The Sledgehammer Approach

First get a test subject, give her a sledgehammer and put a blindfold on her.  Then take some eggs and place them in a circle, at about sledgehammer distance from your test subject.

Spin the test subject around a bit and then tell her to swing down the sledgehammer onto the ground in front of her.  Take the sledgehammer from her, lead her carefully away from the test area, making sure she doesn't trip over any of the eggs.  Then you can remove the blindfold and give her a cup of tea and a biscuit.

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Now you ask her to explain her position regarding whether or not she hit an egg with the sledgehammer.

If I were the test subject, the first thing I'd do is check my clothes to see if I had any egg fragments on me.  Let's say there isn't, or at least she can't see any egg product on her.

Does she know that she hit an egg?  Note that she doesn't have any evidence that she did.

Does she know that she didn't hit an egg?  Note that the absence of evidence is not evidence of absence.  She could have missed it, or hit it in such a way as to not get spattered.  She doesn't have definitive evidence that she missed but she might think that she knows that she didn't hit an egg.  This would make her gnostic with respect to egg-missing.  She might, however, given the impossibility of evidence, remain agnostic on this issue.

Note here that there are three possible epistemic positions she could adopt:

She could claim to know that she hit an egg

She could claim to know that she missed all the eggs

She could state that she doesn't know whether she hit or missed

You could also ask her what she believes with respect to hitting or missing eggs and she could arrive at three similar doxic positions:

She could believe that she hit an egg

She could believe that she missed all the eggs

She could state that she doesn't believe anything with respect to whether she hit or missed

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I'm tempted to replace epistemic with gnoseic, as distinct from gnostic, but it's not a term that has taken off and there's no firm definition of the word from which I have taken it (gnoseology).  Basically I want to distinguish between that which is known (or thought to be known) and that which is believed (including that which is known to be believed and not thought to be known).  There are two Greek roots that could be used for "regarding beliefs": doxa (common belief, opinion or acceptance) and pistis (confidence, faith or trust).  There are problems with both.

Doxology means "a liturgical formula of praise to God" rather than being related to belief and pistiology means "the branch of theology which treats of the place and authority of faith" or "doctrine concerned with faith" while pisteology means "the science of faith"(!) and both of these latter terms are far too close to epistemology to avoid confusion, and replacing epistemology with gnoseology has its own problems.  Just work with me here and accept that by epistemic I mean "about knowledge" and by doxic I mean "about belief (as distinct from faith)".

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So, there are a limited number of workable combinations:

Believing (hit) and claiming to know

Believing (hit) and accepting a lack of knowledge

Not believing either and accepting a lack of knowledge

Believing (miss) and accepting a lack of knowledge

Believing (miss) and claiming to know

It seems unreasonable to claim to know that one hit an egg or missed all of them while also claiming to believe the opposite.  This sort of claim works only as hyperbole, an expression of surprise or shock (as in "I know I just won the lottery, but I have to keep pinching myself.  I still don't believe it!")  If you don't believe, you can't know - even if you use Plantinga's modified epistemology you are still talking about warranted true belief instead of the gold standard, justified true belief.

Similarly, it seems unreasonable to believe something and claim to know the opposite.  You might express such a contradiction as some sort of patriotic or loyal hyperbole - "I believe Moldovia is the best country in the world, although I know it isn't really" or "I believe my baby is the most beautiful ever, although I do know he looks like a miniature, bald version of the later era Elvis", but in reality you believe that Moldovia is, at best, only the second or third most liveable nation in the world and that your child is has been beaten mercilessly with the ugly stick.

Let's put this in a table to make it clear what I mean:

	Know Hit	Not Know	Know Miss
Believe Hit	YES	YES	NO
Not Believe	NO	YES	NO
Believe Miss	NO	YES	YES

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It should be pretty clear that I am talking here about pure agnosticism, weak theism/atheism and strong theism/atheism.  Theists of a certain stripe struggle understanding that not believing in a god and believing that a god does not exist is not the same position (which they label as an epistemic position rather than a doxic position).  Hopefully those theists can understand that the positions work perfectly well if we are talking about smashing (or not smashing) an egg with a sledgehammer.  Let's modify the table, so that we make the hittists theists and the missists atheists:

	Know God	Not Know	Know No God
Believe God	YES	YES	NO
Not Believe	NO	YES	NO
Believe No God	NO	YES	YES

Modifying again to ram the point home:

