Four-flour pancakes

I was watching an old Aunt Jemima pancake mix commercial which touted it as being made from four flours: wheat, corn, rye and rice, and I decided to see what pancakes made them are like. I started with this wheat flour pancake recipe, but tweaked some things, and made them this morning. Pretty good. Perhaps more hearty than standard pancakes, and the texture was a bit more crunchy, which I liked.

• 1/2 cup of wheat flour

• 1/2 cup of whole-grain rye flour

• 1/2 cup of corn flour

• 1/2 cup of (non-glutinous) rice flour

• 4 3/4 teaspoons baking powder

• 4 teaspoons white sugar

• 1/3 teaspoon salt

• 1 2/3 cup milk

• 4 tablespoons melted butter

• 1 large egg

• 4 teaspoons apple sauce (or skip and use 1 1/3 egg, if you have some use for the remaining 2/3 of the egg)

• cooking spray (I used canola spray)

• optional: chocolate chips

Mix dry ingredients. Add wet ingredients. Mix well. Heat pan to medium heat. Spray with oil. Put a big serving spoon of mix on the pan. If you want to add chocolate chips, drop them in on top. Wait until the edges are getting dry. (It was surprisingly fast, about 1-2 minutes, and they would burn easily when I wasn’t fast enough.) Flip and brown the other side (again, it’s fast).

Yields 9-10 not very large pancakes. The frying took half an hour with two pans in simultaneous use. I measured out all the ingredients the night before and pre-mixed the dry ingredients so I could be fast in the morning before a pickleball game.

The value of moral norms

Here is a very odd question that occurred to me: Is it good for there to be moral norms?

Imagine a world just like this one, except that there are no moral norms for its intelligent denizens—but nonetheless they behave as we do. They feel repelled by the idea of murder and torture, and find the life of a Mother Teresa attractive, but there are no moral truths behind these things.

Such a world would have one great advantage over ours: there would be no moral evil. That world’s Hitler and Stalin would cause just as much pain and suffering, but they wouldn’t be wicked in so doing. Given the Socratic insight that it is worse to do than to suffer evil, a vast amount of evil would disappear in such a world. At least a third of the evil in the world would be gone. Our world has three categories of evil:

I. Undergoing of natural evils

2. Undergoing of moral evils, and

3. Performance of moral evils.

The third category would be gone, and it is probably the biggest of the three. Wouldn’t that be worth it?

Here is one answer. For cooperative intelligent social animals, a belief in morality is very useful. But to live one’s life by a belief that is false seems a significant harm. Cooperative intelligent social animals in the alternative world would be constantly deceived by their belief in morality. That is a great evil. But is it as great an evil as all Category III evils taken together? I suspect it is but a small fraction of the sum of all Category III evils.

Here is a second answer. In removing moral norms, one would admittedly remove a vast category of evils, but also a vast category of goods: the performance of moral good. If we have the intuition that having moral norms is a good thing—that it would be a disappointment to learn that moral norms were an illusion—then we have to think that the performances of moral good are a very great thing indeed, one comparable to the sum of all Category III evils.

I am attracted to a combination of the two answers. But I can also see someone saying: “It doesn’t matter whether it’s worth having moral norms or not, but it is simply impossible to have cooperative intelligent social animals that believe in morality without their being under moral norms.” A Platonist may say that on the grounds that moral norms are necessary. A theist may say it on the grounds that it is contrary to the character of a perfect God to manufacture the vast deceit that would be involved in us thinking there are moral norms if there were no moral norms. These aren’t bad answers. But I still feel it’s good that there really are moral norms.

Philosophy and child-raising

Philosophy Departments often try to attract undergraduates by telling them about instrumental benefits of philosophy classes: learning generalizable reading, writing and reasoning skills, doing better on the LSAT, etc.

But here is a very real and much more direct reason why lots of people should take philosophy classes. Most people end up having children. And children ask lots of questions. These questions include philosophical ones. Moreover, as they grow, especially around the teenage years, philosophical questions come to have special existential import: why should I be virtuous, what is the point of life, is there life after death, is there a God, can I be sure of anything?

