Fine-tuning of both physical and bridge laws

A correspondent pointed me to a cool paper by Neil Sinhababu arguing that the theist can’t consistently run a fine-tuning argument on which it is claimed that it is unlikely that the constants in the laws of physics permit intelligent life, because if God exists, then for any constants in the physical laws God can make psychophysical bridge laws that make sure that there is intelligent life. By choosing the right bridge laws, God can make a single electron be conscious, after all. Thus any set of constants in laws of physics is compatible with intelligent life.

A quick response is that in the context of the fine-tuning argument, by “intelligent life” we should probably mean “intelligent biological life”. For instance, angels and conscious electrons don’t count, as they aren’t biological. And in fact, I think, in practice the fine-tuning argument is more about biological life than intelligent life as such. This suggests, however, that proponents of the fine-tuning argument should be clearer here. In particular, we (I am one of the proponents) should emphasize that there is a great value in the existence of biological life, and especially intelligent biological life, and this value is not found in intelligent non-biological life. This value is why a perfect being is not unlikely (or at least not extremely unlikely) to fine-tune the universe to for such life.

Second, I think Sinhababu’s argument points to a more subtle way to formulate the fine-tuning thesis. What’s fine-tuned is not the laws of physics alone, but the combination of the laws of physics and the bridge laws, and they are fine-tuned together in such a way as to ensure that there is neither too little nor too much intelligent life. For instance, a set of psychophysical laws where any computation isomorphic to the kinds of computations our brains results in mental functioning like ours would result not just in panpsychism but omnisapientism—everything around us is sapient. For with some cleverness we can find an isomorphism between the states of a single particle and the states of the brain that preserves causation. But omnisapientism isn’t very good: it damages the significance of morality if everything we do creates and destroys vast numbers of sapient beings.

A compositional fine-tuning argument

Assume naturalism about the human mind. Our best naturalistic account of the human mind is functionalism. But functionalism faces multiple too-many-minds problems. The most famous of these are the Chinese Room and its variants like Schwitzgebel’s consciousness of the United States argument. But a more troubling bevy of problems comes from abundant ontologies. Thus, as Dean Zimmerman noted (building on Unger), where I am there are many clouds of atoms that differ from me in an insignificant way—say, an atom in some insignificant skin cell. On functionalism, each of these clouds should have the same conscious states as I do. Or, as Johnston argued, I have many personites—temporal parts of my life that are intrinsically just like the life of a person could be. On functionalism, they will have the same conscious states as me. The clouds of atoms and personites are not just a consequence of functionalism but also of other naturalistic accounts of mind.

But why are the too-many-minds problems problems, beyond the fact that they are counterintuitive? After all, we have good reason to think that the mind is mysterious enough that the true theory will have some counterintuitive consequences.

I think the best answer is ethics. If a country has a person-level mind, then it would be a murder-suicide for the citizens to vote to dissolve the country. But it is not wrong for the citizens to vote to dissolve a country for, say, economic reasons. If the Zimmerman argument is right, then where there is a person feeling pain, there are many other beings with human-level consciousness feeling the same pain. But the number of being that coincide with a specific person rapidly increases with the size of the person—the more cells they have, the more clouds of atoms there are that differ with respect to a few insignificant atoms. Consequently, if we have a choice between relieving an equal pain in two smaller persons or one much larger person, we should always relieve the pain in the larger one, because the number of conscious atom clouds coinciding with the larger person is likely much larger than the total number of atom clouds coinciding with the smaller ones. In other words, crucial intuitions about equal treatment of people are undercut. Something similar is true on the Johnston arguments if the number of personites is finite, and if it’s infinite we have other ethical problems. On the other hand, there is no immediate serious ethical problem in saying the Chinese Room is conscious.

Given functionalism, I think there is only one way to block the ethically problematic too-many-minds cases: deny that the alleged entities exist. There are no countries. There is only one human-shaped cloud of atoms where I am. There are no personites. But we better not go all the way to blocking all complex objects—we will get other ethical problems if we conclude with the early Unger that humans don’t exist. In other words:

1. If functionalism and ethical realism are true, restricted composition is true.

Restricted composition says that some but not all (proper) pluralities of atoms compose a whole. Note that (1) also applies to some other naturalistic theories than functionalism.

