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).

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.

What else might properties do?

Suppose that we think of properties as the things that fulfill some functional roles: they are had in common by things that are alike, they correspond to fundamental predicates, etc. Then there is no reason to think that these functional roles are the only things properties do. It is prima facie compatible with fulfilling such functional roles that a property do many other things: it might occupy space, sparkle, eat or think.

Can we produce arguments that the things that fulfill the functional roles that properties are defined by cannot occupy space, sparkle, eat or think? It is difficult to do so. What is it about properties that rules out such activity?

Here's one candidate: necessity. The functional roles properties satisfy require properties to exist necessarily. But all things that occupy space are contingent. And all things that sparkle or eat also occupy space. So no property occupies space, sparkles or eats. (Yes, this has nothing to say about thinking.) Yeah, but first of all it's controversial that all properties are necessary. Many trope theorists think that typical tropes are both contingent and properties. Moreover, it may be that my thisness is a property and yet as contingent as I am. Second, it is unclear that everything that occupies space has to be contingent. One might argue as follows: surely, for any possible entity x, it could be that all space is vacant of x. But it does not follow that everything that occupies space has to be contingent. For we still have the epistemic possibility of a necessary being contingently occupying a region space. Christians, for instance, believe that the Second Person of the Trinity contingently occupied some space in the Holy Land in the first century--admittedly, did not occupy it qua God, but qua human, yet nonetheless did occupy it--and yet the standard view is that God is a necessary being. (Also, God is said to be omnipresent; but we can say that omnipresence isn't "occupation" of space, or that all-space isn't a region of space.)

So the modal argument isn't satisfactory. We still haven't ruled out a property's occupying space, sparkling or eating, much less thinking. In general, I think it's going to be really hard to find an argument to rule that out.

Here's another candidate: abstractness. Properties are abstract, and abstracta can't occupy space, sparkle, eat or think. But the difficulty is giving an account of abstracta that lets us be confident both that properties are abstract and that abstract things can't engage in such activities. That's hard. We could, for instance, define abstract things as those that do not stand in spatiotemporal relations. That would rule out occupying space, sparkling or eating--but the question whether all properties are abstracta would now be as difficult as the question whether a property can occupy space. Likewise, we could define abstract things as those that do not stand in causal relations, which would rule out sparkling, eating and thinking, but of course anybody who is open to the possibility that properties can do these activities will be open to properties standing in causal relations. Or we could define abstractness by ostension: abstract things are things like properties, propositions, numbers, etc. Now it's clear that properties are abstracta, but we are no further ahead on the occupying space, sparkling, eating or thinking front--unless perhaps we can make some kind of an inductive argument that the other kinds of abstracta can't do these things, so neither can properties. But whether propositions or numbers can do these things is, I think, just as problematic a question as whether properties can.

All in all, here's what I think: If we think of the Xs (properties, propositions, numbers, etc.) as things that fulfill some functional roles, it's going to be super-hard to rule out the possibility that some or all Xs do things other than fulfilling these functional roles.

For more related discussion, see this old contest.

More remarks on Aristotelian set theory

If we have an Aristotelian picture of abstracta, we should expect that what mathematical objects exist differs between possible worlds.

For the Aristotelian, abstract objects are abstractions from concrete things. So we shouldn’t expect the same full panoply of sets regardless of what concrete things there are. For instance, suppose that the universe contains exactly three point particles, A, B and C. Then we can immediately abstract from these particle positions distance ratios like AB : BC, AC : AB and AC : BC. These ratios are then represented by real numbers. So we are going to have these real numbers. More sophisticated abstractive processes may well generate other real numbers: for instance, we will have a real number representing the ratio of the height of the triangle drawn from A to the base BC. And given a real number, we might be able to use purely abstract processes to generate further real numbers: given a and b, we may generate a + b and ab, say. But there is no reason to think that these abstract processes will generate the same collection of real numbers regardless of what the three particle positions we start with are.

So, what real numbers exist should vary between possible worlds. But every real number defines a subset of the natural numbers (just write the real number in binary, and let the nth bit decide if n is in the subset or not). If the real numbers vary between possible worlds, so do the subsets of the natural numbers. In particular, we should expect that in different possible worlds, a different set counts as ``the power set’’ of the natural numbers.

Furthermore, what bijections there are between sets will vary between possible worlds. Thus, if we see the question of whether two sets have the same count of members as having the same answer in every world where the two sets exist, we cannot take the standard Cantorian account of the size of a set. Instead, we may want to generate the concept of sameness of size from bijections in different worlds. Thus, we may try to say that two sets A and B are the same size at level 0 provided that there is a bijection between A and B. Then we say that A and B are the same size at level n provided that possibly there is a set C that is the same size as A at level p and the same size as B at level q and n ≥ 1 + p + q. Finally, we say that A and B are the same size simpliciter provided that they are the same size at some finite level. This is complicated, and I haven’t checked under what assumptions it generates a transitive relation (it’s plausibly reflexive and symmetric).