	Know God	Not Know	Know No God
Believe God	Strong, Hard or Gnostic Theist	Weak, Soft or Agnostic Theist	NO
Not Believe	NO	Pure or Adoxic Agnostic	NO
Believe No God	NO	Weak, Soft or Agnostic Atheist	Strong, Hard or Gnostic Atheist

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We can make another modification to the second last table to highlight a confusion that might lie behind the theists' problem with the spectrum of atheistic non-belief (and no, I am not going to try to introduce apatheism into the mix.  How much, if at all, you care about the question would introduce another axis - want god to exist (to some extent), not care, want god to not exist (to some extent)):

	Know	Not Know	Know Not
Believe	YES	YES	NO
Not Believe	NO	YES	???
Believe Not	NO	YES	YES

In informal English, it is possible to say that you know that there is no god and also say that you do not believe.  This can trip you up when you look at "Not Believe-Know Not" as a cell - it actually seems vaguely possible.  I don't think this confusion would occur in most other languages because they are structured differently and that it's the auxiliary verb (do) that leads to this confusion in modern English.  Strictly speaking, to know anything you must believe it (albeit not necessarily in the religious sense of believing).  What we also see here is an equivocation on the word "believe" - if you are working with a strong, faith-like definition of "believe" then this cell would be workable.  I, on the other hand, am using "believe" in a purely functional sense - meaning something along the lines of "hold to be true".

I am reasonably sure that a substantial proportion of people, when trying to fill the table in this last form, would at least pause for a moment when filling in the "Not Believe-Know Not" cell, some would fill it with YES and then argue that they are right.  However, I reckon that the very same people would have no problem filling the table out correctly when expressed in terms of hitting or missing an egg with a sledgehammer - because the definition of belief as approaching faith is not triggered in that example.

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Some might wonder how this construct might align with Dawkin's spectrum of belief.  That spectrum is more probabilistic and the table would look something like this:

	Know	Not Know	Know Not
100% god	YES	NO	NO
75-99% god	NO	YES	NO
51-74% god	NO	YES	NO
50% god	NO	YES	NO
25-49% god	NO	YES	NO
1-24% god	NO	YES	NO
0% god	NO	NO	YES

[I had written something here about the "0% god - Not Know" and "100% god - Not Know" cells, but it wasn't particularly important and I added confusion by misusing the term certainty.  Hopefully people can work out why those cells change to NO so I don't really need to attempt a fumbling explanation.]

Note the difference in strength between what could be called "strong atheism" and "weak atheism".  It can be as little as 1% or 0.01%.  Alternatively, you could call "1-24% (chance of) god" strong atheism, or use some other low probability range if you prefer, the higher values would then be weak atheism and the zero point zero recurring percent would be reserved for people that we call "idiots".   (Similarly anyone claiming a 100% certainty with respect to god would be an idiot, but we are generally encouraged to be more diplomatic when this sort of unrealistic certainty is expressed by a believer.)

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Hmmm.  Anyone hungry for an omelette?

A Snide Comment

Remember I wrote that Max Andrews is a philosopher who started out with a Bachelor of Science in Religion from the Liberty University in Virginia (in The Four Mechanisms of the Ignopolypse)?  Unfortunately, I can't resist making a snide comment (or rather another snide comment).

In his thesis, he talks about a mechanism associated with the generation of life permitting universes that cannot be explained, namely the "conversion of the energy of the inflation field to the normal mass/energy we find in our universe", which he chalks up to "Einstein’s E=mc² + the coupling of the inflation field and the matter fields".  He then launches into a potted history of relativity.

Strangely, he uses an article from the Indian Academy of Sciences' journal Resonance that seems to have been commissioned to celebrate the anniversary of Einstein's golden year of 1905, rather than a text book on relativity.  But this article doesn't seem to be the source of Andrews' problems.

Towards the end of this history, Andrews writes:

The square of the speed of light is called the constant of proportionality. It does the job of converting from the unit in which mass is expressed to the unit in which energy is expressed.¹⁵⁵ With this, Einstein’s Special Theory of Relativity was born.

You what?

"Constant of proportionality" is a general term, the gravitational constant G is a constant of proportionality, Planck's constant h is a constant of proportionally, Coulomb's law constant k is a constant of proportionality and, yes, if you are interested in E and m, then c² is a constant of proportionality.  What it isn't is the constant of proportionality.

