Tuesday, July 30, 2013

Re-revisiting my case that the existence of God has no plausibility

Something interesting: an argument I consider to be something of an aside from the main thrust of my thoughts and discussion on God and religion seems to be the one that works people up the most. Of course, it sounds far more contentious than it may be.

My argument, in very short, is that I don't think it is necessarily the case that it is philosophically indefensible to say that the plausibility of the God hypothesis is zero, with the qualifier "almost surely."

Evidently, I have not communicated my case very well, or so I'm told, so I will try again to clarify.

First: "Almost surely" means something very precise in mathematics. It means "off a set of measure zero." Communicating clearly what "measure" means for a lay reader is a bit tricky, but in the sense I'm using it (probability/plausibility), it works out essentially, though very, very loosely, to having an infinite-to-one chance against it.

It must be noted that "one" here could be replaced with any finite value without changing matters. It also should be noted that if I compare two different sizes of infinity, the smaller size is effectively measure zero against the larger size, and, indeed, (as with the Cantor set in the interval [0,1], using Lebesgue measure, or as with the primes in the natural numbers cheating to use the natural density false-measure) measure zero can still occur by comparing two infinite sets of the same cardinality, i.e. same size. If this paragraph was confusing, I'm sorry. Math is hard.

Here are a couple of important points I've argued on this theme so far:
  1. Almost surely not the case is not categorical denial. Because events with probability zero, almost surely, or hypotheses with plausibility zero, almost surely, could conceivably occur, we are not engaging in categorical denial to say zero, almost surely, in this context. Categorical denial is philosophically indefensible. This is not categorical denial, and the only reason I know of standing in the way here is one that is debatable.
  2. I don't think there's any way to justify any nonzero plausibility for God's existence. Because we have absolutely no evidence for the existence of God, we have absolutely no justifiable reason to assign any nonzero plausibility to the hypothesis of God's existence. To summarize this briefly, I'm suggesting that since they have no evidence to go on, believers in God cannot produce anything to justify a claim of any nonzero plausibility for God's existence.
  3. Putting those together lands the God hypothesis in zero, almost surely, plausibility because (a) they cannot justify a nonzero plausibility; (b) categorical denial is not defensible; and (c) plausibility zero, almost surely, is what's left over.
  4. If some "God" exists, then, it is almost surely an abstract concept, not an entity in reality, particularly not an entity with agency that interacts with our universe in any way at all.

This is the essential core of my argument. It gets rather a lot of objection. I'll address a few of those.

"But couldn't people just turn this around and argue for God."

No. Evidence is at the center of this argument, and the burden of proof is on the believer to produce it. There is no credible evidence for an extant God that interacts with our universe in any way. None. I'm willing to concede that "incredible evidence" contributes evidence that has measure zero because it is not credible.

How would someone turn this around? There almost surely is a God because atheists can't produce evidence that there isn't one? But this is shifting the burden of proof. We have no trouble dismissing such an attempt as exactly that.

"But you can't say something has probability/plausibility zero."

Please, understand that prob/plaus. zero, almost surely, is not the same as categorical denial. It's very likely if you're arguing "but you can't..." that you might be missing this point.

If this sounds ridiculous still, I recommend the following litmus test. Replace God with "the Force" from Star Wars, or any other known fiction. Should we hesitate to say that the Force, specifically as it was conceived in Star Wars, has a nonzero probability of being a real thing in our universe? Should we hesitate to argue that the likelihood that Hogwarts School of Witchcraft and Wizardry has a nonzero probability of truly existing somewhere in the north of the UK?

If your immediate thought is "yes, we must," ask yourself why. Is it because George Lucas and J. K. Rowling wrote those ideas down? What about the ideas that other writers have written? Must all of them get a nonzero plausibility? What about ideas that writers haven't written yet? What about ideas that no writer will ever write? Must these all get nonzero plausibilities?

Here's a problem with that: if there are infinitely many conceivable fictions (and why shouldn't there be, using the numbers themselves as suggestive evidence?), then we have a problem. There are two possibilities here: either most of these fictions have zero plausibility, almost surely, or there is some mechanism by which their plausibilities diminish fast enough for the total plausibility to converge. So which is it? The vast majority of fictions have literally negligible plausibility or the vast majority of them have no plausibility off a set of measure zero? Depending on the number of conceivable fictions, this choice may not even be valid, and plausibility zero may be required.

So we have to assign nonzero plausibilities to all possible fictions, and yet we can't really. There are huge swaths of potential fictions that we simply give plausibility zero, almost surely, to without even hesitating: all of the wide, wide majority of them that will never even be thought of.

"But Bayesian reasoning says you can't assign a probability/plausibility zero."

Um, okay. Bayesian rules do assert that we cannot assign certain (zero or one) probabilities and plausibilities. It's possible that Bayesian reasoning, as conceived, has some issues with it--particularly at this point. Perhaps Bayesian reasoning doesn't apply here--to abstract hypotheses. As I repeatedly say, we can't mistake the theory for reality. Bayes's theorem is only as good as the axiomatic assumptions that it rests upon.

Also, I kind of think it could be possible. The reason Bayesian reasoning rejects assigning certain priors is because they could not be overcome by any evidence. But certain, almost surely, evidence should be able to overcome almost certain doubt (by something like L'Hospital's rule, for the math-inclined). Applying almost certain evidence against almost certain doubt produces an indeterminate form in Bayes's theorem (0/0), and if crafted carefully, this can conceivably be gotten around.

Indeed, this happens (theoretically) all the time. Every real number in the interval [0,1] has probability zero, almost surely, of being chosen in any random trial, but invariably, at lease one of those values is chosen. Once we have that (almost) certain evidence of whatever value crops up, it overcomes the almost sure doubt we had against it.

