Avoiding temporal parts of elementary particles

It would be appealing to be able to hold on to all of the following:

1. Four-dimensionalism.

2. Elementary particles are simples.

3. There is only kind of parthood and it is timeless parthood.

4. Uniqueness of fusions: a plurality of parts composes at most one thing.

But (1)–(4) have a problem in cases where one object is transformed into another object made of the same elementary particles. For instance, perhaps, an oak tree dies and then an angel meticulously gathers together all the elementary particles the oak ever has and makes a pine out of them, which he shortly destroys before it can gain any new particles. Then the elementary particles of the oak seem to compose the pine, contrary to (4).

One common solution for four-dimensionalists is to deny (2). Elementary particles have temporal parts, and you can’t make the old temporal parts of the oak’s particles live again in the pine. But there are problems with this solution. First, you might believe in a patchwork principle which should allow the old temporal parts to get re-used again. Second, it is intuitive to think that elementary particles are parts of the oak. But on the temporal part solution, this violates the transitivity of parthood, since the elementary particles will have temporal parts that outlive the oak. Third, the temporal parts of particles seem to be just as physical as the particles, and you might think that it’s the job of physics and not metaphysics to tell us what physical objects there are, so positing the temporal parts steps on the physicist’s toes in a problematic way. Fourth, and I am not fully confident I understand all the ramifications here, we need some kind of primitive relation joining the temporal parts of the particle into a single particle, since otherwise we cannot distinguish the case where two electrons swap properties and positions (and thereby reverse the sign of the wavefunction) from the case where they don’t.

The second common solution is to deny (3), distinguishing parthood from an irreducible parthood-at-t, and then say that trees are merely composed-at-t from elementary particles. I find an irreducible parthood-at-t kind of mysterious, but perhaps it’s not too terrible.

I want to offer a different solution, with an unorthodox four-dimensionalist Aristotelianism. Like orthodox Aristotelianism, the unorthodox version introduces a further entity, a form. And now we deny that a tree is composed of the elementary particles. Instead, we say that a tree is composed of form and elementary particles. One minor unorthodox feature here is that we don’t distinguish the parthood of a form in a substance and the parthood of a particle in a substance: there is just one kind of parthood. The more unorthodox thing will be, however, that we allow elementary particles to outlive their substances. The resulting unorthodox four-dimensionalist Aristotelianism then allows one to accept all of (1)–(4), since the pine is no longer composed of parts that compose the oak, as the oak’s form is not a part of the pine.

But we still have to account for parthood-at-t. After all, it just is true that some electron e is a part of the oak at some but not other times. And this surely matters—it is needed to account for, say, the mass and shape of the oak at different times. How do we that? Well, we might suppose that even if in our unorthodox Aristotelianism particles can outlive their substances, they get something from the substance’s form, even if it’s not identity. Perhaps, for instance, they get their causal powers from the substance’s form. (We then still need to say something about unaffiliated particles—particles not inside a larger substance. Perhaps when a particle, considered as a bit of matter, gets expelled from a larger substance and becomes unaffiliated, it gains its own substantial form. It loses that form when it joins into a larger substance again. At any given time, it gets its causal powers from the substance’s form.) So we can say that e is a part of the oak at t if and only if e gets its causal powers from the oak’s form at t.

The epistemic force of beauty in laws of nature does not reduce to simplicity

Some people think that simplicity of laws of nature is a guide to truth, and some think beauty of laws of nature is. One might ask: Is the beauty of laws of nature a guide that goes beyond simplicity? Are there times when one could make epistemic decisions about the laws of nature on the basis of beauty where simplicity wouldn’t do the job?

I think so. Here is one case. Suppose we live in a Newtonian universe, and we are discovering fundamental forces. The first one has an inverse cube law. The second has an inverse cube law. These two laws account for most phenomena, but a few don’t fit. Scientists think there is a third fundamental force. For the third force law, we have two proposals that fit the data: an inverse square law and a slightly more complicated inverse cube law. It is, I think, quite reasonable to go for an inverse cube law by induction over the laws.

