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.

Heavier objects fall sooner

We like to say that Galileo was right that more massive objects don’t fall any faster than lighter ones, at least if we abstract away from friction.

But it occurred to me that there is a sense in which this is false. Suppose I drop an object from a meter above the moon, and measure the time until impact. If the object is more massive, the time to impact is lower. Why? Because there are two relevant gravitational accelerations that affect the time of impact: the moon pulls the object down, but simultaneously additionally the object pulls the moon up. The impact time is affected by both accelerations, and the more massive the object, the greater the upward acceleration of the moon, even though the object's acceleration is unaffected by its mass.

Of course, if we are dropping a one kilogram ball, the gravitational acceleration it induces on the moon is about 1/10²³ of the gravitational acceleration the moon induces on it. It’s negligible. But it’s still not zero. :-) A heavier object of the same size will impact sooner.

If all this is unclear, think about the extreme case where we are “dropping” a black hole on the moon.

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.

Force-realism and simultaneous causation

If a charged particle is an electromagnetic field, the field exerts a Lorentz force F = qE + qv × B, where q and v are the charge and velocity of the particle, E is the electric field and B is the magnetic field. All of these quantities are taken at one location in spacetime. Thus, if realism about forces is correct, we have simultaneous causation: the electromagnetic field simultaneously causes the force.

Not everyone is a realist about forces, though. One might think that the electromagnetic field directly causes the subsequent change in velocity instead of causing a force which in turn causes the change in velocity.

Is our universe of sets minimal?

Our physics is based on the real numbers. Physicists use the real numbers all over the place: quantum mechanics takes place in a complex Hilbert space, and the complex numbers are isomorphic to pairs of real numbers, while relativity theory takes place in a manifold that is locally isomorphic to a Lorentzian four-dimensional real space.

The real numbers are one of an infinite family of mathematical objects known as real closed fields. Other real closed fields than the real numbers could be used in physics instead—for instance, the hyperreals—and I think we would have the same empirical predictions. But the real numbers are simpler and more elegant: for instance, they are the only Dedekind-complete and the minimal Cauchy-complete real closed field.

At the same time, the mathematics behind our physics lives within a set theoretic universe. That set theoretic universe is generally not assumed to be particularly special. For instance, I know of no one who assumes that our set theoretic universe is isomorphic to Shepherdson’s/Cohen’s minimal model of set theory. On the contrary, it is widely assumed that our set theoretic universe has a standard transitive set model, which implies that it is not minimal, and few people seem to believe the Axiom of Constructibility which would hold in a minimal model.

This seems to me be rationally inconsistent. If we are justified in thinking that the mathematics underlying the physical world is based on a particularly elegant real closed field even though other fields fit our empirical data, we would also be justified in thinking it’s based on a particularly elegant universe of sets even though other universes fit our empirical data.

(According to Shipman, the resulting set theory would be one equivalent to ZF + V=L + “There is no standard model”.)

The incompleteness of current physics

1. There is causation in the physical world.

2. Causation is irreducible.

3. Our fundamental physics does not use the concept of causation.

4. So, our fundamental physics is incomplete as a description of the physical world.

Set theory and physics

Assume the correct physics has precise particle positions (similar questions can be asked in other contexts, but the particle position context is the one I will choose). And suppose we can specify a time t precisely, e.g., in terms of the duration elapsed from the beginning of physical reality, in some precisely defined unit system. Consider two particles, a and b, that exist at t. Let d be the distance between a and b at t in some precisely definable unit system.

Here’s a question that is rarely asked: Is d a real number?

This seems a silly question. How could it not be? What else could it be? A complex number?

Well, there are at least two other things that d could be without any significant change to the equations of physics.

First, d could be a hyperreal number. It could be that particle positions are more fine-grained than the reals.

Second, d could be what I am now calling a “missing number”. A missing number is something that can intuitively be defined by an English (or other meta-language) specification of an approximating “sequence”, but does not correspond to a real number in set theory. For instance, we could suppose for simplicity that d lies between 0 and 1 and imagine a physical measurement procedure that can determine the nth binary digit of d. Then we would have an English predicate M_d(n) which is true just in case that procedure determined the n binary digit to be 1. But it could turn out that in set theory there is no set whose members are the natural numbers n such that M_d(n). For the axioms of set theory only guarantee the existence of a set defined using the predicates of set theory, while M_d is not a predicate of set theory. The idea of such “missing numbers” is coherent, at least if our set theory is coherent.

