Friday, November 29, 2013

"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.

Thursday, November 28, 2013

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.

Tuesday, November 26, 2013

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.

Friday, November 22, 2013

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.

---

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 & xy ((∀z) (Mzz=xz=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.

Thursday, November 21, 2013

Answering for William Lane Craig: A series from notes about infinity, III

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 third post in this series, which I anticipate will span four posts. The visit to his class was 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 in my own fashion the question Evangelical apologist William Lane Craig was asked 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. They can be accessed here:
Notes about infinity I: About Numbers.
Notes about infinity II: Is infinity quality or quantity?

The formatting here, since there are three different voices presented, will have my words in black, Craig's questioner's words offset in block quotes and presented in dark green, and Craig's own words offset in block quotes and and presented in purple. I'll respond as I go through various parts of the question.

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We'll start immediately by looking at the question Craig was asked by a reader named Hargus from South Africa.
Dear Dr Craig
I am a devout Christian and I read your Q&A every week. I have read Q#323 today and I was totally surprised by it because it is basically exactly what I have been thinking about for the last couple of days.
Immediately, then, to do this justice, we should take a moment to examine Craig's Q&A #323, "The Concept of God." In the interest of brevity, I will only address what I considered to be relevant to the topic of infinity and God from Q&A #323. Craig is writing here, addressing a questioner named Ian.
So far so good; now let’s look at your example: the set of natural numbers, which has an actually infinite number of members. You assert, “we know that no possible being can know all of the elements of this set, since it's uncountable.” This inference is mistaken, Ian. First, the set of natural numbers is countable! It’s what mathematicians call a denumerably infinite collection. For an example of a non-denumerable collection, you should have picked, e.g., the set of real numbers, which is so numerous that it can’t even be counted.
I have to take issue immediately. Here we have either a misunderstanding by Craig or a despicable misuse of semantics. Mathematicians call the set of natural numbers “countable” or “denumerable,” but none imply that they could actually all be counted. Indeed, Craig himself makes this point (and does so in the response we’re investigating, numbered Q&A #325, as we will see in the next post in this series).

His questioner, Ian, is using “uncountable” in a colloquial, non-precise way, and Craig has turned to a precise mathematical term. Craig is correct to identify the set of real numbers as “non-denumerable,” or “uncountable,” but the way in which he points this out reveals that he’s a novice, not an expert, with this topic. A simple note that mathematicians call that size "countable," yet another lamentable term, and that he hopes to avoid confusion by making that clear is all that is necessary.

Never mind that minor point, however. Why can’t someone know all the real numbers? There’s nothing incoherent about the universally quantified claim that for any person S and any number n, if n is a real number, then n is known by S. Certainly, we may agree that “if one being knows or can know more natural numbers than another, then that being is more knowledgeable or capable, respectively, in this area.” It follows that a being who knew all the real numbers would be maximally knowledgeable or capable in that area because there aren’t any more real numbers than that. So this is a nice illustration of maximal greatness in this respect.
While Craig is accurate in saying that knowing all of the real numbers would exhibit "maximal greatness in this respect," it's worth noting that it is a poor illustration of "maximal greatness," since "this respect" is itself not maximal. I do not point this out to say that Craig is in error or has misspoken, but I do wish to make it clear that the use of "maximal" here is local to a given context, that of a particular set of potential knowledge. It's easy to misread this to think that he's implying maximal greatness in more generality.

To quickly elaborate, Georg Cantor, who is known to Craig in this context, proved that given any infinite collection, one can create a larger infinite collection from it (known as its power set). Thus, the non-denumerable (uncountable) infinite cardinality of the real numbers is not maximal but rather just another size of infinity that is certainly larger than the so-called “countable” infinity that enumerates the natural numbers. This is a poor illustration of “maximal greatness,” then, as I pointed out in God Doesn't; We Do and again in Dot, Dot, Dot--there is no such thing because given any level of greatness, in this respect, a greater level can be conceived of immediately.

Can someone know any real number?