I refer in other articles to mass-energy (as in "concentration of mass-energy" and "concentrations of mass-energy") because mass and energy are sort of the same thing, and their units are mediated by the relationship between time and space (which is what the speed of light c represents).  But it's a little inaccurate to suggest that the speed of light squared somehow has the duty of converting mass to energy (or the inverse square of the speed of light has the duty of converting energy into mass).  The energy is there in the mass all the time, any conversion of "mass to energy" such as seen in the nuclear reactions in a power station or nuclear explosion is not really a conversion but a release of energy which was previously bound in one form so that it may be expressed in other forms (kinetic and thermal energy plus some radiation).

Perhaps I am being finicky here, but it seems to me that the issue that Andrews is alluding to is that we may not understand the mechanism by which energy from the inflaton field (not "inflation field") could end up being expressed as mass - what might cause this energy to be bound up as mass.  One thing that seems pretty clear is that not all energy ends up as "normal mass/energy" and in fact the indications are that the majority of it ends up as dark matter, dark energy and maybe even (as quite recently suggested) dark radiation.  ("The standard model of cosmology indicates that the total mass–energy of the universe contains 4.9% ordinary matter, 26.8% dark matter and 68.3% dark energy." - wikipedia article on Dark Matter)

And then we get to my snide comment.  Andrews' finishes the sequence with the words "With this, Einstein’s Special Theory of Relativity was born."  Um, excuse me, but that is plain wrong.  The E=mc² relationship might well fall out of Special Relativity (see On Time), but the "birth" of Special Relativity was very much in evidence in a different paper, On the Electrodynamics of Moving Bodies, and not in the article Does the Inertia of a Body Depend upon its Energy-Content? which was written almost three months later.  And the phrasing E=mc² didn't appear until 46 years later.

These sorts of errors are perhaps what one would expect when one's Bachelor of Science is taken in Religion.

Ah, it feels good to get that out of my system.

The Most Valuable Solution to the Four Doors Problem

A few days ago, I posted the Four Doors Problem.  So far, no-one has had a go at solving it.  I'm guessing that some would be far happier critiquing my solution anyway, so here it is!

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Most will recognise the problem as a complicated variation of the Two Doors Problem, for which a solution is readily available.  What are less readily available are the two other solutions to the two doors problem.

Briefly, in the two doors problem, there are only two doors which may be sentient and able to answer questions themselves, or there may be two identical guards who answer on behalf of the doors.  A classic version has one guarding the door to hell, while the other guards the door to heaven.  The heaven guard always tells the truth, while the hell guard always lies, but other than that you know nothing meaning that there are no visual clues as to which is which.  You're given one question to find out which door leads to heaven (which contains magic chocolate, or some such nonsense).

The standard answer is to ask one of the guards which door the other guard would say led to hell.  Then you go through that door, because it goes to heaven.

Another option is to ask a guard whether both doors lead to heaven.  If she says yes, then she's the hell guard so take the other door.  If she says no, then she's the heaven guard so take her door.  This relies on the additional fact that the liar guards the hell door.  If you were told no more than one lies and one tells the truth, then you'd still not know which door was which but you would be able to ask another direct question to the truth telling guard (and thus avoid being stabbed by the third guard).

The third option would risk the ire of any third guard because it's a bit tricky.  You could ask "if I had asked you earlier which door leads to heaven, which door would you have pointed to?"  The heaven guard always tells the truth, so she would have pointed to the heaven door.  The hell guard always lies, so will lie about having lied before and thus will point to the heaven door.  So no matter which guard it is, you can go through the door pointed to.

The benefit of this method (third guards aside) is that not only do you no longer need to worry about whether the guards are located in front of "their" door, the solution will work even if the heaven guard had a night off and an off-duty hell guard had taken her place (so that both lie), or vice versa (so both tell the truth).  In other words, it's the right solution if the problem were to be rephrased as "you only know that guards come in two varieties, they either tell the truth all the time or lie all the time, but you don't know what the guards in front of you are, both truth-tellers, both liars or one of each".

So, with four doors and the type of responses that Monty can give, we can ask a similar question - albeit a little more complex:

If with a third question I asked you which door would you have indicated in response to a first question had I - with the first question - asked which door had the money behind it?

Breaking it down a little:

If it's the truth-telling platform, then the door pointed to will have the money behind it because Monty will tell the truth about having told the truth before.