Anyway, the point is this: It is an interesting and not easy philosophical project to work out the set-theoretic consequences of Aristotelianism. This could make a good dissertation.

Is music a sound?

It seems that music is a sound, and sound is constituted by vibrations of a surrounding medium, i.e., typically air. But you can listen to a musical piece through a bone conduction headset. In that case, you're not listening to vibrations of a surrounding medium. Moreover, we could suppose that the performer is producing electronic music live, which is directly piped to the audience's bone conduction headsets without any speakers anywhere, so it's never in the air (except accidentally). Assuming that the bone conduction headsets produce the same experienced quality as listening to the music the normal way, it seems that the audience wouldn't be losing out on anything musically relevant. Yet, if music is a kind of sound, and sound is vibrations of a surrounding medium, then we have a paradox:

1. There is no music there.
2. But the audience isn't missing out on anything musically relevant.
   
3. So, music need not be musically relevant!

Perhaps, though, music doesn't require vibrations of a surrounding medium. The vibrations of bones might be sufficient to qualify as music. I am not sure, however, whether in the concert that I have imagined the audience counts as hearing vibrations of their bones. Yes, their bones vibrate, but the content of the experience isn't the vibration of bones. Rather, it sounds like sound coming from outside them, so the content is external sound, but in my story that's absent, replaced by a mere illusion of sound.

In any case, we can modify the story. Suppose the piece is performed electronically, and never generates the relevant vibrations. Instead, it is directly piped to the performer's and audience's brains' auditory centers. It seems that musically nothing is lost, even though now there is definitely no sound at all.

The conclusion that music needn't be musically relevant is absurd. So we have to deny the claim that music is a sound. What is it then? A sequence of experiences? Then there is no music when the performer and audience are deaf. Maybe that's a bullet we should bite?

Maybe music--both the music composed by a composer and the music performed by a performer (who may be in part or whole a composer, as in cases of improvisation)--should be seen as an abstract sound type. The composer and performer discover music but don't create it. In order to grasp an abstract sound type, it is not needed that one hear an instance of it, but only that one have an experience as of hearing an instance of it. The performer, thus, causes the audience to have experiences as of hearing instances of it. Those experiences are neither sound nor music. We can then, by extension, call an instance of the abstract sound type--i.e., a concrete sound--"music". But music in this sense is not musically relevant except as a vehicle for music in the Platonic sense.

(In case it's relevant, I should note that I'm largely tone deaf, and I do not speak from experience.)

(One can mount another argument against the thesis that music is a sound on the basis of Cage's 4'33". But if 4'33" is music, we could still say it normally involves sound, in that one could have a more nuanced theory on which sound isn't the vibrations, but a token pattern of vibrations. And no-vibrations counts a pattern of vibrations.)

A problem for easy ontology arguments

Consider this "easy ontology" argument:

1. There are no unicorns.
2. So, there are zero unicorns.
3. So, there is a zero.

This seems fine. Now consider the parallel:

4. Every leprechaun is a fairy.
5. So, the set of leprechauns is a subset of the set of fairies.
6. So, there is a set of leprechauns.
7. If there is a set of leprechauns, it's empty. (There aren't any leprechauns!)
8. So, there is an empty set.

That seems fine as well. So far so good. But now:

9. Every non-self-membered set (set a that isn't one of its own members) is a set.
10. So, the set of non-self-membered sets is a subset of the set of all sets.
11. So, there is a set of non-self-membered sets (the Russell set).

But of course (11) yields a contradiction (just ask if the Russell set is a member of itself).

What to do? One move is to make the easy ontology arguments defeasible. This isn't in the spirit of the game. The other is to add to the premises of the easy ontology argument a coherence premise: that there is a coherent theory of zero, of the empty set and of the Russell and universal sets. The coherence premise will be false in the Russell case but will be true in the other cases. But the point is one that should make us take easy ontology less easily. (I wouldn't be surprised if this was in the easy ontology literature, with which I have little familiarity.)

From necessary abstracta to a necessary concrete being

Start with the Aristotelian thought that abstract entities are grounded in concrete ones. Add this principle:

1. If x is grounded only in the ys, then it is impossible for x exist without at least some of the ys existing.

Consider now a necessarily existing abstract entity, x, that is grounded only in concrete entities. (Some abstract entities may be grounded in other abstract entities, but we want to avoid circularity or regress.) Thus:

2. x is a necessarily existing abstract entity.

Add this premise:

3. There is a possible world in which none of the actual world's contingent concrete entities exist.

This isn't the more controversial assumption that there could be a world with no contingent concrete entities. Rather, it is the less controversial assumption that these particular concrete entities that we have in our world could all fail to exist, perhaps replaced by other contingent concrete entities.

