Probabilities of regresses of chickens

Suppose we have a backwards-infinite sequence of asexually reproducing chickens, ..., c₋₃, c₋₂, c₋₁, c₀ with c_n having a chance p_n of producing a new chicken c_n + 1 (chicken c₀ may or may not have succeeded; the earlier ones have succeeded). Suppose that the p_n are all strictly between 0 and 1, and that the infinite product p₋₁p₋₂p₋₃... equals some number p strictly between 0 and 1.

Intuitively, we should be surprised that chicken c₀ exists if p is low and not surprised if p is high. If we have observed c₀ and are considering theories as to what the chances p_n are, other things being equal, we should prefer the theories on which the product p is high to ones on which it’s low.

But what exactly does p measure? It seems to be some kind of a chance of us getting c₀. But it doesn’t measure the unconditional probability of getting an infinite sequence of chickens leading up to c₀. For that is very tiny indeed, since it is extremely unlikely that the world would contain chickens at all. It seems to be a kind of conditional probability. Let q_n be the proposition that chicken c_n exists. Then P(q₀∣q_n) = p₀p₋₁p₋₂...p_n, and so p is the limit of the conditional probabilities P(q₀∣q_n). It is plausible thus to think of p as a conditional probability of q₀ on q_−∞, which is the infinite disjunction of all the q_n.

But q_−∞ is a rather odd proposition. It is grounded in q_n for every finite n, assuming that a disjunction, even an infinite one, is grounded in its true disjuncts. Thus every one of the q_n is explanatorily prior to q_−∞. But this means that P(q₀∣q_−∞) is actually a conditional probability of q₀ on something that isn’t explanatorily prior to q₀—indeed, that is explanatorily posterior to q₀. This challenges the interpretation of p as a chance of getting chicken c₀.

I am not quite sure what conclusion to draw from the above argument. Maybe it offers some support for causal finitism, by suggesting that things are weird when you have a backwards infinite causal sequence?

Inferentialism and the completeness of geometry

The Quinean criterion for existential commitment is that we incur existential commitment precisely by affirming existentially quantified sentences. But what’s an existential quantifier?

The inferentialist answer is that an existential quantifier is anything that behaves logically like an existential quantifier by obeying the rules of inference associated with quantifiers in classical logic.

Here is a fun little problem with the pairing of the above views. Tarski proved that, with an appropriate axiomatization, Euclidean geometry is complete and consistent, i.e., for every geometric sentence ϕ, exactly one of ϕ and its negation is provable from the axioms. Now let us stipulate a philosophically curious language L^*. Syntactically, the symbols of L^* are the symbols of L but with asterisks added after every logical connective, and the sentences are of L^* are the sentences of L with an asterisk added after every connective and predicate. The semantics of L^* are as follows: the sentence ϕ of L^* means that the sentence of L formed by dropping the asterisks from ϕ is provable from the axioms of Euclidean geometry.

Inferentially, the asterisked connectives of L^* behave exactly like the corresponding non-asterisked connectives of L.

Consider the sentence ϕ of L^* that is written ∃^*x(x=^*x). This sentence, by stipulation, means that ∃x(x=x) is provable from the axioms of Euclidean geometry. According to the Quinean criterion plus inferentialism, it incurs existential commitment, because ∃^*x, since it behaves inferentially just like an existential quantifier, is an existential quantifier. Now, it is intuitively correct that ∃^*x(x=^*x) does incur existential commitment: it claims that there is a proof of ∃x(x=x), so it incurs existential commitment to the existence of a proof. So in this case, the inferentialist Quinean gets right that there is existential commitment. But rather clearly only coincidentally so! For now consider the sentence ψ that is written ∀^*x(x=^*x). Since ∀^*x behaves inferentially just like ∀x, by inferentialist Quineanism it incurs no existential commitment. But ψ means that there is a proof of ∀x(x=x), and hence incurs exactly the same kind of existential commitment as ϕ did, which said that there was a proof of ∃x(x=x).

What can the inferentialist Quinean respond? Perhaps this: The language L^* is syntactically and inferentially compositional, but not semantically so. The meaning of p∨^*q, namely that the unasterisked version of p∨^*q has a proof, is not composed from the meanings of p and of q, which respectively mean that p has a proof and that q has a proof. But that’s not quite right. For meaning-composition is just a function from meanings to meanings, and there is a function from the meanings of p and of q to the meaning of p∨^*q—it’s just a messy function, rather than the nice function we normally associate with disjunction.

Perhaps what the inferentialist Quinean should do is to insist on the intuitive non-inferentialist semantic compositional meanings for the truth-functional connectives, but not for the quantifiers. This feels ad hoc.

Even apart from Quineanism, I think the above constitutes an argument against inferentialism about logical connectives. For the asterisked connectives of L^* do not mean the same thing as their unasterisked variants in L.

Some issues concerning eliminative structuralism for second-order arithmetic

Eliminative structuralist philosophers of mathematics insist that what mathematicians study is structures rather than specific realizations of these structures, like a privileged natural number system would be. One example of such an approach would be to take the axioms of second-order Peano Arithmetic PA2, and say an arithmetical sentence ϕ is true if and only if it is true in every standard model of PA2. Since all such models are well-known to be isomorphic, it follows that for every arithmetical sentence ϕ, either ϕ or  ∼ ϕ is true, which is delightful.

The hitch here is the insistence on standard (rather than Henkin) models, since the concept of a standard depends on something very much like a background set theory—a standard model is a second-order model where every subset of Dⁿ is available as a possible value for the second-order n-ary variables, where D is the first-order domain. Thus, such an eliminative structuralism in order to guarantee that every arithmetical sentence has a truth value seems to have to suppose a privileged selection of subsets, and that’s just not structural.

One way out of this hitch is to make use of a lovely internal categoricity result which implies that if we have any second-order model, standard or not, that contains two structures satisfying PA2, then we can prove that any arithmetical sentence true in one of the two structures is true in the other.

But that still doesn’t get us entirely off the hook. One issue is modal. The point of eliminative structuralism is to escape from dependence on “mathematical objects”. The systems realizing the mathematical structures on eliminative structuralism don’t need to be systems of abstract objects: they can just as well be systems of concrete things like pebbles or points in space or times. But then what systems there are is a contingent matter, while arithmetic is (very plausibly) necessary. If we knew that all possible systems satisfying PA2 would yield the same truth values for arithmetical sentences, life would be great for the PA2-based eliminative structuralist. But the internal categoricity results don’t establish that, unless we have some way of uniting PA2-satisfying systems in different possible worlds in a single model. But such uniting would require there to be relations between objects in different worlds, and that seems quite problematic.

Another issue is the well-known issue that assuming full second-order logic is “too close” to just assuming a background set-theory (and one that spans worlds, if we are to take into account the modal issue). If we could make-do with just monadic second-order logic (i.e., the second-order quantifiers range only over unary entities) in our theory, things would be more satisfying, because monadic second-order logic has the same expressiveness as plural quantification, and we might even be able to make-do with just first-order quantification over fusions of simples. But then we don’t get the internal categoricity result (I am pretty sure it is provable that we don’t get it), and we are stuck with assuming a privileged selection of subsets.

Causal Robinson Arithmetic

Say that a structure N that has a distinguished element 0, a unary function S, and binary operations + and ⋅ is a causal Robinson Arithmetic (CRA) structure iff:

1. The structure N satisfies the axioms of Robinson Arithmetic, and

2. For any x in N, x is a partial cause of the object Sx.

The Fundamental Metaphysical Axiom of CRA is:

• For every sentence ϕ in the language of arithmetic, ϕ is either true in every metaphysically possible CRA structure or false in every metaphysically possible CRA structure.