It seems reasonable to say that d is indeed a real number, and to say similar things about any other quantities that can be similarly physically specified. But what guarantees such a match between set theory and physics? I see four options:

1. Luck: it’s just a coincidence.

2. Our set theory governs physics.

3. Physics governs our set theory.

4. There is a common governor to our set theory and physics.

Option 1 is an unhappy one. Option 4 might be a Cartesian God who freely chooses both mathematics and physics.

Option 2 is interesting. On this story, there is a Platonically true set theory, and then the laws of physics make reference to it. So it’s then a law of physics that distances (say) always correspond to real numbers in the Platonically true set theory.

Option 3 comes in at least two versions. First, one could have an Aristotelian story on which mathematics, including some version of set theory, is an abstraction from the physical world, and any predicates that we can define physically are going to be usable for defining sets. So, physics makes sets. Second, one could have a Platonic multiverse of universes of sets: there are infinitely many universes of sets, and we simply choose to work within those that match our physics. On this view, physics doesn’t make sets, but it chooses between the universes of sets.

Microphysics and philosophy of mind

Much (but not all) contemporary philosophy of mind is written as if microphysics were fundamental physics. But as far as I know, only on those interpretations of quantum mechanics that disallow indeterminacy as to the number of particles can microphysics be fundamental physics. The most prominent such interpretation is Bohmianism. On most other interpretations, the most we can say about the number of particles is that we are in a superposition between states with different numbers of particles. But reality has to have determinate numbers of fundamental entities. The picture of reality we get from both relativity theory and mainstream interpretations of quantum mechanics other than Bohmianism and its close cousins is that fundamental physical reality consists of global entities such as the spacetime manifold or the wavefunction of the universe rather than microscopic entities like particles. (I am attracted to a non-mainstream interpretation on which the fundamental physical entities may include mid-sized things like dogs and trees.)

Sometimes, pretending microphysics is fundamental physics is excusable. For certain discussions, it doesn’t matter what the fundamental physics is—the arguments work equally well for global and local fundamental entities. In other cases, all that matters is relative fundamentality. Thus, facts about chemistry might be held to be more fundamental relative to biology, and facts about microphysics might be fundamental relative to chemistry, even if the microphysics facts themselves are not fundamental simpliciter, being reducible, say, to facts about global fields.

But even when the arguments do not formally rely on fundamental physics being microphysics, it is risky in a field so reliant on intuition to let one’s intuitions be guided by acting as if fundamental physics were microphysics. And doing this is likely to mis-focus one’s critical attention, say focusing one more on the puzzle of why the functioning of various neurons produces a unified consciousness than on the puzzle of how the functioning of a handful of global entities results in the existence of billions of minded persons.

Digital and analog states, consciousness and clock skew

In a computer, we have multiple layers of abstraction. There is an underlying analog hardware level (which itself may be an approximation to a discrete quantum world, for all we know)—all our electronic hardware is, technically, analog hardware. Then there is a digital hardware level which abstracts from the analog hardware level, by counting voltages above a certain threshold as a one, below another—lower—threshold as a zero. And then there are higher layers defined by the software. But it is interesting that there is already semantics present at the digital level: three volts (say) means a one while half a volt (say) means a zero.

At the (single-threaded) software level, we think of the computer as being in a sequence of well-defined discrete states. This sequence unfolds in time. However, it is interesting to note that the time with respect to which this sequence unfolds is not actually real physical time. One reason is this. At the analog hardware level, during state transitions there will be times when the voltage levels are in an area that does not define a digital state. For instance, in 3.3V TTL logic, a voltage below 0.8V is considered a zero, a voltage above 2.0V is considered a one, but in between what we have is “undefined and results in an invalid state”. Since physical changes at the analog hardware level are continuous, whenever there is a change between a zero and a one, there will be a period of physical time at which the voltage is in the “undefined” range.