For children’s scientific questions, there is always Wikipedia. But that won’t be very helpful with the philosophical ones. In a less diverse society, where parents can count on agreeing philosophically with the schools, parents could outsource children’s philosophical questions to a teacher they agree with. Perhaps religious parents can count on such agreement if they send their children to a religious school, but in a public school this is unlikely. (And in any case, outsourcing to schools is still a way of buying into something like universal philosophical education.) So it seems that vast numbers of parents need philosophical education to raise their children well.

Hyperreal infinitesimal probabilities and definability

In order to assign non-zero probabilities to such things as a lottery ticket in an infinite fair lottery or hitting a specific point on a target with a uniformly distributed dart throw, some people have proposed using non-zero infinitesimal probabilities in a hyperreal field. Hajek and Easwaran criticized this on the grounds that we cannot mathematically specify a specific hyperreal field for the infinitesimal probability. If that were right, then if there are hyperreal infinitesimal probabilities for such a situation, nonetheless we would not be able to say what they are. But it’s not quite right: there is a hyperreal field that is "definable", or fully specifiable in the language of ZFC set theory.

However, for Hajek-Easwaran argument against hyperreal infinitesimal probabilities to work, we don’t need that the hyperreal field be non-definable. All we need is that the pair (^*R,α) be non-definable, where ^*R is a hyperreal field and α is the non-zero infinitesimal assigned to something specific (say, a single ticket or the center of the target).

But here is a fun fact, much of the proof of which comes from some remarks that Michael Nielsen sent me:

Theorem: Assume ZFC is consistent. Then ZFC is consistent with there not being any definable pair (^*R,α) where ^*R is a hyperreal field and α is a non-zero infinitesimal in that field.

[Proof: Solovay showed there is a model of ZFC where every definable set is measurable. But every free ultrafilter on the powerset of the naturals is nonmeasurable. However, an infinite integer in a hyperreal field defines a free ultrafilter on the naturals—given an infinite integer M, say that a subset A of the naturals is a member of the ultrafilter iff |M| ∈ ^*A. And a non-zero infinitesimal defines an infinite integer—say, as the floor of its reciprocal.]

Given the Theorem, without going beyond ZFC, we cannot count on being able to define a specific hyperreal non-zero infinitesimal probability for situations like a ticket infinite lottery or hitting the center of a target. Thus, if a friend of hyperreal infinitesimal probabilities wants to be able to define one, they must go beyond ZFC (ZFC plus constructibility will do).

Doxastic moral relativism

Reductive doxastic moral relativism is the view that an action type’s being morally wrong is nothing but an individual or society’s belief that the action type is morally wrong.

But this is viciously circular, since we reduce wrongness to a belief about wrongness. Indeed, it now seems that murder is wrong provided that it is believed that it is believed that it is believed ad infinitum.

A non-reductive biconditional moral relativism fares better. This is a theory on which (a) there is such a property as moral wrongness and (b) necessarily, an action type has that property if and only if it is believed that it does. Compare this: There is such a property as mass, and necessarily an object has mass if and only if God believes that it has mass.

There is a biconditional-explanatory version. On this theory (a) there is such a property as moral wrongness and (b) necessarily, an action type has that property if and only if, and if so then because, it is believed that it does.

While both the biconditional and biconditional-explanatory versions appear logically coherent, I think they are not particularly plausible. If there really is such a property as moral wrongness, and it does not reduce to our beliefs, then it just does not seem particularly plausible to think that it obtains solely because of our beliefs or that it obtains necessarily if and only if we believe it does. The only clear and non-gerrymandered examples we have of properties that obtain solely because of our beliefs or necessarily if and only if we believe they do are properties that reduce to our beliefs.

All this suggests to me that if one wishes to be a relativism, one should base the relativism on a different attitude than belief.

Sacraments and New Testament law

Christians believe that Jesus commanded us to baptize new Christians. However, there is a fundamental division in views: some Christians (such as Catholics and the Orthodox) have a sacramental view of baptism, on which baptism as such leads to an actual supernaturally-produced change in the person baptized, while others hold a symbolic view of it.

Here is an argument for the sacramental view. We learn from Paul that there is a radical change in God’s law from Old to New Testament times. I think our best account of that change is that we are no longer under divinely-commanded ceremonial and symbolic laws, but as we learn from the First Letter of John, we are clearly still under the moral law.