Regarding the existence of God: Show me God, and you can overcome my almost sure doubt. The same is true of the Force and Hogwarts.

A potential way to deal with this is to refuse to assign a prior and only to examine the role of evidence to see if it points uniformly one way or another. On the question of whether or not God exists, as there is no credible evidence to support it and tons of evidence refuting theological claims of all sorts, we can conclude that the evidence points us clearly in one direction: down to lower and lower plausibilities. Given that there are probably infinitely many different mechanisms (various Gods, other supernatural events, etc.) that could be proposed to account for what theologians attempt to pass off as evidence, the situation looks really, really dire for the plausibility that God exists.

"But the universe could be a simulation; we can't even know that."

So? Again, evidence: we have lots of evidence for nature existing. We have no evidence for God's existence. In the language here, I would say that God's existence has plausibility zero, almost surely, because of no evidence, and nature has plausibility one, almost surely, since we have nothing but evidence for it.

Put another way, everything we see suggests the world exists, off a set of measure-zero philosophical games, and nothing we see suggests that God exists, apart from a set of measure-zero philosophical games.

So, the same argument applies, yet again to the case I'm making and without having to change anything about it. The question ultimately comes down to epistemic gaps, which always exist. We have a very, very narrow (measure zero, almost surely, I'd argue) epistemic gap on the claim "nature exists." We have a very, very wide epistemic gap on the claim "God exists." My suggestion is merely that in the total paucity of credible evidence, that gap regarding God's existence is as wide as is philosophically defensible.

This raises an important point--the atheism versus theism argument is misleading. The question isn't really "no God" versus "God" because nature is inextricable from the discussion. The question is "nature does not require a God for its existence, operation, or explanation" versus "nature does require a God for its existence, operation, or explanation." This distinction is important.

Sure, anyone could then argue: "well, why is there nature at all, something rather than nothing?" and however philosophically interesting this question is, it doesn't really go anywhere. If it is true that nothing begets nothing, and yet we have something, the simplest conclusion to draw from this observation is that "nothing" doesn't apply to our universe. Theologians like to call the fundamental something "God," and it appears that cosmologists like to call it a "quantum field." This, other than a bad name on the part of the theologians, a term they're using to sneak across non sequiturs, doesn't really matter much. The theologians can't get to something they really want to call "God" without proving that the primordial something has agency. (Proving does not mean baldly asserting.) Good. luck. with. that.

"Either God exists or God doesn't, so this is inappropriate."

Possibly so. The argument about what it means to attempt to apply probabilistic and plausibilistic reasoning to hypotheses is not settled. Ultimately, I think that such an estimate with regard to the validity of hypotheses has to speak to our degree of certainty with which we can make the conclusion.

For example, take the Higgs boson. Either it exists or it does not, so we can't talk about the probability that it exists. Except that we can--sort of. We have amassed quite a lot of evidence matching the theory now, enough to conclude that when we say "the Higgs boson exists," we mean something. What we mean is that we are very confident that that which we mean by the term "Higgs boson" exists. To be clear, it could be that bosons do not really exist in reality, but whatever it really is that we are calling bosons clearly produces phenomena that can be explained in terms of the standard model including the term "boson." This, as it turns out, only matters a little, and in practice almost none at all.

Further, we can actually quantify how confident we are that the Higgs boson, or whatever it really is that we describe by that idea, is a real thing. We can't know for certain, but before scientists at CERN were willing to claim that it does exist, they had to be quite sure. How sure? The standard is roughly that a false identification, given statistics on the data, has a likelihood of less than one in 3,500,000.

In other words, we have a hypothesis, "the Higgs boson exists," to which we can assign a probability that tells us our degree of confidence in the given hypothesis. This sort of discussion of our level of epistemic confidence in the existence of God is somehow fundamentally invalid, though? The only reason we'd say so is because there is no credible evidence that God exists.

So, hopefully this clears it up some. Ultimately, I don't consider this argument central to what I'm doing or really all that important, even if it seems to gather a lot of attention. I will admit that I'm more than a bit annoyed that non-believers are bullied into saying things like "there's a very, very small chance, short of zero, that God exists," which opportunistic theologians and journalists pounce upon very disingenuously. After having thought about it for a long time, I'm just not convinced that it has to be "short of zero" or that any of us needs to say that anymore.

Monday, July 29, 2013

What's really being said in the FOX Reza Aslan interview

Yesterday, a video of a FOX interview with religious studies professor and author Reza Aslan went viral, being decried widely as FOX's most embarrassing interview yet (which really says something).

Aslan is being interviewed, ostensibly, about his book, Zealot: The Life and Times of Jesus of Nazareth, but in typical FOX style, it appears to have an ulterior motive to it. In it, the interviewer, FOX anchor Lauren Green, continually needles Aslan about his religious identification as a Muslim, repeatedly ignoring Aslan's replies about being a religious studies scholar with multiple degrees, including a Ph.D., a profound interest in the character of Jesus, and over two decades of experience being a religious scholar, including in New Testament studies. It's almost as if the interviewer doesn't really care about Aslan's qualifications at all. See for yourself:

There is a valuable lesson in this very annoying "interview" of Aslan that I think isn't getting the attention it needs, and it is a loud and clear call for secularism. I want to give it some of that attention.

In the interview, Green repeatedly throws variants of "You're a Muslim, so why did you write a book about the founder of Christianity?" at Aslan, despite his repeated assertions that he is a religious studies expert, whose job it is to study religions academically, one "who happens to be a Muslim." Green repeats this essential question over and over, clearly trying to undermine Aslan's credibility by indentifying him as a Muslim first, religious studies expert a distant second. Green knows exactly what she's doing.