Now, more importantly, I’m not sure about this claim that any person S can know any (arbitrary) real number n. Without at least implicitly assuming infinities exist, it may not be correct to say that anyone can know the wide majority of irrational numbers, and the vast bulk of real numbers are irrational. How could we without infinity? Each has an infinitely long decimal expansion that does not repeat periodically, and most exhibit no discernible pattern in their decimal digits. Again, we see inexpert thinking on these matters.

Given some of his other arguments, take a moment to note that Craig explicitly admits that knowing more propositional knowledge confers a degree of greatness of a kind upon a more knowledgeable being.

What about all of them?

While I suspect Craig is merely using this as an example, I'd like to point out that knowing all of the real numbers is hardly short of preposterous. It is very difficult to convey how large the set of real numbers is, but it is so large that when asked this about the natural numbers, my response is "I don't think so," whereas when asked it about the reals, my immediate reaction is incredulous laughter with a loud "no."

To get a sense of it, think about a number with lots of decimal places. Realize that each of those decimal places admits ten possible digits. If there are only a few decimal places, say twelve, that gives us a trillion possibilities. If there are hundreds of decimal places, well, that's a lot of possible real numbers. Now realize that every real number can be expressed with a decimal expansion that contains infinitely many decimal places, each of which can take ten possible digits. Change any one of those anywhere, and the result is a new real number.

Of course, we should wonder, since he brought it up, if God could know all of the real numbers and what it would mean in both cases. Craig wants to deny that God knows infinitely many things, as we'll see, so we must also ask: If not, why not?

I think your confusion is that you’re conflating something’s having no upper bound with a property’s not having a maximal degree. Hence, you say, “Since there are certain areas pertaining to knowledge and power which do not have an upper-bound, it is necessarily the case that any possible being's net knowledge and capabilities are limited and imperfect.” As we’ve seen, that doesn’t follow. A property can have a highest degree, e.g., knowing all the natural numbers, even though there is no highest number.”
First, I don't think Ian is confusing anything. He admits that he feels that there are certain areas, noting knowledge and power, that do not appear to be bounded above. Imagine, for instance, lifting a weight. It could weigh 100 pounds, 200 pounds, or more. At what point would we say that even in the imagination the weight could not weigh any more than that? I'd contend that we can never say that and be both honest and correct about it. We could also wonder how many natural or real numbers someone can know. There is no upper bound to this, and saying that knowing all of them means 'highest degree' doesn't get around that fact.

Additionally, there are some tricky points here. One is the nearest to a sticking point for salvaging the “God’s maximal greatness” argument (due to Anselm, 11th c.). Before addressing it, notice that Craig has directly implied that it may be possible to know all of the natural numbers. This is interesting because (a) he later argues even God doesn’t know them all, and (b) that God’s knowledge doesn’t work propositionally like this anyway. Why make this argument, then?

Maximal God

The big sticky point here, for what it’s worth, is on whether we can conceive of the idea of God as being the sum-total of all possible goodness (possessing all positive properties), and whether this list is finite or not, as possessing all of them implies maximality. This argument is the crux of Alvin Plantinga's ontological argument, which he appears to have adapted from Kurt Gödel. Here, the issue of infinitude of God’s nature can be side-stepped, which may be desirable for apologists since I assert that if God is infinite then God is also abstract (i.e. not real).

The main issue with this “maximal (though finitely) good” definition of God is that it seems highly implausible given our observation of the world (Cf. Problem of Evil). Still, even without appealing to that, it is somewhat problematic because it raises the question of why an all-powerful, all-knowing, perfectly benevolent God isn’t able to conceive of or create a single additional good thing in the universe. Clearly, this question lays bright highlights all over the Problem of Evil, once we go back to it--particularly, our universe is filled with much obvious suffering and what appear to be sub-optimal circumstances for identifying a universal state of perfect goodness.

In effect, taking the position that God is maximally good, though finitely so, also ties God’s hands. In this conception, God has so ordered the universe such that were he to change anything, it would be worse off for his doing so. Obviously, this contradicts or renders pointless other theological objects (like intercessory prayer, for one).

Can someone know all the natural numbers?

I don't think so. The first property of the natural numbers most people should put in their minds in this context is that there is no largest natural number. It is entirely unclear how someone could know a collection of information that is infinite in scope.

