Reducing exact similarity

It is a commonplace that while Platonists need to posit a primitive instantiation relation for a tomato to stand in to the universal redness, trope theorists need an exact similarity relation for the tomato’s redness to stand in to another object’s redness, and hence there is no parsimony advantage to Platonism.

This may be mistaken. For the Platonist needs a degreed or comparative similarity relation, too. It seems to be a given that maroon is more similar to burgundy than blue is to pink, and blue is more similar to pink than green is to bored. But given a degreed or comparative similarity relation, there is hope for defining exact similarity in terms of it. For we can say that x and y are exactly similar provided that it is impossible for two distinct objects to be more similar than x and y are.

That said, comparative similarity is perhaps too weird and mysterious. There are clear cases, as above, but then there are cases which are hard to make sense of. Is maroon more or less similar to burgundy than middle C is to middle B? Is green more or less similar to bored than loud is to quiet?

A privation theory of evil without lacks of entities

Taking the privation theory literally, evil is constituted by the non-existence of something that should exist. This leads to a lot of puzzling questions of what that “something” is in cases such as error and pain.

But I am now wondering whether one couldn’t have a privation theory of evil on which evil is a lack of something, but not of an entity. What do I mean? Well, imagine you’re a thoroughgoing nominalist, believing in neither tropes nor universals. Then you think that there is no such thing as red, but of course you can say that sometimes a red sign fades to gray. It is natural to say that the faded sign is lacking the due color red, and the nominalist should be able to say this, too.

Suppose that in addition to being a thoroughgoing nominalist, you are a classical theist. Then you will want to say this: the sign used to participate in God by being red, but now it no longer thusly participates in God (though it still otherwise participates in God). Even though you can’t be a literal privation theorist, and hold that some entity has perished from the sign, you can be a privation theorist of sorts, by saying that the sign has in one respect stopped participating in God.

A lot of what I said in the previous two paragraphs is fishy. The “thusly” seems to refer to redness, and “one respect” seems to involve a quantification over respects. But presumably nominalists say stuff like that in contexts other than God and evil. So they probably think they have a story to tell about such statements. Why not here, then?

Furthermore, imagine that instead of a nominalist we have a Platonist who does not believe in tropes (not even the trope of participating). Then the problems of the “thusly” and “one respect” and the like can be solved. But it is still the case that there is no entity missing from the sign. Yet we still recognizably have a privation theory.

This makes me wonder: could it be that a privation theory that wasn’t committed to missing entities solve some of the problems that more literal privation theories face?

Theism and abundant theories of properties

On abundant theories of properties (whether Platonic universals or tropes), for every predicate, or at least every predicate satisfied by something, there is a corresponding property expressed by the predicate.

Here is a plausible sounding argument:

1. The predicate “is morally evil” is satisfied by someone.

2. So, on an abundant theory of properties, there exists a property of being morally evil.

3. The property of being morally bad, if it exists, is thoroughly evil.

4. So, on an abundant theory of properties, there exists something that is thoroughly evil.

5. If theism is true, nothing that exists is thoroughly evil (since every entity is the perfect God or created by the perfect God).

6. So if theism is true, an abundant theory of properties is false.

If I accepted an abundant theory of properties, I would question (3). For instance, maybe properties are concepts in the mind of God. A concept of something morally evil is not itself an evil concept.

Still, it does seem to me that this argument provides a theist with a little bit of a reason to be suspicious of abundant theories of properties.

Against an argument against Platonism

Consider this familiar argument:

1. We cannot know about the sorts of things that don’t causally affect us.

2. Abstract objects are the sort of thing that doesn’t causally affect us.

3. So, we cannot know about abstract objects.

But note that if it were possible for something to non-causally affect us, that could well be good enough for us to know about it. So, unless we have independent reason to think that the only way things can affect is is causally, instead of (1) we should only affirm:

4. We cannot know about the sorts of things that don’t affect us.

But to argue against abstract objects, we then need:

5. Abstract objects are the sort of thing that doesn’t affect us.

However, on heavy-weight Platonism, abstract objects do affect us. Coldness makes us cold, being in pain makes us hurt, etc. So, the heavy-weight Platonist will reject (5).