It seems then that the well-defined software state thus can only occur at a proper subset of the physical times. Between these physical times are physical times at which the digital states, and hence the software states that are abstractions from them, are undefined. This is interesting to think about in connection with the hypothesis of a conscious computer. Would a conscious computer be conscious “all the time” or only during the times when software states are well defined?

But things are more complicated than that. The technical means by which undefined states are dealt with is the system clock, which sends a periodic signal to the various parts of the processor. The system is normally so designed that when the clock signal reaches a component of the processor (say, a flip-flop), that component’s electrical states have a well-defined digital value (i.e., are not in the undefined range). There is thus an official time at which a given component’s digital values are defined. But at the analog hardware level, that official time is slightly different for different components, because of “clock skew”, the physical phenomenon that clock signals reach different components at different times. Thus, when we say that component A is in state 1 and component B is in state 0 at the same time, the “at the same time” is not technically defined by a single physical time, but rather by the (normally) different times at which the same clock signal reaches A and B.

In other words, it may not be technically correct to say that the well-defined software state occurs at a proper subset of the physical times. For the software state is defined by the digital state of multiple components, and the physical times at which these digital state “count” is going to be different for different components because of clock skew. In fact, I assume that the following can and does sometimes happen: component B is designed so that the clock signal reaches it after it has reached component A, and by the time component B is reached by the clock signal, component A has started processing new data and no longer has a well-defined digital state. Thus at least in principle (and I don’t know enough about the engineering to know if this happens in practice) it could be that there is no single physical time at which all the digital states that correspond to a software state are defined.

If this is right, then when we go back to our thought experiment of conscious computer, we should say this: The times of the flow of consciousness in that computer are not even a subset of the physical times. They are, rather, an abstraction, what we might call “software time”. If this is right, the question of whether the computer is presently conscious will be literally nonsense. The computer’s software time, which its consciousness is strung out along, has a rather complex relationship to real time.

So what?

I don’t know exactly. But I think there are a few directions one could take this line of thought:

1. Consciousness has to be strung out in a well-defined way along real time, and so computers cannot be conscious.

2. It is likely that similar phenomena occur in our brains, and so either our consciousness is not based on our brains or else it is not strung out along real time. The latter makes the A-theory of time less plausible, because the main motive for the A-theory is to do justice to our experience of temporality. But if our experience of temporality is tied to an abstracted software time rather than real time, then doing justice to our experience of temporality is unlikely to reach the truth about real time. This in turn suggests to me the conditional: If the A-theory of time is true, then some sort of dualism is true.

3. The problem that transitions between meaningful states (say, the ones and zeros of the digital hardware level) involve non-meaningful states between them is likely to afflict any plausible theory on which our mental functioning supervenes on a physical system. In digital computers, the way a sequence of meaningful states is reconstructed is by means of a clock signal. This leads to an empirical prediction: If the mental supervenes on the physical, then our brains have something analogous to a clock signal. Otherwise, the well-defined unity of our consciousness cannot be saved.

Reality is strange

The doctrines of the Trinity, the Incarnation and transubstantiation initially seem contradictory. Elaborate theological/philosophical accounts of the doctrines are available (e.g., from St. Thomas Aquinas), and given these, there is no overt contradiction. But the doctrines still seem very strange and they feel like they border on contradiction, with the accounts that remove contradiction sometimes looking like they are ad hoc designed to remove the contradiction from the doctrine. This may seem like a good reason to reject the doctrines.

But to reject the doctrines for this reason alone would be mistaken. For similar points can be made about Relativity Theory and Quantum Mechanics. To say that simultaneity is relative or that a physical object has no position but rather a probability distribution over positions borders on contradiction, and the philosophical moves needed to defend these seem ad hoc designed to save the theories. If we’ve learned one thing from physics in the 20th century, it is that the true physics of the world is very strange indeed.

Nor are theology and science the only places where things are strange. Similar things can be said about the mathematics of infinity, or even just common sense claims such as that there is change (think of Zeno’s paradoxes) or that material objects persist over time (think of the Ship of Theseus and the paradoxes of material composition).

We can, thus, be very confident that created reality is very strange indeed. And hence, shouldn’t we expect similar strangeness—indeed, mystery—in the Creator and his relationship to us?