But it’s not enough that restricted composition be true. What we need is a carefully fine-tuned restricted composition. If we restrict composition too much, there will be no humans—and that’s ethically unacceptable. If we don’t restrict composition enough, there will be too many minds of an ethically problematic sort. In other words, restricted composition must be fine-tuned to fit with human ethics.

That’s difficult to do. For instance, van Inwagen’s life-account—that a plurality composes a whole if and only if it has a life together—has the problem that clouds of atoms that differ from me insignificantly have a life together just as I do.

Given naturalism, I think any restricted composition account that fits with ethics will involve seemingly arbitrary choices. Thus, one might start with van Inwagen’s account, but have an incredibly fine-grained account of what counts as “a life together” such that only one of the clouds of atoms nearly coinciding with me has a life together—namely, the cloud constituting me. But such a fine-grained account will have a ton of free parameters, and will be an implausible candidate for a metaphysically necessary account of restricted composition. Thus, the account will not only be fine-tuned but will likely be contingent.

How do we explain the fine-tuning of restricted composition for ethics? It’s hard to see how to do it other than by supposing that fundamental reality is value-driven. There are two main value-driven theories of fundamental reality: theism and axiarchism, where the latter is something like the view that reality must be for the best. Thus we have an argument for theism or axiarchism. And axiarchism, as Rescher noted, plausibly implies theism, since it’s for the best that there be a perfect being. So, either way, we get theism.

We can also run this argument in a Bayesian way. Assume naturalism about the earth ecosystem as a background belief, and assume as part of the background that the physical simples are arranged as they are. On atheism, it is extremely unlikely that composition is fine-tuned for ethics. On theism, it is at least moderately likely. So, we have significant evidence for theism.

Objection: God can’t control which cloud of atoms composes a whole, because whatever is the answer, the answer is metaphysically necessary.

Response: First, as noted above, it is likely that any ethically fine-tuned restricted composition theory has a bunch of parameters that appear contingent, and hence is likely contigent. Second, God is creator and has power over being itself. It seems quite plausible that where there is a bunch of particles God can lend his power to create an entity composed of the particles. Third, if God exists, likely modality itself is grounded in God—all reality necessarily reflects the goodness of God. But if so, then divine goodness may help to explain surprisingly good features of necessary truths, such as a fine-tuned but necessary theory of composition. Fourth, we don’t need to be certain of any of the above. All we need is that one of these stories is an order of magnitude more likely on theism than the fine-tuning of restricted composition is given naturalism (where the probabilities are all epistemic).

If my argument succeeds, it yields a dilemma:

2. Either naturalism about humans is false or God exists.

One may ask whether some variant of the above fine-tuning argument applies if naturalism about humans is true. I expect it does, but the exact shape of the bump under the rug will be different for different non-naturalistic stories. For instance, on Cartesian theories, there will be the question of why there is exactly one soul per human body. On strong emergence, we can ask why consciousness arises in exactly one of the human-shaped clouds of atoms where I am.

A sloppy fine-tuning argument

This argument is an intuition-pump. I don’t know if it can be made rigorous.

Start with some observations. Let Q₀ be the nomic parameters of our universe—the exact values of all the constants in the laws of nature. To avoid serious problems with higher infinities and probability, I will make a technical assumption, which I will assume to be neutral be theism and atheism:

0. There are at most countably many universes.

Now:

1. For no non-zero countable cardinality n does theism have a bias against the hypothesis that there are countable many universes with cardinality at least n.

2. The parameters Q₀ are life-permitting.

3. For any fixed countable cardinality n of universes, theism has a significant bias in favor of distributions of parameters that include more universes with life-permitting parameters.

4. If (2) and (3), then for any countable cardinality n of universes, theism has a significant bias in favor of at least one of them having the parameters given by Q₀.

5. Thus, theism has a bias in favor of a universe with Q₀.

6. Thus, the obtaining of Q₀ is evidence for theism.

Some thoughts on the premises.

Regarding 1: Theism actually seems to have a bias in favor of the hypothesis that there are at least n universes. After all, theism has a bias in favor of the hypothesis that there is at least one universe: that there is a universe is quite surprising on atheism, but not so on theism, given that God is by definition perfectly good, and the good tends to spread. But the same reasoning suggests a bias on theism in favor of larger numbers of universes.