There is something indeed beautiful about the idea that the same power law applies to all the forces of nature. But if we just go with simplicity, we will go for an inverse square law. However, going for the inverse cube law seems clearly reasonable, and it is what beauty suggests—but not simplicity.

Here is another thought. Sometimes a fundamental law has some particularly lovely mathematical implications. For instance, a conservative force law is connected in a lovely way with a potential. But it need not be the case that a conservative force law is simpler than a non-conservative alternative. (It is true that a conservative force is the gradient of a potential. If the potential can be particularly simply expressed, this makes it easier to express the conservative force law. But we can have a case where the potential is harder to express than the force itself.)

Simplicity and Newton's inverse square law

When I give talks about the way modern science is based on beauty, I give the example of how everyone will think Newton’s Law of Gravitation

1. F = Gm₁m₂/r²

is more plausible than what one might call “Pruss’s Law of Gravitation”

2. F = Gm₁m₂/r^{2.00000000000000000000000001}

even if they fit the observation data equally, and even if (2) fits the data slightly better.

I like the example, but I’ve been pressed on this example at least once, because I think people find the exponent 2 especially plausible in light of the idea of gravity “spreading out” from a source in concentric shells whose surface areas are proportional to r². Hence, it seems that we have an explanation of the superiority of (1) to (2) in physical terms, rather than in terms of beauty.

But I now think I’ve come to realize why this is not a good response to my example. I am talking of Newtonian gravity here. The “spreading out” intuition is based on the idea of a field of force as something energetic coming out of a source and spreading out into space around it. But that picture makes little sense in the Newtonian context where the theory says we have instantaneous action at a distance. The “spreading out” intuition makes sense when the field of force is emanating at a uniform rate from the source. But there is no sense to the idea of emanation at a uniform rate when we have instantaneous action at a distance.

The instantaneous action at a distance is just that: action at a distance—one thing attracting another at a distance. And the force law can then have any exponent we like.

With General Relativity, we’ve gotten rid of the instantaneous action at a distance of Newton’s theory. But my point is that in the Newtonian context, (1) is very much to be preferred to (2).

Beauty and simplicity in equations

Often, the kind of beauty that scientists, and especially physicists, look for in the equations that describe nature is taken to have simplicity as a primary component.

While simplicity is important, I wonder if we shouldn’t be careful not to overestimate its role. Consider two theories about some fundamental force F between particles with parameters α₁ and α₂ and distance r between them:

1. F = 0.8846583561447518148493143571151840833168115852975428057361124296α₁α₂/r²

2. F = 0.88465835614475181484931435711518α₁α₂/r^{2 + 2−64}.

In both theories, the constants up front are meant to be exact and (I suppose) have no significantly more economical expression. By standard measures of simplicity where simplicity is understood in terms of the brevity of expression, (2) is a much simpler theory. But my intuition is that unless there is some special story about the significance of the 2 + 2⁻⁶⁴ exponent, (1) is the preferable theory.

Why? I think it’s because of the beauty in the exponent 2 in (1) as opposed to the nasty 2 + 2⁻⁶⁴ exponent in (2). And while the constant in (2) is simpler by about 106 bits, that additional simplicity does not make for significantly greater beauty.

Simplicity and gravity

I like to illustrate the evidential force of simplicity by noting that for about two hundred years people justifiably believed that the force of gravity was Gm₁m₂/r² even though Gm₁m₂/r^2 + ϵ fit the observational data better if a small enough but non-zero ϵ. A minor point about this struck me yesterday. There is doubtless some p ≠ 2 such that Gm₁m₂/r^p would have fit the observational data better. For in general when you make sufficiently high precision measurements, you never find exactly the correct value. So if someone bothered to collate all the observational data and figure out exactly which p is the best fit (e.g., which one is exactly in the middle of the normal distribution that best fits all the observations), the chance that that number would be 2 up to the requisite number of significant figures would be vanishingly small, even if in fact the true value is p = 2. So simplicity is not merely a tie-breaker.

Note that our preference for simplicity here is actually infinite. For if we were to collate the data, there would not just be one real number that fits the data better than 2 does, but a range J of real numbers that fits the data better than 2. And J contains uncountably many real numbers. Yet we rightly think that 2 is more likely than the claim that the true exponent is in J, so 2 must be infinitely more likely than most of the numbers in J.