Causal Finitism—the doctrine that nothing can have infinitely many things causally prior to it—implies that any CRA is order isomorphic to the standard natural numbers (for any element in the CRA structure other than zero, the sequence of predecessors will be causally prior to it, and so by Causal Finitism must be finite, and hence the number can be mapped to a standard natural number), and hence implies the Fundamental Metaphysical Axiom of CRA.

Given the Fundamental Metaphysical Axiom of CRA, we have a causal-structuralist foundation for arithmetic, and hence for meta-mathematics: We say that a sentence ϕ of arithmetic is true if and only if it is true in all metaphysically possible CRA structures.

Sensory-based hacking and consent

Suppose human beings are deterministic systems.

Then quite likely there are many cases where the complex play of associations combined with a specific sensory input deterministically results in a behavior in a way where the connection to the input doesn’t make rational sense. Perhaps I offer you a business deal, and you are determined to accept the deal when I wear a specific color of shirt because that shirt unconsciously reminds you of an excellent and now deceased business partner you once had, while you have found the deal dubious if I wore any other color. Or, worse, I am determined to reject a deal offered by some person under some circumstances where the difference-maker is that the person is a member of a group I have an implicit and irrational bias against. Or perhaps I accept the deal precisely because I am well fed.

If this is true, then we are subject to sensory-based hacking: by manipulating our sensory inputs, we can be determined to engage in specific behaviors that we wouldn’t have engaged in were those sensory inputs somewhat different in a way that has no rational connection with the justification of the behavior.

Question: Suppose a person consents to something (e.g., a contract or a medical procedure) due to deliberate deterministic sensory-based hacking, but otherwise all the conditions for valid consent are satisfied. Is that consent valid?

It is tempting to answer in the negative. But if one answers in the negative, then quite a lot of our consent is in question. For even if we are not victims of deliberate sensory-based hacking, we are likely often impacted by random environmental sensory-based hacking—people around us wear certain colors of shirts or have certain shades of skin. So the question of whether determinism is true impacts first-order questions about the validity of our consents.

Perhaps we should distinguish three kinds of cases of consent. First, we have cases where one gives consent in a way that is rational given the reasons available to one. Second, we have cases where one gives consent in a way that is not rational but not irrational. Third, we have cases of irrational consent.

In cases where the consent is rational, perhaps it doesn’t matter much that we were subject to sensory-based hacking.

In cases where the consent is neither rational nor irrational, however, it seems that the consent may be undermined by the hacking.

In cases where the consent is irrational, one might worry that the irrationality undercuts validity of consent anyway. But that’s not in general true. It may be irrational to want to have a very painful surgery that extends one’s life by a day, but the consent is not invalidated by the irrationality. And in cases where one irrationally gives consent it seems even more plausible that sensory-based hacking undercuts the consent.

I wonder how much difference determinism makes to the above. I think it makes at least some difference.

We have systematic overdetermination in our movements

The causal exclusion argument requires us to deny that there is systematic overdetermination between mental and physical causes.

But it is interesting to note that in the real world there is systematic overdetermination of physical movements. Suppose I raise my arm. My muscle contraction is caused by a bunch of electrons moving in the nerves between the brain and the muscle. Suppose there are N electrons involved in the electrical flow, for some large number N. But now note that except in extremely rare marginal cases, any N − 1 of the electrons are sufficient to produce the same muscle contraction. Thus, my muscle contraction is overdetermined by at least N groups of electrons. Each of these groups differs from the original N electron group by omitting one of the electrons. And each group is sufficient to produce the effect.

One might try to defend the no-systematic-overdetermination view by saying that what doesn’t happen is systematic overdetermination by non-overlapping causes. There are two problems with this approach. First, it is not empirically clear that there isn’t systematic overdetermination by non-overlapping causes. It could turn out that typically twice as many electrons are involved in nerve impulses as are needed, in which case there are two non-overlapping groups each of which is sufficient. Second, the anti-physicalist can just say that there is overlap between the mental cause and the physical cause—the mental cause is not entirely physical, but is partially so.

Alternately, one might say that there may be systematic overdetermination of physical events by physical events, but not of physical events by physical and mental events. This would need an argument.

More on God causing infinite regresses

In my previous two posts I focused on the difficulty of God creating an infinite causal regress of indeterministic causes as part of an argument from theism to causal finitism. In this post, I want to drop the indeterministic assumption.

Suppose God creates a backwards infinite causal regress of (say) chickens, where each chicken is caused by parent chickens, the parent chickens by grandparent chickens, and so on. Now, I take it that the classical theist tradition is right that no creaturely causation can function without divine cooperation. Thus, every case where a chicken is caused by parent chickens is a case of divine cooperation.

Could God’s creative role here be limited to divine cooperation? This is absurd. For then God would be creating chickens by cooperating with chickens!

So what else is there? One doubtless correct thing to say is this: God also sustains each chicken between its first moment of life and its time of death. But this sustenance doesn’t seem to solve the problem, because the sustenance is not productive of the chickens—it is what keeps each chicken in existence after it has come on the scene. So while there is sustenance, it isn’t enough. God cannot create chickens by cooperating with chickens and by sustaining them.

Thus God needs to have some special creative role in the production of at least some of the chickens, fulfilling a task over and beyond cooperation and sustenance. Furthermore, this special task must be done by God in the case of an infinite number of the chickens, since otherwise there would be a time before which that task was not fulfilled—and yet God created infinitely the chickens before that time, too, since we’re assuming an infinite regress of chickens.

What happens in these cases? One might say is that in these special cases, God doesn’t cooperate with the parent chickens. But since no creaturely causation happens without divine cooperation, in these cases the parent chickens don’t produce their offspring, which contradicts our assumption of the chickens forming a causal regress. So that won’t do.

So in these cases, we seem to have two things happening: divine cooperation with chicken reproduction and divine creation of the chicken. Since divine cooperation with chicken reproduction is sufficient to produce the offspring, and divine creation of the chicken is also sufficient, it follows that in these cases we have causal overdetermination.

Now, we have some problems. First, does this overdetermination happen in all cases of chicken reproduction or only in some? It doesn’t need to happen in all of them, since it is overdetermination after all. But if it happens only in some, then it is puzzling to ask how God chooses which cases he overdetermines and which he does not.

Second, when there is overdetermination, the overdetermination is not needed for the effect. So it seems that if God’s additional role is that of overdetermining the outcome, that role is an unnecessary role, and the chickens could be produced by mere divine cooperation, which we saw is absurd. This isn’t perhaps the strongest of arguments. One might say that while in each particular case the overdetermining divine creative action is not needed, it is needed that it occur in some (indeed, infinitely many) cases.

Third, just as it is obviously absurd if God creates chickens merely by cooperating with chickens, it seems problematic, and perhaps absurd, that God creates chickens merely by cooperating with chickens and overdetermining that cooperation.

Famously, Aquinas thinks that God could have created an infinite regress of fathers and sons, and hence presumably of chickens as well. At this point, I can think of only one plausible way of getting Aquinas out of the above arguments, and it’s not a very attractive way. Instead of saying that God cooperates with the production of offspring, we can say that occasionalism holds in every case of substantial causation, that all causation of one substance’s existence by another is a case of direct divine non-cooperative causation, with the creaturely causation perhaps only limited to the transmission of accidents. Like all occasionalism, an occasionalism about substance causation is unappealing philosophically and theologically.

God and chancy infinite causal regresses

Suppose that a dod is a critter that chancily, with probability 1/2, causes one offspring during its life. The lifespan of a dod is one year. Further, imagine that like Sith, there are only ever one or two dods at a time, because each dod dies not long after reproducing, and if there were two or more mature dods at once, they’d fight to the death.