Regarding 2: Obvious.

Regarding 3: I think the main way to challenge (3) is to say that God would only care about having one universe with life-permitting parameters, and wouldn’t care about having a larger number. But I think this is implausible given that the good tends to spread. In fact, it seems likely that God would create only universes with life-permitting parameters, which would induce a strong bias in favor of such parameters.

Regarding 4: This is a very substantial assumption. It won’t hold for every set of exact parameters, because some sets of parameters might be life-permitting but would be likely to generate a universe that is really unfortunate in some regard. I don’t think the parameters Q₀ behind our universe are like that, but this is a matter of dispute, and intersects with the problem of evil. Note also that it is important for the “significant” in (4) that even if n is (countably) infinite, the probability getting exactly Q₀ on atheism is low (in fact, infinitesimal).

The big technical difficulty, which makes me doubtful that the argument can be made rigorous, are the infinities involved.

Fields and finetuning

Here is an interesting fine-tuning issue, inspired by a talk I heard from Brian Cutter at the 2023 ACPA meeting.

It seems likely that physical reality will involve one or more fields: objects that assign values to points in space (“ordinary” space or configuration space), which values then govern the evolution of the universe.

The fine-tuning issue is this. A plausible rearrangement principle should allow any mathematical assignment of values of the field to the points in space as metaphysically possible. But intuitively “most” such assignments result in a configuration that cannot meaningfully evolve according to our laws of nature. So we want to have an explanation of the fine-tuning—why are we so lucky as to have an assignment that plays nice with the laws of nature.

For a toy example, consider an electric field, which is a vector field E that generates a force F = qE on a particle of charge q. Intuitively, “most” vector fields will be nonmeasurable. But for a nonmeasurable electric field, we have no hope for a meaningful solution to the differential equations of motion. (OK, I’m ignoring the evolution of the field itself.)

For another example, suppose we think of the quantum wavefunction as a function over configuration space rather than as a vector in Hilbert space (though I prefer the latter formulation). If that function is nonmeasurable—and intuitively “most” are nonmeasurable—then we have no way to use quantum mechanics to predict the further evolution of this wavefunction. And if that function, while measurable, is not square integrable (I don’t know if there is a sense of “most” that applies here), then we have no way to use the Born rule to generate measurement predictions.

The multiverse objection to the fine-tuning argument for theism

Consider a fine-tuning argument like this:

1. On theism, it is moderately likely that there would be a fine-tuned universe.

2. On naturalism, it is extremely unlikely that there would be a fine-tuned universe.

3. So, the existence of a fine-tuned universe is very significant evidence for theism over naturalism.

These days, the main response to this is to invoke a rich multiverse, and to note:

4. On multiverse naturalism, it is nearly certain that there would be a fine-tuned universe.

It follows from (1) and (4) that the existence of a fine-tuned evidence is moderate evidence for multiverse naturalism over theism.

If (4) undercuts anything in the argument (1)–(3), it is (2). How could (4) undercut (2)? It would have to be roughly as follows:

5. On naturalism, prior to the evidence of a fine-tuned universe, it is not very unlikely that there is a multiverse.

When we combine (4) with (5), we do indeed get that it’s not extremely unlikely that there would be a fine-tuned universe.

But (5) is dubious. For prior to the evidence of a fine-tuned universe, the rational credence in a naturalistic multiverse should be extremely small. This is because one of the prior ratioanl constraints on credences is that they should make skeptical hypotheses extremely unlikely. And a naturalistic multiverse is a kind of skeptical hypothesis, for multiple reasons. First, it denies the uniformity of nature (at least if it’s the kind of multiverse relevant to fine-tuning, where the laws of nature vary between universes). Second, it implies intuitively absurd claims, such as that probably there are fairies and Greek gods out of sight of our observation (namely in other universes). Third, on many versions it threatens most of our common-sense knowledge by making Boltzmann brains at least as likely as ordinary brains. Fourth, at least the infinite versions of the multiverse hypotheses endanger probabilistic reasoning, since crazy things happen infinitely many times and non-crazy things happen infinitely many times in a multiverse, and it’s hard to say that the crazy things are less likely.

