Gödel's positive predicates

This is part of my series on debugging the ontological argument.

As explained in the previous post, the basic structure of Gödel's Ontological Argument (GOA) is to first prove the "consistency" of God, and then it follows that God must exist.  In this post, I will how we prove God's consistency.

The GOA introduces a second order predicate, that is, a predicate that is applied to other predicates.  This second order predicate is called "positivity", but the name is more suggestive than meaningful.  For example, we could say that the property of being red is a "positive" property, or we could say that being red is not a positive property.  Symbolically, we would write this as $$P(R)\tag{1}\label{1}$$ where R means "is red" and P means "is positive".

GOA at first appears to stipulate a definition of "positivity", and of course it is fine to stipulate a new definition for a new concept.  But my objection is that the definition is poorly formed.

Too many premises

The GOA takes the following premises about "positivity":

$\ref{P1}$: If positive predicate Z entails¹ predicate Y, then Y is also positive.
$\ref{P2}$: Given any predicate and its negation, exactly one of them is positive.
$\ref{P3}$: The conjunction of all positive predicates (called "God-like") is itself a positive predicate.
$\ref{P4}$: If a predicate is positive, then it is necessarily positive.
$\ref{P5}$: Necessary existence is positive.  Necessary existence basically means that in every possible world there is a copy of the given object.

And in symbolic logic:² $$\forall Z \forall Y~ (P(Z) \wedge (Z \rightarrow Y) ) \Rightarrow P(Y)\tag{P1}\label{P1}$$ $$\forall Z~ P(\lnot Z) \Leftrightarrow \lnot P(Z)\tag{P2}\label{P2}$$ $$P(G);\qquad \text{Definition of G:}~ \forall Z~ P(Z) \Rightarrow (G \rightarrow Z)\tag{P3}\label{P3}$$ $$\forall Z~ P(Z) \Rightarrow \square P(Z)\tag{P4}\label{P4}$$ $$P(NE)\tag{P5}\label{P5}$$ I leave out the logical definition of necessary existence because it's very technical and not relevant yet.

The GOA is not an especially popular form of the ontological argument, and perhaps now you can see why.  It gets rid of some of the questionable premises in other ontological arguments, but it replaces them with five whole new ones.  Five!  It's easy enough to just say, one of those premises must be wrong.

Indeed, I can't think of any reason we should think that any of the premises are true.  As far as I can tell, the proof is not referring to any natural concept of "positivity".³  In particular, I don't see why every property either needs to be positive or its negation does.  What if being red is positive in some contexts, but in other contexts it's better to be not red?  Or what if it's positive in this world, but there's a possible world where it's not positive?

Indeed, we can stop here, as far as rebutting the GOA is concerned.  It sure is a fancy argument, too bad its premises aren't remotely persuasive.

But the purpose of this series is to dig deeper.  So we turn to the question: exactly which of the premises is wrong?

Stipulative definitions

Since "positivity" obviously doesn't correspond to any real concept of positivity as far as I know, the only way I can interpret the word is simply as a placeholder.  We're creating a whole new concept, and "positivity" is just the name we gave it.  And since it's a whole new concept, we can stipulate whatever definition we like for it.

Indeed, I believe that none of the premises are individually "wrong".  "Positivity" is a meaningless word, and we're allowed to stipulate certain things about it.  Any of the premises, taken individually, I find acceptable to stipulate as a partial definition.  Taken together, however, there might be an issue.

It is not true that you can stipulate just any definition for a word.  The Stanford Encyclopedia of Philosophy lists two criteria that a stipulative definition must follow, the first of which is relevant here:

A stipulative definition should not enable us to establish essentially new claims—call this the Conservativeness criterion. We should not be able to establish, by means of a mere stipulation, new things about, for example, the moon.

The whole project of establishing the existence of God by stipulative definitions, therefore seems rather quixotic.  The very success of the proof only demonstrates that the stipulative definition that we began with was wrong.

What's happening here, mechanically, is that the definition of positivity is overly constraining.  As a second-order predicate, "positivity" has only so many degrees of freedom.  As we give it partial definitions, we are constraining its degrees of freedom, but at some point we begin also to make constraints on the world (or on the set of possible worlds).⁴

It's quite similar to giving a single word two definitions.  I can define a "foo" as an eight-legged snake, and I can define a "foo" as Socrates, and either of those definitions are fine on their own.  Taken together, they can be used to prove that Socrates is an eight-legged snake, which is a sign the definitions too constraining (even if by a quirk of history it turned out to be true that Socrates was an eight-legged snake).

If I were to pinpoint any of the five premises as particularly problematic, I would say that the $\ref{P1}$ is.  The concept of entailment is based on the material conditional, which as I argued previously is a very counterintuitive concept.  Using this premise, if any impossible predicate is positive then all predicates are positive.  I'm not sure what "positivity" is really intended to signify, but this premise would seem to go against the spirit of it.

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1. As explained in the previous post, entailment means necessary implication.  In the case of predicates, if Z entails Y, then it is necessary that for every object which has predicate Z, it also has predicate Y.

2. Here are some details on how the proof follows from these premises.  Using premises $\ref{P1}$ and $\ref{P2}$, we can prove that all positive predicates are strictly consistent.  If a positive predicate were not strictly consistent, then it would entail a both Z and not Z, which would mean that both Z and not Z are positive predicates, which contradicts $\ref{P2}$.  Using the rest of the premises, we show that being God-like is positive, and therefore strictly consistent.

3. According to Gödel, "Positive means positive in the moral aesthetic sense (independently of the accidental structure of the world)".  Thus it is clear that he intended to capture some sort of natural concept of positivity, although I think the proof fails to live up to these intentions.  I will disregard Gödel's intentions with the hope of exploring the best possible version of the proof.

4. Premises P1, P2, and P5 suffice to make a constraint on the actual world: they prove that something exists.  Just P1 and P2 suffice to constrain the possible worlds: they prove that it is possible that something exists.  Though what they prove is trivial, it is a sign that the definition is already too constrained and cannot be stipulated.

Gods are unimaginably unlikely

The main reason I don't believe in gods now is different from the reason I had when I stopped believing about nine years ago.  Nine years ago, I would have merely said that the arguments for God were lacking.  I am even less charitable today, because I don't think there's any reason to consider God as a hypothesis in the first place.  We do have social reasons to talk about God, but as far as truth-seeking is concerned, the probability of God is so low that by even thinking about it we're tipping the scales.

This is not the argument I would use to try to persuade people of atheism.  My way of thinking of it is too technical.  And it advances a position that is far stronger than is socially necessary for an atheistic society.  What does it matter to me whether you believe the probability of God is unimaginably low, or just extremely low?

But then, I'm not really in the business of directly persuading people to atheism in general.  Just talking about stuff is more fun.

Generally, a good model for thinking about degrees of belief is to speak of Bayesian probabilities.  You might start with "neutral" priors, such as God having a 50% chance to exist.  Then you consider all the evidence and arguments for and against God, and modify the probability accordingly.  An atheist would likely look at the evidence, and think that the problem of evil and problem of divine silence weigh heavily against the existence of God, and that none of the usual arguments in favor of god are effective.  But under such an analysis, how low would you really rate the probability of God?  I think you would rate it very low, but not unimaginably low.

I rate the probability of God even lower than that, basically because I go beyond the basic Bayesian analysis.  I think assigning a 50% prior probability to God is already far too favorable.

As far as theories of the world go, the idea of God is extremely peculiar and narrow.  The ideas of consciousness and intentionality are ordinary to us, because that's the kind of life that matter and evolution produce for us.  But to theorize about consciousness and intentionality which exists prior to the laws of physics is very strange.  And that's before even introducing our even more peculiar ideas of morality.

It reminds me when non-physicists think that quantum mechanics is so strange, that there must be something beyond quantum mechanics.  It is true that quantum mechanics is strange, but strangeness is relative to our own experience in the world that emerges on large scales.  What makes quantum mechanics strange is that it produces a world which looks entirely different on small and large scales.

