Fine-tuning and the objection from very different life-supporting worlds

I enter a room with four walls, three of them red, and the fourth white, except for a small red patch, about 1 cm² in size. I also find a dart stuck in that small red patch. (This is of course a variant of Leslie's story about the wasp and the dart.) What should I think about what happened here?

I don't know. But I know that what I should not think is that the dart was tossed in an unbiased random direction. Rather, I would instead conclude that for some reason whatever process or agency propelled the dart had both a bias in favor of this wall and a bias in favor of red. Here's, very roughly, how one would make a Bayesian model of this. There is the unbiased randomness hypothesis U. Let's give it credence 1/2. And there are four relevant strong bias hypotheses: B₁, B₂, B₃ and B₄, according to which the the dart was tossed with a strong bias for wall 1, 2, 3 or 4, respectively, as well as a strong bias in favor of red. These four bias hypotheses are prima facie roughly equally likely. The probability that at least one of them is true isn't going to be all that high, but also isn't going to be all that low. There may well be reasons beyond our ken for bias in favor of one wall or another. Let's say that the probability that some one of these bias hypotheses is true is about 1/16. Thus the prior probability of B₄ will be about 1/64, as the bias hypotheses are approximately equally likely.

But note that our evidence--the dart in red on wall 4--is much better predicted by B₄ than by U. How much better? Well, if the walls are three by four meters in size (a reasonable set of dimensions for the wall), the probability of hitting our small red patch will be one in 480,000 on U, but relatively high (depending on what we mean by "strong" in "strong bias") on B₄, let's say 1/10. Then Bayes' Theorem tells us that we have extremely strong confirmation of B₄, with posterior probability 99.96%.

Suppose we go a little more extreme. The room has 10,000 walls, each of the same size as before, (it's a giagantic myriagonal room), all but the last being completely red, with the last being white except for a small patch, with the same dimensions as before. Then what happens? Well, our uniform randomness hypothesis has an even smaller probability of predicting hitting the red patch on the 10,000th wall, though it has a very high probability of hitting red somewhere. On the other hand, now our bias hypotheses need to be split between 10,000 walls. Thus, the B₁₀₀₀₀ hypothesis will have a probability of 1/160,000, assuming the probability that some one of the bias hypotheses is true is 1/16 as before. Plugging this into Bayes' Theorem, we get 99.86%, which is roughly the same probability as before! (The reason is pretty simple: as we increase the number of walls, the prior odds and likelihood ratio go up in roughly inverse proportion, leaving the posterior odds roughly unchanged.)

This is, of course, supposed to be a response to the objection to the fine-tuning argument based on the claim that for all we know, if the parameters defining the physics were very different from what they are, life might be quite likely (this is supposed to correspond to the three red walls), even though in the vicinity of the actual values of the parameters, life-permissiveness is rare (this is the white wall with a small red patch). The reasonable conclusion is that whatever cause generated our physics had a bias in favor of both (a) life and (b) the rough vicinity of our place in the space of possible parameter values. And we have an obvious explanation of why a cause might have bias (a): the cause is a morally good agent. But bias (b) is something we may not have an explanation for. Nonetheless, even without an explanation, we can have a good Bayesian argument.

Hair, air and heir?

We sometimes say that someone said something other than they meant to. And we're quite serious about claiming that they said that thing that they misspoke. "He meant to say 'I need a new watertank' but said that he needs a new riverbank."

But suppose I mean to say: "I don't like the hair", but I misspeak and omit the "h". What did I say? Was it that I don't like the air or that I don't like the heir?

It might here depend exactly at which level the mistake is made. If it is a mere mispronunciation, we cannot say whether it was air or heir that I said I didn't like. In fact, I think in that case I didn't say either. In cases of mere mispronunciation it seems accurate to say that I said I don't like the hair but I pronounced it wrong.

But the error could have happened at the level of word choice: I might have chosen "air" in place of "hair", accidentally. I might even have seen a flash of the written representation before my mind's eye (that sometimes happens to me when I hear or say homonyms: I am one of those people for whom the written language is primary). In such cases, in principle, analyzing my brain could reveal that I in fact used "air". This is not a case of mispronunciation but use of the wrong word. Maybe.