We could wonder about God, though, since God is hypothesized to have no normal limitations, like a physical brain with only so much storage space, and is said to be omniscient. If God can know all of the natural numbers, then two things follow: first, God actualizes infinity in his knowledge set, and second, God has propositional knowledge ("n is a natural number" is a proposition). I don't immediately see a problem with these consequences, but Craig denies both of them. The reason I don't see a problem with them, of course, is because I think God is also an abstraction we have imagined, not a real thing.


Now we can move on to Q&A #325, the object of discussion.

I do not aim to tackle Craig's 325th response in this post but rather will answer his questioner's questions in my own way. Hargus writes,
Infinity is a concept that has been driving me nuts as I cannot completely fathom it. A few comments I want to make:
1. Is it possible that God could have counted the whole set of natural numbers? The answer according to me is no. Why? Because the set of natural numbers is infinite. You will never stop counting. So even though God is almighty, He cannot do things like these. It simply doesn't make sense. It's like saying God can create a rock that He cannot lift. It's a logical incoherent statement.
I feel this requires an important note from the outset. The first question, could God count all of the natural numbers, actually doesn’t make sense since we have no clear notion of what “God” means.

This sounds pedantic, but I think it's a core problem with dealing with apologists, theologians, and theology in general: we are simply to assume that the apologist means something that is clearly understood when speaking of God, and that's simply not the case. If we suppose that he means an omni-capable intelligence of some kind, then the question could possibly be addressed. If, instead, we assume only the uncaused first cause Craig defends in the Kalam Argument, I don't think it has any meaning. If we assume Yahweh of the Old Testament, the answer is probably not, seeing as this Yahweh is frequently surprised and angered to find out things he didn't know when there were only two to four people living on earth in the special garden he allegedly created for them. If we mean the laws of physics or a grand sense of universal oneness in the universe, the question is literally nonsense. Can you see how it is a problem? It's a very big one that apologists all expect us to gloss over so that they can make their arguments.

More directly to Hargus's question, that the infinity of the natural numbers cannot be completed by succession (counting) is accurate, so it shouldn’t be the case that any being or entity could “count the whole set of natural numbers.” In other words, I think Hargus is correct in his assessment here. Further, it is my opinion that this creates a problem for an omniscient deity who seemingly must know what is the last number he would ever count, if he tried, and yet then would automatically know the next number by the simple rules of succession (a logical contradiction).

2. Regarding the existence of abstract objects: One thing that has bugged me about this is the whole thing about infinity. Does it exist? I have read some of your comments on this, namely that you say that infinity is a mere concept of the mind that doesn't exist in the real word. This makes sense to me. So if there are an infinite amount of natural numbers, how could they all exist? Well I believe numbers exist in two to three ways:
a. Since numbers are mere concepts, they exist in our minds as we think about them.
b. Numbers exist conceptually as real world objects represent them. I.e. the number of planets in the solar system.
c. They exist conceptually as symbols on a piece of paper (or computer screen).
Because only a finite amount of thoughts, objects and written symbols exist, it follows that only a finite amount of real numbers exist in the real world.
This is complex--the ontology of mathematical objects is its own rich philosophical field with many unsettled debates that I’m not an expert in. I generally fall into the camp that numbers are abstract objects that count other objects, real or abstract. This is closest to Hargus's situation that he labels (a). It would be helpful to read Part I in this series if you want more information about what numbers are and hopefully addresses Hargus's (b).

In that sense, things are still tricky to get right because while it seems clear that there are only finitely many objects (say atoms, protons, or what-have-we) in the observable universe, they can be combined and recombined in various ways, and the combinations thereof can be combined and recombined in various ways again, ad infinitum, providing not only abstract structures of arbitrary size but also ones rooted in physically identifiable structures, though these are eventually quite convoluted. Some physicists suggest that the universe itself may actually be infinite in scope, perhaps containing infinitely many objects (e.g. stars), though I’m not entirely clear on what this even means or how we could find it out.

As to infinity, I’d say that we don’t know if infinity is physically actualized, that we lack reason to claim that it is physically actualized, and that as an abstraction there is no good reason for it not to be “actualized” (as mental “stuff”). I give no real ontology to abstractions, though, and do not identify mental “stuff” as “stuff” at all--it’s just ideas. Indeed, a rich, interesting, and useful abstract framework has been built up around the notion of an actualized (but abstract) infinite.