Velocity and teleportation

Suppose a rock is flying through the air northward, and God miraculously and instantaneously teleports the rock, without changing any of its intrinsic properties other than perhaps position, one meter to the west. Will the rock continue flying northward due to inertia?

If velocity is defined as the rate of change of position, then no. For the rate of change of position is now westward and the magnitude is one meter divided by zero seconds, i.e., infinite. So we cannot expect inertia to propel the rock northward any more. In fact, at this point physics would break down, since the motion of an object with infinite velocity cannot be predicted.

But if velocity (or perhaps momentum) is an intrinsic feature that is logically independent of position, and it is merely a law of physics that the rate of change of position equals the velocity, then even after the miraculous teleportation, the rock will have a northward velocity, and hence by inertia will continue moving northward.

I find the second option to be the more intuitive one. Here is an argument for it. In the ordinary course of physics, the causal impact of physical events at times prior to t₁ on physical events after t₁ is fully mediated by the physical state of things at t₁. Hence whether an object moves after time t₁ must depend on its state at t₁, and only indirectly on its state prior to t₁. But if velocity is the rate of change of position, then whether an object moves via inertia after t₁ would depend on the position of the object prior to t₁ as well as at t₁. So velocity is not the rate of change of position, but rather a quality that it makes sense to attribute to an object just in virtue of how it is at one time.

This would have the very interesting consequence that it is logically possible for an object to have non-zero velocity while not moving: God could just constantly prevent it from moving without changing its velocity.

Approximatable laws

Some people, most notably Robin Collins, have run teleological arguments from the discoverability of the laws of nature.

But I doubt that we know that the laws of nature are discoverable. After all, it seems we haven’t discovered the laws of physics yet.

But the laws of nature are, surely, approximatable: it is within our power to come up with approximations that work pretty well in limited, but often useful, domains. This feature of the laws of nature is hard to deny. At the same time, it seems to be a very anthropocentric feature, since the both the ability to approximate and the usefulness are anthropocentric features. The approximatability of the laws of nature thus suggests a universe whose laws are designed by someone who cares about us.

Objection: Only given approximatable laws is intelligence an advantage, so intelligent beings will only evolve in universes with approximatable laws. Hence, the approximatable laws can be explained in a multiverse by an anthropic principle.

Response: Approximatability is not a zero-one feature. It comes in degrees. I grant that approximatable laws are needed for intelligence to be an advantage. But they only need to be approximatable to the degree that was discovered by our prehistoric ancestors. There is no need for the further approximatability that was central to the scientific revolution. Thus an anthropic principle explanation only explains a part of the extent of approximatability.

A tension in some theistic Aristotelian thinkers

Here is a tension in the views of some theistic Aristotelian philosophers. On the one hand, we argue:

1. The mathematical elegance and discoverability of the laws of physics is evidence for the existence of God

but we also think:

2. There are higher-level (e.g., biological and psychological) laws that do not reduce to the laws of physics.

These higher-level laws, among other things, govern the emergence of higher-level structures from lower-level ones and the control that the higher-level structures exert over the lower-level ones.

The higher-level laws are largely unknown except in the broadest outline. They are thus not discoverable in the way the laws of physics are claimed to be, and since no serious proposals are yet available as to their exact formulation, we have no evidence as to their elegance. But as evidence for the existence of God, the elegance and discoverability of a proper subset of the laws is much less impressive. In other words, (1) is really impressive if all the laws reduce to the laws of physics. But otherwise, (1) is rather less impressive. I’ve never never seen this criticism.

I think, however, there is a way for the Aristotelian to still run a design argument.

Either all the laws reduce to the laws of physics or not.

If they all reduce to the laws of physics, pace Aristotelianism, we have a great elegance and discoverability design argument.