But can a principled distinction be drawn between the case of mispronunciation and the case of using the wrong word? I have my doubts. So perhaps we should put very little weight on the "but said..." in cases of misspeaking, and not even take literally my opening example of "said that that he needs a new riverbank." We can correctly say that he uttered the sounds "I need a new riverbank", but I am not sure we can say that he actually said that he needed a new riverbank. (Almost surely he didn't assert it.)

Maybe where we want to draw the distinctions is in the success of communication. If the listener can correctly tell what the speaker meant to say, then we can say: "He successfully communicated that...". If the listener misunderstands, then we can say: "He tried to communicate that..., but instead what he got across is that..." (note that the listener's misunderstanding may or may not be the speaker's fault in general, so we may want to add: "due to misspeaking" or "due to the listener's mishearing"; or the flaw could be joint; or it could just be due to background noise.) And the final case is where the listener just doesn't get it, and then we simply say that the speaker failed to communicate his message (again, there is the question of who, if anyone, screwed up). This shifts attention away from the one-sided stuff--"What is said"--to the joint activity of communication.

The greatest discovery in the history of human biology

If one searches for "the greatest discovery in the history of biology", the top hits indicate that it was the discovery of DNA. Maybe, though I'm not sure. But least in the history of human biology, the greatest discovery surely was the discovery that pregnancy is caused by coitus. (A discovery presumably made independently in multiple cultures.)

Beginnings

An obvious definition of having a beginning is:

1. x has a beginning provided that x exists at some time but there is an earlier time at which x does not exist.

But this doesn't seem right. After all, it may well be that (a) the universe has a beginning (about 14 billion years ago) but (b) there is no time before the universe. In light of this, I've tended to say something like:

2. x has a beginning provided that x exists at some time at which it has finite age.

There is a somewhat recondite potential counterexample to (2). Suppose that the universe has an infinite past, and object x has a temporally gappy existence, such that last year x existed only for half a year (the other half is the gap), the year before x existed only for a quarter of a year, and you see where that's going. So x's current age is something like 1/2+1/4+1/8+... = 1 year. So x is one year old. By definition (2), x has a beginning. But it doesn't seem like x has a beginning.

But perhaps this case is not fatal to (2). Maybe we should agree that x has a beginning. For the relevant time sequence for saying whether something does or does not have a beginning is internal time. And x has a finite internal time past. If we say this, then we will also say that a person y that has a slowed-down past of the following sort also has a beginning: over the last year, y functioned (in all respects, mental and physical) at half of the speed of a normal person; the year before, at a quarter of the speed of a normal person; the year before, at an eighth. Thus, y experienced 1/2+1/4+1/8+... = 1 years of internal time. Yet y has always existed. While I might tolerate saying that x has a beginning, to say that y has a beginning is very awkward. (Maybe it's true, though? It's worth exploring. But for now I shall dismiss this.)

The above cases show that age in (2) must be reckoned in an external manner. To be more precise we should revise (2) to read:

3. x has a beginning provided that x exists at some time T and there is a number N of years (or other units) of time such that x did not exist more than N years (or other units) before T, where the times are reckoned externally.

Note that this is very much an extrinsic characterization of x's having a beginning. We could imagine two objects whose intrinsic careers are exactly alike, one of which has a beginning and the other does not. Take, for instance, our slowed down y from the last counterexample, and then a person who lives through y's past in one ordinary external year. The slowed down y has no beginning, but the other person does, even though their internal lives could be exactly alike. This seems unsatisfactory.

Also, intuitively, having a beginning is more about the order properties of time rather than metric properties of time. But (3) (as well as (2)) makes it be a feature of the metric properties of time.

I am not quite sure what to do with these thoughts. Maybe this: The notion of a beginning isn't actually all that natural a notion. Perhaps the natural notion in the vicinity is the notion of having a cause?

