they were forced to place God in the sky and make God big enough for both peoples. We've been to the sky; in fact we've been above it. God isn't there either. He did not meet us in low earth orbit; he did not meet us when we walked upon the moon; and our robots have not found him on any planet in our solar system or, now, just outside it. Our telescopes cannot see him, even as they peer out into the apparently edgeless universe, and so people have moved God beyond the universe to an imaginary realm, and as God went, they've had to make him big enough to account for it all.

Within that "“all"” is not only every conception of God that humans have entertained and worshiped, but also every potential notion of God that is conceivable. Monotheism demands it, in lieu of an actual physical God of this world. The potential conceptions of God are infinite, and so God too must be infinite to usurp them all. But as we've seen, God cannot be infinite, unless believers would accept he must also be abstract--—that is, mental stuff. Though apologists may defend this idea of God, a majority of believers do not accept an unreal philosophical deity of this kind. Theirs is a living, breathing, acting agent who will one day judge the living and the dead, and this is a fact too easy to lose sight of when talking with apologists.

A special thanks to Jonathan MS Pearce for providing the voice recording of the text.

Get Dot, Dot, Dot: Infinity Plus God Equals Folly here.

William Lane Craig's Reply: A series from notes about infinity, IV

As previously noted, I'm writing a series of blog posts that are adapted from notes I made as preparation to talk with philosopher and author Peter Boghossian's Atheism class at Portland State on November 19, 2013. This is the fourth and final post in this series. I visited his class to address infinity and God, following from the theme presented in my new book, Dot, Dot, Dot: Infinity Plus God Equals Folly.

In this post, I aim to address the reply given by Evangelical apologist William Lane Craig in his Q&A #325, "Infinity and God" on his Reasonable Faith website. Recall that Craig's Q&A #325 was to be background reading for the discussion with Pete's class.

The first two posts in this series aimed to answer questions that I had posed for background thinking about the topics of numbers and infinity. The third post provides my answers to the questions Craig is asked in Q&A #325. They can be accessed here:
Notes about infinity I: About Numbers.
Notes about infinity II: Is infinity quality or quantity?
Notes about infinity III: Answering for William Lane Craig.

The formatting here, since there are two different voices presented, will have my words in black and Craig's own words offset in block quotes and and presented in purple (to be consistent with the usage in Part III of this series). I'll respond as I go through various parts of Craig's reply.

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I'll start off by immediately getting into Craig's reply. All emphasis in Craig's words are his.

These sorts of questions can keep you awake at night, can’t they, Hardus? Let me address each one.
1. Could God have counted all the natural numbers? It’s fairly widely agreed that God could not begin at 0 and successively count all the natural numbers.

Widely agreed? By who? Anyone that knows? This kind of thing should be considered blasphemy, but it passes for sophisticated apologetics--which is exactly why I call it the art of “making stuff up.”

It is worth recalling that in Part III of this series, we saw Craig (in his Q&A #323, referenced in the question) discussing what would be implied by someone knowing all of the real numbers, which are even more numerous than the natural numbers. He does not categorically deny that such a thing would be possible, which is a bit odd given his stance here.

Of course, for fairness, this question asks if God could count the natural numbers, which is a kind of activity, and not what God might know. And for more fairness, I also suggest that counting all of the natural numbers is impossible and thus would agree with his assessment--if (a) it had a clear enough definition of God so that it meant anything and (b) didn't assume that people know things they don't, including and especially about God.

On the other hand, we might wonder if God, being omniscient, knows the largest natural number that he could count to if he tried, seeing as we're agreeing that he cannot count all of them. As previously noted, an omniscient God perhaps should know this largest quantity and yet would then immediately also know its successor, which is a logical contradiction.

Here it’s helpful to distinguish between a potential infinite and an actual infinite. A potential infinite is a series which has a beginning and is growing indefinitely; infinity serves merely as an ideal limit of the series which it never reaches. An actual infinite is a collection which includes an infinite number of members, that is, the numbers of members in the collection exceeds any natural number.

Craig is correct about the difference between a potential and actual infinity, and it’s worth pausing to make this idea clear (and also to distinguish it from a physical infinite). We've discussed this point previously, so I will not elaborate here.

His definition of an actual infinite is essentially correct but contains an element that is a bit off in that he is talking about a “number of members,” which is fine when not being careful or otherwise speaking loosely but is technically both wrong and misleading. I'm pointing this out only for clarity for my readers and not to pick a nit, and I'm fairly certain Craig knows the distinction. None of the infinite cardinals is a number, even if they represent something quantitative.