If the concrete entities that ground x are all contingent, then we have a violation of the conjunction of (1)-(3), since then all the actual grounders of x could fail to exist and yet x is necessary. So:

4. There is at least one necessary contingent entity among the entities grounding x.

Presentism and abstracta: Two arguments

Here is an argument against presentism:

1. If presentism is true, then to exist is to presently exist.
2. Abstracta exist but do not presently exist.
3. So, presentism is false.

And here is an argument for presentism:

4. The number two existed yesterday and the day before yesterday.
5. Anything that existed yesterday and the day before yesterday persisted.
6. If growing block or eternalism is true, then persisting objects either have temporal parts or have locational properties.
7. The number two does not have temporal parts.
8. The number two does not have locational properties.
9. If presentism is not true, growing block or eternalism is true.
10. So, presentism is true.

My own take on the second argument is to distinguish between two senses of "x exists at t". The first sense is that x has tenseless existence-at-t. This we might call the narrow sense. But there is a broader sense of "x exists at t", which is that either x exists timelessly or x has existence-at-t. Ordinary language tends to use the second sense. If we take the broader sense in (4) and (5), I accept (4) (though with a divine conceptualist reduction) and deny (5). If we take the narrower sense, I deny (4) but accept (5).

Reducing sets

I find sets to be very mysterious candidates for abstract entities. I think it's their extensionality that seems strange to me. And anyway, if one can reduce entities to entities that we anyway want to have in our ontology, ceteris paribus we should. I want to describe a three-step procedure—with some choices at each step—for generating sets. I will use plural quantification quite a lot in this. I am assuming that one can make sense of plural quantification apart from sets.

Step 1: The non-empty candidates. The non-empty candidates, some of which will end up counting as sets in the next step, will be entities that stand in "packaging" relation to a plurality of objects, such that for any plurality, or at least for enough pluralities, there is a candidate that packages that plurality. There are many options for the non-empty candidates and the packaging relation.

Option A: Plural existential propositions, of the form <The Xs exist>, where a plural existential proposition p packages a plurality, the Xs, provided that it attributes existence to the Xs and only to the Xs.

Option B: Plural existential states of affairs (either Armstrong or Plantinga style), i.e., states of affairs of the Xs existing, where a plural existential state of affairs e packages a plurality, the Xs, provided that it is a state of affairs of the Xs existing. I got this option from Rob Koons.

Option C: This family of options generates the candidates in two sub-steps. The first is to have candidates that stand in a packaging relation to individuals, such that each candidates packages precisely one individual. Call these "singleton candidates". For brevity if x is a singleton candidate that packages y, I will say x is a singleton of y. The second step is to take our non-empty candidates to be mereological sums of singleton candidates, and to say that a mereological sum m packages the Xs if and only if m is a mereological sum of Ys such that each of the Ys is a singleton of one of the Xs and each of the Xs is packaged by exactly one of the Ys. We need the singleton packaging relation to satisfy the condition (*) that a mereological sum of singletons of the Xs has no singletons as parts other than the singletons of the Xs. (In particular, no singleton of y can be a part of any singleton of x if x and y are distinct.)

We get different instances of Option C by considering different singleton candidates. For instance, we could have the singleton candidates be individual essences, and a singleton candidate then packages precisely the entity that it is an individual essence of. I got this from Josh Rasmussen. Or we might use variants of Options A and B here: maybe a proposition attributing existence to x or a state of affairs of x existing will be our candidate singleton. (Whether the state of affairs option here differs from Option B depends on whether the state of affairs of a plurality existing is something different from the mereological sum of the states of affairs of the individuals in the plurality existing.)

There are many other ways of packaging pluralities.

Step 2: The empty candidate. We also need an empty candidate, which will be some entity that differs from the non-empty candidates of Step 1. Ideally, this will be an entity of the same sort as the non-empty candidates. For instance, if our non-empty candidates are propositions, we will want our empty candidate to be a proposition, say some contradictory proposition.

Step 3: Pruning the candidates. The basic idea will be that x is a member of a candidate y if and only if y is one of the non-empty candidates and y packages a plurality that has x in it. But the above is apt to give us too many candidates for them all to be sets. There are at least two reasons for this. First, on some of the options, there won't be a unique candidate packaging any given plurality. For instance, there might be more than one proposition attributing existence to the same plurality. Thus, the propositions <The Stagirite and Tully exist> and <Aristotle and Cicero exist> will be different propositions if Millianism is false, but both attribute existence to the same plurality. Second, some of the candidates will be better suited as candidates for proper classes than for sets and some candidates may be unsuitable either as sets or as proper classes. For instance, there might be a proposition that says that the plural existential propositions exist. Such a proposition packages all the candidates, including itself, and will not be a good set or proper class on many axiomatizations.