Narrowly logical necessity

My claim: Logical necessity understood narrowly
(a) violates the very-weak-Brouwer (VWB) axiom, or
(b) is not strong enough to make arithmetical facts be necessary, or
(c) makes every proposition necessarily true.

And of course a modality like that is clearly not what we mean by "necessity/possibility" or even "logical necessity/possibility". This post is an expansion of the brief argument here.

Now it's time to argue for my claim.

By VWB, I mean the following thesis:

• At least one proposition is necessarily possible.

According to the Brouwer axiom (and, a fortiori, given S5), every true proposition is necessarily possible. But VWB is much weaker than that. (In fact, it follows from Axioms T and Necessitation: given T, for any tautology p, Possibly(p) is a theorem; but by Necessitation, Necessarily(Possibly(p)) follows.) It is about as uncontroversial a modal axiom as one can get.

Now to prove my claim. 

A proposition p is narrowly logically possible provided that a contradiction cannot be proved from p while p is narrowly logically necessary provided that p can be proved (within the logical system that defines narrowly logical modality). 

Now suppose VWB is true for our narrow logical necessity. Then it is necessarily true that p is possible for some p. I.e., it can be proved that no contradiction can be proved from p. But if no contradiction can be proved from p, then our logical system is consistent: in an inconsistent classical system, a contradiction can be proved from every claim. And this conditional can be proved. 

Hence, given VWB, it follows that one can prove in the system that the system in question is consistent. It follows by Goedel's Second Incompleteness Theorem that either (i) the system is inconsistent or (ii) the system is not strong enough to contain arithmetic. In case (i), every proposition can be proved (this is classical logic, so we have explosion) and hence we have (c) and in case (ii) we have (b).  So if VWB is true, we have (b) or (c). Thus, in general, we have either (a) or (b) or (c).

Miracles, visible and hidden

We would expect God intervene a lot in our world to prevent misery. But there are at least three things that could give God reason limit his interventions:

1. The value of our trusting in him without being overwhelmed by the obviousness of his interventions.
2. The danger that we would end up counting on miracles, which would undermine our motivations for helping others.
3. The intrinsic value of the world proceeding according to its natural course.

Notice, however, that considerations (1) and (2) only apply to evident miracles, though (3) applies to all. Thus while God always has general reasons to refrain from miracles, those reasons are stronger when the miracles would be evident. Thus if the reasons God has against evident miracles--namely reasons (1), (2) and (3)--are sometimes overcome, we would expect the reason God has against non-evident miracles--namely reason (3) alone--to be overcome as well. (This is not a conclusive argument, because there are also special benefits to evident miracles, namely that God's message spreads.) Hence if there are evident miracles, we would expect that there would likely be hidden miracles as well. I am imagining here cases like this. Francis has early, undetectable cancer. But God has things for Francis to do, plus God doesn't want Francis to suffer, and so he miraculously stops the cancer before anybody knows about it. (Transsubstantiation is also an invisible miracle, but I am not counting it as hidden, since God has made it known that it occurs.)

Given (1)-(3), we could imagine God doing something like this in response to the misery of the world. First, he draws a rough limit n_v on how many visible miracles can happen without seriously endangering (1), (2) and (3). That number might be relatively small. Second, he draws a rough limit n_h on how many hidden miracles can happen without seriously endangering (3). That number is likely to be much larger. So now we have two questions very relevant to the problem of evil:

4. Do we have reason to think there are fewer than about n_v visible miracles?
5. Do we have reason to think there are fewer than about n_h hidden miracles?

There is significant evidence of visible miracles. Not in great numbers, but it is plausible that if the number was significantly higher, then (1) and/or (2) would be endangered. So the answer to (4) is negative. And I think the answer to (5) may be negative as well. For of course we don't know how many hidden miracles there are! We can't do controlled experiments to see how many people and animals develop cancer in the absence of God. Of course, this isn't an answer to the problem of evil. But it's a step in that direction.

(Some people wouldn't count hidden miracles as "miracles", because miracles must be wondrous--they must speak of God and his ways. That's just a terminological point as far as this post is concerned.)