Now it could very well be that there is something underneath quantum mechanics, and that quantum mechanics simply emerges from more fundamental rules.  I think it likely, even.  But why would we ever think that what's underneath would look similar to what's above?  Whatever's under quantum mechanics will not look like classical physics, it will look even stranger than ever.

The idea of a god is basically the theory that what's at the very bottom (a god underlying the entire universe) is similar to what's at the very top (intelligence emergent from complex biological processes) even though the bulk of the middle looks entirely different.  It doesn't make sense to even propose such a thing, and it makes all too much sense that we as humans would propose it anyway.

The other thing you may have noticed is that the existence of God is not at all obvious in our world.   God is intangible, except in our minds where the same feelings could be caused by any number of things.  The only miracles performed are unverifiable, and split across mutually contradictory religions.  This would be overwhelming evidence against a god, except that theists have basically tailored their conception of god to avoid it.

Of course god is intangible.  Of course god only touches our minds.  Of course god is like us since he made us in his image.  Of course god is leery of showing himself directly.  As for the problem of evil, god is just too difficult for our minds to understand (but apparently familiar enough that we can have called god "good" in the first place).

With all these preconceptions built in, gods at least aren't completely eliminated by the evidence.  But when you tailor the god hypothesis like that, you are basically making the god hypothesis even more specific and more strange than ever.  As discussed in a previous post, this is basically an exploitation of the definition of evidence.  By tailoring your theory just right, it is possible to find a theory which is "favored" by the evidence or at least not completely eliminated by it.  But by doing so, you've ultimately chosen a theory which is more unlikely than ever.

Is there any evidence that could make me believe in a god?  Probably--I mean, a complete change in every aspect of the universe would be fairly persuasive.  But the problems with god as an idea come even before we talk about evidence.

Plantinga's Modal Ontological Argument

This is part of my series on debugging the ontological argument.

Finally, after explaining modal logic basics, I'm going to talk about Plantinga's Modal Ontological Argument (MOA).¹  This argument takes as its centerpiece the following premise: $$G \Rightarrow \square G\tag{1}$$ G is the statement "God exists".  This premise is actually quite clever, because it's hardly an assumption at all, but a definition of God.  In order for an object to be God, then the same object must exist in all accessible worlds.²

This definition of God has the funny property that it depends on our choice of frame.  Even if you would call an object God in one frame, you might not be able to call it God in another frame.  As I explained before, frames are simply constructions and it seems odd that whether we call something God should depend on something so arbitrary.  Nonetheless, this difficulty can be overcome if we simply specify a frame as part of God's definition.  I'm not sure if this is what Plantinga does, but I'll propose a reasonable frame.

In this frame, let's say that world w is accessible from u if w is a possible past or future of u, or some combination (such as the possible past of a possible future).  God is taken to be some object which can never start existing, nor stop existing.  Thus if God exists, then God exists in all accessible worlds.  This justifies the premise \ref{ref1}.

Since premise 1 can be justified as a definition, we must look to the rest of the MOA for problems.

Statement of the Modal Ontological Argument

If you understand S5 modal logic well enough, the MOA is actually quite obvious.  If God is possible, then God exists in a world which is accessible from here.  And therefore God is also here.  More formally:
$$G \Rightarrow \square G\label{ref1}\tag{1}$$ $$\Diamond G\label{ref2}\tag{2}$$ $$\therefore G\label{ref3}\tag{3}$$
For those interested, I also include the formal justification for conclusion \ref{ref3}.  In parentheses, I cite the premises and S5 axioms in use.
$$\Diamond \lnot G \Rightarrow \lnot G~(\ref{ref1})\label{3a}\tag{3a}$$ $$\square( \Diamond \lnot G \Rightarrow \lnot G)~(\mathrm{N~and~\ref{3a}})\label{3b}\tag{3b}$$ $$\square\Diamond \lnot G \Rightarrow \square \lnot G~(\mathrm{K~and~\ref{3b}})\label{3c}\tag{3c}$$ $$\Diamond \lnot G \Rightarrow \square \Diamond \lnot G~(\mathrm{s5})\label{3d}\tag{3d}$$ $$\Diamond \lnot G \Rightarrow \square \lnot G~(\mathrm{\ref{3c}~and~\ref{3d}})\label{3e}\tag{3e}$$ $$\Diamond G \Rightarrow \square G~(\mathrm{\ref{3e}})\label{3f}\tag{3f}$$ $$\square G~(\mathrm{\ref{3f}~and~\ref{ref2}})\label{3g}\tag{3g}$$ $$G~(\mathrm{T~and~\ref{3g}})\label{3h}\tag{3h}$$
 Yes, this is logically valid.  All that remains is to justify premise \ref{ref2}, the statement that God is possible.

Is God possible?

Not many people put stock in ontological arguments, and indeed, neither does Alvin Plantinga, even though he formalized the MOA.  Instead, Plantinga merely argues that it is reasonable to think that God is possible, and therefore it is reasonable to believe that God exists.³

The problem is that possibility can mean oh so many different things.  Even if we confine ourselves to Kripke semantics, the meaning of possibility depends on our precise choice of frame.  As for the proof, it requires a very particular meaning of possibility.  Even if it is reasonable to believe God is possible, is it reasonable in the sense required by the proof?

For example, if you take the definition of accessibility that I used in the introduction, then the MOA can be phrased like so: "God existed some time in the past or future.  Since God by nature cannot begin to exist or stop existing, then God also exists now."  This isn't very compelling, because saying that God existed at some point in the past or future is a pretty major assumption.  I am no more likely to believe the assumption than I am to believe the conclusion.

Or suppose we had a broader accessibility relation.  Suppose that world u is accessible from world w if and only if the two worlds obey the same physical laws.  In this case, it's not clear to me how premise \ref{ref1} follows from a natural definition of God.  But let's just take that definition and be careful about the verification of any object as God.  The MOA can be phrased like so: "God exists in some possible world with the same physical laws as ours.  By definition, God must exist in all worlds with those physical laws.  Therefore he exists in our world."

This isn't very compelling because it raises the question, how do we verify that a particular object is God?  In order to check the object against the definition of God, we have to check all accessible worlds to see whether the same object exists in those.  Maybe that's not so bad if we chose a frame where there aren't so many possible worlds, but there is one world we need to check no matter what: our own world.  But by the time that you've verified God exists in our world, what need is there for the MOA?  The MOA is in this interpretation "useless".

While Plantinga argues that the premise of the MOA is "reasonable", we can see that this has no value.  It does not matter whether we believe in gods or not; because the MOA is "useless", it should not advance our belief in God even a little bit.

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1. I'm really considering the simplest possible form of the modal ontological argument, skipping some steps for aesthetic reasons.  Plantinga's full argument can be found here.

2. For the moment we'll just forget about the problem of how to say that two objects in two separate possible worlds are the "same".  We'll also forget about whether existence is a predicate.  These are serious problems, but purely technical ones that will be solved later.

3. One thing you may not have known about Plantinga is that he has written an awful lot about his "reformed epistemology", and I'm sure he has something very particular in mind when he says the premise is "reasonable."  I am not familiar with Plantinga's reformed epistemology and won't address it.

Debugging the ontological argument

I'm introducing a new series on the ontological arguments for the existence of God.  Ontological arguments are basically attempts to prove that God exists using pure reasoning alone.  For example:

God is defined to be the most perfect being.
It is more perfect for something to exist than to not exist.
Therefore, God exists.

However, this is just one ontological argument, and I will cover several more, culminating in Godel's ontological argument.

This is the masterpost for the series, containing a complete list of links, as well as an explanation of my purpose and perspective.

Outline of series:

• The definitional ontological argument
• Is existence a predicate? 
• These things do not exist
• The conceptual ontological argument
• Making sense of existence
• The uselessness of the ontological argument
• Interlude: Aesthetics in the ontological argument
• Plantinga's modal ontological argument
• Modal logic semantics
• Addressing the argument
• Gödel's ontological argument
• Gödel's workarounds to the ontological argument
• Consistency and the material conditional
• Gödel's positive predicates
• Improving on Gödel

Purpose:

The primary purpose is to talk about logic.  When we discuss apparent paradoxes, I find that many beautiful details about logic come out of the woodwork!