I think it is prudent to assume that Green, and her programming directors, are highly intelligent people. They know what they're doing by going back to this probable non-issue (that is an ad hominem attack on Aslan instead of his work), and it is for this reason that people are calling the interview embarrassing and using it to deride FOX. But Green, et al., are teaching us something very important about religion here that we shouldn't miss.

We can see it by recognizing that they aren't stupid. Green knows her audience, and she is doing the interview for her audience more than for Aslan. Since the intended audience of FOX is not a generally unbiased group of well-informed people, she is not aiming the interview to cater to generally unbiased, well-informed people. The usual viewership of FOX is low-information, high-misinformation, conservatively biased people, but that's mostly irrelevant in this particular interview. Green knows her audience is mostly composed of sectarian-minded Christians, and her goal is to illicit distrust of Aslan and his work in that audience. 

Green's message is abundantly clear: "Aslan is different from you. Don't trust him. He's different. He's not one of you. His work can't be trusted. His results can be ignored. He's an enemy, so of course he would say things against your beliefs." This "he would say that" argument, in fact, is the main example of the valid applications of apparent ad hominem arguments.

The key word here is "sectarian," to be distinguished from "secular." Green's approach in this (embarrassing) interview works directly on a key feature that is inherent to sectarianism: sects are designed to overcome trust within the group, and this has the effect of decreasing trust of those outside the group. Other sects, here Muslims as compared to Christians, face a special sort of distrust because it is known that they hold a fundamentally different construction of the world, particularly morally. 

The Muslim worldview, however unknown and mysterious it may be to many FOX-viewing Christians, is not mysterious in the least in the regard that it is known to be not Christianity. Indeed, for black-and-white thinking among low-information viewers, Islam is almost taken to be anti-Christianity, and the trust among these people for Muslims could only be lower if they were atheists. Green is working that angle particularly hard to constantly remind her audience that Aslan, whatever his credentials and whatever he says, is not one of them and is therefore not to be trusted. It seems ridiculous that it would work in this globalizing era, but it works specifically because she knows her audience is sectarian in a sect with profound distrust for Muslims.

Surely, someone will be upset here that I haven't qualified for every kind of Christian. My claim, though, is that it is endemic to Christianity, as a sect, to engage in this kind of behavior, and only the secularization of Christianity has changed that for various Christian groups and individuals. Many Christians often accept secular values ahead of Christian sect-specific ones (and then reverse-engineer them to be "Christian values," but that's another story). Many Christians, then, have embraced secular values, and to the degree that they have secularized, they immunize themselves against the sectarian default of distrust of other groups.

Christians, then, are tolerant and accepting of Muslims to the degree that they are secularized, and FOX's and Green's intended audience are not highly secularized Christians. Indeed, because they are sectarian Christians, whether well-informed or not, they are low-trust viewers when it comes to outsiders. This propensity to distrust is being manipulated effectively and clearly.

Of all the things we can take away from this interview, this is perhaps the most important. It's hardly news that FOX is driving a conservative agenda or attempting to propagandize to its target audience. It's hardly news that they publicly embarrass themselves to everyone else when they do it. It is worthwhile, however, to understand how they're doing it. In this case, it is clear: they are using sect identification specifically to sow distrust for Aslan, his work, and particularly anything like the results of his work, and that distrust clearly must outweigh the damage to their reputation caused by doing such an obvious hatchet-job of an interview. 

These people, FOX, are not visible because they're stupid. They're visible because they are smart enough to know exactly how much stupid they can get away with while increasing their reputation within their viewer base.

FOX engages in this sort of interview because it works with their low-trust viewing base. The reason FOX's propaganda works in this case is sectarianism. Sectarianism is the problem. In a fully secularized society with people who, in the very wide majority, embrace secular values over sectarian ones, whatever their religious identifications, this FOX interview would be too embarrassing to air. The cost to FOX's reputation would be too high to dare do it. Because there is so much adherence to sectarianism, particularly in their viewing base, though, not only can they get away with it, they can get rich and shockingly influential with it.

The demand we have, then, is for the rise of a properly secular age. Sectarianism is inherently divisive, or at least can be driven to be so easily, and that cannot serve as the basis for a strong, healthy nation nor the emergence of a peaceful global society.

Wednesday, July 24, 2013

Does A=A? I'm not so sure

The law of identity, they call it.
It is one of the three classical laws of thought and states, apparently simply enough, "each thing is the same with itself and different from another." Particularly, it urges us to realize that A is not 'not A,' for every anything A.

In some ways it is a big deal and it is important. Notably, it's important in regard to linguistic points. When we use the term firefighter, we mean something specific--which Ray Bradbury used to good effect in Fahrenheit 451. Particularly, if we talk about firefighters in a particular context, it's considered falling afoul of the law of identity to suddenly switch to another meaning for the term. Of course, this kind of equivocation is very common, particularly on terms like "God." There's nothing that bridges the gap, for instance, between the philosopher's abstraction known as "God" and the somewhat more concrete variant of that notion embraced by Christians or Muslims--a point where apologists are too eager to equivocate.

The law of identity seems to get abused heavily in this way, in fact, and it seems to come up a lot when someone with a little philosophical savvy wants to prove the existence of God or, more recently, that Platonism is legitimate. I don't really understand what motivates people to believe that the law of identity somehow counts in philosophical favor for the existence of God, to be honest. I'd welcome a cogent explanation, but I mostly expect that these people just want to be able to say something that sounds superficially smart, using that as a talisman meme to ward off attention from the fact that we don't have any credible evidence that God exists.