Counting generates a potential infinite. To say that someone could count to infinity is to say that a potential infinite could be converted into an actual infinite by adding one member at a time. That’s impossible, since for any natural number n, n+1 is always a finite number.

The property that Craig is referencing here is that the infinite cardinals are each what are called limit cardinals, meaning that you cannot get to them from smaller cardinals by repeated successorship (adding). The infinity that represents the size of the natural numbers is actually a strong limit cardinal, which means one cannot get there from smaller cardinals even by the more powerful operation called exponentiation. Countable infinity is the only example of a strong limit cardinal usually defined or discussed. Each larger infinite cardinal is reached from the smaller ones via exponentiation of the “number to the infinity power” kind. Craig’s assessment here is essentially correct.

The question is, could someone count all the natural numbers one at a time by never beginning but ending at 0?

What does this even mean? By “never beginning”?! And what does it have to do with the question asked?

Observe, though, that Craig’s sloppiness shows up in considering infinity to be something like a number (as opposed to a more general quantity), and this is a source of confusion. This question commits a category error, misinterpreting infinity as being number-like because it is a cardinality. The problem with Craig's hypothesis is that there literally is no starting point, so as he notes, the task could never even be begun.

To my mind that is just as impossible as the first task.

Technically, it is more impossible (if that means anything), since it is also guilty of a category error and “never begins.” The advice of Samwise Gamgee’s old gaffer comes to mind: “the job that takes longest to finish is the one that’s never started.”

If you can’t count to infinity, how could you count down from infinity?

Again: category error. Nonsense. Note that this particular nonsense is a usual line for Craig when defending the Kalam, which is, of course, why he brought it up in the first place.

If I’m right about this, then the series of past events cannot be infinite, either potentially or actually—potentially because it is not growing in a backward direction toward infinity, nor actual because you can’t get through an actually infinite number of items one at a time. So the series of past events must have had a beginning.

Now he reveals where he’s going--he wants to slide in his argument for an ultimate beginning that he can credit to (or blame on?) God. He’s not right about this, though. He’s thinking via a category error that is worth exploring more deeply.

Particularly, Craig is essentially doing what he says can’t be done: jumping over the vastness of the infinitude of the natural numbers to start at a stopping place that doesn’t exist. He is treating infinity like it is a number, claiming it cannot be “gotten to,” and then jumping to it to say you can’t get back (implied: without jumping) once you’re there.

To say that the series of past events must have a beginning by this argument is to assume that there is a beginning that occurs either some finite amount of time ago or infinitely long ago. Craig dismisses the second possibility and concludes the first, but he misses the nuance of the very argument he makes: the series of past events could also be a potential infinite, and epistemically we can say no more. What that would mean is that however far back we look, we could conceivably look back further. This is why Craig talks about the impossibility of a task that is never begun: he's comparing it against the existence of the Universe for which he has assumed a beginning.

Physics, of course, has some things to say about this, but not enough to nail things down. This is why Craig used to hang his hat on the Hawking-Penrose cosmology and no longer does now that he’s realized that the Hawking-Penrose cosmology is not a quantum theory and thus cannot account for what happened before a certain time in what we call “the early universe.” We should think of the early universe, before it cooled enough to be transparent, as a place we don't understand, not a de facto beginning or a period containing a necessary starting place for everything.

2. Does an infinite number of numbers exist? My answer to that question is no, not because I think that the number of numbers is finite but because I think that there are no such things as numbers! Numbers are just useful fictions, like the Equator or the center of mass of the solar system. Do numbers exist in any of the three ways you suggest? Not really.

Perhaps surprisingly, I agree to an extent. I’d prefer the term “abstractions” to “fictions,” although after some fuzzy point it may make sense to make the semantic switch. I’ll add to his list of other useful fictions, or abstractions, if he prefers, though: God. Does it exist in the way he suggests? I'd say, "Not really."

Incidentally, we can actually find the Equator and the center of mass of the solar system... not so with God.

Consider your alternatives:
a. Certainly I have the idea of the number 2, for example. But that thought is not the number 2 itself. The nineteenth century mathematician Gottlob Frege called the view that numbers are ideas in our minds “psychologism” and subjected it mercilessly to criticism. That 2 and my idea of 2 are not identical is evident in that they have different properties; for example, my idea of 2 comes and goes, but 2 itself, if it exists, doesn’t depend upon my thinking about it!