Now, imagine we have an infinite regress of dods, because each dod comes from an earlier dod. This would be hard to believe! After all, at any time at which we have a dod, we should be extremely (infinitely?) surprised that the dods haven’t died out yet. After all the probability that, given a dod at some time that there would be a dod in n years exponentially decreases with n.

Assuming causal finitism is false, it seems God could intentionally create an infinite regress of dods. But what would that look like? Here’s one story. God overrides the chances and directly and intentionally creates a backwards-infinite (and maybe even forwards-infinite, if he so chooses) sequence of dods. In that case, within that sequence the 1/2 chance of dod reproduction plays no explanatory role whatsoever. It seems we have occasionalism or a miracle or both. In any case, it does not appear that we actually have an infinite causal regress of dods in this case—the causation between dods, with its 1/2 chance, seems not to have any explanatory role. So the “overriding” story doesn’t work.

The other option is the Thomistic story. God doesn’t override chances. Instead, through primary causation, God concurs in creaturely causation and makes the finite cause produce its effect in such a way that the finite cause is fully acting as an indeterministic cause (this goes along with a view on which God can make us freely and indeterministically choose things). But this is very strange. For what explanatory role does the 1/2 in the chancy causation play? Assuming God wanted there to be an infinite sequence of dods, he could do exactly the same thing if the chance were 1/10 or 9/10 or even 1. It seems that the dod reproduces if and only if God intends the dod to reproduce, and whether God intends the dod to reproduce seems to have nothing to do with the “1/2” in the dod’s reproductive probabilities—it’s not plausible that God has probability 1/2 of intending each given dod to reproduce. And if God had probability 1/2 of intending each given dod to reproduce, how could he intentionally ensure that there ever are any dods, since the probability that God has infinitely many of these individual-dod-reproduction intentions is zero.

So we have problems. This gives further evidence that theism implies causal finitism.

From theism to causal finitism

Causal Finitism—the thesis that nothing can have an infinite causal history—implies that there is a first cause, and our best hypothesis for what a first cause would be is God. Thus:

1. If Causal Finitism is true, God exists.

But I think one can also argue in the other direction:

2. If God exists, Causal Finitism is true.

Aquinas wouldn’t like this since he thought that God could create a per accidens ordered backwards-infinite causal series.

In this post, I want to sketch an argument for (2). The form of the argument is this.

3. God cannot create a sequence of beings ..., A₋₃, A₋₂, A₋₁, A₀ where each being causes the next one.

4. If God cannot create such a sequence, such a sequence is impossible.

5. The best explanation of the impossibility of such a sequence is Causal Finitism.

Claim (4) comes from omnipotence. Claim (5) is I think the weakest part of the argument. Causal Finitism follows logically from the conjunction of two theses, one ruling out backwards-infinite causal chains and the other ruling out infinite causal cooperation (a precise statement and a proof is given in Chapter 2 of my Infinity book). But I am now coming to think that there is a not crazy view where one accepts the anti-chain part of Causal Finitism but not the anti-cooperation part. However, (a) the main cost of Causal Finitism come from the anti-chain part (the anti-chain part is what forces either a discrete time or a discrete causal reinterpretation of physics), (b) there are significant anti-paradox benefits to maintaining the anti-cooperation part, and (c) the theory may seem more unified in having both parts.

Now let’s move on to (3). Here is an argument. Say that an instance of causation is chancy provided that the outcome has a probability less than one.

6. If God can create a backwards-infinite causal sequence of beings, he can create a backwards-infinite chancy causal sequence of beings as the only thing in creation.

7. Necessarily, if God creates a backwards-infinite chancy causal sequence of beings as the only thing in creation, then there is no creature x such that God determines x to exist.

8. Necessarily, if God creates, he acts in a way that determines that something other than God exists.

9. Necessarily, if God determines that something other than God exists then there is a creature x that God determines x to exist.

10. Necessarily, if God creates a backwards-infinite chancy causal sequence of beings, then there is a creature x such that God determines x to exist. (8,9)

11. Hence, God cannot create a backwards-infinite chancy causal sequence of beings. (7,10)

12. Hence, God cannot create a backwards-infinite causal sequence of beings. (6,11)

The thought behind (6) is an intuition about modal uniformity. I think (6) is probably the most vulnerable part of the argument, but I don’t think it’s the one Aquinas would attack. What I think Aquinas would attack would most likely be (7). I will get to that shortly.

But first a few words about (8). In theory, it is possible to determine that something exists without determining any particular thing to exist. One can imagine a being with a chancy causal power such that if it waves a wand necessarily either a bunny or a pigeon is caused to exist, with the probability of the bunny being 1/2 and the probability of the pigeon being 1/2. But God is not like that. God’s will is essentially efficacious and not chancy. God can play dice with the universe, but only by creating dice. Thus, if God wanted to ensure there is a bunny or a pigeon without ensuring which specific one exists, he would have to create a random system that has chancy propensities for a bunny and for a pigeon and that must exercise one of the two propensities.

In fact, I think divine simplicity may imply this. For by divine simplicity, any two possible worlds that differ must differ in something outside God. Now consider a world w₁ where God determines a bunny to exist, and a world w₂ where God merely determines that a bunny or a pigeon exists and in fact a bunny is what comes about. There seems to be no difference outside God between these two worlds (one might wonder about the relation of being-created: could there be an relation of being-created-chancily and being-created-non-chancily? this seems fishy to me, and suggests a regress—how are the two relations differently related to God? and do we want to multiply such relations, saying there is such a thing as being-created-chancily-with-probability-0.7?). If both worlds are possible, by divine simplicity they must be the same, which is absurd. So at least one must be impsosible. And w₂ is a better candidate for that than w₁.

That still doesn’t establish (8). For I admitted that God can play dice if he creates dice. Thus, it seems that God could determine that something exists without determining where it’s A or B or C (say) by determining there to be dice that decide whether A or B or C are produced. But on this story, God still determines there to be dice, so there is an x—a die—that God determines to exist. I think a bit more could be said here, but as I said, I don’t think this is the main thing Aquinas would object to.

Back to (7). Why can’t God create a chancy backwards-infinite causal sequence while determining some item A_n in it to exist? Well, the sequence is chancy, so the probability that A_n − 1 causes A_n given that A_n − 1 exists is some p < 1. But, necessarily, if one creature causes another, it does so with divine cooperation (Aquinas will not disagree), and conversely if God cooperates with one creature to cause another, the one creature does cause the other. That the probability that God cooperates with A_n − 1 to cause A_n is equal to the probability that A_n − 1 causes A_n, because necessarily one thing happens if and only if the other does. Thus, the probability that God cooperates with A_n − 1 to cause A_n, given that A_n − 1 exists, is p. But p < 1, so it sure doesn’t look like a case of God determining A_n to exist!

But perhaps there is something like overdetermination, but between determination and chanciness (so not exactly over-determination). Perhaps God both determines A_n to exist and chancily cooperates with A_n − 1 to produce A_n. One problem with this hypothesis is with divine simplicity: it does not seem that there is any difference outside God between a world where God does both and God merely cooperates or concurs. But Aquinas may respond: “Yes, exactly. Necessarily, when one creature chancily causes another, God’s primary causation determines which specific outcome results. Thus there is no world where God merely cooperates.” So now the view is that whenever we have chancy causation, necessarily God determines the outcome. But suppose I chancily toss a coin, and it has chance 1/2 of heads and chance 1/2 of tails. Then on this view, I get heads if and only if God determines that I get heads. Hence the chance that God determines I get heads is 1/2. But it seems plausible that God’s determinations are not measured by numerical probabilities, and in any case that they are not measured by numerical probabilities coming from our world’s physics!