Anyway, abuses aside, the law of identity seems entirely straightforward, but is it? I'm not so sure.

There is an easy objection to it by bringing time into the matter. A relatively famous maxim states that "no man jumps into the same river twice, for it is not the same river, and he is not the same man." We have a sense of sameness of object, particularly of person, but it's hardly clear that when A is me or you or somebody that there's any real permanence there. For brevity, I'll skip the almost banal discussion of how quickly our tissues renew, our patterns of thought change, etc., with the sole exception of the tricky issue of whether or not Jane is still the same Jane after she develops dementia.

As the maxim indicates, other complex cases, like rivers for example, are obviously tricky on the law of identity as well--for the same reason. Given any cross-section of any river, the particular collection of molecules (be those part of the water, particulates, bacteria, wood, fish, etc.) flowing through that cross section at any given instant in time is entirely unique. The course of the river changes with time, the depth and current of the river change with every rainfall, even miles away, and so on. Is the Mississippi River the Mississippi River? What is even meant by that?

There are good ways around the problem, for most intents and purposes--clearly we know what is not the Mississippi River, for example. We could define the mighty Mississippi or our friend Jane by a set of properties, Jane's dementia making those properties a bit more difficult to pin down exactly, and those properties seem to make the matter less ambiguous. This problem is a bit too big to wriggle out of, though, just by using "properties." Why? Because almost everything changes almost all the time, and we don't have any real reason to believe that past or future states really exist. It may not mean much to say that something, any something A, is ever A again, at least in the exactly-the-same sense implied by equals.

We might agree to call a particular body of water the Mississippi River, for instance, but the Mississippi River we're talking about at this moment already isn't that Mississippi River and never will be again, nor was it ever that Mississippi River ever before. Jane is still kind of Jane when she develops dementia, but she's already really "not herself." Further, what is she when she dies? A=A implies a certain sense of transcendent eternity--which we literally do not have. So if A=A, then A=A right now, and that's as much as we can say about it, but right now is already over.

This problem of identity being fundamentally ephemeral, even down to the quantum scale, it appears, is a serious issue of incoherence for the law of identity if we want to accept that this law is true (or even better, obviously true). I mean, in a sense it is true (even obvious), but in another sense, it isn't at all true. In fact, in that second sense, it is clearly false. Given this, I'm not entirely convinced that the law of identity is what lovers of classical thought want it to be: a universal truth, often tied up in mathematics since it is usually taken as a fundamental axiom in mathematical systems. Near the end of this little essay, I hope to point out under what conditions it is that this "law" could be considered a universal truth, but I want to clarify this "in one sense true, in one sense false" business first.

Let me allude to Taoist thought to illustrate what I mean. In Taoist thought, there are two fundamental concepts: yin and yang. These concepts are stunningly hard to define on their own because they are polar opposites to one another. Literally, yin refers to the shady side of a hill, and yang refers to the sunny side. Relatively speaking, yin is dark, cold, receptive, earthy, feminine, solid, sinking, etc., while yang is bright, hot, projecting, airy, masculine, rising, etc. On the famous taijitu, what most people call a yin-yang symbol, yin is the black parts, and yang is the white.

Now consider a pool of water, perhaps just off the Mississippi, one where Jane used to dip her toes on hot summer days, though she hardly remembers that now. Is that pool of water yin or is it yang?

It's tempting to say that it is yin, but that is incorrect, as the earth that forms the basin it rests in is more yin, rendering the pool relatively yang in comparison. It's less tempting to say that the pool is yang, and as it turns out, that too is incorrect, as it is far less yang than the sunlight glinting off it. The pool may appear quite still, yin, but it is still presenting movement within it and on its surface, yang. Both answers, yin and yang, are incorrect--and yet both are, in a measure, correct.

What this Taoist construction presents is a third truth value to our familiar Boolean values of true and false. Here, the statement "the pool is yin" is neither strictly true nor strictly false; it's somewhere in between. An argument could be made that the statement is false because the pool is not "pure yin," but "pure yin" doesn't really exist. There's no such thing, and so such a comparison is vacuous and inadmissible. Faced with a third truth value in that system, we could call it "neither" or "both," I suppose, but that's immaterial.

Now, let's revisit the question of whether or not Jane is the same person today as she was yesterday, before or after dementia or death, and if the Mississippi River is still the same Mississippi now as it was five minutes ago, or as it will be in five minutes time. The answer, I think, is that there's something wrong here with the word "same," it applying on two different levels, giving the sense of "yes" and "no" simultaneously being correct. This is just like saying that the pool is yin: yes, but no.

The problem hopefully uncovered, before nailing this down, let's consider something a bit different, a "universal" idea. Consider the number three and the property that is "threeness." Three trees, three pencils, three-letter words, three meals a day--all of these exhibit fundamental threeness. So, if we take A=3, is it accurate to say that 3=3? Well, of course it is. Will it be 3 later? Yes. Was it 3 before? Yes. Would it be 3 even if there were no minds to name it so? In a sense, yes--though this is murky territory. So here, the law of identity works just fine, just like it seems to for Jane or the Mississippi River if we transcend the "mundane" details (in quotes because, actually, they matter quite a lot).

What gives? Why is 3 different than Jane here--but not totally different from Jane? Specifically, why does 3=3 timelessly but Jane=Jane have serious and obvious issues with it?

Timelessness. That answer sounds like a cop-out, though, so we have to ask where timelessness occurs. The answer: in abstractions. Anywhere else? I don't think so. Three is an abstract notion that describes a certain property, and that abstract notion endures beyond the details of what exhibits it. The person of Jane, legally defined or done better by, is also an abstract notion in the minds of everyone who knows her, including herself. That gives us a sense of continuity to Jane, the myriad states that have been Jane, are Jane, and will be Jane, be she a fetus in the womb, toddler with a water sprinkler, firefighter, or dementia-crippled shadow of her former self (or is it better to identify her as an entirely different person in each of these cases, each taking the label "Jane" with a sense of evolving continuity from one state to the other?).