Suppose now that they don’t. Then there is, presumably, a great deal of complex connection between structural levels that is logically contingent. It would be logically possible for minds to arise out of the kinds of arrangements of physical materials we have in stones, but then the minds wouldn’t be able to operate very effectively in the world, at least without massively overriding the physics. Instead, minds arise in brains. The higher-level laws rarely if ever override the lower-level ones. Having higher-level laws that fit so harmoniously with the lower-level laws is very surprising a priori. Indeed, this harmony is so great as to be epistemically suspicious, suspicious enough that the need for such a harmony makes one worry that the higher-level laws are a mere fiction. But if they are a mere fiction, then we go back to the first option, namely reduction. Here we are assuming the higher level stuff is irreducible. And now we have a great design argument from their harmony with the lower-level laws.

Aristotelian metaphysics and global physics

Too much of the contemporary ontological imagination is guided by the idea that the fundamental physical stuff in the world is discrete particles. Yet this is clearly dubious, since quantum mechanics (on non-Bohmian interpretations) suggests that the world is full of superpositions of states with different numbers of particles, while if discrete particles really exist, there had better be a well-defined number of them. Quantum mechanics instead suggests an ontology of the physical world where there is exactly one entity, “the Global Wavefunction”, whose physical state can be aptly represented as a vector in an infinite-dimensional vector space. And even if we didn’t have quantum mechanics’ vector-based approach on the table, we still wouldn’t be in an epistemic position to know that the right physics is based on particles rather than fields.

An ontology of material objects that composes these objects out of particles is held hostage to a particle-based physics that may well not be true. It would be best if one could work on the ontology of material objects without presupposing an answer to the question whether fundamental physical reality is field-like, vector-like or particle-like. I do not know if this is tenable. If it’s not, then the ontology of material objects needs to be done conditionally: If fundamental physical reality is of this sort, then material objects are like this.

Interestingly, some metaphysical problems may become easier given a non-particulate physical substratum. For instance, one of the hardest problems for a contemporary Aristotelian metaphysics has been the problem of what happens to particles that get incorporated into a substance, in light of the axiom that a substance cannot be composed of substances. But if we do not see fundamental physical reality as made of apparently substantial particles, the problem dissolves.

Today I want to sketch two Aristotelian approaches that take globalized vector- and field-approaches seriously. On the vector- and field-approaches, fundamental physical reality consists of a mere handful of entities: a single vector-like entity or several (hopefully no more than a dozen, and ideally only one) field-like entities. But being Aristotelian, we will think there are at least billions of substances: every organism is a substance. If these substances are to be related to fundamental physical entities, billions of them will have to be related to the same fundamental physical entities.

The ordinary substances on my stories will be organisms. There are billions of them. In addition to the ordinary substances, there are extraordinary substances: one for each of the handful of fundamental physical entities (fields or a vector).

My stories now diverge. On the first story, the billions of ordinary substances each encode and ground local features of the global fundamental physical entities. On a field version of the story, you encode and ground the features that the global fields have where you are located and your dog encodes and grounds the features that the global fields have where your dog is located (I am less clear on how to describe the vector version). This is not enough. For there aren’t enough organisms in the universe to ground all of the richness of the global fundamental physical entities: too much of the universe is lifeless. Thus, I propose that there are additional substances located where the organisms are not, and the features of these substances ground the rest of the features of the global fundamental physical entities. One way to run this story is to say that there is one of these additional substances per global fundamental physical entity, and each grounds the features of its corresponding global fundamental phsyical entity away from organisms. These additional substances are like swiss cheese, with the holes being filled with organisms like people and dogs.

On this version of the Aristotelian story—which can be varied in a number of ways—the global fundamental physical entities are not metaphysically fundamental. They are grounded in the many substances of the world.

On the second story, the global fundamental physical entities are substances. They are global substances. These global substances interact with the ordinary substances (there are many ways to spell out this interaction). We can now identify the matter of an ordinary substance x either with x’s powers and liabilities for interaction with the global substances or with the plurality of these global substances qua interacting with x.

There are many options here. Much detail to be worked out. Some options may be inferior to others, but I doubt in the end we will come to a single clearly best option.

Physical possibility

Here is an interesting question: How can one tell from a physics theory whether some event is physically possible according to that theory?

A sufficient condition for physical possibility is that the physics assigns a non-zero chance to it. But this is surely not a necessary condition. After all, it is possible that you will get heads on each of infinitely many tosses of an indeterministic die, while the chance of that is zero.