A new argument for presentism

Here’s an interesting argument favoring presentism that I’ve never seen before:

1. Obviously, a being that fails to exist at some time t is not a necessary being.

2. If presentism is true, we have an elegant explanation of (1): If x fails to exist at t₁, then at t₁ it is true that x does not exist simpliciter, and whatever is true at any time is possibly true, so it is possible that x does not exist simpliciter, and hence x is not a necessary being.

3. If presentism is false, we have no equally good explanation of (1).

4. So, (1) is evidence for presentism.

I don’t know how strong this argument is, but it does present an interesting explanatory puzzle for the eternalist:

5. Why is it that non-existence at a time entails not being necessary?

Here’s my best response to the argument. Consider the spatial parallel to (1):

6. Obviously, a being that fails to exist at some location z is not a necessary being.

It may be true that a being that fails to exist at some location is not a necessary being, since in fact the necessary being is God and God is omnipresent. But even if it’s true, it’s not obvious. If Platonism were true, then numbers would be counterexamples to (6), in that they would be necessary beings that aren’t omnipresent.

But numbers seem to be not only aspatial but also atemporal. And if that’s right, then (1) isn’t obvious either. (In fact, if numbers are atemporal, then they are a counterexample to presentism, since they don’t exist presently but still exist simpliciter.)

What if the presentist insists that numbers would exist at every time but would not be spatial? Well, that may be: but it’s far from obvious.

What if we drop the “Obviously” in (1)? Then I think the eternalist theist can give an explanation of (1): The only necessary being is God, and by omnipresence there is no time at which God isn’t present.

Maybe one can use the above considerations to offer some sort of an argument for presentism-or-theism.

Arity-increase and heavy-weight Platonism

Here is a curious problem. To give a heavy-weight Platonist analysis of an n-ary predication requires an (n + 1)-ary predication:

1. Alice is green [unary]: Alice instantiates greenness [binary].

2. Alice and Bob are friends [binary]: Alice and Bob instantiate friendship [ternary].

But higher arity predication is more puzzling than lower arity predication. Hence, heavy-weight Platonism explains the obscure in terms of the more obscure.

What got me to thinking about this was exploring the idea that Platonists can curry higher arity relations into lower arity ones. But doing so requires a multigrade “instantiates” predicate, and the curried expression of an n-ary predication seems to require an n-ary use of “instantiates”.

On a function- rather than relation-based Platonism, the issue comes up as follows. To say that the value of an n-ary function f at x₁, ..., x_n is y is (n + 2)-ary predication which gets Platonically grounded by the application of the (n + 1)-ary function apply_n such that apply_n(f,x_1,…,x_n) = f(x₁, ..., x_n).

Σ10 alethic Platonism

Here is an interesting metaphysical thesis about mathematics: Σ₁⁰ alethic Platonism. According to Σ₁⁰ alethic Platonism, every sentence about arithmetic with only one unbounded existential quantifier (i.e., an existential quantifier that ranges over all natural numbers, rather than all the natural numbers up to some bound), i.e., every Σ₁⁰ sentence, has an objective truth value. (And we automatically get Π₁⁰ alethic Platonism, as Π₁⁰ sentences are equivalent to negations of Σ₁⁰ sentences.)

Note that Σ₁⁰ alethic Platonism is sufficient to underwrite a weak logicism that says that mathematics is about what statements (narrowly) logically follow from what recursive axiomatizations. For Σ₁⁰ alethic Platonism is equivalent to the thesis that there is always a fact of the matter about what logically follows from what recursive axiomatization.