Nonexplanatory Platonic entities

Benacerraf-style arguments that numbers couldn't be any particular collection of abstract entities (say, some particular set-theoretic construction) because there is a multitude of other constructions that could play the same role will fail if numbers play an explanatory role in the world. And one can imagine metaphysical views on which they play even a physical explanatory role. For instance, charge and mass play an explanatory role, indeed perhaps a causal one, in the world. But a Platonist could think that to have a charge or mass of x units (in the natural respective unit system) is to be charge- or mass-related to the number x. In other words, such determinables are relations, whose second relatum must be a number, and their determinates are cases of that relation for a fixed second relatum.

Now, one can still construct a relation to some set-theoretic isomorph of the numbers that structurally functions just like charge. For instance, if f is an isomorphism from the abstracta relata of charge to some abstract Ss, then we could say that a is related by charge* to y, where y is one of the Ss, precisely when a is related by charge to an x such that y=f(x). But there will be a matter of fact as to whether it is charge or charge* that explains the motion of particles. Surely they both don't—that would be a bogus case of overdetermination.

The point generalizes to other cases of Platonic entities that play an explanatory role—not necessarily a physical one—in the world. For instance, propositions might explain the co-contentfulness of sentences. An isomorph of the system of propositions could play some of the same roles for us, but it would not in fact explain the co-contentfulness of sentences. Compare this case. There is an isomorphism between legal US voters and some set of social security numbers. We can then construct a relation voting* between numbers and candidates such that n votes* for c if and only if the voter with social security number n votes for c. But while one could use facts about voting* to organize our information about elections, it is facts about voting—an action performed by persons, not social security numbers—that in fact explain election outomes.

That said, I think this approach will still tell against the standard set-theoretic constructions of numbers in two ways. First, it will tell against any particular construction. For how likely is it that this construction is the right one? Second, it will tell against anything like the set-theoretic constructions being the numbers. For it seems really unlikely that having a charge of three units is anything like a matter of being related in some way to the set {∅, {∅}, {∅, {∅}}}. So this approach is most plausible if numbers are some kind of sui generis entities.

But, on the other hand, the Benacerraf argument could apply against Platonic entities that play no explanatory role but are merely introduced for our convenience of expression. On some views, possible worlds are like that.

An Aristotelian argument from a necessary being to a necessary concrete being

Suppose that none of the participants in World War II had ever existed. Then it would have been impossible for World War II to occur. Why? Because World War II's existence is solely grounded in the existence, activities, properties and relations of the participants, and

1. If an entity x's existence is solely grounded in the existence, activities, properties and/or relations of the Fs, then it is impossible for x to exist without at least one of the Fs existing.

Now add this Aristotelian axiom:

2. If x is abstract, then x's existence is solely grounded in the existence, activities, properties and/or relations of concreta.

Finally, add this:

3. Every being is either concrete or abstract.
4. There exists a necessary being.
5. There is a world where no one of the contingent concrete beings of our world exists.

One might try to give the number three as an example of a necessary being to support (4).

Now, let N be the necessary being of (4). If N is essentially concrete, we get to conclude that there is a concrete necessary being. If N is essentially abstract, then N is grounded in the existence, activities, properties and/or relations of concreta. If some concreta are necessary, we conclude that there is a concrete necessary being. So suppose all concreta are contingent. Then the beings that N is grounded in don't exist at the world mentioned in (5), which violates the conjunction of (1), (2) and the necessity and abstractness of N. So, no matter what, it follows from (1)-(5) that:

6. There is a necessary concrete being.

Plato might have been a "nominalist"

I was reading the SEP entry on nominalism by Rodriguez-Pereyra. Rodriguez-Pereyra sees nominalism as basically the rejection of causally inert non-spatiotemporal entities. If so, then Plato might have been a nominalist. It seems that Plato did not think the Form of the Good was causally inert--it caused the good arrangement of things in the universe. I don't know if Plato generalized from that case, but he might well have--he might have taken all of the Forms to be capable of causing things to be like them. So, for all I know, Plato was a nominalist.

And Leibniz might have been was a nominalist despite going on and on about abstract objects, because he thought of them as ideas guiding God's deliberation, and hence perhaps we should say that on his view they had a causal role in creation.

This isn't a big deal. Rodriguez-Pereyra's account nicely captures a rejection of modern forms of Platonist.

I wonder, too, whether a belief in Newtonian space is compatible with nominalism by this definition. Newtonian space seems to be causally inert (perhaps unlike the Riemannian manifold of General Relativity). And it may be a category mistake to say that space is spatiotemporal. Though maybe it's fine to say that space is spatiotemporal in some trivial sense.