Annoyances

A week or two ago, Google stopped supporting the authentication protocol used by my commandline tool for posting posts to my blog. I downloaded a new set of commandline tools for posting to blogger and integrated it into my script. Turns out the new tools used OATH, which Google coincidentally phased out in favor of OATH2 a few days later. :-( I still need to find the right tools. The problem is that (a) I prefer editing my posts from the commandline with vim than typing them into Blogger's web interface, and (b) I had a lot of simplified TeX-like math processing in my posting script as well as other conveniences, say for numbered propositions. I considered switching to MathJAX, but MathJAX is annoyingly slow--it first shows the web page with the TeX codes, and then redraws. Moreover, the font doesn't match most of my text. So I reverted from MathJAX. I am hoping I can adapt the b.sh package to do the job--from its source code it looks like it does OATH2.

Why so few kinds for so many particles?

There are something like 10⁸⁰ individual particles and only something like 10² kinds of particles. It seems an incredible coincidence; so many particles, all drawn from so few kinds, even though surely the space of metaphysical possibility contains infinitely many kinds. It's like a country all of whose citizens have names that start with A, B or M.

But perhaps one could explain this by the massively multilocated particle hypothesis (MMPH), namely that to each kind there corresponds only one individual, but highly multilocated, particle (Feynman proposed something like this)? It isn't surprising, after all, if all the names of the villagers in a village start with A, B or M when there are only three villagers.

Still, MMPH does bring in a new mystery: Why are there so very few particles? But perhaps that is a less pressing question?

Lifelikeness of fractals

It's well-known that fractal-type objects can be quite lifelike and easy to generate. I've been scripting Minecraft with Python, in preparation for teaching this to gifted middle- and high-schoolers this summer, and wrote a simple 3D turtle graphics class with pitch/yaw/roll support. Like many kids of my generation, I did 2D turtle graphics programming with LOGO in school, but a 3D turtle just has a load of new possibilities. In particular, the 3D turtle allows for nice 3D fractal generation.

Instructions on how to do this stuff in Minecraft are in my Python coding for Minecraft instructable.

It was very easy to generate the following fairly lifelike tree with a simple bit of recursive code and some randomness.

An L-system does a pretty lifelike job even without randomness (using rules from geeky.blogger):

There is something glorious about a world where structures are mirrored on multiple levels. It makes the different parts and levels of the world be like a work of art, with themes and intertextuality.

You're not killed by the fusion of the Grim Reapers

A grim reaper (GR) is a device that activates at a pre-set time. It checks if Fred--the victim--is alive. If he is, it kills him. If he isn't alive, it does nothing. For the Grim Reaper Paradox, we're supposed to imagine one GR set for 12:30, another for 12:15, another for 12:07.5, and so on. Before each time for which a GR is set, there is an earlier one. But Fred is alive alive at 12:00. Paradox ensues when we notice that Fred must be dead at 12:30 (else that 12:30 GR would have killed him), but no GR could have killed him, since if he were alive at its activation time, he would have been alive when the previous one activated, and hence would have been killed then at least.

John Hawthorne has claimed that Fred is not killed by any one GR, but by them altogether. More precisely, Fred is killed by their mereological sum.

Here's a gruesome way to see the problem with this solution. We can number the GRs in reverse: the 12:30 GR is number 1, the 12:15 one is number 2, and so on. Then suppose that the odd-numbered GRs kill by decapitating and the even-numbered ones kill by stabbing in the heart. Given the setup, Fred is either decapitated or stabbed in the heart but not both. But which one?! If he were decapitated, he would have been first stabbed. If he were stabbed, he would have been decapitated before that.

A quick route from mathematics to metaphysical necessity

The Peano Axioms are consistent. If not, mathematics (and the science resting on it) is overthrown. Moreover, it is absurd to suppose that they are merely contingently consistent: that in some other possible world a contradiction follows logically from them, but in the actual world no contradiction follows from them. So the Peano Axioms are necessarily consistent. But they aren't logically necessarily consistent: the consistency of the Peano Axioms cannot be proved (according to Goedel's second incompleteness theorem, not even if one helps oneself to the Peano Axioms in the proof, at least assuming they really are consistent). So we must suppose a necessity that isn't logical necessity, but is nonetheless very, very strong. We call it metaphysical necessity.