Of course, every alethic Platonist is a Σ₁⁰ alethic Platonist. But I think there is something particularly compelling about Σ₁⁰ alethic Platonism. Any Σ₁⁰ sentence, after all, can be rephrased into a sentence saying that a certain abstract Turing machine will halt. And it does seems like it should be possible to embody an abstract Turing machine as a physical Turing machine in some metaphysically possible world with an infinite future and infinite physical resources, and then there should be a fact of the matter whether that machine would in fact halt.

There is a hitch in this line of thought. We need to worry about worlds with “non-standard” embodiments of the Turing machine, embodiments where the “physical Turing machine” is performing an infinite task (a supertask, in fact an infinitely iterated supertask). To rule those worlds out in a non-arbitrary way requires an account of the finite and the infinite, and that account is apt to presuppose Platonism about the natural numbers (since the standard mathematical definition of the finite is that a finite set is one whose cardinality is a natural number). We causal finitists, however, do not need to worry, as we think that it is impossible for Turing machines to perform infinite tasks. This means that causal finitists—as well as anyone else who has a good account of the difference between the finite and the infinite—have good reason to accept Σ₁⁰ alethic Platonism.

I haven't done any surveys, but I suspect that most mathematicians would be correctly identified as at least being Σ₁⁰ alethic Platonists.

Properties, relations and functions

Many philosophical discussions presuppose a picture of reality on which, fundamentally, there are objects which have properties and stand in relations. But if we look to how science describes the world, it might be more natural to bring (partial) functions in at the ground level.

Objects have attributes like mass, momentum, charge, DNA sequence, size and shape. These attributes associate values, like 3.4kg, 15~kg m/s north-east, 5C, TTCGAAAAG, 5m and sphericity, to the objects. The usual philosophical way of modeling such attributes is through the mechanism of determinables and determinates. Thus, an object may have the determinable property of having mass and its determinate having mass 3.4kg. We then have a metaphysical law that prohibits objects from having multiple same-level determinates of the same determinable.

A special challenge arises from the numerical or vector structure of many of the values of the attributes. I suppose what we would say is that the set of lowest-level determinates of a determinable “naturally” has the mathematical structure of a subset of a complete ordered field (i.e., of something isomorphic to the set of real numbers) or of a vector space over such a field, so that momenta can be added, masses can be multiplied, etc. There is a lot of duplication here, however: there is one addition operator on the space of lowest-level momentum determinates and another addition operator on the space of lowest-level position determinates in the Newtonian picture. Moreover, for science to work, we need to be able to combine the values of various attributes: we need to be able to divide products of masses by squares of distances to make sense of Newton’s laws of gravitation. But it doesn’t seem to make sense to divide mass properties, or their products, by distance properties, or their squares. The operations themselves would have to be modeled as higher level relations, so that momentum addition would be modeled as a ternary relation between momenta, and there would be parallel algebraic laws for momentum addition and position addition. All this can be done, one operation at a time, but it’s not very elegant.

Wouldn’t it be more elegant if instead we thought of the attributes as partial functions? Thus, mass would be a partial function from objects to the positive real numbers (using a natural unit system) and both Newtonian position and momentum will be partial functions from objects to Euclidean three-dimensional space. One doesn’t need separate operations for the addition of positions and of momenta any more. Moreover, one doesn’t need to model addition as a ternary relation but as a function of two arguments.

There is a second reason to admit functions as first-class citizens into our metaphysics, and this reason comes from intuition. Properties make intuitive sense. But I think there is something intuitively metaphysically puzzling about relations that are not merely to be analyzed into a property of a plurality (such as being arranged in a ball, or having a total mass of 5kg), but where the order of the relata matters. I think we can make sense of binary non-symmetric relations in terms of the analogy of agents and patients: x does something to y (e.g. causes it). But ternary relations that don’t reduce to a property of a plurality, but where order matters, seem puzzling. There are two main technical ways to solve this. One is to reduce such relations to properties of tuples, where tuples are special abstract objects formed from concrete objects. The other is Josh Rasmussen’s introduction of structured mereological wholes. Both are clever, but they do complicate the ontology.