I also know there are lots of people out there who are just plain curious about the ontological argument, so my secondary purpose is to satisfy your curiosity.

I am not here to merely show that ontological arguments are wrong.  If I wanted to persuade people, I could simply assert the argument's absurdity--this is essentially the strategy adopted by Richard Dawkins in The God Delusion, for instance. The fact of the matter is that a vanishingly small number of people think ontological arguments are correct.  While I'm sure a few proponents will show up in the comments, most people look at ontological arguments and immediately find them absurd and unpersuasive.

My purpose here is to pinpoint errors in the ontological arguments.  I also call it debugging the argument, as if to go line by line in a computer program and find the missing semicolon. Although unlike debugging a computer program, I will not actually correct the errors, since ontological arguments are fundamentally unfixable.

There are definitely ways to refute ontological arguments without pinpointing the problems.  For example, an argument ad absurdum extrapolates the argument to reach an absurd conclusion:

Ted is defined to be the most excellent being.
It is more excellent for something to exist than to not exist.
Therefore, Ted exists.

But even if you accept the argument ad absurdum, that doesn't tell you what is wrong with the argument.  It's like executing a program and encountering an error--it tells you that there's a bug, but doesn't tell you where.  Thus, in this series, I will refer to, but not rely on argument ad absurdum.

Perspective:

I am a random person on the internet.  I am not here to explain historical refutations of the ontological argument, I am offering my own original vision.  For a more scholarly treatment, I recommend more formal resources such as Wikipedia or the Stanford Encyclopedia of Philosophy.

My perspective is more that of a mathematician than a philosopher.  Most ontological arguments can be framed in logical terms, and my writing will focus on these terms.  Some ontological arguments add rhetorical bells and whistles to the logic, and privately I view these ornaments as unhelpful at best and obscurantist at worst.

Expertise:

Unless you count my minor in math, my expertise here is basically self-taught. I taught myself philosophical concepts as necessary, sometimes consulting my boyfriend, who has a degree in philosophy.  Many years ago, I taught myself modal logic just so I could understand modal ontological arguments.  I wrote a bunch of blog posts about it (see references below), and argued endlessly with a few ontological argument proponents.

Basically, you shouldn't trust me based on my expertise, but based on my arguments and reasoning.

Comment Policy:

If you have comments that do not pertain to any specific post, please comment here on the masterpost.  I will not reply to all comments.  I do not promise to be polite.

Throughout this series I will use MathJax to render LaTeX equations.  This requires that you enable Javascript.  LaTeX will not work for comments, so in the comments I will adopt the following conventions:

E - existence quantifier
A - universal quantifier
~ - negation
^ - and
V - or
=> - implication (will be rendered as =>)
&lt;=&gt; - equivalence (will be rendered as <=>)
N - necessary
P - possible
-&gt; - entailment (will be rendered as ->)
|= - satisfaction relation

I encourage readers to adopt the same conventions to avoid confusion.

Complete list of old posts
This is all the stuff I've previously written on the ontological argument.  I'm afraid some of it is not very clear, and I may have changed my mind on some of the details.  This new series is intended to be my definitive word on the subject.

A simple ontological argument
A modal ontological argument
Godel's ontological argument
Godel's ontological argument, step by step
Modal ontological argument, revisited
Plantinga responds to me
An anecdote on obscurantism
Necessity isn't so necessary

Pascal's wager

On this blog I enjoy criticizing many of the philosophical arguments for God, but it occurs to me that I've never covered Pascal's Wager, despite it being one of the most common arguments out there.  True to my style, I will cover Pascal's Wager with a mathematical bent.

Pascal's Wager is basically a decision theory argument.  You have two choices: believe in God (specifically, the Christian God), or don't.  The world has two possible states, there is a god or there isn't.  Depending on your choice and the state of the world, there are different outcomes.  According to the usual argument, these are the outcomes:

	God exists	God does not exist
Wager for God	Gain all	Status quo
Wager against God	Misery	Status quo

(copied from Stanford Encyclopedia of Philosophy)

A lot of people say that you can't really choose to believe in God or not, but it's possible to influence one's own beliefs.  I'm sure if you made a point to listen to apologetics all the time and avoid discussions of religion on the internet, then you could increase your chances of believing in God.  I think some people have this conclusion in the back of their minds, and use it as a reason to avoid questioning their own religion.

With this outcome matrix, it seems that believing in God is a "dominant" strategy.  That is, the outcome is always better than or equal to not believing in God.  In each column, the outcome in the first row is at least as good as the outcome in the second row.  Dominant strategies are better strategies, therefore we should make an effort to believe in God.

But it isn't really a dominant strategy for several reasons.  Worshipping God costs time and effort (and if any religious practices were genuinely beneficial, you could do them without believing in God).  It would also cost extra if, like I suggested earlier, believing in god means listening to lots of apologetics and avoiding discussions of religion on the internet.

But suppose, for the sake of argument, that believing in God actually costs nothing.  Imagine that no worship is called for, and that choosing to believe is as easy as choosing to speak.  Believing in God is still not a dominant strategy, because the world has more than two possible states.  If you can think of some pathological scenario where people who believe in God are rewarded, then you can also think of a pathological scenario where people who don't believe in God are rewarded.  For example, there's the scenario where a different but equally whimsical God exists, and this god rewards people who don't believe in gods.

	Whimsical God #1 exists	Whimsical God #2 exists	God does not exist
Believe in God	Gain all	Misery	Status quo
Don't believe in God	Misery	Gain all	Status quo

It's not really necessary to come up with a realistic scenario where people who don't believe in God benefit.  As long as there is any such scenario, believing in God is no longer a dominant strategy.  That is to say, it is no longer true that in each column, the first row has at least as good an outcome as the second column.

But strategic dominance isn't the only kind of argument you can make with Pascal's wager.  You can also argue based on what you think are the probabilities of the different states of the world, and how much value you place on the different outcomes.  Many people see salvation as an infinite reward.  If you multiply an infinite reward by a nonzero probability, then it's still infinitely more preferable to any finite outcome.  Therefore, as long as there is a nonzero probability that God exists, one should make every effort to believe in God.

There's already a problem with this argument: if it's at all effective, then it's too effective.  If it's truly the case that a nonzero probability of an infinite reward is infinitely preferable to any other outcome, then all of us should ignore all finite rewards in favor of slightly increasing the probability of satisfying this whimsical god.  More than just spending some time listening to apologetics, you should dedicate your life to brainwashing yourself to maximize the probability that you will believe in God.  And if there's a nonzero probability that God infinitely punishes people who eat shellfish or who mix wool and linen, then one should make every effort to appease this unlikely God.

Given that this is not how Christians behave, and very few Christians believe we should behave that way, we should consider if there's something wrong with the argument.

Pascal's Wager is comparable to the St. Petersburg paradox.  In this paradox, you play a game where you flip a coin repeatedly until you get tails.  If it took N coin flips, then you win 2^N dollars.*  The expected reward for playing this game is infinite, and therefore it is worthwhile to pay any large, finite amount of money to play the game.  And yet, intuitively, it does not seem like a worthwhile investment.

*Dollars have the property that the more you have, the less valuable they are to you.  In a more sophisticated version of the paradox, the reward is in "utils" rather than dollars.

Experts have many different opinions on the resolution to the St. Petersburg paradox.  But very few experts bite the bullet, arguing that it is in fact worth it to invest any finite amount of money to play the game.

Pascal's Wager captures some of the essence of the St. Petersburg paradox, because it also offers a very small chance of a very large reward.  It's a neat trick, harnessing a controversial mathematical paradox to make an argument for God.  This way, even though experts agree that the argument is wrong, nobody agrees on exactly why.  It allows apologists to look at the disagreement among their opponents and believe that they've won.

Myself, I believe that the probability of an outcome with infinite value is precisely zero.  This doesn't necessarily make me a "strong" atheist, it just means that I think it's impossible for any outcome to be infinitely valuable, as impossible as it would be for God to make a square circle.

This makes philosophical sense, because "value" is just what we use to describe what our preferences are.  There is no reward whatsoever that I prefer so strongly that I would risk everything else just to have an infinitesimal chance of winning it.  Therefore there is no infinite value.