Megethology as mathematics and a regress of structuralisms

In his famous “Mathematics is Megethology”, Lewis gives a brilliant reduction of set theory to mereology and plural quantification. A central ingredient of the reduction is a singleton function which assigns to each individual a singleton of which the individual is the only member. Lewis shows that assuming some assumptions on the size of reality (namely, that it’s very big) there exists a singleton function, and that different singleton functions will yield the same set theoretic truths. The result is that the theory is supposed to be structuralist: it doesn’t matter which singleton function one chooses, just as on structuralist theories of natural numbers it doesn’t matter if one uses von Neumann ordinals or Zermelo ordinals or anything else with the same structure. The structuralism counters the obvious objection to Lewis that if you pick out a singleton function, it is implausible that mathematics is the study of that one singleton function, given that any singleton function yields the same structure.

It occurs to me that there is one hole in the structuralism. In order to say “there exists a singleton function”, Lewis needs to quantify over functions. He does this in a brilliant way using recently developed technical tools where ordered pairs of atoms are first defined in terms of unordered pairs and an ordering is defined by a plurality of fusions, relations on atoms are defined next, and so on, until finally we get functions. However, this part can also be done in a multiplicity of ways, and it is not plausible that mathematics is the study of singleton functions in that one sense of function, given that there are many sense of function that yield the same structure.

Now, of course, one might try to give a formal account of what it is for a construction to have the structure of functions, what it is to quantify not over functions but over function-notions, one might say. But I expect a formal account of quantification over function-notions will presumably suffer from exactly the same issue: no one function-notion-notion will appear privileged, and a structuralist will need to find a way to quantify over function-notion-notions.

I suspect this is a general feature with structuralist accounts. Structuralist accounts study things with a common structure, but there are going to be many accounts of common structure that by exactly the same considerations that motivate structuralism require moving to structuralism about structure, and so on. One needs to stop somewhere. Perhaps with an informal and vague notion of structure? But that is not very satisfying for mathematics, the Queen of Rigor.

Metaphysical universism

Here’s a metaphysical view I haven’t seen: the fundamental obejcts (priority version) or the only objects (existence version) are universes, but there can be more than one of these. Call this metaphysical universism (as distinguished from Quisling’s philosophy).

If in fact there is only one universe, metaphysical universism extensionally coincides with monism. But even in that case, metaphysical universism is a different theory, because it has different modal implications. And if we live in a multiverse, metaphysical universism is extensionally different from monism, since monism says that the one fundamental (priority) or one and only (existence) entity is the multiverse as a whole, not the universes.

I can think of two main advantages of metaphysical universism over monism.

First, suppose there is only one universe. It is plausible that there could be another in addition to this one. Metaphysical universism embraces this possibility. Monism only says that The One could have been bigger so as to comprise two spatiotemporally disconnected regions.

Second, there is an old intuition that being and unity are connected. In a multiverse, monism violates this intuition, for in a multiverse it is the universes that have unity, not the multiverse. Indeed quantum entanglement arguments for monism in the context of a non-Everettian multiverse seem to me to point more towards metaphysical universism than monism.

On the other hand, monism has a significant advantage over metaphysical universism insofar as monism solves the problem of truthmakers of negative and universal claims by making The One be the truthmaker of all of them.

Of course, both theories are false.

The ethics of plant care

If someone devotes a significant part of their life to affectionately caring for plastic flowers, there is something wrong with them. Not so in the case of real, living plants. This points to me to the idea that life as such, and not just conscious life or the life of animals, is a valuable thing.

I don’t want to say that it’s always bad to affectionately care for artifacts. When the artifacts have an intimate and significant connection with human beings, as in the case of a chair that grandma made or a work of art, such affectionate care can make sense. But having an affectionate care for plants makes sense even in the absence of a connection to human beings.

What about microscopic forms of life? Can it make sense to fondly feed a bacterium? I think so, but I agree that the case is less clear.

Grim Toe-Cutters

Imagine that Fred has all ten toes at 10 am, and there are infinitely many grim reapers. When a grim reaper wakes up, it looks at Fred, and if he has all his ten toes, it cuts one off and destroys it; otherwise, it does nothing. There are no other toe-cutters around.

Suppose, further, that grim reaper wake-up times can be set by you to any times between 10 and noon, endpoints not included. If you set the activation times to be such that there is a first activation time after 10 am (e.g., the nth reaper wakes up 60/n minutes before noon), there is no paradox of any sort. But if you set the times such that they are all after 10 am, but before every activation time there is another activation time, then… well, then logic guarantees that Fred will get a toe cut off infinitely many times and will regrow a toe infinitely many times! For without toe-regrowing, we get a paradox.

This is, of course, logically and metaphysically possible. Toes can regrow, and it is metaphysically though perhaps not physically possible for them to do so quickly. But what is amazing is that just by setting wake-up times for grim toe-cutters, we can make this miracle happen.

Grim Reapers and logical impossibility

The main objection to the Grim Reaper paradox as an argument against infinite causal sequences is the Unsatisfiable Pair (UP) objection that notes that paradox sets up an impossible situation—and that’s why it’s impossible!

I’m exploring a response that distinguishes metaphysical and (narrowly) logical unsatisfiability. The Grim Reaper situation is not logically unsatisfiable. The UP objection (well, really, Unsatisfiable Quadruple) notes that the following cannot all be true:

1. For all n > 0, the nth reaper wakes up at 60/n minutes after 10 am and kills Fred if and only if Fred is alive.

2. Fred is alive at 10 am.

3. There are no possible causes of Fred’s death other than those described in (1).

4. There are no possible causes of Fred’s resurrection.

But all that’s needed to have these four claims hold is for each reaper to kill Fred and then have Fred causelessly come back to life before the next one kills him. And while I think causeless resurrections are metaphysically impossible, they are (narrowly) logically coherent.

In other words, for the UP objection to work, the unsatisfiability must be metaphysical, not merely narrowly logical. But this, I think, negatively affects the force of the UP objection. For instance, in my Infinity book I consider Grim Reapers with adjustable wake-up times, and I note that for some wake-up time settings (say, the nth reaper wakes up 60/n minutes before noon) there is no paradox, and I ask what metaphysical force prevents the wake-up time settings from being the paradoxical ones. Daniel Rubio in a review of the book responds (in the context of a parody) that “no metaphysical thesis is required to explain this impossibility; the fact that it would lead to a contradiction is enough.” But in fact a metaphysical thesis is required to explain the impossibility, since there is no contradiction (in the narrowly logical sense) in (1)–(4).

Perhaps this is not a big deal. After all the metaphysical thesis here, that causeless events are impossible, is one that I do accept. But nonetheless it is a metaphysical thesis, as such on par with causal finitism, and hence when we consider the explanation of the impossibility of the Grim Reaper story and the impossibility of various other of the causal paradoxes that I discuss, there is something appealing about seeing the case as nonetheless offering support for causal finitism, which explains all of them, while the thesis about causeless events being impossible does not.

Unreliable Grim Reapers

As usual, Fred is alive at 10 am, and there is an infinite sequence of Grim Reapers, where the nth has an alarm set for 60/n minutes after 10 am, and if the alarm goes off, it checks if Fred is dead, and swings its scythe at Fred if and only if Fred is alive. But here’s the twist. These Grim Reapers are unreliable killers. The probability that the nth Reaper’s swing would succeed in killing Fred is 1/n^p, where p is some positive real number, the same for each Reaper, and independently of all other relevant events.