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?

Degrees of freedom

The number of degrees of freedom in a system is the number of numerical parameters that need to be set to fully determine the system. Scientists have an epistemic preference for theories that posit systems with fewer degrees of freedom.

But any system with n real-valued degrees of freedom can be redescribed as a system with only one real-valued degree of freedom, where n is finite or countable. For instance, consider a three-dimensional system which is fully described at any given time by a position (x, y, z) in three-dimensional space. We can redescribe x, y and z by real-valued variable X, Y and Z in the interval from 0 and 1, for instance by letting X = 1/2 + π⁻¹arctan x and so on. Now write out these new variables in decimal:

• X = 0.X₁X₂X₃...
• Y = 0.Y₁Y₂Y₃...
• Z = 0.Z₁Z₂Z₃...

Finally, let:

• W = 0.X₁Y₁Z₁X₂Y₂Z₂X₃Y₃Z₃....

Then W encodes all the information about X, Y and Z, which in turn encode all the information about (x, y, z) and hence about our system at a given time. (This obviously generalizes to any finite number of degrees of freedom. For a countably infinite one, things are slightly more complicated, but can still be done.)

There is a lesson here, even if not a particularly deep one. The epistemic preference for theories that have fewer degrees of freedom cannot be separated from the the epistemic preference for simpler theories. For of course rewriting a theory that made use of (x, y, z) in terms of W is in practice going to make for a significantly messier theory. So we cannot replace a simplicity preference by a preference for a low number of degrees of freedom.

Objection: Instead of a simplicity preference, we may a priori specify that laws of nature be given by differential equations in terms of the variables involved. But when, say, x, y and z vary smoothly over time, it is very unlikely that W will do so as well.

Response: But one can find a replacement for W that is smoothly related to x, y and z up to any desired degree of precision, and hence we can give a differential-equation based theory that fits the experimental data pretty much equally well but has only one degree of freedom.

Haecceitism and degrees of freedom

According to haecceitism as I shall understand it, there are vast numbers (probably infinitely many) possible worlds that are just like ours in all qualitative features but that differ in which particular entities fill which roles in the world. We should, however, prefer theories with fewer fundamental degrees of freedom. And haecceitism introduces many new fundamental degrees of freedom into the theory like the answer to "Who lived Einstein's life?"

This isn't an objection that haecceitism violates the Principle of Sufficient Reason. It might not. It might be that we can explain why Einstein rather than some other (actual or possible) person lived Einstein's life by supposing that the values of different individuals living a life are always incommensurable and that God freely chooses between these incommensurables. But even if (as I have argued) such a divine choice would be an explanation, it wouldn't be a very illuminating one. It would be a choice between vast (probably infinite) numbers of alternatives that are in an important sense on par. A theory that posits that is less attractive.

Elegance and stipulation

Depending on metaphysics, wholes depend on their parts or the parts depend on the wholes. But nothing depends on itself: that would be a vicious circularity. So nothing is a part of itself. On my own preferred story about parts, they are modes of wholes. But perhaps apart from God, nothing is a mode of itself. So, again, nothing is a part of itself (we shouldn't say that God is a part of himself, except trivially if everything is a part of itself).

Yet contemporary usage in mereology makes each thing a part of itself. One is free to stipulate how one wishes. If "part" is the ordinary notion, the contemporary mereologist can stipulate that parthood* is a disjunction of parthood and identity, i.e.,

1. x is a part* of y if and only if x is a part of y or x=y.

However, while one can stipulate how one wishes, one wouldn't expect a disjunctive stipulation to cut reality at its joints.

Why does this matter? Well, one of the interesting questions about parts is what axioms of mereology are true. We have several criteria for what makes a plausible axiom. It's supposed to be intuitive in itself, it's supposed to not lead to paradox, but it's also supposed to be elegant. It seems, however, that one can always ensure the elegance of any axioms with stipulation (just stipulate a zero-place predicate that says that the conjunction of the axioms is true). So it seems we want axioms to be elegant when expressed in terms that cut nature at the joints. In mereology, this would mean that we want axioms to be elegant when expressed in terms of proper parthood* (since proper parthood* is just parthood, the joint-carving natural concept) rather than in terms of parthood*.