But unary partial functions—i.e., unary attributes—are all we need to reduce both properties and relations of arbitrary finate arity. And unary attributes like mass and velocity make perfect intuitive sense.

First, properties can simply be reduced to partial functions to some set with only one object (say, the number “1” or the truth-value “true” or the empty partial function): the property is had by an object provided that the object is in the domain of the partial function.

Second, n-ary relations can be reduced to n-ary partial functions in exactly the same way: x₁, ..., x_n stand in the relation if and only if the n-tuple (x₁, ..., x_n) lies in the domain of the partial function.

Third, n-ary partial functions for finite n > 1 can be reduced to unary partial functions by currying. For instance, a binary partial function f can be modeled as a unary function g that assigns to each object x (or, better, each object x such that f(x, y) is defined for some y) a unary function g(x) such that (g(x))(y)=f(x, y) precisely whenever the latter is defined. Generalizing this lets one reduce n-ary partial functions to (n − 1)-ary ones, and so on down to unary ones.

There is, however, an important possible hitch. It could turn out that a property/relation ontology is more easily amenable to nominalist reduction than a function ontology. If so, then for those of us like me who are suspicious of Platonism, this could be a decisive consideration in favor of the more traditional approach.

Moreover, some people might be suspicious of the idea that purely mathematical objects, like numbers, are so intimately involved in the real world. After all, such involvement does bring up the Benacerraf problem. But maybe we should say: It solves it! What are the genuine real numbers? It's the values that charge and mass can take. And the genuine natural numbers are then the naturals amongst the genuine reals.

Post-Goedelian mathematics as an empirical inquiry

Once one absorbs the lessons of the Goedel incompleteness theorems, a formalist view of mathematics as just about logical relationships such as provability becomes unsupportable (for me the strongest indication of this is the independence of logical validity). Platonism thereby becomes more plausible (but even Platonism is not unproblematic, because mathematical Platonism tends towards plenitude, and given plenitude it is difficult to identify which natural numbers we mean).

But there is another way to see post-Goedelian mathematics, as an empirical and even experimental inquiry into the question of what can be proved by beings like us. While the abstract notion of provability is subject to Goedelian concerns, the notion of provability by beings like us does not seem to be, because it is not mathematically formalizable.

We can mathematically formalize a necessary condition for something to be proved by us which we can call “stepwise validity”: each non-axiomatic step follows from the preceding steps by such-and-such formal rules. To say that something can be proved by beings like us, then, would be to say that beings like us can produce (in speech or writing or some other relevantly similar medium) a stepwise valid sequence of steps that starts with the axioms and ends with the conclusion. This is a question about our causal powers of linguistic production, and hence can be seen as empirical.

Perhaps the surest way to settle the question of provability by beings like us is for us to actually produce the stepwise valid sequence of steps, and check its stepwise validity. But in practice mathematicians usually don’t: they skip obvious steps in the sequence. In doing so, they are producing a meta-argument that makes it plausible that beings like us could produce the stepwise valid sequence if they really wanted to.

This might seem to lead to a non-realist view of mathematics. Whether it does so depends, however, on our epistemology. If in fact provability by beings like us tracks metaphysical necessity—i.e., if B is provable by beings like us from A₁, ..., A_n, then it is not possible to have A₁, ..., A_n without B—then by means of provability by beings like us we discover metaphysical necessities.

Independence of FOL-validity

A sentence ϕ of a dialect of First Order Logic is FOL-valid if and only if ϕ is true in every non-empty model under every interpretation. By the Goedel Completeness Theorem, ϕ is valid if and only if ϕ is a theorem of FOL (i.e., has a proof from no axioms beyond any axioms of FOL). (Note: This does not use the Axiom of Choice since we are dealing with a single sentence.)

Here is a meta-logic fact that I think is not as widely known as it should be.