Here’s the fun thing. It seems possible for Fred to survive the whole ordeal. All it takes is for every Grim Reaper to fail at killing Fred. Nothing absurd happens then. Moreover, it seems this isn’t the only way for absurdity to be avoided in this case. We could also suppose that the nth Reaper kills Fred, while Reapers n + 1, n + 2, … all fail.

Suppose we adopt what seems the best alternative to Causal Finitism, namely the Inconsistent Pair response to the original Grim Reaper paradox, which says that the reason the original paradox is impossible is simply because it embodies an Inconsistent set of propositions—some Reaper has to kill Fred and none can. If that’s what’s wrong with the original Grim Reaper paradox, then it seems we have to accept my Unreliable Reaper story as possible.

But things are a little bit more complicated. The only way to avoid paradox in the Unreliable Reaper story is if there is some n ≥ 0 such that all the Reapers starting with Reaper n + 1 fail. But now suppose that 0 < p ≤ 1. Then the event that all the Reapers starting with Reaper n + 1 fail is less than or equal to (1−1/(n+1)^p)(1−1/(n+2)^p)(1−1/(n+3)^p)... = 0 (this is because Σ_k 1/k^p = ∞ if p ≤ 1). Thus the probability that we have avoided paradox is 0. Hence, if we have to avoid paradox, a specific zero probability event—namely, the event of paradox-avoidance—has to happen (the probability of a countable disjunction of zero probability events is zero). But if it has to happen, it can’t be probability zero, but must be probability one!

Perhaps here we bring back the Inconsistent Pair response. We say that my Unreliable Reaper story is impossible if p ≤ 1, because if p ≤ 1, then a zero probability event has probability one, which is inconsistent. No such problem occurs if p > 1. Thus, on this version of the Inconsistent Pair response, my Unreliable Reaper story is impossible if the success probability of the nth Reaper is 1/n^p for p ≤ 1 but possible if p > 1. And that’s pretty counterintuitive.

On finitistic addition

By a finite alphabet encoding of a set X, such as the real numbers, I mean a one-to-one function ψ from X to countably infinite sequences s₀s₁... taken from some finite alphabet. For instance, standard decimal encoding, with a decision whether to have infinite sequences of trailing nines or not, is a finite alphabet encoding of the reals, with the alphabet consisting of ten digits, a decimal point and a sign. Write ψ_k(x) for the kth symbol in the encoding ψ(x) of x.

A function f from Rⁿ to R is finitistic with respect to a finite alphabet encoding ψ provided that there is a function h from the natural numbers to the natural numbers such that the value of ψ_k(f(x₁,...,x_n)) depends only on the first h(k) symbols in each of ψ(x₁), ..., ψ(x_n).

This concept is related to concepts in “real computation”, but I am not requiring that the finite dependences be all implemented by the same Turing machine.

Theorem: Let X be any infinite divisible commutative group. Then addition on X is not finitistic with respect to any finite alphabet encoding.

A divisible group X is one where for every x ∈ X there is a y such that ny = x. The real numbers under addition are divisible. So are the rationals. So is the set of all rotations in the plane.

This has a somewhat unhappy consequence for Information Processing Finitism. If reality encodes real numbers in a discrete way consistent with IPF, we should not expect each real number to have a uniquely specified encoding.

Proof of Theorem: Suppose addition is finitistic with respect to ψ. Let F be the algebra on X generated by the sets of the form {x : ψ_k(x) = α}. If addition is finitistic, then for any A ∈ F, there is a finite sequence of pairs (A₁,B₁), ..., (A_N,B_N) of sets in F such that

1. {(x,y) : x + y ∈ A} = ⋃_i(A_i×B_i).

Therefore:

2. x + y ∈ A if and only if x ∈ ⋃{A_i : y ∈ B_i}.

Thus:

3.  − y + A =  ∪ {A_i : y ∈ B_i}.

Now as y varies over the members of X, there are at most 2^N different sets generated by the right hand side. Thus,  − y + A can take on only finitely many values. Hence, A has only finitely many translates.

But this is impossible. Let Z be the set of x such that x + A = A. This is an additive subgroup of X. Note that x + Z = y + Z iff x − y ∈ Z iff (x−y) + A = A iff x + A = y + A. Thus, if there are only finitely many x + A, there are finitely many x + Z. Hence X/Z is a finite group. Let n be its order. Then n[x] = 0 for every coset [x] = x + Z in R/Z. For any x ∈ X choose y such that ny = x. Then n[y] = 0, and so [x] = 0, thus Z = X. It follows that A is invariant under every translation, so it must be either ⌀ or X. Hence |F| ≤ 2, which is absurd since F is infinite as X is infinite and ψ is one-to-one.

(I got the main idea for this proof from the answer here.)

Empirical mathematics

Suppose I want to figure out a good approximation to the eigenvalues of a certain Hamiltonian involving a moderately large number of Coulomb potentials. It could well be the case that the best way to do so is to synthesize a molecule with that Hamiltonian and then measure its spectrum. In other words, there are mathematical problems where our best solution to the problem uses scientific methods rather than mathematical proof.

Information Processing Finitism, Part II

In my previous post, I explored information processing finitism (IPF), the idea that nothing can essentially causally depend on an infinite amount of information about contingent things.

Since a real-valued parameter, such as mass or coordinate position, contains an infinite amount of information, a dynamics that fits with IPF needs some non-trivial work. One idea is to encode a real-valued parameter r as a countable sequence of more fundamental discrete parameters r₁, r₂, ... where r_i takes its value in some finite set R_i, and then hope that we can make the dynamics be such that each discrete parameter depends only on a finite number of discrete parameters at earlier times.

In the previous post, I noted that if we encode real numbers as Cauchy sequences of rationals with a certain prescribed convergence rate, then we can do something like this, at least for a toy dynamics involving continuous functions on between 0 and 1 inclusive. However, an unhappy feature of the Cauchy encoding is that it’s not unique: a given real number can have multiple Cauchy encodings. This means that on such an account of physical reality, physical reality has more information in it than is expressed in the real numbers that are observable—for the encodings are themselves a part of reality, and not just the real numbers they encode.

So I’ve been wondering if there is some clever encoding method where each real number, at least between 0 and 1, can be uniquely encoded as a countable sequence of discrete parameters such that for every continuous function f from [0,1] to [0,1], the value of each parameter discrete parameter corresponding to of f(x) depends only on a finite number of discrete parameters corresponding to x.

Sadly, the answer is negative. Here’s why.

Lemma. For any nonempty proper subset A of [0,1], there are uncountably many sets of the form f⁻¹[A] where f is a continuous function from [0,1] to [0,1].

Given the lemma, without loss of generality suppose all the parameters are binary. For the ith parameter, let B_i be the subset of [0,1] where the parameter equals 1. Let F be the algebra of subsets of [0,1] generated by the B_i. This is countable. Any information that can be encoded by a finite number of parameters corresponds to a member of F. Suppose that whether f(x) ∈ A for some A ∈ F depends on a finite number of parameters. Then there is a C ∈ F such that x ∈ C iff f(x) ∈ A. Thus, C = f⁻¹[A]. Thus, F is uncountable by the lemma, a contradiction.

Quick sketch of proof of lemma: The easier case is where either A or its complement is non-dense in [0,1]—then piecewise linear f will do the job. If A and its complement are dense, let (a_n) and (b_n) be a sequence decreasing to 0 such that both a_n and b_n are within 1/2^n + 2 of 1/2ⁿ, but a_n ∈ A and b_n ∉ A. Then for any set U of positive integers, there will be a strictly increasing continuous function f_U such that f_U(a_n) = a_n if n ∈ U and f_U(b_n) = a_n if n ∉ U. Note that f_U⁻¹[A] contains a_n if and only if n ∈ A and contains b_n if and only if n ∉ A. So for different sets U, f_U⁻¹[A] is different, so there are continuum-many sets of the form f_U⁻¹[A].