I suppose it is possible that (a) the rational credence in a naturalistic multiverse is extremely small, but (5) is still true. But the only way that could be is if the prior probability of naturalism is quite low. And while I am happy to say that, I think few naturalists will be. Thus a typical naturalist should, I think, deny (5), and should hold that prior to the evidence of a fine-tuned universe, even on naturalism, a multiverse would be very and maybe even extremely unlikely. The evidence of fine-tuning will greatly raise the probability of a naturalistic multiverse, but given that it started extremely small relative to theism, it is going to stay small.

Humeans laws and constants

On Mill-Ramsey-Lewis accounts of laws of nature, the laws are the propositions that best balance informativeness and brevity (in a language that cuts nature precisely at the joints).

Now, the laws of nature include constants, such as the fine-structure constant whose current best measured value is 1/137.035999206. Now, we might be lucky, and it might turn out that the fine-structure constant will have some neat and elegant precise value. There is a history of speculation that it has such a value—for a while, there was hope it was exactly 1/137, and then other guesses took over. But suppose we don’t get so lucky. Suppose it just is some messy number with no simple expression. That should, after all, be a serious possibility.

In that case, the exact value of the fine-structure constant cannot be a part of the Mill-Ramsey-Lewis “world in a nutshell” system of laws, since the system would then be infinitely long, and we lose our hope of defining laws in terms of brevity.

So we have two options. First, the system of laws might not include any specific information on the value of the fine-structure constant, but might instead be of the form ∃αF(α) where F(α) says nothing about what α is, except maybe that it’s real-valued and positive. If we go for this option, then we have to say that all the things that depend on the actual value of the fine-structure constant—and that apparently includes all of chemistry—are not in fact laws of nature. This will likely fail to yield some counterfactuals that we want, and while the laws will be briefer, they will be far less informative than if they had something to say about the value of α.

So that moves us to the second option, which is that the laws are of the form ∃αF(α) and F(α) includes some constraints on α, such as that it lies between 1/137.04 and 1/137.03. These constraints are sufficiently tight to generate the nomic implications we need for chemistry and biology. But while this result seems a better fit for science, it is metaphysically very strange. For it is very strange to think that the laws allow the fine-structure constant to have any of an infinite number of values, but these values must lie in a narrow range.

Furthermore, the exact narrow range for α would be determined by fine details (I am not sure if the pun is intended) of exactly how informativeness and brevity are balanced in the definition of the laws.

The same issue comes up for other constants in the laws of nature. Either Mill-Ramsey-Lewis laws do not include anything about the values of constants or else they include oddly specific, but not completely specific, ranges.

Two difficulties for wavefunction realism

According to wavefunction realism, we should think of the wavefunction of the universe—considered as a square-integrable function on R³ⁿ where n is the number of particles—as a kind of fundamental physical field.

Here are two interesting consequences of wavefunction realism. First, it seems like it should be logically possible for the fundamental physical field to take any logically coherent combination of values on R³ⁿ. But now imagine that the initial conditions of the wavefunction “field” are have it take a combination of values that is not a square-integrable function, either because it is nonmeasurable or because it is measurable but non-square-integrable. Then the Schroedinger equation “wouldn’t know” what to do with the wavefunction. In other words, for quantum physics to work, given wavefunction realism, we need a very special initial combination of values of the “wavefunction field”. This is not a knockdown argument, but it does suggest an underexplored need for fine-tuning of initial conditions.

Second, the solutions to the Schroedinger equation, understood distributionally, are only defined up to sets of measure zero. In other words, even though the Schroedinger equation is generally considered to be deterministic (any indeterminism in quantum mechanics comes in elsewhere, say in collapse), nonetheless the solutions to the equation are underdetermined when they are considered as square-integrable fields on R³ⁿ—if ψ(⋅,t) is a solution for a given set of initial conditions, so is any function that differs from ψ(⋅,t) only on a set of measure zero. Granted, any two candidates for the wavefunction that differ only on a set of measure zero provide the exact same empirical predictions. However, it is still troubling to think that so much of physical reality would be ungoverned by the laws. (There might be a solution using the lifting theorem mentioned in footnote 6 here, though.)

A way forward on the normalizability problem for the Fine-Tuning Argument

The Fine-Tuning Argument claims that the life-permitting ranges of various parameters are so narrow that, absent theism, we should be surprised that the parameters fall into those ranges.