"Jane," referring to the "same" person, means different things at literally every moment in time, so Jane does not equal Jane literally except when utterly frozen in time. Our abstraction of Jane, mostly composed of memories and guesses, however, is timeless. That Jane=Jane, even as Jane changes throughout her life and eventually after the end of it. The same is true of the Mississippi River, even after the land changes, and the great river is no more. The abstraction of the river is a timeless thing, and it "exists" as do all abstractions in the minds of anyone able to think about them. Certainly, as the details change, the relevant abstraction can be updated continually to encompass the changes, and by considering the problem from this meta position, it makes sense to apply the same term.

Abstractions, yet again, are the special things, in that they are timeless--because they are not real. Here, they are the things that satisfy the law of identity--because they are timeless--because they are not real.

Maybe, then, this is why the apologists and the Platonists are so interested in it? Maybe it is that they want to equate abstractions with reality. Sure, we only experience reality through our abstractions of it, but this is a question that is likely to have more to do with the nature of consciousness than with whether or not abstractions have concrete reality. Religious experience, at least under the Abrahamic faiths, requires a certain degree of Platonism, a belief that ideas have a sort of concrete reality to them. Thus, this law is important to them, perhaps. It's a centerpiece of classical thought about abstract ideas.

This ties right back into the usual use of the law in the first place: avoiding equivocation. When we talk about a firefighter, we're appealing to a certain abstraction--the word not being the thing--and it is disingenuous to imply one abstraction while meaning another.

Does A=A? Sure, as long as A is abstract.

Tuesday, July 23, 2013

A response to a comment about Platonism

A visitor, Marc Missildine, responded to my recent post about why I suspect Platonism is such an alluring way of thinking about mathematics (Link). I expect I'll need more space than a reply box to reply, and I hope to make the response interesting and poignant, so a new post for it. Thanks, Marc, for your comment!

As is customary for me, I will put Marc's commentary in block quote format in green letters. The full text of Marc's comment is here (Link).
"...and nothing except the unfolding of the cards really happens." So.. nothing but EXPLORATION happens?? 
I prefer not to use such charged language as "nothing but" here. In my experience, particularly in philosophy, "nothing but" statements are usually dubious and frequently wrong.

The problem here with "nothing but" is that eventually we meet the seams of what we can say within the axiomatic system we're exploring, e.g. when we follow the Peano Axioms out and realize that they predict the concept of infinity but do not account for it. At that point, we get to decide upon new axioms about the questions raised, which isn't really exploring anymore, at least not in the sense that it's all already there waiting to be found.
This takes us back to the cave example. We can still explore the unknown (in candyland the unknown is what color the next card is going to be), and can reveal truths about the universal system. 
The point here reveals why the cave example isn't any good. A video-game cave is actually a better example than a real physical one. Why? Because it's as if our axioms define a certain digital cave, but the programmer isn't actually able to close off all of the boundaries within that cave. Sure, the boundaries are relatively hidden in most cases, but like playing a glitchy game, there are places where you can kind of fall off the edge of the map. There is an outside to what the axiomatic system is able to describe in exactly the same way that the unprogrammed "void" is a "place" outside of the intended playable area in a glitched video game.

We try to set up the program to close off all the glitches, but we know because of the Incompleteness Theorems that we cannot actually do that. We'll always be able to find glitches that let us "leave the map." At that point, programmers can decide either not to bother with it or to flesh out areas for some of the more accessible, obvious glitches. Outside the metaphor, this is choosing new axioms that define more (or different) cave space to explore--knowing again because of the Incompleteness Theorems that these attempts will never close off all of the seams.

I'm most uncomfortable with your use of the term "universal system" for this reason. We have axiomatic systems that define truth values for a huge variety of statements, but they do not touch those statements that the axiomatic system cannot assess--which by the Incompleteness Theorems are always there (at least if we choose to avoid logical inconsistencies).
Similar to the child, all of humanity does not know what the cards in the deck of mathematics are, nor do we know how big the deck is. 
It's actually worse than that. The Candyland example really breaks down in a sense, it really just being a metaphor to give a sense of what's going on here, and that's hardly the point. We are actually in a situation where not only do we not know that, we also don't know what kinds of cards we'll decide we want to add to the deck until we face those questions, and we're in a situation in which we know that whatever cards we choose, literally no matter what, we aren't guaranteed to be able to "finish the game."

Don't get too caught up on the metaphor, though. It's just an intuition pump.
Nor do we know who 'shuffled' the cards. 
Whoa. Just whoa. I know where this is going, and whoa. Stop.

First, this is the wrong question to be asking, and second, yes we do. "The cards" in this case, really, are the truth values assigned by the axiomatic system, but those values are determined by the choices of axioms and logical framework, which are ultimately choices we have made and continue to make.
This points to a rules-maker who is outside of the boundaries of humanity who created these simple and universal(transcendental) truths(what you call axioms) such as 1=1. 
No it doesn't. First, I just said why it doesn't. Second, inventing a "rules-maker" is choosing an axiom (meaning you choosing an axiom, you being a person, just like all the other axiom-choosers).