This is a bit problematic. For it seems that the standard axioms of mereology get some of their prettiness by using the overlap relation:

2. Oxy iff x and y have a part* in common.

But the overlap relation is a nasty disjunctive mess when expressed in terms of proper parthood*:

3. Oxy iff x=y or x is a proper part* of y or y is a proper part* of x or x and y have a proper part* in common.

This suggests that much of the apparent elegance of the axioms of classical mereology may be spurious. They end up being a mess when you rewrite them in terms of parthood rather than parthood*.

I think the above negative conclusion about the elegance of the axioms of classical mereology is premature, and buys into a mistaken way to measure the elegance of the axioms of a theory. The mistake is to think that one rewrites all the axioms in what Lewis calls "perfectly natural" terms, and then looks at how brief the result is. Mathematicians frequently think that some set of axioms--say, group axioms--are quite "elegant and natural" even when rewriting the axioms in terms of the set membership relation ∈ produces a mess, as it generally does. (Just think of what a mess is produced when anything using the ordered pair (x,y) is rewritten using the set {{x},{x,y}}, and how just about everything in mathematics uses functions and hence ordered pairs.)

One can indeed make any set of axioms brief by careful choice of stipulations. But in some cases the stipulation will itself be very messy (the extreme case is where one replaces all the axioms with a single zero-place predicate) and in other cases there will be many stipulations. But if one can make a set of axioms brief by making a small number of relatively simple stipulations, that is impressive.

A theory can, thus, be elegant even if it is messy and long when all the axioms are written out in perfectly natural terms to the extent that the theory can be elegantly generated from an elegantly small set of elegant stipulations. Classical mereology can satisfy this elegance condition on theories even if I am right that the natural concept of parthood does not allow for proper parts. One just makes the fairly elegant (it's just a disjunction of two natural conditions) disjunctive stipulation (1), and then uses this stipulated notion of parthood* to elegantly stipulate a notion of overlap by means of (2), and then elegantly formulates the rest of the theory in terms of these.

The suggestion I am making is that we measure the complexity of a theory in terms of the brevity of expression in a language that has significant higher-order generative resources that, nonetheless, start with perfectly natural terms. These generative resources allow, in particular, for multiple levels of stipulation. We philosophers have a tendency to simply ignore stipulative definitions. But they do matter. If one takes classical mereology and rewrites the axioms in terms of (proper) parthood, one gets a mess; but the hierarchical stipulative structure of the classical theory is a part of the theory. Furthermore, the generative resources should also allow one to see an axiom schema as simpler and better unified than the sum total of the individual axioms falling under the schema. An axiom schema is not just the sum of the axioms falling under it.

This approach would also let one compare the complexity two different higher-level scientific theories, say in geology or organic chemistry, and say that one is simpler than the other even if both are equally intractable messes when fully expanded out in the vocabulary of fundamental physics. And one can do this even if one does not know how to make the needed stipulations--nobody knows how to define "tectonic plate" in the terminology of fundamental physics, but we can suppose the stipulation to have been made and proceed onward. All this makes it easier to be a reductionist about higher-level theories (I'm not happy about this, mais c'est la vie).

None of this should be news at all to those who are enamoured of computational notions of complexity.

One deep question here is just what generative resources the language should have.

And another deep question should be asked. When we formulate axioms by careful use of stipulation or axiom schemata, what we are really doing is describing the axioms in higher level terms: we are describing a set of sentences formulated in lower level terms. Patterns in reality are sometimes most aptly described not by first-order sentences in fundamental terms, but by describing how to generate those first-order sentences (say, as instances of a schema, or as the result of filling out a sequence of stipulations). We should then ask: How can such patterns be explanatory? I think that if such patterns are explanatory, if they are not mere coincidence, then in an important way reality is suffused with logos, in both of the main sense of the word (language and rationality).

There are, I think, three main options here. One is that we create this reality with our language. Realism forbids that. The second is that we are living in a computer simulation. But this explains the linguistic-type patterns only in contingent reality. But the axioms of mereology or of set theory are not merely contingent. The third is a supernaturalist story like theism, panentheism or pantheism.