On the symbolic view, however, baptism is precisely a ceremonial and symbolic law—precisely the kind of thing that we are no longer under. On the sacramental view, however, it is easy to explain how baptism falls under the moral law. Love of neighbor morally enjoins on us that we provide effective medical treatment to our neighbor, and love of self requires us to seek such treatment for ourselves. Similarly, if baptism is crucial to the provision of grace for moral healing, then love of neighbor morally enjoins on us that we baptize and love of self requires us to seek baptism for ourselves.

The same kind of argument applies to the Eucharist: since it is commanded by God in New Testament times, it is not merely symbolic.

Semantics of syntactically incorrect language

As anyone who has talked with a language-learner knows, syntactically incorrect sentences often succeed in expressing a proposition. This is true even in the case of formal languages.

Formal semantics, say of the Tarski sort, has difficulties with syntactically incorrect sentences. One approach to saving the formal semantics is as follows: Given a syntactically incorrect sentence, we find a contextually appropriate syntactically correct sentence in the vicinity (and what counts as vicinity depends on the pattern of errors made by the language user), and apply the formal semantics to that. For instance, if someone says “The sky are blue”, we replace it with “The sky is blue” in typical contexts and “The skies are blue” in some atypical contexts (e.g., discussion of multiple planets), and then apply formal semantics to that.

Sometimes this is what we actually do when communicating with someone who makes grammatical errors. But typically we don’t bother to translate to a correct sentence: we can just tell what is meant. In fact, in some cases, we might not even ourselves know how to translate to a correct sentence, because the proposition being expressed is such that it is very difficult even for a native speaker to get the grammar right.

There can even be cases where there is no grammatically correct sentence that expresses the exact idea. For instance, English has a simple present and a present continuous, while many other languages have just one present tense. In those languages, we sometimes cannot produce an exact grammatically correct translation of an English sentence. One can use some explicit markers to compensate for the lack of, say, a present continuous, but the semantic value of a sentence using these markers is unlikely to correspond exactly to the meaning of the present continuous (the markers may have a more determinate semantics than the present continuous). But we can imagine a speaker of such a language who imitates the English present continuous by a literal word-by-word translation of “I am” followed by the other language’s closest equivalent to a gerund, even when such translation is grammatically incorrect. In such a case, assuming the listener knows English, the meaning may be grasped, but nobody is capable of expressing the exact meaning in a syntactically correct way. (One might object that one can just express the meaning in English. But that need not be true. The verb in question may be one that does not have a precise equivalent in English.)

Thus we cannot account for the semantics of syntactically incorrect sentences by applying semantics to a syntactically corrected version. We need a semantics that works directly for syntactically incorrect sentences. This suggests that formal semantics are necessarily mere approximate models.

Similar issues, of course, arise with poetry.

Truth-value realisms about arithmetic

Arithmetical truth-value realists hold that any proposition in the language of arithmetic has a fully determined truth value. Arithmetical truth-value necessists add that this truth value is necessary rather than merely contingent. Although we know from the incompleteness theorems that there are alternate non-standard natural number structures, with different truth values (e.g., there is a non-standard natural number structure according to which the Peano Axioms are inconsistent), the realist and necessist hold that when we engage in arithmetical language, we aren’t talking about these structures. (I am assuming either first-order arithmetic or second-order with Henkin semantics.)

Start by assuming arithmetical truth-value necessitism.

There is an interesting decision point for truth-value necessitism about arithmetic: Are these necessary truths twin-earthable? I.e., could there be a world whose denizens who talk arithmetically like we do, and function physically like we do, but whose arithmetical sentences express different propositions, with different and necessary truth values? This would be akin to a world where instead of water there is XYZ, a world whose denizens would be saying something false if they said “Water has hydrogen in it”.