The normalizability objection is that if a parameter ξ can take any real value, then any finite life-permitting range of values of ξ counts as a “narrow range”, since every finite range is an infinitesimal portion of the full range from −∞ to ∞. Another way to put the problem is that there is no uniform probability distribution on the set of real numbers.

There is, however, a natural probability distribution on the set of real numbers that makes sense as a prior probability distribution. It is related to the Solomonoff priors, but rather different.

Start with a language L with a finite symbol set usable for describing mathematical objects. Proceed as follows. Randomly generate finite strings of symbols in L (say, by picking independently and uniformly randomly from the set of symbols in L plus an “end of string” symbol until you generate an end of string symbol). Conditionalize on the string constituting a unique description of a probability measure on the Lebesgue measurable subsets of the real numbers. If you do get a unique description of a probability measure, then choose a real number according to this distribution.

The result is a very natural probability measure P_L (a countable weighted sum of probability measures on the same σ-algebra with weights adding to unity is a probability measure) on the Lebesgue measurable subsets of the real numbers.

We can now in principle evaluate the fine-tuning argument using this measure.

The problem is that this measure is hard to work with.

Note that using this measure, it is false that all narrow ranges have very small probability. For instance, consider the intuitively extremely narrow range from 10¹⁰⁰⁰ to 10¹⁰⁰⁰. Supposing that the language is a fairly standard mathematical language for describing probability distributions, we can specify a uniform distribution on the 0-length interval from 10¹⁰⁰⁰ to 10¹⁰⁰⁰ as U[10¹⁰⁰⁰, 10¹⁰⁰⁰], which is 23 characters of LaTeX, plus an end of string. Using 95 ASCII characters, plus the end of string character, P_L of this interval will be at least 96⁻²⁴ or something like 10⁻⁴⁸. Yet the size of the range is zero. In other words, intuitively narrow ranges around easily describable numbers, like 10¹⁰⁰⁰, get disproportionately high probability.

But that is how it should be, as we learn from the fact that the exponent 2 in Newton’s law of gravitation had better have a non-zero prior, even though the interval from 2 to 2 has zero length.

Whether the Fine-Tuning Argument works with P_L for a reasonable choice of L and for a particular life-permitting range of ξ is thus a hard question. But in any case, for a fixed language L where we can define a map between strings and distributions, we can now make perfectly rigorous sense of the probability of a particular range of possibilities for ξ. We have replaced a conceptual difficulty with a mathematical one. That’s progress.

Further, now that we see that there can be a reasonable fairly canonical probability on infinite sets, the intuitive answer to the normalizability problem—namely, “this range seems really narrow”—could constitute a reasonable judgment as to what answer would be returned by one’s own reasonable priors, even if these are not the same as the probabilities given above.

Oh, and this probability measure solves the tweaked problem of regularity, because it assigns non-zero probability to every describable event. I think this is even better than my modified Solomonoff distribution.

Bohmianism and God

Bohmian mechanics is a rather nice way of side-stepping the measurement problem by having a deterministic dynamics that generates the same experimental predictions as more orthodox interpretations of Quantum Mechanics.

Famously, however, Bohmian mechanics suffers from having to make the quantum equilibrium hypothesis (QEH) that the initial distribution of the particles matches the wavefunction, i.e., that the initial particle density is given by (at least approximately) |ψ|². In other words, Bohmian mechanics requires the initial conditions to be fine-tuned for the theory to work, and we can then think of Bohmian mechanics as deterministic Bohmian dynamics plus QEH.

Can we give a fine-tuning argument for the existence of God on the basis of the QEH, assuming Bohmian dynamics? I think so. Given the QEH, nature becomes predictable at the quantum level, and God would have good reason to provide such predictability. Thus if God were to opt for Bohmian dynamics, he would be likely to make QEH true. On the other hand, in a naturalistic setting, QEH seems to be no better than an exceedingly lucky coincidence. So, given Bohmian dynamics, QEH does support theism over naturalism.

Theism makes it possible to be an intellectually fulfilled Bohmian. But I don’t know that we have good reason to be Bohmian.

Wobbly priors and posteriors

Here’s a problem for Bayesianism and/or our rationality that I am not sure what exactly to do about.