If it's the lying platform, then the door pointed to will have the money behind hit because Monty will lie about having lied.  Now, this needs some clarification, if you asked the first question, rather than asking about the first question, then Monty would have three options to lie.  Then, when answering the third question, he would have three options to lie (by pointing to any door other than the one he had actually pointed at).  But in the abstract, having not actually pointed at a door with an actual first question, Monty is obliged to lie and thus he is obliged to point at the door that he could not have pointed to if there had been a first question.  (This does assume that given an obligation to lie, Monty will choose from the three goat doors at random, making it a little bit equivalent to the Monty Hall problem where he has two goat doors to choose from.  I've imported that assumption on the basis of indifference.  For the purposes of this solution, if you don't like that I've made the assumption, just work on the basis that I had clarified that, when lying, Monty picks a goat door at random.)

I specify first and third questions to account for the flip-flop platform (lie-truth-lie-truth, etc).  In both these questions, Monty will be in the same phase, either telling the truth both times or lying both times and he will therefore point to the money door, following the same logic above.

The last platform introduces an issue because Monty's response will be random and the tricky nature of the question posed to him will not affect how random his response was.  Fundamentally, he's being asked to point at one of four doors, he'll choose one totally at random.  So there's only a 25% probability of his pointing at the money door.

If you asked the question twice, on two different platforms and the answers were the same, you'd have certainty that the door indicated was the money door and you'd have locked in $250,000.  However, you'll only get this if you avoided the random platform (or, while on the random platform, Monty pointed at the money door).  If you didn't avoid the random platform, and Monty didn't point at the money door, then you'll have two different answers and not know which was which, so you're left with a 50-50 decision which has less value (due to risk) than asking another question to be certain of $125,000.  While I've not calculated it, the value of asking two questions (and potentially three) is less than just taking a punt without asking any, namely $250,000.

The optimum is achieved by asking one question and one question only, on one platform only, then taking the door indicated.  25% of the time it'll be the right door because Monty was on the truth-telling platform, 25% of the time it'll be the right door because Monty was on the lying platform, 25% of the time it'll be the right door because Monty was on the flip-flop platform (and he was restrained to either telling the truth twice or lying twice) and 6.25% of the time it'll be the door because Monty was on the random platform and he pointed at the right door at random (25% of 25% = 6.25%).

So the probability that you take the correct door, with half a million behind it, is 81.25% - a value of $406,250.

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If there is a more valuable solution, I'd be interested to hear it.

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If you want to know for certain that you're picking the right door, then you must be prepared to ask the question three times while Monty stands on different platforms although you might only need to ask twice.  When you have two answers that are the same, then you've got the right door - with $250,000 as the prize, about 81.25% of the time.  18.75% of the time Monty will answer randomly and not point to the right door, which means that you'll get two different answers, one right and one wrong, and you will not know which is which.  So you need to ask again with Monty standing on a third platform.  Whichever door is pointed to again is the right door, but after three questions you'll only be winning $125,000.

So, if you are after certainty, you need to ask a maximum of three questions and a minimum of two.

If there's a better way, getting certainty with two questions for example, I'd be keen to know (but I doubt that it's possible with the potential for random answers).

Unbelievable! It's Max Andrews!

In his Unbelievable discussion, Max Andrews stated that he has noted in himself, when shaping apologetic arguments, the tendency to beg the question of god.  You may recall that begging the question is, at base, a fallacy of circular reasoning, even though the circularity might be subtle.  It appears that he and I might have some alignment here in that I have noticed, on the part of apologists, the exact same tendency.  That said, while I agree that many arguments for god presume god, or something attributed to god, I would not, however, have necessarily included fine-tuning based arguments in the list.

These are the apologetic arguments that, off the top of my head, include some element of begging the question:

Moral Argument

Ontological Argument

Argument from First Cause

Argument from Resurrection

Argument from Revelation

Argument via Inference to Best Explanation

A good example of the sort of apologetic question begging involved appears in this video by Dave S at theglobalatheism, which concludes that the ontological argument proves no more than that if god exists, then god exists (which is the law of identity: if X, then X, or X=X).

Another video by Dave S, on the argument from resurrection, also hints at begging the question, pointing out that to get a high-ish likelihood that god exists (Pr≈0.5) you need to assume that claims that god raised people from the dead were true.  Think about this for a moment.  If you assume, up front, that claims that your god raised people from the dead are true, this necessitates the assumption that your god actually exists.  You should find this a little odd, because it would mean that, rather than adding support to the notion that the apologist's god exists, the resurrection argument reduces the likelihood of that god's existence.  This seems odd to me.