Here is a theory on which we have twin-earthability. Suppose that the correct semantics of natural number talk works as follows. Our universe has an infinite future sequence of days, and the truth-values of arithmetical language are fixed by requiring the Peano Axioms (or just the Robinson Axioms) together with the thesis that the natural number ordering is order-isomorphic to our universe’s infinite future sequence of days, and then are rigidified by rigid reference to the actual world’s sequence of future days. But in another world—and perhaps even in another universe in our multiverse if we live in a multiverse—the infinite future sequence of days is different (presumably longer!), and hence the denizens of that world end up rigidifying a different future sequence of days to define the truth values of their arithmetical language. Their propositions expressed by arithmetical sentences sometimes have different truth values from ours, but that’s because they are different propositions—and they’re still as necessary as ours. (This kind of a theory will violate causal finitism.)

One may think of a twin-earthable necessitism about arithmetic as a kind of cheaper version of necessitism.

Should a necessitist go cheap and allow for such twin-earthing?

Here is a reason not to. On such a twin-earthable necessitism, there are possible universes for whose denizens the sentence “The Peano Axioms are consistent” expresses a necessary falsehood and there are possible universes for whose denizens the sentence expresses a necessary truth. Now, in fact, pretty much everybody with great confidence thinks that the sentence “The Peano Axioms are consistent” expresses a truth. But it is difficult to hold on to this confidence on twin-earthable necessitism. Why should we think that the universes the non-standard future sequences of days are less likely?

Here is the only way I can think of answering this question. The standard naturals embed into the non-standard naturals. There is a sense in which they are the simplest possible natural number structure. Simplicity is a guide to truth, and so the universes with simpler future sequences of days are more likely.

But this answer does not lead to a stable view. For if we grant that what I just said makes sense—that the simplest future sequences of days are the ones that correspond to the standard naturals—then we have a non-twin-earthable way of fixing the meaning of arithmetical language: assuming S5, we fix it by the shortest possible future sequence of days that can be made to satisfy the requisite axioms by adding appropriate addition and multiplication operations. And this seems a superior way to fix the meaning of arithmetical language, because it better fits with common intuitions about the “absoluteness” of arithmetical language. Thus it it provides a better theory than twin-earthable necessitism did.

I think the skepticism-based argument against twin-earthable necessitism about arithmetic also applies to non-necessitist truth-value realism about arithmetic. On non-necessitist truth-value realism, why should we think we are so lucky as to live in a world where the Peano Axioms are consistent?

Putting the above together, I think we get an argument like this:

1. Twin-earthable truth-value necessitism about arithmetic leads to skepticism about the consistency of arithmetic or is unstable.

2. Non-necessitist truth-value realism about arithmetic leads to skepticism about the consistency of arithmetic.

3. Thus, probably, if truth-value realism about arithmetic is true, non-twin-earthable truth-value necessitism about arithmetic is true.

The resulting realist view holds arithmetical truth to be fixed along both dimensions of Chalmers’ two-dimensional semantics.

(In the argument I assumed that there is no tenable way to be a truth-value realist only about Σ₁⁰ claims like “Peano Arithmetic is consistent” while resisting realism about higher levels of the hierarchy. If I am wrong about that, then in the above argument and conclusions “truth-value” should be replaced by “Σ₁⁰-truth-value”.)

Possible futures

Given a time t and a world w, possible or not, say that w is t-possible if and only if there is a possible world w_t that matches w in all atemporal respects as well as with respect to all that happens up to and including time t. For instance, a world just like ours but where in 2027 a square circle appears is 2026-possible but not 2028-possible.

Here is an interesting and initially plausible metaphysical thesis:

1. The world w is possible iff it is t-possible for every finite time t.

But (1) seems false. For imagine this:

2. On the first day of creation God creates you and promises you that on some future
   day a butterfly will be created ex nihilo. God never makes any other promises. God never makes butterflies. And nothing else relevant happens.

I assume God’s promises are unbreakable. The world described by (2) seems to be t-possible for every finite time t. For the fact that no butterfly has come into existence by time t does not falsify God’s promise that one day a butterfly will be created. But of course the world described by (2) is impossible.

(It’s interesting that I can’t think of a non-theistic counterexample to (1).)