My discipline is physics, so I have to say that it makes physics sense too.  Suppose we have someone who claims that they'll give you some payoff, if only you provide the initial investment for them to fly over from Nigeria.  If you think there's a million to one chance of getting the payoff, does that mean that they can just offer a million times as much money in order to overcome your doubt?  No, because then you would doubt the story even more!  As they increase the magnitude of their claims, the magnitude of our doubt increases even faster.  Salvation is just the limit of this process of runaway humbug.

There's also the same problem I brought up earlier, that we can also think of whimsical gods which punish rather than reward believers.  For instance, if conservative Christians of a different denomination are right, then your denomination of Christianity may go to hell for heresy.  Is whimsical god #1 really any more likely than whimsical god #2?  I think they are about equally likely, although it's understandable that people are more likely to believe in #1.

I will concede one thing about Pascal's Wager: it may in fact present a problem to a certain set of agnostics.  That is, if you think there's a 25% chance that God exists and 75% chance that God doesn't, then Pascal's Wager applies to you.  However, my understanding, speaking with agnostics, is that this is not what most self-identified agnostics believe.  Most agnostics are not on the fence, probabilistically speaking.  Rather, they think gods are unknowable, or that it is inappropriate to assign probabilities at all.  Further, while many agnostics may be uncertain about gods, they may feel more certain of the nonexistence of salvation.

Missing the point on Fine-Tuning

The Fine-Tuning argument (henceforth the FTA) is the argument that the universe appears fine-tuned to create life, this strongly argues for the existence of a creator with life-friendly intentions.  I've written about the FTA before, in 2009, and I also endorsed a paper written by Ikeda & Jefferys¹ (henceforth I&J) which purports to refute the FTA.  I've now changed my mind, and no longer agree with I&J.

To briefly summarize I&J, they formulate the FTA in bayesian terms.  We already know that life exists, as part of our background knowledge.  The new piece of evidence put forth by the FTA is that the universe is such that life may arise naturalistically (ie it is "life-friendly").  I&J say the problem is that this evidence argues against supernaturalism, not in favor of it.  After all, in a naturalistic world, we must observe that the universe is life-friendly.  In a supernaturalistic world, we might observe otherwise, since supernatural intervention can create or sustain life even when natural laws are not life-friendly.

In formal terms:²

N = universe is naturalistic
L = life exists
F = universe is life-friendly

I&J show that
P(N|F&L) ≥ P(N|L)
Whereas the FTA requires that
 P(N|F&L) << P(N|L)

I think their theorem is basically incontrovertible,³ and I do not contest it. However, I feel it is missing the point.  Here I aim to explain why.

What does the Fine-Tuning argument argue?

A schematic illustration of the Fine-Tuning argument.  The two rows each represent a different meta-theory of the universe.  The colored bars represent the probabilities of different events within those meta-theories.

The FTA asks us to compare two meta-theories of the universe: naturalism and supernaturalism.  In a naturalistic universe, there's a very small probability that life can arise, because the universal constants need to be just right (or so the argument goes).  In a supernatural universe, well, who knows?

I&J make two important points about this argument.  First, they argue that the premise is irrelevant, and the conclusion is a fallacy.  Formally:

Premise: P(F|N) << 1
Fallacious conclusion: P(N|F) << 1

I checked to see if this is in fact the premise and conclusion used by FTA proponents.   The Discovery Institute basically makes the following argument:

Premise: P(F|N) < P(F|~N)
Conclusion: P(N|F) < P(N)

This is similar to the fallacious reasoning identified by I&J, only it's not fallacious.  It's correct.  But it requires a stronger premise.  Namely, we have to assume that the probability of life under naturalism is less than the probability of life under supernaturalism (as illustrated above).

I believe that I&J reject this stronger premise, and I think this is reasonable.  Who is to say that a deity would be particularly likely to arrange the universe such that it contains life?  There are an infinite number of possible deities which each prefer a different configuration of the universe.  If we only speak of deities that have an inclination towards universes with life, then they're likely to create life, but this is just some sort of selection bias on our part.

But even though I tentatively agree with I&J's first point, I will ignore it for the rest of the post, because I wish to evaluate how their second point stands on its own.

I&J's second point is that we need to include all evidence in our Bayesian.  We know two things: life exists, and life can arise.  The probability of naturalism should be conditioned on both these facts, not just one or the other.  Formally:

FTA argues that P(N|F) > P(N)
But the relevant comparison is between the quantitities P(N|F&L) and P(N|L)

With this modification, I&J find that if the universe is life-friendly, this is evidence against the supernatural.  This is easy to see in the above figure.  If we rule out a life-unfriendly universe (the red bar), this decreases the likelihood of supernaturalism.

On the other hand, if we compare the size of the green bars, it seems like the FTA works.  If the universe is life-friendly, that is evidence for naturalism.  But if the universe has life in it, it seems like that's even stronger evidence for the supernatural.  In formal terms, it appears that

P(N|F&L) ≥ P(N|L) << P(N)

And that's why I feel I&J have missed the point.

Life-friendliness vs Life

The "fine tuning" argument isn't "What do you think about God, when you learn that you are alive?" but "What do you think about God, when you learn that the universe is (apparently) fine-tuned or life-friendly?"
--Ikeda & Jefferys

As in the quote above, I&J believe that the only piece of evidence advanced by the FTA is that the universe is life-friendly.  The existence of life is not a piece of evidence, because we already knew life existed.

The thing is, the distinction between "life exists" and "life can arise naturalistically" is a novel idea by I&J.  It is definitely a useful distinction that improves our understanding of the problem.  But I don't think FTA proponents ever make this distinction.  FTA proponents are vague about it, and may not even know themselves whether they are arguing about life-friendliness or life-existence.

Therefore, when I&J say that FTA proponents are using life-friendliness as evidence, and not life-existence, this is totally groundless.

To understand the distinction between life-friendliness and the existence of life, it may help to consider how we might demonstrate life-friendliness.  It's actually very difficult to demonstrate without any sort of presumption of naturalism.  I&J indirectly mention the triple-alpha process, wherein the helium inside of stars fuse to carbon.  Since the triple-alpha process is necessary to naturalistically produce carbon life-forms, Fred Hoyle predicted in the 1950s that there was a particular resonance which greatly increased the probability of the triple-alpha process.  This prediction was later confirmed.

If FTA proponents are truly using life-friendliness as their evidence, they would point to Fred Hoyle's confirmed prediction.  In Biologos treatment, they do discuss the triple-alpha process, but do not mention Hoyle's confirmed prediction.  Instead they talk about how "statistically unusual" the resonance is.

Put in formal terms, I&J say that we should compare the quantities P(N|F&L) and P(N|L), because this is the correct comparison, and because this is the comparison described by FTA proponents themselves.  But I think that FTA proponents are actually comparing P(N|F&L) and P(N).  Furthermore, I think FTA proponents have chosen the correct comparison, whereas I&J have chosen the wrong one.

Trickiness with priors

But what is P(N|L)/P(~N|L)? Why, it is just the prior odds ratio that You assign to describe Your relative belief in N and ~N before You learn that F is true.
--Ikeda & Jefferys

According to I&J, our prior beliefs are based on our knowledge that life exists.  We think, therefore we are.  Therefore, if upon seeing that life exists, you think that supernaturalism is more likely than naturalism, then you have a "prior commitment" to supernaturalism.

"Prior commitment" sounds bad, like you've decided your conclusions before you've considered the evidence.  But generally speaking, there's nothing wrong with coming to conclusions without evidence, if you have an a priori argument.  Since FTA proponents are arguing about the quantity P(N|L), which I&J call a prior probability, we could say that FTA is partly an a priori argument.

Is it really true that the FTA is an a priori argument?  If the only evidence advanced is the existence of life, then the FTA is quite similar to the cosmological argument.  The cosmological argument only advances the evidence that there exists something rather than nothing.  Is the cosmological argument considered an a priori argument?  The internet consensus seems to be not.⁴

If the FTA is not an a priori argument, then I&J have made an error in using P(N|L) as our prior belief, rather than P(N).  If the FTA is an a priori argument, then I&J have basically ignored the a priori component of it, and no wonder that what remains is so unconvincing.