Proposition: Let T be any consistent recursive theory extending Zermelo-Fraenkel set theory. Then there is a sentence ϕ of a dialect of First Order Logic such that according to some models of T, ϕ is FOL-valid (and hence a theorem of FOL) and according to other models of T, ϕ is not FOL-valid (and hence not a theorem of FOL).

Note: The claim that ϕ is FOL-valid according to a model M is shorthand for the claim that a certain complex arithmetical claim involving the Goedel encoding of ϕ is true according to M.

The Proposition is yet another nail in the coffins of formalism and positivism. It tells us that the mere notion of FOL-theoremhood has Platonic commitments, in that it is only relative to a fixed family of universes of sets (or at least a fixed model of the natural numbers or a fixed non-recursive axiomatization) does it make unambiguous sense to predicate FOL-theoremhood and its lack. Likewise, the very notion of valid consequence, even of a finite axiom set, carries such Platonic commitments.

Proof of Proposition: Let G be a Rosser-tweaked Goedel sentence for T with G being Σ₁ (cf. remarks in Section 51.3 here). Then G is independent of T. In ZF, and hence in T, we can prove that there is a Turing machine Q that halts if and only if G holds. (Just make Q iterate over all natural numbers, halting if the number witnesses the existential quantifier at the front of the Σ₁ sentence G.) But one can construct an FOL-sentence ϕ such that one can prove in ZF that ϕ is FOL-valid if and only if Q halts (one can do this for any Turing machine Q, not just the one above). Hence, one can prove in T that ϕ is FOL-valid if and only if I holds.

Thus, in T it is provable that ϕ is FOL-valid if and only if G holds. But T is a consistent theory (otherwise one could formalize in T the proof of its inconsistency). Since G is independent of T, it follows that the FOL-validity of ϕ is as well.

Distinguishing between properties

Some philosophers worry about “principles of individuation” that make two things of one kind be different from another. Suppose we share that worry. Then we should be worried about Platonism. For it is very hard to say what make two fundamental Platonic entities of the same sort different, say being positively charged from being negatively charged, or saltiness from sweetness.

However, the light-weight Platonist, who denies that predication is to be grounded in possession of universals, has a nice story to tell about the above kinds of cases. For here is a qualitative difference between saltiness and sweetness:

• saltiness is necessarily had by all and only salty things, but

• sweetness is not necessarily had by all and only salty things.

But for the heavy-weight Platonist to tell this story would involve circularity, for what it is for a thing to be salty will be to exemplify saltiness.

Of course, this story only works for properties that aren’t necessarily coextensive. But it’s some progress.

Why are there infinitely many abstracta rather than none?

It just hit me how puzzling Platonism is. There are infinitely many abstract objects. These objects are really real, and their existence seems not to be explained by the existence of concreta, as on Aristotelianism. Why is there this infinitude of objects?

Of course, we can say that this is just a necessary fact. And maybe it’s just brute and unexplained why necessarily there is this infinitude of objects. But isn’t it puzzling?

Augustinian Platonism, on which the abstract objects are ideas in the mind of God, offers an explanation of the puzzle: the infinitely many objects exist because God thinks them. That still raises the question of why God thinks them. But maybe there is some hope that there is a story as to why God’s perfection requires him to think these infinitely many ideas, even if the story is beyond our ken.

I suppose a non-theistic Platonist could similarly hope for an explanation. My intuition is that the Augustinian’s hope is more reasonable.

Alethic Platonism

I’ve been thinking about an interesting metaphysical thesis about arithmetic, which we might call alethic Platonism about arithmetic: there is a privileged, complete and objectively correct assignment of truth values to arithmetical sentences, not relative to a particular model or axiomatization.

Prima facie, one can be an alethic Platonist about arithmetic without being an ontological Platonist: one can be an alethic Platonist without thinking that numbers really exist. One might, for instance, be a conceptualist, or think that facts about natural numbers are hypothetical facts about sequences of dashes.