Information Processing Finitism

When I was trying to work out my intuitions about causal paradoxes of infinity, which eventually led to my formulating the thesis of causal finitism (CF)—that nothing can have an infinite causal history—I toyed with views that involved information. I ended up largely abandoning that approach, partly because of my qualms about the concept of information and perhaps partly because of worries about physics that I will discuss below.

But I still think the alternative, which one might call information processing finitism, is something someone should work out in more detail.

• [IPF] Nothing with finite informational content can essentially causally depend on anything with infinite informational content.

Here, informational content is by definition contingent. The “essentially” excludes cases where finite informational content depends on a finite part of something with infinite informational content. How exactly the “essentially” is spelled out is one thing I am not clear on as yet.

The main difficulty with IPF is that our physics seems to violate it. The exact current temperature in Waco depends on the exact temperature, pressure and other facts around the world yesterday. Each of the latter facts involves infinite information—temperature is quantified with a real number, and a real number contains infinite information. Note that here IPF and CF may diverge. An advocate of CF can say that the exact current temperature in Waco depends on a finite number of past events such as “yesterday particle n has parameters P”, even if the parameters P involve real numbers that have infinite energy.

One way to escape this difficulty is to assume that our fundamental physics is actually discrete, and the real numbers in our equations are just an approximation. But I don’t want to stick my neck out so far.

Let’s see if we can make IPF work out with a continuous dynamics. We can suppose that metaphysically speaking, an entity’s having a real-valued parameter is constituted by the entity’s having an infinite sequence of discrete parameters, which parameters are more ontologically fundamental than the real-valued parameter.

For instance, by a one-to-one mapping we can assume our real number is strictly between zero and one, and then define it as an infinite decimal sequence 0.b₁b₂..., specified by an infinite sequence of digits. Unfortunately, then, we have some severe restrictions on what kind of dynamics we can have if we require that each digit of the output depend only on a finite number of digits of the input. For instance, multiplication by 3/4 cannot be defined, because to know whether f(x) starts with 0.24 or 0.25, you’d have to know whether x < 1/3 or x ≥ 1/3, and if the input is 0.333..., then you can’t tell from a finite number of digits which is the case. This kind of problem will occur with any other base.

It would be really nice to find some way of encoding a real number as an infinite sequence of discrete parameters each of which takes on a fixed finite range that escapes this kind of a problem. I am pretty sure this is impossible, but am too tired to prove it right now.

But there is another approach. We can have non-unique (many-to-one) encodings of reals. Here is one such approach, probably not the most natural one. Consider sequences of natural numbers n₁, n₂, ... such that for all k we have n_k ≤ 2k and there exists a real number x between 0 and 1 inclusive with the property that |x−n_k/2k| ≤ 1/k. Say that such a sequence encodes the real number x. In general, there will be more than one sequence encoding x by this rule.

Then if f is a function from [0,1] to [0,1], if we have a sequence n₁, n₂, ... encoding the real number x, to generate an acceptable kth term in a sequence encoding f(x), it suffices to know f(x) to within precision 1/2k, and if f is continuous, then we can do that by knowing a finite number of terms in a sequence encoding x (this is because every continuos function on [0,1] is uniformly continuous).

So any continuous dynamics from [0,1] to [0,1] can be handled in this way. The cost is that fundamental reality has degrees of freedom that are unimportant physically—for fundamental reality distinguishes between different sequences encoding the same real x, but the difference has no physical significance.

I don’t know if there is a way to do this with a unique encoding.

Mereology, plural quantification and free lunches

It is sometimes claimed that arbitrary mereological fusions and plural quantification are a metaphysical free lunch, just a new way of talking without any deep philosophical (or at least metaphysical) commitments.

I think this is false.

Consider this Axiom of Choice schema for mereology:

1. If for every x and y such that ϕ(x) and ϕ(y), either x = y or x and y don’t overlap, and if every x such that ϕ(x) has a part y such that ψ(y), then there is a z such that for every x such that ϕ(x), there is common part y of x and z such that ψ(y).

Or this Axiom of Choice schema for pluralities:

2. If for all xx and yy such that ϕ(xx) and ϕ(yy) either xx and yy are the same or have nothing in common, then there are zz that have exactly one thing in common with every xx such that ϕ(xx).

If arbitrary mereological fusions and plural quantification are a metaphysical free lunch, just a handy way of talking, then whether (1) or (2) is correct is just a verbal question.

But (1) and (2) respectively imply mereological and plural Banach-Tarski paradoxes:

3. If z is a solid ball made of points, then it has five pairwise non-overlapping parts, of which the first two can be rigidly moved to be pairwise non-overlapping and compose another ball of the same size as z, and the last three can likewise be so moved.

4. If the xx are the points of a solid ball, then there are aa, bb, cc, dd and ee which have nothing pairwise in common and such that together they make up xx, and there are rigid motions that allow one to move aa and bb into pluralities that have nothing in common but make up a solid ball of the same size as xx and to move cc, dd and ee into pluralities that have nothing in common and make up another solid ball of the same size.

Conversely, assuming ZF set theory is consistent, there is no way to prove (3) and (4) if we do not have some extension to the standard axioms of mereology or plurals like the Axiom of Choice. The reason is that we can model pluralities and mereological objects with sets of points in three-dimensional space, and either (3) or (4) in that setting will imply the Banach-Tarski paradox for sets, while the Banach-Tarski paradox for sets is known not to be provable from ZF set theory without Choice.

But whether (3) or (4) is true is not a purely verbal question.

One reason it’s not a purely verbal question is intuitive. Banach-Tarski is too paradoxical for it or its negation to be a purely verbal thing.

Another is a reason that I gave in a previous post with a similar argument. Whether the Banach-Tarski paradox holds for sets is not a purely verbal question. But assuming that the Axiom of Separation can take formulas involving mereological terminology or plural quantification, each of (3) and (4) implies the Banach-Tarski paradox for sets.

Some stuff about models of PA+~Con(PA)

Assume Peano Arithmetic (PA) is consistent. Then it can’t prove its own consistency. Thus, there is a model M of PA according to which PA is inconsistent, and hence, according M, there is a proof of a contradiction from a finite set of axioms of PA. This sounds very weird.

But it becomes less weird when we realize what these claims do and do not mean in M.

The model M will indeed contain an M-natural number a that according to M encodes a finite sequence of axioms of PA, and it will also contain an M-natural number p that according to M encodes a proof of a contradiction using the axioms encoded in A.

However, here are some crucial qualifications. Distinguish between the M-natural numbers that are standard, i.e., correspond to an actually natural number, one that from the point of view of the “actual” natural numbers is finite, and those that are not. The latter are infinite from the point of view of the actual natural numbers.

First, the M-natural number a is non-standard. For a standard natural number will only encode a finite number of axioms, and for any finite subtheory of PA, PA can prove its consistency (this is the “reflexivity of PA”, proved by Mostowski in the middle of the last century). Thus, if a were a standard natural number, according to M there would be no contradiction from the axioms in a.

Second, while every item encoded in a is according to M an axiom of PA, this is not actually true. This is because any M-finite sequence of M-natural numbers will either be a standardly finite length sequence of standard natural numbers, or will contain a non-standard number. For let n be the largest element in the sequence. If this is standard, then we have a standardly finite length sequence of standard natural numbers. If not, then the sequence contains a non-standard number. Thus, a contains something that is not axiom of PA.