Of course, it seems like FTA proponents really are trying to advance some sort of evidence, and that's why they talk about universal constants and the triple-alpha process.  I propose that what's happening is that we're actually comparing three meta-theories:

By eliminating the naturalistic theories where fine-tuning is not necessary, it seems we greatly reduce the probability of life existing within naturalism.

Conclusion (TL;DR)

Ikeda and Jefferys refute the Fine-Tuning argument by making an interesting and useful distinction between universes with life, and universes where life can arise naturally.  According to them, it is only necessary to consider the evidence that life can arise naturally.  The problem is that this ignores the main thrust of the Fine-Tuning argument.

To refute the Fine-Tuning argument, we must discuss its many other problems, and not the particular problem put forth by Ikeda and Jefferys.

--------------------------------------------

1. Also see the same paper hosted on a different website with inferior design, but with the extra Appendix 2.

2. If you are unfamiliar with the formalism of conditional probabilities, you can read up on it.  In this post, I will try to minimize formalism, but it is impossible to eliminate it entirely.

3. The only assumption is that P(~F&L&N) = 0, which is true from the definition of F.

4. See here, here, and here, all search results for the cosmological argument.

Getting from First Cause to God

I have this series, "A few things wrong with the cosmological argument", but have ignored it for over six months.  I think that means this should be the last post, even if I didn't have the opportunity to point out every last problem in the cosmological argument.

In previous posts, I discussed lots of little problems in the cosmological argument as presented by William Lane Craig (henceforth WLC).  It's mostly lots of nitpicking of little problems that maybe could have killed the cosmological argument if it weren't already killed by much bigger problems.  But here I will discuss what many people consider to be the biggest problem of all: How do you connect the "First Cause" to God?

This is a major problem with most philosophical arguments for God.  They argue for some ultimate entity, and then just take it for granted that it's God.  And since this argument is typically made by Christian apologists, it's not just any god, it's the Christian God.*  The argument feels like it's relying on the cultural dominance of Christianity.  As soon as someone talks about a powerful entity, God seems like an obvious possibility, only because everyone talks about it so much.  My boyfriend says that because he's played too many fantasy games, his gut feeling is that an epic-level wizard is more likely.

*To be fair, apologists usually supplement their philosophical arguments with historical arguments.

Chris Hallquist made the interesting claim that it wasn't always this way.  Classical theologians used to have arguments for specific traits of God.  Nowadays, people like WLC will only devote a minimal amount of space to the subject, and what's there doesn't make much sense.

I will only briefly address WLC's argument, as presented in his Kalam article:

I think that it can be plausibly argued that the cause of the universe must be a personal Creator. For how else could a temporal effect arise from an eternal cause? If the cause were simply a mechanically operating set of necessary and sufficient conditions existing from eternity, then why would not the effect also exist from eternity? For example, if the cause of water's being frozen is the temperature's being below zero degrees, then if the temperature were below zero degrees from eternity, then any water present would be frozen from eternity. The only way to have an eternal cause but a temporal effect would seem to be if the cause is a personal agent who freely chooses to create an effect in time. For example, a man sitting from eternity may will to stand up; hence, a temporal effect may arise from an eternally existing agent.

I basically disagree with everything:

• I don't think it's strange for a temporal event to arise from some entity that does not have a beginning.  For instance, if that entity consists of two non-interacting particles moving past one another, then this is an eternal entity that has a special point in time (ie the moment of closest approach).  It's also completely mechanistic.
• Perhaps the problem with the two-particle example is that WLC thinks the first cause must be timeless.  I don't think this follows from the Kalam Cosmological argument, which only shows that there is an entity that did not begin.  "Eternal" is not the same as "timeless".
• WLC seems to be imagining "personal agents" as agents with libertarian free will.  (This is odd because in other contexts, Christians appear to believe in compatibilist free will.)  I don't agree that libertarian free will is even a sensible concept.  All personal agents we know of are mechanistic.
• A timeless libertarian agent is even less sensible.  Usually, a libertarian agent is one who may make decisions independent of prior circumstance.  When there is no time, there is no prior circumstance.

Really, there are so many assumptions in WLC's argument, that I have to step back a bit and question more fundamental assumptions.

The whole Kalam cosmological argument rests on an analogy between objects within the universe, which all require causes, and the universe itself.  If the cause of the universe is so dissimilar to the causes of objects within the universe, one wonders why we accepted the whole analogy in the first place!  Therefore, we should ask what causes look like normally.

Causality is a many-faced concept which I've mused about in another blog series.  For example, there's the clinical concept of causation--if you tell a bunch of people to start smoking, they will be more likely to get cancer than a similar group that you told not to smoke, thus smoking causes cancer.  But if we're making philosophical arguments about causation, we might need a more general notion of causation, like from physics.  Here are the key points:

• Every event is caused by all events in its past light-cone.  Without the universe, there will be no light-cones, but it still makes sense that everything is caused by a multiplicity of previous things.
• In another sense, events are "caused" by general physical laws which connect past to future.  Therefore, we might expect that among the universe's causes are abstract principles, or other things that aren't really "things".

So if I were to guess the universe's cause, I'd guess that it was a multiplicity of things, including a few things that aren't really "things".  I would not have guessed that it was a single conscious entity.  I certainly would not have guessed something so outlandish as a timeless libertarian agent.

1. Actual and potential infinities
2. Actual infinities in physics
3. What is real?
4. The "absurdity" of Hilbert's Hotel
5. Interlude: God is infinite
6. Forming Infinity, one by one
7. Uncertain beginnings
8. Entropy: The unsolved problem
9. Kalam as an inductive argument 
10. Getting from First Cause to God

Christian compatibilist free will?

Something interesting came up in the comments on my post about the free will defense against the problem of evil, and I wish to recap it.

First, commenter James brought up the argument that God's omniscience contradicts free will.  Commenter Slightlymetaphysical brushed off the argument.  I would brush it off too.  Pshhh, why should omniscience contradict free will?  Just because you know what someone chose doesn't mean that they didn't choose it.

Of course, the reason I would brush it off is because I'm very used to compatibilist free will.  That is, I believe free will is compatible with determinism.  An act of free will is just something that is arises directly from a set of conscious cognitive processes.  Even if those cognitive processes are deterministic, that doesn't mean our decisions weren't caused by them.

But there are also incompatibilist conceptions of free will.  For example, commenter Larry offered the following definition (without necessarily endorsing it):

1. Assume there is some state of the world at time t.
2. Agent A makes decision D at time t+1.
3. We "roll back" the world precisely to its state at time t
4. Agent A has "free will" if and only if she could make a different decision, D', at time t+1.

Clearly, this kind of "free will" is incompatible with determinism, because in a deterministic world, there is only one possible outcome, given initial conditions.

As a result of the comment discussion, I realized that Christians are implicitly taking a compatibilist view of free will.  As long as they accept the following premises, they must accept the conclusion:

Premise 1: Free will exists.
Premise 2: An omniscient being exists.
Premise 3: The existence of omniscience implies determinism.
Conclusion: Free will is compatible with determinism.

This is very interesting, because it is not in accordance with other Christian views on free will.  And in particular, it's not in accordance with the free will defense against the problem of evil.

The Christian* view on free will is that it is the ability to turn against God.  Eve had the ability to eat that apple (or pomegranate or what have you), and the ability to choose not to.  For reasons infathomable, this kind of free will is so desirable to God that it is worth the evil that proceeds from it.  On the other hand, if we believe in compatibilist free will, then God can have his cake and eat it too.  That is, people can have the ability to choose evil, even when it is predetermined that they will not exercise this ability.

*More accurately, a Christian view, which is not necessarily universal among Christians.

The problem of evil is the question, "Why does God allow evil?"  The free will defense is, "Evil is a necessary consequence of free will."  My proposed counter is, "But free will is compatible with a world where no one chooses evil."