Finitists, a small fringe of people in the philosophy of mathematics, reject the actualization of infinity even as abstractions, and they’ve found that effectively all mathematics can be done without it, but the cost is relatively heavy. Though it may not be satisfying to think of it this way, infinity may very well be a useful fiction of a kind that makes certain kinds of math (like calculus) considerably easier and more accessible. Reading Part II in this series might be of some help here.

As for the third way Hargus feels that numbers may exist, this is an interesting aside (that I won't explore) into the difference between numbers and numerals. Craig actually addresses it in his response, so keep your eyes open for Part IV where I'll present that.

3. Abstract objects aren't limited to numbers. It includes things like music, literature and other forms of art. Now there are an infinite amount of songs that can be composed. Do they all exist? Again the same line of reasoning goes. A tune lives in the mind of the one that composed it and the ones who hear it. It is a concept, an idea which didn't exist before someone thought about it. So there are only a finite number of art forms in the real world and always will be even though the number increases all the time.
This is an interesting question, isn’t it? In the sense that a song could be defined as a concatenation of notes, or a piece of literature a concatenation of words, etc., we could say that at least potentially, yes, the possibilities are infinite. Because these things take time to create, though, ultimately only a very small number of the possible pieces of music or works of literature will ever be made. Interestingly, we could define "rational" songs (which periodically repeat a loop of notes unchangingly forever or that stop eventually) and "irrational" songs (which do not satisfy that definition) in this way, analogously to rational and irrational numbers.

For fun, we might wonder about a machine that, hypothetically, runs indefinitely into the future, choosing notes at random within a particular key (or not within one), creating an unending "song," but this is mostly just something to muse about. By considering the number of ways that each note could occur, we can quickly start to realize that the potential variation in such songs is infinite, although even if they were all explored, at no point would an infinite number of songs have been composed in this way.

4. Now I want to say something about the future. God as we know can foresee the future. This is what makes prophecy possible. We know that the world has a beginning, and it will have an end in this age as the Bible predicts, but the age to come will have no end. So the number of future events counts up to infinity. Now is it possible for God to know everything in the future? Again as explained above this is impossible. One cannot know an infinite amount of things. It is simply an illogical thing to ask from someone like God, even though He is almighty, and I am not trying to limit Him in any way. So God knows the future. How much and how far? Well as much as He chooses or as far He deems necessary.
First, this “as we know” claptrap about God is nonsense that really grates me (and no doubt it grates Boghossian as well). No, you do not know.

We also don't know that the world has a beginning, unless we're talking about something so local as the planet Earth or the human species. The question about whether or not the universe has a beginning is not answered, and no amount of theology-led philosophy on the part of apologists like Craig can make it known.

We also don't know that the world will have an end "in this age" (or even what defines "this age" in this context), again unless we're very local about things, like knowing that the Earth will be destroyed by the sun in a few billion years--which humanity could potentially survive, yet another thing we don't know. The Bible predicting something does not confer knowledge except about whether or not the Bible contains that idea or doesn't, which does not transfer to claims about the real world on the grounds that the Bible is scripture in some religion.

We also don't know that "the age to come" means anything or that it is endless, although that point is discussed in Part II of this series. Hargus goes on, though, to make the same point I do about the idea of an endless future: it seems to commit God to knowing an actually infinite quantity of ideas, which Craig rejects.

The rest of this reveals rather how inane and incoherent the concept of God is, doesn’t it? God is put to an impossible thing by what is considered to be his apparently obvious nature. It looks also as if the concept of God is frightening, otherwise such honest questions seem not to require qualifiers about not wanting to "limit Him in any way."

Hargus closes:
These are my comments. I therefore believe God is almighty, but He cannot do illogical things, or know things that do not exist. Thanks for reading and please let me know if you agree with my views. Also please tell me of articles I can read which might give me more insight into these things.
Finally I want to thank you for the work you're doing. I think you're making a big difference in many people's lives.
Regards.
Hardus                                                
South Africa
I've only included the closing for completeness and don't wish to comment upon it beyond drawing attention to his use of "therefore," after which he makes the right point but only after again reminding us that he believes God is "almighty," for whatever set of reasons he holds in that regard.