So what? Well, here is one applicaiton. Amy Seymour in a nice paper responding to an argument of mine writes about the following proposition about situation where there are infinitely many coin tosses in heaven, one per day:

3. After every heads result, there is another heads result.

She says: “The open futurist can affirm that this propositional content has a nearly certain general probability because almost every possible future is one in which this occurs.” But in doing so, Seymour is helping herself to the idea of a “possible future”, and that is a problematic idea for an open futurist. Intuitively:

4. A possible future is one such that it is possible that it is true that it obtains.

But the open futurist cannot say that, since in the case of contingent futures, there can be no truth about its obtaining. The next attempt at accounting for a possible future may be to say:

5. A future is possible provided it will be true that it is possible that it obtains.

But that doesn’t work, either, since any future with infinitely many coin tosses (spaced out one per day) is such that at any time in the future, it is still not true that it is possible that it obtains, since its obtaining still depends on the then-still-future coin tosses. The last option I can think of is:

6. A future is possible provided that for every future time t it is t-possible.

But that fails for exactly the same reason that the t-possibility of worlds story fails.

Here is one way out: Deny classical theism, say that God is in time, and insist that God has to act at t in order to create something ex nihilo at t. But God, being perfect, can’t make a promise unless he has a way of ensuring the promise to come true. But how can God make sure that he will one day create the butterfly? After all, on any future day, God is free not to create it then. Now, if God promised to create a butterfly by some specific date, then God could be sure that he would follow through, since if he hadn’t done so prior to the specified date, he would be morally obligated to do so on that day, and being perfect he would do so. So since God can’t ensure the promise will come true, he can’t make the promise. (Couldn’t God resolve to create the butterfly on some specific day? On non-classical theism, maybe yes, but the act of resolving violates the clause “nothing else relevant happens” in (2).)

This way out doesn’t work for classical theism, where God is timeless and simple. For given timelessness, God can timelessly issue the promise and “simultaneously” timelessly make a butterfly appear on (say) day 18, without God being intrinsically any different for it. So I think the classical theist has reason to deny (1), and hence has no account of “possible futures” that is compatible with open futurism, and thus probably has to deny open futurism. Which is unsurprising—most classical theists do deny open futurism.

Unrestricted quantification and Tarskian truth

It is well-known—a feature and not a bug—that Tarski’s definition of truth needs to be given in a metalanguage rather than the object language. Here I want to note a feature of this that I haven’t seen before.

Let’s start by considering how Tarski’s definition of truth would work for set theory.

We can define satisfaction as a relation between finite gappy sequences of objects (i.e., sets) and formulas where the variables are x₁, .... We do this by induction on formulas.

How does this work? Following the usual way to formally create an inductive definition, we will do something like this:

1. A satisfaction-like relation is a relation between finite sequences of sets and formulas such that:

   1. the relation gets right the base cases, namely, a sequence s satisfies x_n ∈ x_m if and only if the nth entry of s is a member of the mth entry of s, and satisfies x_n = x_m if and only if the nth entry of s is identical to the mth entry

   2. the relation gets right the inductive cases (e.g., s satisfies ∃x_nϕ if and only if for every sequence s′ that includes an nth place and agrees with s on all the places other than the nth place we have s′ satisfying ϕ, etc.)

2. A sequence s satisfies a formula ϕ provided that every satisfaction-like relation holds between s and ϕ.

The problem is that in (2) we quantify over satisfaction-like relations. A satisfaction-like relation is not a set in ZF, since any satisfaction-like relation includes ((a),ϕ₌) for every set a, where (a) is the sequence whose only entry is a at the first location and ϕ₌ is x₁ = x₁. Thus, a satisfaction-like relation needs to be a proper class, and we are quantifying over these, which suggests ontological commitment to these proper classes. But ZF set theory does not have proper classes. It only has virtual classes, where we identify a class with the formula defining it. And if we do that, then (2) comes down to:

3. A sequence s satisfies ϕ if for every satisfaction-like formula F the sentence F(s,ϕ) is true.

And that presupposes the concept of truth. (Besides which, I don’t know if we can define a satisfaction-like formula.) So that’s a non-starter. We need genuine and not merely virtual classes to give a Tarski-style definition of truth for set theory. In other words, it looks like the meta-language in which we give the Tarski-style definition of truth for set theory not only needs a vocabulary that goes beyond the object-language’s vocabulary, but it needs a domain of quantification that goes beyond the object-language’s domain.