In other words, according to our model M, there is a contradictory collection of axioms of PA, but when we query M as to what that collection is, we find out that some of the things that M included in the collection are not actually axioms of PA. (In fact, they won’t even be well-formed formulas, since they will be infinitely long.) So a crucial part of the reason why M disagrees with the “true” model of the naturals about the consistency of PA is because M disagrees with it about what PA actually says!

Plato and teaching philosophy to the young

In the Republic, Plato says philosophy education shouldn’t start until age 30. I’ve long worried about Plato’s concern about providing young people with tools that, absent intellectual and moral maturity, can just as well be used for sophistry.

Exegetically, however, I think I was missing an important point: Plato is talking about his utopian society, where one can (supposedly) count on society raising the young person to practice the virtues and live by the truth (except for the noble lie). We do not live in such a society. It could well be the case that in our society, young people need the tools.

We might make a judgment like this. Absent the tools of a philosophical education, an intelligent young person set afloat on the currents of our society maybe is 50% likely to be led astray by these currents. The tools are unreliable especially in the hands of the young: perhaps the tools have a 65% chance of leading to the right and 35% of leading to ill. That’s still better than letting the young person navigate society without the tools. But if our society were better—as Plato thinks is the case in his Republic—then the unreliable tools might be worse than just letting society form one.

A puzzle about consistency

Let T₀ be ZFC. Let T_n be T_n − 1 plus the claim Con(T_n − 1) that T_n − 1 is consistent. Let T_ω be the union of all the T_n for finite n.

Here’s a fun puzzle. It seems that T_ω should be able to prove its own consistency by the following reasoning:

If T_ω is inconsistent, then for some finite n we have T_n inconsistent. But Con(T_n) is true for every finite n.

This sure sounds convincing! It took me a while to think through what’s wrong here. The problem is that although for every finite n, T_ω can prove Con(T_n), it does not follow that T_ω can prove that for every finite n we have Con(T_n).

To make this point perhaps more clear, assume T_n is consistent for all n. Then Con(T_n) cannot be proved from T_n. Thus any finite subset of T_ω is consistent with the claim that for some finite n the theory T_n is inconsistent. Hence by compactness there is a model of T_ω according to which for some finite n the theory T_n is inconsistent. This model will have a non-standard natural number sequence, and “finite” of course will be understood according to that sequence.

Here’s another way to make the point. The theory T_ω proves T_ω consistent if and only if T_ω is consistent according to every model M. But the sentence “T_ω is consistent according to M” is ambiguous between understanding “T_ω” internally and externally to M. If we understand it internally to M, we mean that the set that M thinks consists of the axioms of ZFC together with the ω-iteration of consistency claims is consistent. And this cannot be proved if T_ω is consistent. But if we understand “T_ω” externally to M, we mean that upon stipulating that S is the object in M’s universe whose members^M correspond naturally to the members^V of T_ω (where V is “our true set theory”), according to M, it will be provable that the set S is consistent. But there is a serious problems: there just may be no such object as S in the domain of M and the stipulation may fail. (E.g., in non-standard analysis, the set of finite naturals is never an internal set.)

(One may think a second option is possible: There is such an object as S in M’s universe, but it can’t be referred to in M, in the sense that there is no formula ϕ(x) such that ϕ is satisfied by S and only by S. This option is not actually possible, however, in this case.)

Or so it looks to me. But all this is immensely confusing to me.

Existential inertia and spacetime

According to the principle of existential inertia:

1. If x exists at t₁ and t₂ > t₁ and there is no cause of x’s not existing at t₂, then x exists at t₂.

This sounds weird, and one way to get at the weirdness for me is to put it in terms of relativity theory. Times are spacelike hypersurfaces. So, then:

2. If x exists somewhere on a spacelike hypersurface H₁ and H₂ is a later spacelike hypersurface and there is no cause of x’s not existing on H₂, then x exists on H₂.

This seems weird to me. Why should being in one specific area of spacetime metaphysically push one to exist in another specific area of spacetime? I can see how existing in one area of spacetime could physically push one to exist in another. But metaphysically? That seems odd.

Non-formal provability

A simplified version of Goedel’s first incompleteness theorem (it’s really just a special case of Tarski’s indefinability of truth) goes like this:

• Given a sound semidecidable system of proof that is sufficiently rich for arithmetic, there is a true sentence g that is not provable.

Here:

• sound: if s is provable, s is true

• semidecidable: there is an algorithm that given any provable sentence verifies in a finite number of steps that it is provable.

The idea is that we start with a precisely defined ‘formal’ notion of proof that yields semidecidability of provably, and show that this concept of proof is incomplete—there are truths that can’t be proved.

But I am thinking there is another way of thinking about this stuff. Suppose that instead of working with a precisely defined concept of proof, we have something more like a non-formal or intuitive notion of proof, which itself is governed by some plausible axioms—if you can prove this, you can prove that, etc. That’s kind of how intuitionists think, but we don’t need to be intuitionists to find this approach attractive.

Note that I am not explicitly distinguishing axioms.

The idea is going to be this. The predicate P is not formally defined, but it still satisfies some formal constraints or axioms. These can be formulated in a formal language (Brouwer wouldn’t like this) that has a way of talking about strings of symbols and their concatenation and allows one to define a quotation function that given a string of symbols returns a string of symbols that refers to the first string.

One way to do this is to have a symbol ′α′ for any symbol α in the original language which refers to α, and a concatenation operator +, so one can then quote αβγ as ′α′ + ′β′ + ′γ′. I assume the language is rich enough to define a quotation function Q such that Q(x) is the quotation of a string x.

To formulate my axioms, I will employ some sloppy quotation mark shorthand, partly to compensate for the difficulty of dealing with corner quotes on the web. Thus, ′αβγ′ is shorthand for ′α′ + ′β′ + ′γ′, and as needed I will allow substitution inside the quotation marks. If there are nested quotation marks, the inner substitutions are resolved first.

1. For all sentences ϕ and ψ, if P(′ϕ↔︎∼ψ′) and P(′ϕ′), then P(′∼ϕ′).

2. For all sentences ϕ and ψ, if P(′ϕ↔︎∼ψ′) and P(′ψ→ϕ′), then P(′ϕ′).

3. For all sentences ϕ, we have P(′P(′ϕ′)→ϕ′).

4. If ϕ has a formal intuitionistic proof from sufficiently rich axioms of concatenation theory, then P(′ϕ′).

Here, (1) and (2) embody a little bit of facts about proof, both of which facts are intuitionistically and classically acceptable. Assumption (3) is the philosophically heaviest one, but it follows from its being axiom that if ϕ is provable, then ϕ, together with the fact that all axioms count as provable. That a formal intuitionistic proof is sufficient for provability is uncontroversial.

Using similar methods to those used to prove Goedel’s first incompleteness theorem, I think we should now be able to construct a sentence g and the prove, in a formal intuitionistic proof in a sufficiently rich concatenation theory, that:

5. g ↔︎  ∼ P(′g′).

But these facts imply a contradiction. Since 5 can be proved in our formal way, we have:

6. P(′g↔︎∼P(′g′)′). By 4.

7. P(′P(′g′)→g′). By 3.

8. P(′g′). By 6, 7 and 2.

9. P(′∼g′). By 6, 8 and 1.

Hence the system P is inconsistent in the sense that it makes both g and  ∼ g are provable.

This seems to me to be quite a paradox. I gave four very plausible assumptions about a provability property, and got the unacceptable conclusion that the provability property allows contradictions to be proved.

I expect the problem lies with 3: it lets one ‘cross levels’.

The lesson, I think, is that just as truth is itself something where we have to be very careful with the meta- and object-language distinction, the same is true of proof if we have something other than a formal notion.