There's still more to come! In the next, and final, post in this series, I plan to address Craig's answers to Hargus, which can be found at Craig's Q&A #325, "Infinity and God".

If you like how this reads or think this kind of subject matter is interesting, it's very likely that you'll enjoy reading Dot, Dot, Dot, so please pick it up.


Edit: The fourth and final post in this series is available here.

Wednesday, November 20, 2013

Is infinity a quality or quantity: A series from notes about infinity, II

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 second post in this series, which I anticipate will span four posts. The visit to his class was 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 hope to answer questions 3, 4, and 5 from my original list of five prepared background questions for his class to ponder.

3. How can infinity be thought of as a quality and as a quantity?
4. Can the quantitative aspect of infinity be removed from the concept of infinity?
5. How is William Lane Craig's thinking muddled on the idea of "potential" infinities on a future timeline if God is supposed to be separate from the universe and eternal? Particularly, how is Craig treating God in a local frame when his God hypothesis requires a global one? (See Craig's Q&A #325 for context.)

As a note, Evangelical apologist William Lane Craig's Q&A #325, "Infinity and God," was to be background reading for the discussion, and my notes assume that and make use of it.

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3. How can infinity be thought of as a quality and as a quantity?

Infinity is both a quality--being inexhaustible, being able to remove elements without changing the size--and a quantity--a sense of the “number of things” present in infinite collections like the set of (all) natural numbers, {1,2,3,...}. Because infinity answers the question “how many?” at a very intrinsic level, it is immediately and inextricably tied to the notion of being a quantity as well as a quality.

Admittedly, this question seems a little weird, but it is key if we are to untangle the thoughts of apologists like William Lane Craig who wish to think of the infinite only as a quality. Of course, it's important to note that apologists and theologians, Craig in particular, are often unabashed about letting their theology lead their philosophy, even on topics like the philosophy of science or the foundations of mathematics. In this case, Craig's theology, or really his defense of the multiply-failed Kalam Cosmological Argument for the existence of an "uncaused first cause" demands that he reject the infinite as a quantity and consider it only qualitatively, as an abstraction that captures the notion of being inexhaustible.

Potential and actual infinities (and physical ones too)

Craig talks about this typically in the context of "potential" and "actual" infinities. A potential infinity is seen when we have some kind of a process that should go on interminably such that if it were completed, it would result in an infinite collection. An "actual" infinity is the completed infinite collection itself--which may still only be abstract in nature. We see a potential infinity when we think of counting: one, then two, then three, and so on, knowing that we need never stop. The set of numbers that would be generated by "completing" that task, called the natural numbers, is an example of an "actually" infinite set.

These should be distinguished with physical infinities, like if we were to have good reason to believe that the universe itself is infinite in scope and containing an infinite quantity of stars or other materials. A physical infinity would imply an actual infinity, but this implication does not go the other way around because we can have actual infinities of abstract objects like numbers, including many that do not count anything physical at all. Potential infinities merely suggest an actual infinity may be meaningful but need not imply one, and they certainly do not imply a physical infinity.

Infinity as abstraction

Craig takes the position that actual infinities (yet another lamentable name for something) are abstract at best and essentially non-existent. He is in some company in doing so. I take the tack personally that infinite cardinalities, unless proved otherwise by finding out more about our universe, are abstractions and "exist" as mental stuff, not to be confused with Platonic idealism which gives more real ontology to abstract objects than I'm willing to confer.

A small group of mathematicians called finitists (who might be right about this) agree with Craig's stronger position that actual infinities do not exist even as abstractions. I feel that this position may go too far, as I am comfortable with letting abstractions be abstractions, even if that renders some or all of them "useful fictions" and nothing more.

For my part, I'd argue that like numbers, infinite cardinals are abstractions that we use to make models (also abstractions) with which we try to understand the world. Indeed, I even say that infinite collections themselves are abstractions as well since we don't have any reasons to believe that they enumerate anything physical. And, even if they did count a real, physical infinity of things--because of the fact uncovered by Georg Cantor in the 19th century that given any collection, we can use it to define a larger collection--we'd soon be able to get to larger infinite cardinals that, for all intents and purposes, could only be seen as abstractions.