Now, suppose that we try to give such a Tarskian definition of truth for a language with unrestricted quantification, namely quantification over literally everything. This is very problematic. For now the satisfaction-like relation includes the pair ((a),ϕ₌) for literally every object a. This relation, then, can neither be a set, nor a class, nor a proper superclass, nor a supersuperclass, etc.

I wonder if there is a way of getting around this difficulty by having some kind of a primitive “inductive definition” operator instead of quantifying over satisfaction-like relations.

Another option would be to be a realist about sets but a non-realist about classes, and have some non-realist story about quantification over classes.

I bet people have written on this stuff, as it’s a well-explored area. Anybody here know?

Immortality of the soul and the soul's proper operation

This is an attempt to make an argument for the natural immortality of the soul from the premise that the soul has a proper operation that is independent of the body. The argument is going to be rather odd, because it depends on my rather eccentric four-dimensionalist version of Aristotelian metaphysics.

Start with the thought of how substances typically grow in space. They do this by causing themselves to have accidents in new locations, and they come to exist where these new accidents are. Thus, if I eat and my stomach becomes distended, I now have an accident of stomachness in a location where previously I didn’t, and normally I come to be partly located where my accidents are.

It is plausible (at least to a four-dimensionalist) that spatiotemporal substances grow in time like they grow in space. Thus, they produce accidents in a new temporal location, a future one, and typically come to be located where the accidents are—maybe they come to be there by being active in and through the accidents. (There are exceptions: in transsubstantiation, the bread and wine don’t follow their accidents. But I am focusing on what naturally happens, not on miracles.)

Suppose now that the soul has a proper operation that is independent of the body. Given the fact that my intellectual function is temporal in nature, it is plausible that in this proper operation, my soul is producing a future accident of mine—say, a future accident of grasping some abstract fact—and does so regardless of how sorry and near-to-death a state my body has. But a substance normally stretches both spatially and temporally to become partly located where its accidents are. So by producing a future accident of mine the soul normally ensures that I will be there in that future to be active in and through that accident. Thus the soul, in exercising that future-directed proper activity, makes me exist in the future.

Now that I’ve written this down, I see a gap. The fact that the soul has a proper operation independent of the body does not imply that the soul always engages in that operation. If it does not always engage in that operation, then there is the danger that if my body should perish at a time when the operation is not engaged in, the soul would fail to extend my existence futureward, and I would perish entirely.

On this version of the proper function argument, we thus need a proper operation that the soul normally or naturally always engages in. We might worry, however, that the intellectual operations all cease when we are in dreamless sleep. However, we might suppose that the soul by its nature always carries forward in time some aspect of the understandings or abstractions that it has gained, and this carrying forward in time is indeed a proper operation that occurs even in dreamless sleep, since we do not lose our intellectual gains when we are asleep. (We should distinguish this carrying forward of an aspect of the intellectual gains from the aspects of memory that are mediated by the brain. The need to do this is a weakness of the argument.)

The above depends on my idiosyncratic picture of persistence over time: substances cause their future existence. Divine sustenance is divine cooperation with this causation. The argument has holes. But I feel I may be on to something.

The argument does not establish that we necessarily are immortal. We are only naturally immortal, in that normally we do not perish. It is possible, as far as the argument goes, that the proper operation should fail to succeed in extending us into the future, if only because God might choose to stop cooperating in the way that constitutes sustenance (but I trust he won’t).

Aquinas' argument for the immortality of the soul

Aquinas argues that because the human soul has a proper operation—abstract thought—that does not depend on the body, the soul would survive the destruction of the body.

I’ve never quite understood this argument. It seems to show that there could be a point to the soul surviving the destruction of the body, but that doesn’t show that it will.

It seems that by the same token one could say that because my fingers have an operation independent of my toes, my fingers would survive the destruction of the toes. But that need not be true. I could simultaneously have my toes and fingers crushed, and the fingers’ having an operation independent of the toes would do nothing to save them. In fact, in most cases, fingers perish at the same time as toes do. For in most, though not all, human lives, fingers perish when a person dies, and the toes do so as well. So the argument can’t be that strong.

Still, on reflection, there may be something we can learn from the fingers and toes analogy. We shouldn’t expect the fingers to perish simply as a metaphysical consequence of my toes perishing. By analogy, then, we shouldn’t expect the soul to perish simply as a metaphysical consequence of the body perishing. That’s not the immortality of the soul, but it’s some progress in that direction. After all, the main reason for thinking the soul to perish at death is precisely because one thinks this is a metaphysical consequence of the body perishing.