Take a proposition that we are now pretty confident of, but which was highly counterintuitive so our priors were tiny. This will be a case where we were really surprised. Examples:

1. Simultaneity is relative

2. Physical reality is indeterministic.

Let’s say our current level of credence is 0.95, but our priors were 0.001. Now, here is the problem. Currently we (let’s assume) believe the proposition. But if our priors were 0.0001, our credence would have been only 0.65, given the same evidence, and so we wouldn’t believe the claim. (Whatever the cut-off for belief is, it’s clearly higher than 2/3: nobody should believe on tossing a die that they will get 4 or less.)

Here is the problem. It’s really hard for us to tell the difference in counterintuitiveness between 0.001 and 0.0001. Such differences are psychologically wobbly. If we just squint a little differently when looking mentally a priori at (1) and (2), our credence can go up or down by an order of magnitude. And when our priors are even lower, say 0.00001, then an order of magnitude difference in counterintuitiveness is even harder to distinguish—yet an order of magnitude difference in priors is what makes the difference between a believable 0.95 posterior and an unbelievable 0.65 posterior. And yet our posteriors, I assume, don’t wobble between the two.

In other words, the problem is this: it seems that the tiny priors have an order of magnitude wobble, but our moderate posteriors don’t exhibit a correspnding wobble.

If our posteriors were higher, this wouldn’t be a problem. At a posterior of 0.9999, an order of magnitude wobble in priors results in a wobble between 0.9999 and 0.999, and that isn’t very psychologically noticeable (except maybe when we have really high payoffs).

There is a solution to this problem. Perhaps our priors in claims aren’t tiny just because the claims are counterintuitive. It makes perfect sense to have tiny priors for reasons of indifference. My prior in winning a lottery with a million tickets and one winner is about one in a million, but my intuitive wobbliness on the prior is less than an order of magnitude (I might have some uncertainty about whether the lottery is fair, etc.) But mere counterintuitiveness should not lead to such tiny priors. The counterintuitive happens all too often! So, perhaps, our priors in (1) and (2) were, or should have been, more like 0.10. And now perhaps the wobble in the priors will probably be rather less: it might vary between 0.05 and 0.15, which will result in a less noticeable wobble, namely between 0.90 and 0.97.

Simple hypotheses like (1) and (2), thus, will have at worst moderately low priors, even if they are quite counterintuitive.

And here is an interesting corollary. The God hypothesis is a simple hypothesis—it says that there is something that has all perfections. Thus even if it is counterintuitive (as it is to many atheists), it still doesn’t have really tiny priors.

But perhaps we are irrational in not having our posteriors wobble in cases like (1) and (2).

Objection: When we apply our intuitions, we generate posteriors, not priors. So our priors in (1) and (2) can be moderate, maybe even 1/2, but then when we updated on the counterintuitiveness of (1) and (2), we got something small. And then when we updated on the physics data, we got to 0.95.

Response: This objection is based on a merely verbal disagreement. For whatever wobble there is in the priors on the account I gave in the post will correspond to a similar wobble in the counterintuitiveness-based update in the objection.

An unimpressive fine-tuning argument

One of the forces of nature that the physicists don’t talk about is the flexi force, whose value between two particles of mass m₁ and m₂ and distance r apart is given by F = km₁m₂r and which is radial. If k were too positive the universe would fall apart and if k were too negative the universe would collapse. There is a sweet spot of life-permissivity where k is very close to zero. And, in fact, as far as we know, k is exactly zero. :-)

Indeed, there are infinitely many forces like this, all of which have a “narrow” life-permitting range around zero, and where as far as we know the force constant is zero. But somehow this fine-tuning does not impress as much as the more standard examples of fine-tuning. Why not?

Probably it’s this: For any force, we have a high prior probability, independent of theism, that it has a strength of zero. This is a part of our epistemic preference for simpler theories. Similarly, if k is a constant in the laws of nature expressed in a natural unit system, we have a relatively high prior probability that k is exactly 1 or exactly 2 (thought experiment: in the lab you measure k up to six decimal places and get 2.000000; you will now think that it’s probably exactly 2; but if you had uniform priors, your posterior that it’s exactly 2 would be zero).

But his in turn leads to a different explanatory question: Why is it the case that we ought to—as surely we ought, pace subjective Bayesianism—have such a preference, and such oddly non-uniform priors?