Conversely, one can be an ontological Platonist without being an alethic Platonist about arithmetic: one can, for instance, think there really are infinitely many pluralities of abstracta each of which is equally well qualified to count as “the natural numbers”, with different such candidates for “the natural numbers” disagreeing on some of the truths of arithmetic.

Alethic Platonism is, thus, orthogonal to ontological Platonism. Similar orthogonal pairs of Platonist claims can be made about sets as about naturals.

One might also call alethic Platonism “alethic absolutism”.

I suspect causal finitism commits one to alethic Platonism.

Something close to alethic Platonism about arithmetic is required if one thinks that there is a privileged, complete and objectively correct assignment of truth values to claims about what sentence can be proved from what sentence. Specifically, it seems to me that such an absolutism about proof-existence commits one to alethic Platonism about the Σ₁⁰ sentences of arithmetic.

A regress of qualitative difference

According to heavyweight Platonism, qualitative differences arise from differences between the universals being instantiated. There is a qualitative difference between my seeing yellow and your smelling a rose. This difference has to come from the difference between the universals seeing yellow (Y) and smelling a rose (R). But one doesn’t get a qualitative difference from being related in the same way to numerically but not qualitatively different things (compare: being taller than Alice is not qualitatively different from being taller than Bea if Alice and Bea are qualitatively the same—and in particular, of the same height). Thus, if the qualitative difference between my seeing yellow and your smelling a rose comes from being related by instantiation to different things, namely Y and R, then this presupposes that the two things are themselves qualitatively different. But this qualitative difference between Y and R depends on Y and R exemplifying different—and indeed qualitatively different—properties. And so on, in a regress!

Intrinsic attribution

1. If heavyweight Platonism is true, all attribution of attributes to a subject is grounded in facts relating the subject to abstracta.

2. Intrinsic attribution is never grounded in facts relating a subject to something distinct from itself.

3. There are cases of intrinsic attribution with a non-abstract subject.

4. If heavyweight Platonism is true, each case of intrinsic attribution to a non-abstract subject is grounded in facts relating that object to something other than itself. (By 1 and 2)

5. So, if heavyweight Platonism is true, there are no cases of intrinsic attribution to a non-abstract subject. (2 and 4)

6. So, heavyweight Platonism is not true. (By 2 and 5)

Here, however, is a problem with 3. All cases of attribution to a creature are grounded in the creature’s participation in God. Hence, no creature is a subject of intrinsic attribution. And God’s attributes are grounded in a relation between God and the Godhead. But by divine simplicity, God is the Godhead. Since the Godhead is abstract, God is abstract (as well as being concrete) and hence God does not provide an example of intrinsic attribution with a non-abstract subject.

I still feel that there is something to the above argument. Maybe the sense in which a creature’s attributes are grounded in the creature’s participation in God is different from the sense of grounding in 2.

What are properties?

A difficult metaphysical question is what makes something be a property rather than a particular.

In general, heavy-weight Platonism answers the question of what makes x be F, when being F is fundamental, as follows: x instantiates the property of Fness.

It is hard to see what could be more fundamental on Platonism than being a property. So, a heavy-weight Platonist has an elegant answer as to what makes something be a property: it instantiates the second-order property of propertyhood.

Mathematical Platonist Universalism, consistency, and causal finitism

Mathematical Platonists say that sets and numbers exist. But there is a standard epistemological problem: How do we have epistemic access to the sets to the extent of knowing some of the axioms they satisfy? There is a solution to this epistemological problem, mathematical Platonist universalism (MPU): for any consistent collection of mathematical axioms, there are Platonic objects that satisfy these axioms. MPU looks to be a great solution to the epistemological problems surrounding mathematical Platonism. How did evolved creatures like us get lucky enough to have axioms of set theory or arithmetic that are actually true of the sets? It didn’t take much luck: As soon as we had consistent axioms, it was guaranteed that there would be a plurality of objects that satisfied them, and if the axioms fit with our “set intuitions”, we could call the members of any such plurality “sets” while if they fit with our “number intuitions”, we could call them “natural numbers”. And the difficult questions about whether things like the Axiom of Choice are true are also easily resolved: the Axiom of Choice is true of some pluralities of Platonic objects and is false of others, and unless we settle the matter by stipulation, no one of these pluralities is the sets. (The story here is somewhat similar to Joel Hamkins’ set theoretic multiverse, but I don’t know if Hamkins has the kind of far-reaching epistemological application in mind that I am thinking about.)