So, it's both quantitative and qualitative

So, we can think of the infinite as a quality of being beyond any real-world reckoning, true inexhaustibility, or we can think of the infinite as being a quantity, the "number of" objects in some set with infinite cardinality. The contemporary understanding of infinity uses both ideas.

In the context of the set theory that underlies modern mathematics, the infinities act more like quantities that enumerate different kinds of infinite sets, and in context of its use in fields like calculus, it is probably more accurate to think of infinity as a quality: say, for instance, going out sufficiently far in a certain kind of process to get arbitrarily close to something nice to work with. Still, even in this last sense, infinity acts like a quantity in important ways, e.g. having enough iterations of the process to get arbitrarily close or, for "divergent" processes, a comment that the quantity obtained by repeating the process indefinitely returns an indefinitely large return.


4. Can the quantitative aspect of infinity be removed from the concept of infinity?

As hopefully is implied above, I do not think so, at least not meaningfully. To do so requires us, at the least, to reject what is called the Axiom of Infinity, which guarantees as a consequence that at least one infinite set exists. Mathematics can be done without it, but the set of axioms that serve as the backbone of most of the mathematics done in the last century include that axiom.

As noted, removing the quantitative aspect of infinity binds us to a philosophical position in mathematics known as “finitism,” which asserts that the infinite is only a quality of inexhaustibility (sometimes replacing the term “infinite” with “indefinite”). I haven’t deeply explored finitist mathematics, but finitists claim that any math that can be done with infinity (even calculus) can be done without it. Of course, this comes at a cost. From what I have seen, a great deal more mathematical machinery is needed to do things this way. In that regard, infinity may be a useful fiction in the sense that it greatly simplifies some conceptual matters in highly useful mathematics.

Committing ourselves to finitism also raises uncomfortable and unanswerable questions about large numbers, particularly like “when do larger numbers stop possessing meaning? And why?” As noted in the previous post in this series, it also renders the idea of a perfect circle nonexistent, which may well be how things are but feels very uncomfortable. It could be that these questions are themselves meaningless, but they are intuitive. Further, transfinite mathematics has proved to be a rich, interesting field that has shown usefulness in laying theoretical foundations under other practical fields like differential equations.

Rolling back the clock

For fairness, the finitists might be right. That said, to reject the quantitative notion of the infinite would be to roll back the clock about 120 years on our understanding of set theory and other mathematical objects, particularly those related to infinity. I feel like this is a common theme with Evangelical apologists like William Lane Craig--it seems they want to do it with certain results from cosmological physics as well, at the least.

Of course, more fundamentalist apologists than Craig want to do it with biology as well, seeking to undo the core theorem of that field: evolution by natural selection. Since we know that these apologists and theologians are led by their theology, including on their philosophy and even their science, it is my opinion that we are not presented with good reasons to roll back the clock on any of our thinking, to a time when God was easier to defend or otherwise. The right theology could lead us to conclude just about anything, so it is not a reliable method for determining how we should view or think about the world. 


5. Can you see how Craig's thinking is muddled on the idea of "potential" infinities on a future timeline if God is supposed to be separate from the universe and eternal? Particularly, how is Craig treating God in a local frame when his God hypothesis requires a global one?

This question will be addressed more fully as we explore the text of Craig’s Q&A response in more detail in future posts, but as I laid it out separately here, I'll take a moment to comment on it.

A general note about Craig and mathematics

On the whole, although some of his mathematics is a little sloppy, I do not really disagree with much of what Craig thinks about numbers and about infinity, though I do disagree with his uses of it and his conclusions. It's worth noting that for a lay person in mathematics, some of his use of mathematical topics is quite savvy, though the rough edges are plainly visible when one knows where to look. Particularly, his use of arguments about infinity in order to employ the infamous Kalam argument is not only weak but quite poor (committing both a category error and circularity).

Further, as noted, Craig appears to be a finitist--though largely because it seems to be the only way he can maintain a defense of some of the philosophical arguments he wishes to make for his God. This is important to note because it appears to reveal yet again that his theology leads his philosophy, even his philosophy of mathematics. This reeks of a kind of circularity that shouldn't be allowed to pass unexamined. Of course, assuming his God in the first place is a major problem all apologists appear to be guilty of.