Of course, this is one of those purely philosophical arguments, and we all know how unpersuasive those are.  Here are a couple responses:

• Omniscience does not imply determinism. An omniscient being could simply be aware of all possibilities and their outcomes, without knowing which possibilities will come to pass.  One could counter that this kind of omniscience is hardly a proper omniscience at all.  But I for one would still be very impressed by such a being, whether it's proper omniscience or not.
• It is not free will itself that God finds so desirable.  Rather, God desires that people have the ability to go against him, and that they actually exercise this ability.  God's desire does apply to heaven, where he is basically okay with the fact that everyone is choosing good.  God sure has some strange desires, but the truth can be strange.

The burden of proof and God

One of the more tedious arguments concerning gods is the argument over who has the burden of proof.  Whereas many atheists argue that the theist must first make the argument for the existence of gods, their opponents argue that this is a cop out.  For example, on NY Times:

Contemporary atheists often assert that there is no need for them to provide arguments showing that religious claims are false. Rather, they say, the very lack of good arguments for religious claims provides a solid basis for rejecting them. The case against God is, as they frequently put it, the same as the case against Santa Claus, the Easter Bunny or the Tooth Fairy. This is what we might call the “no-arguments” argument for atheism.

I take the side of atheists; I think theists have the burden of proof.  This is not about giving atheists an unfair advantage in the debate, nor is it about making a "no-arguments" argument.  In fact, I do not believe it is an advantage, fair or otherwise, at all.  It's simply about who takes which role.

I'm somewhat influenced by the role of burden of proof in law, though law does not necessarily provide a paragon of debate format.  In a court of law, attorneys set out to prove certain facts.  But producing the arguments and evidence costs money, so the law must specify whose role it is to prove the facts, and to what extent they must be proven.  In most criminal cases, the plaintiff must prove "beyond reasonable doubt", and in most civil cases, the burden of proof is a "preponderance of the evidence".  If you're wondering how to interpret that, it depends whether you ask the defendant's or plaintiff's attorney.

(Please correct me if I made any error with regard to law.)

In the case of proving God, it is only sensible that the theist has the burden of proof.  The atheist doesn't know beforehand what particular god or gods the theist believes in, and doesn't know what arguments to use.  Theists keep on telling us that not all religious people believe the same things, and I wholeheartedly agree.  That's why it's the theist's role to explain what they believe and why.

But this does not confer an advantage to atheists, any more than it confers an advantage to the defendant in a court of law.  It just means that the theist makes the first argument.  If the argument, taken at face value, meets the burden of proof, then it is up to the atheist to counter it.  In other words, the burden of proof shifts to the atheists.  The burden of proof shifts back and forth indefinitely, and does not give an advantage to either side.

By saying theists have the burden of proof, I only mean that they have the initial burden of proof.  It is not meant as a "get out of an argument free" card.

Kalam as an inductive argument

This post continues "A few things wrong about the cosmological argument", a looong series about a few minor problems with the cosmological argument.

One of my favorite worst arguments ever comes from What the Bleep Do We Know!?, a movie which is about how you can use quantum mechanics to create your own reality, reduce crime, walk on water, etc.  Anyway, the argument goes as follows:

1. Only conscious things can be quantum observers.
2. Human cells are quantum observers.
3. Therefore, human cells are conscious things.

(Queue CG-animated human cells having parties.  No seriously, that's what happens in the movie.)

You gotta hand to it to them, they make a valid argument--the conclusion follows from the premises.  But the premises are bullshit.  It simply isn't true that only conscious things can be quantum observers.

But even if they didn't know that before, shouldn't the second premise have cast doubt on the first?  If I said a piece of paper is a quantum observer (this is true), does that mean you should conclude that the piece of paper is conscious?  Or should you go back to the drawing board, because it means that not every quantum observer need be conscious?

On the other hand, consider another argument in the same form:

1.  Particles can never travel faster than light.
2.  An OPERA experiment showed that neutrinos are traveling faster than the speed of light.
3. That experiment must have a flaw somewhere.

And of course, it turned out that there was indeed a loose cable somewhere which caused the experimental error.  But even before we knew that, most experts would have guessed that there was an experimental flaw based on the currently available evidence.

Why is the neutrino example different from the conscious human cell example?  In both cases, the second premise casts doubt on the first premise.

Here is a key point to understand about the neutrino argument: it appears to be a deductive argument, but it is actually an inductive argument.

We know that all the experiments we have done are most consistent with a General Relativity view of space-time (in which particles cannot carry information faster than light), and inconsistent with anything simpler than that.  But we cannot know that particles can never travel faster than light in any situation unless we actually check every situation, including the situation of the OPERA experiment.

In the case of quantum observers, there is no inductive argument that all quantum observers must be conscious.  Because that's not even close to being true.

-------------------------

The basic form of the Kalam cosmological argument is similar:

1. Everything that has a beginning has a cause.
2. The universe has a beginning.
3. Therefore the universe has a cause.

In my view, the second premise calls into question the first.  Because here we have an object, the universe, which does not have a cause we know of.  And so if we assume it has a beginning, maybe it is a counterexample to our supposed rule that things that begin have causes?  So we must ask, how do we know that everything that begins has a cause?  Is it because we've looked at a lot of things that have beginnings, and found that all of them have causes?  Or is there some other argument that doesn't refer to empirical observations?

Note that it is easy* to come up with alternate forms of the first premise which explain our observations just as well.  For example, "Every event has a cause."  "Every object in the universe has a cause."  "Everything that has existed for a finite amount of time has a cause."**  "If something exists at one point in time, something must have existed before that point in time." Some of these allow us to conclude that the universe has a cause, and some do not.

*By contrast, it takes a theoretical physicist to think up possible laws of physics which look like General Relativity in every experiment we've done with the single exception of the OPERA experiment.
**Under some views of the universe, it has existed for a finite amount of time, but does not have a beginning.

In the future, I will discuss a bit about causation, but for now I want to point out a more antiquated argument for god that appears in similar form.

1. Everything that is real must have a mind to observe it.
2. Things continue to be real even when no humans or other material beings are observing them.
3. Therefore, there is a non-material being to observe things.

This argument may seem a little silly, but it has some personal significance to me.  It's one of the classical arguments I was taught in my Catholic education (we were not necessarily taught to think it was a good argument).  It was around this time that I knew a deist who eventually persuaded me that Catholicism was wrong.  The deist used the above argument, roughly, though I didn't buy it.

Since then I have learned that hardly anyone uses this argument or takes it seriously.  I guess it was just that one deist I knew!  But far be it from me to use an argument from popularity.  You should decide for yourself whether the argument is any good, regardless of how popular or unpopular it is!

"A few things wrong about the cosmological argument"
1. Actual and potential infinities
2. Actual infinities in physics
3. What is real?
4. The "absurdity" of Hilbert's Hotel
5. Interlude: God is infinite
6. Forming Infinity, one by one
7. Uncertain beginnings
8. Entropy: The unsolved problem
9. Kalam as an inductive argument
10. Getting from First Cause to God 

The evil of theodicy

Earlier a commenter told me I should stop bashing religion. This left me wondering, where did they see me bashing religion?  I feel like I've mostly said neutral things about it lately.  I should do more religion bashing!

The problem of evil asks: How can there be a all-powerful and all-good god if there is evil in the world?  Obviously this only applies to religions with an all-powerful and all-good god, and I might as well say that I'm thinking of Christianity in particular.

I'm not sure I've ever talked about the problem of evil before.  I don't really like it, because there's no math involved.  And the argument is too sprawling, with a multitude of rebuttals.  In fact, we even have the word "theodicy", which means a defense against the problem of evil.

Most theodicies are not very compelling, but that's not what I want to talk about.  I want to talk about how theodicies, above and beyond being bad arguments, are also evil arguments.  That is, many theodicies involve defending evil, or denying the existence of certain kinds of evils.

The free will defense

The free will defense says that evil exists in the world because of human free will.  The obvious problem with the free will defense is that it ignores natural evils, like hurricanes and disease.  But let me tell you about some of the other kinds of evils it ignores.