Simplicity and beauty

Consider these two candidates for fundamental physical equations:

1. G=8πT
2. G=(8+π)T.

These two equations are equally simple. (The second has three extra characters in the above inscription. But that's just an artifact of the fact that we abbreviate "(8·π)" as "8π".) But the first equation is much more elegant. For it is elegant to multiply π by a power of two while it is inelegant to add a positive integer to π. The former just feels like a much natural expression.

There are other kinds of beauty of physical hypothesis that do not have much to do with simplicity. Sometimes, for instance, a given physical hypothesis can be characterized in two different ways: say, using a variational principle and a mechanistic story (Leibniz often talks about this). Physicists and mathematicians love this sort of thing. It definitely contributes to the felt beauty of the theory, and a theory that has such a dual characterization will, I think, be preferred to one that does not.

We like theories that tell a compelling story. There was something very compelling about Newton's idea that force is the rate of change of momentum and that the force of gravity drops off precisely in proportion to how "spread out" it is over a spherical shell at a given distance (i.e., the force of gravity is inversely proportional to the distance).

These are all aesthetic judgments, ones like those we employ when judging a piece of art or literature. "This really goes with that." "That's just a pointless plot twist."

This could lead us to non-realism about science. But I think it is better to see a tie between the physical world and our aesthetic judgments. It is, for instance, exactly the kind of tie we would expect if the world were the work of an artist whose tastes are not utterly alien to us.

Presentism and theoretical simplicity

It's oft stated that Ockham's razor favors the B-theory over the A-theory, other things being equal. But the theoretical gain here is small: the A-theorist need only add one more thing to her ideology over what the A-theorist has, namely an absolute "now", and it wouldn't be hard to offset this loss of parsimony by explanatory gains. But I want to argue that the gain in theoretical simplicity by adopting B-theoretic eternalism over presentism is much, much larger than that. In fact, it could be one of the larger gains in theoretical simplicity in human history.

Why? Well, when we consider the simplicity of a proposed law of nature, we need to look at the law as formulated in joint-carving terms. Any law can be formulated very simply if we allow gerrymandered predicates. (Think of "grue" and "bleen".) Now, if presentism is true, then a transtemporally universally quantified statement like:

1. All electrons (ever) are negatively charged

should be seen as a conjunction of three statements:

2. All electrons have always been negatively charged, all electrons are negatively charged and all electrons will always be positively charged.

But every fundamental law of nature is transtemporally universally quantified, and even many non-fundamental laws, like the laws of chemistry and astronomy, are transtemporally universally quantified. The fundamental laws of nature, and many of the non-fundamental ones as well, look much simpler on B-theoretic eternalism. This escapes us, because we have compact formulations like (1). But if presentism is true, such compact formulations are mere shorthand for the complex formulations, and having convenient shorthand does not escape a charge of theoretical complexity.

In fact, the above story seems to give us an account of how it is that we have scientifically discovered that eternalist B-theory is true. It's not relativity theory, as some think. Rather it is that we have discovered that there are transtemporally quantified fundamental laws of nature, which are insensitive to the distinction between past, present and future and hence capable of a great theoretical simplification on the hypothesis that eternalist B-theory is true. It is the opposite of what happened with jade, where we discovered that in fact we achieve simplification by splitting jade into two natural kinds, jadeite and nephrite.

Technical notes: My paraphrase (2) fits best with something like Prior's temporal logic. A competitor to this are ersatz times, as in Crisp's theory. Ersatz time theories allow a paraphrase of (1) that seems very eternalist:

3. For all times t, at t every electron is negatively charged.

However, first, the machinery of ersatz times is complex and so while (3) looks relatively simple (it just has one extra quantifier beyond (1)), if we expand out what "times" means for the ersatzist, it becomes very complex. Moreover on standard ersatzist views, the laws of nature become disjunctive in form, and that is quite objectionable. For a standard approach is to take abstract times to be maximal consistent tensed propositions, and then to distinguish actual times as times that were, are or will be true.