This story has a serious problem. It is surely only the consistent axioms that are satisfied by a plurality of objects. Axioms are consistent, by definition, provided that there is no proof of a contradiction from them. But proofs are themselves mathematical objects. In fact, we’ve learned from Goedel that proofs can be thought of as just numbers. (Just write your proof in ASCII, and encode it as a binary number.) Hence, a plurality of axioms is consistent if and only if there does not exist a number with a certain property, namely the property of encoding a proof of a contradiction from these axioms. But on MPU there is no unique plurality of mathematical objects deserving to be called “the numbers”. So now MPU faces a very serious problem. It said that any consistent plurality of axioms is true of some plurality of Platonic objects, and there are no privileged pluralities of “numbers” or “sets”. But consistency is itself defined by means of “the numbers”. And the old epistemological problems for Platonism resurface at this level. How do we have access to “the numbers” and the axioms they satisfy so as to have reason to think that the facts about consistency of axioms are as we think they are?

One could try making the same move again. There is no privileged notion of consistency. There are many notions of consistency, and for any axioms that are consistent with respect to any notion of consistency there exists a plurality of Platonic satisfiers. But now this literally threatens incoherence. But unless we specify some boundaries on the notion of consistency, this is going to literally let square circles into Platonic universalism. And if we specify the boundaries, then epistemological problems that MPU was trying to solve will come back.

At my dissertation defense, Robert Brandom offered a very clever suggestion for how to use my causal powers account of modality to account for provability: q can be proved from p provided that it is causally possible for someone to write down a proof of q from p. This can be used to account for consistency: axioms are consistent provided that it is not causally possible to write down a proof of a contradiction from them. There is a bit of a problem here, in that proofs must be finite strings of symbols, so one needs an account of the finite, and a plurality is finite if and only if its count is a natural number, and so this account seems to get us back to needing privileged numbers.

But if one adds causal finitism (the doctrine that only finite pluralities can together cause something) to the mix, we get a cool account of proof and consistency. Add the stipulation that the parts of a “written proof” need to have causal powers such that they are capable of together causing something (e.g., causing someone to understand the proof). Causal finitism then guarantees that any plurality of things that can work together to cause an effect is finite.

So, causal finitism together with the causal powers account of modality gives us a metaphysical account of consistency: axioms are consistent provided that it is not causally possible for someone to produce a written proof of a contradiction from them.

Could God be divinity?

Here's a plausible thesis:

1. If it is of x's essence to be F, then Fness is prior to x.

This thesis yields a fairly standard argument against the version of divine simplicity which identifies God with the property of divinity. For if God is divinity, then divinity is prior to divinity by (1), which is absurd.

But (1) is false. For, surely:

2. It is of a property's essence to be a property.

But propertyhood is a property, so it is of propertyhood's essence to be a property, and so propertyhood is prior to propertyhood if (1) is true, which is absurd. So, given (2), we need to reject (1), and this argument against the God=divinity version of divine simplicity fails.

Thoughts on theistic Platonism

Platonists hold that properties exist independently of their instances. Heavy-weight Platonists add the further thesis that the characterization of objects is grounded in or explained by the instantiation of a property, at least in fundamental cases. Thus, a blade of grass is green because the blade of grass instantiates greenness (at least assuming greenness is one of the fundamental properties).