Our Baylor alligator

A year or two ago, some juvenile alligators moved into our area or were moved in (we are right on the edge of the alligator zone in Waco). I've been hoping to see one of them. Finally, today, I did, right by Baylor's marina, as you can see from the reflection of the Baylor logo.

A new argument for causal finitism

I will give an argument for causal finitism from a premise I don’t accept:

1. Necessary Arithmetical Alethic Incompleteness (NAAI): Necessarily, there is an arithmetical sentence that is neither true nor false.

While I don’t accept NAAI, some thinkers (e.g., likely all intuitionists) accept it.

Here’s the argument:

2. If infinite causal histories are possible, supertasks are possible.

3. If supertasks are possible, for every arithmetical sentence, there is a possible world where someone knows whether the sentence is true or false by means of a supertask.

4. If for every arithmetical sentence there is a possible world where someone knows whether the sentence is true or false by means of a supertask, there is a possible world where for every arithmetical sentence someone knows whether it is true or false.

5. Necessarily, if someone knows whether p is true or false, then p is true or false.

6. So, if infinite causal histories are possibly, possibly all arithmetical sentences are true or false. (2-5)

7. So, infinite causal histories are impossible. (1, 6)

The thought behind (3) is that if for every n it is possible to check the truth value of ϕ(n) by a finite task or supertask, then by an iterated supertask it is possible to check the truth values of ∀xϕ(x) (and equivalently ∃xϕ(x)). Since every arithmetical sequence can be written in the form Q₁x₁...Q_kx_kϕ(x₁,...,x_k), where the truth value of ϕ(n₁,...,n_k) is finitely checkable, it follows that every arithmetical sequence can have its truth value checked by a supertask.

The thought behind (4) is that one can imagine an infinite world (say, a multiverse) where for every arithmetical sentence ϕ the relevant supertask is run and hence the truth value of the sentence is known.

Provability and truth

The most common argument that mathematical truth is not provability uses Tarski’s indefinability of truth theorem or Goedel’s first incompleteness theorem. But while this is a powerful argument, it won’t convince an intuitionist who rejects the law of excluded middle. Plus it’s interesting to see if a different argument can be constructed.

Here is one. It’s much less conclusive than the Tarski-Goedel approach. But it does seem to have at least a little bit of force. Sometimes we have experimental evidence (at least of the computer-based kind) for a mathematical claim. For instance, perhaps, you have defined some probabilistic setup, and you wonder what the expected value of some quantity Q is. You now set up an apparatus that implements the probabilistic setup, and you calculate the average value of your observations of Q. After a billion runs, the average value is 3.141597. It’s very reasonable to conclude that the last digit is a random deviation, and that the mathematically expected value of Q is actually π.

But is it reasonable to conclude that it’s likely provable that the expected value of Q is π? I don’t see why it would be. Or, at least, we should be much less confident that it’s provable than that the expected value is π. Hence, provability is not truth.

Reducing promises to assertions

To promise something, I need to communicate something to you. What is that thing that I need to communicate to you? To a first approximation, what I need to communicate to you is that I am promising. But that’s circular: it says that promising is communicating that I am promising. This circularity is vicious, because it doesn’t distinguish promising from asking: asking is communicating that I am asking.

But now imagine I have a voice-controlled robot named Robby, and I have programmed him in such a way that I command him by asserting that Robby will do something because I have said he will do it. Thus, to get him to vacuum the living room, I assert “Robby will immediately vacuum the living room because I say so.” As long as what I say is within the range of Robby’s abilities, any statement I make in Robby’s vicinity about what he will do because I say he will do it is automatically true. This is all easily imaginable.

Now, back to promises. Perhaps it works like this. I have a limited power to control the normative sphere. This normative power generates an effect in normative space precisely when I communicate that I am generating that effect. Thus, I can promise to buy you lunch by asserting “I will be obligated to you to buy you lunch.” And I permit you to perform heart surgery by asserting “You will cease to have a duty of respect for my autonomy not to perform heart surgery on me.” As long as what I say is within my normative capabilities, by communicating that I am making it true by communicating it, I make it be true, just as Robby will do what I assert he will do because of my say-so, as long as it is within his physical capabilities.

This solves the circularity problem for promising because what I am communicating is not that I am promising, but the normative effect of the promising:

1. x promises to ϕ to y if and only if x successfully exercises a communicative normative power to gain an obligation-to-y by ϕing

2. a communicative normative power for a normative effect F is a normative power whose object is F and whose successful exercise requires the circumstance that one express that one is producing F by communicating that one is so doing.

There are probably some further tweaks to be made.

Of course, in practice, we communicate the normative effect not by describing it explicitly, but by using set phrases, contextual cues, etc.

This technique allows us to reduce promising, consenting, requesting, commanding and other illocutionary forces to normative power and communicating, which is basically a generalized version of assertion. But we cannot account for communicating or asserting in this way—if we try to do that, we do get vicious circularity.

A curious poker variant

In some games like Mafia, uttering falsehoods is a part of the game mechanic. These falsehoods are no more lies than falsehoods uttered by an actor in a performance are lies.

Now consider a variant of poker where a player is permitted to utter falsehoods when and only when they have a Joker in hand. In this case when the player utters a falsehood with Joker in hand, there is no lie. The basic communicative effect of uttering s is equivalent to asserting “s or I have a Joker in hand (or both)”, though there may be additional information conveyed by bodily expression, tone of voice, or context.

If this analysis of poker variant is correct, then the following seems to follow by analogy. Suppose, as many people think, that it is morally permissible to utter falsehoods in “assertoric contexts” to save innocent lives. (An assertoric context is roughly one where the speaker is appropriately taken to be asserting.) Given that we are always playing the “morality game”, by analogy this would mean that in paradigm instances when we utter a declarative sentence s, we are actually communicating something like “s or I am speaking to save innocent lives.” If this is right, then it is impossible to lie to save innocent lives, just as in my poker variant it is impossible to lie when one knows one has the Joker in hand (unless maybe one is really bad at logic).

The above argument supports this premise:

1. If it is morally permissible to utter falsehoods in assertoric contexts to save innocent lives, it is not possible to lie to save innocent lives.

But:

2. It is possible to lie to save innocent lives.

I conclude:

3. It is not morally permissible to utter falsehoods in assertoric contexts to save innocent lives.

In short: lying is wrong, even to save innocent lives.

Evolution of my views on mathematics

I have for a long time inclined towards ifthenism in mathematics: the idea that mathematics discovers truths of the form "If these axioms are true, then this thing is true as well."

Two things have weakened my inclination to ifthenism.

The first is that there really seems to be a privileged natural number structure. For any consistent sufficiently rich recursive axiomatization A of the natural numbers, by Goedel’s Second Incompleteness Theorem (plus Completeness) there is a natural number structure satisfying A accordingto which A is inconsistent and there is a natural number structure satisfying A according to which A is consistent. These two structures can’t be on par—one of them needs to be privileged.

The second is an insight I got from Linnebo’s philosophy of mathematics book: humans did mathematics before they did axiomatic mathematics. Babylonian apparently non-axiomatic but sophisticated mathematics came before Greek axiomatic geometry. It is awkward to think that the Babylonians were discovering ifthenist truths, given that they didn’t have a clear idea of the antecedents of the ifthenist conditionals.

I am now toying with the idea that there is a metaphysically privileged natural number structure but we have ifthenism for everything else in mathematics.

How is the natural number structure privileged? I think as follows: the order structure of the natural numbers is a possible order structure for a causal sequence. Causal finitism, by requiring all initial segments under the causal relation to be finite, requires the order type of the natural numbers to be ω. But once we have fixed the order type to be ω, we have fixed the natural number structure to be standard.