Preference structures had by no possible agent

Say that a preference structure is a total, transitive and reflexive relation (i.e., a total preorder) on centered worlds--i.e., world-agent pairs <w,x>. Then there is a preference structure had by no possible agent. This is in fact just an easy adaptation of the proof of Cantor's Theorem.

Let c be my own centered world <@,Pruss>. We now define a preference structure Q as follows. If agent x at world w, where <w,x> is not the same as <@,Pruss>, prefers her own centered world <w,x> to c, then we say that c is Q-preferable to <w,x>; otherwise, we say that <w,x> is Q-preferable to c. Then we say that all the centered worlds that according to the preceding are Q-preferable to c are Q-equivalent and all the centered worlds we said to be less Q-preferable than c are also Q-equivalent. Thus, Q ranks centered worlds into three classes: those less good than c, those better than c and finally c itself.

But now note that no possible agent has Q as her preference structure. First of all, I at the the actual world do not have Q as my preference structure--that's empirically obvious, in that the worlds do not fall into three equipreferability classes for me. And if <w,x> is different from <@,Pruss>, then x's preference-order at w (if any) between c and <w,x> differs from what Q says about the order.

So what? Well, I think this provides a slight bit of evidence for the idea that agents choose under the guise of the good.

Limitations, art and evil

It's a standard thought that art thrives on limitations. These may be imposed by the technical capacities of the medium (I was reading this today) or by repressive authorities (think here of communist-era Eastern European literature), or they may be limitations imposed by the artist or her artistic community. In this regard art is like sport, where there are rules that constrain one from what might otherwise be thought of as efficient ways to achieve the goal, such as using a car to "run" a marathon.

Let's not think of God as setting out to create the best possible work of art. The idea of a best possible work of art divorced from model on which God "first" institutes for himself a set of limitations which both constrain and constitutively make possible a particular kind of artistic achievement, and "then" tries to produce the best work within those limitations. For instance, among these limitations there might be a small number of laws of nature and of fundamental kinds of things (compare pixel artists who limit their palette), perhaps with a limited number of self-allowed deviations from the laws. But in addition to such "technical" restrictions, there might be restrictions coming from the content of an artistic vision: what kind of thing it is that God is trying to say in the work.

If we have this sort of a model, then two things happen. The first is that the worry that a perfect being couldn't create since there is no best of all possible worlds disappears. For it is not so hard to think that within certain genre constraints there could be an optimal work (after all, some genre constraints may constrain a work to a finite size; see also this).

The second is that some progress is made on the problem of evil--though by no means is this a solution. For we can answer some "Why did God not do it this way instead?" questions by pointing to the self-imposed artistic limitations. Nonetheless, caution is required. One is very uncomfortable with the thought of God allowing horrendous undeserved suffering for art's sake. Though maybe if the sufferers eventually fully appreciate the art...?

Epistemology, anthropocentrism and Natural Law

The central problem for Bayesian epistemology is where we get our prior probabilities from. Here are three solutions that have something in common:

1. Set our priors based on our common intuitions as to the priors.
2. Set our priors in such a way as to best model our human everyday and scientific inductive reasoning.
3. Use Solomonoff priors, using an idealization of the human mind as the Turing machine.

The obvious commonality in all three solutions is that they are anthropocentric. They just take for granted that our human ways of reasoning are in some way doxastically normative.

Is that bad? I was once teaching Philosophy of Love and Sex, and one of the students complained that the ethics we were talking about was only applicable to humans. I think his worry was that the ethics wouldn't apply to aliens. That's a Kantian concern. But it's off-base: of course sexual ethics will be different for asexually reproducing plasma beings. However, while it's off-base for sexual ethics, it sounds very reasonable to say that our epistemology should apply to all agents, or at least all embodied discursive agents. Unfortunately, there is little hope of solving the problem of priors subject to that assumption.