Positing non-epistemic vagueness doesn't solve a puzzle

Suppose we want to explain why one tortoise doesn't fall down, and we explain this by saying that it's standing on two tortoises. And then we explain why the two lower tortoises doesn't fall down, we suppose that each stands on two tortoises. And so on. That's terrible: we're constantly explaining one puzzling thing by two that are just as puzzling.

Now suppose we try to explain the puzzle of the transition from bald to non-bald in a Sorites sequences of heads of hair (no hair, one hair, two hairs, etc.). We do this by saying that there are going to be vague cases of baldness. But this is just as the case of tortoises. For while previously we had one puzzling transition, from bald to non-bald, now we have two puzzling transitions, from definitely bald to vaguely bald and from vaguely bald to definitely bald. So, we repeat with higher levels of vagueness. The transition from definitely bald to vaguely bald yields a transition from definitely bald to vaguely vaguely bald and a transition from vaguely vaguely bald to definitely vaguely bald, and similarly for the transition from vaguely bald to definitely bald. At each stage, each transition is replaced with two. We're constantly explaining one puzzling thing by two that are just as puzzling.

That said, it is possible with care to stand a tortoise on two tortoises, and we could have evidence that a particular tortoise is doing that. In that case, the two tortoises aren't posited to solve a puzzle, but simply because we have evidence that they are there. A similar thing could be the case with baldness. We might just have direct evidence that there is vagueness in the sequence. But as we go a level deeper, I suspect the evidence peters out. After all, in ordinary discourse we don't talk of vague vagueness and the like. So perhaps we might have a view on which there is one level of vagueness--and then epistemicism, i.e., there is a sharp transition from definitely non-bald to vaguely bald, and another from vaguely bald to definitely bald. But the more levels we posit, the more we offend against parsimony.

Platonism and Ockham's razor

One of the main objections against Platonism is that it offends against Ockham's razor by positing a large number of fundamental entities. But the Platonist can give the following response: By positing these fundamental entities, I can reduce the number of fundamental predicates to one, namely instantiation. I don't need fundamental predicates like "... is charged" or "... loves ...". All I need is a single multigrade fundamental predicate "... instantiate(s) ...", and I can just reduce the claim that Jones is charged to the claim that Jones instantiates charge, and the Juliet loves Romeo to the claim that Julie and Romeo instantiates loving. In other words, the Platonist's offenses against Ockham's razor in respect of ontology are largely compensated for by a corresponding reduction of ideology.

Largely, but so far not entirely. For the Platonist does need to introduce the "... instantiate(s) ..." predicate which the nominalist has no need for. On pain of a Bradley-type regress, the Platonist cannot handle that predicate using her general schema.

(But maybe Platonist can go one step further. She can eliminate single quantifiers from her ideology, too, using the Fregean move of replacing, say, ∃xF(x) with Instantiates(Fness, instantiatedness). Extending this to nested quantifiers is hard, but perhaps not impossible. If that task can be completed, then it seems that our Platonist has gained a decisive advantage over the nominalist: she has only one fundamental predicate and no quantifiers other than names (if names count as quantifiers). Not so, though! For this move needs to be able to handle complex predicates F, and the property Fness corresponding to such a complex predicate will probably have to stand in various structural relations to other properties, and we have complication.)

Aesthetics is epistemically central

Inference to best explanation is central to our epistemic lives. Aesthetic judgments about theories are central to inference to best explanation. Hence, aesthetic judgments are central to our epistemic lives. Thus we should be objectivists about at least a part of aesthetics.

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...?

Divine Belief Simplicity

Divine Belief Simplicity is the thesis that all of God's acts of belief are the same act of belief, the same belief token. While my belief that 2+2=4 seems distinct from my belief that the sky is blue, God's believings are all one. This is a special case of divine simplicity.

Here is an argument for Divine Belief Simplicity. The primary alternative to Divine Belief Simplicity is:

• Divine Belief Diversity: God's act of believing p is distinct from God's act of believing q whenever p and q are different.