It may be a matter of semantics here, but I prefer to think of “two” as the abstract entity that describes the property of “twoness”: Any set of objects with the property of “twoness” will be said to be enumerated by the value “two.” This property is apparently eternal and immutable, as mathematical properties and other abstractions are, but I doubt it has meaning without minds that have the idea of it. Perhaps in this sense, talking about the teleology of mathematical objects is more appropriate than about their ontology.

b. Certainly, things that exist in the world can be numbered; for example, Mars has two moons. But this adjectival use of number terms doesn’t require that numbers themselves exist. As Frege showed, we can express that Mars has two moons without using any number terms at all by saying that there is some entity x which is a moon of Mars and some entity y which is a moon of Mars and x is not identical to y, and for any other object z, if z is a moon of mars, then either x is identical to z or y is identical to z. (In logical notation: (∃x) (∃y) (Mx & My & x≠y ((∀z) (Mz ⊃ z=x ∨ z=y)).) Pretty slick, eh?

Slick? I guess. Observe that in the slick part we still identifiably have a set of things (x and y) that together exhibit the property of twoness. As often is the case, things have progressed substantially since the nineteenth century, where Craig finds an apparently surprising number of his arguments for God (perhaps because God was easier to defend then than now?).

The hotly disputed question is whether the use of number words as referring terms commits us ontologically to the existence of numbers. For example, the statement “Two is the number of Mars’ moons” is thought by many philosophers to commit its user to the reality of the number 2. This strikes me as perverse. Metaphysics is surely a lot more difficult than that! These sorts of ontological commitments happen, as Wittgenstein said, when “language goes on holiday.” You can’t read metaphysical commitments off language—at least so I think.

This is why I lament that the term for "existence" and "existence in the abstract sense" happen to be the same word. I do wonder what commits Craig to the reality of God if he finds this kind of thing a perversion of metaphysics, though.

c. Obviously, we make marks on paper in doing mathematics. But here we need to distinguish between numbers and numerals. There are many different ways to represent the number two: 2, II, úú , and so on. But these are numerals, not numbers. There are many numerals but only one number 2, if such a thing exists. So numbers do not exist on paper or computer screens. So what you should say, I think, is that there is at most only a finite number of numerals in the world, not that there is a finite number of numbers in the world. There are no numbers anywhere.

A rather fun "philosophical" question that rolls around in some mathematics departments asks the question “Does 2=2?” where the two numerals are written in different fonts.

It’s an interesting matter as well to marvel at the confidence with which Craig can assert “there are no numbers anywhere” while maintaining an adamant belief in God. I mean, I do agree with him about the numbers, but I am currently looking at a collection of four pencils on my desk and have an immediate sense of fourness but no matter where I look, I have no sense of deity.

3. What about other kinds of abstract objects? You are absolutely right that philosophers who believe in the existence of abstract objects think that novels, plays, musical compositions, fictional characters, and so forth, exist as abstract objects. What is disputed is whether these are created by their writers and composers or whether these people just happened to stumble upon these pre-existing objects. Many people feel quite uncomfortable in saying, for example, that Leo Tolstoy did not create Anna Karenina but just found it. This view seems to seriously depreciate the creative genius of authors and composers. So many want to say that people created these abstract objects. Still, it’s hard to see why, once you grant that such abstract objects exist, these collections of words or notes did not pre-exist their discovery by these folks. I think it’s better to just deny that such abstract entities exist and maintain that our ability to talk truthfully about them (e.g., “Sherlock Holmes is the most famous detective in English fiction”) doesn’t entail their existence.

This topic may be interesting within philosophy, but the only commentary I have for it at present is a remark by Richard Dawkins in The God Delusion: “The God of the Old Testament is arguably the most unpleasant character in all fiction….”

4. Does God have complete foreknowledge of the future? Yes, why not? Your statement that “the number of future events counts up to infinity” is ambiguous. We’ve already agreed that it’s impossible to “count up” to infinity. So the series of future events “counts up to infinity” only in the sense of a potential infinite: infinity is the limit to which the series of events strives but never reaches. There will never be an actually infinite number of events. From any point in time that you pick the number of events future from that point is always finite and always increasing. If you pick the present event as your point of reference, the number of future events is 0! That’s because temporal becoming is a real and objective feature of the world.

Again! A question about God, and Craig's answer is an absolutely solid and unqualified "yes," as if he knows. This point can hardly be overstated. He doesn't know; in fact, he can't. He's making it up--with all the confidence of an apologist.

This segment of his reply kind of goes all over the place, but his point is solid, though I suspect it misses the mark. His point is that at no point in the future, from now or any reference point, will an infinite number of events have taken place. I'd say that's correct.

I got the impression, however, that Hargus's question was about whether or not God’s mind actualizes this potential infinite. Craig does not address this question, and he is wise not to. To actualize infinity in the mind of God is to render God abstract by Craig’s own argument (or is it a petard, ready to hoist?).