Anyway, what about the fine-tuning argument.  Does this include an element of begging the question?

Perhaps there is.  The fine-tuning argument first suggests that the universe is fine-tuned (arguing the reality of fine-tuning is a speciality of Andrews' partner in the discussion, Luke Barnes).  But that's insufficient to show god.  To show god, the apologist has to go one step further and argue that nothing else explains fine-tuning other than their god.

This is where a question can be comprehensively begged, on two levels.

Firstly, there is the definition of "god".  As indicated above, it can be argued that the ontological argument can fail based on the fact that it tries to define a god into existence.  The fine-tuning argument fails on the same grounds, because of begging the question.  The apologist pressing this argument is effectively trying to define their god into existence via the claim that their god can do anything that is necessary to prove that existence.  Their god can make a universe that looks like ours and seems incredibly unlikely, but you can bet your bottom dollar that if it ever starts to look like the universe is incredibly likely, the same apologist will argue that their god can make an incredibly likely universe as well.  Or an eternally inflating multiverse.  Or a quantum multiverse.  Therefore, their argument collapses into "if our universe were created by the sort of being that can make universes of the sort that we live in, then our universe was created by the sort of being that can make universes of the sort that we live in, let's call it god".

Secondly, there is the sheer unlikeliness of our universe versus typicality.  In this argument, there is no assumption of a god followed by a conclusion of a god, it's more an assumption of atypicality followed by a conclusion of atypicality which feeds into a "this is so surprising, therefore god" argument.  Ignoring for a moment the fact that multiverse theory arises totally independently of counter-apologetics, if we were to posit a multiverse to solve the unlikeliness of our universe, the counter-counter-apologists then rush in to say that, if there were many different kinds of universes in this multiverse, then our type of universe would be atypical - and, therefore, we have an obligation to explain why we are in this atypical universe.

This atypicality is often expressed in terms of a Boltzmann multiverse, the sort that fails on the basis of Boltzmann brains.  In a Boltzmann multiverse, each universe is a fluctuation of order in a vast, enduring region of comprehensive disorder.  In such a regime, small regions of order would be far more likely than regions like our observable universe and we would indeed be unusual.  However, all this does is put a kibosh on a Boltzmann multiverse.  But no-one is seriously positing a Boltzmann multiverse (I will do so semi-seriously in a later article though).

Nevertheless, when people regurgitate WLC's argument against the multiverse (including people like Barnes) they assume a Boltzmann multiverse and conclude that a multiverse is impossible.  Thus they are arguing that an impossible form of multiverse is impossible, which is trivially true.

The other element of the argument is based on the assumption that we must be typical.  This seems similar to the argument that leads to (counter-apologetically derived) multiverses, but there is a subtle but important difference between very highly unlikely and very highly atypical.  When you bring the multiverse to bear on the problem, the likelihood of a very highly unlikely universe existing increases, towards 100% as N (the number of universes) approaches infinity.  At the same time, as N approaches infinity, the number of atypical universes increases, the number of distinct chances of being atypical increases.

Think of it in terms of jellybeans with the probabilities being indicative, rather than rigorous.  If one in every 100 jellybeans was atypical, then with a sample of 100 jellybeans, you would expect on average that one would be atypical and there would be only chance to be atypical.  If there were 10,000 jellybeans, then we'd expect there to be 100 chances to be atypical.  Agreed, there would still be a lot more typical jellybeans than atypical ones, but the increased number of atypical jellybeans would increase commensurate with sample size.  If we introduced a filter that only permits a certain shape of atypical jellybean to pass through, and used a very large, approaching infinite sample size, then so long as the shape is possible, then we should not be surprised to see one.

This is the situation that we find ourselves in.  We are in a universe which, although atypical, must be the way it is for us to find ourselves in it - because it's precisely the sort of universe in which we could develop as persistent intelligent life (rather than Boltzmann brains).

In other words, our atypicality is very much the point, rather than being a failure of the multiverse to explain our fortuitous existence.

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Curiously enough, Dave S includes, as further reading for the resurrection video, which was posted a year ago, both my Sweet Probability and Max Andrews' dissertation on Bayes Theorem and the resurrection.  Some nice recognition there for both of us, even though I at least hadn't noticed it.