But Divine Belief Diversity is false. The argument may be based on an anonymous referee's objection to a paper by Josh Rasmussen—I can't remember very well now—or to some comments by Josh Rasmussen. Here are some assumptions we'll need:

1. For any plurality, the Fs, there is a distinct proposition that the Fs exist or don't exist.

For instance, there is the proposition that the world's dogs exist or don't exist, and the proposition that the French exist or don't exist, and so on. Next:

2. Separation: Given any plurality, the Fs, and a predicate, P, that is satisfied by at least one of the Fs, there is a plurality of all and only the Fs satisfying P.
3. Plurality of Believings: If Divine Belief Diversity holds, then there is a plurality of all divine acts of believing.

But this is enough to run a Russell paradox.

Say that a divine believing b is settish provided that there is a plurality, the Fs, such that b is a believing that the Fs exist or don't exist. For any settish divine belief b, there is the plurality of things that b affirms the existence or nonexistence of. Say that a divine believing b is nonselfmembered provided that b is settish and is not in the plurality of things that b affirms the existence or nonexistence of. By (1), Separation and Plurality of Believings, let p be the proposition that affirms existence-or-nonexistence of the nonselfmembered believings. Now p is true. So there is a divine believing b in p. This is settish. Moreover, this b either is among the nonselfmembered believings or not. If it is, then it's not. If it's not, then it is. So we have a contradiction.

Moreover, this argument does not need to take propositions ontologically seriously. It only needs divine believings to be taken ontologically seriously.

Denying Divine Belief Diversity, however, denies that there is such a thing as the plurality of things that b affirms the existence or nonexistence of.

Simplicity, language-independence and laws

One measure of the simplicity of a proposition is the length of the shortest sentence expressing the proposition. Unfortunately, this measure is badly dependent on the choice of language. Normally, we think of the proposed law of nature

• F=Gmm'/r²

as simpler than:

• F=Gmm'/r^{2.000000000000000000000000000000000000001},

but if my language has a name "H" for the number in the exponent, then the second law is as brief as the first:

• F=Gmm'/r^H.

One common move is to employ theorems to the effect that given some assumptions, measures of simplicity using different languages are going to be asymptotically equivalent. These theorems look roughly like this: if c_L is the measure of complexity with respect to language L, then c_L(p_n)/c_M(p_n) converges to 1 whenever p_n is a sequence of propositions (or bit-strings or situations) such that either the numerator or the denominator goes to infinity. I.e., for sufficiently complex propositions, it doesn't matter which language we choose.

Unfortunately, one of the places we want to engage in simplicity reasoning in is with respect to choosing between different candidates for laws of nature. But it may very well turn out that the fundamental laws of physics—and maybe even a number of non-fundamental laws—are sufficiently simple that theorems about asymptotic behavior of complexity measures are of no help at all, since these theorems only tell us that for sufficiently complex cases the choice of language doesn't matter.

Simplicity, language and design

1. Simplicity is best understood linguistically (e.g., brevity of expression in the right kind of language).
2. Simplicity is a successful (though fallible) guide to truth.
3. If (1) and (2), then probably the universe was made for language users or by a language user.
4. If the universe was made for language users, it was made by an intelligent being.
5. If the universe was made by a language user, it was made by an intelligent being.
6. So, probably, the universe was made by an intelligent being.

Even more on simplicity and theism

Some naturalists say that theism needlessly complicates our view of the world by positing that

• in addition to the material concrete contingent things, there is something immaterial, necessary and concrete.

But the naturalist needs to say that the naturalist needlessly complicates our view of the world by being committed to the claim that

• in addition to the dependent concrete contingent things, there is something independent, concrete and contingent.

(Say, the Big Bang or the universe as a whole.)

Does talking of needless complication get us ahead here?

Simplicity and theism

I have argued elsewhere, as my colleague Trent Dougherty also has and earlier, that when we understand simplicity rightly, theism makes for a simpler theory than naturalism. However, suppose I am wrong, and naturalism is the simpler theory. Is that a reason to think naturalism true? I suspect not. For it is theism that explains how simplicity can be a guide to truth (say, because of God's beauty and God's desire to produce an elegant universe), while on naturalism we should not think of simplicity as a guide to truth, but at most as a pragmatic benefit of a theory. Thus to accept naturalism for the sake of simplicity is to cut the branch one is sitting on.