Heavy-weight Platonism has a significant attraction. After all, according to Platonism (and assuming greenness is a property),

1. Necessarily (i) an object is green if and only if (ii) it instantiates greenness.

The necessary connection between (i) and (ii) shouldn’t just be a coincidence. Heavy-weight Platonism explains this connection by making (ii) explain or ground (i). Light-weight Platonism, which makes no claims about an explanatory connection between (i) and (ii), makes it seem like the connection is a coincidence.

Still, I think it’s worth thinking about some other ways one could explain the coincidence (1). There are three obvious formal options:

2. (ii) explains (i)
3. (i) explains (ii)
4. Something else explains both (i) and (ii).

Option (2) is heavy-weight Platonism. But what about (2) and (3)? It’s worth noting that there are available theories of both sorts.

Here’s a base theory that can lead to any one of (2)–(4). Properties are conceptions in the mind of God. Furthermore, instantiation is divine classification: x’s instantiating a property P just is God classifying x under conception P. It is natural, given this base theory, to affirm (3): x’s instantiating greenness just is God’s classifying x under greenness, and God classifies x under greenness because x is green. Thus, x instantiates greenness because x is green.

But, interestingly, this base theory can give other explanatory directions. For instance, Thomists think that God’s knowledge is the cause of creation. This suggests a view like this: God’s classifying x under greenness (which on the base theory just is x’s instantiating greenness) causes x to be green. On this view, x is green because x instantiates greenness. If the “because” here involves grounding, and not just causation, this is heavy-weight Platonism, with a Thomistic underpinning. Either way, we get (2).

And here is a third option. God wills x to be green. God’s willing x to be green explains both x’s being green and God’s classifying x as green. The latter comes from God’s willing as an instance of what Anscombe calls intentional knowledge. This yields (4).

So, interestingly, a theistic conceptual Platonism can yield any one of the three options (2)–(4). I think the version that yields (3)—interestingly, not the Thomistic one—is the one that best fits with divine simplicity.

Some weird languages

Platonism would allow one to reduce the number of predicates to a single multigrade predicate Instantiates(x₁, ..., x_n, p), by introducing a name p for every property. The resulting language could have one fundamental quantifier ∃, one fundamental predicate Instantiates(x₁, ..., x_n, p), and lots of names. One could then introduce a “for a, which exists” existential quantifier ∃_a in place of every name a, and get a language with one fundamental multigrade predicate, Instantiates(x₁, ..., x_n, p), and lots of fundamental quantifiers. In this language, we could say that Jim is tall as follows: ∃_Jimx Instantiates(x, tallness).

On the other hand, once we allow for a large plurality of quantifiers we could reduce the number of predicates to one in a different way by introducing a new n-ary existential quantifier ∃_F(x₁, …, x_n) (with the corresponding ∀_P defined by De Morgan duality) in place of each n-ary predicate F other than identity. The remaining fundamental predicate is identity. Then instead of saying F(a), one would say ∃_Fx(x = a). One could then remove names from the language by introducing quantifiers for them as before. The resulting language would have many fundamental quantifiers, but only only one fundamental binary predicate, identity. In this language we would say that Jim is tall as follows: ∃_Jimx∃_Tally(x = y).

We have two languages, in each of which there is one fundamental predicate and many quantifiers. In the Platonic language, the fundamental predicate is multigrade but the quantifiers are all unary. In the identity language, the fundamental predicate is binary but the quantifiers have many arities.

And of course we have standard First Order Logic: one fundamental quantifier (say, ∃), many predicates and many names. We can then get rid of names by introducing an IsX(x) unary predicate for each name X. The resulting language has one quantifier and many predicates.

So in our search for fundamental parsimony in our language we have a choice:

• one quantifier and many predicates
• one predicate and many quantifiers.

Are these more parsimonious than many quantifiers and many predicates? I think so: for if there is only one quantifier or only one predicate, then we can collapse levels—to be a (fundamental) quantifier just is to be ∃ and to be a (fundamental) predicate just is to be Instantiates or identity.

I wonder what metaphysical case one could make for some of these weird fundamental language proposals.