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.

Not a finetuning argument

In The Impiety... (1624), as part of the 6th argument for the existence of God, Mersenne writes:

The proportion found between all the bodies of the world also shows that there is a God who has made all the universe in weight, in number and in measure: for the earth has no other ratio with the sun than 1:140, with the moon than 40:1, ... (pp. 98-99)

(I don't know off hand what the ratios are exactly meant to be; if they are ratios of volume, the moon is within 25% of the truth but the sun is off several orders of magnitude; if they are ratios of diameter, the sun is within an order of magnitude of the truth but the moon is an order of magnitude off.)

Mersenne's argument is full of such numerical (claimed) facts (the sun goes around the earth in 365.241 days, the moon traverses the Zodiac in 27 days, etc., etc.) and claims that God is needed to explain these facts. Now, I'm right now teaching on the fine-tuning argument, so I am sensitized to seeing such numbers in an argument for the existence of God. But it's striking that nowhere can I see Mersenne saying why these numbers are at all better than others, especially since surely some tuning facts seem very close at hand--surely, for instance, if the sun were much bigger or much smaller than it is, it would be too hot or too cold for life.

Mersenne explicitly insists that the numbers aren't explained by the essential natures of the objects, just before the above quote:

For the sun wouldn't be any the less the sun if it were closer or further from the earth, just as the stars could still be stars if they absented themselves from us by more than 14,000 earth radii.

Mersenne's argument seems to be a pure application of the idea that all contingent facts need explanation, and the arbitrariness of the numbers in the numerical statements seems to be cited precisely in order to show the contingency of the numerical statements. The argument suggests a strikingly strong commitment to a Principle of Sufficient Reason for contingent facts: all he needs to argue for a cosmic cause is to argue that there are contingent cosmic facts. Mersenne is confident that God has "many reasons" (as he says in the case of one of the numerical claims) for making the numbers be what they are, but these are reasons "which we aren't going to know except in Paradise" (101-102).

Mersenne's argument isn't a design argument--it doesn't advert to value-laden features that a God would have good reason to actualize. I think it's a kind of cosmological argument, but an eccentric one. Rather than arguing from generic features like motion or causation as Aquinas did, it focuses on very particular features.

The focus on these very particular features seems to have two benefits. The first is that it makes any appeal to necessity as the explanation implausible. Maybe it's necessary that there is motion, but it is incredible that it be necessary that the ratio of the diameter of the earth to that of the moon have to be 3.665:1 (to use modern numbers). So we get contingency very easily. The second feature is one I didn't notice right away. The astronomical features cited by Mersenne are ones that would reasonably be thought to be permanent features. They are thus prime candidates to be dismissed by it is so, as it has always been so. Mersenne's focus on the seeming arbitrariness of these features makes it very clear that would be no explanation. Thus Mersenne's cosmological argument works whether or not the past is finite. It is not disturbed by an infinite regress but does not need one either.

Of course, we no longer think that these particular features are permanent in the same way--the earth and sun changed in size in the formation of the solar system. But impermanent features are no better explained by an infinite regress than permanent ones--the permanence of the features in Mersenne's argument is only heuristic (and I don't see him explicitly drawing the reader's attention to the permanence). Plus we could run the argument on the basis of the apparently permanent but seemingly arbitrary elements in the laws of nature, such as precise values of constants.

The downside of Mersenne's argument, however, is that unless it is explained why the features are desirable, it is difficult to show that the cause of these features of the universe must be intelligent.

Infinity

As I walking to class this morning, I was struck by this thought. Historically, the most popular answer to the cosmological argument has been an infinite regress of causes. (That's why the Kalaam argument has such crowd appeal.) And currently the most popular answer to the fine-tuning argument is a multiverse, and probably an infinite one. This shows a kind of inevitability of infinity. Either you believe in an infinite being (assuming one can argue that the First Cause or Designer is God) or you believe in an infinite number of finite beings. In any case, it is an impressive fact about our dim and finite intellectual faculties that they can show us the reality of infinitude. There is a kind of paradoxicality here: learning that there is infinitude (whether an infinite being or an infinite number of finite beings) shows us how limited we are in comparison to that which is not us, and yet it is an impressive feat that we can know of infinitude.

Maybe there is a design argument here, too.