Plausibly, a necessary condition is that the event should be describable within the state space of the theory. Thus, the state space of classical mechanics simply cannot describe an electron being in a superposition of two position states, and hence such a superposition is physically impossible. But this necessary condition is not sufficient, as Newtonian mechanics bans various transitions that can be described within the state space of classical mechanics.

So, we have a necessary condition and a sufficient condition for physical possibility relative to a physics theory. It would be nice to have a necessary and sufficient condition.

Neo-Aristotelian Perspectives on Contemporary Science

The collection Neo-Aristotelian Perspectives on Contemporary Science (eds: Simpson, Koons and Teh) is now available. It's divided into a physical sciences and a life sciences part.

My piece on the Traveling Forms interpretation is in the physical sciences part (interestingly, though, that interpretation is more about us than about physics).

Life and non-life

Assume a particle-based fundamental physics. Then the non-living things in the universe outnumber the living by many orders of magnitude. But here is a striking fact given a restricted compositionality like van Inwagen’s, Toner’s or mine on which all there are is in the universe are particles and organisms: the number of kinds of living things outnumbers the number of kinds of non-living things by several orders of magnitude. The number of kinds of particles is of the order of 100, but there are millions of biological species (they may not all correspond to metaphysical species, of course).

Counting by individuals, living things are exceptional. But counting by kinds, physical things are exceptional. Only a tiny portion of the universe is occupied by life. But on the other hand, only a tiny portion of the space of kinds of entities is occupied by non-life.

I am not sure what to make of these observations. Maybe it is gives some credence to an Aristotelian rather than Humean way of seeing the world by putting the the kinds of features as teleology that are found in living things at the center of metaphysics.

Life science and physical science

I've been thinking that in a nutshell one could put much of the distinctiveness of Aristotelian philosophy as follows: life science is at least as fundamental as physical science.

The Axiom of Separation

The Axiom of Separation in Zermelo-Fraenkel (ZF) set theory implies that, roughly, for any set A and any unary predicate F(x), there is a subset B of all the x in A such that F(x). But only roughly. Technically the axiom only implies this for predicates definable in the language of set theory. We philosophers tend to forget that technical fact when we use set theory, much as we tend to blithely extend set theory to allow for ur-elements (elements that are not themselves sets). But if we are going to be realists about sets (which I am not saying we should be), we should have a real worry about what predicates can be legitimately used in the Axiom of Separation. (That's one of the lessons of this post.)

Consider the predicate L(x) which holds if and only if someone likes x. This is definitely not formulated in the language of set theory, so ZF set theory gives us no guarantee that there is, say, the set of all real numbers that satisfy L(x). If it turns out that there are only finitely many numbers that are liked, then we have no worries: for any real numbers x₁,...,x_n, there is a set that contains them and only them (this follows from the Axiom of Pairs plus the Axiom of Union). There will be other special cases where things work out, say when all but finitely many numbers are liked. But in general there is no guarantee from the axioms of ZF that there is a set of all liked numbers.

One might use this to try to get out of some paradoxes of infinity, by limiting the applicability of set theory. That's a strategy worth exploring further, but risky. For the above observations also severely limit the physical applicability of set theory. Suppose, for instance, that at each of infinitely many points in spacetime there is a well-defined temperature. It is usual then to suppose that there is a function T from the spacetime manifold to the real numbers such that T(z)=u if and only if the temperature at z (or, more precisely, at the point of spacetime corresponding to the point z in the mathematical manifold that models spacetime) is u. And we need there to be such a function T to be able to make physical predictions.

One solution is to extend the Axiom of Separation to include some or even all predicates not in the language of set theory. This is the solution that is typically implicitly used by philosophers. The Axiom of Separation has a lot of intuitive force thus extended, but we need to be careful since we know that the incoherent Axiom of Comprehension also had a lot of intuitive force.

A second option would be to have physics make set-theoretic claims. Thus, a theory positing that at each point of spacetime there is a temperature would also posit that there exists a corresponding function from the mathematical manifold that models spacetime to the real numbers. I think this would be quite an interesting option: it would mean that physics actually places constraints on what the universe of set theory is like.

Perhaps if we are not Platonists about sets, things are easier. But I am not sure. Things might just be murkier rather than easier.