I don't call these things axioms. "Axioms" is the word for these things. I use that word, but it's not the same as your connotative implication. Also, "axioms" aren't "universal(transcendental) truths," they are baldly asserted statements that we agree are sufficiently parsimonious and basic to be getting on with getting started, i.e. to be willing to baldly assert as sufficiently self-evident*. The asterisk here indicates a constant willingness to reassess them at a later date: Cf. the Peano Axioms predicting the axiom of infinity but being unable to handle it and so the Zermelo-Fraenkel axioms of set theory taking over the job.
This points to an ordered universe in which laws and rules are discoverable, because if 1=3 sometimes according to some axioms, and 1=red in other axioms, then the universe loses all meaning.
Incredibly, you missed my entire point, then. It doesn't. It points to an ordered axiomatic system that humans have developed in order to attempt to make sense of the universe we live in. The point isn't that the universe "has meaning," it's that humans, as thinkers, can find meaning in it, which as often as not means defining for ourselves the constructs (languages, as it turns out) in which meaning resides. The central point is that we invent these logical systems and language to attempt to understand what we observe and to communicate those attempts with other people. The logical systems are not the universe itself.

Incidentally, under some definitions, 1 does equal 3 (e.g. in any algebraic system over the field with two elements). "1=red" is just silly--a category error, to let you know.

Again, thank you for your time and your comment.

Sunday, July 7, 2013

The Believer's Triangle: A point about the religious mind

When writing God Doesn't; We Do, I made a particular point a handful of times about the religious mind. I claimed specifically that all religious believers, particularly apologists, live their mental lives somewhere in the "borderlands" between being deluded, deceived, and dishonest. My ultimate points with it were that we may not possess the tools or the ethics to pull apart which is which, and so we really needn't bother. That believers think within the triangle, at least about their religion, is enough. I came up with the name The Believer's Triangle today and kind of like it.

I suppose I should clarify the shades of the meanings I'm using for those three terms, since much of the problem is that they overlap considerably. Indeed, "deluded" and "deceived" are considered synonyms, except in connotation, and both are the result, in a sense, of dishonesty.
  1. Deluded: Believing incorrectly with a sense of confusion about it.
  2. Deceived: Misled, operatively by others.
  3. Dishonest: Intentionally lying to others, especially, and to oneself.
To clarify further, I'll use a few examples: Mary is deluded in that she believes God takes care of her and everyone she loves, particularly because she wants this to be true and hasn't seen through the cognitive biases that might disabuse her of it. Mary has been deceived into thinking that she will have a more fruitful life if she tithes well since her pastor at church tells her so every week, a message reinforced by her family and friends--Mary may not have arrived at this notion on her own, however, as a result of her general belief structure. Mary's pastor is dishonest because he knows that tithing will not actually improve Mary's life in any meaningful way, but he wants her money and so tells her to tithe well anyway.

So, in my use, delusion operates primarily behind one's own cognitive biases, being deceived is a state of being misled by others intentionally or unintentionally, and dishonesty is meant in the strict sense where someone is intentionally saying something he or she knows or strongly suspects not to be true. Part of the point of the whole thing is not to split hairs about where lying to oneself lands in the triangle, etc. They're in here.

I thought of calling it the "Believer's Triangle" having in mind a mental image of the Bermuda Triangle, into which there are associated mysterious phenomena, like believing that the Holy Trinity is more true because it doesn't make any sense at all.

Incidentally, I could probably have called it the "Believer's Uncertainty Principle," in homage to Heisenberg instead--"we cannot determine the accurate state of a believer's degree of being deluded, deceived, and dishonest in any of his or her apologetic-style arguments or thinking." The general point is the same: language isn't doing us a great job here at being able to nail down exactly which is which with them, and we don't possess the kind of knowledge in most cases to tell the difference, even if we often can smell a bullshit artist at work.

Friday, July 5, 2013

The blog posts that inspired Dot, Dot, Dot: Infinity Plus God Equals Folly

A release date for my next little project, kind of an intermission to address an important topic, is drawing nearer, and I do hope that many people will take enough interest in Dot, Dot, Dot: Infinity Plus God Equals Folly to give it a good look. As I mentioned previously when I blogged about it, it has several posts on this blog as a backbone. For those that are interested, I'm writing this post to point you to all of those to give you a sense of what the book will be like and the various themes it presents.

As I noted previously, every essay that has been included in the book has been modified, some rather substantially and some minimally (just cleaned up a bit for print publication), so these essays on the blog are to be taken mostly as first drafts and prototypes of what they evolved into. In many cases, new pieces have been added to these essays to flesh them out, given that blogging puts a premium on brevity. The book, though, is also kept brief--coming in at only around 120 pages (about 37,000 words).