And I am not denying that there are good arguments for the immortality of the human soul. I think there may be an argument from proper operation that makes even more progress towards immortality, but I’ll leave that for another occasion. Moreover, I think the immortality of the human soul follows from the existence of God and the structure of human flourishing.

Snakes and finitude

For years I have thought the finite to be mysterious, and needs something metaphysical like divine illumination or causal finitism to pick it out. Now I am not sure. I think snakes and exact duplicates can help. And if that’s right, then the argument in my other post from today can be fixed.

Here are some definitions, where the first one is supposed to work for snakes that may be in the same or in different worlds:

• Snake a is vertebrally equal to snake b provided that there is a possible world with exact duplicates of a and b such that in that world it would be possible to line up the two snakes vertebra by vertebra, stretching or compressing as necessary but neither destroying nor introducing vertebra.

• Snake a is the vertebral successor of snake b provided there is a possble world with exact duplicates of a and b such that in that world it is possible to line up the two snakes vertebra by vertebra with exactly one vertebra of a outside the lineup, again stretching or compressing as necessary but neither destroying nor introducing vertebra.

• A world w is abundant in snakes provided that w has a snake with no vertebrae (say, an embryonic snake) and every snake in w has a vertebral successor in w.

• A snake a is vertebrally finite provided that in every world in which snakes are abundant there is a snake vertebrally equal to a.

• A plurality is finite provided that it is possible to put it in one-to-one correspondence with the vertebrae of a vertebrally finite snake.

These definitions require, of course, that one take metaphysical possibility seriously.

A dialectically failing argument for truth-value realism about arithmetic

Truth-value realism about (first-order) arithmetic is the thesis that for any first-order logic sentence in the language of arithmetic (i.e., using the successor, addition and multiplication functions along with the name “0”), there is a definite truth value, either true or false.

Now, consider the following argument for truth-value realism about arithmetic.

Assume eternalism.

Imagine a world with an infinite space and infinite future that contains an ever-growing list of mathematical equations.

At the beginning the equation “S0 = 1” is written down.

Then a machine begins an endless cycle of alternation between three operations:

1. Scan the equations already written down, and find the smallest numeral n that occurs in the list but does not occur in an equation that starts with “Sn=”. Then add to the bottom of the list the equation “Sn = m” where m is the numeral coming after n.

2. Scan the equations already written down, and find the smallest pair of numerals n, m (ordered lexicographically) such that n + m= does not occur in the list of equations, and write at the bottom of the list n + m = r where r is the numeral representing the sum of the numbers represented by n and m.

3. Scan the equations already written down, and find the smallest pair of numerals n, m (ordered lexicographically) such that n ⋅ m= does not occur in the list of equations, and write at the bottom of the list n ⋅ m = r where r is the numeral representing the product of the numbers represented by n and m.

No other numerals are ever written down in that world, and no equations disappear from the list. We assume that all tokens of a given numeral count as “alike” and no tokens of different numerals count as “alike”. The procedure of producing numerals representing sums and products of numbers represented by numerals can be given entirely mechanically.

Now, if ϕ is an arithmetical sentence, then we say that ϕ is true provided that ϕ would be true in a world such as above under the following interpretation of its basic terms:

4. The domain consists of the first occuring token numerals in the giant list of equations (i.e., a token numeral in the list of equations is in the domain if and only if no token alike to it occurs earlier in the list).

5. 0 refers to the zero token in the first equation.

6. The value of Sn for a token numeral n is the token in the domain alike to a token appearing after the equal sign in an equation whose left-side consists of a capital S token followed by a token alike to n.

7. The value of n + m for token numerals n and m is the token in the domain alike to a token appearing after the equal sign in an equation whose left-side consists of a token alike to n follow by a plus sign followed by a token alike to m.

8. The value of n ⋅ m for token numerals n and m is the token in the domain alike to a token appearing after the equal sign in an equation whose left-side consists of a token alike to n follow by a multiplication sign followed by a token alike to m.