There is the kind of evil which is caused by ignorance of the consequences of a free action.  To use a silly example, someone could open a door, not knowing someone was there to be smacked in the face.  Or, someone could buy a diamond, not knowing that the money is used to fund a terrible war.  Or if someone votes for X political party, not knowing that God truly endorses the opposing party (har har).  This is evil not caused by free will, but by ignorance, which is arguably a hindrance to true freedom.

There are various kinds of responses to the problem of natural evil.  One of them is that natural evil was released into the world by Adam and Eve's original sin.  Or, if you're Pat Robertson, natural evil was released into the world by modern actions such as homosexuality.  Or if you're Alvin Plantinga, natural evil was caused by the free will of nonhuman beings, like angels or spirits.  Presumably, the means of causation are magical (divine), but it's kind of funny that earthquakes and hurricanes mostly occur where you would expect them if they were caused by chaotic physical processes.

Now here's another problem being ignored: the sheer injustice and cruelty of this situation.  We deserve to be punished for the sin of a couple ancestors?  Or for the sins of other people within our society?  Or for the sins of some otherworldly beings we don't even know about?  The causal chain leading from sin to natural evil isn't very clear, but it's hard to imagine that God really doesn't have any control over it.  Is it interfering with the free will of gay people to change ocean temperatures and prevent a hurricane?  As far as I'm concerned, this theodicy only succeeds by conceding that God is not actually all-good.

The greater good

There are a variety of other theodicies which claim that natural evils are necessarily to achieve the greater good.  Suffering builds character!  (Except when it kills you, then it builds character among your relatives.)

For some reason, theodicy makes me think of classic Calvin and Hobbes

Or, if it's not character building, perhaps the lesser evil prevents some other greater evil.  Like the story about the guy who breaks his leg, and thus avoids a car accident.  God couldn't think of another way, that's not in his omnibenevolent nature.

I see this explanation as rather awful, and not just in the lacking-evidence sense.  It's also awful because now we're all supposed to see the silver lining in our suffering.  Our suffering is for a greater good!  Screw that.  It seems to me that evil comes about by completely natural means, irrespective of what ultimate good will come it.  People with terminal illnesses aren't all dying for the greater good, and it's awful to suggest that they are, or that this is how it should be.

Other theodicies off the top of my head include: "Good cannot exist without evil, just as black and white cannot exist without each other," and, "All wrongs are righted in the afterlife."  I've decided to cut this post short, so I leave it to the reader to decide if there is anything evil about these theodicies.  Are they denying some particular kind of evil?  Are they being callous to people in suffering?  Or perhaps they are not evil arguments at all, just uncompelling?

Interlude: God is infinite

This is a continuation of "A few things wrong about the cosmological argument," an ongoing series. This will be a lighter post.

In William Lane Craig's formulation of the Kalam Cosmological argument, he argues that infinities cannot exist, therefore the universe began, therefore God exists.

But... surely God is infinite?  You know... actually infinite.

This seems like such a jarring inconsistency in the argument.  I give William Lane Craig more credit than that, and I'm sure he has a functional response somewhere.  Let's just pretend that I've put in the effort to find his response, and that I've been suitably convinced that of all the problems with the cosmological argument, this is not one of them.  *wink*

As many of you know I went to a Catholic high school.  It wasn't so bad.  Actually a really good high school, run by Jesuits.  But yes, it was a religious education.  Morning prayers.  Religiously infused mission statement.  Monthly services.  Retreats.  A few teachers who were Jesuits.  I was Catholic at the time, so I didn't mind. Except for the monthly services.  Mass is boring!

I also took some religious classes, learning about the Bible, social justice, and apologetics.  They didn't call it apologetics, but that's what it was.  The teacher for that class was pretty good.  He had a flair for the dramatic and comedic.  He used lots of creative teaching techniques, and frequently showed videos.  From a high schooler's perspective, showing videos in class was like the best thing ever.

I think he was a fan of William Lane Craig.  When I later encountered Lane Craig's arguments, they seemed utterly familiar to me.  I consider them to be classic apologetic arguments, though this is perhaps only my perception.  "Classic" is just whatever I learned about in high school.

Along the same lines, I also believe that one of the "classic" understandings of God is The Infinite.  God isn't merely infinite, God is The Infinite.  You can't possibly comprehend God with your finite mind!  That would be like trying to comprehend the integers, or quantum field theory!  (Silly physicists...)

I took all this stuff with just a grain of salt, but he said one thing that really bothered me.  Paraphrased from memory fragments:

God is defined as The Infinite.  Suppose there were two gods.  Then there would be two infinites.  But there can only be one infinity, because all infinities are overlapping and identical.  Therefore, there can only be one God.

Arrrgh!  I mean, I don't mind arguing for a single God.  Whatever.  But man, that is some bad math!  The set of even numbers and the set of odd numbers are both infinite but they don't overlap!  Also, neither odd nor even numbers are identical to God.  This got me to thinking along other lines, like what is God's cardinality?  Does God's power set have greater cardinality than God?

This series on the cosmological argument is in about the same spirit.  I don't particularly care if people argue for the existence of God.  I definitely don't care if people argue about whether God is single or multiple.  At least, I don't care in the abstract.  But sometimes I see these arguments, and wow!  Bad math, bad physics, bad argumental form.  If someone argues that we're justified in believing an invisible conscious being that can only touch us in our "hearts", that's okay on some level.  (Well, no, it's ridiculous.)  But when you start talking bad math?  Blasphemy!

A few people have accused me of hiding behind lots of complicated logic, math, and physics.  It's all an attempt to dazzle people in to agreeing with me.  But that's not the point.  The point is to trick you all into reading about logic, math, and physics, when you just wanted a refutation of the cosmological argument.  Ha!  Ha ha ha!

...
Tune in next time for more of...

"A few things wrong about the cosmological argument"
1. Actual and potential infinities
2. Actual infinities in physics
3. What is real?
4. The "absurdity" of Hilbert's Hotel
5. Interlude: God is infinite
6. Forming Infinity, one by one 
7. Uncertain beginnings
8. Entropy: The unsolved problem
9. Kalam as an inductive argument
10. Getting from First Cause to God  

Necessity isn't so necessary

Following my retrospective on the ontological argument, I just remembered another thing I always thought was funny about it.  According to the argument, part of God's definition is that God necessarily exists.  But why?

I can just imagine, what if there were some powerful being which shared every property with God, except for the necessary existence.  That is, this being is exactly like God in every way, except that unlike God, it only exists in our world, not every possible world.  By definition, this being would not be God.  After all, it's not the greatest being imaginable.  We can imagine a being that is greater: one that exists in every possible world, not just ours.

But a fine distinction that would be if in our world, we're being subjugated/loved/ignored by an all-powerful and vengeful/benevolent/passive being!  I don't know about you, but I'd call that thing a god, even if it doesn't quite fit the definition in the ontological argument.  As for those other possible worlds where the being doesn't exist, who cares about 'em?  Depending on who you ask, there isn't even any metaphysical significance to the other possible worlds, they're just ideas.

That leaves the ontological argument in a funny position.  The god it argues for does not necessarily have any of the properties we normally assign to a god.  But it does have this extra property, necessary existence, which I do not think is necessary to qualify a being as a god.  Does it really have anything to do with gods, or is it just a logical game, as I've been treating it?  (Also applies to nearly every other philosophical argument for gods.)

And yes, I do have some idea of how ontological argument proponents would respond.  A transcendent being such as a god must also transcend all possible worlds, thus necessary existence is an inseparable quality of God.  But if we understand other possible worlds as mere ideas, this doesn't make any sense.

An anecdote on obscurantism

You know what I realized earlier?  It's been over a year since I've talked about the ontological arguments for a god.  Perhaps it is because I think the last post I wrote summarizing the flaws in the modal ontological argument is about as clear as I can manage.

In the course of blogging about the ontological arguments for years, I attracted a handful of self-described philosophers who argued with me rather persistently, perhaps because I am one of the few people on the internet who is willing to discuss what is wrong with the ontological arguments in detail, rather than what is wrong with them in the big picture.  Of course, I found out that most of these self-described philosophers were unable to speak of logic without making the most basic of errors, even when it was irrelevant to the argument.