The essays from the blog that made it into Dot, Dot, Dot are:
  1. "On Reality and Logic," an exploration of the difference between logical constructs and the reality we use them to attempt to make sense of.
  2. The five "Down the Rabbit Hole" essays, which were this blog's first real exploration of the weirdness and difficulty with infinity. These five essays followed me a bit as I explored the topic of infinity deeply for the first time (most mathematicians have no need to do this to the level I was digging) and are the real inspiration, together with "On Reality and Logic" that inspired this work. Incidentally, all of this exploration followed an email discussion I was having with Richard Carrier about uses of infinity in theological-style arguments. Links to follow: DtRH-I, in which I lay out some of the basic sense of the axioms that lead us to consider infinity--and paradoxes that arise already.
  3. DtRH-II, in which I explore the psychological reasons that we go awry with infinity so easily--kind of the essence of the Dot, Dot, Dot title I chose for the whole book.
  4. DtRH-III, in which I explore some of the paradoxes related to infinity that arise from accepting some of the strange fringe mathematics that it has led to--particularly leading us to believe that infinity is really just an abstraction.
  5. DtRH-IV, in which I develop more of the nitty-gritties about probability theory and infinity, particularly focusing on the alluring but problematic concept of "choosing any number at random."
  6. DtRH-V, in which I pull the series together with a reiteration and deeper development of the ideas at the end of essays I and III in the series, really digging into the axiomatic construction and abstract nature of the beast here.
  7. "Coming Clean About an Error that Is but Isn't," an admission of a common misuse of infinity that I presented in God Doesn't; We Do, why it matters, and why it doesn't matter for my purposes there.
  8. "Revisiting My Case that the Existence of God is Infinitely Unlikely," a continuation and clarification of the discussion in "Coming Clean" with a reiteration of my primary point with the argument I made in the fifth chapter of God Doesn't; We Do that God's existence may be said to be infinitely unlikely.
  9. "About Gödel's Ontological Argument," a discussion pointing out the underlying flaws in modal logic "proofs" of the existence of some "God."
Also as I noted previously, there are sixteen essays plus an introduction and conclusion in total in Dot, Dot, Dot, so there are seven (plus two) essays in the book that do not appear anywhere else. This should give interested readers plenty to chew on, while these somewhat scattered blog essays are tamed into expressing a series of themes that can be summarized ultimately by the title itself: Literally all of what is going on with infinity is wrapped up in the ellipsis, the "dot, dot, dot," and thus there is a lot of room for error and confusion there.

Indeed, at this point, I'm quite convinced that the human mind is not capable of truly grasping the infinite except via the "dot, dot, dot" that brushes almost all of it under the rug--and this is utterly fascinating.

Monday, July 1, 2013

Reader question about Bayes's Theorem and the alleged resurrection of Jesus

I got the following email today from a fellow named Thomas. He wants to know about Bayes's theorem and its applications to historical questions, notably the alleged resurrection of Jesus. Thomas writes:
Dear Mr Lindsay
I wonder whether I can ask you a question about the application of Bayes' theorem to miracles. I shall start by saying that I certainly don't believe in miracles. However, it seems to me that there is a problem in using Bayes' theorem to argue against miracles. Consider, for example, the probability of having the evidence that we do concerning Jesus' alleged resurrection if Jesus wasn't resurrected. It seems to me that the probability of having that evidence is infinitesimal. The reason is that the probability of any event in history is infinitesimal. If history could be rewound to 100BC and replayed a billion times there would be no Christianity, no belief in a resurrection and, in fact, no Jesus. The same could be said about the Second World War. If you could rewind history to 1800 and replay it a billion times there would be no Adolf Hitler. By that I don't mean that there would be Adolf Hitler but he would live a different life, I mean that there would be no such individual at all.
Is this the right way of looking at it, or have I missed something?
I'm putting this here because I wrote Thomas a rather long response that actually covers a topic I've been meaning to blog about for a while concerning Bayes's theorem. My response follows the dashed lines.

- - -

Hi Thomas,
In general, I'd say that Bayesian reasoning has stood up quite well in practice, so there is some pretty solid reasoning behind it and its application. Applying Bayes's Theorem to the historical process is a reasonable thing to do, but it has to be done carefully.

Let me start with one of your assumptions (I get the impression you've been reading Richard Carrier, maybe?). "The probability of any event in history is infinitesimal." This is problematic on a number of fronts.

First, infinitesimals are kind of a fringe branch of mathematics except as a formalism. In other words, I don't know if they mean anything, really. This may or may not be a huge strike against using them in Bayesian reasoning. If it is a strike against them, I will have to deal with the fact that it does some damage to one of my arguments about God's non-existence, which is fine. This question is a rather big one for me lately, and I don't think it's resolved in general yet. The problem could actually be with Bayesian reasoning in certain cases, though, which is another matter.

Second, which probability are you talking about? In Bayesian reasoning, you're having to consider two separate probabilities (really, for hypotheses like this, plausibility is the right word, not probability): the prior and the posterior. The prior plausibility is how likely we estimate the hypothesis to be just on whatever background knowledge we have, whereas the posterior plausibility is how likely we estimate it to be after we consider the evidence for or against it. Which, the prior or the posterior, are you saying is infinitesimal? It seems you're talking about the prior.

Third, this raises the question about how we assign prior plausibilities in the first place. Ultimately, they don't matter, and so a decent case can be made that in situations where we are trying to answer questions about one-time events or broad hypotheses, we simply shouldn't assign a prior at all. All we need to do is examine the evidence and see where it is leading the posterior. In the case of the resurrection of Jesus, the evidence points almost uniformly away from such an occurrence, so whatever prior plausibility we assign it, unless we assume it is certainly true or false (or all but, i.e. infinitesimally different from one or zero), will be swamped by the evidence that we have that suggests that no such thing ever occurred. Some of this is by examining what evidence we have, much of it is examining expected evidence we do not have, and still a lot is based upon Hume's standard for such claims. The famous maxim now known as "Hitchens's razor: That which can be asserted without evidence can be dismissed without evidence" could be rendered in Bayesian speak to capture this very notion along with your observation. Essentially, what it's saying is that the prior plausibility for certain kinds of hypotheses (e.g. historical events, one-time occurrences, ontological hypotheses) is ultimately completely arbitrary and therefore may be meaningless.