The question I suspect Hargus meant presses, though, particularly since one of the properties attributed to God is being eternal and standing outside of time. Is God bound up with the unfolding of time in our universe or not? If so, in what way is God eternal or outside of time? If not, then why shouldn't God be able to see the totality of the past, present, and future and have proper knowledge of it? And why shouldn’t this constitute an actual infinity if time presents a potential infinity--again, per Christian theology, at least in the “world to come”?

The only escape from this pen for Craig is to reject some Christian theology. It could be about the hereafter, and perhaps the reality of cosmology, depending on how that goes. Maybe he could claim that at some point in the future, God ends everything that is a kind of universe in which there is knowledge of events. Such a claim contradicts the idea of eternally keeping one’s personality in heaven, at the least, though.

You say that it’s impossible for God to know everything in the future. That doesn’t follow from anything we’ve said.

Except apparently it does. Craig has claimed that an actual infinity is an abstract thing only, and yet if God were to know everything in a potentially infinite future, which is what Hargus asked, then God should know an actually infinite number of things--even in an astoundingly simple universe. Craig can’t have God knowing an actual infinity of things because that would render God an abstraction as well, so it would seem to follow directly from Craig’s rejection of the actual infinity that God cannot know an actually infinite set of things.

To get an objectionable, actually infinite number of things out of this, you have to think that God’s knowledge is broken up into propositional bits that actually exist. But such a view of God’s knowledge is not obligatory for the theist (and traditionally has been denied by theists). Suppose God’s knowledge of reality, including the future, is non-propositional in nature, and we finite cognizers represent what God knows non-propositionally by breaking it up into propositional bits. (For an analogy think of your unbroken visual field, which someone could represent by breaking it up into pixels.) Then there is no actual infinity of ideas, thoughts, propositions, or what have you.

This part gets a fairly big “uh, what?” attached to it. God’s knowledge is not propositional in nature? Not only does this appear to contradict what Craig himself wrote in Q&A #323 (referred to by the questioner and noted above), it seems to contradict essentially everything people believe about God. For example, here's a proposition: “Charles murdered Jane.” This is precisely the kind of knowledge required for God to judge Charles (and perhaps Jane).

Perhaps more relevantly, "Charles thought x (at 11:46) and then thought y (at 11:48)," which is also precisely the kind of knowledge that the Abrahamic religions are so utterly concerned with having God judging. If time runs ever on and on with sentient minds under God's dominion all throughout, once they began, then someone is going to be thinking something at every moment in the potentially infinite future, and God traditionally is taken to be cognizant of all of it.

This strikes me, then, as being intellectual gerrymandering because it insists that this perfect and omniscient being is, indeed, required to be ignorant of simple propositional knowledge for no better reason than that apparently otherwise it would be too hard to defend the being from philosophical analysis.

To pick a nit, with regard to Craig’s analogy about the visual field, someone should perhaps tell him that there are only finitely many rods and cones in the retinas of the eyes, and so our “unbroken visual field” is indeed pixelated in a sense before the brain interprets it. It is not yet known if reality is continuously smooth or discrete at very small scales. If discrete, his analogy is even more utterly bogus. If continuous, infinity is actualized physically everywhere, contradicting his claim that infinity isn't actualized.

So do not limit God by denying His complete foreknowledge of the future. There is no good reason to adopt such a view and it impugns God’s greatness.

Does this not feel like talking out of both sides of his mouth now? Perhaps this is how God's greatness is kept unimpugned: by saying one thing about God now and another thing later. God can't, but must, know infinitely much, apparently, but that shall not be said in the same sentence.

It is interesting and worth commenting on that Craig chides Hargus for limiting God after Hargus expressed clear discomfort and unwillingness to do that. Perhaps this also is how God's alleged greatness is maintained: utter fear of challenging the idea even accidentally and sidelong.

The most pressing commentary to make here, though, I think is on the nature of these kinds of questions in the first place. Hargus admits that he is a committed and devout believer, but these are questions for doubters, questions that exist at the fraying edges of the fabric of belief. It is incredibly unlikely that Craig will ever convince a skeptic to believe in God based upon this kind of waffling about infinity and highly abstract philosophy, though I suppose he can reassure the devout and faltering by folding under the loose edges.


This finishes the series!
If you have enjoyed reading it or find this kind of material interesting, I think it is very likely that you'll enjoy Dot, Dot, Dot: Infinity Plus God Equals Folly, so do pick it up.
More importantly, do yourself a big favor by joining me in thanking Peter Boghossian for initiating this entire series by picking up his Manual For Creating Atheists.