It seems we now have well-defined truth-value assignments to all arithmetical sentences. Moreover, it is plausible that these assignments would be correct and hence truth-value realism about arithmetic is correct.

But there is one serious hole in this argument. What if there are two worlds w₁ and w₂ with lists of equations both of which satisfy my description above, but ϕ gets different truth values in them? This is difficult to wrap one’s mind around initially, but we can make the worry concrete as follows: What if the two worlds have different lengths of “infinite future”, so that if we were to line up the lists of equations of the two worlds, with equal heights of lines, one of the two lists would have an equation that comes after all of the equations of the other list?

This may seem an absurd worry. But it’s not. What I’ve just said in the worry can be coherently mathematically described (just take a non-standard model of arithmetic and imagine the equations in one of the lists to have the order-type of that model).

We need a way to rule out such a hypothesis. To do that, what we need is a privileged notion of the finite, so that we can specify that for each equation in the list there is only a finite number of equations before it, or (equivalently) that for each operation of the list-making machine, there are only finitely many operations.

I think there are two options here: a notion of the finite based on the arrangement of stuff in our universe and a metaphysically privileged notion of the finite.

There are multiple ways to try to realize the first option. For instance, we might say that a finite sequence is one that would fit in the future of our universe with each item in the sequence being realized on a different day and there being a day that comes after the whole sequence. (Or, less attractively, we can try to use space.) One may worry about having to make an empirical presupposition that the universe’s future is infinite, but perhaps this isn’t so bad (and we have some scientific reason for it). Or, more directly in the context of the above argument, we can suppose that the list-making machine functions in a universe whose future is like our world’s future.

But I think this option only yields what one might call “realism lite”. For all we’ve said, there is a possible world whose future days have the order structure of a non-standard model of arithmetic, and the analogue to the mathematicians of our world who employed the same approach as we just did to fix the notion of the finite end up with a different, “more expansive”, notion of the finite, and a different arithmetic. Thus while we can rigidify our universe’s “finite” and or the length of our universe’s future and use that to fix arithmetic, there is nothing privileged about this, except in relation to the actual world. We have simply rigidified the contingent, and the necessity of arithmetical truths is just like the necessity of “Water is H₂O”—the denial is metaphysically impossible but conceivable in the two-dimensionalist sematics sense. And I feel that better than this is needed for arithmetic.

So, I think we need a metaphysically privileged notion of the finite to make the above argument go. Various finitism provide such a notion. For instance, finitism simpliciter (necessarily, there are only finitely many things), finitism about the past (necessarily, there are always only finitely many past items), causal finitism (necessarily, each item has only finitely many causal antecedents), and compositional finitism (necessarily, each item has at most finitely many parts). Finitism simpliciter, while giving a notion of the finite, doesn’t work with my argument, since my argument requires eternalism, an infinite future and an ever-growing list. Finitism about the past is an option, though it has the disadvantage that it requires time to be discrete.

I think causal finitism is the best option for what to plug into the argument, but even if it’s the best option, it’s not a dialectically good option, because it’s more controversial than the truth-value realism about arithmetic that is the conclusion of the argument.

Alas.

Causation and counterfactuals

Suppose that an extremely reliable cannon is loaded with a rock, and pointed at a window, and the extremely reliable timer on the cannon is set for two minutes. Two minutes later, the cannon shoots out the rock causing the window to break.

The Lewisian counterfactual account of causation accounts for the causation by the counterfactual:

1. Were the cannon not to have fired the rock, the window wouldn’t have broken.

But imagine that a risk-taking undersupervised kid was walking by towards the end of the the two minutes, and on a whim considered swapping the rock in the cannon for their steel water bottle. The decision whether to do the swap was an extremely conflicted one, and a single neuron’s made the difference, and resulted in the swap not happening.

We can set up the story in such a way that on Lewis’s way of measuring the closeness of worlds, a world where the kid swapped the rock for the water bottle is closer than any worlds where the timer wasn’t set or where the cannon misfired or where the cannon wasn’t loaded or anything like that. In that case on a Lewisian analysis of counterfactuals:

2. Were the cannon not to have fired the rock, the window would still have broken.

But surely whether the kid walks by or not, the cannon’s firing the rock caused the window to break.