Let me try to soften you up in favor of anthropocentrism about priors with an ethics analogy. If sharks developed rationality, we wouldn't expect their flourishing to involve quite as much friendship as our flourishing does. Autonomy and friendship are both of value, and yet are in tension, and we would expect different species to resolve that tension differently based on the different ways that they are characteristically adapted to their environment. This is, indeed, an argument for a significant Natural Law component in ethics: even if values are kind-independent, the appropriate resolution of tensions between them is something that may well be relative to a kind.

But there are similar kinds of tensions in the doxastic life. For instance, there is a value to quickly grasping patterns in nature and generalizing them and a value to being more doxastically cautious. We can imagine that agents with one characteristic way of life might flourish in their doxastic lives better if they are eager patterners—they see three tigers and conclude all tigers are dangerous—and agents with a different characteristic way of life might do better to be more cautious in generalizing. Moreover, the appropriate resolution of the tension is likely to be dependent on the subject matter.

This particular tension is nicely modeled within the priors: the particular balance is determined by how the priors are for "nice patterns" (and what counts) versus how the priors are for "mess".

So one way of living with the anthropocentrism of proposals like (1)-(3) (which are not, of course, all of a piece: there is a spectrum there, with more and more idealization as one goes down the list) is to accept a Natural Law epistemology. For each kind of rational agent, there is a natural way for the minds of agents of that kind to think. This natural way yields decisions between competing doxastic values. In a Bayesian setting, this is most prominently embodied in the choice of priors. There are, literally, such things as natural and unnatural priors for a rational agent of a given kind.

There is, however, something disquieting. What about truth? Don't we want our reasoning to get us to truth? What if our kind-relative norms don't get us there?

Well, first of all, we want to both get to truth and avoid falsehood. As William James famously notes, the two desiderata are in tension—you can get all truth by believing every proposition and you can avoid all falsehood by believing none—and there are different ways of resolving the tension. James's own solution was to relativize to the individual. But my Natural Law suggestion is that the resolution of the tension is to be relativized to the kind (and perhaps subject matter). (There may be some absolute constraints, of course.)

But the worry remains. What if our priors are just not conducive to getting to the truth? (It won't help to say that in the limit we get convergence, because we don't want to wait for the limit!) What if our epistemic procedures, appropriate as they are to our kind, fail to get at the truth?

After all, we can imagine eager pattern identifiers in Humean worlds, where the patterns they identify are always spurious, and cautious agents in extremely nicely arranged worlds who keep on missing out on the order around them, as well as less extreme cases.

Here is where theism can help. God put us, with the natures we have, in a certain environment. It is reasonable to think that there would then be a fit between truth-conduciveness (and falsehood-avoidingness) and the characteristic ways of reasoning normative for our kind. This role for God in Natural Law epistemology is somewhat similar to the role of God in Kantian ethics. Kant has this deep concern: "What if in fact doing the right thing doesn't lead to happiness?" To feel the concern, make it a universal concern: "What if everybody's doing the right thing didn't actually lead to anybody being happy?" And although he thinks in a scenario like this people should still do the right thing, despite the cost for everyone, he thinks that we should postulate God to rule out such unhappy thoughts.

I suppose one might hope that evolution could help relieve the worry. Our characteristic doxastic ways of life evolved for our environment, so we would expect some fit between our priors and our environment (interesting question: if our individual priors have evolved, are they still priors?). I agree, but only in a limited way. The evolutionary argument is only going to help in those areas of doxastic life that were important to our fitness where the ways of life were evolving. It's not going to help us much in modern physics or metaphysics. It will help with those areas of doxastic life where the level of abstraction and complexity is much less.

While in the above I held out for a Natural Law epistemology (or metaepistemology), I could also see someone defending a Divine Command epistemology.

Quick puzzle

You will be paid oodles of money if and only if your next decision is irrational. What should you do?

(There are standard cases in the literature where you lose a lot unless you become irrational, say by taking an irrationality potion. The case at hand differs significantly from those, because it concerns the next decision. You can't just rationally decide to take an irrationality potion, because doing so would constitute your next decision and that would be rational.)