Confusion on potential and actual infinities

Now, to address the question directly, I must briefly summarize Craig's position. Craig is quick to talk about potential infinities in the future timeline of the universe, say when asked (as in his Q&A #325). He is also quick to dismiss a potential infinity in the past timeline of the universe (again as in Q&A #325) by arguing about an actual infinity there.

This is important. For him to accept a potential infinity in the future but not in the past reveals a bias. A potential infinity in the future means that however far forward in time we look (or wait), we eventually will be able to look (or wait) a little further, this property existing without a boundary--one might say that "time keeps on ticking, ticking, ticking into the future," for instance. On the other hand, a potential infinity in the past should mean just the same: however far back we look, we should conceivably be able to look back a little further. A potential infinity in the past does not assume that we must measure time from infinitely long ago.

Craig dismisses this by saying that if the timeline in the past were infinite, then an infinite number of moments must have passed to get to the present moment. Here, though, we see him confusing a potential infinite in the past, described above, with an actual infinite in the past. In essence, what Craig is saying with regard to the past is that we either have a beginning of time finitely long ago or infinitely long ago, and then he dismisses the infinite case as nonsense (which it actually is--if we have an infinite past timeline, then there was no beginning). But this jumps to an actual infinity in the past and assumes a beginning, which is what Craig wants to establish via this argument in the first place. I raise this point in Dot, Dot, Dot.

God's independence from time

Now, to get to the meat of my study question, note that God is supposed to be both eternal and outside of the universe, including both time and space. Thus, God should be able to take a global view of the timeline of the entire universe. To the best I can tell, this idea is the God that Craig defends, and it seems to be integral to the general understanding of the deity, among a few other things. If God is outside of time and eternal, omniscience should imply that our timeline (inside of time) should be simultaneously and entirely visible to him all at once. This is a global view, available only from outside, instead of the local view we have on the inside.

The key point here is that from God's perspective, the same argument Craig gives to dismiss an infinite past timeline should also prohibit an infinite future timeline. It seems easy to sidestep this point by saying that the universe must therefore also have an end, but it isn't so. Christian theology binds him there. Even if the universe as we know it comes to an end, Christian theology proposes an eternal (infinite) hereafter. Presumably, God would have full knowledge or awareness of this infinite timeline in the afterlife, and thus God is committed to an actual infinity in the future time direction. That removes, from the global perspective, any qualified reason to reject an infinite timeline in the past time direction.

Some rebuttals rebutted

The rebuttal, of course, is to say that the hereafter is a different realm, and that the universe has actually ended, so time stopped. The problem with this claim is that every description of heaven that I am aware of ever having heard about contains the idea that there will at least be sentient minds/souls present that are capable of experiencing things, presumably in time. People will "get to spend eternity with Jesus." People will "get to live forever with their friends and relatives." Etc. Those kinds of statements only make sense in the context of a future timeline, either in our universe or in some (imaginary) meta-universe where God and heaven are, and that timeline is always described as being eternal, i.e. infinite in scope. To me, that suggests the only escape from this issue is either to admit actual infinities, which breaks Craig's avenue to the Kalam, or to revise Christian theology with a finite future timeline.

A final point to defend here is the case where we could have a finite past and infinite future. That's technically true: it is a possibility. That's not the point, though. The point is that Craig defends that the past must be finite because infinity doesn't exist. He's employing a double-standard on the past and future timelines based upon a question-begging assumption that the universe had a beginning (either finitely long or infinitely long ago)--and all this to establish an uncaused first cause, which is less even than a Deistic creator!


There's still more to come! In the next post in this series, I plan to start addressing Craig's Q&A #325, "Infinity and God". I don't think I'll cover all of it in a single post because it is fairly long and requires a diversion into another of Craig's Q&A's (#323, "The Concept of God") since it was the basis for the question posed in #325.

If you like how this reads or think this kind of subject matter is interesting, it's very likely that you'll enjoy reading Dot, Dot, Dot, so please give it some consideration.


Edit: Part III in this series is now available here.