Fourth, according to Bayesian reasoning, whatever Richard Carrier and a few others assert, assigning an infinitesimal prior is probably not allowed. In this case, this fact renders your claim incoherent. You simply cannot claim that the (prior) plausibility of any event in history is infinitesimal. This opens wide vistas of questions regarding how we are to handle hypotheses with Bayes's Theorem since there are infinitely many potential hypotheses. This book, I'm told, is quite good for this problem, but I haven't done more than skim it: http://books.google.com/books?id=jPpyGPJNn28C

Fifth, the entire way that most people (including me in the past) use Bayes's Theorem to evaluate hypotheses is probably incorrect. This is not the same as saying that it is incorrect to engage in Bayesian reasoning, but it urges us to be more cautious in how we do it. Frequently, we're given an equation that allows us to calculate an estimate of the posterior probability by considering three numbers: (1) the prior, (2) the likelihood we would see the evidence we see given the validity of the hypothesis in question, (3) the likelihood we would see the evidence we see under alternative hypotheses. In practice, here's the problem: (1) the prior itself may be meaningless in many cases, see above. (2) How can we estimate this number except in relatively simple cases? Particularly, we should expect a lot of error in this number. (3) This number is probably actually impossible to estimate since there are infinitely many potential alternative hypotheses, each with their own plausibilities--and these may be problematic in a deeper sense (next paragraph). Since (1) is subjective and ultimately pretty unimportant, (2) is error-prone particularly to biases like confirmation bias, and (3) is probably impossible to estimate and might actually be meaningless, this application of Bayes's theorem (popular with Richard Carrier, blogged about by me here before following Carrier) is probably not very informative at all.

The deeper problem alluded to in (3) above is that we could have two different hypotheses that have enormous amounts of explanatory power. For example, consider the hypothesis "general relativity is correct." What are the alternative hypotheses here? What are their plausibilities? What if the alternative hypothesis subsumes general relativity in the same way general relativity subsumed Newtonian mechanics? What if we reach a point with our theoretical frameworks, e.g. with hypotheses like string theory, etc., where we are no longer able to experimentally verify that one has more explanatory and predictive power than another. I don't care about the physics here--my question is how do we estimate the probability required for (3)? I don't think we can do it at all.

A better way to use Bayes's theorem, which I've been meaning to blog about for months, is to directly compare two hypotheses, using (admittedly subjective) estimates for the prior and degree to which the evidence is predicted by that hypothesis. This gives us a sense of the relative weights of two hypotheses proposed to explain the same phenomenon. For example, let's consider the resurrection of Jesus as one of our hypotheses, and let's consider "his desperate followers lied, hallucinated, or otherwise mistook the situation as is typical in cult situations."

If we use Bayes's theorem on both of these hypotheses and then divide one by the other, we can see their relative weights. I put "Resurrection" on top and "Cult Behavior" on bottom and see it like this:

P(Resurrection given evidence)     Prior(Resurrection)     P(Evidence on Resurrection) 
---------------------------------------------- = ----------------------------- x --------------------------------------------
P(Cult behavior given evidence)    Prior(Cult Behavior)    P(Evidence on Cult Behavior)

Now, while I don't necessarily think applying a prior probability to a hypothesis like "God exists" is valid, doing so regarding the alleged resurrection of Jesus is probably not only valid but strongly supported on "both sides" of the discussion. I don't think it's controversial in the least to suggest that (if we're talking in general) that the prior plausibility of the resurrection is very low. Indeed, the potency of the Christian story depends upon that fact, so we can all reasonably agree that it is very unlikely that someone tortured to death on a cross rose from the dead. This kind of thing just never apparently happens. I don't think it's the least bit controversial, given the number of them that have and still arise exhibiting identical forms of behavior, to suggest that the prior plausibility of cult behavior is very high (nearly 1).

The examination of evidence is trickier. My general contention is that faith is a cognitive bias that distorts how we evaluate evidence, shifting our evaluation to support what we already believe. The most argumentation is needed here, particularly since we have already presented reasons that people may try to dismiss the already overwhelmingly damning case that the prior plausibilities present. I would argue this: had there been a genuine resurrection, Jesus' alleged promises would be recognizably verified, i.e. there would be unambiguous evidence in favor of Christianity. That such evidence does not exist is powerfully damaging to the case for the resurrection. Furthermore, the overwhelming fracturing of the Christian faith into over 40,000 denominations suggests strongly that there's more not to believe than there is to believe with this whole story. Furthermore, this is exactly what we expect to see in cult behavior that takes on a life of its own outside of the original charismatic groups and leaders: the untenable, unsupported claims lead to differences in interpretation and thus schisms. On that, without trying to estimate either number in the fraction on the far right, I suggest that the number on bottom must be considerably larger than the number on top, not least because everything we see in light of the resurrection can be equally well explained in light of cult behavior, but not vice-versa.

Given that, we have that the comparative weight of the two hypotheses in question tilts very, very heavily toward "cult behavior" and away from "resurrection." If we admit the priors in this case, one in which background knowledge is pretty extensive (we've seen no one else, literally no one, not any animal even, come back to life as in the Jesus story), then the question is almost beneath consideration since the relative prior for the resurrection against the prior for cult behavior is necessarily pitifully low and everything else in the equation lowers it.

To really beat this dead horse, the entire Christian story hinges upon the existence of God. There, I do not think we can assess priors meaningfully, but the evidence of the world has consistently pointed away from any god other than some decreasingly interesting Deist God (current state of affairs: "God" is the underlying field of quantum states and the fluctuation that allowed the Big Bang to occur). This is, very importantly, not the God of Christianity. Since the entire Christian story hinges upon a specific God hypothesis that isn't this one, and since the evidence has uniformly pointed away from that God (with "superstitious people exist and believe superstitions" as a reasonable comparative hypothesis) for as long as we've been able to collect evidence, the whole house of cards has really fallen on Christianity--without having to examine even a single one of its questionable, reprehensible, and bankrupt supposedly divine teachings.

I hope that answers your question. Since I put a lot of effort into this and don't think it belittles you in any way, I intend to publish it on my blog. Just wanted to let you know. Thank you for the question and your patience with my long reply.