Buried somewhere in the archives, buried in dozens of long comments, buried in jargon and symbols, is a joke I thought so funny that it's stuck with me over a year later.  It starts like this:

What can you do to make an already obscure argument even more difficult to understand?

As I explained before, the basic modal ontological argument has just a few steps:

Premise 1: If God exists, then God necessarily exists.
Lemma: If God possibly exists, then God exists. (proof omitted, as it is irrelevant to this post)
Premise 2: God possibly exists.
Conclusion: God exists.

But what if we were to make this argument instead? The difference is in bold.

Premise 1: If God exists, then God necessarily exists.
Lemma: If God does not exist, then God does not possibly exist.
Premise 2: God possibly exists.
Conclusion: God exists.

It's the same argument, only the lemma has been replaced with its contrapositive.  Every if-then statement has a logically equivalent contrapositive statement.  For example:

If Socrates is a man, then Socrates is mortal.
Contrapositive: If Socrates is not mortal, then Socrates is not a man.

The contrapositive of the contrapositive statement is the original statement.  So whenever we make an if-then statement, we have two choices in how to say it.  Why not choose the one that is most clear and intuitive to lay audiences?  Often times the two statements can each be unclear for different reasons, but that's not the case here.  The lemma as originally stated is shorter, allows the conclusion to use modus ponens rather than modus tollens, not to mention that it flows more naturally from the proof I omitted.  So why replace it with its contrapositive?

That's more or less what one of the philosophers did.  It's a very small thing that hardly matters, but I couldn't help but think... why?  Why take these tiny steps to make an obscure argument just a tiny bit more obscure?  I asked him, and he said it was the simplest way to state the argument.  He also seemed to have trouble understanding whenever I stated the argument the other way.  I found all of this hilarious.

Some might say this is to be expected, since obscurantism is what philosophers are trained to do.  I suspect that the person simply didn't understand the argument well enough to spot a purely unnecessary step that was added in.

Full rejection with an open mind

Some months ago, PZ Myers expressed the opinion that no evidence could convince him of the existence of a god.  But this caused much disagreement, since no critical thinker likes the idea of being closed to any evidence.  And so, this debate has been echoing around the atheosphere, continuing until today.

Put me on PZ's side... sort of.  The thing is, I can imagine a few scenarios where I'd become convinced of a theistic universe.

Like if I stumbled into another universe through a wardrobe, and found myself adventuring with a talking lion-god.  Or if I had a compass which would always answer any question truthfully, including moral questions, and do it with attitude.  On the other hand, lucid dreams are fairly common, so I'd need it to be repeatedly verifiable by multiple people.  Once verified, I'd believe it, at least for the remainder of the dream or hallucination.

Mind you, these scenario would only convince me of a supernatural world, and not convince me of any specific version of supernaturalism.  In particular, they would fail to convince me of any of the major religions.  In fact, nothing could convince me of the major religions, because any such fantastic show of evidence would be out of character for them, and thus argue for something completely different.

Also, that will never happen.  Seriously, just because I can imagine it doesn't mean it's possible.

What my imagination can do is privilege a hypothesis.  The probability of a god (as conceived by religion) is extremely small, and we can't even conceive of numbers that small.  So by merely imagining the god hypothesis, we've tricked our minds into overestimating its likelihood.  And before we know it, we start thinking that evidence like this is anything other than pathetic.

In short, I agree with PZ Myers that no evidence could convince me that gods exist.  But how do I reconcile this with my strong principles of allowing for the possibility that I may be wrong?  Easy.  To convince me that gods exist, you need two things:

1. A strong philosophical argument showing that my current thinking is completely off-base (not just a little off-base).
2. Lots and lots of evidence.

Not only is the god hypothesis lacking evidence, it's philosophically unsound.  The major arguments for gods are not about evidence, they're about explaining why no evidence is necessary.  This is a fundamentally problematic argument, and fundamentally problematic attitude.  No amount of evidence will fix it.

And that doesn't mean I'm totally closed-minded.  That just means that I first need to be convinced that these flaws are not so fundamental after all.

Pascal's swindle

Blaise wanted to borrow some dough
For an investment he said would grow.
"I'll pay back tenfold,
Before fall takes hold!"
But I frowned, and let my doubt show.

To every concern he'd reply,
"How can you ignore stakes so high?"
Then he'd up the reward
Infinite! Absurd!
All I could say was, "Nice try."

Suddenly I think we should turn all arguments for and against God into limericks.  Give them the level of seriousness they deserve.  But I don't like poetry, so someone else would have to do it.

Entropy and God

Following my discussion of Boltzmann Brains, I wanted to mention its relation to yet another argument for God.

The argument goes that a low-entropy initial condition is extremely unlikely unless God exists.  Therefore, God probably exists.

(This argument is not to be confused with the argument that evolution contradicts the Second Law, a mistake once made by PZ Myers.)

My response is that we don't know that the initial condition is extremely unlikely.  It's only extremely unlikely if we assume all microstates are equally likely initial conditions.  Also, saying that we do not have an explanation is not the same as saying we do have an explanation and we call that explanation God.

Let's say that I have a differential equation that I can't solve (a common scenario in physics).  A common practice is to posit the existence of a solution and call that solution something like the Bessel Function or the Legendre Function.  But I can't posit a solution and call that solution God.  Got the distinction? The analogy isn't perfect though; in differential equations we have existence theorems, in metaphysics we do not.

Rationalist atheism does not claim that science has all the answers, but rather, that religion has none of the answers.

God as quantum observer

Today we will look at the intersection of quantum nonsense and terrible arguments for God.  How fun!

This argument is based on something Michio Kaku said to Deepak Chopra in an interview. It wasn't very clearly stated, so I'm filling in the blanks.

1. Quantum mechanics says that things do not exist unless there is an observer to observe them.
2. The universe exists.
3. There must be an omniscient conscious being to observe the universe.
4. That being is God.

 As with most arguments for God, the last step is troublesome, because there are all sorts of things the being could be.  There's no compelling reason to think that the object in question can appropriately be called "God", much less the Judeo-Christian god.  For example, why should we think that it's a single object observing the entire universe rather than multiple objects?  But since Deepak Chopra is not part of any western religion, he would skip step 4, thus avoiding the problem.  Unfortunately for him, the rest of the argument is crap too.

Rebutting the rest of the argument is really an exercise in applying the advice in my presentation on Quantum Mechanics for Skeptics.

Error #1: Elementary misunderstanding of what Quantum Mechanics actually says

According to step 1, things don't exist until we observe them.  This is incorrect.   Observation does not cause things to come into existence, it just causes a mixed state to become a definite state.  For example, if we have an electron in two places at once, and we observe its position, then it "collapses" into a single position.  The electron existed before you observed it.  The mixed state existed before you observed it.

Getting a little more advanced: Even after you observe an object, in some sense it's still in a mixed state.  This is because all states are mixed states.  So even if you mistakenly believed that "mixed state" = "does not exist", then you would be forced to conclude that nothing exists with or without observers.

Error #2: Misunderstanding observers

Step 1 just talks about observers, while step 3 suddenly jumps to conscious observers.  Observers do not need to be conscious.  The chair I'm sitting on functions as an observer.  A piece of white paper functions as an observer.  Anything with enough atoms to be visible to the naked eye probably functions as an observer.  When you realize the banality of observers, it makes much more sense to conclude that the universe is full of non-conscious observers, none of which need be omniscient.

Error #3: Misunderstanding quantum interpretations

This whole time we've been talking about observers.  But observers are specific to one interpretation of quantum mechanics, the Copenhagen interpretation.  Under many other interpretations, there is no such thing as an observer.

The thing is with quantum interpretations, they're all or nothing.  All interpretations make the same predictions.  Therefore, all of them predict God, or none of them do.  If you've formulated an argument that only works for the Copenhagen interpretation, but not for others, then this is a sign that you made an error somewhere.  I already showed errors in the argument, so the problem is resolved.

A summary:
#1: The universe does not need to be observed to exist.
#2: There are probably non-conscious observers all over the universe.
#3: A valid argument would make sense under all quantum interpretations, but this one does not.

I've also heard an argument from quantum mechanics that God does not exist.  It is also a crap argument.  I will cover it next time.